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Computational aspects of optimal information revelation

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Computational aspects of optimal information revelation

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Content COMPUTATIONAL ASPECTS OF OPTIMAL INFORMATION REVELATION by Yu Cheng A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) December 2017 Copyright 2017 Yu Cheng Acknowledgments I first want to thank my Ph.D. advisor Shang-Hua Teng. A few months before IbecameShang-Hua’sstudent, IwasathistalkonLaplacianParadigm atTsinghua University. The elegance of the materials, together with his research vision, moti- vated me to pursue graduate study in theoretical computer science. I am grateful to him for seeing the potential in me, and taking me in as his student when I was a teaching assistant at University of Southern California (USC). During my Ph.D., Shang-Hua has been a source of constant encouragement, support, and guidance, for both research and life. I thank him for his patience, and for the flexibility he offers me to work on anything that excites me. Shang-Hua is happy to let me take the wheel, but he is also always there when I need his advice. He often embeds his philosophies in his stories — stories that I understand better as time goes by. I would like to thank Shaddin Dughmi and David Kempe for many enjoyable discussions, and for sharing their knowledge and experience with me. I am also grateful for their guidance on writing and presentations. I thank Shaddin for intro- ducing me to mechanism design and convex optimization, which eventually lead to the works in this thesis. I thank David for teaching me randomized algorithms, especially the principle of deferred decisions; for his advice on running and hiking, and for his invaluable help in preparing this thesis. i I would also like to thank the rest of my thesis committee, Yan Liu and Ben Reichardt, forprovidingfeedbacktomythesisandgivingmeadviceoncareerpaths. The bulk of this thesis occurred in 2014–2015 at USC. Shaddin Dughmi, in particular,mademanyinvaluablecontributionstotheresultsinthisthesis,including but not limited to, the mixture selection framework. I would like to thank Umang Bhaskar and Chaitanya Swamy for our long-distance collaboration, and for their contributions to the results on signaling in routing games. I would like to thank Ilias Diakonikolas for introducing me to the world of learning and testing distributions, and Alistair Stewart for teaching me how to play with the sum of independent random vectors. I thank Ilias and Alistair for their contribution to the results on computing Nash equilibria in anonymous games. I thank Ho Yee Cheung, Ehsan Emamjomeh-Zadeh, and Li Han for many casual discussions on theorems, proofs, and puzzles; and for their contributions to the results on mixture selection and persuading voters presented in this thesis. I am very fortunate to have the opportunities to learn from Xi Chen. I thank himforintroducing metoequilibrium computationandthecomplexityclassPPAD. I wish I could be as calm as him, and I deeply treasure the times when we worked on “random” open questions at the Simons Institute for the Theory of Computing. I would like to thank Richard Peng for our discussions on faster algorithms for large-scale graph and matrix problems. I thank Dehua Cheng for showing me the power of singular value decomposition. I would also like to thank the rest of my collaboratorsduringmygraduatestudy,fromwhomIlearnedsomuch: WadeHann- Caruthers, Daniel Kane, Robert Kleinberg, Young Kun Ko, Abhishek Samanta, AaronSidford, XiaoruiSun, RaviSundaram, OmerTamuz, BoTang,AdrianVladu, Di Wang, and Haifeng Xu. ii I am grateful to Leana Golubchik for her advice and support, particularly during the first and last year of my Ph.D. I would like to thank Leonard Adleman for sharing his stories and being a constant source of inspiration; I admire Len’s passion for mathematics and boxing. IthankmyfellowstudentsinthetheorygroupatUSC:BrendanAvent, Joseph Bebel, Hsing-Hau Chen, Xinran He, Ho Yee Cheung, Ehsan Emamjomeh-Zadeh, Lian Liu, Anand Kumar Narayanan, Ruixin Qiang, Alana Shine, and Haifeng Xu. All of them together made Theoroom a wonderful place to be in during my time. I am grateful to Duke University for offering me my first official job, and to Rong Ge, Kamesh Munagala, and Debmalya Panigrahi for making it possible. I thank the Department of Computer Science at USC for partially supporting my graduate studies through teaching assistantships; and I thank the wonderful staff in the department, especially Lizsl De Leon, Lifeng (Mai) Lee and Kusum Shori, for always being so friendly and helpful to me and many other students. I would like to thank Chin-Yew Lin, Konstantin Voevodski, and Wen Xu, for offering the opportunity to experience industry and work with real data, and for their help and guidance during my internships. I thank Yong Yu for building up the ACM Class at Shanghai Jiao Tong University, which has enabled a generation of young people, including me, to pursue their dreams. I thank Fei Fang for her time. I am indebted to my parents, Ningqiu Cheng and Xiaoli Lyu, for encouraging me to learn more about maths and programming at an early age. I thank them for allowing me to play computer games as much as I want as a kid, for respecting my life choices, and for their support and unconditional love. I thank Peihan Miao for her company during my good and bad days. iii Abstract Strategic interactions often take place in environments rife with uncertainty and information asymmetry. Understanding the role of information in strategic interactions is becoming more and more important in the age of information we live in today. This dissertation is motivated by the following question: What is the optimal way to reveal information, and how hard is it computationally to find an optimum? We study the optimization problem faced by an informed principal, who must choose how to reveal information in order to induce a desirable equilibrium, a task often referred to as information structure design, signaling or persuasion. Our exploration of optimal signaling begins with Bayesian network routing games. This widely studied class of games arises in several real-world settings. For example, millions of people use navigation services like Google Maps every day. Is it possible for Google Maps (the principal) to partly reveal the traffic conditions to reduce the latency experienced by selfish drivers? We show that the answer to this question is two-fold: (1) There are scenarios where the principal can improve selfish routing, and sometimes through the careful provision of information, the principal can achieve the best-coordinated outcome; (2) Optimal signaling is computationally hard in routing games. Assuming P 6= NP, there is no polynomial-time algorithm that does better than full revelation in the worst case. iv We next study the optimal signaling problem in one of the most fundamental classesofgames: Bayesiannormalformgames. Wesettlethecomplexityof(approx- imately) optimal signaling in normal form games: We give the first quasipolynomial time approximation scheme for signaling in normal formgames; and complementing this, weshowthatafullypolynomialtimeapproximationscheme foroptimalsignal- ing is NP-hard, and rule out any polynomial time approximation scheme assuming theplanted clique conjecture. Itisworth noting thatour algorithmworks forgames withaconstantnumberofplayers, andforalargeandnaturalclassofobjectivefunc- tions including social welfare, while our hardness results hold even in the simplest Bayesian two-player zero-sum games. Complementing our results for signaling in normal form games, we continue to investigate the optimal signaling problem in two special cases of succinct games: (1) Second-price auctions in which the auctioneer wants to maximize revenue by revealing partial information about the item for sale to the bidders before running the auction; and (2) Majority voting when the voters have uncertainty regarding their utilities for the two possible outcomes, and the principal seeks to influence the outcome of the election by signaling. We give efficient approximation schemes for all these problems under one unified algorithmic framework, by identifying and solvingacommonoptimizationproblemthatliesatthecoreofalltheseapplications. Finally, we present the currently best algorithm (asymptotically) for computing Nash equilibria in complete-information anonymous games. Compared to all other games we study in this thesis, anonymous games are the only class of games whose complexity of equilibrium computation is still open. We present the currently best algorithm for computing Nash equilibria in anonymous games, and we also provide some evidence suggesting our algorithm is essentially tight. v Contents Acknowledgments i Abstract iv Contents vi 1 Introduction 1 1.1 Information Structure Design . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 An Informational Braess’ Paradox . . . . . . . . . . . . . . . . 3 1.2.2 Prisoner’s Dilemma of Incomplete Information . . . . . . . . . 5 1.2.3 A Probabilistic Second-Price Auction . . . . . . . . . . . . . . 6 1.3 A Frontier of Computational Game Theory . . . . . . . . . . . . . . . 8 1.4 Our Contributions and Thesis Organization . . . . . . . . . . . . . . 10 2 Background and Notation 14 2.1 Bayesian Games and Signaling . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Signaling Schemes. . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.2 Normal Form Games . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.4 Network Routing Games . . . . . . . . . . . . . . . . . . . . . 22 2.1.5 Second-Price Auctions . . . . . . . . . . . . . . . . . . . . . . 24 2.1.6 Majority Voting . . . . . . . . . . . . . . . . . . . . . . . . . . 26 vi 2.1.7 Anonymous Games . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 The Posterior Selection Problem . . . . . . . . . . . . . . . . . . . . . 29 2.3 Planted Clique Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 The Ellipsoid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Signaling in Network Routing Games 34 3.1 Informational Braess’ Paradox Revisited . . . . . . . . . . . . . . . . 35 3.2 Full Revelation Is a (4/3)-Approximation . . . . . . . . . . . . . . . . 37 3.3 NP-hard to Approximate Better Than 4/3 . . . . . . . . . . . . . . . 38 4 Signaling in Normal Form Games 42 4.1 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.1 Bayesian Prisoner’s Dilemma Revisited . . . . . . . . . . . . . 42 4.1.2 Helping a Friend in a Poker Game . . . . . . . . . . . . . . . . 44 4.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 QPTAS for Signaling in Normal Form Games . . . . . . . . . . . . . 49 4.4 Hardness Results for Signaling in Normal Form Games . . . . . . . . 56 4.4.1 NP-hardness of Signaling with Exact Equilibria . . . . . . . . 57 4.4.2 NP-hardness of an FPTAS . . . . . . . . . . . . . . . . . . . . 59 4.4.3 Planted-Clique Hardness of a PTAS . . . . . . . . . . . . . . . 63 5 Mixture Selection: An Algorithmic Framework 78 5.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Noise Stability and Lipschitz Continuity . . . . . . . . . . . . . . . . 82 5.2.1 Consequences of Stability and Continuity . . . . . . . . . . . . 85 5.3 A Meta-Algorithm for Signaling . . . . . . . . . . . . . . . . . . . . . 89 5.3.1 A New QPTAS for Signaling in Normal Form Games . . . . . 93 5.4 Hardness Results for Mixture Selection . . . . . . . . . . . . . . . . . 96 5.4.1 NP-hardness in the Absence of Lipschitz Continuity . . . . . . 97 5.4.2 Planted Clique Hardness in the Absence of Stability . . . . . . 100 vii 6 Signaling in Anonymous Games 104 6.1 Signaling in Second-Price Auctions . . . . . . . . . . . . . . . . . . . 105 6.1.1 PTAS from Mixture Selection: Revenue is Stable . . . . . . . 105 6.1.2 NP-hardness of an Additive FPTAS . . . . . . . . . . . . . . . 108 6.2 Persuasion in Voting . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2.1 Maximizing Expected Number of Votes . . . . . . . . . . . . . 109 6.2.2 Maximizing the Probability of a Majority Vote . . . . . . . . . 111 6.2.3 Hardness Results for Persuading Voters . . . . . . . . . . . . . 113 6.3 Computing Equilibria in Anonymous Games . . . . . . . . . . . . . . 113 6.3.1 Summary of Results and Techniques . . . . . . . . . . . . . . 116 6.3.2 Searching Fewer Moments . . . . . . . . . . . . . . . . . . . . 119 6.3.3 A New Moment Matching Lemma . . . . . . . . . . . . . . . . 131 6.3.4 Slight Improvement Gives FPTAS . . . . . . . . . . . . . . . . 136 7 Conclusion and Open Questions 140 Reference List 145 viii Chapter 1 Introduction 1.1 Information Structure Design What is the best way to reveal information to other strategic players? This is a question wewant tosolve during apoker gameathome, aswell asin billion-dollar industries like online ad auctions. The strategic decisions of the players depend crucially on the information available to them, and the act of exploiting an infor- mational advantage to influence the behavior of others is indeed universal. In Bayesian games, players’ payoffs often depend on the state of nature, which may be hidden from the players. Instead, players receive a signal regarding the state of nature which they use to form beliefs about their payoffs, and choose their strategies. Thus the strategic decisions and payoffs of the players depend crucially on the information available from the signal they receive. In this thesis, we study the optimization problem faced by an informed prin- cipal, who must choose how to reveal information in order to induce a desirable equilibrium, a task often referred to as information structure design, signaling or persuasion. Similar to classic mechanism design 1 , we have a principal who is inter- ested in the outcome of the game, but the difference is that the principal influences 1 Mechanism design is a field in economics and game theory that studies the design of mech- anisms or incentives in strategic settings: A “principal” may choose the rules/structures of the game to induce a desirable outcome, given that other players act rationally. 1 the players by designing the information structure, rather than through designing the game structure. We focus on the design of public information structures that reveal the same information to all players. The study of private signaling schemes is interesting in its own right, but falls beyond the scope of this thesis. Likemechanism design, theinformationstructuredesignquestionisinherently algorithmic: How hard is it computationally to find the optimal information struc- ture? In this thesis, we settle the computational complexity of optimal signaling in several fundamental game-theoretic settings: Bayesian normal form games, and Bayesian succinct games including network routing games, second-price auctions and voting with threshold rule. 1.2 Motivating Examples To motivate the questions we investigate in this thesis, we first give three examples that are incomplete-information variants of classic examples studied in game theory. The first example, presented in [47], is a Bayesian network routing game adapted from Braess’ paradox. The second example, presented in [46], is a Bayesian normal form game adapted from the prisoner’s dilemma. The third example is a variation of a probabilistic single-item second-price auction in [52]. These three examples raise several interesting observations. First, designing theoptimal signaling scheme isanimportant task, because revealing therightinfor- mationcanleadtomuchbetterresultscomparedtotrivialschemeslikenorevelation and full revelation. Second, as opposed to the “Market for Lemons” [2] example 2 , 2 Akerlof[2]usesthemarketforusedcarsasanexampletoillustratethatinformationdeficiency can lead to worse outcomes. In his example, the market degrades in the presence of information 2 sometimes more information can also degrade the payoffs of all players and/or the principal, and the optimal information structure may reveal some but not all the information available. Third, they put forward the modeling question of how we should formulate the signaling problem, as well as the algorithmic question of how hard it is computationally to find the optimal signaling scheme. 1.2.1 An Informational Braess’ Paradox v s t w ℓ(x) =x ℓ(x) =1 ℓ(x) =θ ℓ(x) =1 ℓ(x) =x Figure 1.1: The informational Braess’ paradox. Consider the informational Braess’ paradox given in Figure 1.1. It is a non- atomic Bayesian network routing game of incomplete information, a variant of the classic Braess’ paradox (See, e.g. [19, 82, 85]). We call it informational Braess’ paradoxbecauseinthisexample,somewhatcounter-intuitively, allthedriverscando worse if they have more information about the uncertainty in the traffic conditions. In Figure 1.1, one unit of flow wants to travel from the source node s to the sink node t in the network, and the latency function ℓ(x) on each edge describes the delay experienced by drivers on that edge as a function of the fraction of overall asymmetry between buyers and sellers, if the owner of the good used cars (“peaches”) cannot distinguish himself from the owner of defective used cars (“lemons”). 3 traffic using that road. The latency of the vertical edge is parameterized by a state of nature θ, which is drawn uniformly from{0,1}. When θ = 0, this reverts back to the traditional Braess’ paradox. Selfish drivers would all take the zig-zag paths→v→w→t, where all drivers experience a latency of 2. If the drivers can cooperate rather than being selfish, the socially optimal routing is to send half of the drivers on the top path s → v → t, and the otherhalfonthebottompaths→w→t. Thisway, alldriversexperience alatency of 1.5, but then each individual would benefit from switching to the zig-zag path, and hence Braess’ paradox occurs. Now suppose that every morning nature flips a coin and decides the latency of the edgev→w to be either θ = 0 orθ =1. The principal (e.g., navigation services like Google Maps) knows the exact value of θ, but the drivers only know the prior distributionofθ. Supposethattheprincipalwantstominimizetheexpectedlatency experienced by the selfish drivers at equilibrium, assuming that the agents are risk neutral 3 . Observe that all players would drive along the zig-zag path when E[θ] ≤ 0.5. WhenE[θ]≥ 0.5, the delay on the edge v→w is enough to deter the drivers away from using the edge. Consider the following signaling schemes: • Full information: When θ = 0 the drivers experience latency 2, and when θ =1 the drivers experience latency 1.5. The expected latency is 1.75. 3 The decision of a risk neutral player is not affected by the degree of uncertainty in a set of outcomes. A rational risk-neutral player always chooses the strategy with the highest expected payoff. 4 • No information(Optimal signaling scheme): Risk neutral players treatθ asits expectationE[θ] = 0.5, which is enough to incentivize the drivers to split and play the socially optimal solution, so the latency is always 1.5. 1.2.2 Prisoner’s Dilemma of Incomplete Information Cooperate Defect Cooperate −1+θ 0 −1+θ −5+θ Defect −5+θ −4 0 −4 Figure 1.2: Prisoner’s dilemma of Incomplete Information. Consider a two-player game with incomplete information given by its normal- formrepresentationinFigure1.2,avariantoftheprisoner’sdilemma. Twomembers of a criminal gang are arrested and imprisoned. Without communicating with each other, they must choose to either cooperate with the other by remaining silent, or to defect by betraying the other and testifying. The two players (row player and column player) move simultaneously, and receive payoffs given in the cell specified by the combination of their actions. In each cell, the lower-left number represents the payoff of the row player, and the upper-right number represents the payoff of the column player. The payoffs are also parameterized by a state of natureθ, which affects players’ payoffs together with their actions. Whenθ =0, the game reverts back to the classic prisoner’s dilemma, in which the socially optimal outcome is for both players to cooperate. However, the only 5 Nash equilibrium of the game is when both players defect, because it is strictly better for each player to defect, no matter what the other player does. Suppose nature flips a coin and decides to give the prisoners θ extra reward for cooperating, where θ can be negative and is drawn uniformly from {0,2}. The principal knows the exact value of θ, but both players only know the prior distri- bution. Suppose the principal cares about maximizing social welfare, which in this game is equivalent to maximizing the probability of both players cooperating, again assuming the players are risk neutral. Observe thatriskneutralplayers wouldplaycooperateasanequilibrium ifthe expected value of θ was at least 1. Consider the following signaling schemes: • Full information: Players would defect when θ = 0. When θ = 2, players would prefer to cooperate and therefore they cooperate 1/2 of the time. • No information (Optimal signaling scheme): Both players treatθ as its expec- tationE[θ] = 1, which is just enough to incentivize them to cooperate. So by revealing no information, the principal gets the players to always cooperate and obtain the highest expected social welfare. 1.2.3 A Probabilistic Second-Price Auction Consider a single-item second-price auction of a probabilistic good whose actualinstantiationisknowntotheauctioneer butnottothebidders. Thisproblem was considered in [52] and [21], and Example 1.3 is adapted from [21]. We use θ to denote the possible types of the probabilistic good. As shown in Figure 1.3, the item has four possible types and we have four bidders participating in the auction. 6 Item type 1 Item type 2 Item type 3 Item type 4 Bidder 1 1 0 0 0 Bidder 2 0 1 0 0 Bidder 3 0 0 1 0 Bidder 4 0 0 0 1 Figure 1.3: An example of a probabilistic single-item auction. Each bidder isonlyinterested inoneofthefourpossible itemtypes with avaluation of 1, and not interested in any other types. Assume that the auction format is a second-price auction, that is, the per- son who bids highest wins the item, but he needs to pay only the second highest bid. Since we are running a second-price auction, risk neutral players play the dominant-strategy truth-telling equilibrium, by bidding the expected value for the item. The principal (auctioneer) can choose to reveal some information about the actual realization θ, with the goal of maximizing her revenue. Consider the following signaling schemes: • Full information: If the principal reveals full information about θ, then only one bidder bids 1 and everyone else bids 0. Since the principal is running a second-price auction, the revenue of the auction is 0. • No information: Without further information, the expected value of the item is 1/4 to all bidders, so everyone bids 1/4, and the revenue of the auction is 1/4. • Optimal signaling scheme: One of the possible signaling schemes that max- imizes revenue is to reveal whether θ ∈ {1,2} or θ ∈ {3,4}. Upon learning 7 this information, two out of four bidders remain interested in the item, and their expected value for the item increases from 1/4 to 1/2, because the item now has only two possible types. The revenue of the auction is 1/2 since the principal receives two bids of 1/2 and two bids of 0. 1.3 A Frontier of Computational Game Theory Information revelation has been widely studied by game theorists and economists, exploring how to reveal information strategically to other selfish agents (e.g., see [2, 5, 13, 14, 16, 64, 73, 77, 83]). However, most of the studies on infor- mation disclosure have been non-algorithmic, even though the demand for efficient algorithms has never been higher in the age of information we live in today. Whereas understanding the role of information in influencing strategies is a classical problem in game theory, the computational problem of designing optimal information structures for Bayesian games, commonly called the signaling problem, has received mostly recent attention [7, 21, 46, 48, 49, 50, 52, 62]. These exciting developments during the past decade have brought new insights towards a much better understanding of the role of information through a computational lens. Complexity theory, through concepts like NP-Completeness, aims to distin- guish between the problems that admit efficient algorithms and those that are intractable. In this thesis, we focus on the computational aspects of information structure design. Our goals are to mathematically formulate the signaling problem, to develop algorithms and prove matching lower bounds, and to characterize the exact time complexity of optimal information revelation in different game-theoretic scenarios. 8 For designing efficient algorithms, we analyze the structural properties of the optimal signaling scheme, identify an optimization problem that arises naturally when one seeks to craft posterior beliefs, and develop a powerful algorithmic frame- work that solves the (approximately) optimal signaling problem in a variety of games. For deriving hardness results, our goal is to design the information asym- metry of the game to encode computation in the optimal information structure. For example, if we have an oracle for optimal signaling, can we use it to find maximum independent set ofagraph, ortorecover ahidden clique inarandom graph? Exam- ining the computational complexity of optimal signaling in different games helps us recognize the essence of these problems. Besides improving our understanding about the role of information in game theory, the investigation of optimal information revelation has also led to powerful algorithmic frameworks and basic open questions. For example, in Chapter 5 we extract a common optimization problem that lies at the core of several signaling problems,identifytwo“smoothness”propertieswhichseemtogovernthecomplexity of near-optimal signaling schemes, and resolve a number of open problems under one algorithmic framework. Another example is in Section 4.4, where we utilize the equivalence between separation and optimization to show hardness of signaling in normal form games. However, the proof would be much more elegant if we had a better understanding of this equivalence in the approximate sense, in particular, if we could resolve Open Problem 7.2 in the positive. 9 1.4 Our Contributions and Thesis Organization Westudytheoptimalsignalingprobleminseveralfundamentalgame-theoretic settings: Bayesian network routing games, Bayesian normalform games, probabilis- tic second-price auctions, and majority voting with incomplete information. We focus on public signaling schemes where the principal reveals the same information toallplayers. Ourmaincontributionistoderiveefficient approximationalgorithms, aswellashardnessresultsfortheseclassesofgamesthatclosethegapbetweenwhat is achievable in polynomial time (or quasipolynomial time) and what is intractable. We start from an example of using information to battle selfishness in routing games. We then continue to systematically study the signaling problem in both normal form games with a bounded number of players, as well as succinct games with many players. Signaling in network routing games: In Chapter 3, we consider the signal- ing problem in (nonatomic, selfish) Bayesian network routing games, wherein the principal seeks to reveal partial information to minimize the average latency of the equilibrium flow. We show that it is NP-hard to obtain any multiplicative approxi- mation better than 4 3 , even with linear latency functions (Theorem 3.2). This yields an optimal inapproximability result for linear latencies, since we show that full rev- elation obtains the price of anarchy of the routing game as its approximation ratio (Theorem3.1),which is 4 3 forlinearlatencyfunctions[85]. Thesearethefirstresults for the complexity of signaling in Bayesian network routing games. Signaling in normal form games: In Chapter 4, we consider signaling in Bayesiannormalform games,wherethepayoffentriesareparametrizedbyastateof 10 nature. Dughmi[46]initiatedthecomputationalstudyofthisproblemandobtained various hardness results. On the algorithms side, we give two different approaches (Theorem 4.1 and Theorem 5.12) for obtaining a bi-criteria QPTAS for normal- form games with a constant number of players, and for a large and natural class of objective functions like social welfare and weighted combination of players’ util- ities [26, 27]. In other words, we can in quasipolynomial time approximate the optimal reward from signaling while losing an additive ǫ in the objective as well as in the incentive constraints. For hardness results, [46] considered the special case of signaling in Bayesian (two-player) zero-sum games, in which the principal seeks to maximize the equilib- rium payoff of one of the players, and ruled out a fully polynomial time approxima- tion scheme (FPTAS) for this problem assuming planted clique hardness. We show thatitisNP-hardtoobtainanadditiveFPTAS(Theorem4.7),settlingthecomplex- ity of the problem with respect to NP-hardness. Moreover, we show that assuming the planted clique conjecture (Conjecture 2.6), there does not exist a polynomial time approximation scheme (PTAS) for the signaling problem (Theorem 4.10). Mixture selection framework In Chapter 5, we pose and study an algorithmic problem which we term mixture selection, a problem that arises naturally in the design of optimal information structures. The mixture selection problem is closely related to the optimal signaling problem. We identify two “smoothness” property of Bayesian games that seem to dictate the complexity of mixture selection and optimal signaling: Lipschitz continuity and a noise stability notion that we define. We present an algorithmic framework that (approximately) solves mixture selec- tion (Theorem 5.6) and optimal signaling (Theorem 5.10) in a number of different 11 Bayesian games. The approximation guarantee of our algorithm degrades grace- fully as a function of the two smoothness parameters, in particular, when the game is O(1)-Lipschitz continuous and O(1)-stable, we obtain an additive PTAS optimal signaling. Wealsoshowthatneitherassumption suffices byitself foraPTAS(Theo- rems5.18and5.19). WegiveanewQPTASforsignalinginnormalformgameusing our algorithmic framework (Theorem 5.12). Moreover, our algorithms for signaling in multi-player games also follow from the powerful mixture selection framework. Signaling in anonymous games In Chapter 6, we consider signaling in anony- mous games. In contrast to the normal form games we study in Chapter 4, anony- mous games form an important class of succinct games, capturing a wide range of game-theoretic scenarios, including auctions and voting. We start with two special cases of anonymous games, both admitting a PTAS. In Section 6.1, we consider signaling in the context of a probabilistic second- price auction. In this setting, the item being auctioned is probabilistic, and the instantiation of the item is known to the auctioneer but not to the bidders. The auctioneer must decide what information toreveal in order tomaximize her revenue in this auction. Emek et al. [52] and Miltersen and Sheffet [21] considered several special casesofthisproblemandpresented polynomial-timealgorithmswhen bidder types are fixed. [52] showed that in the general setting, where the auctioneer holds probabilistic knowledge on thebidders’ valuations, anFPTAS forsignaling becomes NP-hard. We resolve the approximation complexity of optimal signaling in the Bayesian setting by giving an additive PTAS (Theorem 6.2). In Section 6.2, we study the persuasion in voting problem proposed by Alonso andCâmara[5]. Considerabinaryoutcomeelection—saywhetheraballotmeasure 12 is passed — when voters are not fully informed of the consequences of the measure, and hence of their utilities. Each voter casts a Yes/No vote, and the measure passes if the fraction of Yes votes exceeds a certain pre-specified threshold. We consider a principal who has control over which information regarding the measure is gathered and shared with voters, and looks to maximize the probability of the measure passing. Wepresent a multi-criteria PTAS for thisproblem (Theorem 6.5). Section 6.3 takes a detour and studies anonymous games with complete infor- mation. We give the first polynomial time algorithm for computing Nash equilibria of inverse polynomial precision in anonymous games with more than two strategies (Theorem 6.6), and present evidence suggesting that our algorithm is essentially tight (Theorem 6.7). This gets us closer to pinning down the computational com- plexityofNashequilibriainanonymousgames, andtheonlyquestionleftiswhether there is a FPTAS for computing equilibria or not. 13 Chapter 2 Background and Notation We use R + for the set of nonnegative reals. For an integer n, let [n] def = {1,2,...,n}. If n ≥ 1, we use ∆ n to denote the (n−1)-dimensional simplex {x ∈ R n + : P i x i = 1}. We refer to a distribution y ∈ ∆ n as s-uniform if and only if it is the average of a multiset of s standard basis vectors in n-dimensional space. Let 1 n ∈R n bethevector with1inallitsentries,I n×n bethen×nidentity matrix, and e i be the i-th standard basis vector containing 1 as its i-th entry and 0 elsewhere. Weuse“iff”toabbreviate“ifandonlyif”. Fortwofunctionsf(n)andg(n),we writef(n)=O(g(n)) iff there exists constantsC andn 0 such that|f(x)|≤C|g(x)| for all n≥n 0 ; we write g(n) = Ω( f(n)) iff f(n) =O(g(n)), and g(n) = Θ(f(n)) iff f(n) = O(g(n)) and f(n) = Ω( g(n)). We say f(n) = o(g(n)) iff lim n→∞ f(n) g(n) = 0, and g(n) = ω(f(n)) iff f(n) = o(g(n)). We use poly(n) to denote a polynomial function ofn. When wesaywith highprobability, wemean with probabilityatleast 1− 1 n α for some constantα> 0; and the parametern will be clear from the context. Let|I| denote the description size of the instanceI. An (additive) polynomial time approximation scheme(PTAS) isan algorithmthatruns in timepoly(|I|), and returns a solution of value at least OPT(I)−ǫ for every instance I and constant ǫ> 0. An fully polynomial time approximation scheme (FPTAS) is a PTAS whose running time for an instanceI and parameterǫ is poly |I|, 1 ǫ . An quasipolynomial time approximation scheme (QPTAS) is an algorithm that runs in time |I| O(log|I|) and returns an ǫ-optimal solution for every instance I and constant ǫ> 0. 14 2.1 Bayesian Games and Signaling A Bayesian game is a game in which the players have incomplete information on the payoffs of the game. In this thesis, we consider a number of Bayesian games in which payoffs are parametrized byθ, the state of nature. We use Θ to denote the set of all states of nature, and assume θ ∈ Θ is drawn from a common-knowledge prior distribution which we denote by λ. We consider Bayesian games given in the explicit representation: • An integer M denoting the number of states of nature. We index states of nature by the set Θ= [M] ={1,...,M}. • A common-knowledge prior distribution λ∈∆ M on the states of nature. • A set of M games of complete information, one for each state of nature θ, describing the payoff structure of the game. Note that a game of complete information is the special case withM = 1, i.e., the state of nature is fixed and known to all. In all our applications, we assume players a priori know nothing aboutθ other than its prior distribution λ, and examine policies whereby a principal with access to the realized value ofθ may commit to a policy revealing information to the play- ers regarding θ. The goal of the principal is then to commit to revealing certain information aboutθ — i.e., a signaling scheme — to induce a favorable equilibrium over the resulting Bayesian subgames. This is often referred to as signaling, persua- sion, or information structure design. The recent survey by Dughmi [47] contains a nice summary of the work on information structure design in the algorithmic game theory community. 15 2.1.1 Signaling Schemes Asignalingschemeisapolicybywhichaprincipalreveals(partial)information about the state of nature. We call a signaling scheme public if it reveals the same information to all the players, and private when different signals are sent to the players through private channels. In this thesis, we focus on the design of public signaling schemes. Let M = |Θ|. A signaling scheme specifies a set of signals Σ and a (possibly randomized) map ϕ : Θ → ∆ |Σ| from the states of nature Θ to distributions over the signals in Σ. Abusing notation, we use ϕ(θ,σ) to denote the probability of announcing signal σ∈Σ conditioned on the state of nature being θ∈Θ. Each signal σ yields a posterior distribution µ σ ∈ ∆ M . It was observed by Kamenica and Gentzkow [69] that signaling schemes are in one-to-one correspon- dence with convex decompositions ofthe prior distributionλ∈∆ M : Formally, a sig- nalingschemeϕ: Θ→ Σcorrespondstotheconvexdecompositionλ= P σ∈Σ p σ ·µ σ , where (1) p σ = Pr θ∼Θ [ϕ(θ) = σ] = P θ∈Θ λ(θ)ϕ(θ,σ) is the probability of announc- ing signal σ, and (2) µ σ (θ) = Pr θ∼Θ [θ|ϕ(θ) = σ] = λ(θ)ϕ(θ,σ) pσ is the posterior belief distribution of θ conditioned on signal σ. The converse is also true: every convex decomposition of λ ∈ ∆ M corresponds to a signaling scheme. Alternatively, the reader can view a signaling schemeϕastheM×|Σ| matrixofpairwise probabilities ϕ(θ,σ)satisfying conditions(1)and(2)with respect toλ∈∆ M . Sometimes wealso describe a signaling scheme as{p µ } µ ∈Δ M , where P µ ∈Δ M p µ µ =λ 1 . The signals Σ in such a signaling scheme are described implicitly, and correspond to the posteriorsµ for which p µ >0. 1 We only deal with signaling schemes with finitely many signals in all of our algorithms and analyses, so we use P even though we are summing over the uncountable simplex. 16 Note that each posterior distribution µ ∈ ∆ M defines a complete-information subgame: for every outcome s of the game (i.e., every pure strategy profile), risk neutral players take the expected payoff over θ ∼µ as their expected payoff under s. Theprincipal’s utilitydepends on theoutcomeofthesubgames. Given asuitable equilibrium concept and selection rule, we let f : ∆ M → R denote the principal’s utilityasafunctionoftheposteriordistributionµ . Forexample, inanauctiongame f(µ ) may be the social welfare or principal’s revenue at the induced equilibrium, or any weighted combination of players’ utilities, or something else entirely. The principal’s objective as a function of the signaling scheme ϕ can be mathematically expressed by F(ϕ,λ)= P σ p σ ·f(µ σ ). Fix a Bayesian game, and let f + (λ) denote the value of the optimal signaling scheme when the prior is λ. We note that f + (λ) is a concave function of the prior λ, since ifλ 1 andλ 2 form a convex decomposition ofλ, so do the optimal posteriors for λ 1 and λ 2 . Therefore, the optimal choice of a signaling scheme is related to the concave envelope f + of the function f ([46, 69]). Definition 2.1. The concave envelope f + of a function f is the point-wise lowest concave function h for whichh(x)≥f(x) for all x in the domain. Equivalently, the hypograph of f + is the convex hull of the hypograph of f. Specifically, such a signaling scheme achieves P σ p σ ·f(µ σ ) = f + (λ). Thus, there exists a signaling scheme with (M +1) signals that maximizes the principal’s objective, by applying Carathéodory’s theorem 2 to the hypograph of f. 2 In convex geometry, Carathéodory’s theorem [22] states that if a point x ∈ R d lies in the convex hull of a set P, then there exists a subset P ′ ⊆ P with |P ′ | ≤ d+1 such that x is in the convex hull of P ′ . 17 2.1.2 Normal Form Games A normal form game is defined by the following parameters: • An integer k denoting the number of players, indexed by the set [k] = {1,...,k}. • An integer n bounding the number of pure strategies of each player. Without loss of generality, we assume each player has exactly n pure strategies, and index them by the set [n] ={1,...,n}. • A family of payoff tensors A = {A 1 ,...,A k } with A i : [n] k → [−1,1], where A i (s 1 ,...,s k ) is the payoff to player i when each player j plays strategy s j . A Bayesian normal form game is described by payoff tensors A θ i : [n] k → [−1,1], one per player i and state of nature θ, where A θ i (s 1 ,...,s k ) is the pay- off to player i when the state of nature is θ and each player j plays strat- egy s j . For a mixed strategy profile x 1 ,...,x k ∈ ∆ n , we use A i (x 1 ,...,x k ) = P s 1 ,...,s k ∈[n] T(s 1 ,...,s k )· Q k i=1 x i (s i ) to denote player i’s expected payoff over the pure strategy profiles drawn from (x 1 ,...,x k ). In a general Bayesian normal form game, absent any information about the stateofnaturebeyondthepriorλ,riskneutralplayerswillbehaveasinthecomplete information game E θ∼λ h A θ i . We consider signaling schemes which partially and symmetrically inform players by publicly announcing a signal σ, correlated with θ; this induces a common posterior belief on the state of nature for each value of σ. When players’ posterior beliefs over θ are given by µ ∈ ∆ M , we use A µ to denote the equivalent complete information gameE θ∼µ h A θ i . As shorthand, we use A µ i (x 1 ,...,x k ) to denoteE h A θ i (s 1 ,...,s k ) i when θ∼µ ∈∆ M and s i ∼x i ∈∆ n . 18 Theprincipal’sobjectiveisdescribedbyafamilyoftensorsA θ 0 :[n] k →[−1,1], one for each state of nature θ ∈ Θ. Equivalently, we may think of the objective as describing the payoffs of an additional player in the game. For a distributionµ over states of nature, we useA µ 0 =E θ∼µ h A θ 0 i to denote the principal’s expected utility in a subgame with posterior beliefs µ , as a function of players’ strategies. Extended Security Games AnextendedsecuritygameisafamilyofBayesianzero-sumgames. ABayesian zero-sum game is specified by a tuple Θ,{A θ } θ∈Θ ,λ . For each state of nature θ∈Θ, A θ ∈ [−1,1] n×n specifies the payoffs of the row player in a zero-sum game. Let Row and Col denote the row player and the column player respectively. An extended security game can be viewed as a polymatrix game between three players: Nature, Row, and Col. Formally, the payoff matrix for stateθ is given by A θ def = A+b θ 1 T n + 1 n (d θ ) T , where A∈R n×n ,b θ ∈R n ,d θ ∈R n . (2.1) Let B and D be matrices having columns {b 1 ,...,b M }, and {d 1 ,...,d M } respec- tively. The payoff of the row player is the sum of her payoffs in three separate games: a gameA betweenRow andCol, a gameB between Row and Nature, and a gameD between Nature andCol. Weobtain the following expressions forA µ and f(µ ) for µ ∈ ∆ M . A µ =A+(Bµ ) 1 T n + 1 n (µ T D T ), f(µ )= max x∈Δn x T Bµ +min j∈[n] x T A+µ T D T j . (2.2) 19 A special case of an extended security game (and the reason for this termi- nology) is the network security game defined by [46]. Given an undirected graph G = (V,E) with n = |V| and a parameter ρ ≥ 0, the states of nature correspond to the vertices of the graph. The row and column players are called attacker and defender respectively. The attacker and defender’s pure strategies correspond to nodes ofG. LetB be the adjacency matrix ofG, and setA =D T =−ρI n×n . Then, for a given state of nature θ ∈V, and pure strategies a,d ∈V of the attacker and defender, the payoff of the attacker is given bye T a Be θ −ρ(e T a +e T θ )e d . The interpre- tation is that the attacker gets a payoff of 1 if he selects a vertex a that is adjacent to θ. This payoff is reduced by ρ if the defender’s vertex d lies in {θ,a}, and by 2ρ if d=θ =a. 2.1.3 Nash Equilibria The celebrated theorem of Nash [80] states that every finite game has an equilibrium point. Thesolution concept ofNash equilibrium (NE) hasbeen tremen- dously influential in economics and social sciences ever since (e.g., see [65]). For a game with k players and n strategies per player, a mixed strategy is an element of ∆ [n] , and a mixed strategy profile x= (x 1 ,...,x k ) maps every player i to her mixed strategy x i ∈ ∆ [n] . Throughout this thesis, we adopt the (approximate) Nash equilibrium as our equilibrium concept. There are two variants. We define them below in normal form games and note that the concept of Nash equilibria is universal in games — it simply states that each player plays a best response to other players’ strategies and has no incentive to deviate. We use x −i to denote the strategies of players other than i in x. The support of a vector, supp(x), is the set of indices i such that x i > 0. 20 Definition 2.2. Let ǫ≥0. In a k-player n-action normal form game with expected payoffs in [−1,1] given by tensors A 1 ,...,A k , a mixed strategy profile x 1 ,...,x k ∈ ∆ n is an ǫ-Nash Equilibrium (ǫ-NE) if A i (x 1 ,...,x k )≥A i (t i ,x −i )−ǫ for every player i and alternative pure strategy t i ∈[n]. Definition 2.3. Let ǫ≥0. In a k-player n-action normal form game with expected payoffs in [−1,1] given by tensors A 1 ,...,A k , a mixed strategy profile x 1 ,...,x k ∈ ∆ n is an ǫ-well-supported Nash equilibrium (ǫ-WSNE) if A i (s i ,x −i )≥A i (t i ,x −i )−ǫ for every player i, strategy s i in the support of x i , and alternative pure strategy t i ∈[n]. Clearly, every ǫ-WSNE is also an ǫ-NE. When ǫ = 0, both correspond to the exact Nash Equilibrium. Note that we omitted reference to the state of nature in the above definitions — in a subgame corresponding to posterior beliefs µ ∈ ∆ M , we naturally use tensors A µ 1 ,...,A µ k instead. Fixing an equilibrium concept (NE, ǫ-NE, or ǫ-WSNE), a Bayesian game (A,λ), and a signaling scheme ϕ : Θ → Σ, an equilibrium selection rule distin- guishes an equilibrium strategy profile (x σ 1 ,...,x σ k ) to be played in each subgameσ. Togetherwiththepriorλ,theBayesianequilibriaX ={x σ i :σ∈Σ,i∈[k]}inducea distribution Γ∈ ∆ Θ×[n] k over statesof play—werefer toΓ asa distribution of play. We say Γ is implemented by signaling scheme ϕ in the chosen equilibrium concept 21 (be it NE, ǫ-NE, or ǫ-WSNE). This is analogous to implementation of allocation rules in traditional mechanism design. For a signaling scheme ϕ and associated (approximate) equilibria X = {x σ i :σ∈ Σ,i∈[k]}, our objective function can be written as F(ϕ,X)= E θ∼λ " E σ∼ϕ(θ) E s∼x σ [A 0 (θ,s)] # . When ϕ corresponds to a convex decomposition {(µ σ ,p σ )} σ∈Σ of the prior dis- tribution, this can be equivalently written as F(ϕ,X) = P σ∈Σ p σ A µ σ 0 (x σ ). Let OPT = OPT(A,λ,A 0 ) denote the maximum value of F(ϕ ∗ ,X ∗ ) over signaling schemesϕ ∗ and (exact) Nash equilibriaX ∗ . We seek a signaling schemeϕ: Θ→Σ, as well as corresponding Bayesian ǫ-equilibria X such that F(ϕ,X)≥OPT −ǫ. 2.1.4 Network Routing Games A network routing game is a tuple G = (V,E),{ℓ e } e∈E ,{(s i ,t i ,d i )} i∈[k] , where G is a directed graph with latency function ℓ e : R + → R + on each edge e. Each (s i ,t i ,d i ) denotes a commodity; d i specifies the volume of flow routed from s i to t i by self-interested agents, each of whom controls an infinitesimal amount of flow and selects an s i -t i path as her strategy. A strategy profile thus corresponds to a multicommodity flow composed ofs i -t i flows of volume d i for all i; we call any such flow a feasible flow. The latency on edge e due to a flow x is given by ℓ e (x e ), where x e is the total flow on e. The latency of a path P is ℓ P (x) def = P e∈P ℓ e (x e ). The total latency of a flow x is L(ℓ,x) def = P e∈E x e ℓ e (x e ). An optimal flow x OPT is a feasible flow with minimum latency. A Nash flow (also called a Wardrop flow) x NE is a feasible flow 22 where every player chooses a minimum latency path; that is, for alli, alls i -t i paths P, Q with x NE (e) > 0 for all e ∈ P, we have ℓ P (x NE ) ≤ ℓ Q (x NE ). All Nash flows have the same total latency (see, e.g., [85]). The price of anarchy (PoA) measures how the efficiency of a system degrades due to selfish behavior of its agents. In network routing games, the price of anarchy is defined as the ratio between latencies of the Nash flow and the optimal flow: PoA =L(ℓ,x NE )/L(ℓ,x OPT ). The price of anarchy for a class of latency functions is the maximum ratio over all instances involving these latency functions. In a Bayesian network routing game, the edge latency functions {ℓ θ e } e∈E may depend on the state of nature θ ∈ Θ (and, as before, we have a prior λ ∈ ∆ Θ ). The principal seeks to minimize the latency of the Nash flow. Given µ ∈ ∆ Θ , the expected latency function on each edge e is ℓ µ e (x e ) def = P θ∈Θ µ (θ)ℓ θ e (x e ). Define f(µ ) def = L(ℓ µ ,x µ NE ), where x µ NE is a Nash flow for latency functions {ℓ µ e }. The signaling problem in a Bayesian routing game is to determine (p µ ) µ ∈Δ M ≥ 0 of finite support specifying a convex decomposition of λ (i.e., P µ ∈Δ M p µ µ = λ) that minimizes the expected latency of the Nash flow, P µ ∈Δ M p µ f(µ ). In Bayesian network routing games, Vasserman et al. [90] study the problem of signaling to reduce the average latency. They define the mediation ratio as the average latency at equilibrium for the best private signaling scheme, to the average latency for the social optimum, and give tight bounds on the mediation ratio for a special family of Bayesian routing games. In contrast, we study the computational complexity of obtaining the best public signaling scheme, and conclude that finding an ( 4 3 −ǫ)-approximation is NP-hard. 23 2.1.5 Second-Price Auctions The following parameters describe a variant of an single-item auction 3 : • An integer n denoting the number of bidders, indexed by the set [n]. • A common-knowledge prior distribution D on bidders’ valuations v ∈ [0,1] n , where v i denotes the value of player i for the item. A probabilistic auction has M possible states of nature θ ∈ Θ, and each θ represents a possible instantiation of the item being sold. Instead of having an n-dimensional vector as bidders’ valuations, we now have valuation matrices V ∈ [0,1] n×M , where V i,j denotes the value of player i for the item corresponding to the state of nature j. Again, we have a common-knowledge prior distribution D on V, given either explicitly or as a “black-box” sampling oracle. We examine signaling in probabilistic second-price auctions, as considered by Emek et al. [52] and Miltersen and Sheffet [21]. In this setting, the item being auc- tioned is probabilistic, and the instantiation of the item is known to the auctioneer but not to the bidders. The auctioneer commits to a signaling scheme for (par- tially) revealing information about the item for sale before subsequently running a second-price auction. As an example, consider an auction for an umbrella: the state of natureθ can be the weather tomorrow, which determines the utilityV i,θ of an umbrella to player i. We assume that λ and D are independent. We also emphasize that a bidder 3 In this thesis, for normal form games, we use n to denote the number of strategies and k for the number of players; for multi-player succinct games, we use n to denote the number of players and k for the number of strategies. For both families of games, we are interested in games with k =O(1) and we want to bound the running time as a function of n. 24 knows nothing about θ other than its distribution λ and the public signal σ, and the auctioneer knows nothing about V besides its distribution D prior to running the auction. The game being played is the following: 1. The auctioneer first commits to a signaling scheme ϕ : Θ→Σ; 2. Astateofnatureθ∈Θisdrawn accordingtoλandrevealed totheauctioneer but not the bidders; 3. The auctioneer reveals a public signal σ∼ϕ(θ) to all the bidders; 4. A valuation matrix V ∈ [0,1] n×M is drawn according to D, and each player i learns his value V i,j for each potential item j; 5. Finally, a second-price auction for the item is run. Notethat step (4)is independent ofsteps (1-3), so they can happen in no particular order. Weadoptthe(unique)dominant-strategytruth-tellingequilibriumasoursolu- tion concept. Specifically, given a signaling scheme ϕ : Θ→ Σ and a signal σ ∈ Σ, in the subgame corresponding to σ it is a dominant strategy for player i to bid E θ∼λ [V iθ |ϕ(θ) =σ] — his posterior expected value for the item conditioned on the received signal σ. Therefore the item goes to the player with maximum posterior expected value, at a price equal to thesecond-highest posterior expected value. The algorithmic problem we consider is the one faced by the auctioneer in step (a) — namely computing an optimal signaling scheme — assuming the auctioneer looks to maximize expected revenue. 25 2.1.6 Majority Voting Consider an election with n voters and two possible outcomes, ‘Yes’ and ‘No’. Forexample, votersmayneed tochoosewhethertoadoptanewlaworsocialpolicy; boardmembersofacompanymayneedtodecidewhethertoinvest inanewproject; and members of a jury must decide whether a defendant is declared guilty or not guilty. The new policy passes if the fraction of ‘Yes’ votes exceeds a certain pre- specified threshold. We index the voters by the set [n] = {1,...,n}. Without loss of generality, we assume utilities are normalized: voter i has a utility ofu i ∈[−1,1] for the ‘Yes’ outcome, and 0 for the ‘No’ outcome. In most voting systems with a binary outcome, including threshold voting rules, it is a dominant strategy for voter i to vote ‘Yes’ if the utility u i is at least 0. We study the signaling problems encountered in the context of voting intro- duced by Alonso and Câmara [5]. In this setting, voters have uncertainty regarding their utilities for the two possible outcomes (e.g., the risks and rewards of the new project). Specifically, voters’ utilities are parameterized by an a priori unknown state of nature θ drawn from a common-knowledge prior distribution. We assume voters’ preferences are given by a matrix U ∈ [−1,1] n×M , where U i,j denotes voter i’s utility in the event of a ‘Yes’ outcome in state of nature j. A voter i who believes that the state of nature follows a distribution µ ∈∆ M has expected utility u(i,µ )= P j∈Θ U i,j µ j for a ‘Yes’ outcome. We adopt the perspective of a principal — say a moderator of a political debate—withaccesstotherealizationofθ,whocandeterminethesignalingscheme throughwhichinformationregardingthemeasureisgatheredandsharedwithvoters. Alonso andCâmara[5]consider aprincipal interested inmaximizing theprobability that at least 50% (or some given threshold) of the voters vote ‘Yes’, in expectation 26 over states of nature. They characterize optimal signaling schemes analytically, though stop short of prescribing an algorithm for signaling. 2.1.7 Anonymous Games We study anonymous games (n,k,{u i a } i∈[n],a∈[k] ) with n players labeled by [n] ={1,...,n}, andk common strategies labeled by [k] for each player. The payoff of a player depends on her own strategy, and how many of her peers choose which strategy, but not on their identities. When player i ∈ [n] plays strategy a ∈ [k], her payoffs are given by a function u i a that maps the possible outcomes (partitions of all other players) Π k n−1 to the interval [0,1], where Π k n−1 = {(x 1 ,...,x k ) | x j ∈ Z + ∧ P k j=1 x j =n−1}. Approximate Equilibria in Anonymous Games. We define ǫ-approximate Nash equilibrium for anonymous games. The definition is essentially equivalent to ǫ-equilibrium in normal form games (Definition 2.2), except now the game has a succinct representation. A mixed strategy profile s is an ǫ-approximate Nash equilibrium in an anonymous game if and only if ∀i∈[n],∀a ′ ∈[k], E x∼s −i h u i a ′(x) i ≤ E x∼s −i ,a∼s i h u i a (x) i +ǫ, where x ∈ Π k n−1 is the partition formed by n−1 random samples (independently) drawn from [k] according to the distributionss −i . Note that given a mixed strategy profile s, we can compute a player’s expected payoff by straightforward dynamic programming (see, e.g., [84]). 27 Poisson Multinomial Distributions. A k-Categorical Random Variable (k- CRV)isavectorrandomvariablesupportedonthesetofk-dimensionalbasisvectors {e 1 ,...,e k }. A k-CRV is i-maximal if e i is its most likely outcome (break ties by taking the smallest indexi). Ak-Poisson multinomial distribution of ordern, or an (n,k)-PMD,isavectorrandomvariableoftheformX = P n i=1 X i wheretheX i ’sare independent k-CRVs. The case of k = 2 is usually referred to as Poisson Binomial Distribution (PBD). Note that a mixed strategy profile s = (s 1 ,...,s n ) of an n-player k-strategy anonymous game corresponds to the k-CRVs {X 1 ,...,X n } where Pr[X i = e a ] = s i (a). The expected payoff of player i ∈ [n] for playing pure strategy a ∈ [k] can also be written asE[u i a (X −i )] =E h u i a P j6=i,j∈[n] X j i . Let X = P n i=1 X i be an (n,k)-PMD such that for i ∈ [n] and j ∈ [k] we denote p i,j = Pr[X i = e j ], where P k j=1 p i,j = 1. For m = (m 1 ,...,m k ) ∈ Z k + , we define the m th -parameter moments of X to be M m (X) def = P n i=1 Q k j=1 p m j i,j . We refer to kmk 1 = P k j=1 m j as the degree of the parameter moment M m (X). Total Variation Distance and Covers. The total variation distance between two distributions P and Q supported on a finite domainA is d TV (P,Q) def = max S⊆A |P(S)−Q(S)|=(1/2)·kP −Qk 1 . If X and Y are two random variables ranging over a finite set, their total variation distance d TV (X,Y) is defined as the total variation distance between their distribu- tions. Forconvenience, we will often blur the distinction between a random variable and its distribution. 28 Let (X,d) be a metric space. Given ǫ > 0, a subset Y ⊆ X is said to be a properǫ-coverofX withrespecttothemetricd:X 2 →R + ,ifforeveryX ∈X there exists some Y ∈ Y such that d(X,Y) ≤ ǫ. We will be interested in constructing ǫ-covers for high-variance PMDs under the total variation distance metric. Multidimensional Fourier Transform. For x ∈ R, we will denote e(x) def = exp(−2πix). The (continuous) Fourier Transform of a function F :Z k →C is the function b F : [0,1] k →C defined as b F(ξ) = P x∈Z ke(ξ·x)F(x). For the case that F is a probability mass function, we can equivalently write b F(ξ)=E x∼F [e(ξ·x)]. Let X = P n i=1 X i be an (n,k)-PMD with p i,j def = Pr[X i = e j ]. To avoid clutter in the notation, we will sometimes use the symbol X to denote the cor- responding probability mass function. With this convention, we can write that c X(ξ)= Q n i=1 c X i (ξ)= Q n i=1 P k j=1 e(ξ j )p i,j . 2.2 The Posterior Selection Problem The signaling problem can be formulated as the mathematical program (P), max X µ ∈Δ M α µ f(µ ) s.t. X µ ∈Δ M α µ µ (θ) =λ θ for all θ∈Θ, α≥0. (P) Notice that any feasible α must also satisfy P µ ∈Δ M α µ = 1; hence, α is indeed a distribution over ∆ M , and a feasible solution to (P) yields a signaling scheme. Let opt(λ) denote the optimal value of (P), and note that this is a concave function of λ. Although (P) has a linear objective and linear constraints, it is not quite a linear program (LP) since there are an infinite number of variables. Ignoring this issue for now, we consider the following dual of (P). 29 min w T λ s.t. w T µ ≥f(µ ) for all µ ∈∆ M , w∈R M . (D) The separation problem for (D) motivates the following dual signaling problem. Definition 2.4 (Dual signaling with precision parameter ǫ). Given an objective function f : ∆ M →[−1,1], w∈R M , and ǫ>0, distinguish between: (i) f(µ )≥w T µ +ǫ for some µ ∈∆ M ; if so return µ ∈∆ M s.t. f(µ )≥w T µ −ǫ; (ii) f(µ )<w T µ −ǫ for all µ ∈∆ M . The posterior selection problem is the special case of dual signaling where w =η 1 M for some η∈R. Definition 2.5 (Posterior selection with precision parameter ǫ). Given an objec- tive function f : ∆ M → [−1,1] and ǫ > 0, find µ ∗ ∈ ∆ M such that f(µ ∗ ) ≥ max µ ∈Δ M f(µ )−ǫ. 2.3 Planted Clique Conjecture Some of our hardness results are based on the hardness of the planted-clique and planted clique cover problems. In the planted clique problem PClique(n,p,k), one must distinguish the n- node Erdős-Rényi random graph G(n, 1 2 ) in which each edge is included indepen- dently with probability 1 2 , from the graph G(n, 1 2 ,k) formed by “planting” a clique inG(n, 1 2 ) ata randomly(or, equivalently, adversarially) chosen set ofk nodes. This problem was first considered by Jerrum [66] and Ku˘ cera [72], and has been the sub- ject of a large body of work since. A quasipolynomial time algorithm exists when k ≥ 2logn, and the best polynomial-time algorithms only succeed for k = Ω( √ n) 30 (see, e.g., [4, 30, 40, 56]). Several papers suggest that the problem is hard for k =o( √ n)byrulingoutnaturalclassesofalgorithmicapproaches(e.g., [55,57,66]). The planted clique problem has therefore found use as a hardness assumption in a variety of applications (e.g., [3, 46, 63, 67, 78]). Conjecture 2.6 (Planted-clique conjecture). For some k = k(n) satisfying k = ω(logn) and k = o( √ n), there is no probabilistic polynomial time algorithm that solves PClique n, 1 2 ,k with constant success probability. The planted clique cover problem was introduced in [46]. Multiple cliques are now planted and one seeks to recover a constant fraction of them. Definition 2.7 (Planted clique cover problem PCover(n,p,k,r) [46]). Let G ∼ G(n,p,k,r) be a random graph generated by: (1) including every edge independently with probability p; and (2) for i = 1,...,r, picking a set S i of k vertices uniformly at random, adding all edges having both endpoints in S i . We call theS i ’s the planted cliques andp the background density. We seek to recover a constant fraction of the planted cliques S 1 ,...,S r , given G∼G(n,p,k,r). Dughmi [46] showed that the planted clique cover problem is at least as hard as the planted clique problem. Given an instance G of PClique(n,p,k), we can generate an instance G ′ of PCover(n,p,k,r) by planting r−1 additional random k-cliques into G (as in step (2) of Definition 2.7). Because the cliques S 1 ,...,S r are indistinguishable, recovering a constant fraction of the planted cliques from G ′ would recover each of S 1 ,...,S r with constant probability. In particular, doing so would recover the original planted clique with constant probability. 31 2.4 The Ellipsoid Method The ellipsoid method is an iterative method for minimizing convex functions, and it does so by generating a sequence of ellipsoids whose volume decreases in each step. The shallow-cut ellipsoid method is a variant of the ellipsoid method, which is useful when we only have access to an approximate separation oracle. The classic ellipsoidmethod,ineachiteration,cutsthecurrentellipsoidusingahyperplanethat goes through the center of the ellipsoid. At a high level, the shallow-cut method takes a “shallower” cut that is close to the center of the ellipsoid, removing slightly less than half of the current ellipsoid. It turns out that this is still sufficient for finding an (approximately) optimal solution. We utilize the shallow-cut ellipsoid method to translate hardness results for the posterior selection problem to hardness results for optimal signaling. Formally, we use the following lemma. Lemma 2.8 (Chapters 4, 6 in [61]; Section 9.2 in [81]). Let X ⊆R n be a polytope described by constraints having encoding length at most L. Suppose that for each y ∈ R n , we can determine in time poly(size of y,L) if y / ∈ X and if so, return a hyperplane of encoding length at most L separating y from X. (i) The ellipsoid method can find a point x∈X or determine that X =∅ in time poly(n,L). (ii) Let h : R n → R be a concave function and K = sup x∈X h(x)−inf x∈X h(x). Suppose we have a value oracle for h that for every x ∈ X returns ψ(x) satisfying |ψ(x)−h(x)| ≤ δ. There exists a polynomial p(n) such that for any ǫ ≥ p(n)δ, we can use the shallow-cut ellipsoid method to find x ∗ ∈ X 32 such that h(x ∗ ) ≥ max x∈X h(x) − 2ǫ (or determine X = ∅) in time T = poly n,L,log( K ǫ ) and using at most T queries to the value oracle for h. 33 Chapter 3 Signaling in Network Routing Games Navigationservices (e.g. GoogleMaps)onsmartphoneshavechanged people’s livesduringthepastfewyears. WhileusingGoogleMapsisaseasyastypinginyour destination and following its navigation, there are three key aspects of this real-life scenariothatarerelevanttothedesignofinformationstructures: (1)Drivingtowork is a game with incomplete information. Traffic has uncertainty, and the congestion of every road is different each day. (2) The principal (Google Maps) knows more aboutthereal-timetrafficthanthedrivers. Oneofthemainreasonswhydriversuse Google Maps is to learn more about real-time traffic conditions so they can choose a better route. (3) Drivers are selfish and prefer to take the shortest paths to their destination. It is well known that efficiency of network routing degrades due to the selfish behavior of the drivers (see, e.g., [85]). These three aspects motivate us to study signaling in routing games. At a high level, we view the traffic conditions learned by Google Maps as an informational advantage, and we ask if (and how) Google Maps can utilize this advantage to help selfish drivers. In this chapter, we consider information revelation in (non-atomic, selfish) Bayesian network routing games. We are interested in the most natural setting in which theprincipal seeks to minimize theaverage latency experienced by adriver in the system, knowing that the players would act selfishly after learning more about 34 the traffic. Is it possible for the principal to carefully reveal information to reduce the latency of equilibrium flow? And if so, how much can information revelation help selfish routing? We show that the answer to this question is two-fold. (i) Therearescenarioswheretheprincipalcanimproveselfishrouting. Sometimes through the careful provision of information, the principal can achieve the best-coordinated outcome. (ii) Optimal information revelation is hard in routing games in the worst case: Assuming P 6= NP, there is no polynomial-time algorithm that does better than full revelation. More specifically, we show that full revelation obtains the price of anarchy (defined in Section 2.1.4) of the routing game as its approximation ratio (Theorem 3.1), which is 4 3 for linear latency functions [85]. We then settle the approximability of the problem by showing thatit is NP-hard to obtaina multiplicative approximation better than 4 3 , even with linear latency functions (Theorem 3.2). Theresultsinthischapterappearedin[15],andmycoauthorsUmangBhaskar and Chaitanya Swamy did most of the work. 3.1 Informational Braess’ Paradox Revisited WeuseaslightvariationoftheinformationalBraess’paradox(Example1.1)to illustrate the problem we are trying to solve, and show that sometimes the principal must reveal some but not all information to minimize the latency of selfish routing. In Example 3.1, the states of nature θ 1 ,θ 2 are independent random variables, and 35 v s u t w ℓ(x) =θ 1 ℓ(x) = 1−θ 1 ℓ(x) =x ℓ(x) =1 ℓ(x) =θ 2 ℓ(x) =1 ℓ(x) =x Figure 3.1: A Bayesian network routing game adapted from Example 1.1. both are drawn uniformly from the set{0,1}. There are two edges froms tou, and exactly one of these two edges is going to have latency 0, while the other edge has latency 1. The edge v→w has latency 0 half of the time, and latency 1 otherwise. In theoptimal signaling scheme, theprincipal reveals therealization ofθ 1 , but hides the value of θ 2 . This is because the drivers have no externality froms tou, so they can all take the shorter edge. But for driving from u to t, the drivers are better off notlearning thevalueofθ 2 , otherwisewhenθ 2 = 0allofthem deviate tothezig-zag path and experience a longer travel time. In the example above, by revealing no information about the edge v → w, it is as if the principal can remove the edge from the graph. It turns out that this intuition isthe key to showing hardness forthis problem. Weshow that theoptimal signaling problem in routinggamesisashardasthenetwork design problem studied by Cole et al. [31], where theprincipal puts taxes on the edges to minimize the total drivers’disutility(latency+tax). Itwasshownin[31]thattheproblemisequivalent to deciding which edges to remove to minimize the latency of the Nash flow, so we call it the network design problem. Our reduction constructs a Bayesian routing game from an instance of the network design problem, in which we can translate principal’s signaling scheme back to aset of taxes on theedges —the principal puts more taxes on an edge if he reveals less information about it (and vice versa). 36 3.2 Full Revelation Is a (4/3)-Approximation In this section, we prove that full revelation is a 4 3 -approximation for signaling in Bayesian network routing games with linear latency functions. Recall that the price of anarchy (PoA) for a class of latency functions is the maximum ratio, over all instances involving these latency functions, of the latencies of the Nash flow and optimal flow. For linear latency functions, the PoA is 4 3 [85]. Intuitively,theresultfollowsbecausefullrevelationisthebestsignalingscheme if one seeks to minimize the expected latency of the optimal flow, and the multi- plicative error that results from this change in objective (from the latency of the Nashflowtothatoftheoptimalflow)cannotexceed thepriceofanarchy. Ourresult directly generalizes to arbitrary latency functions and multi-commodities, and the approximation ratio of full revelation is bounded by the PoA of the set of allowable latency functions. Theorem 3.1. In Bayesian routing games, the full revelation signaling scheme has the price of anarchy for the underlying latency functions as its approximation ratio. In particular, for linear latencies, full revelation achieves a 4 3 -approximation. Proof. Given a state of natureθ∈Θ, we usex θ NE andx θ OPT to denote the Nash flow and the optimal flow with respect to the latency functions {ℓ θ e } respectively. Let ρ be the price of anarchy for the collection {ℓ θ e } e∈E,θ∈Θ of latency functions, so we have L(ℓ θ ,x θ OPT )≥L(ℓ θ ,x θ NE )/ρ) for all θ∈Θ. The full revelation signaling scheme has latency L def = P θ∈Θ λ θ L(ℓ θ ,x θ NE ). We show that the average latency of any signaling scheme {p µ } is at least (L/ρ). 37 X µ ∈Δ M p µ f(µ )= X µ ∈Δ M p µ L(ℓ µ ,x µ NE ) = X µ ∈Δ M p µ X θ∈Θ µ (θ)L(ℓ θ ,x µ NE ) ≥ X µ ∈Δ M p µ X θ∈Θ µ (θ)L(ℓ θ ,x θ OPT ) = X θ∈Θ λ θ L(ℓ θ ,x θ OPT ) ≥ X θ λ θ L(ℓ θ ,x θ NE )/ρ. 3.3 NP-hard to Approximate Better Than 4/3 We now prove matching hardness result to show that Theorem 3.1 is tight. The proof of Theorem 3.2 is a direct reduction from the problem of computing edge tolls that minimize the total (latency + toll)-cost of the resulting equilibrium flow. Theorem 3.2. For any ǫ> 0, obtaining a ( 4 3 −ǫ)-approximation for the signaling problem in Bayesian routing games is NP-hard, even in single-commodity games with linear latency functions. Let G,s,t,d be a single-commodity routing game. By scaling latency func- tions suitably, we may assume that d = 1 and omit it from now on. We reduce from the problem of determining edge tolls τ ∈ R E + that minimize L(ℓ+τ,x ℓ+τ NE ), whereℓ+τ denotes the collection of latency functions{ℓ e (x)+τ e } e , andx ℓ+τ NE is the Nash flow for ℓ+τ. Note that L(ℓ+τ,x) = P e x e (ℓ e (x e )+τ e ) takes into account the contribution from tolls; we refer to this as the total cost of x. The problem of computing optimal tolls that minimizes (latency + toll) is inapproximable within a factor of 4 3 . 38 Theorem 3.3 ([31]). There are optimal tolls where the toll on every edge is 0 or ∞. For everyǫ> 0, there is no 4 3 −ǫ -approximation algorithm for the problem of computing optimal tolls in networks with linear latency functions, unless P = NP. Let G =(V,E),ℓ,s,t beaninstanceofanetworkdesignproblemwithlinear latencies. Let m =|E|≥ 5. Let L 0 =L(ℓ,x ℓ NE ) be the latency of the Nash flow for the original graph ℓ. Let τ ∗ be optimal {0,∞}-tolls, L ∗ = L(ℓ+τ ∗ ,x ℓ+τ ∗ NE ) be the optimal cost, and B ∗ def = {e∈E :τ ∗ e =∞}. We can view τ ∗ as removing the edges in B ∗ . We create a Bayesian routing game as follows. Let G 1 = (V 1 ,E 1 ),s 1 ,t 1 and G 2 = (V 2 ,E 2 ),s 2 ,t 2 be two copies of (G,ℓ,s,t). Add vertices s, t, and edges (s,s 1 ), (s,s 2 ) and (t 1 ,t), (t 2 ,t). Call the graph thus created H. For e ∈ E 1 ∪E 2 with corresponding edgee ′ ∈E, we set the latency function in the new graph to be h e (x) =ℓ e ′(x), and we set h e (x) =0 for e∈{(s,s 1 ),(s,s 2 ),(t 1 ,t),(t 2 ,t)}. Next, we introduce uncertainty to this game by randomly removing one edge in H. Each state of nature θ corresponds to an edge e in H (i.e., Θ =E H ), which is going to be effectively removed from the graph. Formally, we set: λ θ =                    1 m 2 if θ∈E 1 ∪E 2 , 1 2 − 1 m if e =(s,s 1 ) or (s,s 2 ), 0 if e =(t 1 ,t) or (t 2 ,t). h θ e (x) =          h e (x)+m 2 L 0 if θ =e, h e (x) otherwise. Our Bayesian routing game is (H,{h θ e } θ,e ,s,t),λ . Theideaisthatstateθencodestheremovalofedgeθ: specifically, ifµ (θ)≥ 1 m 2 for a posterior µ , then h µ simulates the edge θ breaking down due to the large constant termm 2 L 0 . LetB 1 ,B 2 be theedge-sets corresponding toB ∗ inG 1 andG 2 39 respectively. The priorλ is set up to satisfy two important properties: (1) it admits a convex decomposition into posteriors µ 1 and µ 2 , where µ 1 simulates that G 1 \B 1 is connected to s and G 2 is disconnected from s (similarly for µ 2 ); and (2) any reasonable signaling scheme must put most of the probability mass into posteriors µ , where h µ connects only one of G i to s, so that {µ e m 2 L 0 } e∈E i yields tolls τ for edgesinthenetworkdesignproblem withL(ℓ+τ,x ℓ+τ NE )≤f(µ ). Lemma3.4and3.5 make the statements in (1) and (2) precise, and Theorem 3.2 follows immediately from Lemmas 3.4, 3.5 and Theorem 3.3. Lemma 3.4. Let L ∗ be the cost of optimal tolls for a network design instance (G,ℓ,s,t). There is a signaling scheme for the above Bayesian routing game (H,{h θ e } θ,e ,s,t),λ with expected latency L ∗ . Proof. WepartitiontheedgesofH intotwosets: B 1 ∪(E 2 \B 2 )∪{(s,s 2 )}asoneset, and the remaining edges as the other. We claim the signaling scheme that reveals which set contains the broken edge θ has expected latency L ∗ . Formally, define posterior µ 1 ∈∆ E H as: µ 1 (θ) =                    2 m 2 if θ∈B 1 or θ∈(E 2 \B 2 ), 1− 2 m if θ =(s,s 2 ), 0 otherwise. We can defineµ 2 symmetrically and check thatλ= (µ 1 +µ 2 )/2. We will show that f(µ 1 ) =f(µ 2 ) =L ∗ , proving the lemma. Consider distribution µ 1 . The argument for µ 2 is symmetrical. The idea is that an edgee withµ 1 (e)> 0 hash µ 1 e (x)≥ 2L 0 , which effectively deletese fromH; other edges have h µ 1 e (x) = h e (x). So µ 1 retains edges in G 1 \B 1 , and disconnects 40 G 2 froms. The remaining graph corresponds to the optimal solution ofthe network designproblemonG 1 , removingthebadedgesB 1 andaddingtwoextraedges(s,s 1 ) and (t 1 ,t) both with latency 0. Therefore, the latency of the Nash flow under µ 1 is exactly L ∗ . Lemma 3.5. Given a signaling scheme {p µ } µ ∈Δ M for the Bayesian routing game (H,{h θ e } θ,e ,s,t),λ with expected latency L, one can obtain tolls τ such that the routing game (G,ℓ+τ,s,t) has Nash latency at most L 1−3/m . Proof. Assume L < L 0 , otherwise τ = 0 suffices. By Markov’s inequality, at least 1− 2 m−1 oftheprobabilitymassofp must beonposteriorsµ withµ (s,s 1 ) +µ (s,s 2 ) ≥ 1/m. There must exist such a posterior µ with f(µ )≤ L 1−2/(m−1) ≤ L 1−3/m . Fix such a posterior µ . Without loss of generality, we assume µ (s,s 1 ) ≥ 1 2m . Let x = x µ NE be the Nash flow for latency functions h µ . Since m ≥ 5 and L < L 0 , we have h µ (s,s 1 ) ≥ mL 0 /2 > L 1−3/m ≥ f(µ ), so we must have x (s,s 1 ) = 0, i.e., x is supported onG 2 . For e ∈ E 2 , we also use e to denote the corresponding edge in E. For every e∈E 2 , we haveh µ e (x) =ℓ e (x)+µ e m 2 L 0 . Thus, defining τ e =µ e m 2 L 0 for alle∈E, we obtain that x restricted to E 2 (with s 2 corresponding to s) is a Nash flow for (G,ℓ+τ,s,t). The latency of the restricted flow is equal tof(µ ), because under the posterior µ , every s-t path in G corresponds to an s 2 -t 2 path in H with the same latency. 41 Chapter 4 Signaling in Normal Form Games In theprevious chapter, westudied howinformationrevelation canhelp selfish routing, and we showed that the principal must solve NP-hard problems to do even slightly better than full revelation. In this chapter, we examine the complexity of optimal signaling in one of the most fundamental classes of games: normal form games. As we will see, the problem of (approximately) optimal signaling in normal formgamesiscomputationallyeasierthanoptimalsignalinginroutinggames—the principal can obtain anǫ-additive optimal signaling scheme in quasipolynomial time for any constant ǫ> 0. and this cannot be improved to polynomial time assuming the planted clique conjecture. Recall that in Bayesian normal form games, we have a principal and a game whosepayoffentriesdependonthestateofnatureθ. Playersonlyknowthecommon prior distribution of θ, while the principal knows the realization of θ and seeks to reveal partial information aboutθ to induce a desirable equilibrium. For the formal definition of signaling in Bayesian normal form games, see Section 2.1.2. 4.1 Two Examples 4.1.1 Bayesian Prisoner’s Dilemma Revisited We start with a variation of the Prisoner’s dilemma given in Example 1.2. 42 Cooperate Defect Cooperate −2,−2 −5,0 Defect 0,−5 −4,−4 (a) Payoff when θ = 1. Cooperate Defect Cooperate 1,1 −5,0 Defect 0,−5 −4,−4 (b) Payoff when θ = 2. Figure 4.1: A Bayesian normal form game adapted from the Prisoner’s dilemma. In Figure 4.1, the payoff of the game depends on the state of nature θ, which is drawn from {1,2} uniformly by nature. We are given two normal form games as input, one for each possible state of nature. In each cell, the first number repre- sents the payoff of the row player, and the second number represents the payoff of the column player. The principal is interested in maximizing the (expected) social welfare. If the principal reveals full information, the players defect when θ = 1 and cooperate when θ = 2. The expected social welfare is (−8+2)/2=−3. Consider asignalσ andthecorrespondingposteriorbeliefµ overθ. Letµ (1)= Pr[θ = 1] and µ (2) = Pr[θ = 2]. The expected payoff for both players to cooperate is Pr[θ =1]·(−2)+Pr[θ =2]·1=µ (2)−2µ (1)= 1−3µ (1). Whenµ (1)≤1/3, the payoff is enough to incentivize risk neutral players to cooperate. The optimal signaling scheme uses two signals, σ 1 (the “defect” signal) and σ 2 (the “cooperate” signal). The principal announces σ 2 whenever θ = 2. When θ = 1, the principal announcesσ 2 with probability 1/2 and announcesσ 1 otherwise. Conditioned on the signal beingσ 2 , we have Pr[θ = 1]= 1/3. Based on the analysis above, playerswillcooperateunderσ 2 andtheexpected socialwelfareoftheoptimal signaling scheme is 2 Pr[θ =1,σ 1 ]·(−4)+Pr[θ =1,σ 2 ]·(−2)+Pr[θ =2]·1 = 2 1 4 ·(−4)+ 1 4 ·(−2)+ 1 2 ·1 =−2. 43 Example 3.1 is similar to the informational Braess’ paradox (Example 1.1), in which the principal tries to help the players fight their selfishness through careful provision of information. This is merely one of the many facets of optimal informa- tion revelation, and we will see a different perspective in the next example. 4.1.2 Helping a Friend in a Poker Game Fold Call Check 1 1 Bet 1 2 (a)Row’s payoff when θ = 1. Fold Call Check −1 −1 Bet 1 −2 (b)Row’s payoff when θ = 2. Figure 4.2: A Bayesian poker game. Consider a Bayesian zero-sum game given in Figure 4.2 1 . We use Row and Col to denote the row and column players respectively. The game proceeds as follows: 1. Each player puts $1 on the table, then gets a card. Players do not get to see the cards (including their own). 2. Row goes first, and she can choose to bet another $1 (Bet) or not (Check). 3. If Row bets, Col can choose to put in $1 as well (Call) or to give up (Fold). If Row bets and Col folds, Row wins automatically. Otherwise both cards are revealed, and the player with the higher card wins and takes all the money. Note that in this game, Col does not have the option to bet if Row checks. 1 This example is inspired by Will Ma’s talk on Poker at Google NYC in summer 2013. 44 To simplify the problem, we assume nature flips a coin to decide who has the higher card. If θ = 1 then Row has a higher card, and if θ = 2 then Col has a higher card. This game can be represented in normal form as in Figure 4.2. When RowbetsandColfolds,thevalueofthecardsareirrelevant;Rowtakesthemoney, winning $1 from Col. In all other cases, the cards are revealed and the player with higher card wins and gets either $1 or $2 from her opponent. If Row and Col play this game without extra information aboutθ, the value of the game is 0, because Col will never fold and neither player has an advantage. Now suppose there is a principal who knows the exact value of θ and wants to help Row in this game by sending a signal to both players. If the principal reveals full information,thenRowwins$1whenθ = 1andloses$1whenθ =2,sotheexpected payoff of Row is again 0. Consider a signal σ and the corresponding posterior belief µ . Let µ (1) denote the probability that θ = 1. • Ifµ (1)< 1/2,thenColhasahigherchanceofwinningifthecardsarerevealed, so Col always calls and Row has no incentive to bet. The equilibrium of the game is (Check, Call), and Row’s expected payoff is Pr[θ = 1]−Pr[θ = 2]=µ (1)−(1−µ (1))= 2µ (1)−1. • For 1/2 ≤ µ (1) < 3/4, Row has a higher chance of winning if the cards are revealed, so she prefers to bet. IfCol folds, she loses $1 for sure. IfCol calls, her expected payoff is 2(Pr[θ = 2]−Pr[θ = 1])= 2−4µ (1)>−1, 45 so the equilibrium of the game is (Bet, Call), and Row’s expected payoff is 4µ (1)−2. • When µ (1)≥ 3/4, the analysis in similar to that in the previous case, except that the probability of Row winning the showdown (θ = 1) is large enough that it is better for Col to fold. The equilibrium of the game is (Bet, Fold), and Row’s payoff is 1. Recall that f(µ ) is the principal’s objective (in this case, Row’s expected payoff)undertheposteriorµ . Wecanplotoutf(µ )asafunctionofµ (1)(Figure4.3). The optimal signaling scheme (shown in decomposition form in Figure4.3) uses two signals,onerevealingRowhasthelowercard(µ (1)=0,Rowloses$1),andonejust enough to force Col to fold when Row bets (µ (1) = 3/4, Row wins $1). The first signal appears with probability 1/3, and the second signal appears with probability 2/3, so Row and the principal’s expected utility is 1/3. 1 −1 0.25 0.50 0.75 1.00 µ (1) f(µ )= Row’s expected payoff × × × Figure 4.3: Row’s expected payoff as a function of µ (1) = Pr[θ = 1]. The dashed line represents the optimal signaling scheme in the prior decomposition form. In thepoker game, depending on theposterior beliefs, there arethreedifferent outcomes of the game. This can be observed in the three sections of the piecewise 46 linear function in Figure 4.3. In order to decompose the prior optimally, there is no need for more than one signal per outcome, due to the fact that the function is linear in each section. This observation is crucial for our algorithm in Section 4.3. We hope that these two examples illustrate the importance and complexity of optimal signaling in normal form games. We will now proceed to state our algorith- mic and hardness results. 4.2 Summary of Results In this chapter, we investigate the computational complexity of optimal sig- naling in Bayesian zero-sum games. In Section 4.3, we develop the first (bi-criteria) quasipolynomial time approx- imation scheme (QPTAS) for signaling in normal form games. In other words, for every constant ǫ> 0, we can in quasipolynomial time compute a near-optimal sig- naling scheme, losing an additive ǫ in the objective as well as in the equilibrium constraints (Theorem 4.1). InSection4.4,wefirstshowthattherelaxationinplayers’incentiveconstraints isnecessary, otherwisetheproblembecomesNP-hard(Theorem4.5). Wethensettle thecomplexityofthesignalingproblemwithrespecttoNP-hardnessbyshowingthat it is NP-hard to obtain an additive FPTAS (Theorem 4.7). Finally, we show that assuming the planted clique conjecture (Conjecture 2.6), the QPTAS in Section 4.3 is essentially optimal (Theorem 4.10). It is worth noting that our algorithm applies to general sum normal form games with any constant number of players, and an abstract class of objectives which includes the social welfare and weighted combinations of player utilities as 47 a special case; while all of our hardness results hold for Bayesian two-player zero- sum games. Zero-sum games admit a canonical and tractable notion of equilibrium, which allows us to study the complexity of optimal signaling without equilibrium computation concerns. Theworkpresentedinthissectionappearedpreviouslyasresearchpapers. The algorithm in Section 4.3 appeared in [27], and the hardness results in Section 4.4 appeared in [15]. Previous and Recent Work Dughmi [46] initiated the computational study of signaling in Bayesian zero- sum games, and obtained various hardness results. Specifically, it was shown that no FPTAS is possible for the signaling problem for zero sum games, assuming the planted clique conjecture. In Section 4.4.2, westrengthen theresult of[46] byruling out an FPTAS assuming P 6= NP, and ruling out a PTAS based on the planted clique conjecture. Recently, Rubinstein[86]showedthatourQPTASinTheorem4.3isessentially tightassumingtheExponentialTimeHypothesis(ETH).Comparedtoourhardness results in Section 4.4.3, [86] replaced our average-case assumption of planted clique hardness with a more conventional worst-case assumption (ETH). 48 4.3 QPTAS for Signaling in Normal Form Games In this section, we consider signaling in normal form games when the adopted solution concept is the ǫ-well-supported Nash equilibrium (ǫ-WSNE) 2 , and give the first QPTAS for this problem. Our approach consists of two main steps. 1. Construct a Small Dictionary of Equilibria: Thisisadiscretefamilyofobjects which indexes the potential equilibria ofa signaling scheme, with the property that they form an ǫ-cover of the space of all equilibria with respect to the space of signaling schemes and the design objective. 2. Construct a near-optimal Signaling Scheme: We then compute a near-optimal signaling scheme which induces subgames with equilibria from our dictio- nary. This involves solving a nontrivial optimization problem which optimally decomposes the prior distribution into posterior beliefs inducing equilibria in our dictionary. Our dictionary is based on the work of Lipton et al. [74]. Specifically, [74] showstheexistenceofaquasipolynomial-sizefamilyofmixedstrategyprofileswhich, simultaneously for all games and equilibria of those games, includes a profile which approximates the payoffs of the equilibrium to within an additiveǫ, and itself forms an ǫ-equilibrium. To combine these approximate equilibria into a signaling scheme, we make two observations: First, the space of posterior beliefs which induce a par- ticular equilibrium forms a convex polytope; second, the optimization problem of optimally partitioning the prior belief into a quasipolynomial number of posterior 2 Since every ǫ-WSNE is also an ǫ-approximate Nash equilibrium (ǫ-NE), our results apply to ǫ-NE as well. 49 beliefs, onein each polytopecorresponding to anequilibrium, can beformulated via a linear program after an appropriate change of variables. Formally, we prove the following bi-criteria result. Recall that F(ϕ,X) is the principal’s objective value for the signaling scheme ϕ and associated (approximate) equilibria X; and OPT(A,λ) is the maximum reward for the principal over all possible signaling schemes and (exact) equilibria. Theorem 4.1. Fix ǫ > 0. Given as input an explicitly-described Bayesian nor- mal form game (A,λ) with k = O(1) players, n actions, and M states of nature, and an objective A 0 : [M]× [n] k → [−1,1], there is an algorithm with runtime poly(M,n logn/ǫ 2 ) which outputs a signaling scheme ϕ and corresponding Bayesian ǫ-well-supported Nash equilibria X satisfying F(ϕ,X)≥ OPT(A,λ)−ǫ. The proof of Theorem 4.1 hinges on three main lemmas. The first lemma (Lemma 4.2) allows us to restrict attention to equilibria with small support, which follows easily from the results of [74] 3 . The second lemma (Lemma 4.3) states that the posterior beliefs implementing a particular approximate equilibrium form a simple polytope, in doing so reducing the signaling problem to optimization over convex decompositions of λ into a family of posteriors, each belonging to a given polytope. The third lemma (Lemma 4.4) shows that optimization over such convex decompositions reduces to a linear program. Lemma 4.2. Let tensors A 1 ,...,A k : [n] k → [−1,1] describe a k-player game of complete information with n pure strategies per player, and let A 0 : [n] k → [−1,1] 3 Babichenko et al. [8] later improved the parameters of [74]. For a normal form game with k players and n strategies and for any constant ǫ > 0, [8] proves there exists an ǫ-cover of Nash equilibriainwhicheachplayerrandomizesuniformlyamongasetofsizeO(logk+logn); incontrast [74] requires a set of size O(k 2 (logk + logn)). Since we are interested in the setting where the number of players k =O(1), the two results are asymptotically the same for us. 50 be a tensor describing an objective function on pure strategy profiles. For each ǫ> 0, there exists an integer s =s(ǫ) =O(k 2 log(kn)/ǫ), such that for every mixed strategy profile x = (x 1 ,...,x k ), there is a profile e x = (e x 1 ,...,e x k ) of s-uniform mixed strategies such that |A i (x)−A i (e x)|≤ǫ for all players i, |A 0 (x)−A 0 (e x)|≤ǫ, and if x is a Nash equilibrium of A ={A 1 ,...,A k } then e x is an ǫ-WSNE of A. Lemma 4.3. Fix a normal form game of incomplete information {A θ i ∈ [−1,1] n k : i∈[k],θ∈[M]}withk players,nactions,andM statesofnature. Consideramixed strategy profile x = (x 1 ,...,x k ) with x i ∈ ∆ n . For each ǫ ≥ 0, the set of posterior beliefs inducing x as an ǫ-WSNE is a convex polytope described by L = poly(k,n) linear inequalities. Lemma 4.4. Given a family of non-empty polytopes P 1 ,...,P t ⊆∆ M described by L inequalities each, a point λ ∈ ∆ M , and linear objectives w 1 ,...,w t ∈ R M , the non-linear optimization problem (4.1) can be solved in poly(t,L,M) time. maximize P t σ=1 p σ w σ ·µ σ subject to P t σ=1 p σ = 1 P t σ=1 p σ µ σ =λ µ σ ∈P σ , for σ = 1,...,t. (4.1) Before proving each of these lemmas, we first elaborate on how they imply Theorem 4.1. 51 Proof of Theorem 4.1. Given a signaling scheme ϕ with decomposition form (p,µ ), and an (approximate) equilibrium x σ for each subgame corresponding to σ, the principal’s objective value is F(ϕ,x)= X σ∈Σ p σ A 0 (µ σ ,x σ ) where A 0 (µ,x ) denotesE θ∼µ [E s∼x [A 0 (θ,s)]]. Let N n,ǫ ⊆ ∆ n denote the set of all s-uniform mixed strategies. Lemma 4.2 impliesthat,inordertocompletetheproofofTheorem4.1,itsufficestoshowhowto exactly optimize, in the claimed time, over signaling schemes in which x σ ∈(N n,ǫ ) k for each signal σ ∈ Σ. We may restrict attention to signaling scheme/equilibrium combinations for which each mixed strategy profile x ∈ (N n,ǫ ) k is selected for at most one subgame: when x is the equilibrium for both the subgames A σ 1 and A σ 2 , we can “merge” the two signals σ 1 and σ 2 into a signal (σ 1 ,σ 2 ), giving rise to a new subgame A (σ 1 ,σ 2 ) with posterior belief µ (σ 1 ,σ 2 ) = pσ 1 pσ 1 +pσ 2 µ σ 1 + pσ 2 pσ 1 +pσ 2 µ σ 2 and probabilityp (σ 1 ,σ 2 ) =p σ 1 +p σ 2 . Lemma 4.3 implies thatx remains an (approximate) equilibrium of the merged subgame. Moreover, the objective is unchanged because A 0 (µ,x ) is linear in its first argument. Wefirstdiscardstrategyprofilesin(N n,ǫ ) k whichcannotbeinducedasequilib- riaofanyposteriorbelief. Thiscanbedoneinpolynomialtime,bycheckingwhether the corresponding polytope (as given by Lemma 4.3) is empty. Let N ⊆ (N n,ǫ ) k denotethesetofs-uniformǫ-equilibriathatcanbeinduced bysomeposteriorbelief. For notational convenience we assume that eachx∈N is induced asan equilibrium of exactly one subgame, by allowing signals which occur with probability 0. Since the number of players k is a constant, we can index N ⊆(N n,ǫ ) k as{x 1 ,...,x t } for t =|N|≤|N n,ǫ | k =n O(logn/ǫ 2 ) , and we can write our optimization task as follows. 52 maximize P t σ=1 p σ A 0 (µ σ ,x σ ) subject to P t σ=1 p σ = 1 P t σ=1 p σ µ σ =λ x σ is an equilibrium of A µ σ , for σ =1,...,t. (4.2) Lemma 4.3, and thelinearity ofA 0 (µ,x )in its first argument, imply that opti- mization problem (4.2) is of the form given in (4.1) withL= poly(k,n) constraints. Lemma 4.4, and our assumption that k = O(1), imply that (4.2) can be solved in time poly(M,n logn/ǫ 2 ). This completes the proof of Theorem 4.1. We now prove Lemmas 4.2, 4.3, and 4.4. Proof of Lemma 4.2 We can think of the objective tensor A µ 0 : [n] k → [−1,1] as describing the utility of an additional player (the principal) in the game with a trivial strategy set. The rest follows from [74, Theorem 2]. Proof of Lemma 4.3 Forx to be anǫ-WSNE ofA µ = P M θ=1 µ (θ)A θ , the following set of inequalities must hold for µ ∈∆ M : P M θ=1 µ (θ)A θ i (j,x −i )≥ P M θ=1 µ (θ)A θ i (k,x −i )−ǫ, for i∈[k],j ∈supp(x i ),k∈[n]. Since x is fixed, we have a system of poly(k,n) linear inequalities in µ . 53 Proof of Lemma 4.4 We write an equivalent linear program via a change of variables. Specifically, we let ν σ =p σ µ σ . Observe that after this change (4.1) becomes: maximize P t σ=1 w σ ·ν σ subject to P t σ=1 p σ = 1 P t σ=1 ν σ =λ νσ pσ ∈P σ , for σ = 1,...,t. (4.3) (4.3) isnot yet a linear program. However, sinceP σ isdescribed by an explicit set of inequalities C σ y b σ , the non-linear inequalities C σ ν σ /p σ b σ can be re- written as C σ ν σ p σ b σ . Moreover, note that ν σ /p σ ∈P σ ⊆ ∆ M , so p σ = P θ ν σ (θ) holds for every feasible solution. This results in an equivalent linear program with variables ν 1 ,...,ν t ∈R M + , from which p σ = P θ ν σ (θ) and µ σ =ν σ /p σ can be recov- ered efficiently. Remarks Zero-sum games When applied to two-player zero-sum games with the objec- tive to maximize the row-player’s payoff, our signaling scheme provides a stronger guarantee. In such settings, both players retain the same payoff in any exact Nash equilibrium. Also, any ǫ-equilibria give a payoff that is ǫ-close to the payoff of any exact equilibrium. Thus, the signaling scheme provided in Theorem 4.1 can be directlycomparedwiththequalityoftheoptimalsignalingschemewithoutworrying about equilibrium selection, instead of a bi-criteria guarantee. 54 Reducing the number of signals Although the signaling scheme provided in Theorem 4.1 might use a quasipolynomial number of signals, we can reduce the number of signals to M +1. Let f λ be the objective value of the signaling scheme, and consider the set oft signals used{µ 1 ,...,µ t } and their corresponding objective values {f 1 ,...,f t }. Observe that the (M +1)-dimensional point (f λ ,λ) is a convex combination of the set of points P ={(w 1 ,µ 1 ),...,(w t ,µ t )}. Since f λ is the objec- tive value of the best signaling scheme that uses only the posteriors {µ 1 ,...,µ t }, (f λ ,λ) belongs to some facet of the convex hull of P. Hence by Carathéodory’s theorem, (f λ ,λ) can be written as a convex combination of M +1 points from P, and such a decomposition can be computed in time polynomial in the size of P. This decomposition gives a valid signaling scheme with the same objective value, using only M +1 signals. Extending the bicriteria guarantee Our algorithm can extend beyond exact Nash equilibria. For every 0 ≤ δ < ǫ, we can compute a signaling scheme with ǫ-equilibria that are competitive with the optimal signaling scheme that uses δ- equilibria. Formally, let OPT δ (A,λ) denote the maximum reward for the principal over all possible signaling schemes and δ-equilibria. We can compute a signaling schemeϕ and corresponding Bayesian (ǫ+δ)-equilibriaX in time poly(M,n logn/ǫ 2 ), and the value of the signaling scheme satisfies F(ϕ,X) ≥ OPT δ (A,λ)−ǫ. Theo- rem 4.1 is a special case of this result with δ =0. Stackelberg games Ourresult can beextended toStackelberg gameswhich often arise in security games. Recall that in a Stackelberg game [91], one player (the leader) first commits to a (mixed) strategy, and then all other players (followers) simultaneously play their strategies upon learning the leader’s strategy. Our result 55 can be readily extended to Bayesian Stackelberg games when the objective of the signaling scheme is to maximize the leader’s payoff. In this case, we can simply drop the constraints regarding the leader in the polytopes defined in Lemma 4.3, and only require the followers to play an approximate equilibrium in our algorithm presented in Theorem 4.1. Equilibrium selectionrules Ouralgorithmcomputesasignalingscheme aswell as the associated (approximate) equilibria for the subgames. We assume that the principalcanimplementanyequilibrium(i.e.,thebestequilibriumshecancompute) in each subgame. It remains open whether one can find a near-optimal signaling scheme independent of equilibrium selection. For example, when the (real) players always choose the worst equilibrium for the principal after a signal is revealed. 4.4 Hardness Results for Signaling in Normal Form Games In this section, we prove hardness results for approximately optimal signaling in normal form games. In Section 4.4.1, we show that relaxing the incentive constraints is necessary if the principal’s objective can be a tensor over the state of play (Theorem 4.5). In Section 4.4.2, we show that it is NP-hard to obtain an additive FPTAS (The- orem 4.7). In Section 4.4.3, we show that assuming the planted clique conjecture (Conjecture 2.6), there is no PTAS for signaling in zero-sum games (Theorem 4.10). 56 4.4.1 NP-hardness of Signaling with Exact Equilibria Our bicriteria QPTAS in Theorem 4.1 allows the principal’s payoff to depend on the specific strategies the players take. We show that the relaxation in play- ers’ incentive constraints is necessary if we want general objective functions, even for signaling in zero-sum games. More specifically, we show that it is NP-hard to distinguish whether the optimal signaling scheme has value 0 or at least 1/2. Theorem 4.5. Given a Bayesian zero-sum game Θ,{A θ } θ∈Θ ,λ and a principal objectivetensorA 0 ,itisNP-hardtodistinguishwhethertheoptimalsignalingscheme has value 0 or at least 1 2 . The NP-hardness proofuses a reduction fromthe balanced vertex cover(BVC) problem proposed by Conitzer and Sandholm [32]. In BVC, we are given a graph G = (V,E), and we want to know if G has a vertex cover of size |V| 2 . Given an instance of BVC with n nodes, we construct the following Bayesian zero-sum game where the states of nature correspond to nodes of G (i.e., Θ = V) and the prior is uniform, i.e., λ = 1 n /n. We useRow andCol to denote the row player and the column player respec- tively. Row’s pure strategies correspond to picking a node u ∈ V, and Col’s strategies correspond to either picking a vertexv, an edgee, or a special strategys. The payoff of Col when she plays v is      n n−2 if v / ∈{θ,u}, 0 otherwise. e is      n n−2 if e is not incident with θ, 0 otherwise. s is 1. The principal is only interested in getting Col to play the strategy s, that is, A θ 0 (v,s)=1 for allθ,v∈V; all other entries of A 0 are 0. 57 The idea behind our construction is the following: nature and Row pick two nodes θ,u∈V to “protect”, but only nature “protects” all the edges incident to θ. Col can choose to “attack” either a nodev, an edgee, or to “give up” and play the strategy s. The principal wants Col to give up, so he has to coordinate the state of nature and Row’s strategy to protect different nodes. Because we do not relax the incentive constraints, the principal must find a vertex cover of size n/2. which becomes NP-hard. Lemma 4.6. The Bayesian zero-sum game defined above has a signaling scheme of value at least 1 2 if and only if G has a vertex cover of size n 2 . Proof. First, suppose G has a vertex cover C with |C| = n 2 . The principal simply signals if θ ∈C or not. That is, λ is decomposed as (µ 1 +µ 2 )/2, where µ 1 (v) = 2 n for allv∈C (and 0 otherwise), andµ 2 (v)= 2 n for allv / ∈C. For posteriorµ 1 , there is a Nash equilibrium where Row picks u ∈ V \C uniformly at random and Col chooses strategy s; thus, the principal gets a value of 1. This is because every node isprotected with probability 2 n , and every edgeisprotected with probabilityatleast 2 n ; so the payoff of Col for a pure strategy v or e is at most n n−2 1− 2 n = 1. Since µ 1 is announced with probability 1 2 , this signaling scheme achieves value at least 1 2 . On the other hand, we show that if µ is a posterior with f(µ ) > 0, then G has a BVC solution. Recall that f(µ ) is the principal’s objective value under the posterior µ . Let (x,y) be a Nash equilibrium that attains value f(µ ), that is, f(µ )=x T P θ µ (θ)A θ 0 y. Since f(µ )>0, we must have y s >0, so every node inV must be protected with probability at least 2 n . This requires nature andRow to be perfectly negatively correlated and never protect the same node. Formally, we must have n n−2 (1 − x(u))(1 − µ (u)) ≤ 1 for all u ∈ V, and summing up over all nodes we have P u (1−x(u))(1−µ (u))≤n−2, which implies 58 P u x(u)µ (u)≤ 0. So it must be that for all u∈V, exactly one of µ (u) and x(u) is equal to 2 n , and this induces a natural bisection of the graph. LetC ={v :µ v > 0}; we know|C|= n 2 . The payoff ofCol for an edgee = (s,t) is n n−2 (1−µ s −µ t ), which must be at most 1, so we haveµ s +µ t ≥ 2 n for all edges (s,t). It follows thatC is a vertex cover of G. In light of the hardness result with general objective functions (Theorem 4.5), for the rest of this chapter, we focus on signaling in two-player zero-sum games and the simplest principal’s objective function — maximizing the row player’s expected payoff at equilibrium. In zero-sum games, both players retain the same payoff in any exact Nash equilibrium, and any ǫ-equilibrium gives a payoff that is ǫ-close to the playoff of any exact equilibrium. Thus, for signaling in zero-sum games with the objective to maximize therow-player’s payoff, wecanabsorb thelossin theincentive constraints into the loss in objectives. Signaling schemes with bicriteria guarantees (e.g., our QPTAS in Section 4.3) can be directly compared with the quality of the optimal signaling scheme. In this setting, we can study the complexity of optimal signaling withoutworryingaboutequilibriumselectionorbicriteria/single-criteriaguarantees. 4.4.2 NP-hardness of an FPTAS In this section, we prove signaling in normal form games does not admit an FPTAS unless P = NP (Theorem 4.7). Theorem 4.7. There is no FPTAS for the signaling problem, even for network security games, unless P = NP. 59 Theorem 4.7follows byconsidering thesignaling problem fromadual perspec- tive. The signaling problem can be written as a mathematical program (P) with linear objective and constraints, but an infinite number of variables. Ignoring this issue, we can consider the dual problem (D). Motivated by the separation problem for the dual, we consider the posterior selection problem (Definition 2.5). Our key insight is that the posterior selection problem is a useful tool for deriving hardness results. This usefulness stems from the equivalence of separation and optimization [61], which shows that an algorithm for the separation problem can be used to solve the optimization problem and vice versa. We exploit and build upon this equivalence. We prove that this equivalence holds despite the infinite- dimensionality of (P), and furthermore, isapproximation preserving: an FPTAS for signaling yields an FPTAS for the posterior selection problem (Theorem 4.8). Whereas, typically, an (approximate) separation or membership oracle is used to(approximately) solvetheoptimizationproblem, weexploit thisequivalence inan unorthodoxfashion byleveraging thehardness oftheoptimization problem toprove hardness results for themembership problem. We show thatit is NP-hard to obtain an FPTAS for the posterior selection problem in normal form games (Lemma 4.9), and thus it is NP-hard to obtain an FPTAS for optimal signaling in normal form games. ItisworthnotingthatweobtainourNP-hardnessresultwithminimaleffort, a fact that underscores the benefits of moving to the posterior selection problem. Theorem 4.7 follows immediately from Theorem 4.8 and Lemma 4.9. Theorem 4.8. An FPTAS for the signaling problem yields an FPTAS for the posterior selection problem. 60 Proof. Recall thatf :∆ M → [−1,1] maps a posterior distribution to the principal’s objective value, and f + is the concave envelope of f (Definition 2.1). Observe that f is decided by the Bayesian game, and does not depend on the prior distribution λ. The optimal signaling scheme has value f + (λ) for a given prior λ. Consider the hypograph P ⊆ R M+1 of f + . An algorithm B for (approxi- mately) optimal signaling gives a membership oracle for P: a point (µ,η ) ∈R M+1 belongs to P if and only if η ≤ f + (µ ). For the posterior selection problem we want to compute max µ f(µ ) = max µ f + (µ ). Let w = (0,...,0,1) ∈ R M+1 . The posterior selection problem can be viewed as maximizing a linear function over P: max (µ,η )∈P η = max x∈P w T x. At a high level, the theorem statement can be inter- preted as membership oracle is sufficient for optimization [61, 81], and we need to quantify to what extent it is approximation preserving. Formally, we show that if we have a polynomial time ǫ poly(M) -approximation algorithm B for optimal signaling, then we can convert it into a polynomial time ǫ-approximation algorithm for the posterior selection problem. Given a posterior selection instance and a precision parameterǫ> 0, we invoke part (ii) ofLemma 2.8 with X = ∆ M , h(·) = f + (·), K = 2, and B as the imperfect value oracle with precisionδ = ǫ 2p(M) . Note thatp(·) is the polynomial given in Lemma 2.8, andf + is concave asneeded. Lemma 2.8will return apointx ∗ ∈∆ M in polynomial timewith f + (x ∗ )≥ max µ ∈Δ M f + (µ )−2p(M)δ = max µ ∈Δ M f(µ )−ǫ, anǫ-optimal solution for the posterior selection problem. Lemma 4.9. There is no FPTAS for the posterior selection problem, even for net- work security games, unless P = NP. Proof. The proof follows via a reduction from the balanced complete bipartite sub- graph (BCBS) problem. In BCBS, given as input a bipartite graphG= (V ∪W,E) 61 and an integer r ≥ 0, we want to determine if G contains an r×r biclique. Garey and Johnson [58] showed that the BCBS problem is NP-complete. Given a BCBS instance, we set ǫ = 1 n 8 where n = |V|+|W|. We create a Bayesian network security game on G (defined in Section 2.1.2) and set ρ = 2rnǫ. This means that states of nature correspond to nodes of G (Θ = V ∪W) and the payoff matrix for a distribution µ ∈ ∆ Θ is given by Equation (2.1) where B is the adjacency matrix of G and A = D T = −ρI n×n . Intuitively, the principal and the rowplayer want tobeadjacent toeach other, while atthesame timetheyareforced to randomize because of thelargepenalty termρif the column player catches either of them. We show that solving this posterior selection instance to precisionǫ would decide the BCBS-instance. We first show that ifG has ar×r biclique V ′ ×W ′ , then there exists someµ with f(µ )≥ 1−2nǫ. Set µ (v) = 1/r for all v ∈V ′ and x(v) = 1/r for all v ∈W ′ . Then, by Equation (2.2), we have f(µ )≥x T Bµ −ρkµ +xk ∞ = 1−ρ/r = 1−2nǫ, where x T Bµ = 1 because V ′ , W ′ form a biclique. On the other hand, if there exists µ ∈ ∆ M with f(µ )≥1−(2n+2)ǫ, then G contains anr×r biclique. Letx be the row player’s mixed strategy at equilibrium, sof(µ )=x T Bµ −ρkµ +xk ∞ . LetV ′ def = {v∈V ∪W :µ (v)≥1/n 3 }andW ′ def = {v∈ V ∪W :x(v)≥1/n 3 }. Every vertex in V ′ must be adjacent to every vertex in W ′ , otherwise x T Bµ ≤1−1/n 6 <1−(2n+2)ǫ. Thus, V ′ and W ′ must be in different partitions and form a biclique. It remains to show that |V ′ | ≥ r and |W ′ | ≥ r. By the definition of V ′ we have P v∈V ′µ (v) = 1− P v6∈V ′µ (v) > 1− 1/n 2 . Since kµ +xk ∞ = x T Bµ −f(µ ) ρ ≤ (2n+2)ǫ ρ = 1+1/n r , |V ′ |≥ P v∈V ′ µ (v) (1+1/n)/r >r 1−1/n 2 (1+1/n) =r(1−1/n). Hence |V ′ |≥r, and similarly |W ′ |≥r. 62 To our best knowledge, it remains open whether Theorem 4.8 can bestrength- ened to show that an ǫ-approximation for signaling yields an O(ǫ)-approximation for posterior selection, so that a PTAS for signaling yields a PTAS for posterior selection. We leave this as an intriguing open question (Problem 7.2). Below, we rule out a PTAS for signaling under an orthogonal hardness assumption. 4.4.3 Planted-Clique Hardness of a PTAS In this section, we rule out a PTAS for signaling in normal form games assuming the planted-clique conjecture. Theorem 4.10. There is a constant ǫ 0 > 0 such that, assuming the planted-clique conjecture (Conjecture 2.6), there is no ǫ 0 -approximation for the signaling problem in Bayesian zero-sum games. We follow the intuition behind the proof of Lemma 4.9 and construct a game wheretheprincipal needstoidentifydensesubgraphs. Themaindifferences are: (1) the error parameter ǫ 0 in this section is a constant, which is too large for detecting a few missing edges in a clique; and (2) Theorem 4.8 does not hold for translating PTAS hardness results. We resolve (1) by reducing from gap/promise problems (planted clique), where the densest large subgraphs either have density 1 or close to 1/2; and we handle (2) by using a “direct” reduction from the planted clique cover problem (Definition 2.7). Intuitively, a clique corresponds to a good poste- rior distribution; and optimal signaling decomposes the prior distribution into good posteriors, which corresponds to partitioning an input graph into dense subgraphs. The proof of Theorem 4.10 combines and strengthens techniques from [6, 46]. The idea is to set up a Bayesian zero-sum game where both the principal and the 63 rowplayermustrandomizeoverΩ(log n)-sizehigh-densitynodesetsforthesignaling schemetoachievelargevalue;recoveringtheselarge-densitysetsfromanear-optimal signaling scheme allows one to solve the planted-clique cover problem. Dughmi [46] used payoffs of magnitude Ω(log 2 n) to enforce the above property, which is insuffi- cient to rule out a PTAS. We instead leverage a device by Althöfer [6] to ensure the above “large-spreading” property. This device is used to show planted-clique hard- ness for computing the ǫ-best Nash equilibrium by Hazan and Krauthgamer [63]; and also used to show thatǫ-approximate equilibrium requires Ω(log n) support size by Feder et al. [54] (both results are for constant ǫ). One crucial technical issue is that we need to strengthen the planted-clique recovery result in [46]. To recover a specific planted clique S of size k = ω(log 2 n) with high probability (in the presence of other such planted cliques), [46] requires a set R ⊆ S with |R| = ω(log 2 n), whereas we only require a set R ⊆ S with |R| = Ω(log n) (which is asymptotically tight). This is important because we can only ensure that spreading takes place over Ω(log n)-size sets. We reduce from the planted clique cover problem with k = k(n) such that k =ω(logn) and k =o( √ n), and r = 5n k , which we omit for the rest of the section. We use G − and G + to denote the background edges and the clique edges added in steps (1) and (2) of Definition 2.7 respectively. Note that G − and G + may contain the same edges. We use bi-density G (S,T) to denote the density of the bipartite graph S×T in G: bi-density G (S,T) def = |{(u,v)∈S×T :{u,v}∈E}| |S||T| . We require the planted clique instance to satisfy Lemma 4.11. 64 Lemma 4.11. A graph G∼G n, 1 2 ,k,r with planted cliques {S 1 ,...,S r } satisfies the following properties with high probability (for sufficiently large n), (i) All large bipartite subgraphs have density about 1 2 before planting the cliques: For all S,T ⊆V with |S|,|T|≥c 2 logn, bi-density G −(S,T)≤ 1 2 + 1 20 . (ii) Almost all nodes are in some clique: LetS def = V \ S i S i . We have|S|≤e −4.9 n. (iii) All cliques are robustly recoverable: For every planted clique S i and every subset R ⊆ S i with |R| = c 3 logn, there is a polynomial time algorithm that recovers S i from G given R. Theorem 4.10 follows immediately from Lemmas 4.11 and 4.12. Lemma 4.12. LetG∼G n, 1 2 ,k,r be a planted clique cover instance that satisfies Lemma 4.11. There is a polynomial-time randomized reduction that takes the graph G as input and outputs a Bayesian zero-sum game such that the following hold with high probability. (Completeness) There is a signaling scheme of value at least 0.99. (Soundness) Given a signaling scheme of value at least 0.97, one can recover a constant fraction of the cliques planted in G. It is worth pointing out that the Bayesian zero-sum game we construct always admits a signaling scheme of large value; however finding a near-optimal signaling scheme in polynomial time would refute the planted-clique conjecture. In the rest of this section, we prove Lemma 4.12. We use the following param- eters. ǫ =0.03, Z = 20, c 3 =10 3 , c 2 =10 5 , c 1 =c 2 log(4Z/3)+2, N =n c 1 . 65 To keep the presentation simple, we give a construction where A θ i,j ∈ [−Z,Z] (as opposed to [−1,1]). LetA G denote the (n×n) adjacency matrix ofG = (V,E), and let A − G and A + G denote the adjacency matrices of G − and G + respectively. We useRow andCol to denote the row and column players respectively. The states of natureandRow’sstrategies correspond tothenodesofG. Thepriorλis 1 n /n, i.e., each state of nature (each vertex) is equally likely to occur. For every θ ∈ Θ =V, the payoff matrix A θ ∈ [−Z,Z] n×(2N+1) is given by [a θ B 1 n (d θ ) T ], which are defined as follows: (1) a θ is the θ-th column of the adjacency matrix A G , so a θ i = 1 if (i,θ)∈E and is 0 otherwise. (2) B is an n×N matrix where each B i,j is set independently to 2−Z with probability 3 4Z , and 2 otherwise. (3) d θ ∈ [−Z,Z] N is the θ-th row of B. Equivalently, if we use D to denote the n×N matrix having rows (d θ ) T for θ∈Θ, we have D =B. To gain some intuition, observe that for a posteriorµ andRow’s mixed strat- egy x, the row vector x T A µ yielding Col’s payoffs is [x T A G µ x T B µ T D]. Thus, if Col plays action 1 (with probability 1), the expected payoff of Row is equal to x T A G µ . If µ and x are uniform over S,T ⊆ V, the expected payoff is exactly bi-density G (S,T). The remaining 2N pure strategies of Col are used to force the principal and Row to choose a posterior µ and mixed strategy x respectively that are “well spread out”. TheaverageoftheentriesinanycolumnofB andDis 5 4 > max i a θ i . Exploiting this, part (i) of Lemma 4.13 implies that if x and µ both randomize uniformly over a large set of vertices, Col plays column 1. The completeness proof follows 66 from (roughly speaking) choosing posteriors and Row’s strategies that randomize uniformly over the planted cliques. For the completeness proof, ifRow’s strategyx (in some subgame) hassupport ofsize at mostc 2 logn, then part (ii) of Lemma 4.13 implies that there exists a column of B that Col can play to make f(µ ) negative. Similarly, for a posterior µ with small support, Col can play some column of D to makef(µ )negative. Thus, inordertoobtainvaluecloseto1, bothµ andRowhave to randomize over Ω(log n)-size sets of nodes. Using this, one can carefully extract a collection of node-sets which can then be used to recover the planted cliques. Intuitively, B is used to force Row (x) to randomize over a large set, and D is used to force Nature (µ ) to randomize. Formally, we prove the following lemma about the matrix B (and D). Lemma4.13. Forthen×N matrixB i,j whereeachB i,j isindependentlysetto2−Z with probability 3 4Z and set to 2 otherwise, the following hold with high probability. (i) Randomizing over a large set is good: For a fixed set R ⊆ V with |R| = ω(logn), we have 1 |R| P i∈R B i,j > 1 for every j ∈[N]. (ii) Any distribution supported on a small set is bad: For every R ⊆ V with |R|≤c 2 logn, there exists some j ∈[N] such that B i,j = 2−Z for all i∈R. Proof. We first prove (i). The proof is a standard application of Chernoff bounds. Fix a column j ∈[N]. We haveE P i∈R B i,j |R| = 5 4 , where the expectation is over the random construction of B. Since |R| = ω(logn), the size of R is large enough so that Chernoff bounds imply that Pr P i∈R B i,j |R| < 9 8 ≤ 1 2N poly(n) . The union bound over all N columns yields the claim. 67 We now prove (ii). It is sufficient to show the claim for all R ⊆ V with |R| = c 2 logn. Fix some R with |R| = c 2 logn. For a given j ∈ [N], we have Pr[∃i∈R s.t. B i,j 6= 2−Z] =1− 3 4Z |R| . So Pr[∀j ∈[N],∃i∈R s.t. B i,j 6=2−Z]= 1− 3 4Z |R| N ≤ exp −N 3 4Z |R| =exp(−n 2 ). In other words, the probability thatB failsto “catch” a small setR isexponentially small. Taking the union bound over allR⊆V with |R|=c 2 logn, we obtain Pr ∃R⊆V with |R|=c 2 logn s.t. no j ∈[N] satisfies B i,j = 2−Z for all i∈R ≤ n c 2 logn ! exp(−n 2 )≤exp c 2 log 2 n−n 2 ≤ 1 exp(n) . Therefore, the probability that there exists a j ∈ [N] to “catch” every R with |R|=c 2 logn is at least 1− 1 exp(n) as claimed. Completeness proof in Lemma 4.12 We use a deterministic signaling scheme that groups together states of nature in the same planted clique. We first partition the graph into disjoint large cliques and a small number of remaining nodes. Let S ′ i = S i \ S 1≤j<i S j for i ∈ [r] be the set of vertices in S i that do not appear in earlier cliques. Define S def = V \ S j S j as the remaining vertices. Finally, let S ′ 0 def = S∪ n v∈S ′ i :|S ′ i |< k 10 4 o . Our signaling scheme is (Σ,p,µ ) where the set of signals is Σ = {0}∪ n i ∈ [r] : |S ′ i | ≥ k 10 4 o . For each signal σ, p σ = |S ′ σ | n and µ σ is the uniform distribution over S ′ σ . For posterior µ σ , where σ 6= 0, consider the strategy x σ where Row plays 68 the uniform distribution on S ′ σ . Part (i) of Lemma 4.13 implies that Col’s best response tox σ is to play column 1, with high probability over the randomness inB. Therefore, f(µ σ )≥bi-density(S ′ σ ,S ′ σ ) =1− 1 |S ′ σ | ≥1− 10 4 k . Recall that G is a good planted clique instance, thus part (ii) of Lemma 4.11 guarantees that |S| ≤ e −4.9 n. With r = 5n k , we have |S ′ 0 | ≤ |S|+ 5n k · k 10 4 ≤ e −4.7 n. So the signaling scheme has value at least X σ∈(Σ∩[r]) p σ f(µ σ )≥ X σ∈(Σ∩[r]) p σ 1− 10 4 k ≥(1−e −4.7 ) 1− 10 4 k ≥0.99. Soundness proof in Lemma 4.12 For a signal σ ∈ Σ with corresponding posterior µ σ , let x σ denote Row’s equilibrium strategy for A µ σ . We first filter out the set of “useful” signals with relatively high values. Let Σ 1 = {σ ∈ Σ : f(µ σ ) ≥ 1− √ ǫ}. The value of the signaling scheme is P σ∈Σ p σ f(µ σ ) ≥ 1−ǫ. Noting that f(µ ) ≤ 1 for all µ , by a simple counting argument, we have p Σ 1 ≥ 1− √ ǫ. We show that for all σ ∈ Σ 1 , µ σ and x σ place a significant mass over a large set of nodes, and use this insight to extract clusters. Recall that ǫ = 0.03 and Z = 20. For every signal σ ∈ Σ 1 , define T σ = n i :e T i A G µ σ ≥1− Z √ ǫ Z−2 o , and let e x σ be the uniform distribution on T σ . Intuitively, T σ is the set of good strategies for Row under the signal σ. Let T ={T σ :σ∈Σ 1 } denote the collection of these node-sets. As we shall see, T is going to play an important role for recovering a constant fraction of the planted cliques. Fix σ ∈ Σ 1 with 1− √ ǫ ≤ f(µ σ ) ≤ 1. We first show that T σ cannot be too small. Otherwise, Col can punish Row for concentrating on a few strategies. By 69 the definition of T σ , f(µ σ ) ≥ 1− √ ǫ and a simple counting argument, we have x σ (T σ )≥ 2 Z . It follows that|T σ |>c 2 logn, because everyR⊆V with|R|≤c 2 logn must satisfy x σ (R) < 2 Z . Otherwise, suppose x σ (R) ≥ 2 Z , then by part (ii) of Lemma 4.13, there exists a column j of B having B i,j = 2−Z for all i ∈ R. We have P i∈[n] (x σ (i))B i,j ≤ (2−Z)x σ (R) + 2 1−x σ (R) ≤ 0, which implies that f(µ σ )≤0, a contradiction. We now switch from x σ to e x σ in order to relate the value of the signaling scheme to bi-density. As before, G − are the background edges and G + are the clique edges, and A − G and A + G are the corresponding adjacency matrices. Let A i G be the adjacency matrix of the clique S i . Note that A G ≤A − G +A + G ≤ A − G + P r i=1 A i G (since the planted cliques contain existing edges of G − and may overlap). Let R denote the c 2 logn largest entries in e x T σ A − G , and let e µ σ be the uniform distribution on R. Recall that G is a good planted clique instance that satisfies Lemma 4.11. Since both T σ and R have size at least c 2 logn, and e µ σ and e x σ are uniform distributions over them, part (i) of Lemma 4.11 guarantees that e x T σ A − G e µ σ = bi-density(T σ ,R)≤ 11 20 . Moreover, we have µ σ (R)< 2 Z = 1 10 because otherwise Col can choose a strategy in D to punish the concentration in µ σ . Since the maximum entry of e x T σ A − G outside of R is at most the average entry in R, we have e x T σ A − G µ σ ≤ 1 10 + 9 10 ·e x T σ A − G e µ σ ≤ 1 10 + 9 10 · 11 20 < 3 5 . This tells us that the background density does not contribute enough to make f(µ σ ) close to 1. Therefore, the reason behind the high value of µ σ must be that T σ ×R overlaps with some of the planted cliques. 70 Because P σ∈Σ 1 p σ (e x T σ A G µ σ ) ≥ (1− √ ǫ) 1− Z √ ǫ Z−2 > 2 3 (substituting in our choice of ǫ= 0.03 andZ = 20), we have 1 15 = 2 3 − 3 5 < X σ∈Σ 1 p σ (e x T σ A G µ σ )−max σ∈Σ 1 e x T σ A − G µ σ ≤ X σ∈Σ 1 p σ e x T σ (A G −A − G )µ σ ≤ X σ∈Σ 1 p σ r X i=1 e x T σ A i G µ σ = r X i=1 X σ∈Σ 1 p σ µ σ (S i ) |T σ ∩S i | |T σ | ≤ r X i=1 X σ∈Σ 1 p σ µ σ (S i ) max T∈T |T ∩S i | |T| ! (∗) ≤ r X i=1 |S i | n max T∈T |T ∩S i | |T| ! = 5 r r X i=1 max T∈T |T ∩S i | |T| ! . Inequality (∗) follows since for every v ∈ Θ, we have P σ∈Σ 1 p σ (µ σ ) v is at most P σ∈Σ p σ (µ σ ) v =λ v = 1 n . We have 1 r P r i=1 max T∈T |T∩S i | |T| ≥ 1 75 . By a simple counting argument, at least a 1 297 -fraction ofS 1 ,...,S r satisfy max T∈T |T∩S i | |T| ≥ 1 100 . Therefore, to recover a constantfractionofthecliques,itissufficienttoshowthatwecanrecoveranyS =S i from a set T with |T|≥c 2 logn and |S∩T|≥ |T| 100 ≥c 3 logn. Observe that we can assume without loss ofgeneralitythat|T|≤ 2c 2 logn. Otherwise we canpartitionT into disjoint subsets of size between c 2 logn and 2c 2 logn, and one of these subsets T ′ would have |S∩T ′ | |T ′ | ≥ |S∩T| |T| . Part (iii) of Lemma 4.11 states that we can, in polynomial time, recover any planted clique S i given an arbitrary subset R ⊆ S i with size |R|≥c 3 logn. With |T|≤ 2c 2 logn and |S∩T|≥c 3 logn, we can simply enumerate allsubsetsR⊆T ofsizec 3 lognandrun theclique recovery algorithmin 71 Lemma 4.11. Moreover, the enumeration can be done in time 2c 2 logn c 3 logn = poly(n). So by iterating through every T ∈T, partitioning and running the clique recovery algorithm, we can recover all the S i ’s that satisfy max T∈T |T∩S i | |T| ≥ 1 100 , which is at least a constant fraction of all the planted cliques. This concludes the soundness proof. A Tighter Amplification Lemma for Planted Clique The rest of this chapter is devoted to prove Lemma 4.11. Lemma 4.11 gives three properties that a planted clique cover instanceG∼G(n, 1 2 ,k,r) should satisfy with high probability. Recall that k = k(n) satisfies k = ω(logn) and k = o( √ n), r = Θ(n/k), and c 3 = 10 3 . As before, we use G − to denote the background edges and G + to denote the clique edges (see Definition 2.7). For part (i), we need to show that, with high probability, all large bipartite subgraphs have density close to 1 2 in G − ∼ G(n, 1 2 ). This is a direct corollary of Lemma 4.14 with c = c 2 = 10 5 and ǫ = 0.1. Lemma 4.14 follows from a standard application of the Chernoff bound and the union bound. Lemma 4.14 (Proposition B.2 in [46] quantified). Let 0<ǫ< 1 and c≥ 50· 1+ǫ ǫ 2 . For all n≥2, we have Pr h ∃S,T ⊆V with |S|,|T|≥clnn, bi-density G −(S,T)> 1+ǫ 2 i ≤ 1 n 4 . Part (ii) claims that all except a small constant fraction of the nodes are covered by some clique. Recall that r = 5n k , and S def = V \ S j S j is the set of uncovered nodes. We have E h |S| i = n·Pr h v∈S i = n(1− k n ) r ≤ e −5 n, and |S| ≤ 72 e 0.1 ·E[|S|]≤ e −4.9 n with high probability due to standard Chernoff bounds (since the events {v∈|S|} v∈V are negatively correlated). Our main technical contribution in Lemma 4.11 is part (iii). The claim is the following: For G ∼ G(n,p,k,r) with planted cliques {S 1 ,...,S r }, with high probabiltiy over the randomness in G, every planted clique S i can be recovered in polynomial time given an arbitrary subset R⊆S i with |R|≥c 3 lnn. It is well known that for a planted clique instance G ∼ G(n, 1 2 ,k), with high probabilityovertherandomnessinG,onecanrecovertheplantedcliqueS givenR⊆ S with |R|≥c 3 lnn. We generalize this result and show that, despite the presence of Θ( n k ) other planted cliques, every clique S i can still be recovered from c 3 lnn nodes. 4 It is important that our recovery algorithm works for any R ⊆ S (rather than with high probability for a fixed R). This is because the set R is obtained from a near-optimal signaling scheme ϕ. Since ϕ is produced by an algorithm after examining the planted clique cover instance G, the choice of R can depend on the realization of G. Fix some i∈[r] and let S =S i . We use the following algorithm to recover S: (1) Let S ′ be all the common neighbors of R. (2) Let b S be the vertices in S ′ with at least k−1 neighbors in S ′ . Part (iii) follows immediately from Lemmas 4.15 and 4.16, and a union bound over all i ∈ [r]. We use E, E − and E + to denote the edges of G, G − and G + respectively. 4 We tighten the recovery algorithm in [46], which requires |R| = ω(log 2 n) nodes from the planted clique. The difference in the magnitude of |R| poses certain challenges and necessitates some key changes to the analysis in [46]. 73 Lemma 4.15. With high probability, we have |S ′ |≤|S|+c 3 lnn. Lemma 4.16. With high probability, for all v / ∈S we have |E(v,S)|≤0.7|S|. We first elaborate how they imply part (iii) of Lemma 4.11. Observe that all the nodes in S will survive Step (1) and (2), so S ⊆ b S. We show that, with high probability, no other vertices survive Step (1) and (2). Lemma 4.15 states that at most c 3 lnn nodes outside of S survive Step (1). Lemma 4.16, together with the assumption that |S| = ω(logn), implies that E(v,S ′ ) ≤ E(v,S) + (|S ′ |−|S|) ≤ 0.7|S|+c 3 lnn < 0.71|S| < k−1 for all v ∈ S ′ \S (and for sufficiently large n). Therefore, allnodesinS ′ \S getsfiltered inStep(2), andwehave b S =S asclaimed. Before we continue to prove Lemmas 4.15 and 4.16, we state the following lemma which is crucial for our analysis. Lemma 4.17 is the main reason why our clique recovery result is asymptotically tight and better than that of [46]. Lemma 4.17. With high probability, we have |E + (v,S)|≤12lnn for all v / ∈S. Proof of Lemma 4.15 We first look at bad vertices in S ′ that is due to the background edges. Let A = {v / ∈ R : |E − (v,R)| ≥ 0.8|R|}. Then, bi-density G −(R,A) ≥ 0.8. Because this density is much higher than 1 2 , it cannot be the case that both A and R are large. Formally, Lemma 4.14 holds for c = c 3 = 10 3 and ǫ = 0.6 with high probability. Since |R|≥c 3 lnn, we have |A|<c 3 lnn. 74 We next show that the clique edges do not introduce extra vertices to S ′ . By Claim 4.17, for everyv / ∈S, the clique edges will not increaseE(v,S) too much. For all but at most c 3 lnn nodes v / ∈S, we have |E(v,R)|=|E − (v,R)|+|E + (v,R)|≤ 0.8|R|+|E + (v,S)|≤0.8|R|+12lnn≤0.82|R|. Hence, with high probability, at most c 3 lnn nodes outside of S survive Step (1). Proof of Lemma 4.16 We prove the claim for a fixed v / ∈ S and then take a union bound over all v / ∈S. We upper bound|E(v,S)| by inspecting the edges inE − andE + separately. With high probability, we have|E − (v,S)|≤0.6|S| for allv / ∈S. This is a standard application of the Chernoff bound and the union bound, since |S|=ω(logn). ForedgesinE + (v,S)weuseLemma4.17. Withhighprobability, forallv / ∈S, we have |E + (v,S)|≤12lnn =o(|S|), and therefore for allv / ∈S, |E(v,S)|=|E − (v,S)|+|E + (v,S)|≤0.6|S|+o(|S|)≤0.7|S|. Proof of Lemma 4.17 Lemma 4.17 bounds the number of edges between a set S (of size k) and a vertex v / ∈ S, when the graph is exactly the union of Θ( n k ) random k-cliques. The expected number of times v gets covered by these cliques is Θ(1), so with high probability,v is coveredO(logn) times. On theother hand, the expected size of the overlap between some clique S i and S is k 2 n = o(1), so with high probability, every 75 S i overlaps with S on O(logn) vertices. If we simply combined these two bounds (as in [46]), we get a weaker version of Lemma 4.17 with E + (v,S)≤O(log 2 n). The key observation (by David Kempe) is that we can use the principle of deferred decisions to improve this analysis. We ask all the cliques first to decide whether to include v or not, and defer their choices on other nodes. Roughly speaking, there are O(logn) cliques that include v, and they contain O(klogn) random vertices; The expected total size of the overlap between these cliques and S is at most O(klogn· k n ) = o(1), so tail bounds and the union bound imply that E + (v,S)≤O(logn). The following lemma will be useful in proving Lemma 4.17. Intuitively it says that, to upper bound E + (v,S), we can pretend the O(klogn) vertices (from the O(logn) cliques that containv) arechosen independently and uniformlyat random. Lemma 4.18(seeEx. 1.13in[45],Lemma1.19in[44]). LetX 1 ,...,X n bearbitrary binary random variables. Suppose for every i, and every x 1 ,...,x i−1 ∈ {0,1}, we have Pr[X i = 1|X 1 = x 1 ,X 2 = x 2 ,...,X i−1 = x i−1 ] ≤ p i . Let Y 1 ,...,Y n be independent binary random variables with Pr[Y i = 1] =p i for all i∈ [n]. Then, for any M, we can upper bound Pr[ P n i=1 X i > M] using the upper-tail Chernoff bound for Pr[ P n i=1 Y i >M]. In particular, for any ǫ ∈ (0,1) and µ ≥ P n i=1 p i , we have Pr[ P n i=1 X i > (1+ǫ)µ ]≤e −ǫ 2 µ/ 3 . Proof of Lemma 4.17. Fixv / ∈S, and letX denote the random variable|E + (v,S)|. LetS 1 ,...,S r−1 be the planted cliques other thanS. LetI be therandom index-set of cliques that contain v; that is, I ⊆ [r−1] is such that v ∈ S i for all i ∈ I, and 76 v / ∈ S i for all i / ∈I. Notice that the events {i ∈I} for i∈ [r−1] are independent Bernoulli trials with probability k n . So we have Pr[|I|> 6logn]≤ 1 n 2 . Fix an index set J ⊆ [r − 1] with |J| ≤ 6logn and consider Pr[X > 12logn|I =J]. We use Pr ′ to denote probabilities conditioned on the event I =J. Conditioned onI =J, we haveX ≤ P i∈J,u∈S Y i,u , whereY i,u is the random variable indicating if u ∈ S i . Fix an ordering of the Y i,u random variables. If we consider the random variable Y i,u , and any realization σ of the random variables appear- ing before Y i,u , we have Pr ′ [Y i,u = 1|realizationσ of the variables before Y i,u ] ≤ k n . Since |J|k 2 n <6logn, we can now use Lemma 4.18 and infer that Pr ′ [X >12logn]≤ e − 6logn 3 . Finally, we have Pr[X >12logn] = X J⊆[r−1] Pr[I =J]·Pr[X >12logn|I =J] ≤ X J⊆[r−1]: |J|>6logn Pr[I =J]+ X J⊆[r−1]: |J|≤6logn Pr[I =J]·Pr[X >12logn|I =J] ≤ Pr[|I|> 6logn]+ X J⊆[r−1]: |J|≤6logn Pr[I =J]· 1 n 2 ≤ 2 n 2 . 77 Chapter 5 Mixture Selection: An Algorithmic Framework In Chapter 3 and Chapter 4, we settled the computational complexity of opti- mal signaling in Bayesian network routing games and normal form games. The best approximationalgorithmwehadforthesetwoclassesofgamesarefairlydifferent. In network routing games we simply reveal full information; and in normal form games we use one signal for each approximate equilibrium and solve a quasipolynomial size linear program. There are many other interesting game-theoretic applications that involves the design of information structures. Do we need to come up with a different approximation algorithm for every new class of game we encounter, or is there a building block that many of these signaling problems have in common? Inthischapter, weidentifytwoparametersthatseemtodictatethecomplexity of optimal signaling, and present an algorithmic framework that (approximately) solves the optimal signaling problem in a number of different Bayesian games. We poseandstudyafundamentalalgorithmicproblemwhichwetermmixture selection, a problem that arises naturally in the design of optimal information structures: Definition 5.1 (Mixture Selection). For a function g : [−1,1] n → [−1,1] and a positive integer M, M-dimensional mixture selection for g is the following opti- mization problem: Given an n×M matrix A with entries in [−1,1], find x in the M-dimensional simplex ∆ M maximizing f(x) def = g(Ax). 78 Themixtureselectionproblemiscloselyrelatedtotheposteriorselectionprob- lem (Definition 2.5) and the optimal signaling problem. At a high level, mixture selection and posterior selection ask for the best posterior distribution for the prin- cipal’s objective, while the optimal signaling problem asks for the optimal way to decompose a prior into “good” posteriors that maximizes the principal’s expected objective value. Recall thatf(µ ) denotes the principal’s objective value under the posteriorµ . To connect mixture selection to the posterior selection problem, we consider signal- ingproblemswheref(µ )canbewrittenasg(Aµ )forafunctiong : [−1,1] n →[−1,1] and a matrix A∈ [−1,1] n×M . The mixture selection problem is more general, and it captures the posterior selection problem with a fixed g and arbitrary A. Mix- ture selection focuses on how the complexity ofg affects the complexity of posterior selection. InSection4.4.2, wehavealreadyseen thattheposteriorselection problem can beuseful when proving hardness results ofsignaling. Inthischapter, we present a meta-algorithm that works for both signaling and posterior selection, where the running time of our algorithm depends only on the “complexity” of g. The work presented in this chapter appeared in [26]. 5.1 Summary of Results We investigate how the complexity of mixture selection (and optimal signal- ing) depends on the complexity of the function g. We identify two “smoothness” parameters of the function g which tightly control the complexity of mixture selec- tion. The first smoothness quantity is a familiar one, namely Lipschitz continuity in the L ∞ metric. The second quantity, which we define and term noise stability 79 (Definition 5.3), borrows ideas from related definitions of stability in other contexts (e.g., [68, 79]), though it is importantly different. Informally, noise stability con- trols the degree to which low-probability — and possibly correlated — errors in the inputs of g can impact its output. The approximation guarantee of our algorithm degrades gracefully as a func- tion of the Lipschitz continuity and noise stability of g (Theorem 5.6). Moreover, the same conditions — noise stability and Lipschitz continuity — on the functiong alsolead toasimilar approximation scheme forthecorresponding signaling problem (Theorem 5.10). In particular, wheng is bothO(1)-Lipschitz continuous andO(1)- stable, we obtain an (additive) polynomial-time approximation scheme (PTAS) for mixture selection and optimal signaling. We also show that neither assumption suffices by itself for an additive PTAS (Theorems 5.18 and 5.19). Our results for mixture selection can be viewed as generalizing the main insightsofLiptonetal.[74]. First,weshowthatwheng isnoisestableandLipschitz continuous, and x ∈ ∆ M is arbitrary, there is a sparse vector e x for which g(Ae x) is not much smaller than g(Ax). The proof of this fact proceeds by sampling from x and letting e x be the empirical distribution, as in [74]. However, when g is suffi- ciently noise stable and Lipschitz continuous, we obtain a better tradeoff between the number of samples required and the error introduced into the objective than does [74], and this is crucial for our applications. Our analysis bounds the expected difference between g(Ax) and g(Ae x) as the sum of two terms: The first term rep- resents the error in the output of g caused by the low-probability “large errors” in itsn inputs, and the second term represents the error in the output ofg introduced by the higher-probability “small errors” in its n inputs. The first term is bounded using noise stability, and the second is bounded using Lipschitz continuity. 80 Second, we instantiate the above insight algorithmically, as does [74]. Specif- ically, our algorithm enumerates vectors e x of the desired sparsity in order to find an approximately optimal solution to our mixture selection problem. We note that our guarantees are all parametrized by the Lipschitz continuity c and the noise sta- bility β of the function g. Most notably, we obtain an additive polynomial-time approximation scheme (PTAS) whenever bothβ and c are constants. Despite the simplicity of our framework, we find that it has powerful impli- cations for optimal signaling in games. Notably, we find that we resolve or make progressonanumberofknownopenproblems,andsomenewones,usingoneunified algorithmic framework. 1. Optimal signaling in Bayesian normal form games (defined in Sec- tion 2.1.2): In Section 5.3.1, we derive a new QPTAS for this problem using the mixture selection framework. We use the fact that every function isO(n)- stable, and the fact that the function measuring the quality of equilibria sat- isfies a bi-criteria notion of Lipschitz continuity which we define. 2. Revenue-maximizing signaling inprobabilisticsecond-price auctions (defined in Section 2.1.5): A PTAS for this problem follows easily from our framework. We use the fact that the function max2(·), the second largest entry of a vector, is Lipschitz continuous and noise stable. 3. Persuasion in voting (defined in Section 2.1.6): We design a multi- criteria PTAS for this problem using our framework, using the fact that the functiong (vote-sum) (t) = 1 n |{i :t i ≥0}|isnoisestableandLipschitzcontinuous in a bi-criteria sense. 81 We present the results for auctions and voting together in the next chapter (Chapter6),wherewesystematicallyexploreoptimalsignalinginanonymousgames. 5.2 Noise Stability and Lipschitz Continuity In this section, we present our notion of noise stability, and derive approxima- tion algorithms for this problem when the function g is simultaneously noise stable and Lipschitz continuous with respect to the L ∞ metric. Our approximation guarantees will be additive — i.e., an ǫ-approximation algorithm for mixture selection outputsx∈∆ M withf(x)≥ max y∈Δ M f(y)−ǫ. To illustrate our techniques, we use the following function g (mid) : [−1,1] n → [−1,1], which averages all but the top and bottom quartiles of its inputs, as a running example. g (mid) (t) = 1 ⌈3n/4⌉−⌊n/4⌋ ⌈3n/4⌉ X i=⌊n/4⌋+1 t [i] , where t [i] denotes the i th largest entry of t. Throughout this chapter, we use t i to denote the i th entry of t, and use t [i] to denote the i th largest entry of t. Though we present our framework for functions g : [−1,1] n → [−1,1], we define mixture selection similarly for functions g : [0,1] n → [0,1]. The two defini- tions are equivalent up to normalization, and it is easy to verify that all our results and bounds for mixture selection carry through unchanged to either definition. OurmainresultappliestofunctionsgwhicharebothnoisestableandLipschitz continuous with respect to the L ∞ metric. We now formalize these two conditions. 82 Lipschitz Continuity A function g : [−1,1] n → [−1,1] is c-Lipschitz continuous in L ∞ — or c-Lipschitz for short — if and only if for all t,t ′ in the domain of g, |g(t)−g(t ′ )|≤ c||t−t ′ || ∞ . To illustrate, our example function g (mid) is 1-Lipschitz. We note that Lipschitz continuity in L ∞ is a stronger assumption than in any otherL p norm. Noise Stability Our notion of noise stability captures the following desirable property of a function g : [−1,1] n →[−1,1]: if a random process corrupts (i.e., modifies arbitrar- ily) some of the inputs to g, with no individual input disproportionately likely to be corrupted, then the output ofg does not decrease by much in expectation. Such random corruption patterns are captured by our notion of a light distribution over subsets of [n], defined below. Definition 5.2 (Light Distribution). Let D be a distribution supported on subsets of [n]. For α ∈ (0,1], we say D is α-light if and only if Pr R∼D [i ∈ R] ≤ α for all i∈[n]. In other words, a light distribution bounds the marginal probability of any individual element of [n]. When corrupted inputs follow a light distribution, no individual input is too likely to be corrupted. However, we note that our notion of light distribution allows arbitrary correlations between the corruption events of variousinputs. Wedefineanoisestablefunctionasonewhichisrobust,inanaverage sense, tocorruptingasubsetR ofitsninputswhenR follows alightdistributionD. Our notion of robustness is one-sided: we only require that our function’s output not decrease substantially in expectation. This one-sided guarantee suffices for all 83 our applications, and is necessitated by some. We note that the light distribution D, as well as the (corrupted) inputs, are chosen adversarially. We make use of the following notation in our definition: Given vectors t,t ′ ∈[−1,1] n and a set R⊆[n], we say t ′ ≈ R t if t i = t ′ i for all i 6∈ R. In other words, if t ′ ≈ R t, then t ′ is a result of corrupting only the entries of t corresponding to R. Definition 5.3 (Noise Stability). Given a function g : [−1,1] n → [−1,1] and a real number β ≥ 0, we say g is β-stable if and only if the following holds for all t∈[−1,1] n , α∈(0,1], and α-light distributions D over subsets of [n]: E R∼D min{g(t ′ ):t ′ ≈ R t} ≥g(t)−αβ. To illustrate this definition, we show that our example function g (mid) is 4-stable. To see this, observe that changing k entries of the input to g (mid) can decrease its output by at most 4k n . This is because each of thek entries can go from 1 to −1 in the worst case, causing a change of 2k, and then we normalize by n/2. WhenR isdrawnfromanα-lightdistribution andtisanarbitraryinput, 4-stability therefore follows from the linearity of expectations: E R∼D min{g (mid) (t ′ ):t ′ ≈ R t} ≥ E R∼D " g (mid) (t)− 4|R| n # ≥g (mid) (t)−4α. We note that every function g : [−1,1] n → [−1,1] is 2n-stable, which follows from the union bound. As a useful building block for proving some of our functions stable, we show that stable functions can be combined to yield other stable functions if composed with a convex, nondecreasing, and Lipschitz continuous function. 84 Proposition 5.4. Fix β,c≥0, and let g 1 ,g 2 ,...,g k : [−1,1] n → [−1,1] be β-stable functions. For every convex function h : [−1,1] k → [−1,1] which is nondecreasing in each of its arguments and c-Lipschitz continuous in L ∞ , the function g(t) def = h(g 1 (t),...,g k (t)) is (βc)-stable. Proof. For all t∈[−1,1] n and all α-light distributions D, E R∼D   min t ′ ≈ R t g(t ′ )   = E R∼D   min t ′ ≈ R t h(g 1 (t ′ ),...,g k (t ′ ))   ≥ E R∼D   h(min t ′ ≈ R t g 1 (t ′ ),...,min t ′ ≈ R t g k (t ′ ))   (Since h is nondecreasing) ≥h( E R∼D   min t ′ ≈ R t g 1 (t ′ )   ,..., E R∼D   min t ′ ≈ R t g k (t ′ )   ) (Jensen’s inequality) ≥h(g 1 (t)−αβ,...,g k (t)−αβ) (Stability of each g i ) ≥h(g 1 (t),...,g k (t))−αβc (Lipschitz continuity of h) =g(t)−αβc. As a consequence of the above proposition, a convex combination of β-stable functions isβ-stable, and the point-wise maximum ofβ-stable functions isβ-stable. 5.2.1 Consequences of Stability and Continuity We now state the two main results of our framework. Both results apply to functions g : [−1,1] n → [−1,1] which are simultaneously Lipschitz continuous and noise stable, and n×M matrices A with entries in [−1,1]. Given a vector x∈∆ M 85 and integer s > 0, we view x as a probability distribution over [M], and use the randomvariable e x∈∆ M todenotetheempiricaldistributionofsi.i.d.samplesfrom x. Formally, e x= 1 s P s i=1 e k i , where k 1 ,...,k s ∈[M] are drawn i.i.d. according to x. Ourfirstresultshowsthatwhenthenumberofsamplessischosenasasuitable functionoftheLipschitzcontinuityandnoisestabilityparameters,g(Ae x)isnotmuch smaller thang(Ax) in expectation over e x. At a high level, we bound this difference asasumoftwoerrorterms: oneaccountsfortheeffectoflow-probabilitylargeerrors in the inputs e t = Ae x to g, and the other accounts for effect of higher-probability small errors in the inputs e t. The former error term is bounded using noise stability, and the latter error term is bounded using Lipschitz continuity. Theorem 5.5. Let g : [−1,1] n → [−1,1] be β-stable and c-Lipschitz in L ∞ , let A be ann×M matrix with entries in [−1,1], letα,δ>0, and lets≥2ln( 2 α )/δ 2 be an integer. Fix a vector x ∈ ∆ M , and let the random variable e x denote the empirical distribution of s i.i.d. samples from the probability distribution x. The following then holds: E[g(Ae x))]≥g(Ax)−αβ−cδ. Proof. Denote t = Ax and e t = Ae x. Note that e t is a random variable. Also note that t i and e t i can be viewed as the mean and empirical mean, respectively, of a distribution supported on A i,1 ,...,A i,M ∈ [−1,1]. We say the i th entry of t is approximately preserved if |t i − e t i | ≤ δ, and we say it is corrupted otherwise. Let R ⊆ [n] denote the set of corrupted entries. Hoeffding’s inequality, and our choice of the number of samples s, imply that R follows an α-light distribution. Let t ′ be such that (1) t ′ i = e t i for i ∈ R, and (2) t ′ i = t i otherwise. Observe that t ′ ≈ R t, and||t ′ − e t|| ∞ ≤δ. We can now bound the expected difference between g(t) and g( e t) as a sum of the error introduced by corrupted entries and the error introduced by the approximately preserved entries of t: 86 g(t)−E h g( e t) i =E[g(t)−g(t ′ )]+E h g(t ′ )−g( e t) i ≤αβ +cδ. Notice that if we fix the desired approximation errorǫ, the minimum required number of samples s in Theorem 5.5 to guarantee that E[g(Ae x))] ≥ g(Ax)−ǫ is obtained by minimizing ⌈2ln( 2 α )/δ 2 ⌉ over α,δ > 0 satisfying αβ +δc ≤ ǫ. There- fore, the required number of samples depends only on the error term ǫ, the noise stability parameter β, and the Lipschitz continuity parameter c; in particular, it is independent of n and M. As a corollary of Theorem 5.5, we derive the following algorithmic result. Theorem 5.6. Letg :[−1,1] n →[−1,1] beβ-stable andc-Lipschitz, and letM > 0 be an integer. For every δ,α> 0, the M-dimensional mixture selection problem for g admits an (αβ+cδ)-approximation algorithm in the additive sense, with runtime n·M O(δ −2 log(1/α)) ·T, where T denotes the time needed to evaluate g on a single input. Proof. Let s ≥ 2ln(2/α)/δ 2 be an integer. Our algorithm simply enumerates all s-uniform distributions, and outputs the one maximizing g(Ax). This takes time n · M O(s) · T. The approximation guarantee follows from Theorem 5.5 and the probabilistic method. As a consequence of Theorem 5.6, the mixture selection problem for g (mid) admits apolynomial-time approximation scheme (PTAS) in theadditive sense. The same holds for every functiong which isO(1)-stableandO(1)-Lipschitz continuous. Specifically,bysettingα = ǫ 2β andδ = ǫ 2c ,anǫ-approximationalgorithmrunsintime n·m O(c 2 log(β/ǫ)/ǫ 2 ) ·T. Interestingly, neither noise stability nor Lipschitz continuity alone suffices for such a PTAS, as we argue in Section 5.4. 87 5.2.1.1 A Bi-criteria Extension of the Framework Motivated by two of our applications, namely Optimal signaling in normal form games and Persuasion in voting, we extend our framework to the design of approximation algorithms for mixture selection with a bi-criteria guarantee when the function in question is stable but not Lipschitz continuous. We first define a (δ,ρ)-relaxation of a function. Definition 5.7. Given two functionsg,h: [−1,1] n →[−1,1] and parametersδ,ρ≥ 0, we say h is a (δ,ρ)-relaxation of g if for all t 1 ,t 2 ∈[−1,1] n with ||t 1 −t 2 || ∞ ≤δ, h(t 2 )≥g(t 1 )−ρ. Note that Lipschitz continuous functions are their own relaxations. In lieu of the Lipschitz continuity condition, we prove our bounds for a relaxation of the function. Theorem 5.8. Letg : [−1,1] n →[−1,1] beβ-stable, letA be ann×M matrix with entries in [−1,1], let α > 0 and δ,ρ≥ 0, and let s≥ 2ln( 2 α )/δ 2 be an integer. Fix a vector x ∈ ∆ M , and let the random variable e x denote the empirical distribution of s i.i.d. samples from probability distribution x. The following then holds for any (δ,ρ)-relaxation h of g, E[h(Ae x))]≥g(Ax)−αβ−ρ. Proof. Because the proof is almost identical to the proof of Theorem 5.5, we just mention the necessary modifications. Again, let t = Ax, let e t = Ae x, let R ⊆ [n] denote the set of corrupted inputs, and let t ′ be such that t ′ i = e t i for i ∈ R and t ′ i =t i otherwise. Then 88 g(t)−E h h( e t) i =E[g(t)−g(t ′ )]+E h g(t ′ )−h( e t) i ≤αβ+E h g(t ′ )−h( e t) i ≤αβ+ρ, where the first inequality follows by noise stability of g, and the last inequality follows from the fact that h is a (δ,ρ)-relaxation of g. Having replaced Theorem 5.5 by Theorem 5.8, a similar computational result as Theorem 5.6 can be inferred in the bi-criteria sense. 5.3 A Meta-Algorithm for Signaling In this section, we use our framework to define an abstract signaling problem and characterize its approximation complexity. This abstract problem captures all of the signaling problems considered in this thesis. Toconnecttoourmixtureselectionframework, weconsidersignalingproblems in which the principal’s utility f(µ ) from a posterior distribution µ ∈ ∆ M can be written as g(Aµ ) for a function g : [−1,1] n → [−1,1] and a matrix A∈[−1,1] n×M . As described in Section 2.1.1, a signaling scheme ϕ with signals Σ corresponds to a family of probability-posterior pairs {(p σ ,µ σ )} σ∈Σ decomposing the prior λ ∈ ∆ M into a convex combination of posterior distributions (one per signal): λ = P σ∈Σ p σ µ σ . The objective of our signaling problem is then F(ϕ)= X σ∈Σ p σ f(µ σ ) = X σ∈Σ p σ g(Aµ σ ). 89 Wenotethatthissignalingproblemcanalternativelybewrittenasan(infinite- dimensional) linear program which searches over probability measures supported on ∆ M with expectationλ. The separation oracle for the dual of this linear program is a mixture selection problem. Whereas we do not use this infinite-dimensional for- mulation oritsdualdirectly, wenevertheless show thatthesameconditions—noise stability and Lipschitz continuity — on the functiong which lead to an approxima- tion scheme for mixture selection also lead to a similar approximation scheme for our signaling problem with f(µ )=g(Aµ ). Lemma 5.9. If g is β-stable and c-Lipschitz, then for any constants α,δ > 0, and for any integer s≥ 2δ −2 ln(2/α), there exists a signaling scheme e ϕ for which every posteriordistributioniss-uniform, andF(e ϕ)≥ OPT−(αβ+cδ) whereOPT denotes the value of the optimal signaling scheme. Proof. Let s ≥ 2δ −2 ln(2/α), and let τ ∈ [M s ] index all s-uniform posteriors, with e µ τ denoting the τ’th such posterior. For an arbitrary signaling scheme ϕ = (Σ,{(p σ ,µ σ )} σ∈Σ ), we show that each posterior µ σ can be decomposed into s-uniform posteriors without degrading the objective by more than αβ +cδ: 1. µ σ canbeexpressedasaconvexcombinationofs-uniformposteriorsasfollows. µ σ = X τ∈[M s ] e p σ,τ e µ τ with e p σ ∈ ∆ M s. (5.1) 2. The value of objective function, i.e., g(Aµ σ ), is decreased by no more than αβ +cδ through this decomposition, X τ∈[M s ] e p σ,τ ·g(Ae µ τ )≥g(Aµ σ )−(αβ+cδ). (5.2) 90 The existence ofsuch a decomposition follows fromTheorem 5.5: Fixσ, and let e µ ∈ ∆ M betheempiricaldistributionofsi.i.d. samplesfromdistributionµ σ ∈∆ M . The vectore µ isitselfarandomvariablesupportedons-uniformposteriors,itsexpectation is µ σ , and by Theorem 5.5 we have E[g(Ae µ )]≥g(Aµ σ )−(αβ +cδ). Therefore, by taking e p σ,τ = Pr[e µ = e µ τ ] for each τ ∈[M s ] we get the desired decomposition of µ σ . The lemma follows by composing the decomposition ϕ with the decomposi- tions of the posterior beliefs µ σ to yield a signaling scheme e ϕ with only s-uniform posteriors andF(e ϕ)≥F(ϕ)−(αβ+cδ). Specifically, the signals of e ϕ are Σ×[M s ], where signal (σ,τ) has probability p σ · e p σ,τ and induces the posterior e µ τ . 1 Using Equations (5.1) and (5.2), it is easy to verify that this describes a valid signaling scheme with F(e ϕ)≥F(ϕ)−(αβ+cδ). Lemma 5.9 permits us to restrict attention to s-uniform posteriors without much loss in our objective. Since there are onlyM s such posteriors, a simple linear program with M s variables computes an approximately optimal signaling scheme. Theorem 5.10 (Polynomial-Time Signaling). If g is β-stable and c-Lipschitz, then for any constant α,δ > 0, there exists a deterministic algorithm that constructs a signaling scheme with objective value at least OPT− (αβ +cδ), where OPT is the value of the optimal signaling scheme. Moreover, the algorithm runs in time poly(M δ −2 ln(1/α) )·n·T, where T is the time needed to evaluate g on a single input. Proof. Let s be an integer with s ≥ (2δ −2 ln(2/α)), and let τ ∈ [M s ] index all s- uniform posteriors. Lemma 5.9 shows that restricting to s-uniform posteriors only introduces anαβ+cδ additive loss in the objective. Thus it suffices to compute the 1 Note, however, that we can also “merge” all signals with the same posteriore µ τ without loss. 91 optimal signaling scheme supported only ons-uniform posteriors. This can be done using the following linear program: maximize P τ∈[M] p τ ·g(Aµ τ ) subject to P τ∈[M] p τ µ τ =λ p∈∆ M (5.3) Note µ τ is the τ’th s-uniform posterior — the only variables in this LP are p 1 ,...,p M s. Our proofs can be adapted to obtain a bi-criteria guarantee in the absence of Lipschitz continuity, as in Section 5.2. The following theorem follows easily, and we omit the details. Theorem 5.11 (Polynomial-Time Signaling (Bi-criteria)). Let g,h : [−1,1] n → [−1,1] be such that g is β-stable and h is a (δ,ρ)-relaxation of g, and let α > 0 be a parameter. There exists a deterministic algorithm which, when given as input a matrix A ∈ [−1,1] n×m and a prior distribution λ ∈ ∆ M , constructs a signaling scheme ϕ ={(p σ ,µ σ )} σ∈Σ such that X σ∈Σ p σ h(Aµ σ )≥OPT −αβ−ρ, where OPT is the maximum value of F(ϕ ∗ ) = P σ∈Σ ∗p ∗ σ g(Aµ ∗ σ ) over signal- ing schemes ϕ ∗ = {(p ∗ σ ,µ ∗ σ )} σ∈Σ ∗. Moreover, the algorithm runs in time poly(M δ −2 ln(1/α) )·n·T, where T denotes the time needed to evaluate h on a single input. 92 Remarks We note that our proof suggests an extension of the result in Theorem 5.10 to cases in whichf is given by a “black box” oracle, so long as we are promised thatitisoftheformf(µ )=g(Aµ ). Inthismodel theruntimeofouralgorithmdoes not depend onn, but instead depends on the cost of querying f. We also point out that even though we precompute the quality of allM s posteriors, we can guarantee that our output signaling scheme uses at most M +1 signals; this is because LP (5.3) has only M +1 constraints, and therefore admits an optimal solution where at most M +1 variables are non-zero. 5.3.1 A New QPTAS for Signaling in Normal Form Games In this section, we present an approach different from the one in Section 4.3, which also gives a quasipolynomial-time approximation scheme for the problem of optimalsignalinginBayesian normalformgames. Weprovethefollowingbi-criteria result. Theorem 5.12. Let ǫ > 0 denote an approximation parameter, let (A,λ) be a Bayesian normal form game with k = O(1) players, n actions, and M states of nature, and let A 0 : [M] × [n] k → [−1,1] be an objective function given as a tensor. There is an algorithm with runtime poly(M ln(n/ǫ) ǫ 2 ,n lnn ǫ 2 ) which out- puts a signaling scheme ϕ and corresponding Bayesian ǫ-equilibria X satisfying F(ϕ,X) ≥ OPT(A,λ,A 0 )−ǫ. This holds for both approximate NE and approx- imate WSNE. In other words, when the number of players is a constant we can in quasipoly- nomial timeapproximatetheoptimal reward fromsignaling while losing anadditive ǫ in the objective as well as in the incentive constraints. Compared to our result 93 in Section 4.3, the running time is slightly worse for general sum games, but not directly comparable for zero-sum games (depending on which one of n and M is larger). More specifically, for zero-sum games and constant ǫ > 0, the QPTAS in Section 4.3 builds an ǫ-cover over Nash equilibria and runs in time n O(logn) , while the QPTAS in this section builds an ǫ-cover over posterior beliefs and runs in time M O(logn) . Fix ǫ > 0. To prove this theorem, we define functions g and g ǫ which each take as input a k-player n-action game of complete information B, given as payoff tensors B 1 ...,B k : [n] k → [−1,1], and an objective tensor B 0 : [n] k → [−1,1], and output a number in [−1,1]. Specifically, g(B,B 0 ) = max{B 0 (x) :x∈ EQ(B)} and g ǫ (B,B 0 ) = max{B 0 (x) : x∈ EQ ǫ (B)}, where EQ(B) denotes the set of Nash equilibria of the game B, and EQ ǫ (B) denotes the (non-empty) set of ⌈s(ǫ/4)⌉- uniform ǫ-Nash equilibria (or ǫ-WSNE) for s as given in Lemma 4.2. Recall that B 0 (x) denotes evaluating the multilinear map described by tensor B 0 at the mixed strategy profile x∈∆ k n . Now suppose we fix a Bayesian game (A,λ) and objective tensor A 0 as in the statement of Theorem 5.12. For a subgame with a posterior distribution µ ∈ ∆ M over states of nature, the principal’s expected utility at the “best” Nash equi- librium of this subgame can be written as g(A µ ,A µ 0 ). Similarly, the principal’s expected utility at the “best” ⌈s(ǫ/4)⌉-uniform ǫ-NE (or ǫ-WSNE) can be written as g ǫ (A µ ,A µ 0 ). Observe that the input to both g and g ǫ is a linear function of µ , as needed to apply the results in Section 5.3. For a signaling scheme ϕ corre- sponding to a decomposition λ = P σ∈Σ p σ µ σ of the prior distribution λ into poste- rior distributions (see Section 2.1.1), we can write the principal’s expected utility as F(ϕ) = P σ∈Σ p σ g(F µ σ ,A µ σ ) assuming that the players reach the “best” Nash 94 equilibrium in each subgame, and F ǫ (ϕ) = P σ∈Σ p σ g ǫ (F µ σ ,A µ σ ) assuming that the players reach the “best” ⌈s(ǫ/4)⌉-uniform ǫ-equilibria. We use OPT to denote the maximum value of F over all signaling schemes. We prove Theorem 5.12 by exhibiting an algorithm for computing a signaling scheme ϕ such that F ǫ (ϕ)≥OPT−ǫ. The proof hinges on two main lemmas. Lemma 5.13. The function g is 2(k+1)n k -stable. Proof. As noted in Section 5.2, any function mapping a hypercube [−1,1] N to the interval [−1,1] is 2N stable. The function g is such a function with N = (k + 1)n k . Lemma 5.14. The function g ǫ is an (ǫ/4,ǫ/2)-relaxation of g. Proof. Consider tensors B 0 , f B 0 : [n] k → [−1,1] with |B 0 (s)− f B 0 (s)| ≤ ǫ/4 for all s ∈ [n] k , and two k-player n-action games B = (B 1 ,...,B k ) and e B = ( e B 1 ,..., e B k ) with |B i (s)− e B i (s)| ≤ ǫ/4 for all s ∈ [n] k . It suffices to show that g ǫ ( e B, f B 0 ) ≥ g(B,B 0 )−ǫ/2. Let x ∈ ∆ k n be the Bayesian equilibrium of B for which B 0 (x) = g(B,B 0 ). By Lemma 4.2, there is a profile e x of ⌈s(ǫ/4)⌉-uniform mixed strategies such that e x is an ǫ/4-equilibrium of B, and B 0 (e x) ≥ B 0 (x)−ǫ/4. Since e B differs from B by at most ǫ/4 everywhere, it follows that e x is an ǫ-equilibrium of e B, i.e., e x ∈ EQ ǫ ( e B). Similarly, since f B 0 differs from B 0 by at most ǫ/4 everywhere, it follows that f B 0 (e x) ≥ B 0 (e x)−ǫ/4 ≥ B 0 (x)−ǫ/2. We conclude that g ǫ ( e B, f B 0 ) ≥ f B 0 (e x)≥g(B,B 0 )−ǫ/2. We now complete the proof of Theorem 5.12 by instantiating Theorem 5.11 withg,h =g ǫ ,andα = ǫ 4(k+1)n k . Theruntimeispoly(M ln(1/α) ǫ 2 ,(k+1)n k ,T),whereT is the time needed to evaluateg ǫ (and compute the corresponding ⌈s(ǫ/4)⌉-uniform 95 ǫ-equilibrium) on a given input. Recall that k = O(1) and α = ǫ poly(n) . Moreover, using brute-force enumeration of all ⌈s(ǫ/4)⌉-uniform mixed strategy profiles we conclude thatT is bounded by a polynomial inn lnn ǫ 2 . Therefore our total runtime is poly(M ln(n/ǫ) ǫ 2 ,n lnn ǫ 2 ), as needed. Remarks Similar to our results in Section 4.3, in the special case of two-player zero-sum games and a principal interested in maximizing one player’s utility, our techniques lead to a more efficient approximation scheme and a uni-criteria guaran- tee. Thisisbecausetheprincipal’spayofftensorB 0 equalsthepayofftensorB ofone of the players (say, player 1), and consequently the function g(B,B 0 ) =g(B,B) = max x min y x T By is n 2 -stable and 2-Lipschitz. Its Lipschitz continuity follows from the fact that an ǫ-equilibrium of a zero-sum game leads to utilities within ǫ of the equilibrium utilities. Moreover, evaluatingg now takes timeT = poly(M,n). Theo- rem 5.10 instantiated withα = ǫ 4n 2 andδ =ǫ/4, leads to an algorithm with runtime poly(M ln(n/ǫ) ǫ 2 ,n), which outputs a signaling scheme ϕ and corresponding Bayesian (exact) Nash-equilibria X satisfying F(ϕ,X)≥OPT(A,λ,A 0 )−ǫ. 5.4 Hardness Results for Mixture Selection We now present evidence that both our assumptions — Noise stability and Lipschitz continuity — appear necessary for general positive results along the lines of those in Theorem 5.6. Noise stability alone is not sufficient for a PTAS. In Section 5.4.1, we define a function g (slope) : [0,1] n → [0,1] which is 1-stable. Furthermore, g (slope) is O(1)-Lipschitz with respect to the L 1 metric, which is a weaker property than 96 Lipschitz continuity with respect to L ∞ . We show in Theorem 5.18 that there is a polynomial-time reduction from the maximum independent set problem on n-node graphs to the n-dimensional mixture selection for g (slope) . The reduction precludes a polynomial-time (additive) ǫ-approximation algorithm for some constant ǫ> 0. Lipschitz continuity alone is not sufficient for a PTAS. One might hope to prove NP-hardness of mixture selection in the absence of stability. However, since every function g : [−1,1] n → [−1,1] is 2n-stable, Theorem 5.6 implies a quasipolynomial-time approximation scheme in the additive sense whenever g is O(1)-Lipschitz. Nevertheless, we prove hardness of approximation assuming the planted clique conjecture ([66] and [72]). More specifically, in Section 5.4.2, we exhibit a reduction from the planted k-clique problem to mixture selection for the 3-Lipschitz function g (clique) k (t) = t [k] − t [k+1] + t [n] . When k = ω(log 2 n) and A is the adjacency matrix of an n-node undirected graph G, we show that max x g (clique) k (Ax)≈ 1 2 with highprobabilityifG∼G(n, 1 2 ), andmax x g (clique) k (Ax)≈ 1 with high probability if G∼G(n, 1 2 ,k) (defined in Section 2.3). 5.4.1 NP-hardness in the Absence of Lipschitz Continuity We now show that stability alone does not suffice for an additive PTAS for mixture selection, in general. First, we show that mixture selection for a 1-stable function g (vote-sum) (t) does not admit a (uni-criteria) additive PTAS unless P = NP. g (vote-sum) is motivated by the application of persuading voters presented in Section 6.2, and simply returns the fraction of nonnegative entries of t =Ax, i.e., g (vote-sum) (t) def = X i∈[n] 1 n I[t i ≥0]. 97 Inaddition,sinceg (vote-sum) isnotcontinuousinanymetric,weexhibita“smoothed” function g (slope) which is 1-stable and O(1)-Lipschitz with respect to L 1 , but not O(1)-Lipschitz with respect to L ∞ , and show that mixture selection for g (slope) still does not admit an additive PTAS unless P = NP. Both NP-hardness results share a similar reduction from the maximum inde- pendent set problem. We use a consequence of the result by [71], namely that there exists a constant ǫ such that it is NP-hard to approximate maximum independent set to within an additive error of ǫn, where n denotes the number of vertices. Given an n-node undirected graph G, let OPT IS = OPT IS (G) be the size of its largest independent set. We define the n×n matrix A=A(G) as follows: • Diagonal entries of A are all 1 2 (A i,i = 1 2 for all 1≤i≤n). • When vertices i and j share an edge in G, both A i,j and A j,i are −1. • All other entries of A, namely A i,j for non-adjacent distinct vertices i and j, are − 1 4n . We relate OPT IS to convex combinations of the columns of A as follows. Observation 5.15. Let I be an independent set of G with |I| = k. There exists x ∈ ∆ n such that k entries of Ax are at least 1 4n , and all remaining entries are strictly negative. Proof. Let x ∈ ∆ n be the normalized indicator vector of I — i.e., x i = 1 k if i ∈ I and x i = 0 otherwise. By construction (Ax) i = 1 k ( 1 2 −(k−1) 1 4n ) ≥ 1 4n whenever i∈I, and (Ax) i ≤− 1 4n otherwise. 98 Observation 5.16. For any x ∈ ∆ n , nonnegative entries of Ax correspond to an independent set of G. Consequently, Ax can have at most OPT IS nonnegative entries. Proof. Let t = Ax. Consider an edge {i,j} of graph G. We have t i ≤ x i 2 −x j − 1 4n (1−x i −x j ) and a similar inequality for t j , so t i +t j ≤− x i +x j 2 − 1−x i −x j 2n < 0. Therefore,t i andt j cannot be both nonnegative. We conclude that the nonnegative coordinates of t correspond to an independent set of G. Observations 5.15 and 5.16 imply that max x∈Δn g (vote-sum) (Ax) = OPT IS n . Com- bined with the fact that obtaining an additive PTAS for the maximum independent set problem is NP-hard, we get the following theorem. Theorem 5.17. Mixture selection for the 1-stable function g (vote-sum) admits no additive PTAS unless P = NP. Noting that g (vote-sum) is a discontinuous function, for emphasis we exhibit a function g (slope) which is Lipschitz continuous in L 1 (but not in L ∞ ) and 1-noise stable, but for which the same impossibility result holds by an identical reduction. Informally, g (slope) “smoothes” the threshold behavior of g (vote-sum) as follows: each input t i contributes 0 to g (slope) (t) when t i ≤ 0, contributes 1 n when t i ≥ 1 4n , and the contribution is a linear function of t i increasing from 0 to 1 n for t i ∈ [0, 1 4n ]. Formally, we define g (slope) (t) = P n i=1 min n 4max{0,t i }, 1 n o . Since each entry of t contributes at most 1 n to g (slope) (t), it is easy to verify that g (slope) is 1-stable. Moreover, since the partial derivatives of g (slope) (t) are upper-bounded by 4, it is 99 4-Lipschitz continuous with respect to the L 1 metric. Observations 5.15 and 5.16 imply that max x∈Δn g (slope) (Ax) = OPT IS n , ruling out an additive PTAS for mixture selection for g (slope) . Theorem 5.18. The function g (slope) is 1-stable and O(1)-Lipschitz with respect to L 1 , and yet mixture selection for g (slope) admits no additive PTAS unless P = NP. 5.4.2 Planted Clique Hardness in the Absence of Stability We now present evidence that Lipschitz continuity alone does not suffice for a PTAS for mixture selection. Recalling that a quasipolynomial time algorithm follows from our framework whenever a function is O(1)-Lipschitz, we reduce from the planted clique problem—for which a quasipolynomial time algorithm exists, and yet a polynomial-time algorithm is conjectured not to exist—rather than from an NP-hard problem. We let k = k(n) be as in Conjecture 2.6, and consider mixture selection for the function g (clique) k : [0,1] n → [0,1] with g (clique) k (t) = t [k] −t [k+1] +t [n] , where t [i] denotes the i’th largest entry of the vector t. It is easy to verify that g (clique) k is 3-Lipschitz with respect to the L ∞ metric, yet is not O(1)-stable. We prove the following theorem. Theorem 5.19. Conjecture 2.6 implies that there is no additive PTAS for mixture selection for g (clique) k . To prove Theorem 5.19, we show that max x∈Δn g (clique) k (Ax) is arbitrarily close to 1 with high probability when A is the adjacency matrix of G ∼ G(n, 1 2 ,k), and is bounded away from 1 with high probability when A is the adjacency matrix of G∼G(n, 1 2 ). For convenience, and without loss of generality, we assume that both 100 random graphs include each self-loop with probability 1 2 — i.e., diagonal entries of the adjacency matrix A are independent uniform draws from {0,1} in both cases. Our argument is captured by the following two lemmas. Lemma 5.20. Fix a constant ǫ> 0. Let G∼G(n, 1 2 ,k), and let A be its adjacency matrix. With probability 1−o(1), there exists an x ∈ ∆ n such that g (clique) k (Ax) ≥ 1−ǫ. Proof. LetC denotetheverticesoftheplantedk-clique. Wesetx i = 1 k ifi∈C and0 otherwise. Lett=Ax. Fori∈C,t i ≥1− 1 k . On the other hand, all other entries of t concentrate around 1 2 with high probability. Fori / ∈C,t i issimply theaverageofk independent BernoullirandomvariablesbydefinitionofG(n, 1 2 ,k); usingHoeffding’s inequality, we bound the probability that t i deviates from its expectation by more than a constant δ> 0, to be chosen later: Pr t i − 1 2 >δ ≤2e −2δ 2 k . By the union bound, t i ∈ [ 1 2 −δ, 1 2 +δ] simultaneously for all i / ∈ C with probability at least 1−n2 −Ω(k) = 1−o(1). Thus t [k+1] −t [n] ≤ 2δ and g (clique) k (t) = t [k] −(t [k+1] −t [n] ) ≥ 1− 1 k −2δ with probability 1−o(1). Choosing δ = ǫ/3, we conclude that g (clique) k (t)≥1−ǫ with probability 1−o(1). Lemma 5.21. Fix a constant ǫ > 0. Let G ∼ G(n, 1 2 ), and let A be its adjacency matrix. With probability 1−o(1), g (clique) k (Ax)≤ 3 4 +ǫ for all x∈∆ n . Proof. Recall that g (clique) k is O(1)-Lipschitz and — like any other function from the hypercube to the bounded interval — O(n)-stable. If there exists x ∗ such that g (clique) k (Ax ∗ )≥ 3 4 +ǫ, then Theorem 5.5implies that there isan integers =O(logn) 101 and an s-uniform vector e x such that g (clique) k (Ae x) > 3 4 . There are n s such vec- tors. We next show that for an arbitrary fixed vector x ∈ ∆ n the probability that g (clique) k (Ax)> 3 4 isatmost 2 −Ω(k) . This will complete theproofby theunion bound, since 1−n s ·2 −Ω(k) =1−o(1). Fix x∈ ∆ n , and let t =Ax. Define D as the distribution supported on [0,1] which is sampled as follows: drawa uniformly from{0,1} n , and output a·x. Since A is the adjacency matrix of G ∼ G(n, 1 2 ), each entry t i of t can be viewed as an independent draw fromD. We exploit a key property ofD in our proof, namely the fact that D is symmetric about 1 2 . Formally we mean that Pr D [r] = Pr D [1−r] for all r∈[0,1], and this follows easily from the definition ofD. Symmetry ofD implies that Pr r∼D [r≥ 1 2 ]= Pr r∼D [r≤ 1 2 ]≥ 1 2 . Recalling that k = o(n) and that entries of t are independent draws from D, the Chernoff bound implies that the following holds with probability at least 1−2 −Ω(n) : t [n] ≤ 1 2 ≤t [k+1] . (5.4) If g (clique) k (t)> 3 4 , then the following two conditions must hold: 1. t [k] > 3 4 , and 2. t [k+1] −t [n] < 1 4 . Condition 1 implies that the k largest entries of t are all at least 3 4 . Furthermore, unless Inequality (5.4) is violated — which happens with probability 2 −Ω(n) — Con- dition2impliesthattheremainingentriesoftareallstrictlybetween 1 4 and 3 4 . Letp denote Pr r∼D [r≤ 1 4 ], also equal to Pr r∼D [r≥ 3 4 ] by symmetry ofD. The probability that k entries of t are at least 3 4 and all remaining entries are in ( 1 4 , 3 4 ) is given by 102 n k p k (1−2p) n−k ,whichismaximizedatp = k 2n ,withmaximumvalue2 −Ω(k) . Insum- mary, the probability that g (clique) k (Ax)> 3 4 is at most 2 −Ω(k) +2 −Ω(n) = 2 −Ω(k) . ETH-Hardness in the Absence of Stability For mixture selection in the absence of noise stability, we can also show that a QPTAS is the best-possible approximation scheme, assuming the Exponential Time Hypothesis (ETH) [15]. Our proof follows from a clean reduction from the best- Nash problem, for which Braverman et al. [20] showed that a QPTAS is essentially optimal. We choose to present the planted clique hardness result in this thesis because it is more elementary, and gives a simple function g (clique) k . 103 Chapter 6 Signaling in Anonymous Games Anonymous games are multiplayer games in which the utility of each player depends on her own strategy, as well as the number (as opposed to the identity) of other players who play each of the strategies. Anonymous games comprise an important class of succinct games — well-studied in the economics literature (see, e.g., [17, 18, 76]) — capturing a wide range of phenomena that frequently arise in practice, including auctions, voting systems, and congestion games. In this chapter, we study the complexity of optimal signaling in anonymous games. We start with two special cases: probabilistic second price auctions, and majority voting with uncertainty. We give the first polynomial time approximation schemes (PTAS) for both problems (Theorem 6.2 and 6.5), which follow from the powerful mixture selection framework presented in Chapter 5. We then take a slight detour to present the currently (asymptotically) best algorithm for computing Nash equilibria in anonymous games (Theorem 6.6); and we also present some evidence suggesting our algorithm might be essentially tight (Theorem 6.7). Anonymous games have a unique property compared to all other games we study in this thesis (e.g., network routing games, normal form games, sec- ond price auctions and majority voting); the computational complexity of (approx- imate) Nash equilibria in complete-information anonymous games is still open. The work presented in this chapter appeared in [26] and [28]. 104 6.1 Signaling in Second-Price Auctions In this section, we examine signaling in probabilistic second-price auctions as defined in Section 2.1.5. Recall that in this setting, a probabilistic item is being auctioned, and the instantiation of the item is known to the auctioneer but not to the bidders. This is particularly relevant in advertising auctions, where items are impressions associated with demographics that are a priori unknown to the advertisers bidding in the auction. We consider the algorithmic problem faced by an auctioneer, who seeks to reveal partial information to maximize the expected revenue before subsequently running a second-price auction. It was shown in [21, 52] that polynomial-time algo- rithms exist for several special cases of this problem. However, the general problem was shown to be NP-hard even with 3 bidders — specifically, no additive FPTAS exists unless P = NP. In this section, we resolve the approximation complexity of this basic signaling problem by giving an additive PTAS. We note that variations of this problem were considered in [59, 62], with different constraints on the signaling scheme — the results in these works are not directly relevant to our model. 6.1.1 PTAS from Mixture Selection: Revenue is Stable Given a probabilistic auction with valuation distribution D, and a signaling scheme ϕ expressed as a decomposition {p σ ,µ σ } σ∈Σ of the prior distribution λ, we can express the auctioneer’s expected revenue as X σ∈Σ p σ E V∼D [max2(Vµ σ )], 105 wherethefunctionmax2returnsthesecondlargestentryofagivenvector. Toapply our main theorem, we need to show that the revenue in a subgame with posterior distribution µ ∈ ∆ M — namely E V∼D [max2(Vµ )] — can be written in the form g(Wµ ) for a matrix W. To facilitate our discussion we assume that the valuation distribution D has finite support size C, though this is without loss of generality. Imagine we form a large matrixW by stacking matrices in the support ofD on top of each other. Formally, W = [V T 1 ,V T 2 ,...,V T C ] T where V i is the ith matrix in the supportofD. WhenmatrixV i isdrawnfromD,wetakethesecond-highestbidfrom the rows ofW corresponding toV i (rows(i−1)·n+1 toi·n, wheren is thenumber players). For S ⊆ [nC] and t ∈ [0,1] nC , let max2 S (t) denote the second-highest value among entries oft indexed byS. Then we can write the auctioneer’s expected revenue as g (rev) (Wµ )= E V∼D h max2 S(V) (Wµ ) i where S(V) is the set of rows in W corresponding to V. Lemma 6.1 (Smooth and Stable Revenue). The function g (rev) (t) = E V∼D h max2 S(V) (t) i is 1-Lipschitz and 2-stable. Proof. Becausemax2 S is1-Lipschitzforafixed setofindicesS, itfollowsthatg (rev) , which is a convex combination of these 1-Lipschitz functions, is also 1-Lipschitz. To show that g (rev) is stable, we first show that the function max2 : [0,1] n → [0,1] is stable. Given t ∈ [0,1] n and a random set R ⊆ [n] drawn from an α-light distribution D, the union bound implies that R includes neither of the two largest entries of t with probability at least 1−2α. In this case, the value of max2 is not affected by corruption of the entries indexed by R. Hence 106 E R∼D min{max2(t ′ ) :t ′ ≈ R t} ≥(1−2α)·max2(t)+2α·0≥max2(t)−2α. Therefore max2 is 2-stable, which implies that max2 S : [0,1] nC → [0,1] is also 2- stable for any fixed set of indices S. The function g (rev) is a convex combination of functions of the form max2 S , and is therefore also 2-stable by Proposition 5.4. Theorem 6.2. The revenue-maximizing signaling problem in second-price auctions admits an additive PTAS when the valuation distribution is given explicitly, and an additive PRAS when the valuation distribution is given by a sampling oracle. Proof. Lemma 6.1 shows that the function g (rev) is 2-stable and 1-Lipschitz. If the valuation distribution D is explicitly given with support size C, the function g (rev) can be evaluated in poly(n,M,C) time. Then for any ǫ > 0, it follows from Theorem 5.10 by setting α = ǫ/4 and δ = ǫ/2 that there is a deterministic algo- rithm that computes a signaling scheme with expected revenue (OPT−ǫ), in time poly(n,M ǫ −2 ln(1/ǫ) ,C). IfD is given via a sampling oracle, standard tail bounds and the union bound imply that C = Θ((slogm+log(γ −1 ))/ǫ 2 ) samples from D suffice to estimate to withinO(ǫ) the revenue associated with everys-uniform posterior in ∆ M , with suc- cess probability 1−γ. Since revenue isO(1)-stable andO(1)-Lipschitz, Lemma 5.9 implies thatwecan restrict attention tosignaling schemes withs-uniformposteriors for s = poly( 1 ǫ ). Proceeding as in Theorem 5.10, using the revenue estimates from Monte-Carlo sampling in lieu of exact values, we can construct a signaling scheme with revenue (OPT−ǫ) in time poly(n,M ǫ −2 ln(1/ǫ) ,log( 1 γ )), with success probability 1−γ. 107 6.1.2 NP-hardness of an Additive FPTAS Emek et al. [52] proved that revenue-maximizing signaling in probabilistic sec- ond price auctions is NP-hard, via a reduction from MAX-CUT. More specifically, given a graph G with n nodes and m edges, they can construct a Bayesian sec- ond price auction such that the value of the optimal signaling scheme is roughly m+C ∗ poly(n) , where C ∗ is the size of the maximum cut of G. Since MAX-CUT is APX- hard and their reduction is gap preserving up to a multiplicative factor of poly(n), Emek et al. [52] implicitly ruled out an additive FPTAS for this problem. 6.2 Persuasion in Voting In this section, we study persuasion problem in voting as defined in Sec- tion 2.1.6. Recall that we have an election with two possible outcomes. Each voter casts a‘Yes’/‘No’ vote, and theballotmeasure ispassed ifthefractionof‘Yes’ votes exceedsacertainpre-specified threshold. Asin[5],wefocusonthescenarioinwhich voters have uncertainty regarding their utilities for the two possible outcomes. We consider a principal looking to influence the outcome of the election by signaling, who wants to maximize the probability of the measure passing. For our approximation algorithms, we also allow implementation in approxi- mately dominant strategies — i.e., we sometimes assume a voter votes ‘Yes’ if his utility u(i,µ ) is at least −δ for a small parameter δ. 1 We assume that the state of natureθ∈Θ is drawn from a common priorλ∈∆ M , and a principal with access to θ reveals apublicsignalσ priortovoters casting theirvotes. Asusual, weadoptthe 1 Such relaxations seem necessary for our results. Moreover, depending on the context, modes of intervention for shifting the votes of voters who are close to being indifferent may be realistic. 108 perspectiveofaprincipallookingtocommittoapublicsignalingschemeϕ: Θ→Σ, for some set of signals Σ. Alonso and Câmara[5]consider aprincipal interested inmaximizing theprob- ability that at least 50% (or some given threshold) of the voters vote ‘Yes’, in expectation over states of nature. Theirs is the natural objective when the elec- tion employs a majority (or threshold) voting rule, and the principal is interested in influencingtheoutcomeofthevote. Approximatingthisobjectiverequiresnontrivial modifications to our framework, and therefore we begin this section by examining a different, yet also natural, objective: the expected number of ‘Yes’ votes. We design a bi-criteria approximation scheme for this objective, then describe the necessary modifications for the threshold function objective of [5]. 6.2.1 Maximizing Expected Number of Votes We now examine bi-criteria approximation algorithms for maximizing the expected number of ‘Yes’ votes. For our benchmark, we use the function g (vote-sum) (t) def = P i∈[n] 1 n I[t i ≥0], whereI[E] denotes the indicator function for event E. Assuming voters vote ‘Yes’ precisely when their posterior expected utility for a ‘Yes’ outcome is nonnegative, the number of ‘Yes’ votes when voters have prefer- ences U ∈[−1,1] n×m and posterior belief µ ∈∆ M equals g (vote-sum) (Uµ ). When the stateofnatureisdistributedaccordingtoacommonpriorλ,andvotersareinformed accordingtoasignalingschemeϕ={µ σ ,p σ } σ∈Σ ,theexpected numberof‘Yes’votes equals F (vote-sum) (ϕ,U,λ) def = P σ∈Σ p σ g (vote-sum) (Uµ σ ). We use OPT (vote-sum) (U,λ) to denote the maximum value of F (vote-sum) (ϕ,U,λ) over public signaling schemes ϕ. As the first step to apply our framework, we prove that g (vote-sum) is stable. 109 Lemma 6.3. The function g (vote-sum) is 1-stable. Proof. For each voter i ∈ [n], let g i : [−1,1] n → {0,1} be the function indicating whether voter i prefers the ‘Yes’ outcome, i.e., g i (t) = I[t i ≥ 0]. Each individual g i is 1-stable, because as long as the i’th input t i is not corrupted the output of g i does not change. Therefore g (vote-sum) (t) = 1 n n P i=1 g i (t), being a convex combination of 1-stable functions, is 1-stable by Proposition 5.4. Unfortunately, g (vote-sum) is not O(1)-Lipschitz. We therefore employ the bi- criteria extension to our framework from Definition 5.7. Specifically, for a param- eter δ > 0, we assume a voter votes ‘Yes’ as long as his expected utility from a ‘Yes’ outcome is at least −δ. Correspondingly, we define the relaxed function g (vote-sum) δ (t) def = P i∈[n] 1 n I[t i ≥ −δ]; the expected number of ‘Yes’ votes from a sig- naling scheme ϕ ={µ σ ,p σ } σ∈Σ can analogously be written as F (vote-sum) δ (ϕ,U,λ) def = P σ∈Σ p σ g (vote-sum) δ (Uµ σ ). We can verify that g (vote-sum) δ is a (δ,0)-relaxation of g (vote-sum) ; combining this fact with Theorem 5.11 yields a bi-criteria approximation scheme for the problem of maximizing the expected number of ‘Yes’ votes. Theorem 6.4. Let ǫ,δ > 0 be parameters, let U ∈ [−1,1] n×M describe the pref- erences of n voters in M states of nature, and let λ ∈ ∆ M be the prior of states of nature. There is an algorithm with runtime poly(M δ −2 ln(1/ǫ) ,n) for computing a signaling scheme ϕ such that F (vote-sum) δ (ϕ,U,λ)≥ OPT (vote-sum) (U,λ)−ǫ. 110 6.2.2 Maximizing the Probability of a Majority Vote We now sketch the necessary modifications when the principal is interested in maximizing theprobability ofa‘Yes’ outcome, assuming amajorityvoting rule. We make two relaxations: we assume a voter votes ‘Yes’ as long as his expected utility from a ‘Yes’ outcome is at least −δ, and assume that the ‘Yes’ outcome is attained when at least a (0.5−δ) fraction of voters vote ‘Yes’. Our benchmark will be the maximum probability of a ‘Yes’ outcome in the absence of these two relaxations. We note that [5] do not require these relaxations. They focus on character- izing the structures of the optimal signaling scheme, and fall short at providing an algorithm for (approximately) optimal signaling. In their analysis for persuading multiple voters, they make use of (the convex hull of) the set of posteriors that induce more than 50% of the voters to vote ‘Yes’; this set is in general non-convex and may have exponentially many disconnected regions, making it difficult to con- vert their insights into efficient algorithms. We define our benchmark using the function g (vote-thresh) (t) =I[g (vote-sum) (t)≥ 0.5]whichevaluatesto1ifatleasthalfofitsninputsarenonnegative,andto0other- wise. This function is notO(1)-stable, so we work with a more stringent benchmark which is. Specifically, for a parameter δ > 0, we use the function g (vote-smooth-thresh) δ which is pointwise greater than or equal to g (vote-thresh) , defined as follows: g (vote-smooth-thresh) δ (t)=          1 δ g (vote-sum) (t)−0.5+δ if g (vote-sum) (t)∈[0.5−δ,0.5] g (vote-thresh) (t) otherwise. 111 Observe that g (vote-smooth-thresh) δ applies a continuous piecewise-linear function to the output ofg (vote-sum) ; it is easy to verify thatg (vote-smooth-thresh) δ is 1 δ -stable, and upper bounds g (vote-thresh) . Finally, to measure the quality of our output we define the relaxed function g (vote-thresh) δ : [−1,1] n → {0,1}, which outputs 1 if at least a (0.5−δ) fraction of its inputs exceed −δ, and outputs 0 otherwise. By Definition 5.7, g (vote-thresh) δ is a (δ,0)-relaxation of g (vote-smooth-thresh) δ (and, consequently, also of g (vote-thresh) ). Let F (vote-thresh) (ϕ,U,λ) and F (vote-thresh) δ (ϕ,U,λ) denote the functions which evaluate the quality of a signaling scheme ϕ using g (vote-thresh) and g (vote-thresh) δ , respectively. Moreover, let OPT (vote-thresh) (U,λ) be the maximum value of F (vote-thresh) (ϕ,U,λ)oversignalingschemesϕ. WeapplyTheorem5.11tog (vote-thresh) δ andg (vote-smooth-thresh) , settingα =ǫδ, and use the fact thatg (vote-smooth-thresh) upper- bounds our true benchmark g (vote-thresh) , to conclude the following. Theorem 6.5. Let ǫ,δ > 0 be parameters, let U ∈ [−1,1] n×M describe the prefer- ences of n voters in M states of nature, and let λ ∈ ∆ M be the prior of states of nature. There is an algorithm with runtime poly(n,M δ −2 ln(1/ǫδ) ) for computing a signaling scheme ϕ such that F (vote-thresh) δ (ϕ,U,λ)≥ OPT (vote-thresh) (U,λ)−ǫ. Connection to Maximum Feasible Subsystem of Linear Inequalities Turning our attention away from signaling, we notethatg (vote-sum) (Ax) simply counts the number of satisfied inequalities in the system Ax 0. Mixture selec- tion for g (vote-sum) is therefore the problem of maximizing the number of satisfied inequalities over the simplex. Using our framework from Section 5.2, we obtain a bi-criteria PTAS for this problem. Moreover, using Monte-Carlo sampling, our bi- criteria PTAS extends to the model in which A is given implicitly; specifically, the 112 rows ofA correspond to the sample space of a distributionD over [−1,1] m , and are weighted accordingly. In this implicit model, we can think of mixture selection for g (vote-sum) as the problem of finding x ∈ ∆ M which maximizes the probability that a·x≥ 0 for a∼D. 6.2.3 Hardness Results for Persuading Voters In Section 5.4.1, we showed that the posterior selection problem for g (vote-sum) does not admit a (uni-criteria) additive PTAS unless P = NP. Inspired by our reduction, Dughmi and Xu [51] ruled out a (uni-criteria) PTAS for the problem of signaling to maximize the expected number of votes. Both reductions construct a Bayesian voting instance I from a graph G. At a high level, we showed that a good posterior of I corresponds to a large independent set of G; and [51] showed that a near-optimal signaling scheme of I corresponds to covering G using large independent sets. Conceptually, the idea of switching from maximum independent set to graph-coloring is equivalent to moving from planted clique to planted clique cover. 6.3 Computing Equilibria in Anonymous Games The complexity andefficient approximation ofNash equilibria have been stud- ied intensively during the past decade, and much progress has been made (e.g., see [1, 9, 10, 11, 12, 23, 24, 25, 33, 35, 39, 53, 70, 74, 75, 87, 88, 89]). Despite much effort, the computational complexity of approximate Nash equilibria in anonymous games remains open. 113 In recent years, equilibrium computation in anonymous games has attracted significant attention in TCS [25, 33, 36, 37, 38, 39, 41, 60]. Consider the family of anonymous games where the number of players, n, is large and the number of strategies, k, is bounded. It was recently shown by Chen et al. [25] that computing an ǫ-approximate Nash equilibrium of such games is PPAD-complete when ǫ is exponentially small, even for anonymous games with 5 strategies 2 . On the algorithmic side, Daskalakis and Papadimitriou [36, 37] presented the first polynomial-time approximation scheme (PTAS) for this problem with running time n (1/ǫ) Ω(k) . For the case of 2-strategies, this bound was improved [34, 38, 39] to poly(n)·(1/ǫ) O(log 2 (1/ǫ)) , and subsequently sharpened to poly(n)·(1/ǫ) O(log(1/ǫ)) in [42]. In recent work, Daskalakis et al. [33] and Diakonikolas et al. [41] generalized the aforementioned results [39, 42] to any fixed number k of strategies, obtaining algorithms for computing ǫ-well-supported Nash equilibria (see Definition 2.3) with runtime of the form n poly(k) ·(1/ǫ) klog(1/ǫ) O(k) . That is, the problem of computing approximate Nash equilibria in anonymous games with a fixed number of strate- giesadmitsanefficientpolynomial-timeapproximationscheme(EPTAS).Moreover, the dependence of the running time on the parameter 1/ǫ is quasipolynomial — as opposed to exponential. We note that all the aforementioned algorithmic results are obtained by exploiting a connection between Nash equilibria in anonymous games and Pois- son multinomial distributions (PMDs). This connection — formalized in [36, 37] — translates constructive upper bounds on ǫ-covers for PMDs to upper bounds on 2 [25]showedthatcomputinganequilibriumof7-strategyanonymousgamesisPPAD-complete, but 3 of the 7 strategies in their construction can be merged, resulting in a 5-strategy anonymous game. 114 computing ǫ-Nash equilibria in anonymous games (see Section 2.1.7 for formal def- initions). Unfortunately, as shown in [33, 41], this “cover-based” approach cannot lead to qualitatively faster algorithms, due to a matching existential lower bound on the size of the corresponding ǫ-covers. In related algorithmic work, Goldberg and Turchetta [60] studied two-strategy anonymous games (k = 2) and designed a polynomial-time algorithm that computes anǫ-approximate Nash equilibrium for ǫ= Ω( n −1/4 ). The aforementioned discussion prompts the following natural question: What is the precise approximability of computing Nash equilibria in anonymous games? In this chapter, we make progress on this question by establishing the following result: Foranyδ >0, andanyn-player anonymous gamewith a constant number of strategies, there exists a poly δ (n) time algorithm that computes an ǫ-approximate Nash equilibrium of the game, for ǫ = 1/n 1−δ . 3 Moreover, we show that the exis- tence of a polynomial-time algorithm that computes an ǫ-approximate Nash equi- librium for ǫ = 1/n 1+δ , for any small constant δ > 0 — i.e., slightly better than the approximation guarantee of our algorithm — would imply the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem. That is, we essentially show that the value ǫ = 1/n is the threshold for the polynomial-time approximability of Nash equilibria in anonymous games, unless there is an FPTAS. Inthefollowingsubsection, wedescribeourresultsindetailandprovideanoverview of our techniques. 3 The runtime of our algorithm depends exponentially on 1/δ. We remind the reader that the algorithms of [33, 41] run in quasipolynomial time for any value of ǫ inverse polynomial in n. 115 6.3.1 Summary of Results and Techniques We study the following question: Forn-playerk-strategyanonymousgames,howsmallcanǫbe(asafunc- tion of n), so that an ǫ-approximate Nash equilibrium can be computed in polynomial time? Upper Bounds. We present a polynomial time algorithm that computes ǫ- approximate equilibria in anonymous games for an inverse polynomial ǫ above a certain threshold. Theorem 6.6. For any δ >0, and any n-player k-strategy anonymous game, there is a poly δ,k (n) time algorithm that computes a (1/n 1−δ )-approximate equilibrium of the game. This is the first polynomial time ǫ-approximation for some k > 2 strategies and some inverse polynomial ǫ. Overview of Techniques. The high-level idea of our approach is this: If the desired accuracy ǫ is above a certain threshold, we do not need to enumerate over an ǫ-cover for the set of all PMDs. Our approach is in part inspired by [60], who design an algorithm (for k = 2 and ǫ = Ω( n −1/4 )) in which all players use one of the two pre-selected mixed strategies. The [60] algorithm can be equivalently interpreted as guessing a PBD from an appropriately small set. One reason this idea succeeds is the following: If every player randomizes, then the variance of the resulting PBD must be relatively high, and (as a result) the corresponding subset of PBDs has a smaller cover. 116 Ourquantitativeimprovementforthek = 2caseisobtainedasfollows: Instead of forcing players to selected specific mixed strategies — as in [60] — we show that there always exists an ǫ-approximate equilibrium where the associated PBD has variance at least Θ(nǫ). Whenǫ=n −c for somec<1, thevariance is polynomial in n. Wethenconstruct apolynomial-sizeǫ-cover forthesubset ofPBDswithvariance at least this much, which leads to a polynomial-time algorithm for computing ǫ- approximate equilibria in 2-strategy anonymous games. The idea for the general case of k > 2 is similar, but the details are more elaborate, since the structure of PMDs is more complicated for k > 2. We proceed as follows: we start by showing that there is an ǫ-approximate equilibrium whose corresponding PMDhaslargevariancein each direction. Ourmain structural result is a robust moment-matching lemma (Lemma 6.11), which states that the close- ness in low-degree moments of two PMDs, with large variance in each direction, implies their closeness in total variation distance. The proof of this lemma uses Fourier analytic techniques, building on and strengthening previous work [41]. As a consequence of our moment-matching lemma, we can construct a polynomial-size (ǫ/5)-coverforPMDswithsuchlargevariance. Wetheniteratethroughthiscoverto find anǫ-approximate equilibrium, using a dynamic programming approach similar to the one in [39]. We now provide a brief intuition of our moment-matching lemma. Intuitively, ifthetwoPMDsinquestionarebothveryclosetodiscreteGaussians, thencloseness in thefirst two moments issufficient. Lemma 6.11 can beviewed asa generalization ofthisintuition, whichgivesaquantitativetradeoffbetweenthenumberofmoments we need to approximately match and the size of the variance. The proof of Lemma 6.11 exploits the sparsity of the Fourier transform of our PMDs, and the fact that 117 higher variance allows us to take fewer terms in the Taylor expansion when we use moments to approximate the logarithmic Fourier transform. This completes the proof sketch of Theorem 6.6. Lower Bounds. When ǫ = 1/n, we can show that there is an ǫ-approximate equilibrium wheretheassociatedPMDhasavarianceatleast1/k ineverydirection. Unfortunately, the PMDs in theexplicit quasipolynomial-size lower bounds given in [33,41]satisfythisproperty. Thus,weneedadifferentapproachtogetapolynomial- time algorithm for ǫ=1/n or smaller. In fact, we prove the following result, which states that even a slight improve- ment of our upper bound in Theorem 6.6 would imply an FPTAS for computing Nash equilibria in anonymous games. It is important to note that Theorem 6.7 applies to all algorithms, not only the ones that leverage the structure of PMDs. Theorem 6.7. For n-player k-strategy anonymous games with k =O(1), if we can compute an O(n −c )-approximate equilibrium in polynomial time for some constant c > 1, then there is an FPTAS for computing (well-supported) Nash equilibria of k-strategy anonymous games. Remark. As observed in [33], because there is a quasipolynomial time algorithm for computing an (n −c )-approximate equilibrium in anonymous games, the problem cannot be PPAD-complete unless PPAD ⊆ Quasi-PTIME. On the other hand, we do not know how to improve the quasipolynomial-time upper bounds of [33, 41] when ǫ< 1/n. Recall that computing an ǫ-approximate equilibrium of a two-player general- sum n×n game (2-NASH) for constant ǫ also admits a quasipolynomial-time algo- rithm [74]. Very recently, Rubinstein [88] showed that, assuming the exponential 118 time hypothesis (ETH) for PPAD, for some sufficiently small universal constant ǫ> 0, quasipolynomial-time is necessary to compute an ǫ-approximate equilibrium of 2-NASH. It is a plausible conjecture that quasipolynomial-time is also required for ǫ-Nash equilibria in anonymous games, when ǫ = n −c for some constant c > 1. In particular, this would imply that there is no FPTAS for computing approxi- mate Nash equilibria in anonymous games, and consequently the upper bound of Theorem 6.6 is essentially tight. 6.3.2 Searching Fewer Moments In this section, we present a polynomial-time algorithm that, for n-player anonymousgameswithaboundednumberofstrategies, computesanǫ-approximate equilibrium withǫ=n −c foranyconstantc<1(Theorem6.6). Theorem6.6applies to general k-strategy anonymous games for any constant k ≥ 2. As a warm-up, we start by describing the simpler setting of two-strategy anonymous games (k = 2). Lemma 6.8. For an n-player k-strategy anonymous game, there always exists an ǫ-approximate equilibrium where every player plays each strategy with probability at least ǫ k−1 . Proof. Given an anonymous game G = (n,k,{u i a } i∈[n],a∈[k] ), we smooth players’ utility functions by requiring every player to randomize. Fix ǫ > 0. We define an ǫ-perturbed gameG ǫ as follows. When a player plays some pure strategy a∈[k] in G ǫ , we map it back to the original game as if she played strategyj with probability 1−ǫ, and played some other strategya ′ 6=a uniformly at random (i.e., she playsa ′ with probability ǫ k−1 ). Her payoff in G ǫ also accounts for such perturbation, and is 119 defined to beher expected payoff given thatall theplayers (including herself) would deviate to other strategies uniformly at random with probability ǫ. Formally, let X ǫ (e j ) denote the k-CRV that takes value e j with probability 1−ǫ, and takes valuee j ′ with probability ǫ k−1 for each j ′ 6=j. The payoff structure of G ǫ is given by u ′i a (x) def = (1−ǫ)E h u i a (M ǫ (x)) i + ǫ k−1 X a ′ 6=a E h u i a ′(M ǫ (x)) i , ∀i∈ [n],a∈[k],x∈Π k n−1 , where M ǫ (x) = P j∈[k] x j X ǫ (e j ) is an (n−1,k)-PMD that corresponds to the per- turbed outcome of the partitionx∈Π k n−1 of all other players. Lets ′ = (s ′ 1 ,...,s ′ n )denoteanyexactNashequilibriumofG ǫ . Wecaninterpret this mixed strategy profile in G equivalently as s = (s 1 ,...,s n ), where s i = (1− kǫ k−1 )s ′ i + ǫ k−1 1, where 1 = (1,...,1). We know that under s each player has no incentive to deviate to the mixed strategies X ǫ (e j ) for all j ∈[k], therefore a player can gain at most ǫ by deviating to pure strategies in G, so s is an ǫ-approximate equilibrium with s i (j)≥ ǫ k−1 for all i∈[n], j ∈[k]. Warm-up: The Case of k = 2 Strategies. For two-strategy anonymous games (k = 2), if all the players put at least ǫ probability mass on both strategies, the resulting PBD is going to have variance at least nǫ(1−ǫ). When ǫ =n −c for some constant c < 1, the variance is at least Θ(n 1−c ) = n Θ(1) . We can now use the following lemma from [43], which states that if two PBDs P and Q are close in the first few moments, then P and Q are ǫ-close in total variation distance. Note that without any assumption on the variance of the PBDs, we would need to check the first O(log(1/ǫ)) moments, but when the variance is n Ω(1) , which is the case in our application, we only need the first constant number of moments to match. 120 Recallthatann-PBDisthesumofnindependent Bernoullirandomvariables. An n-PBD P can be represented by its n parameters p 1 ,...,p n , where p i is the probability of the i-th Bernoulli takes the value of 1. In the following lemma, for technical reasons, these parameters are partitioned into two sets with s and s ′ elements (s+s ′ =n), depending on whether they are greater than 1/2 or not. Lemma 6.9 ([43]). Let ǫ> 0. Let P and Q be n-PBDs with P having parameters p 1 ,...,p s ≤ 1/2 and p ′ 1 ,...,p ′ s ′ > 1/2, and Q having parameters q 1 ,...,q s ≤ 1/2 and q ′ 1 ,...,q ′ s ′ > 1/2. Suppose that V = Var[P]+1 = Θ(Var[Q]+1) and let C > 0 be a sufficiently large constant. Suppose furthermore that the following holds for A =C q log(1/ǫ)/V and for all positive integers ℓ, A ℓ   s X i=1 p ℓ i − s X i=1 q ℓ i + s ′ X i=1 (1−p ′ i ) ℓ − s ′ X i=1 (1−q ′ i ) ℓ   < ǫ Clog(1/ǫ) (6.1) Then d TV (P,Q)<ǫ. Let ǫ = n −c . For Lemma 6.9 we have V ≥ nǫ(1 − ǫ) and A = Θ q log(1/ǫ)/V = O q logn n 1−c . The difference in the moments of parameters of P and Q in Equation (6.1) is bounded from above by n, so whenever ℓ> 2+2c 1−c , the condition in Lemma 6.9 is automatically satisfied for sufficiently large n because A ℓ n =O log ℓ/2 n n (1−c)ℓ/2 n ! < 1 C·n c ·clogn = ǫ Clog(1/ǫ) . So it is enough to search over the first ℓ = Θ 1 1−c moments when each player puts probability at least Ω( n −c ) on both strategies. The algorithm for finding such an ǫ-approximate equilibrium uses moment search and dynamic programming, and is given for the case of general k in the remainder of this section. 121 The General Case: k Strategies. We now present our algorithm for n-player anonymous games with k > 2 strategies and prove Theorem 6.6. The intuition of the k = 2 case carries over to the general case, but the details are more elaborate. First, we show (Claim 6.10) that there exists an ǫ-approximate equilibrium whose corresponding PMD has variance (nǫ/k) in all directions orthogonal to the vector 1 = (1,...,1). Then, we prove (Lemma 6.11) that when two PMDs have such high variances, the closeness in their constant-degree parameter moments trans- lates to their closeness in total variation distance. This structural result allows us to construct a polynomial-size (ǫ/5)-cover for set subset of all PMDs with large variance. We then iterate through this cover to find an ǫ-approximate equilibrium (Algorithm 6.2). We first prove that when all players put probability at least ǫ k−1 on each strat- egy, the covariance matrix of the resulting PMD has relatively large eigenvalues, exceptthezeroeigenvalueassociatedwiththeall-oneeigenvector. Theall-oneeigen- vector has eigenvalue zero because the coordinates of X always sum to n. Claim 6.10. LetX = P n i=1 X i be an (n,k)-PMD and let Σ be the covariance matrix of X. If p i,j = Pr[X i =e j ]≥ ǫ k−1 for all i∈ [n] and j ∈ [k], then all eigenvalues of Σ but one are at least nǫ k−1 . 122 Proof. Fix any unit vector v ∈ R k that is orthogonal to the all-one vector 1, i.e., P j v j = 0 and P j v 2 j = 1. Together with the assumption that p i,j ≥ ǫ k−1 , we have Var[v T X i ] =E v T X i −E h (v T X i ) i 2 = n X j=1 p i,j   v j − n X j ′ =1 p i,j ′v j ′   2 ≥min j {p i,j }· n X j=1   v 2 j +   n X j ′ =1 p i,j ′v j ′   2 −2v j   n X j ′ =1 p i,j ′v j ′      = min j {p i,j }·   1+n   n X j ′ =1 p i,j ′v j ′   2    ≥ ǫ k−1 . Therefore, v T Σv = Var[v T X]= n X i=1 Var[v T X i ]≥ nǫ k−1 . So, for all eigenvectors v orthogonal to 1, we have v T Σv = λv T v = λ ≥ nǫ k−1 as claimed. Werecallsomeofthenotationsforreadabilitybeforewedescribetheconstruc- tionofourǫ-coverofhigh-variancePMDs. WeuseX todenoteageneric(ℓ,k)-PMD for some ℓ ∈ [n], and we denote p i,j = Pr[X i = e j ]. We use A t ⊆ [ℓ] to denote the set of t-maximal CRVs in X, where a k-CRV is t-maximal if e t is its most likely outcome, and we use X t = P i∈At X i to denote the t-maximal component PMD of X. For a vectorm = (m 1 ,...,m k )∈Z k + , we definem th parameter moment ofX t to beM m (X t ) = P i∈At Q k j=1 p m j i,j .Werefertokmk 1 = P k j=1 m j asthedegree ofM m (X). We use S to denote the set of all k-CRVs whose probabilities are multiples of ǫ 20kn . 123 The following robust moment-matching lemma provides a bound on how close degree-ℓ moments need to be so that two (n,k)-PMDs are ǫ-close to each other, under the assumption that n≫k (the anonymous game has many players and few strategies) andp i,j ≥ ǫ k−1 (every player randomizes). Lemma 6.11 allows us to build a polynomial-size (ǫ/5)-cover for PMDs with high variance, and since we know that there is an ǫ-approximate equilibrium with a high variance, we are guaranteed to find one in our cover. Lemma 6.11. Fix 0<c<1 and letǫ=n −c . Assume thatn≥k Θ(k) for some suffi- ciently large constant in the exponent. LetX, Y be (n,k)-PMDs withX = P k i=1 X i , Y = P k i=1 Y i where each X i , Y i is an i-maximal PMD. Let Σ X and Σ Y denote the covariance matrices of X and Y respectively. Suppose all non-zero eigenvalues of Σ X ,Σ Y are at least ǫn/k, and all the parameter moments m of degree ℓ ≤ 2+2c 1−c satisfy that M m (X i )−M m (Y i ) ≤ǫ. Then, we have that d TV (X,Y)≤ǫ. Lemma 6.11 follows from Proposition 6.12. Proposition 6.12. Let ǫ > 0. Let X, Y be (n,k)-PMDs with X = P k i=1 X i , Y = P k i=1 Y i where each X i , Y i is an i-maximal PMD. Let Σ X and Σ Y denote the covariance matrices of X and Y respectively, where all eigenvalues of Σ X and Σ Y but one are at least σ 2 , where σ ≥ poly(klog(1/ǫ)). Suppose that for 1 ≤ i ≤ k, ℓ≥1, for all moments m of degree ℓ with m i =0, we have that M m (X i )−M m (Y i ) ≤ ǫ·σ ℓ C ′k+ℓ ·k 3ℓ/2+1 ·log k+ℓ/2 (1/ǫ) for a sufficiently large constant C ′ . Then d TV (X,Y)≤ǫ. 124 The proof of Proposition 6.12 exploits the sparsity of the continuous Fourier transform of our PMDs, as well as careful Taylor approximations of the logarithm of theFourier transform. We defer theproofof Proposition6.12 to thenext section. Proof of Lemma 6.11 from Proposition 6.12. To guarantee that d TV (X,Y) ≤ ǫ, Proposition 6.12 requires the following condition to hold for a sufficiently large constant C ′ : M m (X i )−M m (Y i ) ≤ ǫ k(C ′ log(1/ǫ)) k ·   q ǫn/k C ′ k 3/2 log 1/2 (1/ǫ)   ℓ , ∀i∈[k],ℓ≥1. (6.2) To prove the lemma, we use the fact thatn≫k and essentially ignore all the terms except polynomials of n. Formally, we first need to show that ǫ≤ ǫ k(C ′ log(1/ǫ)) k ·   q ǫn/k C ′ k 3/2 log 1/2 (1/ǫ)   ℓ , ∀ℓ≥ 1, under the assumption that c < 1, ǫ = n −c and n ≥ k O(k/(1−c)) . After substituting ǫ=n −c , observe thatn 1−c ≥C ′2 k 4 logn, so theterm insidetheℓ-thpower isgreater than 1. Thus, we only need to check this inequality for ℓ = 1, which simplifies to n 1−c ≥C ′2k+2 k 6 (logn) 2k and holds true. In addition, we need to show that condition (6.2) holds automatically for ℓ> 2+2c 1−c . This follows from the fact that the difference in parameter moments is at most n and n≫k, M m (X i )−M m (Y i ) ≤n≤ ǫ k(C ′ log(1/ǫ)) k ·   q ǫn/k C ′ k 3/2 log 1/2 (1/ǫ)   ℓ , ∀ℓ> 2+2c 1−c . 125 Lemma 6.11 states that the high-degree parameter moments match automat- ically, which allows us to impose an appropriate grid on the low-degree moments to cover the set of high-variance PMDs. The size of this cover can be bounded by a simple counting argument: We have at most k O( 1 1−c ) moments with degree at most O( 1 1−c ), and we need to approximate these moments for eacht-maximal component PMDs, so there are at most k·k O( 1 1−c ) = k O( 1 1−c ) moments M m (X t ) that we care about. We approximate these moments to precisionǫ=n −c , and the moments have value at most n, so the size of the cover is n n −c k O( 1 1−c ) =n k O(1/1−c) . We define this grid on low-degree moments formally in the following lemma. For every (ℓ,k)-PMD X with ℓ∈ [n], we associate some data D(X) with X, which is a vector of the approximate values of the low-degree moments M m (X t ) of X. Lemma 6.13. Fix 0 < c < 1 and let ǫ = n −c . Assume that n ≥ k Θ(k) for some sufficiently large constant in the exponent. We define the data D(W) of a k-CRV W as follows: D(W) m,t =                    M m (W) rounded to the nearest if W is t-maximal. integer multiple of ǫn, 0, otherwise. For ℓ∈ [n], we define the data of an (ℓ,k)-PMD X = P ℓ i=1 X i to be the sum of the data of its k-CRVs: D(X)= P ℓ i=1 D(X i ). The data D(X) satisfies two properties: 1. (Representative) If D(X) = D(Y) for two (n,k)-PMDs (or two (n− 1,k)- PMDs) X and Y, then d TV (X,Y)≤ǫ. 2. (Extensible) For independent PMDs X and Y, we have that D(X +Y) = D(X)+D(Y). 126 Proof. The “extensible” property follows directly from the definition of D(X). To see the“representative” property, notethatwe roundM m (W)tothenearest integer multiple of ǫn, so the error in the moments of W is at most ǫn/2. When we add up the data of an (n,k)-PMD or (n−1,k)-PMD, the error in the moments of each t-maximal component PMDs is at most ǫ/2. So if two PMDs X and Y have the same data, their low-degree moments differ by at most ǫ, and then by Lemma 6.11 we have d TV (X,Y)≤ǫ. Algorithm 6.1: GenerateData Input :{S i } n i=1 , ǫ> 0. Output: The set of all possible data D of (n,k)-PMDs X = P n i=1 X i where X i ∈S i . 1 D 0 ={}; 2 for ℓ =1...n do 3 forall the D∈D ℓ−1 do 4 forall theW ∈S ℓ do 5 Add D+D(W) to D ℓ if it is not in D ℓ already; 6 Keep track of an (ℓ,k)-PMD whose data is D+D(W); 7 returnD =D n ; Ouralgorithm(Algorithm6.2)forcomputingapproximateequilibria issimilar to the approach used in [39] and [41]. We start by constructing a polynomial-size (ǫ/5)-cover ofhigh-variancePMDs(Algorithm6.1), andthen iterateover thiscover. For each element in the cover, we compute the set of (3ǫ/5)-best-responses for each player, and then run the cover construction algorithm again, but this time we only allow each player to choose from her (3ǫ/5)-best-responses. If we can reconstruct a PMD whose moments are close enough to the one we started with, then we have found an ǫ-approximate Nash equilibrium. 127 Algorithm 6.2: Moment Search Input : An n-player k-strategy anonymous gameG, ǫ=n −c for some c< 1. Output: An ǫ-approximate Nash equilibrium of G. 1 S ={all k-CRVs whose probabilities are multiples of ǫ 20kn }; 2 D n = GenerateData({S i =S} n i=1 , ǫ/5); 3 D n−1 = GenerateData({S i =S} n−1 i=1 , ǫ/5); 4 forall theD∈D n do 5 Set S i =∅ for all i; 6 forall the X i ∈S do 7 Let D −i =D−D(X i ); 8 if ∃Y D −i ∈D n−1 with D(Y D −i ) =D −i and X i is a (3ǫ/5)-best response to Y D −i then 9 Add X i to S i ; 10 D ′ n = GenerateData({S i } n i=1 , ǫ/5); 11 if D∈D ′ n then 12 return (X 1 ,...,X n ) inD ′ n with D( P n i=1 X i ) =D Recall that a mixed strategy profile for a k-strategy anonymous game can be represented as a list ofk-CRVs (X 1 ,...,X n ), whereX i describes the mixed strategy ofplayeri. Recallthat(X 1 ,...,X n )isanǫ-approximateNashequilibriumifforeach playeri we haveE h u i X i (X −i ) i ≥E[u i a (X −i )]−ǫ for alla∈[k], whereX −i = P j6=i X j is the distribution of the sum of other players strategies. Lemma 6.14. Fix an anonymous game G = (n,k,{u i a } i∈[n],a∈[k] ) with payoffs nor- malized to [0,1]. Let (X 1 ,...,X n ) and (Y 1 ,...,Y n ) be two lists of k-CRVs. If X i is a δ-best response to X −i , and d TV (X −i ,Y −i ) ≤ ǫ, then X i is a (δ + 2ǫ)-best response to Y −i . Moreover, if (X 1 ,...,X n ) is a δ-approximate equilibrium, and d TV (X i ,Y i ) + d TV (X −i ,Y −i ) ≤ ǫ for all i ∈ [n], then (Y 1 ,...,Y n ) is a (δ + 2ǫ)- approximate equilibrium. 128 Proof. Since u i a (x)∈ [0,1] for all a∈[k] and x∈Π k n−1 , we have that E h u i a (X −i ) i −E h u i a (Y −i ) i ≤d TV (X −i ,Y −i ), ∀i∈[n],a∈[k]. Therefore, if d TV (X −i ,Y −i ) ≤ ǫ, and player i cannot deviate and gain more than δ when other players play X −i , then she cannot gain more than (δ+2ǫ) when other players play Y −i instead of X −i . The second claim combines the inequality above with thefactthat, ifplayeriplaysY i instead ofX i andthemixed strategiesofother players remain the same, her payoff changes by at most d TV (X i ,Y i ). Formally, E h u i X i (Z −i ) i −E h u i Y i (Z −i ) i ≤d TV (X i ,Y i ), ∀k-CRV X i ,Y i ,∀(n−1,k)-PMD Z −i . The next lemma states that there exists an (ǫ/5)-approximate equilibrium whose probabilities are all integer multiples of ǫ 20kn . Claim 6.15. There is an (ǫ/5)-approximate Nash equilibrium (X 1 ,...,X n ), such that for all i ∈ [n] and j ∈ [k], the probabilities p i,j = Pr[X i = e j ] are multiples of ǫ 20kn , and also p i,j ≥ ǫ 10k . Proof. We start with an (ǫ/10)-approximate Nash equilibrium (Y 1 ,...,Y n ) from Lemma 6.8 with p i,j ≥ ǫ 10k , and then round the probabilities to integer mul- tiples of ǫ 10kn . We construct X i from Y i as follows: for every j < k, we set Pr[X i = e j ] to be Pr[Y i = e j ] rounded down to a multiple of ǫ 20kn and we set Pr[X i = e k ] = 1− P j<k Pr[X i = e j ] so the probabilities sum to 1. By triangle inequality of total variation distance, for every i we have d TV (X i ,Y i ) ≤ ǫ 20n and d TV (X −i ,Y −i ) ≤ ǫ(n−1) 20n . An application of Lemma 6.14 shows that (X 1 ,...,X n ) is an (ǫ/5)-approximate equilibrium. 129 We are now ready to prove Theorem 6.6. Proof of Theorem 6.6. We show that for any n-player k-strategy anonymous game, if both c > 0 and k are constants, then there is a poly(n) time algorithm that computes an ǫ-approximate equilibrium for ǫ = 1/n 1−c . If n = k O(k) = O(1), we use the algorithm in [36] which runs in timen (1/ǫ) Ω(k) =O(1). So for the rest of the proof, we assume that n ≥ k Θ(k) as required in Lemma 6.11 and 6.13, and prove that Algorithm 6.2 always outputs an ǫ-approximate Nash equilibrium, and bound the running time. We first show that the output (X 1 ,...,X n ) is an ǫ-approximate equilibrium. Recall that S is the set of all k-CRVs whose probabilities are multiples of ǫ 20kn , and S i ⊆ S is the set of approximate best-responses of player i. When we put X i in S i , we checked that X i is a (3ǫ/5)-best response to Y D −i . Note that D(Y D −i ) = D−D(X i )=D(X −i ), sobyLemma 6.13d TV X −i ,Y D −i ≤ǫ/5foralli. ByLemma 6.14, X i is indeed an ǫ-best response to X −i for all i. Next we show the algorithm must always output something. By Claim 6.15 thereexistsan(ǫ/5)-approximateequilibriumX ′ i witheachX ′ i ∈S. Ifthealgorithm does not terminate successfully first, it eventually considers D(X ′ ). Because X ′ −i is an (n−1,k)-PMD, the algorithm can find some Y D −i with D(Y D −i ) = D(X ′ )− D(X ′ i ) = D(X ′ −i ), and by Lemma 6.13 we have d TV X ′ −i ,Y D −i ≤ ǫ/5 for all i. Since X ′ i is an (ǫ/5)-best response to X ′ −i , Lemma 6.14 yields that X ′ i is a (3ǫ/5)- best response toY D −i , so we would add each X ′ i toS i . Then our cover construction algorithmisguaranteedtogenerateasetofdatathatincludesD(X ′ ),andAlgorithm 6.2 would produce an output. 130 Finally, we bound the running time of Algorithm 6.2. Let N = O n k O(1/1−c) denote the size of the (ǫ/5)-cover for the high-variance PMDs. The cover can be constructed in time O(n·N ·|S|) as we try to add onek-CRV fromS in each step. We iterate through the cover, and for each element in the cover, we need to find the subset S i ⊆ S of (3ǫ/5)-best responses for player i, and then run the cover construction algorithm again using only the best responses {S i } n i=1 . So the overall running time of the algorithm isO(nN|S|)· poly(n k )|S|+O(nN|S|) =n k O(1/1−c) . When both 0<c<1 andk are constants, the running time is polynomial inn. 6.3.3 A New Moment Matching Lemma This subsection is devoted to the proof of Proposition 6.12. For two (n,k)- PMDs with variance at leastσ 2 in each direction, Proposition 6.12 gives a quantita- tive bound on how close degree-ℓ moments need to be (as a function of ǫ, σ, k and ℓ, but independent of n), in order for the two PMDs to beǫ-close in total variation distance. The proof of Proposition 6.12 exploits the sparsity of the continuous Fourier transforms of our PMDs, as well as careful Taylor approximations of the logarithm of the Fourier transform. The fact that our PMDs have large variance enables us to take fewer low-degree terms in the Taylor approximation. For technical reasons, we split our PMD as the sum of k independent component PMDs, X = P k i=1 X i , whereallthek-CRVsinthecomponentPMDX i arei-maximal. BecausetheFourier transform of X is the product of the Fourier transforms of X i , we can just bound the pointwise difference between the logarithms of the Fourier transforms of each component PMD. One technicality is that since we have no assumption on the variances of the component PMDs X i , their Fourier transforms may not be sparse, 131 so it is crucial that we bound this difference only on the effective support of the Fourier transform of the entire PMD. We start by considering a set S that includes the effective support of X (and Y when we show that the means are close): Lemma 6.16(EssentiallyCorollary5.3of[41]). LetX bean(n,k)-PMD withmean µ and covariance matrix Σ, such that all the non-zero eigenvalues of Σ are at least σ 2 where σ≥poly(1/ǫ). Let S be the set of points x∈Z k where (x−µ ) T 1 = 0 and (x−µ ) T (Σ+I) −1 (x−µ )≤(Cklog(1/ǫ)), for some sufficiently large constantC. Then,X ∈S with probability at least 1−ǫ/2, and |S|= q det(Σ+I)·O(log(1/ǫ)) k/2 . Proof. Applying Lemma 5.2 of [41], we have that (X −µ ) T (Σ +I) −1 (X −µ ) = O(klog(k/ǫ)) with probability at least 1 − ǫ. The set of integer coordinate points in this ellipsoid is the set S. Note that |S| is equal to the volume of S ′ = n y∈R k : ∃x∈S with ky−xk ∞ ≤1/2 o , because S ′ is the disjoint union of cubes of volume 1, one for each integer point. But S ′ is again contained in an ellipsoid with (y −µ ) T (Σ +I) −1 (y −µ ) = O(klog(k/ǫ)), so |S| = Vol(S ′ ) = q det(Σ+I)·O(log(1/ǫ)) k/2 . Next we show that c X, the Fourier transform of X, has a relatively small effective support. We fold the effective support onto [0,1] k to obtain the set T. We use [x] to denote the additive distance of x ∈ R to the closest integer, i.e., [x] = min x ′ ∈Z |x−x ′ |. 132 Lemma 6.17. Let X be an (n,k)-PMD with mean µ and covariance matrix Σ, such that all the non-zero eigenvalues of Σ are at least σ 2 where σ ≥ poly(klog(1/ǫ)). Let S be as above. Let c X be the Fourier transform of X. Let T def = n ξ ∈[0,1] k : ∃ξ ′ ∈ξ+Z k with ξ ′T Σξ ′ ≤Cklog(1/ǫ) o , for some sufficiently large constant C. Then, we have that (i) For ξ ∈T, and for all 1≤i,j ≤k, [ξ i −ξ j ]≤2 q Cklog(1/ǫ)/σ. (ii) Vol(T)|S|=O(Clog(1/ǫ)) k . (iii) R [0,1] k \T c X(ξ) dξ ≤ǫ/(2|S|). Lemma 6.17 is a technical generalization of Lemma 5.5 of [41]. This lemma establishes that the contribution to the Fourier transform c X coming from points outside of T is negligibly small. We then use the sparsity of the Fourier transform to show that, if two PMDs have Fourier transforms that are pointwise sufficiently close within the effective supportT, then the two PMDs are close in total variation distance. Lemma 6.18. Let X, Y, S, T be as above. If c X(ξ)− b Y(ξ) ≤ǫ(C ′ log(1/ǫ)) −k for all ξ ∈T and a sufficiently large constant C ′ , then d TV (X,Y)≤ǫ. 133 Proof. For any x∈Z k , taking the inverse Fourier transform, we have that Pr[X = x] = R ξ∈[0,1] ke(−ξ ·x) c X(ξ)dξ and similarly Pr[Y = x] = R ξ∈[0,1] ke(−ξ ·x) b Y(ξ)dξ. Thus, |Pr[X =x]−Pr[Y =x]| = Z ξ∈[0,1] k e(−ξ·x) c X(ξ)− b Y(ξ) dξ ≤ Z ξ∈[0,1] k c X(ξ)− b Y(ξ) dξ = Z ξ∈T c X(ξ)− b Y(ξ) dξ+ Z ξ∈[0,1] k \T c X(ξ)− b Y(ξ) dξ ≤Vol(T)·ǫ(C ′ log(1/ǫ)) −k + ǫ 2|S| ≤ O(Clog(1/ǫ)) k |S| ·ǫ(C ′ log(1/ǫ)) −k + ǫ 2|S| ≤ ǫ |S| . Since X and Y are outside of S each with probability less than ǫ/2, we have that d TV (X,Y)≤ǫ/2+ 1 2 P x∈S |Pr[X =x]−Pr[Y =x]|≤ǫ. We now have all the ingredients to prove Proposition 6.12. For two PMDs X and Y that are close in their low-degree moments, we show that their Fourier transforms c X and b Y are pointwise close on T, and then by Lemma 6.18, X and Y are close in total variation distance. Proof of Proposition 6.12. LetX,Y,S,T beasabove. Given Lemma 6.18, weonly need to show that∀ξ ∈T, c X(ξ)− b Y(ξ) ≤ǫ(C ′ log(1/ǫ)) −k . 134 Fix ξ ∈T. We first examine, without loss of generality, the Fourier transform d X k of the k-maximal component PMD. Let A k ⊆ [n] denote the set of k-maximal CRVs. d X k (ξ)= Y i∈A k k X j=1 e(ξ j )p i,j =e(|A k |ξ k ) Y i∈A k   1− k−1 X j=1 (1−e(ξ j −ξ k ))p i,j )   =e(|A k |ξ k )exp   X i∈A k log   1− k−1 X j=1 (1−e(ξ j −ξ k ))p i,j )     =e(|A k |ξ k )exp   − X i∈A k ∞ X ℓ=1 1 ℓ   k−1 X j=1 (1−e(ξ j −ξ k ))p i,j )     =e(|A k |ξ k )exp   − X m∈Z k−1 + kmk 1 m ! 1 kmk 1 M m (X k ) k−1 Y j=1 (1−e(ξ j −ξ k )) m j    (6.3) For notational convenience, we use Ψ k X to denote the expression inside exp(·) in Equation (6.3). A similar formula holds for the Fourier transforms c X i and c Y i of other i-maximal PMDs, and we use Ψ i X and Ψ i Y to denote the corresponding expressions inside exp(·). Since the Fourier transform of a PMD is the product of the Fourier transform of its component PMDs, we have c X(ξ)− b Y(ξ) = k Y t=1 c X t (ξ)− k Y t=1 c Y t (ξ) = e k X t=1 |A t |ξ t ! k Y t=1 exp Ψ t X −exp Ψ t Y ≤2π k X t=1 Ψ t X −Ψ t Y , 135 wherethelastinequalityisduetoe( P k t=1 |A t |ξ t )= 1,and|exp(a)−exp(b)|≤|a−b| if the real parts of a and b satisfy Re(a),Re(b)≤0. So to prove that c X(ξ) and b Y(ξ) are pointwise close for all ξ ∈T, it is enough to bound from above 2π P k t=1 |Ψ t X −Ψ t Y |. We use the fact that |1−e(ξ j −ξ k )| = O([ξ j −ξ k ]), and recall that [ξ i −ξ j ] ≤ 2 q Cklog(1/ǫ)/σ by Lemma 6.17. We also usethemultinomialidentity P m∈Z k−1 + ,kmk 1 =ℓ ℓ m = (k−1) ℓ . WhenC ′ isasufficiently large constant, we have c X(ξ)− b Y(ξ) ≤2π k X t=1 Ψ t X −Ψ t Y = 2π k X t=1 X m∈Z k−1 + kmk 1 m ! 1 kmk 1 M m (X t )−M m (Y t ) k−1 Y j=1 (1−e(ξ j −ξ k )) m j ≤2π ∞ X ℓ=1 (k−1) ℓ ℓ   O   q klog(1/ǫ) σ     ℓ k X t=1 max m∈Z k−1 + ,kmk 1 =ℓ M m (X t )−M m (Y t ) ≤ ∞ X ℓ=1 k ℓ   C ′ q klog(1/ǫ) 2σ   ℓ k· ǫσ ℓ C ′k+ℓ ·k 3ℓ/2+1 ·log k+ℓ/2 (1/ǫ) = ∞ X ℓ=1 2 −ℓ ǫ(C ′ log(1/ǫ)) −k =ǫ(C ′ log(1/ǫ)) −k . 6.3.4 Slight Improvement Gives FPTAS In this section, we show that even a slight improvement of our upper bound would imply an FPTAS for computing (well-supported) Nash equilibria in anony- mous games (Theorem 6.7). It is a plausible conjecture that assuming the ETH for 136 PPAD, there is no such FPTAS, in which case our upper bound (Theorem 6.6) is essentially tight. Theorem 6.7followsdirectlyfromthefollowingtwo lemmas. Lemma 6.19con- vertsan ǫ 2 4n -approximateNashequilibriumintoanǫ-well-supportedNashequilibrium (seeDefinition2.3), byreallocatingeach player’s probabilitiesonstrategieswith low expected payoffs to the best-response strategy (first observed in [35]). Lemma 6.20 thenusesapaddingargumenttoshowthat,forǫ-well-supported Nashequilibria, the question of whether there is a polynomial-time algorithm for ǫ = n −c is equivalent for all constants c> 0. Lemma 6.19. For any n-player game whose payoffs are normalized to be between [0,1], if we have an oracle for computing players’ payoffs, we can efficiently convert an ǫ 2 4n -approximate equilibrium into an ǫ-well-supported equilibrium. Proof. Take an ǫ 2 4n -approximate equilibrium of the game. We call a strategy “good” for a player if the strategy is an ǫ 2 -best response for the player, and we call it “bad” otherwise. A player can put at most probability ǫ 2n on the “bad” strategies without violating the ǫ 2 4n -approximate equilibrium condition. We move all the probabilities on“bad”strategiesforallplayersto(anyoneof)theirbestresponsessimultaneously. After moving the probabilities, every player assigns non-zero probabilities only to the “good” strategies. Since the total probability we moved is at most ǫ 2 and the payoffs are in [0,1], the previously “good” strategies ( ǫ 2 -best responses) are now ǫ-best responses. Lemma 6.20. For n-player k-strategy anonymous games with k =O(1), if an 1 n γ - well-supported equilibrium can be computed in time O(n d ) for constants γ,d > 0, then there is an FPTAS for computing approximate-well-supported Nash equilibria in anonymous games. 137 Proof. Let ǫ be the desired quality of the well-supported equilibrium. If 1 n γ ≤ ǫ we are done, so we assume n is smaller. We set n ′ = (1/ǫ) 1/γ , so that 1 n ′γ = ǫ. Given an n-player anonymous game G, we build an n ′ -player anonymous game G ′ as follows: we add n ′ −n dummy players, and give the dummy players utility 1 on strategy 1, and 0 on any other strategies so in any ǫ-well-supported equilibria, the dummy player must all play strategy 1 with probability 1. (Note that this is only true for ǫ-well-supported Nash equilibrium; in an ǫ-approximate Nash equilibrium, the dummy players can putǫ probability elsewhere.) Weshift the utility function of the actual players to ignore the dummy players on strategy 1. Formally, the payoff structure of G ′ is given by: • For each i>n, u ′i a (x) =          1 if a = 1 0 otherwise • For each i ≤ n, we subtract the number of players on strategy 1 by n ′ −n and then apply the original utility function. We define φ : Z k → Z k as φ(x 1 ,...,x k ) =(x 1 −(n ′ −n),x 2 ,...,x k ), u ′i a (x) =          u i a (φ(x)) if x 1 ≥n ′ −n 0 otherwise Since ǫ = 1 n ′γ , by assumption we can compute an ǫ-well-supported equilibrium of G ′ in time O(n ′d ), and we can simply remove the dummy players to obtain an ǫ- equilibrium of theoriginalgameG. The running timeisO(n ′d ) = poly(n,1/ǫ)when γ =Θ(1). 138 Proof of Theorem 6.7. Assume that we can compute an O(n −c )-approximate equi- libriuminpolynomialtimeforsomeconstantc>1. Letγ =c−1,sowecancompute an O 1 n 1+γ -approximate equilibrium in polynomial time. By Lemma 6.19, we can convert itinto anO 1 n γ/2 -well-supported equilibrium. Lemma 6.20then statesthat any polynomial-time algorithm that computes a well-supported Nash equilibrium of aninverse polynomialprecision givesanFPTAS forcomputing well-supported Nash equilibria in anonymous games. 139 Chapter 7 Conclusion and Open Questions Algorithmic game theory is rife with strategic interactions with uncertainty and information asymmetry. In this thesis, we examined the following question through a computational lens: What is the best way to reveal information to other strategic players, and how hard is it to find the optimal information structure? We studied the design of information structures — a principal who is privy to private information must choose how to reveal information to induce a better outcome. We developed algorithms and proved matching hardness results for sig- naling in many important classes of games: normal form games, and succinct games including network routing games, second price auctions and majority voting. We saw the role of information revelation changes from chapter to chapter. In informational variants of Braess’ paradox and prisoner’s dilemma, a principal tries to hide information to help the players fight their selfishness. In normal form games, a principal who wants to help his friend must identify which portion of the information helps one of the players but not the other; which may require her to identify dense subgraphs in a given graph. In second price auctions, a principal who seeks to maximize her revenue must reveal some but not all information to induce the right amount of competition in the market. 140 The computational complexity ofoptimal signaling also changes, and becomes easier from chapter to chapter. For network routing games, in the worst case, the principalhastosolveNP-hardproblemstodobetterthanrevealingfullinformation. Innormalformgames,theprincipalcancomputeanear-optimalsignalingschemein quasipolynomial time. As we move to anonymous games like second price auctions and voting, the principal can signal approximately optimally in polynomial time. By settling the computational complexity of these signaling problems, we improved our understanding of information asymmetry in games, as well as the power and limitations of strategic information revelation. The investigation of optimal information revelation has also led to powerful algorithmicframeworks. Drivenbythedesireforfundamentalinsights, weidentified themixtureselection problem—analgorithmicproblem thatarisesnaturallyinthe design of optimal information structures. We presented two complexity measures that seem to dictate the complexity of mixture selection and optimal signaling, and solved a number of signaling problems near-optimally under the mixture selection framework. The design of information structures is emerging as a new area in algorithmic game theory, an area that is still largely unexplored. This thesis addresses the optimal signaling in several basic families of Bayesian games, and there are many exciting problems to be discovered and solved. We list a few open questions below. Open Questions Problem 7.1 (Private signaling). How does the computational complexity change if the principal is allowed to reveal different information to different players? 141 In this thesis, we study public signaling schemes, where the principal must reveal the same information to all players. Does private communication make the principal more powerful, and how does the complexity of optimal signaling change? Dughmi and Xu [51] showed that, for multi-player games with n players, the gap between the value of the optimal public and private signaling schemes is at least Θ(n). They also settle the complexity of public and private signaling when there arenoexternalities 1 . Itremainsaninterestingopenquestionhowtosignalefficiently in games with externalities, and whether the interaction between the players makes the signaling problem harder or not. Problem 7.2 (Equivalence of optimization and separation). For a polytope P con- tained in the simplex, if we are given a PTAS for the separation (or membership) oracle of P — an oracle that runs in polynomial time for any constant ǫ > 0 and has ǫ-additive error — can we obtain a PTAS for optimization over P? In other words, do we need a much more precise membership oracle to be able to optimize approximately? In Section 4.4, we ruled out an FPTAS for optimal signaling using FPTAS hardness of posterior selection. Recall that the posterior selection problem asks for the best posterior distribution, while the signaling prob- lem asks for the best decomposition (of the prior distribution) into posteriors. It is often easier to show the posterior selection problem is hard, and then use the same intuition to derive a direct reduction for the hardness of signaling. For exam- ple, finding a planted clique in a random graph is hard, and for similar reasons finding a constant fraction of a planted clique cover (i.e., decomposing into dense subgraphs) is also hard; approximating the size of the maximum independent set is 1 In games with no externalities, each player’s payoff depends only on his own action (and also on the state of nature for Bayesian games), but not on the actions of other players. 142 hard; similarly approximating the chromatic number (i.e., decomposing into inde- pendent sets) is also hard. These ideas are used implicitly in [15, 29, 46, 51] to show PTAS hardness results for different signaling problems, and these results can be unified if Problem 7.2 can be resolved in the positive. Problem 7.3 (Nash equilibria in anonymous games). Is there an FPTAS for com- puting Nash equilibria in anonymous games? Almost all the algorithmic results for equilibrium computation in anonymous games can be viewed as first guessing the outcome of the game, and then trying to reconstruct this outcome using only the best response of each player. New ideas seem to be needed for qualitatively faster algorithms. On the other hand, for ruling out an FPTAS, it is unlikely that the approach in [25] can work directly. This is because 1/poly(n) precision is only enough to de-anonymize O(logn) players, but O(logn)-player O(1)-strategy games can be solved in time n O(loglogn) (rather than quasipolynomial time) due to the existence theory of the reals. Problem 7.4 (Routing games with non-linear latencies). Is there a better signaling scheme than full revelation for Bayesian routing games with non-linear latencies? We showed that no polynomial time algorithm can do better than 4/3 in the worst case for signaling in network routing games. The best signaling algorithm we know of, which simply reveals full information, is a multiplicative approximation withtheratioequaltothepriceofanarchy. Itremainsopenwhatisthebestpossible ratio we can obtain in polynomial time for non-linear latency functions. Problem 7.5 (Planted clique conjecture). Is there a formal connection between planted clique and widely used worst-case hardness assumptions, e.g., the Exponen- tial Time Hypothesis (ETH)? 143 It was shown that computing ǫ-best Nash equilibrium in two-player normal formgamesrequiresquasipolynomialtimeforasmallenoughconstantǫ>0,assum- ing either the planted clique conjecture [63] or the ETH [20]. Two of the hardness results in this thesis, optimal signaling in normal form games, and mixture selec- tion in the absence of noise stability, can both be obtained by assuming either the planted clique conjecture [15] ortheETH [86]. Isthereaformalconnection between the planted clique conjecture and the ETH? 144 Reference List [1] Abbott, T., Kane, D., and Valiant, P. (2005). On the complexity of two-player win-lose games. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS), pages 113–122. [2] Akerlof, G. A. (1970). The market for “lemons”: Quality uncertainty and the market mechanism. The quarterly journal of economics, pages 488–500. [3] Alon, N., Andoni, A., Kaufman, T., Matulef, K., Rubinfeld, R., and Xie, N. (2007). Testing k-wise and almost k-wise independence. In Proceedings of the 38th ACM Symposium on Theory of Computing (STOC), pages 496–505. [4] Alon, N., Krivelevich, M., andSudakov, B.(1998). Findingalargehidden clique in a random graph. In Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 594–598. [5] Alonso, R. and Câmara, O. (2016). Persuading voters. The American Economic Review, 106(11):3590–3605. [6] Althöfer, I. (1994). On sparse approximations to randomized strategies and convex combinations. Linear Algebra and its Applications, 199:339–355. [7] Babichenko, Y. and Barman, S. (2016). Computational aspects of private Bayesian persuasion. arXiv preprint 1603.01444. [8] Babichenko, Y., Barman, S., and Peretz, R. (2014). Simple approximate equi- libria in large games. In Proceedings of the 16th ACM Conference on Economics and Computation (EC), pages 753–770. [9] Babichenko, Y., Papadimitriou, C., and Rubinstein, A. (2015). Can almost everybody be almost happy? PCP for PPAD and the inapproximability of Nash. In Proceedings of the 7th Conference on Innovations in Theoretical Computer Science, pages 1–9. 145 [10] Babichenko, Y. and Rubinstein, A. (2017). Communication complexity of approximate Nash equilibria. In Proceedings of the 48th ACM Symposium on Theory of Computing (STOC), pages 878–889. [11] Bárány,I.,Vempala,S.,andVetta,A.(2005). Nashequilibriainrandomgames. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS), pages 123–131. [12] Barman, S. (2015). Approximating Nash equilibria and dense bipartite sub- graphs via an approximate version of Caratheodory’s theorem. In Proceedings of the 47th Annual ACM on Symposium on Theory of Computing, pages 361–369. [13] Bergemann, D., Brooks, B., and Morris, S. (2015). The limits of price discrim- ination. The American Economic Review, 105(3):921–957. [14] Bergemann, D. and Morris, S. (2017). Information design: A unified perspec- tive. Cowles Foundation Discussion Paper No. 2075. [15] Bhaskar, U., Cheng, Y., Ko, Y. K., and Swamy, C. (2016). Hardness results for signaling in Bayesian zero-sum and network routing games. In Proceedings of the 18th ACM Conference on Economics and Computation (EC), pages 479–496. [16] Blackwell, D. (1951). Comparison of experiments. In Second Berkeley Sympo- sium on Mathematical Statistics and Probability, volume 1, pages 93–102. [17] Blonski, M. (1999). Anonymous games with binary actions. Games and Eco- nomic Behavior, 28(2):171–180. [18] Blonski, M.(2005). ThewomenofCairo: Equilibria inlargeanonymousgames. Journal of Mathematical Economics, 41(3):253–264. [19] Braess, D. (1968). Über ein paradoxon aus der verkehrsplanung. Unternehmensforschung, 12(1):258–268. [20] Braverman, M., Ko, Y. K., and Weinstein, O. (2015). Approximating the best Nash equilibrium inn o(logn) -time breaks the exponential time hypothesis. In Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 970–982. [21] Bro Miltersen, P. and Sheffet, O. (2012). Send mixed signals: earn more, work less. In Proceedings of the 14th ACM Conference on Electronic Commerce (EC), pages 234–247. [22] Carathéodory, C. (1907). Über den variabilitätsbereich der koeffizienten von potenzreihen, die gegebene werte nicht annehmen. Mathematische Annalen, 64(1):95–115. 146 [23] Chen, X., Cheng, Y., and Tang, B. (2017). Well-supported versus approximate Nash equilibria: Query complexity of large games. In Proceedings of the 8th Conference on Innovations in Theoretical Computer Science. [24] Chen, X., Deng, X., and Teng, S.-H. (2009). Settling the complexity of com- puting two-player Nash equilibria. Journal of the ACM, 56(3):1–57. [25] Chen, X., Durfee, D., and Orfanou, A. (2015). On the complexity of Nash equilibria in anonymous games. In Proceedings of the 46th ACM Symposium on Theory of Computing (STOC), pages 381–390. [26] Cheng, Y., Cheung, H. Y., Dughmi, S., Emamjomeh-Zadeh, E., Han, L., and Teng, S.-H. (2015). Mixture selection, mechanism design, and signaling. In Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS), pages 1426–1445. [27] Cheng, Y., Cheung, H. Y., Dughmi, S., and Teng, S.-H. (2014). Signaling in quasipolynomial time. arXiv preprint 1410.3033. [28] Cheng, Y., Diakonikolas, I., and Stewart, A. (2017). Playing anonymous games using simple strategies. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 616–631. [29] Cheng, Y. and Ko, Y. K. (2017). Finding best mixture of Nash is hard. Manuscript. [30] Coja-Oghlan, A. (2010). Graph partitioning via adaptive spectral techniques. Combinatorics, Probability & Computing, 19(2):227. [31] Cole, R., Dodis, Y., and Roughgarden, T. (2006). How much can taxes help selfish routing? Journal of Computer and System Sciences, 72(3):444–467. [32] Conitzer, V. and Sandholm, T. (2006). Computing the optimal strategy to commit to. In Proceedings of the 8th ACM Conference on Electronic Commerce (EC), pages 82–90. [33] Daskalakis, C., De, A., Kamath, G., and Tzamos, C. (2016). A size-free CLT for Poisson multinomials and its applications. In Proceedings of the 47th ACM Symposium on Theory of Computing (STOC), pages 1074–1086. [34] Daskalakis, C., Diakonikolas, I., and Servedio, R. A. (2012). Learning Poisson binomial distributions. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pages 709–728. 147 [35] Daskalakis, C., Goldberg, P. W., and Papadimitriou, C. H. (2009). The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195–259. [36] Daskalakis,C.andPapadimitriou,C.H.(2007).Computingequilibriainanony- mous games. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 83–93. [37] Daskalakis, C. and Papadimitriou, C. H. (2008). Discretized multinomial dis- tributions and Nash equilibria in anonymous games. In Proceedings of the 49th IEEE Symposium on Foundations of Computer Science (FOCS), pages 25–34. [38] Daskalakis, C. and Papadimitriou, C. H.(2009). Onoblivious PTAS’s forNash equilibrium. InProceedingsof the 40th ACM Symposium on Theory of Computing (STOC), pages 75–84. [39] Daskalakis, C. and Papadimitriou, C. H. (2015). Approximate Nash equilibria in anonymous games. Journal of Economic Theory, 156:207–245. [40] Dekel, Y., Gurel-Gurevich, O., and Peres, Y. (2011). Finding hidden cliques in linear time with high probability. In Proceedings of the 8th Workshop on Analytic Algorithmics and Combinatorics, pages 67–75. [41] Diakonikolas, I., Kane, D. M., and Stewart, A. (2016a). The Fourier transform of Poisson multinomial distributions and its algorithmic applications. In Pro- ceedings of the 47th ACM Symposium on Theory of Computing (STOC), pages 1060–1073. [42] Diakonikolas, I., Kane, D. M., and Stewart, A. (2016b). Optimal learning via the Fourier transform for sums of independent integer random variables. In Proceedings of the 29th Conference on Learning Theory, pages 831–849. [43] Diakonikolas, I., Kane, D. M., and Stewart, A. (2016c). Properly learning Poisson binomial distributions in almost polynomial time. In Proceedings of the 29th Conference on Learning Theory, pages 850–878. [44] Doerr, B. (2011). Analyzing randomized search heuristics: tools from proba- bility theory. In Theory of randomized search heuristics. World Scientific. [45] Dubhashi, D. P. and Panconesi, A. (2009). Concentration of measure for the analysis of randomized algorithms. Cambridge University Press. [46] Dughmi, S. (2014). On the hardness of signaling. In Proceedings of the 55th IEEE Symposium on Foundations of Computer Science (FOCS), pages 354–363. 148 [47] Dughmi, S. (2017). Algorithmic information structure design: a survey. ACM SIGecom Exchanges, 15(2):2–24. [48] Dughmi, S., Immorlica, N., and Roth, A. (2014). Constrained signaling in auction design. In Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1341–1357. [49] Dughmi, S., Kempe, D., and Qiang, R. (2016). Persuasion with limited com- munication. In Proceedings of the 18th ACM Conference on Economics and Com- putation (EC), pages 663–680. [50] Dughmi,S.andXu,H.(2016). AlgorithmicBayesianpersuasion. InProceedings of the 47th ACM Symposium on Theory of Computing (STOC), pages 412–425. [51] Dughmi,S.andXu, H.(2017). Algorithmicpersuasion withnoexternalities. In Proceedings of the 19th ACM Conference on Economics and Computation (EC), pages 351–368. [52] Emek,Y.,Feldman,M.,Gamzu,I.,PaesLeme,R.,andTennenholtz,M.(2012). Signaling schemes for revenue maximization. In Proceedings of the 14th ACM Conference on Electronic Commerce (EC), pages 514–531. [53] Etessami, K. and Yannakakis, M. (2010). On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing, 39(6):2531–2597. [54] Feder, T., Nazerzadeh, H., and Saberi, A. (2007). Approximating Nash equi- libria using small-support strategies. In Proceedings of the 9th ACM Conference on Electronic Commerce (EC), pages 352–354. [55] Feige, U. and Krauthgamer, R. (2003). The probable value of the Lovász– Schrijverrelaxationsformaximumindependentset. SIAMJournalonComputing, 32(2):345–370. [56] Feige, U. and Ron, D. (2010). Finding hidden cliques in linear time. In Pro- ceedings of the 21st Conference on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, pages 189–204. [57] Feldman,V., Grigorescu, E.,Reyzin, L.,Vempala, S., andXiao,Y.(2013). Sta- tisticalalgorithmsandalowerboundfordetectingplantedcliques. InProceedings of the 44th ACM Symposium on Theory of Computing (STOC), pages 655–664. [58] Garey,M.R.andJohnson,D.S.(1979). ComputersandIntractability: AGuide to the Theory of NP-Completeness. W. H. Freeman & Co. 149 [59] Ghosh, A., Nazerzadeh, H., and Sundararajan, M. (2007). Computing optimal bundles forsponsored search. InProceedings of the 3rd Workshop on Internet and Network Economics (WINE), pages 576–583. [60] Goldberg, P. W. and Turchetta, S. (2015). Query complexity of approximate equilibria in anonymous games. In Proceedings of the 11th Workshop on Internet and Network Economics (WINE), pages 357–369. [61] Grötschel, M., Lovász, L., and Schrijver, L. (1993). Geometric algorithms and combinatorial optimization. Algorithms and Combinatorics, 2:1–362. [62] Guo, M. and Deligkas, A. (2013). Revenue maximization via hiding item attributes. In Proceedings of the 23rd international joint conference on Artifi- cial Intelligence, pages 157–163. [63] Hazan, E.andKrauthgamer, R.(2011). Howhardisittoapproximatethebest Nash equilibrium? SIAM Journal on Computing, 40(1):79–91. [64] Hirshleifer, J. (1971). The private and social value of information and the reward to inventive activity. The American Economic Review, 61(4):561–574. [65] Holt,C.andRoth,A.(2004).TheNashequilibrium: Aperspective. Proceedings of the National Academy of Sciences, 101(12):3999–4002. [66] Jerrum, M. (1992). Large cliques elude the Metropolis process. Random Struc- tures & Algorithms, 3(4):347–359. [67] Juels, A. and Peinado, M. (2000). Hiding cliques for cryptographic security. Designs, Codes and Cryptography, 20(3):269–280. [68] Kahn,J.,Kalai,G.,andLinial,N.(1988). Theinfluenceofvariablesonboolean functions. In Proceedings of the 29th Annual Symposium on Foundations of Com- puter Science, pages 68–80. [69] Kamenica,E.andGentzkow, M.(2009). Bayesianpersuasion. Technical report, National Bureau of Economic Research. [70] Kannan, R. and Theobald, T. (2007). Games of fixed rank: a hierarchy of bimatrix games. In Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1124–1132. [71] Khot, S. and Saket, R. (2012). Hardness of finding independent sets in almost q-colorable graphs. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS), pages 380–389. 150 [72] Kučera, L. (1995). Expected complexity of graph partitioning problems. Dis- crete Applied Mathematics, 57(2):193–212. [73] Lehrer, E., Rosenberg, D., and Shmaya, E. (2010). Signaling and mediation in games with common interests. Games and Economic Behavior, 68(2):670–682. [74] Lipton, R. J., Markakis, E., and Mehta, A. (2003). Playing large games using simple strategies. In Proceedings of the 5th ACM Conference on Electronic Com- merce (EC), pages 36–41. [75] Mehta, R. (2014). Constant rank bimatrix games are PPAD-hard. In Pro- ceedings of the 45th ACM Symposium on Theory of Computing (STOC), pages 545–554. [76] Milchtaich, I. (1996). Congestion games with player-specific payoff functions. Games and Economic Behavior, 13(1):111–124. [77] Milgrom, P. R. and Weber, R. J. (1982). A theory of auctions and competitive bidding. Econometrica, 50(5). [78] Minder, L. and Vilenchik, D. (2009). Small clique detection and approximate Nashequilibria. InApproximation, Randomization, and Combinatorial Optimiza- tion. Algorithms and Techniques, pages 673–685. [79] Mossel, E., O’Donnell, R., and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality. Annals of Mathematics, 171(1):295–341. [80] Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1):48–49. [81] Nemirovski, A. S. and Yudin, D. B. (1983). Problem complexity and method efficiency in optimization. Wiley. [82] Nisan,N.,Roughgarden,T.,Tardos,E.,andVazirani,V.V.(2007).Algorithmic Game Theory. Cambridge University Press. [83] Okuno-Fujiwara,M.,Postlewaite, A.,andSuzumura,K.(1990).Strategicinfor- mation revelation. The Review of Economic Studies, 57(1):25–47. [84] Papadimitriou, C. H. (2005). Computing correlated equilibria in multi-player games. In Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), pages 49–56. [85] Roughgarden, T. and Tardos, E. (2002). How bad is selfish routing? Journal of the ACM, 49(2):236–259. 151 [86] Rubinstein, A. (2015a). ETH-hardness for signaling in symmetric zero-sum games. arXiv preprint 1510.04991. [87] Rubinstein, A. (2015b). Inapproximability of Nash equilibrium. In Proceedings of the 46th ACM Symposium on Theory of Computing (STOC), pages 409–418. [88] Rubinstein, A. (2016). Settling the complexity of computing approximate two- player Nash equilibria. In Proceedings of the 57th IEEE Symposium on Founda- tions of Computer Science (FOCS), pages 258–265. [89] Tsaknakis, H. and Spirakis, P. (2007). An optimization approach for approxi- mateNashequilibria. InProceedingsofthe3rdWorkshoponInternetandNetwork Economics (WINE), pages 42–56. [90] Vasserman, S., Feldman, M., and Hassidim, A. (2015). Implementing the wis- dom of Waze. In Proceedings of the 30th International Joint Conference on Arti- ficial Intelligence (IJCAI), pages 660–666. [91] Von Stackelberg, H. (1934). Marktform und gleichgewicht. Springer. 152

Abstract (if available)

Abstract Strategic interactions often take place in environments rife with uncertainty and information asymmetry. Understanding the role of information in strategic interactions is becoming more and more important in the age of information we live in today. This dissertation is motivated by the following question: What is the optimal way to reveal information, and how hard is it computationally to find an optimum? We study the optimization problem faced by an informed principal, who must choose how to reveal information in order to induce a desirable equilibrium, a task often referred to as information structure design, signaling or persuasion. ❧ Our exploration of optimal signaling begins with Bayesian network routing games. This widely studied class of games arises in several real-world settings. For example, millions of people use navigation services like Google Maps every day. Is it possible for Google Maps (the principal) to partly reveal the traffic conditions to reduce the latency experienced by selfish drivers? We show that the answer to this question is two-fold: (1) There are scenarios where the principal can improve selfish routing, and sometimes through the careful provision of information, the principal can achieve the best-coordinated outcome

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Asset Metadata

Creator Cheng, Yu (author)

Core Title Computational aspects of optimal information revelation

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Computer Science

Publication Date 08/28/2017

Defense Date 05/08/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag computation complexity,game theory,information asymmetry,OAI-PMH Harvest,signaling

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Teng, Shang-Hua (committee chair), Dughmi, Shaddin (committee member), Kempe, David (committee member), Liu, Yan (committee member), Reichardt, Ben (committee member)

Creator Email chycharlie@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-425458

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