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University of Southern California Dissertations and Theses
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Three essays on industrial organization
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Three essays on industrial organization
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Three Essays on Industrial Organization by Zhen Chen 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 (ECONOMICS) May 2022 Copyright 2022 Zhen Chen Acknowledgements I want to express my gratitude to many people who have helped me make this thesis a reality. Guofu Tan’s guidance throughout my whole Ph.D. program is incredible. He is my role model as an economist. Our meetings and discussions are the key highlights of my time at USC. I am particularly fascinated by his economic insights and his great personality. Since my second year, we have had great conversations about industrial organization, antitrust economics, auctions, and many more. He always impresses me with critical thinking, strong logic, and extensive knowledge of literature. Guofu is also a thoughtful advisor. My Ph.D. was more challenging than I had previously anticipated, and there were many times that I felt depressed and had to re-evaluate my decision to pursue the Ph.D. degree. Guofu is always the first professor from whom I ask for help. I appreciate how understanding and supportive Guofu has been. Whenever I chat with other USC Ph.D. students about committee members, I always encourage them to invite Guofu. Jonathan Libgober is a wonderful professor, mentor, and fantastic friend. He is diligent and de- voted to economics. Whenever I have any questions or ideas about economics, he can immediately direct me to plenty of papers he had read that studied related issues. Jonathan spends tremendous time and effort mentoring his students. He attended most of my presentations and had many chats with me on improving my paper and presentation skills. Our conversations have always inspired me to explore my research further. Jonathan’s perspectives on economic theory and his encourage- ments greatly affected my research. Many other professors at USC and other institutions offered valuable guidance during my Ph.D. program. I want to thank Feng Zhu from Harvard Business School for the two remarkable summer ii internships. He’s super friendly and humorous, and he inspired my interest in understanding mar- keting strategies. Giorgio Coricelli, Geert Ridder, Jeffery Weaver, Paulina Oliva, Matthew Kahn, and many other professors at USC also offered me great comments on my projects or gave me instructions on life as a Ph.D. student. I am thankful for all their help. Several of my classmates became great friends. Yejia (Richard) Xu is the best classmate and friend anyone could hope for. He is kind, knowledgeable, and always helpful. Whenever I had dif- ficulty with rigorous proofs, I always turned to him, and he never let me down. I benefited greatly from our discussions about the industrial organization and game theory. Yinqi (Mike) Zhang is a superb friend. His seat is next to mine in my office, and we had countless daily communications about our research. He also shared his experience in all aspects of life. I couldn’t imagine what my time at graduate school would be without him. I want to thank my parents, Bin Chen and Yueqi Xu, who have been my biggest supporters. Their love, guidance, and advice are an enormous part of my journey at USC. Although we are thousands of miles away, I can always feel our connection, which gives me real strength. Lastly, I want to thank my wife, Lu Tang. We support each other when we face challenges and celebrate together when things go well. I am very sure this thesis would be impossible without her. iii Table of Contents Acknowledgements ii List of Tables vii List of Figures viii Abstract ix Chapter 1: Information Asymmetry in Online English Auctions: the Winner’s Curse and the Winner’s Blessing 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Information Asymmetry and Bidding Strategy . . . . . . . . . . . . . . . . 7 1.3.2 No Experts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Having Experts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3.1 The Winner’s Curse and the Winner’s Blessing . . . . . . . . . . 10 1.3.3.2 Properties of the Bidding Functions . . . . . . . . . . . . . . . . 11 1.3.4 An Example: Power Distribution . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.4.1 Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.5 Comparison of the Willingness to Bid . . . . . . . . . . . . . . . . . . . . 16 1.4 Auction Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.1 With One Expert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 More Than One Expert . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.1 Chinese Government Used Car Auction . . . . . . . . . . . . . . . . . . . 20 1.5.2 Bidder Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.2.1 Asymmetric valuation and information . . . . . . . . . . . . . . 22 1.5.2.2 Timing of Bids . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.3 Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.4 Bids and Participation of Experts . . . . . . . . . . . . . . . . . . . . . . 27 1.5.5 The Effect of Experts’ Participation on Individuals’ Final Bids . . . . . . . 28 1.5.5.1 OLS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.5.2 Endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5.6 Auction Revenues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 iv 1.6 Robustness Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.1 Unobservable Potential Bidders . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.2 Experts vs. Individual Bidders . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6.3 License Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.4 Definition of Experts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 2: Quality Competition among Free Digital Platforms 36 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.2 Background: Merger of Huya and Douyu . . . . . . . . . . . . . . . . . . 40 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.1 Platform Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.2 Participation Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.3 Tipping Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.4 Three stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.1 Tipping Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.2 Participation Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.3 Quality Investment Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 49 2.4 Social Optimal Quality Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4.1 Equilibrium and Social Optimum . . . . . . . . . . . . . . . . . . . . . . 52 2.5 Other Revenue-Generating Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 53 2.5.1 All-pay Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.5.1.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5.2 Advertising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6 Discussion: Merger and Downward Quality Pressure (DQP) . . . . . . . . . . . . 57 2.6.1 DQP and the equilibrium quality level . . . . . . . . . . . . . . . . . . . . 59 2.6.2 Example of DQP: Advertising . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Chapter 3: Frenemies in the music streaming market: exclusive contract and sub-licensing 63 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Background: Music Piracy and Copyright War in China . . . . . . . . . . . . . . . 67 3.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.1.1 Insufficient Investment Level . . . . . . . . . . . . . . . . . . . 70 3.4.2 Sub-licensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4.2.1 Direct Effect of Revenue Sharing on Investment . . . . . . . . . 72 3.4.2.2 Revenue Sharing and Pricing . . . . . . . . . . . . . . . . . . . 73 3.4.2.3 The Effects of Higher Prices on Investments . . . . . . . . . . . 74 3.4.2.4 The Effects of Investments on Prices . . . . . . . . . . . . . . . 74 3.5 Contracts and Payment to Content Provider . . . . . . . . . . . . . . . . . . . . . 75 3.6 A Linear Demand System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 v 3.6.1 Integrated Platforms under Non-exclusion . . . . . . . . . . . . . . . . . . 82 3.6.2 Competitive Platforms under Non-exclusion . . . . . . . . . . . . . . . . . 82 3.6.3 Exclusive Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.6.4 Sub-licensing Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.6.4.1 Fixed-fee Sub-licensing Contract . . . . . . . . . . . . . . . . . 85 3.6.4.2 Revenue Sharing Sub-licensing Contract . . . . . . . . . . . . . 86 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 References 89 Appendices 93 H Appendix of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 H.1 Proof of Proposition 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 H.2 Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 H.3 Proof of Corollary 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 H.4 Proof of Corollary 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 H.5 Proof of Corollary 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 H.6 Proof of Proposition 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 H.7 Proof of Proposition 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 I Appendix of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 I.1 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 I.2 Proof of Corollary 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 I.3 Proof of Corollary 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 I.4 Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 I.5 Proof of Corollary 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 I.6 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 I.7 Proof of Corollary 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 I.8 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 I.9 Proof of Proposition 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 J Appendices of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 J.1 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 J.2 Proof of Proposition 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 J.3 Proof of proposition 12 and Corollary 12 . . . . . . . . . . . . . . . . . . 126 J.4 More details about the proof of Proposition 13 . . . . . . . . . . . . . . . 127 J.4.1 Price increases with r . . . . . . . . . . . . . . . . . . . . . . . 127 J.4.2 Price increases with total investment s . . . . . . . . . . . . . . 129 J.5 Extension of lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 J.6 More details about the Salop circle demand system . . . . . . . . . . . . . 131 J.7 Simulations on social welfare . . . . . . . . . . . . . . . . . . . . . . . . 134 vi List of Tables 1.1 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 log(Final Bids) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 log(Auction Revenue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3 log(Final Bids): Potential Bidders . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.4 log(Final Bids): Compare Experts and Individuals . . . . . . . . . . . . . . . . . . 105 3.5 log(Final Bids) with a new control: Department With Power . . . . . . . . . . . . 106 3.6 log(Final Bids): Dealer Deposits 5 Cars . . . . . . . . . . . . . . . . . . . . . . 107 3.7 log(Final Bids): Dealer Deposits 4 Cars . . . . . . . . . . . . . . . . . . . . . . 108 3.8 log(Final Bids): Dealer Deposits 2 Cars . . . . . . . . . . . . . . . . . . . . . . 109 vii List of Figures 1.1 Bidding Parameter K(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Bidding Parameter K(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Bidding time (24h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Bidding time (2h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 Video-game Streaming Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 The Network Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1 Direct Relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2 Direct Relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Decision Process of the Content Provider . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Non-exclusive and Exclusive Demand . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Comparative Statics about the Dominance of SP 1 . . . . . . . . . . . . . . . . . . 136 viii Abstract This thesis is a collection of three independent essays exploring research topics related to industrial organization. The three essays investigate information asymmetry in English auctions, quality competition among digital platforms, and sublicensing/exclusive contracts. The first essay (“Information Asymmetry in Online English Auctions: the Winner’s Curse and the Winner’s Blessing”) is my job market paper. It studies the competition between experts and individual bidders in an online open auction where bidders do not know who drops out. The expert, who has superior information, focuses on the common value (e.g., quality) because her goal is resale. The individual bidder, who only knows the private taste, is concerned about both the common and private values. We use a theoretical model to show that the participation of experts may (i) reduce an individual bidder’s willingness to bid due to the winner’s curse, and (ii) increase the individual bidder’s willingness to bid due to the revelation of common value (which we refer to as the winner’s blessing). The winner’s blessing vanishes when the number of individual bidders is large. In addition to the bidding strategies, we discuss how the number of experts affects the auction revenues. We then examine the used car auctions from 2015 to 2019 to verify our theoretical predictions. The empirical results suggest that the presence of experts decreases individuals’ final bids and the auction revenues. The second essay (“Quality Competition among Free Digital Platforms”) is co-authored with my advisor Guofu Tan and my classmate Yejia (Richard) Xu. We build a model of a three-stage game to analyze quality competition among free (zero-pricing) digital platforms. In the first stage, platforms invest in the quality of their basic services. In the second stage, observing the quality levels of the platforms, users make participation decisions. In the last stage, users choose their ix tipping amounts, and the platforms use a fraction of the tips to provide club goods and lotteries with a fixed prize. We explore users’ tipping and participation strategies and derive conditions under which the equilibrium quality level is insufficient, optimal, or excessive, relative to the socially optimal quality level. We also analyze other revenue-generating mechanisms. If platforms adopt an all-pay auction mechanism, we show that the equilibrium quality level is always insufficient. If platforms use an advertisement mechanism, the equilibrium quality level is insufficient when the marginal revenues from an additional user are low. We then discuss the impact of mergers and policy implications. The last essay (“Frenemies in the music streaming market: exclusive contract and sub-licensing”) explores sub-licensing and exclusive contracts’ effects on boosting total investments in public goods (e.g., fighting music piracy) and increasing the payment to an upstream firm (e.g., con- tent providers). Using a general demand system, we show that the free-riding issue makes the total investment level insufficient compared to the optimal market level. We argue that downstream firms (e.g., streaming platforms) can use revenue-sharing sub-licensing contracts to boost invest- ment. We show that with prices fixed, revenue sharing is irrelevant to the total investment level. However, revenue sharing induces higher prices and boosts the total investment level through the price channel. We also show that the content provider will give up non-exclusivity if sub-licensing or exclusive contracts result in higher total market profits. We adopt a specific demand system to verify that firms will price higher and invest more by switching to sub-licensing or exclusive contracts. Under certain conditions of market dominance, the content provider has an incentive to require a revenue-sharing contract or an exclusive contract. x Chapter 1 Information Asymmetry in Online English Auctions: the Winner’s Curse and the Winner’s Blessing 1.1 Introduction Auction literature often focuses on either (independent) private value auctions or common value auctions. 1 In practice, bidders often have different weights on private and common values. Due to the divergent interests, bidders usually have asymmetric information regarding the value of the goods. A bidder may shade or raise her bids based on her information and her expectation of other bidders’ information. The asymmetry in valuations and information arises naturally in many large markets such as government procurement, security markets, and automobile sales. For instance, in oil drainage lease auctions conducted by the U.S. government, studies find firms having drilled for oil in regions close to the location of the auctioned lease to behave in ways consistent with being better informed about the profitability than non-neighbor firms (Hendricks and Porter (1988); Hendricks, Porter, and Wilson (1994)). The security prices and trading strategies when some traders have insider information have been analyzed extensively in security markets. In the equilibrium, all private information is incorporated into prices by the end of trading. (Kyle (1985); Glosten and Milgrom 1 In a private value auction, each bidder knows how much she values the object for sale, but this value is private information and is independent across bidders. In a common value auction, the value of an item for sale is identical among bidders. 1 (1985); Back and Baruch (2004)). In car auctions, asymmetric information about cars’ quality could arise between sellers and buyers (Lewis (2011)) or between individual buyers and resellers (Tadelis and Zettelmeyer (2015); P. A. Haile and Tamer (2003)). Despite economists’ pursuits in understanding the value of superior information, few researchers have analyzed auctions with both asymmetric valuation and asymmetric information. This paper considers the following scenario in used car auctions as a motivating example. If a bidder wants to purchase a car for personal use, she will consider both her personal preferences (such as brand, model, color) and the common (market) value. However, suppose a bidder is a dealer who intends to resell the car in another market. The dealer may have a more accurate estimation about the market value, 2 and may solely care about it because he wants to maximize the expected revenue from a resale. To understand the effects of the informed resellers (hereafter “experts” or “dealers” ), we build a theoretical model to analyze the bidding behaviors of individual bidders and experts in English auctions. We show that the presence of the experts has two effects on an individual bidder’s willingness to bid. We call these two effects the winner’s curse and the winner’s blessing. Our notion of “the winner’s curse” is slightly different from the one in auction literature. In the literature, the winner’s curse is a phenomenon that may occur in common value auctions, where the winner is the bidder with the most optimistic evaluation of the asset and therefore will tend to overestimate and overpay. However, the overpayment does not generally occur because bidders will rationally expect the “curse” and shade their bid accordingly. We focus on the bid shading due to the rational expectation of the winner’s curse. In our paper, the bid shading happens because individual bidders believe the experts have better information about the common value. Whenever an individual bidder wins an auction, she knows the experts have dropped out, and thus she has suffered from her inaccurate estimation of the common value. Therefore, she will adjust her estimation of common value accordingly and reduce her bid. 2 In practice, the common value of a car is not only determined by its quality, but also by market demand. For example, it might be easier to sell a silver Toyota than a blue sports car. Dealers, intuitively, are more experienced in estimating the market demand. 2 The winner’s blessing, however, occurs because the experts bid up to the common value. If an individual bidder wins the auction, it may be the case that the expert submitted the second-highest bid (thus the payment), and the second-highest bid perfectly reveals the common value. Compared to only considering the winner’s curse, when considering this possible revelation into account, the individual bidder increases her estimation of the common value, thus bidding more aggressively. We decompose these two effects in the equilibrium conditions that determine the optimal cutoff strategies. 3 Next, we analyze how the participation of experts affects auction revenues. We derive the expected revenue of an auction with multiple individual bidders and experts. In addition to the winner’s curse and the winner’s blessing, the participation of experts has two more effects: the crowding out effect and Bertrand competition. If we fix the total number of bidders, substitut- ing individual bidders with experts results in fewer individual bidders (and fewer private values), leading to lower revenues. Bertrand competition between experts, however, has a positive impact on the revenues. When the number of individual bidders is large, both the effects of the winner’s blessing and Bertrand competition effect vanish. Therefore, auctions with experts tend to have lower revenue than those without experts. We test three hypotheses that are derived from the above theoretical model by analyzing the government used-car auctions in China. In total, nearly 5,000 cars were auctioned off from Febru- ary 2016 to October 2019. These auctions were held on the same auction website. We view large buyers as experts with superior information on the market value, and identify those experts from their pre-auction deposits. The empirical results are consistent with our theoretical predictions. We find that the participation of these experts significantly reduces the individual bidders’ willingness to bid. In addition to bids, we analyze the auction revenues and show that the participation of the experts also negatively affects revenues. We show that the total number of bidders does not 3 Our paper shows that, unlike the first-price auction, (anonymous) English auction may weaken the winner’s curse, because the winner pays the second-highest bid, possibly from the experts. Zillow Group, Inc. is a good example. In the fourth quarter of 2021, Zillow reported an unexpected huge loss and announced that it is exiting its iBuyer business. Home buying is like a first-price auction. Many economists believe Zillow’s loss is because homeowners have superior information, and Zillow suffers from the winner’s curse. Our paper suggests that Zillow’s mistake would have been less deadly if the home buying had used second-price or English auctions. 3 affect an individual bidder’s final bid but has a significant positive impact on the auction revenues. We introduce two instrumental variables to reduce the concern of endogeneity, and the regression results are robust. Lastly, we perform four robustness checks and discuss the policy implications. Our contributions to the literature are as follows. We present a simple model that combines both asymmetric valuation and asymmetric information in a general English auction. 4 We discover a new effect in the auction with experts called “the winner’s blessing”. To the best of our knowledge, the winner’s blessing has not been studied by other economists. In addition to the theoretical contribution, we collect the used-car auction data ourselves and test several hypotheses derived from our theoretical model. Our paper is the first to use this data set. The paper is organized as follows: Section 1.2 reviews the relevant literature and our con- tributions. Section 1.3 describes our model and the effect of experts on an individual bidder’s willingness to bid. Section 1.4 discusses the impact of the experts on auction revenues. Section 1.5 provides empirical analyses that test our model. Section 1.6 checks the robustness of our empirical strategies. Section 1.7 concludes and discusses two policy implications. The appendix contains the proofs. 1.2 Literature Review Auction theory has traditionally been divided into two categories: private value auctions and com- mon value auctions. Vickrey (1961) provides an early strategic analysis of private value auctions when the values are independent across bidders. Many modern auction theories stem from My- erson (1981), which proves that Vickrey’s results about the equivalence in expected revenue of different auctions apply generally. His finding of the virtual surplus is useful in understanding our results. Following Myerson, the attention is mostly on private value auctions. Common value auctions, which mean the value of an item for sale is identical among bidders, do generate some 4 We believe the ”button auction” assumption, which is commonly used, is too restrictive for online auctions. Hence, we allow users to reenter an auction whenever they choose. 4 research. Literature of common value auctions usually investigates the winner’s curse, both theo- retically and experimentally. (See P. Klemperer (1998); J. Bulow and P. Klemperer (2002); Eyster and Rabin (2005); Charness and D. Levin (2009), etc.) One of the most popular formats of auctions is the English auction. Milgrom and Weber (1982) explore one kind of English auction. They propose the “button English auction,” where active bid- ders must depress a button. If a bidder does not want to overbid the current price, she releases the button and never reenters the auction. Milgrom and Weber (ibid.) show that revealing information raises the expected price. Our paper allows for a more general setting where bidders can submit a bid whenever they want. We show that having more information does not necessarily increase rev- enue. J. I. Bulow and P. D. Klemperer (1994) also focus on the revenue of English auctions. They demonstrate that if bidders have symmetric private values, an additional bidder in an ascending auction is worth more to the seller than the ability to choose the optimal standard mechanism to sell the object. Our paper shows that adding more bidders does not necessarily increase revenues when some bidders have more information and only care about the common value. While auction theorists have explored private and common value auctions quite extensively, economists exert relatively less effort on potentially asymmetric bidders. Regarding asymmetry, a few economists have studied asymmetric information, asymmetric valuation, and resale. For example, Piccione and Tan (1996) studied asymmetric information. They consider the bidding behavior of experts and non-experts in a first-price auction and show that revenue may decrease with the probability of informed bidders, suggesting that more information may not always be better. Maskin and Riley (2000) explored bidders with asymmetric valuations. They argue that revenue equivalence fails to hold when we drop the symmetric assumption. Lastly, Hafalir and Krishna (2008) study asymmetric bidders with independent private value in auctions with resale and how the post-auction trade alleviates inefficiencies. Cheng and Tan (2010) follow Hafalir and Krishna (2008) and extend the revenue-ranking comparison to auctions with common values. The research on mixed value auctions is also relatively less common. For example, Perry and Reny (1999) show that when we have asymmetric bidders and mixed values, the release of public 5 information may cause revenues to decrease. Their results are similar to our theoretical results, but the experts release public information in our model. Bobkova (2021) approaches the auction theory from the information choice perspective. That paper discusses the incentives of bidders, who have mixed values, to learn private or common values in different formats of auctions. We believe more rigorous research is needed to understand the auctions in which bidders have mixed values, asymmetric information, and asymmetric valuations. In addition to the theoretical auction literature, there is abundant empirical analysis on auc- tions. Hendricks and Porter (1988) pioneer the empirical estimations of auctions with asymmetric information. They investigate the federal offshore oil and gas drainage lease sales and find that adjacent firms (similar to the “experts” in our paper) are better informed about the value of a lease and coordinate their bidding decisions. The non-neighbor firms (similar to the “uninformed indi- viduals” in our paper) are left with zero profits. Lewis (2011) analyzes the eBay motor auctions and shows that pictures posted by the sellers on the auction web page are strong positive signals of quality because of sellers’ adverse selection. Our reduced-form empirical strategy is closely re- lated to his method. Although our empirical analysis adopts the reduced form analysis, economists have developed advanced structural estimation methods for auctions. P. A. Haile and Tamer (2003) propose an incomplete model to estimate bidder valuations nonparametrically. Athey and P. Haile (2006) provide an extensive review of structural estimations in the literature on auctions. Athey, J. Levin, and Seira (2011) study sealed bid and open auction data from the U.S. Forest Service timber auctions and show that sealed bid auctions attract more small bidders than do open. 1.3 Model We consider environments with risk-neutral bidders pursuing an indivisible object. Among the bidders, n are individual bidders who care about both their independent tastes of the good v i and the common value (e.g., quality) of the good S. Thus, we assume individual bidder i’s valuation of the good is V i =rv i +(1r)S, with r2[0,1]. The rest of the bidders are resellers (or experts, 6 as we will discuss below), who are only interested in the quality of the good, i.e., V s = S. The individual bidders know if there are experts in the auction but do not know who the experts are. We assume that ex-ante v i s are independently drawn from a cumulative distribution F(), with probability density function f(). Individual bidder i observes the realization of v i . The common value S, which has cumulative distribution G() and probability density g(), is the same for every bidder, but only experts can observe the realization of S. Both distributions have support [a,b], witha 0. The bidders participate in a standard English auction (open auction, ascending auction). In our auction, the auctioneer announces the start, and bidders continuously raise the current price beginning at 0. The highest bidder at any given moment is considered to have the standing bid, which can only be displaced by a higher bid from a competing buyer. If no competing bidder challenges the standing bid within a given time frame, the standing bid becomes the winner, and the item is sold to the highest bidder at a price equal to the bid. No bidder needs to announce her drop-out. 5 Because the individual bidders also care about the common value of the good, the experts’ information about the common value is valuable to individual bidders, and this information is partly disclosed during the bidding process by the standing bid. We will discuss how information affects bidders’ bidding strategy in the next section. 1.3.1 Information Asymmetry and Bidding Strategy Similar to the dominant strategy of valuation-bidding in the second-price auction, each bidder also has a dominant strategy in an English auction with independent values, which is to remain in the bidding process until the price reaches the bidder’s expected valuation. 6 The intuition still holds when we consider English auctions with asymmetric bidders with mixed values. 5 Unlike Milgrom and Weber, we do not require the bidders to signal their drop-out decision. One complexity of our setting is that the bidders do not know whether the informed expert has dropped out or not. 6 If we allow for an asymmetric auction with resale, it turns out that it is not a weakly dominant strategy to bid one’s value in a second-price auction. However, bidding true values is still an equilibrium. See Hafalir and Krishna (2008) for a thorough discussion. 7 When bidders are asymmetric in both information and valuations, the bidders bid up to their expected values. We assume that all individual bidders use the same cutoff strategy b I (). One can understand the cutoff b I () as the maximum amount of bids that the bidder is willing to submit. Similarly, we assume experts use a cutoff strategy b E (). If the current standing bid is below their cutoff, bidders continue to participate. Once the standing bid exceeds their cutoff, they stop bidding. Thus, the experts who observe the realization of the common value will bid until the standing price reaches their estimates, b(S)= S. 7 The individual bidders, however, do not know the common value. Hence, based on the current bid, they will update their estimations of the object’s expected values. 1.3.2 No Experts First, we consider the simple case where all bidders are individual bidders, and no bidders have any information about the common value. Hence bidder i’s expected value of the goods is V(v i )=rv i +(1r) Z b a Sg(S)dS (1.1) The second term is the ex-ante expectation of the common value. It is a weakly dominant strategy for individual bidders to bid up to their expected values in an open auction. Thus, with no experts, bidder i’s optimal strategy is to stay in the auction until the standing bid reaches V(v i ): b NE (v i )=rv i +(1r)E(S) (1.2) 7 We provide a more rigorous characterization of an expert’s bidding strategy when his signal about the common value is noisy in the proof of Proposition 1, which can be found in the appendix. 8 1.3.3 Having Experts Suppose the individual bidders know for sure that an expert is participating in an auction. In that case, the standing bid will reveal information about the common value, as experts have perfect information about it. Having more than one expert will have the same effect on an individual’s bidding strategies as having one expert because all experts have the same precise signal on common value and behave the same. 8 In fact, the current standing bid could come from either another individual bidder or an expert, leading to different interpretations. To illustrate the difference, let’s suppose the current standing bid is c. One key change that the experts bring to the auction is that if an individual bidder wins the auction at bid c, he knows for sure that the common value will be less than c. Otherwise, the expert would have raised the price to S > c. This is known as the winner’s curse. Another crucial effect is that when an individual bidder wins the auction, she is unsure who submitted the second-highest bid. The second-highest bid could have come from an expert with a positive probability. In this case, the second-highest bid (or the standing bid) fully reveals information on the common value. To the best of our knowledge, literature has not seriously analyzed the latter effect. Therefore, if an individual bidder wins the auction at standing bid c, her ex-post belief on the distribution of common value will (i) account for the fact that the common value is less than c and (ii) have a mass on the common value being c. Analyzing dynamic belief updating is challenging. However, if we assume bidders use cutoff strategies, we can solve for bidding functions using a static approach. The intuition for the cutoff strategy is similar to the one used in English auctions with independent values. The cutoff is the ex- pected value. If the standing bid is below the expected value, an individual bidder should continue to bid. The individual bidder should drop out when the standing bid reaches her expected value, as winning the object will yield a negative expected payoff. We only consider the semi-symmetric equilibrium where individual bidders use the same cutoff strategy, and experts use another cutoff strategy. 9 8 The number of experts, however, would affect the revenue, as we will discuss in Section 1.4. 9 As the standing bid increases, the individual bidder’s estimated valuation also increases, leading to the possibility of non-cutoff strategies. We leave the analyses of other types of strategies to future studies. 9 To rigorously derive the cutoff, we differentiate an individual bidder’s expected payoff with respect to her cutoff bid B and derive bidders’ optimal cutoff strategies. We make the following assumption on distributions: Assumption 1. Cumulative distribution F() and G() are both log-concave. The log-concavity of the distribution functions is helpful for us to check the second-order condition and prove that the candidate cutoff strategy is indeed a global maximum. The following proposition summarizes our main finding on the optimal cutoff strategy, given other bidders are also using a cutoff strategy. 1.3.3.1 The Winner’s Curse and the Winner’s Blessing Proposition 1. (Main result) When there are experts in an English auction, in equilibrium: (i) Experts’ maximum bidding function is b E (S)= S; and (ii) Individual bidders’ maximum bidding function b I (v i ) is solved implicitly from the follow- ing equation: B=rv i +(1r)E(SjS< B)+ r(v i B) n 1 F(v i ) f(v i ) g(B) G(B) (1.3) where b I (v i )= B. Proof. See Appendix. Proposition 1 separates the two effects we discussed earlier. When we take the derivative of the expected revenue, two terms appear. First,(1r)E(SjS< B) describes the case when the second-highest bid comes from an indi- vidual bidder. If we compare equation (3) to the equation with no experts (2), we can see that this term demonstrates the winner’s curse. When estimating common value, individual bidders now account for the fact that experts may have a less optimistic signal than the winning bid. 10 Second, r(v i B) n1 F(v i ) f(v i ) g(B) G(B) originates from the case when the second-highest bid comes from experts, and thus winning bidder pays S. This captures the winner’s blessing. In this case, the winning individual bidder will update her estimation of the common value as the second-highest bid. Intuitively, the common value estimation is higher than before, and she is more likely to bid higher. An interesting observation from the proof of Proposition 1 is that when there is only one individual bidder (n= 1), we directly have b I (v i )= v i . 10 The intuition is that the winner’s blessing dominates. The second-highest bid could only come from the expert, so the individual bidder knows when she wins the auction, she is paying the common value of the object. Hence if she wins the auction by bidding more than v i (which means v i < S), she will incur a loss as, as rv i +(1r)S S< 0. Similarly, if she wins at a bid lower than v i (which means v i > S), she will make a profit becauserv i +(1r)S S> 0. In this case, b I (v i ) could be higher than b NE (v i ) if v i > E(S). 1.3.3.2 Properties of the Bidding Functions It is helpful to discuss two extreme cases when r = 0 and r = 1, which resemble purely private and common values auctions. Corollary 1. Whenr = 0, b I (v i )=a. Whenr = 1, b I (v i )= v i . Proof. See appendix. Ifr= 0, we are in a purely common value auction. Individual bidders are at a disadvantage because they do not know the common value of the item. Knowing that they will have negative payoffs due to the winner’s curse whenever they win, individual bidders will quit from the beginning by setting a maximum bid as the lower bound of the support. When r = 1, individual bidders only care about private value. The second and third terms of the right-hand side of equation (3) vanish together as there is no winner’s curse and the revelation of common value (winner’s blessing) is of no interest to individual bidders now. 10 Readers could get this result by setting n= 1 in equation (H.3) in the proof of Proposition 1. 11 In auction theory, We are interested in finding a monotone increasing maximum bidding func- tion, b I (v i ). The following corollary shows this result. Corollary 2. The maximum bidding function b I (v i ) strictly increases with v i . Proof. See appendix. The cutoff strategy is well-behaved. For each realization of private value v i , there is a unique cutoff v i , and the monotonicity ensures that, among the individual bidders, only the bidder with the highest valuation can win the object. 11 After proving the uniqueness of the cutoff bidding strategy, we are interested in exploring the properties of the strategy. First, we show that the winner’s blessing always has positive impact on individual bidders’ cutoff bids, which is by proving that the winner’s blessing r(v i b I (v i )) n1 F(v i ) f(v i ) g(B) G(B) is always positive. Corollary 3. The maximum bidding function b I (v i )< v i . Proof. See appendix. This corollary supplements the analysis of the winner’s blessing. Because b I (v i ) < v i , we know the third term on the right-hand side of equation (3) must be positive. The intuition is that when the second-highest bidder turns out to be an expert, the winning bidder knows that the common value is “not that bad” compared to the case where the second-highest bid comes from an individual bidder. Thus this possibility of revelation induces individual bidders to bid higher. 1.3.4 An Example: Power Distribution When experts participate in an auction, the maximum bidding function of individual bidders b I (v i ) is implicitly solved. For illustration, we provide a simple example to solve for the explicit cutoff strategy. Assumption 2. The common value S follows uniform distribution U[0,1]. The private value v i follows power distribution with F(v i )= v g i , with v i 2[0,1] andg2[0,1]. 11 However, as illustrated by Krishna (2003), an English auction with asymmetric bidders is not necessarily efficient. Our auction is not efficient. We will show this in Section 5. 12 Figure 1.1: Bidding Parameter K(r) With this assumption, we can derive the explicit cutoff strategy for individual bidders. Interest- ingly, the bidding function is an affine function of the private valuation v i . Corollary 4. Under Assumption 2, b I (v i )= K(r,g)v i , with K(r,g)= r 1+r (1 1 (n 1)g )+ r 1+r s (1 1 (n 1)g ) 2 + 1+r r 2 (n 1)g K(r,g) is increasing withr and decreasing withg. Proof. See appendix. Settingg = 1 and n= 10, we plot the relationship between K(r) andr. The graph is as follows: 13 Figure 1.2: Bidding Parameter K(n) As discussed in Corollary 1, whenr = 0, individual bidders are only concerned with the com- mon value, but they know experts have perfect information about it. Therefore, they will bid 0 to avoid winning the auction. When r = 1, individual bidders only care about their private values, and thus their weakly dominant strategy is to bid up to their private value v i . When r2(0,1), K is strictly increasing with r. With a higher r, the auction is more “private-value.” Thus, the individual bidders suffer less from information disadvantage and increase the multiplier K. Note that uniform distribution is a special case of power distribution wheng = 1. Hence, our result also applies when both distributions of v i and S are uniform. Similarly, we plot the relationship between K(n) and n, wheng = 1 andr = 0.5: 14 When n= 1, choosing K = 1 is a weakly dominant strategy. This property is discussed in our Proposition 1. As n continues to increase, the bidding parameter K decreases, illustrating our idea that the positive effect of the winner’s blessing decreases with n. When n goes to infinite, the parameter K goes to 2r 1+r . The maximum bidding function b I (v i )= 2r 1+r v i is the solution of maximum bid when individual bidders ignore the winner’s blessing completely. 12 Hence, if the number of individual bidders n is sufficiently large, individual bidders know the second-highest bid is likely from another uninformed individual bidder. Consequently, they will ignore the winner’s blessing, and shade their bids due to the concern of the winner’s curse. 1.3.4.1 Inefficiency It is known that English auctions with asymmetric valuation could be inefficient, meaning that the goods might not be awarded to the bidder with the highest valuation. Our auction could be inefficient as well. If the distributions follow Assumption 2, we know b I (v i )= Kv i . The experts are going to bid until b E (S)= S. Intuitively, if Kv i < S, the object is assigned to an expert. However, if v i > S, individual bidder i’s valuation rv i +(1r)S is higher than the expert’s valuation S. Because K 1, it is possible that both conditions Kv i < S and v i > S hold simultaneously. Here we give an example. Suppose there exists r and g such that K < 1 1+2e , where e is very small. The existence of suchr andg should be straightforward, as K= 0 whenr= 0. In addition, we assume ex post valuations are S= 1 2 and v i = 1 2 +e. The individual bidder has higher valuation, as r 1 2 +(1r)( 1 2 +e) > 1 2 = S. So in an efficient auction, the individual bidder would win the auction. However, the individual bidder’s maximum bid b I (v i )= Kv i < 1 1+2e ( 1 2 +e)= 1 2 = S= b E (S). Thus, the object is awarded to the expert who has a lower valuation. As a result, our auction with asymmetric information could be inefficient. 12 b(v i )= 2r 1+r v i is the unique solution of the equation b(v i )=rv i +(1r)E(SjS< b(v i )). The right-hand side of the equation is the expected value of the object conditional on the common value being less than the winning bid b(v i ). 15 1.3.5 Comparison of the Willingness to Bid As we discussed in the last section, there are two opposing effects introduced by experts. The winner’s curse encourages individual bidders to bid more conservatively since they worry that their bids will win only if the experts’ estimates are low. The winner’s blessing encourages individual bidders to bid more aggressively. Thus, the total effect is ambiguous. However, the winner’s blessing diminishes with the number of individual bidders n, as illus- trated in our Proposition 1. The intuition is that when there are more individual bidders, the win- ning bidder is more convinced that the second-highest bid comes from another individual bidder, and thus will ignore the winner’s blessing. Consequently, the winner’s curse dominates, and the bidder’s maximum bid will be lower than when no expert is in the auction. Corollary 5. When n is sufficiently large, if individual bidders know experts are attending the same auction, they will submit lower bids than they would do in an auction without experts. b NE (v i )> b I (v i ). The takeaway is that when an auction is not a purely private value auction, we should not naively believe having more bidders always leads to more competition and higher bids. This is in sharp contrast with purely private value auctions, where having more bidders always leads to weakly higher revenues. Nevertheless, it is not correct to say experts always lead to lower bids. When the number of individual bidders is small, the winner’s blessing could encourage bidders to bid more aggressively. 1.4 Auction Revenue This section discusses how auction revenue is affected by the number of experts when the total number of bidders remains fixed. Since the maximum bidding function changes substantially from the “no expert” case to the “with experts” one, it is challenging to compare the revenue directly. However, we will discuss the comparative statics of the number of experts when bidders know at least one expert is in the auction. Consider the following scenarios. The first is the competition 16 between n individual bidders and 1 expert. The second is n m+ 1 individual bidders and m experts, where m 2. 1.4.1 With One Expert After solving our maximum bidding functions b E (S) and b I (v i ), the revenue of the auctioneer is equivalent to an English auction with purely independent values, in which individual bidders are endowed with value b I (v i ) and experts are endowed with value S. In the equivalent auction, individual bidder i receives independent values b I (v i ), which follows a cumulative distribution ˆ F() and probability density ˆ f(), with support [ˆ a, ˆ b]. A group of experts have an independent value S, which still follows cumulative distribution G() and probability density function g(). We extend the analysis of Maskin and Riley (2000) to derive the expected revenue of the auc- tion. The idea is to calculate the expected revenue from individual bidders and experts separately. 13 Our analysis starts with the experts. The experts win the auction when S> max i fb I (v i )g. So the expected payment of an expert with S= v E is Z v E a bd ˆ F n (b) (1.4) Similarly, individual bidder i wins the auction if b(v i ) > maxfb I (v i ),Sg. Therefore her ex- pected payment of the auction is Z v i a bd ˆ F n1 (b)G(b) (1.5) 13 Maskin and Riley (2000) considered two bidders with asymmetric distribution of values in auctions with indepen- dent valuations. We extend their analysis with multiple bidders with asymmetric values. 17 To derive the expected revenue of the auction on each type of bidders, we need to integrate each type of bidders’ payment over the support of their “valuations.” The total revenue is summarized in the following proposition. Proposition 2. In an English auction with n individual bidders and one expert bidder, the total expected revenue of the auction is R E = Z ˆ b ˆ a n ˆ F n1 (v)G(v) ˆ f(v)(v 1 ˆ F(v) ˆ f(v) )dv + Z b a ˆ F n (v)g(v)(v 1 G(v) g(v) )dv (1.6) Proof. See appendix. Proposition 2 is a direct application of Myerson (1981)’s Lemma 3 . The terms v 1 ˆ F(v) ˆ f(v) and v 1G(v) g(v) are the virtual surpluses that an auctioneer could extract from a individual bidders and experts in a standard auction. The ˆ F n1 (v)G(v) and F n (v) are the probability of winning for individual bidders and an expert with valuation v. Hence, the total revenue of an auction is the expectation of the virtual surplus extracted from all bidders. 14 1.4.2 More Than One Expert Now consider the case where there are m 2 experts and n m+ 1 individual bidders. Since experts have perfect information about the common value of an item, without collusion, the experts engage in Bertrand competition, where they overbid each other until the standing bid reaches the realization of common value. Thus, experts’ expected payment increases when there are more than 14 Similar results are discussed in Lemma 2 of Bulow and Klemperer (1994) and equation (3.8) of Maskin and Riley (2000). Bulow and Klemperer’s Lemma 2 (1994) suggests that, in a standard auction with N bidders, the expected revenue has the form E t få i2N p i (t)MR i (t)g, where MR i = t i 1F i (t i ) f i (t i ) is the marginal revenue of bidder i and p i is the probability that bidder i wins the auction. However, J. I. Bulow and P. D. Klemperer (1994) only considers symmetric bidders, while Maskin and Riley (2000) only considers two asymmetric bidders. 18 one expert. On the other hand, an individual bidder’s strategy does not change. Whether there is one expert or multiple experts, the individual bidder wins the object when the standing bid exceeds the common value. Similar to the one expert case, experts win the auction if and only if S> max i fb(v i )g. However, they pay S instead of the second-highest bid submitted by individual bidders when they win the auction, as shown in equation (1.4). The expected payment of the experts when common value S= v E is Z v E a v E d ˆ F nm+1 (b) (1.7) Using a similar method derived from Proposition 2, we have the expected revenue for more than one expert as follows: Proposition 3. In an English auction with n m+ 1 individual bidders m 2 expert, the total expected revenue of the auction is R ME =(n m+ 1) Z ˆ b ˆ a ˆ F nm (v)G(v) ˆ f(v)(v 1 ˆ F(v) ˆ f(v) )dv + Z b a ˆ F nm+1 (v)g(v)vdv (1.8) Compared to Proposition 2, Proposition 3 only has two differences. First, comparing the one- expert and the two-expert case, we notice that there will be Bertrand competition in the case with more than one expert. The second term of the revenue function illustrates this effect. When there is more than one expert, the auctioneer can leave no profit (rent) to experts. Hence, the virtual surplus of experts v 1G(v) g(v) is replaced by their true valuation v. So going from one expert to two, the revenue from the experts increases. Nonetheless, adding more experts does not bring any 19 additional benefits, since they have identical signals. Second, when we fix the number of total bidders, having more experts leads to fewer individual bidders. The experts’ crowding out effect results in fewer random draws of private value. Thus the revenue extracted from individual bidders is lower. This effect is reflected by the fact that the first term is strictly decreasing in m. 15 To summarize, there are four effects involved in the increase of the number of experts: (i) the winner’s curse, as individual bidders are more cautious on bidding, (ii) the winner’s blessing, which encourages bidders to bid more aggressively, (iii) Bertrand competition, reflected by the intensified competition between bidders, and (iv) the crowding-out effect, meaning that as the number of experts increases, less random private values are drawn, and their maximum declines. The first and last effects negatively impact revenue, while the second and third effects are positively correlated with revenue. It is worth noting that when there are many individual bidders, the second and third effects vanish, as bidders believe the second-highest bid is more likely to come from individual bidders and an expert’s maximum bid is less likely to be the winning bid. Thus, more experts only lower the auction revenue when there are many individual bidders (or if bidders believe there are many individual bidders). 1.5 Data and Methodology In this section, we conduct a series of empirical estimations to verify and illustrate our theoretical propositions. 1.5.1 Chinese Government Used Car Auction Starting in 2015, the anti-extravagance campaign driven by President Xi Jinping has meant that all ranks of officials below the deputy minister have been banned from commuting in government- owned cars, except for law enforcement and emergency response team. These used cars are sold in online public auctions, with all earnings turned over to the national treasury. These auctions 15 The monotonicity is clear when we do an integral by parts. 20 are English auctions with soft close time (Ockenfels and Roth (2006)). The regular time for each auction is 24 hours. After 24 hours, the auctions enter soft closes. The auction’s close time is extended by 5 minutes when a bid is received in the last 5 minutes. The auction ends if no one submits new bids within the last 5 minutes. We obtain the auction data from web pages and directly from officials in the government of the Sichuan Province and the city of Chengdu from 2015 to 2019. We exclude observations with non- standard or missing data and also those related to medium and heavy trucks. We also drop cars with an engine displacement greater than 7 liters. 16 For each vehicle, we observe car characteristics, in- cluding make, mileage, year, and a lengthy description of its condition. We also obtain information on the bidding process, i.e., who submitted what bid when. Table 1 summarizes the variables in the data set, which contains information about cars (4,634), registered bidders (18,959), and success- ful bids (657,068). During the five years, the auctioneer sold the cars on 57 separate days, with an average of about 100 cars being auctioned off each day. We report the statistical descriptions of the used cars in Table 1.1. Table 1.1: Descriptive statistics . count mean sd min max Final price 4633 3.22049 3.710006 .03 42.86 Total Bidder # 4633 51.90114 31.01146 1 245 Mileage 4479 10.32318 5.973997 .0497098 47.41801 Year 4627 10.40782 2.935148 2 18 Displacement 4633 2.008237 .5367628 .8 6.2 Length of Description 4633 276.9823 211.6488 0 1641 Text: No Collision 4633 .0306497 .1723853 0 1 Text: Engine Issue 4633 .1074897 .3097683 0 1 Text: Scratched 4633 .3218217 .4672254 0 1 16 We exclude observations about trucks because we believe very few individual bidders are interested in buying them. The second reason is that medium and heavy truck plates follow a separate plate system than cars. Sometimes trucks and cars have the same license plate number with different colors. Dropping observations of trucks makes plates a unique identifier of vehicles so that we can merge data sets easily. 21 The cars are old (about ten years on average) and well-traveled (nearly 100,000 miles on the odometer). The total number of bidders in each auction is roughly 54, suggesting strong compe- tition. “No collision” is considered a positive phrase, while “scratched” and “engine issue” are considered negative phrases. One of the auctions’ features is that there is a unique rule for the deposits. The rule is that, if a bidder deposits$1,500 before the auction, she is allowed to stay as the highest bidder for one car’s auction at a time. If she deposits $1,500 A, where A 2, she is allowed to simultaneously stay as the highest bidder for at most A auctions. When this bidder has been the highest bidder for A auctions, she is not allowed to submit new bids until other bidders overbid her. We use this deposit information to identify individual bidders and dealers. Intuitively, individual bidders looking for one vehicle for their personal use should only deposit for one car. If they deposit for more than one car and are the highest bidder in more than one auction simultaneously, they risk winning more cars than desired. Winners can pull out after an auction, but it comes with losing the deposit. On the other hand, dealers or large companies (hereafter “dealers”) want to buy multiple cars. They will deposit a considerable amount for access to bidding on good deals. 1.5.2 Bidder Heterogeneity 1.5.2.1 Asymmetric valuation and information Individual bidders generally take into account not only their individual preferences, but also the common value as they may sell the car in the future. Compared with individual bidders, dealers only care about a car’s common value, as their participation in this competition is solely for resale (or commercial use). Because they are more experienced with used car trading, they usually have better information on a car’s market value than individual bidders have. Thus dealers are similar to the experts defined in our theoretical model, and we will call them experts hereafter. In this paper, 22 we define experts as bidders who deposit more money for at least three cars. 17 We try different definitions of experts in the robustness check section. 1.5.2.2 Timing of Bids Besides asymmetry in information and valuations, experts understand auctions better than indi- vidual bidders. The timing of bids may reveal this knowledge. All of the auctions have 24 hours regular time and a soft close time, so that if there is a new bid submitted within the last 5 minutes, the system will extend the auction close time by 5 minutes. We plot the empirical distribution of the bidding times of individual bidders and experts in Figure 1.3. From the empirical distribution, we observe that individual bidders submit more early bids than experts. In an English auction, early bidding reveals information, which experienced bidders want to avoid. Most individual bidders are not professional bidders and may not have time to bid at the final minute. As a result, some individual bidders submit their bids early. The experts, however, are disciplined. Relatively very few experts engage in early bidding. Figure 1.4 illustrates the empirical distribution of bidding times within 2 hours of the soft close. Figure 1.3 and Figure 1.4 show that experts mostly submit bids late around the soft close and drop out earlier than individual bidders. This strategy of late bidding is called “sniping.” The “snipers” aim at submitting bids right before the end of the auctions, giving other bidders no time to outbid. More-informed buyers may delay bidding until the last minutes of the auction to avoid creating competition for their bids, leading to a lower winning bid. Figure 1.3 and Figure 1.4 demonstrate that individual bidders and experts have very different bidding styles. Experts submit bids late, possibly because they don’t want to reveal their superior information to individual 17 There are three reasons why we define experts this way. Intuitively, individual bidders may buy one or two cars for themselves and relatives. If they are buying many cars, they may care less about private preference. The second reason is that for bidders who identify them as “car service companies,” their average deposit is 4.68 cars. Additionally, we find that bidders who are identified as experts are more likely to enter the auction in the last hour and drop out of the auction earlier than individual bidders, which suggests that they may possess superior information or are more experienced with the English auction. See section 1.5.2.2 for more information. We perform a robustness test in section 6, where we change three to five, four, and two. 23 Figure 1.3: Bidding time (24h) 24 Figure 1.4: Bidding time (2h) 25 bidders. They also drop out early, possibly because the market values they care about are less extreme than individual bidders’ personal preferences. 1.5.3 Model Predictions We focus on the information transmission from experts to individual bidders during an English auction. An individual bidder can not accurately identify experts. However, since auctions are almost simultaneous, individual bidders can observe who has submitted the standing bid. If an in- dividual bidder sees someone leading the bids in multiple auctions simultaneously, she will realize that this bidder demands multiple units and thus is an expert. A rational individual bidder would consider experts’ presence and carefully observe who si- multaneously submitted the standing bid in multiple auctions. When the individual bidders suc- cessfully detect an expert, they will update their belief that experts are participating in the auction and bid more cautiously. When an expert is detected, the individual bidder knows the expert’s ID number, but she will also anticipate other unidentified experts in this auction. So the setting is similar to our theoretical setting where experts are anonymous. With more experts in one auction, it is easier for an individual bidder to identify experts and submit a lower bid due to the winner’s curse. Thus, Proposition 1 leads to the following hypothesis: Hypothesis 1. If we fix the total number of bidders, the final bids of individual bidders decrease as the proportion of experts increases. Most bidders are individual bidders. Intuitively, if there are many individual bidders, it is less likely for experts to win. Thus, more individual bidders lead to a weaker winner’s blessing and a weaker Bertrand competition effect. When we substitute individual bidders with experts, the negative impacts of the winner’s curse and crowding out effect are likely to dominate the winner’s blessing and the Bertrand competition among experts. Fixing the number of bidders, we expect the total revenue (the sale price) to be lower as the proportion of experts increases. Thus we have the following hypothesis from Proposition 1 and 3: 26 Hypothesis 2. If we fix the total number of bidders, the auction revenue decreases with the per- centage of experts. From the discussion of Proposition 3, the number of bidders n will positively impact the total revenue. It is unclear how the number of bidders will affect an individual’s final bid due to the win- ner’s curse and the winner’s blessing. However, the winner’s blessing vanishes when the number of individual bidders is large. Therefore, based on Proposition 1, we hypothesize the following: Hypothesis 3. As the total number of bidders increases, the auction revenue increases. The total number of bidders does not affect individual bidders’ final bids. 1.5.4 Bids and Participation of Experts In this part of the analysis, we examine the relationship between individual bidders’ highest bids (winning or losing) and the percentage of experts. We run a log-linear regression model of the following form: log(b i j )=s j k+ x j b+e i j (1.9) where b i j is bidder i’s final bid in the English auction for vehicle j. 18 We use her final bid as a proxy for her maximum bidding value. 19 s j is the percentage of experts in the auction j. We expect the coefficient k to be negative due to the winner’s curse (and crowding out effect for the regressions of auction revenue). x j is a vector of characteristics of automobile j, car model fixed effect, and month fixed effect. e i j is an error term capturing the idiosyncratic taste of bidder i for 18 In section 5.5, the dependent variable is the logarithm of auction revenue. 19 While the use of the jump bidding strategy may cast doubt on the validity of the proxy, we choose this proxy because of the following two reasons. First, the minimum incremental bid is just $15, allowing bidders to overbid their competitors by a small amount, which results in multiple interactions between bidders. In the auctions, on average, each bidder bids 8.4 times. Continuous participation of the bidders supports the effectiveness of our proxy. On the other hand, the literature shows that jump bidding is more likely to occur earlier in an auction (Easley and Tenorio (2004)). Based on these two reasons, we believe that the final bid is a good proxy for bidders’ maximum willingness to bid. 27 the vehicle j. For the moment, we assume thats j ande i j are uncorrelated. In the next section, we are going to discuss various ways to deal with the endogeneity issue. 1.5.5 The Effect of Experts’ Participation on Individuals’ Final Bids We use the logarithm of individual bidders’ final bids as the dependent variable for the four regres- sions. In our data, there are in total 18,959 potential bidders, of which only 17.8% are identified as experts. The relatively large number of individual bidders supports our conjecture that the winner’s curse and crowding out effect will dominate, and our hypotheses are plausible. We will adjust this expert identification criterion in our robustness check section. 1.5.5.1 OLS Results We report our results of four specifications in Table 3.1. All regression tables are put into the appendix. In specification (1), the vector of covariates includes the percentage of experts in the same auction, car characteristics (including mileage, model, year, car model, and displacement), and month fixed effect for seasonality. The coefficients generally have the expected signs. Our variable of interest is the “% of experts”, which is calculated from the bidders who submitted bids in each auction. The regression shows that, with each percentage increase of experts in an auction, individual bidders’ final bid will decrease by 0.379%. This is consistent with the prediction of our theoretical model that the participation of experts (or large buyers) encourages other bidders to bid more cautiously as experts may possess more information about the car’s market value. Specification (2) does text analysis by including control variables regarding the description of the vehicles, which includes the length of the description, whether the description contains certain keywords such as “apparent repair,” “no collision,” and “engine issue.” The signs are also as expected. We add dummies for the presence of certain phrases that may signal the quality of the vehicles. Cars with negative descriptions like “apparent repair,” “engine issue,” and “scratched” generally receive lower bids. 28 Interestingly, a longer description of the vehicle increases the final bid, which seems counter- intuitive since a longer description could mean the car has more issues. However, as Lewis (2011) finds in eBay used-car auctions, sellers of high-quality cars will describe their cars carefully on the web page. Those selling low-quality cars provide minimally descriptive web pages and receive lower bids. We believe the same reason applies here. A car with a minimal description is not necessarily a good quality signal but may cause bidders to be alerted that the auctioneer is hiding something. 1.5.5.2 Endogeneity The OLS estimations potentially suffer from endogeneity, as the correlation between the percent- age of experts and individual bidders’ final bids may not be causal due to omitted variable bias or selection issues. Experts’ participation might be affected by the demand for used cars. They are more likely to participate in auctions with specific car models with specific colors and other car characteristics. We have the car model as a control variable in our regressions, but there might be unobserved car characteristics that are correlated with experts’ participation. To deal with the endogeneity issue, we propose two instrumental variables (IV). The first is the percentage of experts in all other auctions on the same date and the same platform. For the validity of the instrument, one should check two conditions: the relevance condition and the ex- clusion restriction. 20 On the one hand, if an expert has participated in one online used-car auction on the platform, it is more likely that he will participate in other used-car auctions on the same platform. Thus, the relevance condition is satisfied. On the other, the number of experts in other auctions is not correlated with the unobserved car characteristics in this auction. The only channel for experts’ participation in other auctions to affect final bids in this auction is through affecting experts’ participation in this auction. 21 20 Relevance condition means the instrument should correlate with the endogenous variable, and the exclusion re- striction states that the instrument should not correlate with the error term. 21 One possible endogeneity issue is that experts’ participation decisions are affected by market demand for certain cars during a certain period. We controlled car models and time fixed effect. Therefore, this is less of a concern. 29 Specification (3) reports the 2SLS regression results. The first stage has F-statistics 88.3, in- dicating that our instrument is strong. The signs of the variables that are significant at 1% do not change. The percentage of experts remains significantly negatively correlated with the final bid of individual bidders, with a one % increase of experts resulting in a 0.817 % decrease in individ- ual bidders’ final bids. The coefficients of the total number of bidders are insignificant, which is expected as a bidder’s maximum bidding function is not related to the total number of bidders. One concern is that the quality of cars sold on the same platform within the same day may correlate with each other, thus making the instrumental variable endogenous. While the auctioneer claims that the cars sold on the same day were randomly chosen, we would like to double-check the robustness. Therefore, specification (4) uses another instrumental variable: the percentage of experts in the auctions on the same platform held on the closest previous date. The percentage of experts in another sequence of auctions should by no means affect the bids submitted in that day’s auctions. Still, it may correlate with the experts’ participation in today’s auctions. If an expert participated in the auction held two weeks ago, he would likely attend the current auction. The first stage has F-statistics 122.69, which also suggests that no weak-IV issue presents here. The signs of the coefficients remain mostly unchanged. The new instrumental variable regression suggests that a one % increase in experts’ percentages results in a 0.841 % decrease in individual bidders’ final bids, which is consistent with our theory. 1.5.6 Auction Revenues As discussed in our theory, increasing the number of experts has multiple effects. The winner’s blessing and Bertrand competition raise the revenues, but these effects vanish when there are many individual bidders. Oppositely, the winner’s curse discourages individual bidders from bidding aggressively, which will drive the auction revenues down. In addition to the winner’s curse, when we have more experts, random draws of private value from individual bidders are replaced by a fixed common value of the expert, which drives the expected revenues down as well because of 30 lower expected maximum independent private values. To conclude, when the number of individual bidders is large (which is satisfied by our data), having more experts leads to lower revenues. Table 3.2 summarizes the results. The four specifications demonstrate a negative correlation between the percentage of experts and the logarithm of the auction revenues. The coefficients are significant at a level of 1%. The negative correlation suggests that the winner’s curse and crowding out effect dominate the winner’s blessing and the Bertrand competition between experts. Thus if the auctioneer wants to increase revenue, he should limit the participation of experts. Another interesting result is that now the total number of bidders has a significant positive impact on auction revenues, while when we look at the regression of individuals’ final bids, the coefficients are mostly small and insignificant. These are consistent with our theory because the number of bidders should not affect individual bidders’ willingness to bid, but should positively impact the total revenues. The other coefficients are mostly significant and have expected signs. 1.6 Robustness Check 1.6.1 Unobservable Potential Bidders In section 5, the variable “% of experts” is measured by the bidders who actually submitted bids in the auctions. We only use the data of bidders who submitted bids because they are the only observable bidders in English Auctions. Unlike sealed bid auctions, we usually do not observe all the potential bidders, as some may drop out without submitting a bid when the standing bid reaches their valuation. While the unobservability of some bidders raises concerns for measurement error, the auction data set we obtained has an interesting survey that may help us alleviate this concern. The survey asks each bidder which vehicles she intends to bid. 22 Using this data set, we can define a new variable of interest as “potential % of experts”, which measures what percentage of bidders who 22 The survey is not mandatory for bidders to bid in the auctions. At that time, however, bidders were not informed of this. Consequently, we believe this survey is a good measure of the potential bidders. 31 signaled interests in buying this car are experts. We perform similar regressions as we did in Table 3.1 and summarize the results in Table 3.3. The regression results show that using the survey to compute the potential proportion of dealers yields similar results to the observed bidders in the auctions. The four specifications demonstrate that individual bidders’ final bids are significantly lower with a higher percentage of potential experts. The magnitude of these effects is even larger with potential bidders than observed bidders. 1.6.2 Experts vs. Individual Bidders Section 1.5.2.2 shows that the bidders identified as experts behave differently from the individual bidders. Experts usually submit bids close to the soft close time and drop out faster after the soft close time than the individual bidders. In this section, we show that the existence of experts has different impacts on the bidding strategies of individual bidders and experts. We run OLS and 2SLS regressions (2) and (4) for either individual bidders or experts. The results are summarized in Table 3.4. The regressions reveal that both individual bidders’ and experts’ final bids decrease as experts’ proportion increases. However, The impact on individual bidders is larger than the impact on the experts. The OLS results indicate that the impact on Experts is 31.7% smaller than the one on individual bidders, and the 2SLS results indicate that the impact on Experts is 25.6% smaller than individuals. If experts do have perfect information on the value of the cars, they should not take into account the presence of other experts. Nevertheless, experts may have different noisy signals about the market values, and thus they may also experience the winner’s curse and thus will shade their bids strategically. The regression results show that experts shade their bids less than individual bidders, possibly due to having better information. 32 1.6.3 License Plates Since the used cars in our dataset are from the provincial government or government-affiliated firms, the license plate sometimes has special meanings. It is well-known among bidders the last digits of the plate could indicate whether the car is coming from a department with political power. If the plate ends with A,B,C,D, bidders know that the car is from the “Provincial Party Committee”, “Provincial People’s Congress”, “Provincial government”, and “Provincial Political Consultative Conference”, respectively. There are at least two reasons why those special license plates would encourage bidders to bid more aggressively. First, those departments with power usually have a special maintenance team for their cars. Hence the license plates are correlated with the unobservable quality of the cars. Second, there is a so-called “official rank standard” ideology in Chinese culture, which means people usually worship high-rank officials in government. Individual bidders might be willing to pay more for cars from the department with the most political power. We add a dummy variable, “Department with power”, as a control variable to the regression, which indicates whether the car has a license plate from the four departments mentioned above or not. The regression results are summarized in Table 3.5 in the appendix. We see that the special license plate does have a significant positive effect on individual bidders’ final bids. The coefficients of other variables remain almost the same. 1.6.4 Definition of Experts One may have concerns about our method to identify an expert in our data set. Instead of claiming bidders who deposit for at least three cars are experts that possess superior information, we adjust the criterion to five cars, four cars, and two cars. The regression results are summarized in Tables 3.6, 3.7, and 3.8. These tables are put into the appendix. As long as we define multi-unit buyers as experts, the coefficients of the percentage of experts are all negative and significant at a level of 1%. The results suggest that we have a robust definition. The coefficients are less negative when we identify more bidders as experts. The change is due to 33 an overestimated percentage of experts in each auction. If we deem bidders who deposit for two cars as experts, we may falsely identify many individual bidders as experts, thus underestimating the effect of experts. In addition, the coefficients of other variables remain stable and highly significant, showing that our regression model is robust in predicting the average final bids of individual bidders. 1.7 Conclusion Our theory and data indicate that when the number of bidders is large, the participation of informed bidders may reduce individual bidders’ willingness to bid and the final auction revenues. These findings raise several policy implications on how to increase the expected revenue of an auction. Intuitively, we might want to restrict the participation of bidders with superior information. One way to restrict the participation of experts is to spend more resources on attracting individual bidders. Forbidding experts from participating in the auctions is an intuitive method. For example, E-commerce companies like eBay may want to limit the participation of resellers. In practice, however, experts could pretend to be individual bidders by registering for auctions using individual IDs. Therefore the first method may not be practical. The second method is to limit the number of auctions in which a bidder can participate. On the one hand, limiting simultaneous participation in multiple auctions could reduce an individual bidder’s ability to identify an expert. On the other, this constraint would reduce the percentage of experts in each auction, as experts who aim for multiple cars are more constrained by this policy than individual bidders. From an individual bidder’s perspective, it is crucial to determine whether she is bidding against experts who possess superior information. If so, she should consider the trade-off of the winner’s curse and the winner’s blessing and adjust her bidding strategy accordingly. There are several avenues for our future research. First, our theoretical model assumes experts have perfect information about the common value. In practice, their estimations could be more accurate than individual bidders but may not be perfect. Second, it may be helpful to derive the 34 conditions under which the winner’s curse dominates the winner’s blessing. Third, our empirical analysis does not take bidders’ characteristics into account. Individual bidders’ ages, genders, and incomes should affect their willingness to bid. We leave these extensions for future research. 35 Chapter 2 Quality Competition among Free Digital Platforms 2.1 Introduction Digital platforms are at the heart of online economic activity. On the one hand, they bring un- precedented convenience to users. On the other, they also arouse antitrust agencies’ concerns about whether the platforms engage in anti-competitive practices that may reduce social welfare. In fact, regulators worldwide have closely scrutinized big online platforms due to potential anti- competitive concerns. Since 2019, the Department of Justice (DOJ) has conducted an antitrust investigation into Big Tech companies like Google, Facebook, Amazon, and Apple. The DOJ has recently concluded that U.S. antitrust law needs an overhaul to allow more competition in the U.S. internet market. China, too, has released Guidelines for Anti-monopoly in the Field of Plat- form Economy in late 2020, which was aimed at stopping the anti-competitive practices of digital platforms. A major problem these and many other countries face is that plenty of online platforms do not directly charge users, and thus many classic conclusions from literature in the fields of law and economics about price competition do not apply. Video platforms like YouTube and TikTok, search engines like Google, social media like Facebook make their services freely available but collect revenues from advertisements. Video-game streaming platforms like Huya, Douyu, Twitch, donation-ware like Wikipedia generate revenue from various types of digital tipping. The Chinese game streaming giants Huya and Douyu make $30 billion annually, and surprisingly, 93% of their 36 revenues come from digital tipping. 1 These successful platforms which rely on digital tipping serve as a key motivation for our analyses in this paper. Because there are various zero-pricing business models in the platform economy, economists usually do case-by-case studies and do not have a general economic theory to understand these business models. In addition to the challenges in understanding the digital markets, the lack of pricing also leads to practical issues. For instance, a special concern from an antitrust perspective is the definition of the relevant market. When firms compete in price, the standard way is to use the test of a small but significant and non-transitory increase in price (SSNIP) to examine whether a slight price increase (usually 5%) is profitable for a hypothetical monopolist. If this is the case, then we define the current market as a relevant market. Otherwise, we add another most closely correlated basket of products to the concerned hypothetical monopoly market and do the test again. Since some online platforms do not charge users directly (i.e., zero-pricing), the definition of a relevant market becomes unclear. For example, Huya and Douyu proposed a merger in 2020 and drew a heated debate about the boundary of the video-game live-streaming market. Google, Facebook, among many online platforms under antitrust investigations, also offer zero-pricing services to their users. Therefore, the traditional market definition does not work, and it needs to be revised. In general, in the case of online platforms, the definition of the relevant market is unavoidable. In the absence of price, users inevitably make their decisions to use a product or service based on some form of quality consideration. A promising yet controversial test, a small but significant non-transitory decrease in quality (SSNDQ), is deemed a possible alternative to the SSNIP test. However, the theoretical justification and how to rigorously apply the test to online platforms are yet to be studied. This paper aims at building a general model to analyze the quality competition among zero- pricing platforms, especially to understand the competition in the Chinese video-game streaming market. In our baseline model, symmetric platforms offer free basic services to their user bases and generate revenue from collecting tips. We analyze a three-stage game. In the first stage, platforms 1 For more information on Huya and Douyu’s revenues, see https://ir.huya.com/index.php?s=120 and https://ir.douyu.com/Annual-Reports. 37 decide how much they will invest in improving the quality of their basic service. After observing the quality of each platform, users decide which platform they will use. In the last stage, these users choose the tipping amounts, and the platforms use a fraction of the total tips to offer club goods and lotteries with fixed prizes. We first explore tipping as platforms’ methods to generate revenues. We show that the revenue function is concave and increases with the number of users under general assumptions and that the externality among users within the same platform is positive. We prove that there is a unique participation equilibrium if the user externality is not too strong. This paper’s main finding is that the equilibrium quality level is either insufficient or excessive compared with the socially optimal quality level, depending on the marginal revenue of an addi- tional user, the user externality, and the platforms’ horizontal differentiation. We identify sufficient conditions under which the equilibrium quality level is socially insufficient. Intuitively, if user ex- ternality is significantly negative, attracting more users may backfire, and the quality provision is insufficient. Moreover, if the marginal revenue from an additional user is low, platforms also provide inadequate investment in quality due to the lack of incentive to compete for more users. Lastly, if platforms are very horizontally differentiated, the effect of quality investment is ineffi- cient in attracting users, thus making the equilibrium quality level socially insufficient. Whether we need regulations on the zero-pricing platform market or not depends on the relation between the marginal revenue from an additional user, the magnitude of user externality, and platforms’ horizontal differentiation. The following section studies other prominent revenue-generating business models adopted by zero-pricing platforms, including using all-pay auctions and advertisements. For each business model, we look into the properties of the revenue functions and user externality. We derive neces- sary and sufficient conditions for these business models to result in insufficient or excessive quality provisions. We show that, if platforms generate revenue from an all-pay auction, the equilibrium quality level is always insufficient compared with the socially optimal quality level. On the other 38 hand, if the platforms rely on advertising for revenue, then the equilibrium quality level is insuf- ficient when the marginal revenue from an additional user is low. Lastly, we show that a merger between two platforms is likely to have downward quality pressure. Under regularity conditions, the equilibrium quality level after a merger is lower. However, if the pre-merger quality provision is socially excessive, whether a merger will reduce or increase social welfare is unclear. During the analysis, we propose an ”effective price” notation for zero-pricing platforms. We construct the effective price from the quality competition model and discuss its properties. The introduction of the effective price will help bridge the gap between the two well-known tests for market definition, i.e., the SSNIP test and SSNDQ test, from which we may determine what per- centage of quality reduction is comparable to a 5% price increase. This paper is organized as follows. Section 2.1.1 briefly reviews related literature on platform competition, all-pay auctions, and lotteries with club goods provisions. Section 2.2 proposes a general model to describe the interaction between platforms and users. In Section 2.3, we perform an equilibrium analysis of users’ tips, participation, and platforms’ quality investments. In Section 2.4, We derive conditions to determine whether the quality competition is insufficient or excessive. Section 2.5 discusses two other business models that lead to different revenue and externality functions and their implications on quality competitions. We discuss the impact of a merger in Section 2.6 and conclude in Section 2.7. Most of the proofs are put into the appendix. 2.1.1 Literature review Classic literature on two-sided markets, for example, Rochet and Tirole (2003), Rochet and Tirole (2006), Armstrong (2006), have investigated the pricing and subsidizing strategy of two-sided plat- forms. These papers provide basic frameworks for modeling externalities (network effects), either indirectly across two sides or directly from the same side. We focus on the same side externality. In our setting, zero-pricing platforms compete in quality. Most of this literature assumes that the user side is single-homing so that users only buy from one seller (Reisinger (2012), Armstrong and Vickers (2019) and many others). For simplicity, in our paper, we also assume users of online 39 platforms are single-homing. Tan and Zhou (2021) study the price competition and optimal entry problem of multi-sided platforms. Our paper adopts a similar framework but focuses on whether quality competition is socially desirable. We analyze several revenue-generating business models, including lottery with club goods, all-pay auctions, and advertising. For example, Morgan and Sefton (2000) studies how lotteries could be used to induce higher public goods provision. Basically, the all-pay auction and lottery are two ways to add competition (negative externality) into public goods provision, thus inducing the bidders or wagers to pay more for the public goods. Goeree, Maasland, Onderstal, and Turner (2005) analyze auction mechanisms like first-price, second-price, and all-pay auctions when users also care about the auctioneer’s revenue. They conclude that an all-pay auction is the optimal fund-raising mechanism. Our paper is motivated by the business practices observed in the video-game streaming markets in the U.S. and China. The leading platforms in the markets allow users to give digital tips to streamers. Several pieces of empirical literature have obtained data from Twitch and Douyu to analyze the tipping behavior of the users (Jia, Shen, Epema, and Iosup (2016), Z. Zhu, Yang, and Dai (2017), M. Tang and Huang (2019)). Theoretically, P. Tang, Zeng, and Zuo (2017) use an all- pay auction with a proportional allocation rule to explain users’ tipping behavior. In a more recent paper, Jain and Qian (2021) discuss how the nature of competition among products, the number of users, and the type of users affect platforms’ profit from advertisements and tips. 2.1.2 Background: Merger of Huya and Douyu The motivating example we have in mind is the video-game streaming market. The two leading game-centric live streaming platforms in China, Huya and Douyu, proposed a merger in October 2020. The merger immediately drew the Chinese antitrust agency’s attention. However, antitrust enforcement for this case is complicated by the absence of prices. Douyu and Huya do not charge users, which makes it hard to apply any price test to define the boundary of the relevant market. 40 Figure 2.1: Video-game Streaming Industry Interestingly, 93% of the two platforms’ $3.1 billion annual revenue is digital tipping, and the other 7% comes from advertising. In July 2021, China’s State Administration of Market Regulation (SAMR) said it would block the deal on antitrust grounds. SAMR said the block is because Huya and DouYu’s combined market share in the video game live streaming industry would be over 70%. SAMR claimed that the video- game live-streaming market constitutes one relevant market because the live-streaming service ”is substantially different from entertainment live streaming and E-commerce live broadcast in content, users, streamers’ skills, barriers of entry, source of income and the main competitors.” 2 Nevertheless, SAMR does not provide any formal test on why these live-streaming services are not substitutable. In a market that relies on tipping as its main source of income, it is also not clear whether any conduct in this market is anti-competitive or not. This paper discusses the nature 2 https://www.samr.gov.cn/fldj/tzgg/ftjpz/202107/t20210708 332421.html 41 of quality competition, and how a merger affects social welfare. We also consider other business models such as auctions and advertising. Thus, we believe our results can be applied to industries other than the video-game streaming market. 2.2 Model The baseline model features K platforms competing for users of total mass N by providing high- quality basic services. The platforms do not charge prices to users directly but generate revenues from users’ participation and activities. In the baseline model, we assume platforms encourage users to tip by providing club goods with lotteries, but our model could be extended to other revenue-generating mechanisms. 2.2.1 Platform Revenue Platform k only chooses quality investment q k . Denote Q=fq 1 ,q 2 ,...,q K g as the quality invested by the platforms. Knowing the quality investment profile, the demand of the platform n k (Q) would be determined in the subsequent participation game where n =fn 1 ,n 2 ,...,n K g is users’ participation profile on each platform. We initially focus on symmetric revenue function R() and constant marginal cost c. p k (Q)= R(n k (Q)) c q k (2.1) While investments are costly, a higher investment level could attract more users to the platform and indirectly help the platform profit. 42 2.2.2 Participation Game Each (infinitesimal) user is endowed with an individual taste profilee =fe 0 ,e 1 ,...,e K g. Eache k of the entire population is independent and forms an identical distribution G(e). 3 In this paper, we assume G(e) follows the Gumbel distribution with parameter b. A higher b indicates a stronger horizontal differentiation among platforms. Given the quality investments Q and anticipating a participation profile n, a user’s utility from joining platform k is u k = B(q k )+ A(n k )+e k (2.2) and the outside option yields utility level u 0 = v 0 +e 0 . Each user is single-homing and will choose the platform (or the outside option) which gives him the highest utility. We assume v 0 is sufficiently small that in equilibrium, users must choose one platform. They may not opt out. Allowing a relevant outside option would make our analysis intractable. In equilibrium, the number of users who join each platform will be equal to the anticipated participation profile. A user’s utility on platform k consists of three parts: (i) Utility from basic service, B(q k ), de- pends on the quality level q k provided by platform k. We assume that more quality investment yields higher utility for users, and the investment has a diminishing marginal return: B 0 (q) > 0, B”(q)< 0. We also assume that the Inada conditions hold: lim q!0 B 0 (q)=¥, lim q!¥ B 0 (q)= 0. (ii) Utility from interacting with other users, A(n k ), for simplicity, is assumed to depend on the number of users on the same platform. (iii) Individual taste,e k , measures the horizontal differenti- ation among platforms. 3 An alternative interpretation of the model is to treat it as a random utility model so thate is a realization of K+ 1 random variables,e 0 ,e 1 ,...,e K , independently and identically distributed as G(e) across users and platforms (Anderson et al. ,1995). The number of users choosing each platform will be deterministic in our way of interpretation, but will be random in the alternative interpretation. The two interpretations are thus equivalent if the profit function is linear in demand. However, the alternative interpretation will be troublesome as we have a profit function that is nonlinear in demand. 43 2.2.3 Tipping Game The baseline model assumes that the platform’s revenue R(n) and the utility from interaction A(n) are generated from the following tipping game. Suppose the number of users on the platform is n. Although we regard n as continuous in the main model, with a sufficiently large total number of users N, conceptually n can always be viewed as taking integer values. Moreover, we assume n 1 in the tipping game. Observing n, user i who uses the platform chooses his tips x i to maximize his utility: A(n,x i )= 8 > > > > < > > > > : h(a( n å j=1 x j Z))+ x i å n j=1 x j Z x i , if n å j=1 x j Z 0, if n å j=1 x j < Z (2.3) Ifå n j=1 x j Z, i.e., the total tips are sufficient to cover the lottery expense, the lottery will be held, and user i wins the fair lottery with a fixed prize Z with probability x i å j x j . In this case,å j x j Z depicts the excess of total tips over the cost of prize Z. A fraction a of excess tips is used to produce club goods. In this section, we let a be exogenously fixed. 4 Consequently, the platform retains(1a)(å j x j Z). The production function of club goods h() has a diminishing marginal return, hence h(0)= 0, h 0 (Y)> 0 and h 00 (Y)< 0. 5 If the platform receives insufficient tips to cover the cost of lottery prizes, we assume the lottery is called off, and all tips are returned. Therefore, no budget is used on club goods, and both users and platforms receive 0 payoff. 6 4 We also analyzed the case where the platform chooses optimala to maximize its revenue R(). It turns out that, if h(Y) = gY , for some fixed g2[ 1 na ,1], all the results discussed in this paper for fixed a case also apply to the endogenousa case. Moreover, we show that the optimala is an interior solution (i.e.,a 2(0,1)). The intuition is that it is never optimal for the platform to take all the tips or use all the tips to produce club goods. 5 More precisely, h is not just a production function, as it also represents consumer’s utility for the club good. It is actually a composite function. 6 Our tipping game is similar to the public goods provision with lotteries game in Morgan (2000). Unlike Morgan (2000), our focus is on analyzing the revenue-generating mechanism and users’ utility. Unlike Morgan’s benevolent public goods provider, who aims to maximize total contribution, in our model, the platform only uses a fraction of the excess tips for club goods production and retains the rest as revenue. 44 Intuitively, there are three effects of a highera on the platform’s revenue. Firstly, it means that the platform decides to spend more of the total contributions on club goods and retain a smaller fraction. Secondly, a higher a induces each user to tip more because he gets more club goods in return. Lastly, a higher a also means a stronger direct network effect, making it easier for the platforms to attract users. 2.2.4 Three stages Based on all the arguments above, we now present a three-stage game. Stage 1: Platforms compete by investing in the quality of their basic services. The investment is costly but will increase users’ utility. Stage 2: Each user first learns the quality levels and his taste for the platforms and then chooses which platform to join. Stage 3: After choosing which platform to join, a user observes how many users are on the same platform and then decides his tips. 7 The three-stage model is solved by backward induction. The equilibrium concept is the Sub- game Perfect Equilibrium. 2.3 Equilibrium Analysis 2.3.1 Tipping Equilibrium We first focus on the third stage, where users decide their tips. For tractability, we assume h 0 (0)> 3 2a and lim Y!¥ h 0 (Y)< 1 a . 8 7 In the Chinese video-game streaming market, the platforms provide information about the ”popularity”, which is a garbled signal of the true number of users. A user knows this information before he gives tips. Twitch also reveals how many people are watching. We are inspired by these features and assume each user knows the number of users before deciding his activity. 8 This is a weaker version of the Inada conditions. We impose lim Y!¥ h 0 (Y)< 1 a to ensure the revenue of a platform does not go to infinity. See the proof of Proposition 4 for more details. We are interested in showing R(n) to be finite 45 Proposition 4. There exists a unique tipping equilibrium in which users tip the same amount. Proof. See appendix. The symmetric tipping equilibrium is attributed to users’ symmetric tastes over the club goods. The uniqueness of the symmetric tipping equilibrium helps us derive the comparative statics. We don’t need to worry about a coordination problem where a user only tips when others tip. In equilibrium, each user’s tip x(n) is the solution to ah 0 (a(nx Z))+ n 1 n 2 x Z 1= 0 (2.4) Define excess tips over the lottery prize as T(n)= nx(n) Z in equilibrium. The revenue of the platform is thus R(n)=(1a)T(n) and each user has utility A(n)= T(n) n + h(aT(n)) Corollary 6. T(n), R(n) and A(n) are positive and increase with the number of users. Proof. See appendix. Corollary 6 demonstrates that the revenue of a platform is positive, finite, and is increasing in the user base. Another result is that the network effect is always positive. There are two effects from one more user entering the platform. On the one hand, the user’s entry has a positive externality by increasing the total tips and the level of the club goods. On the other hand, the entry lowers current for the sake of realistic concerns and future analysis of the quality investment game. Morgan (2000) does not impose a similar condition on the limiting behavior of h 0 (), and thus the total contributions could possibly go to infinity. 46 users’ chances of winning the lottery, which is a negative externality. Under our assumptions, the positive externality dominates the negative externality. As we are interested in zero-pricing platforms that do not charge prices to users, it is essential to connect our analysis to standard price competition. Using our model, we can propose the revenue- per-user R(n) n as an ”effective price” that resembles the regular price. 9 Corollary 7. If h 0 () is concave, then R(n) is concave, and there exists a unique ˆ n 1 such that, the revenue per user R(n) n is increasing with n for 1 n ˆ n, while decreasing with n for n> ˆ n. Proof. See appendix. Corollary 7 states that we should not take it for granted that the effective price is downward sloping. Unlike a standard demand system, the effective price (revenue per user) may increase and then decrease with the demand. Corollary 8. If h 0 () is concave, then A 0 (n)< 2t if h 0 (at)< 1 a . Proof. See Appendix. In Corollary 8 we provide conditions under which the positive network effect can be bounded from above. We shall assume h 0 () is concave throughout the rest of the paper. 2.3.2 Participation Equilibrium Let’s go back to the second stage and characterize users’ simultaneous participation decisions. We analyze the Participation Equilibrium n(Q) defined as follows: Definition 2.3.1 (Participation equilibrium). Users’ participation profile n(Q) is a Participation Equilibrium if and only if it solves the following system of equations: 9 The effective price is important in the antitrust analysis because it can help us bridge the gap between SSNIP test and SSNDQ test, which are commonly used for market definition. SSNIP test requires a 5% price increase, which is a well-defined change. SSNDQ test requires a 25% quality decrease (see Hartman, Teece, Mitchell, and Jorde (1993)). However, quality is very hard to measure, and it is unclear why a 25% quality decrease is equivalent to a 5% increase in price. We should analyze how much quality reduction is quantitatively equivalent to a 5% increase in the effective price R(n) n , which approximates a 5% increase in the real price. 47 For all k2f1,2,...,Kg, n k = N Z e:k=argmax t2f1,2,...,Kg fB(q t )+A(n t )+e t g dG(e) (2.5) The quality investment q k has two effects on users’ participation decisions. A higher-quality q k will increase a user’s utility of the basic service B(q k ), directly attracting users to the platform. On the other hand, if there is a positive (negative) externality, a higher q k will have an indirect positive (negative) effect on attracting users. Because we assume e follows Gumbel distribution with parameter b, the share of users who choose platform k follows the Logit specification: s k = exp( B(q k )+A(n k ) b ) å K l=1 exp( B(q l )+A(n l ) b ) (2.6) and the mass of users n k who choose platform k satisfies the following equation: n k = Ns k = N exp( B(q k )+A(n k ) b ) å K l=1 exp( B(q l )+A(n l ) b ) (2.7) Proposition 5. Suppose A 0 (n) < 2b N holds. Then for any quality profile Q, there exists a unique participation equilibrium. Proof. See Appendix. Under the condition h 0 ( ab N )< 1 a , we show that A 0 (n)< 2b N for all n. Proposition 5 proves the uniqueness result to the case with relatively small positive externality. 10 Note thatb measures the horizontal differentiation of users’ heterogeneity. The proposition states that, as long as the degree 10 It is easy to show that the participation equilibrium is unique if the negative externality (A 0 (n)< 0) holds for all platforms. To see this, suppose for a quality profile Q 0 , n 0 =fn 0 1 ,n 0 2 ,...,n 0 K g is a participation equilibrium. Without loss of generality, suppose there exists another participation equilibrium n 00 and indexes i and j, such that n 00 i > n 0 i and n 00 j < n 0 j . Because there is a negative externality, users on platform i would deviate to platform j, and n 00 can not be a participation equilibrium. A contradiction. 48 of externality is small relative to the degree of user heterogeneity (or product differentiation), system (2.5) forms a contraction mapping and leads to the unique participation equilibrium. The intuition is relatively straightforward. Positive externality attracts users to join the same platform, but horizontal differentiation of the platforms induces users to choose their preferred platforms. When horizontal differentiation dominates the positive externality, users don’t cluster on one platform, and the participation equilibrium is unique. We will focus on symmetric quality investment equilibrium in the next section, and thus we are interested in symmetric (semi-symmetric) participation equilibrium which arises naturally. Corollary 9. If the quality profile is symmetric (semi-symmetric 11 ), the unique participation equi- librium is also symmetric (semi-symmetric). Proof. See Appendix. 2.3.3 Quality Investment Equilibrium We now determine the equilibrium quality of the platforms. Proposition 6. If for n2(0,N), ¶ ¶n R 0 (n) b( 1 n + 1 Nn ) A 0 (n) 1 K1 A 0 ( Nn K1 ) ! < 0 there exists a unique symmetric equilibrium for the quality competition, where the equilibrium quality level q e is defined by B 0 (q e )= cK K 1 b n e A 0 (n e ) n e R 0 (n e ) (2.8) where n e = N K . Proof. See Appendix. 11 Being semi-symmetric means all platforms but one are symmetric. 49 Following Proposition 6, since A 0 (n e )< 2b N , we know that for K 2 b n e A 0 (n e )>b N K 2b N =(1 2 K )b 0 (2.9) Thus B 0 (q e ) > 0 and we have an interior solution for the quality level. 12 Moreover, we have the following observations. Corollary 10. (i) The quality q e decreases with c. (ii) The quality q e increases with N if nR 0 (n) and A 0 (n) R 0 (n) weakly increasing with n. Proof. See Appendix. To gain some intuition about the quality level in equilibrium, we rewrite the equation in the following form. B 0 (q e ) n e K1 K b n e A 0 (n e ) R 0 (n e )= c (2.10) The left-hand side of equation (2.10) is the marginal benefits from increasing the quality level, and the right-hand side is the marginal cost of quality. The intuition is simply that in equilibrium, the platforms will choose the quality that makes marginal benefit equal to marginal cost so that the profits are maximized. Figure 2.2: The Network Effect 12 If K= 1, we have B 0 (q e )=¥, which tells us that when there is no competition, the platform has no incentive to invest in quality, and as a result, the quality level under competition q e = 0. It is because quality only helps attract users but does not bring direct revenue to the platform. 50 As illustrated from Figure 2.2, the improvement of quality has two paths to attract users to the platform. Thus it affects the marginal benefit of quality improvement in two ways. First, it attracts users directly to the platform (the red part). Second, when users enter the platform, they will consider the externality (the blue part), which is the indirect effect. Note that in equilibrium, q e is increasing with the network externality A 0 (n). Intuitively, if the positive externality is more significant, the marginal benefit of the quality provision is higher, and platforms will compete fiercely on quality, making the quality level higher. 2.4 Social Optimal Quality Level Suppose a social planner wants to choose symmetric quality levels q to maximize social welfare, denoted by W(q). Social welfare function is W(q)= K(R( N K ) cq)+ N(A( N K )+ B(q)+b(ln(K)+g e )) (2.11) The first part is the total producer surplus, and the second part is the total consumer surplus. Note that b(ln(K)+g e )=E[max i=1,2,...,k e k ] is the expectation of the maximum of users’ tastes, given that users’ tastes follow Gumbel distribution with parameter b, where g e 0.577 is the Euler–Mascheroni constant. By taking the first-order condition, it’s straightforward to see the social optimal q s is the solu- tion to B 0 (q)= Kc N = c n e (2.12) which is not a function of A() or R(). Intuitively, since we assume platforms are symmetric, each platform has the same market share n= N K . The user externality function A() and the revenue 51 function R() only depend on the fixed market share n and will not affect the social optimal quality level. Similarly to the quality equilibrium, the social optimal quality level is chosen to make marginal social benefit MR= NB 0 (q s ) equal to the social marginal cost MC= Kc. The intuition is that if all the platforms have fewer users or have higher costs, it is socially optimal for them to offer a relatively lower quality level. 2.4.1 Equilibrium and Social Optimum By comparing the socially optimal quality q s with equilibrium quality q e , we have Proposition 7. Quality level under competition q e is lower than socially optimal quality level q s if and only if b n e A 0 (n e ) R 0 (n e ) > K 1 K (2.13) where n e = N K . This is the main result of this paper. It states that whether the quality competition is socially insufficient, optimal, or excessive depends on the revenue function R() and the user externality function A(). Suppose the marginal revenue R 0 () of each user is low in the equilibrium. In that sense, the competition on users will be relatively weak, and the equilibrium quality level will be lower than the socially optimal quality level. On the other side of the coin, if the externality is relatively negative, platforms have inadequate incentives to compete for more users. Thus the equilibrium quality level will also be lower than the socially optimal quality level. Finally, with a large Gumbel parameter b, the horizontal differentiation between platforms is strong. As a result, the competition is softened, and the equilibrium quality level shall be lower than the socially optimal level. It turns out that the key to determining whether we should regulate an online platform market featuring quality competition is to analyze its business model. To be more specific, the key data 52 we need to gather is the marginal revenue R 0 () of an additional user, the externality A 0 (), and also the extent of horizontal differentiation of the services. Corollary 11. If the platforms collect tips from users, equilibrium quality level q e is lower than socially optimal quality level q s when either N orb is sufficiently large. 2.5 Other Revenue-Generating Mechanisms This section considers other revenue-generating mechanisms platforms could use to make profits from their user bases. We consider the following two cases: A platform could use a sealed bid all-pay auction and offer an add-on service to be auctioned off to the users. For example, in the video game streaming market, the streamers usually express their gratitude or send a gift to the highest tipper. The platform does not directly get money from users but sells users’ attention to advertisers. We will study these cases to derive properties on the revenue function R() and user externality functions A(), and provide discussions about whether these business models lead to insufficient or excessive quality provision. 2.5.1 All-pay Auction Suppose instead of providing club goods and lottery, each platform is endowed with one unit of add-on service to be auctioned off. 13 After users choose a platform, they will learn their indepen- dent private valuation for the add-on service, drawn from a distribution with probability density 13 In the context of the video game streaming platform, streamers may auction off ‘ streamer’s attention’. For example, streamers may offer special gratitude to the highest tipper or invite the highest tipper to play a game together. This scenario is very common in the video game-streaming market. 53 function f(v), cumulative density function F(v), and support[v,¯ v]. 14 We consider the users par- ticipating in a sealed-bid standard auction where the prize is awarded to the highest bidder. The most applicable auction format is an all-pay auction in the video-streaming market. By the revenue equivalence theorem, the revenue of an all-pay auction is the same as that in any standard auction. The revenue of the auctioneer is R(n)= Z ¯ v v J(v)dF n (v) (2.14) where J(v)= v 1F(v) f(v) is the virtual valuation. Denote d(n)= R vdF(v) as the expected max- imum value of the object with n users. When a user chooses which platform to join, he does not know his true valuation for the add-on service. Thus the ex-ante expected utility of a user, as a function of the total number of users on this platform is A(n)= 1 n (d(n) R(n)) (2.15) Lemma 1. Under the all-pay auction mechanism, the following hold: (i) Expected utility A(n)=d(n)d(n 1); (ii) A(n) is decreasing with the number of users n; (iii) Platform’s revenue R(n)=d(n) nA(n); (iv) R() and A() have the relation: R 0 (n)+ nA 0 (n)= 0. Proof. See Appendix. The intuition for Lemma 1 is as follows. (i) Every user’s expected utility from the add-on service is his marginal contribution to the expected highest valuation. (ii) When the platform uses an all-pay auction to generate revenue, the competition on the prize results in a negative externality. The unique participation equilibrium arises naturally. (iii) The platform collects the total surplus d(n) net of the expected surplus taken by the users nA(n). (iv) Since an additional user ”earns” his 14 We assume that users only learn the realization of the valuation of add-on service after joining the platform. In the video-game streaming market, this is like users only know how much he likes a steamer after watching the streaming. 54 marginal contribution, the total expected payoff of the auctioneer R(n) and the originally existing users nA(n) is not affected by the additional user. Proposition 8. If platforms generate revenues through all-pay auctions, then the equilibrium qual- ity level is lower than the socially optimal level. Proof. Substituting R 0 (n) =nA 0 (n) from Lemma 1 into equation (2.13) of Proposition 7, we have b nA 0 (n) R 0 (n) = b R 0 (n) + 1> K 1 K (2.16) By 7, q e < q s . If platforms adopt all-pay auctions to generate revenue, the quality level in equilibrium is al- ways socially insufficient. We provide a simple example to elaborate the proposition and provide more discussions. 2.5.1.1 An Example For purpose of illustration, let the valuation for the add-on service v follow the exponential dis- tribution with parameter l, and let B(q)= ln(q). Then we could derive A(n)= 1 nl and R(n)= å n i=1 1 il 1 l ln(n)1 l , where R(n)/n ln(n)1 nl is the effective price. The effective price increases in n if n < ˆ n, and decreases in n if n ˆ n, where ˆ n= e 2 . This property is similar to what we have in the main model. We could solve for the optimal q as q c = N(K 1) cK(Nbl+ K) In this special case, if we fix the number of firms K, we have q increasing with the number of users N. So with more users, the platforms have higher incentives to offer better quality levels. Op- timal q is increasing with the number of firms K iff K 1+ p 1+ Nbl. On the one hand, having 55 more platforms in the market makes acquiring users more difficult, but on the other hand, it means fewer users on a single platform, and thus the marginal revenue from each user is higher. When K is relatively small, the latter effect dominates the first effect and induces a higher equilibrium quality level. The equilibrium quality level q is decreasing with exponential parameter l. Intuitively, with a higherl, users’ private valuations are lower Platforms have lower revenue, and thus have lower incentives to compete for more users. The Left-hand side expression (2.16) can be simplified to nbl+ 1. nbl+ 1> 1> K 1 K from which we always have q s > q c . This again verifies that Proposition 8 holds. 2.5.2 Advertising The last business model we will discuss in this paper is advertising. Video platforms such as YouTube and TikTok, search engines such as Google usually use this business model. We assume an advertiser’s willingness to pay is proportional to how many users have been reached by the ads. We assume users are single-homing. 15 Following Armstrong (2006), we assume there are mass 1 heterogeneous advertisers, who have private information about their profitq from each reader who sees their ads. q has CDF G a () and PDF g a (). Advertisers could join multiple platforms. Platform k charges a fixed lump sum price p i to each advertiser. Then a typeq advertiser will place an ad on the platform ifqn k > p k . As a result, 1 G a ( p k n k ) advertisers will join platform k, for all k2f1,2,...,Kg. We can then derive platform i’s revenue function, given its user base n k , the platform chooses the price to advertisers: 15 If there are many multi-homing users, platforms may have weaker incentives to engage in quality competition. This will be an interesting extension of our model. See Athey et al. (2017) for comparative statics on the fraction of multi-homing users. 56 R(p k )= p k (1 G a ( p k n k )) The optimal price is solved by p k n k = 1G a ( p k n k ) g a ( p k n k ) . So knowing the distribution of advertiser’s profit per view, the optimal p k is p k = n k q , whereq is the solution of the equationq = 1G a (q) g a (q) , and is independent of n k . The solution is unique if we assume increasing hazard rate function. Then the platform k’s profit as a function of its user base is R(n k )= n k q (1 G a (q )) From this profit function, we know a platform’s profit from advertisement is proportional to its user base. We assume A(n)= 0. Then we simply have A 0 (n)= 0 and R 0 (n)=q (1 G a (q )). Define the q (1 G a (q )) as q a , which measures the marginal revenue of the platform from an additional user. The left-hand side of equation (2.13) is b q a . Proposition 9. Compared with the socially optimal quality level, the equilibrium quality level in the advertisement model is insufficient ifq a < Kb K1 , and is excessive ifq a > Kb K1 . A large q a means the marginal revenue from an additional viewer is high. Intuitively, if the advertising payment to the platform for each user is high, the marginal benefit of improving the quality level is also high. The platforms will engage in fierce competition, which is too costly for society. 2.6 Discussion: Merger and Downward Quality Pressure (DQP) In the main sections, we analyze the quality competition between platforms. We conclude that the quality provision of platforms may be insufficient or excessive. A legit question to ask is how a horizontal merger will affect the quality provision. A common concern for a horizontal merger is that it may substantially lessen competition. The 2010 U.S. horizontal merger guideline states, ”A merger between firms selling differentiated 57 products may diminish competition by enabling the merged firm to profit by unilaterally raising the price of one or both products above the pre-merger level.” 16 Nonetheless, the effect of a merge on the provision of quality is much less clear and is not well-studied by researchers. For simplicity, we consider two platforms 1 and 2 compete by investing in quality. The profit function of platform i is 17 p i (q 1 ,q 2 )= R i (q 1 ,q 2 ) c q i (2.17) Consider the merger’s effect on platform 1’s quality level q 1 . Let q 1 denote the pre-merger optimal quality level, and let q 1 be the after-merger optimal quality level. Before the merger, the optimal quality q 1 satisfies: ¶R 1 (q 1 ,q 2 ) ¶q 1 q c= 0 (2.18) After the merger, the optimal quality q 1 satisfies: ¶R 1 (q 1 ,q 2 ) ¶q 1 q + ¶R 2 (q 1 ,q 2 ) ¶q 1 q c= 0 (2.19) Assumption 3. (i) ¶R i (q i ,q j ) ¶q i > 0, ¶R i (q i ,q j ) ¶q j < 0, ¶ 2 R i (q i ,q j ) ¶q 2 i < 08i= 1,2. The difference between equation (2.19) and equation (2.18) gives us: ¶R 1 (q 1 ,q 2 ) ¶q 1 q ¶R 1 (q 1 ,q 2 ) ¶q 1 q = ¶R 2 (q 1 ,q 2 ) ¶q 1 q (2.20) 16 Horizontal merger guideline, section 6.1, https://www.justice.gov/atr/horizontalmergerguidelines08192010#2 17 In the main paper, we use R(n k (Q)), where Q=fq 1 ,q 2 ,...,q K g, see equation 2.1. R(q 1 ,q 2 ) is a little bit abuse of notation. 58 Suppose that q 2 =q 2 . 18 If we have ¶ 2 R i (q i ,q j ) ¶q 2 i < 0 being true for all q i , then because the right hand side ¶R 2 (q 1 ,q 2 ) ¶q 1 q > 0, we know q 1 q 1 < 0. This is called downward quality pressure. We can also approximate the quality change using the definition of derivative: q 1 q 1 ¶R 2 (q 1 ,q 2 ) ¶q 1 q ¶ 2 R 1 (q 1 ,q 2 ) ¶q 2 1 q (2.21) Lemma 2. If the revenue function R 1 (q 1 ,q 2 ) and R 2 (q 1 ,q 2 ) are concave, then the merger has a negative impact on the quality provision. There is a downward quality pressure. 2.6.1 DQP and the equilibrium quality level DQP only suggests that both platforms tend to reduce their quality provisions after the merger. However, it does not imply that the equilibrium quality level is certainly lower after the merger. Note that Lemma 1 only considers the situation where q 2 is fixed. We know the merger would negatively pressure platform 2’s quality provision using the same reasoning. Then we need to ensure the two quality movements will reinforce each other. Assumption 4. (i) ¶ 2 R i (q 1 ,q 2 ) ¶q 1 ¶q 2 > 0,8i= 1,2. (ii) ¶R 2 1 ¶q 2 1 ¶R 2 2 ¶q 2 2 > ¶R 2 1 ¶q 1 ¶q 2 ¶R 2 2 ¶q 1 ¶q 2 . Assumption 4(i) ensures that q 1 and q 2 tends to move in the same direction. 19 Assumption 4(ii) ensures that the cross-partials are relatively small than the own second-order derivatives so that it is impossible to have both q 1 and q 2 increasing after the merger. It is worth noting that if platforms are symmetric, Assumption 4(ii) is a necessary condition for the Hessian matrix to be negative definite. 18 We are interested in determining the impact of a merger on platform 1’s quality provision. Therefore, we hold platform 2’s quality level fixed. See Cheung (2016). 19 This is derived by applying implicit function theorem to equation (2.18) to make sure ¶q i ¶q j > 0. The intuition is similar to supermodularity. 59 Proposition 10. Under Assumptions 3 and 4, the equilibrium quality level after the merger is lower than the equilibrium quality level before the merger. Proof. See appendix. 2.6.2 Example of DQP: Advertising We consider the advertising model as an example. We assume the quality investment has two effects (instead of one in the previous sections). First, it induces more users to choose this platform, and second, it increases each advertiser’s willingness to pay for each user. The profit function is thus: p 1 = p 1 (q 1 )D 1 (q 1 ,q 2 ) cq 1 (2.22) The function p 1 () denotes each advertisers’ payment on platform 1. This payment increases with platform 1’s quality. The p 1 () function relaxes our main model’s assumption of fixed per-user valuation. The optimal quality level before merger can be derived from FOC: p 0 1 (q 1 )D 1 (q )+ p 1 (q 1 ) ¶D 1 (q) ¶q 1 q c= 0 (2.23) After the merger, the FOC is p 0 1 (q 1 )D 1 (q )+ p(q 1 ) ¶D 1 (q) ¶q 1 q + p 2 (q 2 ) ¶D 2 (q) ¶q 1 q c= 0 (2.24) 60 Following Cheung (2016), 20 by assuming that D 1 (q ) = D 1 (q ), q 2 = q 2 , ¶D 1 (q) ¶q 1 q = ¶D 1 (q) ¶q 1 q , and ¶D 2 (q) ¶q 1 q = ¶D 2 (q) ¶q 1 q , we have q 1 q 1 p 2 (q 2 ) ¶D 2 (q) ¶q 1 q p 00 1 (q 1 )D 1 (q )+ p 0 1 (q 1 ) ¶D 1 (q) ¶q 1 q (2.25) = p 2 (q 2 )D 12 p 00 1 (q 1 )q 1 h 1 + p 0 1 (q 1 )) (2.26) This expression is similar to what we had in equation 2.21. The numerator is the derivative of platform 2’s revenue with respect to q 1 , and the denominator is the second-order derivative of platform 1’s revenue evaluated at the after merger equilibrium. Additionally, for advertising model, we can simplify the downward quality pressure in terms of the transition ratio D 12 = ¶D 2 (q) ¶q 1 ¶D 1 (q) ¶q 1 and the own price elasticityh 1 . 2.7 Conclusion This article provides a framework to study platform quality competition in an oligopolistic market. We show that the determination of whether the equilibrium quality level is insufficient or exces- sive depends on platforms’ marginal revenues, user externality, horizontal differentiation, and the number of platforms. We also show that if platforms adopt an all-pay auction, the equilibrium quality level is always insufficient. Regulators might have incentives to subsidize these platforms to induce a higher investment in various kinds of quality. Our model relies on several crucial assumptions. We assume that the users can only join one platform. For some applications, this assumption is generally not valid. It would be interesting to study the quality competition when some users are multi-homing on more than one platform. Further, we assume the outside option is not attractive. It would be more realistic to include an attractive outside option. Last but not least, our model on advertisement is oversimplified. We 20 See Page 711 of Cheung (2016). 61 would like to consider a more realistic model to understand this crucial business model. We leave these and other important research questions for the future. 62 Chapter 3 Frenemies in the music streaming market: exclusive contract and sub-licensing 3.1 Introduction A common practice that emerged in the music streaming market is to sign exclusive contracts be- tween content providers (music conglomerates, indie musicians) and streaming platforms. Music streaming platforms need to distinguish themselves from their competitors by securing exclusives to attract more subscribers. Top artists (including Taylor Swift, Drake, and Britney Spears) de- cided to leverage their popularity when negotiating exclusive deals with music streaming service providers like Apple music (Jihui Chen and Fu (2017)). The war of exclusivity is also taking place in China. While exclusive contents in the Chinese music market are standard, platforms adopt an interesting strategy to resell their exclusive content to other platforms by revenue-sharing contracts, which we will refer to as “sub-licensing” in this paper. 1 One argument supporting the exclusive and sub-licensing contracts is that it may give the plat- forms a stronger incentive to fight the common enemy, music piracy. There has been a long history 1 On August 12, 2019, the largest music streaming platform in China, Tencent music entertainment (TME), was reported to be under large-scale Chinese probe over anti-competitive copyright agreements with global music labels, including Universal Music Group, Sony Music Entertainment, and Warner Music Group, even when quite a few sub-licensing contracts have been in effect. The probe was suspended on February 2020, possibly because TME sub-licensed its music to more platforms and the authorities deem this behavior pro-competitive. 63 of music piracy in the Chinese streaming market. According to IFPI, a leading organization repre- senting the interest of the recording industry, 99% of the music in china was pirated music in 2011. Top artists in China are usually underpaid. One of the most popular singers, Wang Feng, admits that he earned only $100,000 from copyrights during his more than 20 years musician career, let alone those less popular singers. Several questions puzzle us. Should sub-licensing or exclusivity be considered anti-competitive? How would such contracts affect anti-piracy investments? Who benefits from sub-licensing and exclusivity? To answer these questions, in this paper, we investigate the potential effect of sub- licensing and exclusive contracts on the incentives for anti-piracy investment, pricing, and their effects on the payment to the content provider. We assume platforms compete both on subscription fees and investments in public goods (Like investment in fighting music piracy). Firstly, we demonstrate that under non-exclusive contracts the platforms prefer free-riding on other platforms’ anti-piracy investment, 2 resulting in a low investment level compared to the optimal market level. One possible remedy suggested by content providers and the leading platform is to use an exclusive contract with revenue sharing between platforms. They claim that, by the Coase theorem, revenue sharing would result in well-defined property rights and thus solve the issue of externality. We show that if we keep prices fixed, the revenue sharing contract is irrelevant to the total investment level. However, the irrelevant result fails if we consider corner solutions or allow platforms to choose prices. Revenue sharing induces firms to price higher, thus giving them more incentives to invest. Besides, we discuss the content provider’s preference over different types of contracts (non-exclusive, exclusive, or sub-licensing). We find that content providers prefer revenue sharing or exclusive contracts over non-exclusive contracts when such contracts could generate high profit for the total market. Moreover, using a specific linear demand system, we show that an exclusive contract induces a higher anti-piracy investment and a higher subscription fee than a non-exclusive contract. The 2 The argument could be extended to other investments in public goods including hosting music events, sponsoring talent shows, etc. 64 content provider will prefer an exclusive contract to a non-exclusive contract when market dom- inance is strong. We also show that a fixed-fee sub-licensing contract is irrelevant to the invest- ment, pricing, and fixed payment to the content provider. On the other hand, a revenue-sharing sub-licensing contract can be profitable for the music industry and induce higher investments. The content provider has an incentive to choose a revenue-sharing contract over a non-exclusive con- tract. Our main contribution to the literature is that we provide a new perspective in understanding the sub-licensing contract in the music market and call for a more careful evaluation of this practice. We discuss the invalidity of using some classic economic results in analyzing this issue. In addition, we consider a model with more than one platform engaging in anti-piracy decisions, where the literature mostly focuses on discussing the comparative statics of the anti-piracy investment from a single platform directly (Reavis Conner and Rumelt (1991); Bae and Choi (2006)). In the next section, we discuss a few papers that inspire us. 3.2 Related literature The exclusive contract has attracted the most attention and controversy in the antitrust field. A few papers have discussed the impact of exclusive contracts. Here we list some of them. Exclu- sive contracts raise anti-competitive issues since they may deter entry or foreclose rivals (Aghion, Bolton, et al. (1987); Mathewson and Winter (1987); Bernheim and Whinston (1998); Calzolari and Denicol` o (2015)). In addition to these anti-competitive effects, the exclusive contracts in the music market also prevent consumers from other streaming platforms from accessing exclusive content, which serves as a common criticism on the usage of exclusive contracts. Interestingly, economic literature also states that exclusive contracts can serve as a pro-competitive mechanism. If we take investment into consideration, exclusive contracts can encourage invest- ment and effort provision when investments have an externality (Segal and Whinston, 2000). Ex- clusive contracts also benefit consumer welfare in that the upstream firm will under-price the unit 65 price of its product under exclusivity if it adopts a two-part tariff, and the exclusivity intensifies competition in the primary goods market (Jihui Chen and Fu (2017)). Because exclusive contracts prevent some consumers from getting access to exclusive content, the government has proposed compatibility between two-sided markets. 3 But some economists argued that when horizontal differences between platforms are small, incompatibility may generate larger total welfare than compatibility (Casadesus-Masanell and Ruiz-Aliseda (2008)). Another argument is that resale may increase prices and make consumers worse off (H¨ offler and Schmidt (2008)). Music sub-licensing is a special form of compatibility in the music streaming market. It remains an open question for us to see what impact sub-licensing could have on profit distribution, intermediate prices, and social welfare. In this paper, we extend the previous literature on exclusive contracts to a new form of contract format: Exclusive contract with revenue sharing sub-licensing. In addition to analyzing its effect on pricing, we focus on the implication on the public goods investment and the surplus extracted by the content provider. The rest of the paper is organized as follows. Section 3.3 reviews the history of exclusive contracts in the Chinese music streaming market. Section 3.4 provides a general baseline model to understand the effects of revenue-sharing sub-licensing in restoring market optimal pricing and level of investment. Section 3.5 discusses when content providers are restricted to a fixed fee contract, under what condition will content providers choose exclusive contracts (with or without sub-licensing). In Section 3.6, we use a Salop circle demand system to solve for prices and invest- ment level in equilibrium when: (i) platforms are integrated, and the contract is non-exclusive, (ii) platforms are competitive, and the contract is non-exclusive, and (iii) the contract is exclusive or sub-licensing. We conclude and provide discussions in Section 3.7. 3 For instance, United States v. Microsoft Corp., 253 F.3d 34 (D.C. Cir. 2001)) 66 3.3 Background: Music Piracy and Copyright War in China Piracy in the music market has been an infamous issue in China for many years. 4 According to a report by the International Federation of the Phonographic Industry, the pirate rate of the Chinese digital music market was 99% in 2011. The prevailing trend of listening to pirated music was reversed in 2015, when the Chinese government launched an operation named “Sword Net”, which demanded the digital streaming outlets take down unlicensed music. The tech giants who had previously been running music streaming services without proper license started to cooperate with content providers to resolve copyright issues, which triggered a copyright war between the major music platforms. Thanks to the anti-piracy operation and the competition between the major music platforms, the music industry soared. In 2017, China saw revenues grow by 35.3%, driven by a 26.5% rise in streaming revenues. In 2018, The music market in China rose to the seventh position from the tenth position. (IFPI 2018, 2019). Tencent Music Entertainment Group (TME), which was founded in 2016 with three leading music platforms (QQ, Kugou and Kuwo) is undoubtedly the biggest streaming platform with 800 million Monthly Active Users. 5 The most competitive alternative is NetEase Music, which claims to have 600 million registered users and more than 100 million monthly active users. 6 To better take advantage of copyright in the competition, major music streaming platforms started to engage in a bidding war for exclusive rights to songs, a trend that mirrors similar devel- opments in the television and film industries. In addition to driving up licensing fees, users often had to download multiple apps to access a more extensive selection of songs, and music platforms frequently sued each other for copyright infringement. The exclusive contract in the music industry has aroused the attention of the copyright author- ity. In September of 2015, the copyright authority met with the music platforms and warned them 4 See a detailed discussion about music piracy issue in Liu (2009) 5 https://www.tencentmusic.com/zh-cn/company.html#p1 6 Date from Quest Mobile 67 against buying exclusive rights to music content, saying that this act will “harm the wider dissemi- nation of music.” To alleviate the concerns of the authority and the public, major platforms reached a music sharing contract. For example, TME and NetEase signed an agreement to share over 99% of the songs to which they hold exclusive licenses. 7 Starting from January of 2020, Tencent music will sub-license its exclusive music to Tiktok, another giant competitor who’s specialized in short videos. 3.4 Model We consider the case with a general demand system and the investment in public goods (for ex- ample, anti-piracy). We will turn to a Salop circle for the demand system later. The idea of the model follows from Winter (1993). We assume that there are one content provider and two music platforms. While Winter assumes that upstream seller uses a two-part tariff to induce investment, we focus on the case that the content provider only requires a fixed payment. 8 We assume the cost of composing music is sunken, and there is zero marginal cost for serving every customer, which is common in the digital market. In this paper, we ignore the competition between content providers and assume there is only one content provider who aims at maximizing the fixed payment it charges. 3.4.1 Model The two platforms maximize: 7 https://www.caixinglobal.com/2018-02-12/tencent-netease-press-play-on-music-sharing-agreement- 101210887.html 8 Fixed fee contracts are common in the Chinese music market between content providers and streaming platforms, because it is costly for content providers to verify how many times some music has been played via a platform. Content providers have to rely on music platforms to be truth-telling. However, since monitoring the stream volume is costly, music platforms have incentives to under-report. This is especially true in the Chinese music streaming market because many deals are made between foreign content providers and domestic streaming platforms. Under this circumstance, the streaming fee is not optimal. Content providers may charge a high fixed fee and a low streaming fee to music platforms to extract more surplus. 68 fp 1 ,s 1 g=argmax p 1 ,s 1 p 1 = p 1 D 1 (p 1 , p 2 ,s 1 + s 2 ) s 1 g(s 1 + s 2 ) fp 2 ,s 2 g=argmax p 2 ,s 2 p 2 = p 2 D 2 (p 1 , p 2 ,s 1 + s 2 ) s 2 g(s 1 + s 2 ) Platforms choose both subscription fees and investments to maximize profits. The demand of platform i is given by D i (p i , p j ,s i + s j ), and g(s i + s j ) is the average cost of investment, where s 1 and s 2 are the investment in public goods of the two platforms. That is to say, the average cost of investment depends on the sum of both platforms’ investments. Investment is difficult to observe. So it is relatively hard for content providers to contract on investment. Define s= s 1 + s 2 as the total investment level of the market. We assume diminishing returns to the investments, and this is thus reflected as increasingly costly investments for both platforms. Unlike traditional literature discussing the consumption of public goods with a fixed price, the investment in this paper is assumed to be increasingly costly. We claim this is a more reasonable assumption. If we consider anti-piracy investment, additional investment in anti-piracy may not be as efficient. The investment is mainly in lawsuits. Then what matters to anti-piracy is the total amount of lawsuits, irrespective of who initiated it. If the two platforms are integrated, The integrated platform wants to maximize P=p 1 +p 2 = p 1 D 1 (p 1 , p 2 ,s 1 + s 2 )+ p 2 D 2 (p 1 , p 2 ,s 1 + s 2 )(s 1 + s 2 )g(s 1 + s 2 ) We make the following assumption: Assumption 5. The feasible set of prices and investment levels are compact. The Demand func- tions satisfies: (i) ¶D i ¶ p i < 0, ¶D i ¶ p j > 0, ¶D i ¶s > 0; (ii) ¶ 2 D i ¶ p 2 i < 0 ¶ 2 D i ¶s 2 < 0; (iii) ¶ 2 D i ¶s¶ p i > 0 ¶ 2 D j ¶s¶ p i > 0; (iv) j ¶D i ¶ p i j>j ¶D i ¶ p j j,j ¶D i ¶ p i j>j ¶D j ¶ p i j andj ¶ 2 D i ¶ p 2 i j>j ¶ 2 D i ¶ p i ¶ p j j; (v) the cost of investment satisfies: ¶g(s) ¶s > 0, ¶ 2 g(s) ¶s 2 08i, j2fi,2g 69 The compactness assumption is reasonable since prices and investment levels have a lower bound of 0, and we should not expect an infinite price or investment level. (i) and (ii) of Assumption 5 are standard assumptions of monotonicity and concavity for demand function. Assumption 5(iii) says that, with an increase in the anti-piracy investments, consumers have a less preferred outside option, and thus have a less elastic demand for the legal platforms. 9 Assumption 5(iv) is not very straightforward. It serves as a sufficient condition for profit functions to have a negative semi- definite Hessian matrix with respect to prices. We show in the appendix that this assumption is satisfied in both linear and Logit demand systems. This assumption is only used in the directions marked with “”s in figure 1. Assumption 5(v) ensures that the cost function is increasing and weakly convex. This assumption means that it is increasingly costly for society to fight piracy. Note that a linear increasing function g(s) = as for a positive a obviously satisfies the above assumption. A constant cost function g(s)= a will result in corner solutions and does not satisfy the assumption. With the two assumptions, we are ready to prove the existence. Proofs are put into the appendix. Lemma 3. The best response function of each firm is single-peaked, and a market equilibrium exists. 3.4.1.1 Insufficient Investment Level We will show that, due to the free-riding issue of anti-piracy investment, the investment level of anti-piracy is lower than the optimal investment level that maximizes joint surplus. Proposition 11. The anti-piracy investment level under competition is insufficient to maximize the total producer surplus. 9 For example, ¶ 2 D i ¶s¶ p i > 0 means that with higher anti-piracy investments, if platform i raises its price, it is hard for consumers to opt-out for pirated music. So they are more likely to accept the price. Similarly, ¶ 2 D i ¶s¶ p j > 0 means that if the opponent j raises its price, it’s more likely that the consumers will choose platform i as a substitute. The arguments could be extended to general investment in public goods. The idea of the proof is simple. Since we have four decision variables, we decompose the question into four sub-questions, and then conclude on the total effects. The following graph can illustrate the proof. 10 Integration Price Total investment Positive Positive Positive Positive* Figure 3.1: Direct Relationship. The two platforms’ integration reduces pricing competition and solves the free-riding problem. As a result, in a competitive market, we have lower prices and lower total investment levels than integrated platforms. Winter (1993) draws a similar conclusion, but our model setting is different. We have a more general demand system than Winter’s two-dimensional Hotelling setting. In China, music piracy, among the piracy of other digital products, has been a big issue for many years. It remains an interesting question for us to solve the free-riding issue of anti-piracy investment. Possible vertical restraints used by many content providers include selling the stream- ing rights to one platform and requiring the licensed platform to sub-license the content to the other streaming platforms. A sub-licensing contract is usually a revenue-sharing contract plus a fixed fee. Some experts claim that revenue sharing makes the property rights of the music bet- ter defined and may result in a higher level of anti-piracy investment. We will show that revenue sharing does not directly affect the total investment level but may indirectly induce a higher total investment level. 10 Sign * means we need assumption 5(iv), which are satisfied in Linear and Logit demand systems. 71 3.4.2 Sub-licensing Suppose now one music platform (say platform 1) gets exclusive content from a content provider and is required by the content provider to sub-license the music to its rival platform. This sub- licensing contract is a revenue-sharing contract. 11 Suppose the revenue sharing parameter is r, which is decided by platform 1 and platform 2 from negotiation and potentially follows the advice from the content provider. We assume the two music streaming platforms determine the revenue sharing rate. The two platforms choose investment levels by maximizing: fp 1 ,s 1 g=argmax p 1 ,s 1 p 1 D 1 (p 1 , p 2 ,s 1 + s 2 )+ rp 2 D 2 (p 1 , p 2 ,s 1 + s 2 ) s 1 g(s 1 + s 2 ) fp 2 ,s 2 g=argmax p 2 ,s 2 (1 r)p 2 D 2 (p 1 , p 2 ,s 1 + s 2 ) s 2 g(s 1 + s 2 ) where s 1 and s 2 reflect the units purchased (or contribution) of public goods by platform 1 and platform 2. We are interested in knowing how the rate of sharing r would affect s 1 + s 2 . 3.4.2.1 Direct Effect of Revenue Sharing on Investment Determining ¶s 1 +s 2 ¶r requires calculating determinants of many 4 by 4 matrices, which are hard to do by hand. Since the effect of revenue sharing on the total investment level is ambiguous. We would like to understand what causes the change in the total investment level. In this section, following the proof of proposition 11, we consider a simplified case where p 1 and p 2 are fixed. The fixed prices may result from a high cost of price change, government restrictions, etc. So this section still entails some applications. Proposition 12. (The irrelevant result) Assume platforms can not change their prices. A revenue- sharing contract changes the investment of each platform but is irrelevant to the total investment level unless the solution is a corner solution. 11 A fixed fee is irrelevant to the decision on pricing or investment 72 This result indicates that the revenue sharing sub-licensing contract does not directly enhance the incentive for investment. It tells us that it is impossible to use a revenue-sharing contract to directly solve the free-riding issue. We find similar neutrality theorems in public goods literature (Warr, 1983; Bergstrom and Varian, 1985). Especially, Bergstrom and Varian (1985) provide a general result that if agents’ characteristics sum up to a constant, and the first-order conditions satisfy certain properties, then the provision of public goods is independent of the distribution of characteristics. Our first-order conditions surprisingly satisfy the requirement of their general result. However, our model pro- vides a realistic context and more economic insights into their general results, which we think are of some interest. We also provide some discussions about the total investment level of corner solutions. A corner solution arises when the revenue sharing rate r is large or the cost function is linear. We have the following corollary. Corollary 12. If one SP has a corner solution s 2 = 0, total investment level is increasing with r. So if one of the investment levels is at a corner solution, the “Irrelevant result” fails. 3.4.2.2 Revenue Sharing and Pricing However, the interior total investment level may still increase if we do not fix prices. The effect mainly comes from the price channel. We can do a similar exercise regarding the effect of revenue sharing on prices and keep the total investment level fixed. Proving both prices increase with r under a general demand needs extra assumptions on the cross partials, which turns out to be not very intuitive. 12 We can show that without considering investment, p 1 and p 2 increases in r for linear demand or Logit demand system. Both proofs use similar tricks as we did with investment, holding price fixed. Linear case is simple because the cross partial of ¶ 2 D i ¶ p i ¶ p j = 0. The proof for the Logit demand system is put into the appendix. 12 For example, ¶ 2 D i ¶ p i ¶ p j 0 would be a sufficient but not necessary condition for our results to hold. Generally, this is saying that given my opponent’s price being higher, the demand for my products will be less affected if I raise my price. 73 Lemma 4. If the investment level is fixed, under linear or Logit demand, revenue sharing will induce both platforms to raise their prices. 3.4.2.3 The Effects of Higher Prices on Investments Given that r raises prices of both platforms and is irrelevant to the investments directly, we will analyze the effect of price increases. We solve this part by taking FOC with respect to investment, and total differentiate with respect to p 1 and p 2 . Use Cramer’s rule to solve partial derivatives ¶s 1 ¶ p 1 , ¶s 2 ¶ p 1 , ¶s 1 ¶ p 2 , ¶s 2 ¶ p 2 . Then we could derive the partial derivative of s 1 + s 2 with respect to p 1 and p 2 , holding r fixed. ¶s 1 + s 2 ¶ p 1 = g 0 (s)( ¶D 1 ¶s + p 1 ¶ 2 D 1 ¶s¶ p 1 + p 2 ¶ 2 D 2 ¶s¶ p 1 ) jAj > 0 ¶s 1 + s 2 ¶ p 2 = g 0 (s)( ¶D 2 ¶s + p 1 ¶ 2 D 1 ¶s¶ p 2 + p 2 ¶ 2 D 2 ¶s¶ p 2 ) jAj > 0 wherejAj is a positive determinant and is defined in the proof of Proposition 12 in the appendix. That is to say, holding all other factors fixed, an increase in the prices will induce a higher total investment level. 3.4.2.4 The Effects of Investments on Prices The last step to complete the analyses is to show that increasing the total investment level will induce the platforms to price higher. Again, this requires that the cross partials of the demand system satisfy certain complicated conditions. Similar to section 4.3, we can show that keeping r fixed, p 1 and p 2 increases with s for linear demand or Logit demand system. 13 We conjecture that, although revenue sharing does not directly affect the total investment level, it softens the competition between platforms, thus inducing higher prices and higher profit. Due 13 Again, ¶ 2 D i ¶ p i ¶ p j 0 would be a sufficient but not necessary condition for our results to hold. We omit the proof for linear demand (its cross partials are 0 and satisfy the requirement), and the proof for Logit demand is put into the appendix. to higher profits, the platforms have more incentives to invest in anti-piracy. Therefore, we ex- pect to see an increase in anti-piracy investment level after seeing revenue-sharing sub-licensing contracts. 14 Rate of sharing Price Total investment Positive* Irrelevant Positive Positive* Figure 3.2: Direct Relationship. Proposition 13. (A relevant result) Revenue sharing will induce a higher total investment level through the price channel without fixing prices. Proposition 13 tells us that it is incorrect to use the Coase theorem or Coase contracting directly to conclude that revenue sharing results in a better-defined property right and will induce a higher investment level. It is also incorrect to use Warr (1983) or Bergstrom and Varian (1985) to conclude that revenue sharing is irrelevant. More careful analyses are needed. 3.5 Contracts and Payment to Content Provider In the previous sections, we discussed the effects of revenue sharing on the total investment level of the market. This section will talk about the content provider’s incentive in choosing contract formats. For illustration purposes, we will refer to the content provider as CP and streaming platform as SP in the rest of the paper. 14 The sign means that we need an assumption that ¶ 2 D i ¶ p i ¶ p j is not too negative, which is satisfied under linear and Logit demand system. 75 Content provider NE Ex Sub Figure 3.3: Decision Process of the Content Provider Being restricted to a fixed-price contract, CP has three options for the market structure: non- exclusive, exclusive, and sub-licensing contracts. It is helpful to make notations clear. If CP exclusively licenses to SP 1 without sub-licensing, we define the total profit of platforms asP(1,0), withp 1 (1,0) andp 2 (1,0) representing the profit of two platforms; If CP licenses to both SPs, the total profit isP(1,1), withp 1 (1,1) andp 2 (1,1) representing the profit of two platforms; If CP decides to exclusively license to SP 1 but requires SP 1 to sub-license the music to SP 2 , the joint profit isP s (1,0), andp s 1 (1,0),p s 2 (1,0), are similarly defined, respectively. The idea follows from Tan and Yu (2018), and we add discussion of sub-licensing to make it more complete. We assume perfect commitment for licensed SP on the types of contracts. In other words, If CP decides that the contract is exclusive, the licensed SP cannot sell the music content to the other SP even if it is profitable. On the other hand, if CP decides that the contract is sub-licensing, the licensed SP cannot keep the content from its rivals. 15 The decision process is assumed to be sequential. CP first decides whether the contract is exclusive, sub-licensing, or non-exclusive. If CP decides that the contract is non-exclusive, the decision process ends, and both streaming platforms get the music. If CP decides that the contract is exclusive/sub-licensing, she will then decide whether to require the licensed platform to sub- license the music or not. 15 This is what we observe in real life since Sony Music Entertainment, Universal Music Group and Warner Music Group actually add sub-licensing clauses to their contract with Tencent. We also explored the case with no commit- ment, where the licensed SP can decide to sub-license or not on its own. It is interesting that Lemma 3 is still robust. CP will still decide the exclusivity based on the total profit of the market. We put the discussion in the appendix. 76 We do a backward induction. First, consider the second stage where CP chooses between the exclusive contract and sub-licensing contract. For either format of contract, the content provider can use an auction with a reserve price. Under exclusive contract, the reserve price would be p 1 (1,0)p 1 (0,1) for SP 1 andp 2 (0,1)p 2 (1,0) for SP 2 ; Under sub-licensing contract, the reserve price would be p s 1 (1,0)p s 1 (0,1) for SP 1 and p s 2 (0,1)p s 2 (1,0) for SP 2 . Both platforms have incentives to submit a bid, and the platform with the highest bid would win the auction. This outcome turns out to be the best that CP could achieve. Without loss of generality, we assume SP 1 will win the auction. The conditions for SP 1 to win the content is p 1 (1,0)p 1 (0,1)>p 2 (0,1)p 2 (1,0) p s 1 (1,0)p s 1 (0,1)>p s 2 (0,1)p s 2 (1,0) In other words, p 1 (1,0)+p 2 (1,0)>p 1 (0,1)+p 2 (0,1) p s 1 (1,0)+p s 2 (1,0)>p s 1 (0,1)+p s 2 (0,1) That is to say, the content provider will assign the exclusive (or sub-licensing) contract to the platform that generates total profit for the market. For the first stage, we discuss two cases: If p s 1 (1,0)p s 1 (0,1) >p 1 (1,0)p 1 (0,1). SPs expect CP to choose sub-licensing in the second stage. As a result, it is common knowledge that the disagreement point is the payoff of getting the music from its opponent. The content provider will offer take it or leave it offer p 1 (1,1)p s 1 (0,1) andp 2 (1,1)p s 2 (1,0) to SP 1 and SP 2 . Then she chooses non-exclusive offer if and only if 77 p s 1 (1,0)p s 1 (0,1) 0 and c> 0. 16 The model has three periods. Content provider selects the form of contracts from non-exclusive, exclusive, and sub-licensing. Given the content provider’s selection, platforms choose the investment level. Platforms then choose subscription prices, and consumers choose which platform to adopt. Because investment is usually a long-run plan, whereas pricing is usually a short-run decision, I let platforms decide their investments first and then decide the subscription fees. 16 There are two reasons why c might be positive. First, pirated music may not be a perfect substitute for legal music and typically entails lower quality. Some pirated music is directly recorded from a concert or streaming service. Even with digital copying, pirated music may lack technical support, whereas the quality of authorized music increases over time. Another interpretation is the searching cost. Searching or reproducing pirated music is time-consuming, negatively affecting social welfare. 80 Following Bae and Choi (2006)’s idea, we assume the anti-piracy investment increases the cost of using pirated music. The incremental cost is assumed to be s 1 + s 2 , where s 1 and s 2 represents the marginal contribution to each platform. Suppose a consumer located at point x is subscribed to SP j (SP 3 for pirated music) at a price p j (price is 0 for pirated music). His utility is given by 17 U 1 = v+ v 1 p 1 tD(0,x) U 2 = v p 2 tD( 2 3 ,x) U 3 = v c s 1 s 2 tD( 1 3 ,x) The profit of the two platforms are: p 1 = p 1 D 1 (p 1 , p 2 ,s 1 + s 2 ) s 1 g(s 1 + s 2 ) p 2 = p 2 D 2 (p 1 , p 2 ,s 1 + s 2 ) s 2 g(s 1 + s 2 ) For simplicity, the total investment level to achieve such incremental cost is assumed to be g(s 1 + s 2 )= a(s 1 + s 2 ). 18 This unit cost function satisfies our Assumption 5. One simple intuition for anti-piracy is that it may decrease the quality of pirated music or make it more time-consuming for consumers to find the proper music. 19 The convexity of cost function captures the idea that it is increasingly costly to search for more pirated music. The equilibrium concept that we will be using in this paper is Subgame Perfect Equilibrium. We make the following assumption to ensure that the function is concave in investment so that the optimal investment level is unique and non-negative. 17 Compared to Bae and Choi (2006), which assumes that consumers have a distribution of value towards pirate software, our Salop circle actually makes a stronger assumption that the distribution is uniform. They assume there is only one technology firm in the market, whereas I have two platforms. 18 Since the anti-piracy investment is usually the cost on litigation cases or the cost that makes it harder for people to produce pirated music, it is natural to assume the cost function to be convex to ensure a unique solution. 19 For instance, the investment in lawsuits may deter people from making pirated music, thus making pirated music less accessible. Investment in anti-piracy technology may also decrease the quality of pirated music. 81 Assumption 6. The cost function is sufficiently convex so that the profit function is concave, in other words, a> 1 4t so that all the investments are positive. 3.6.1 Integrated Platforms under Non-exclusion Firstly, we consider the optimal pricing and service level that maximizes the total profit of the music industry, i.e., the total profit of SP 1 and SP 2 . Under a fully collusive setting, the integration maximizes P=p 1 +p 2 = p 1 D 1 + p 2 D 2 a(s 1 + s 2 ) 2 In the third period, the optimal prices are obtained by first order condition ¶P ¶ p i = 08i2f1,2g, given s 1 and s 2 . After getting the best response p i (s 1 ,s 2 ), I substitute the best response back into the joint profit function and use first order condition again to solve the optimal s 1 and s 2 . All the explicit solutions and proofs of corollaries are put into appendix. 3.6.2 Competitive Platforms under Non-exclusion Let’s consider the case in which the content provider signs a non-exclusive contract with both platforms(SP 1 and SP 2 ). The two SPs now compete on prices and investments. In the third period, given the investments in the first period, the platforms solve the following profit function by choosing the best subscription fee p 1 , p 2 . p NE 1 = p 1 D 1 (p 1 , p 2 ,s 1 ,s 2 ) as 1 (s 1 + s 2 ) p NE 2 = p 2 D 2 (p 1 , p 2 ,s 1 ,s 2 ) as 2 (s 1 + s 2 ) Corollary 13. The investment level under the competitive setting is lower compared to a fully integrated platform. The dominant firm has a higher incentive to invest in anti-piracy. 82 The total profit of the market is P NE = p NE 1 +p NE 2 . Corollary 13 verifies our Proposition 11 that anti-piracy investment has positive externality and both platforms want to free-ride on the other platform’s investment. Consequently, both platforms exert insufficient effort to maximize joint profit in equilibrium. Intuitively, the dominant platform suffers less from the free-riding issue because it has a bigger market to protect and relatively worries less about the rival platform free- riding on the investment. 3.6.3 Exclusive Contract Now suppose the content provider signs an exclusive contract with SP 1 . In this case, only SP 1 has an incentive to invest in anti-piracy. SP 2 can no longer get the content from the platform and thus is excluded from the market. This setting represents the current Chinese music market situation, where TME, who’s undoubtedly the largest streaming platform in China, exclusively obtained most of the popular music content. On the contrary, NetEase, which is the most competitive alternative, realized the importance of copyright a little bit late and turned to collect exclusive copyrights from indie musicians. Without a sub-licensing contract from TME, it is hard for Netease to survive. However, the exclusive contract does not directly exclude pirated music because illegal copies of the music are still available. The profit function of the (surviving) platforms is P EX =p EX 1 = p 1 D EX 1 (p 1 ,s 1 ) as 2 1 After comparing the total investment level under the exclusive and non-exclusive contracts, the following results arise. Corollary 14. Compared with a non-exclusive contract: (1) An exclusive contract induces platform 1 to invest more than the total investment under a non-exclusive contract. 83 (2) An exclusive contract always induces a higher subscription fee than a non-exclusive con- tract. The differences in investment and price are more significant if the dominance of platform 1 is stronger. (3) The total profit of the market is ambiguous, but P EX P NE is increasing with market dominance. Intuitively, exclusive contract forecloses SP 2 , thus alleviating the concern about anti-piracy investment’s free-riding issue. This result is in line with Proposition 2 in Segal and Whinston (2000) that if the seller (one of the platforms) invests and the investment has positive spillover, the investment level increases with exclusivity. While they only consider a single-dimensional investment from one player, both platforms can invest in our model. Interestingly, with the dominance of the exclusively licensed platform (v 1 ) being stronger, the gap between investment under the exclusive contract and the non-exclusive contract is more sig- nificant. Previously, Tan and Yu (2018) showed that, without considering investments, the content provider has an incentive to sign an exclusive contract with the streaming platform with higher quality. On top of that, our model suggests that the content provider wants to sign an exclusive contract with a stronger platform to induce a higher anti-piracy investment level. Combining the two ideas, we may have a better idea of why most foreign content providers sign exclusive contracts with TME. Combining Corollary 14(3) and Lemma 5, we can conclude that if market dominance is suf- ficiently strong, the total profit of the market under an exclusive contract will be higher than the one under a non-exclusive contract. So the content provider will prefer an exclusive contract to a non-exclusive contract. Consumers are worse off since (i) the prices are higher; (ii) it is more costly for users to con- sume pirated music. (iii) Some consumers who switch from pirate to streaming platforms are worse 84 off due to a revealed preference argument. 20 However, in the long run, musicians have higher in- centives to compose better music. The long-run social welfare is unclear because more consumers enjoy higher quality music and spend less time searching pirated music. We will provide discus- sions on short-run social welfare and leave long-run social welfare for future research. 3.6.4 Sub-licensing Contract This section considers a new scenario in which the content provider can require a licensed platform to sub-license the music to the other music platforms that are not assigned copyrights. We assume SP 1 signs an exclusive contract with the content provider and is authorized by the content provider to sub-license the music content to other platforms, in the model, SP 2 . Note that traditional literature models resale as changing the quality of downstream platforms. One essential aspect that literature has not seriously considered is the intensified competition due to less horizontal differentiation after sub-licensing. Our Salop model can address some of the concerns by introducing a new platform SP 2 back to the model, and reducing the average distance from a consumer to the nearest platform, i.e., a smaller horizontal differentiation. 3.6.4.1 Fixed-fee Sub-licensing Contract Consider the following situation. Platform SP 1 gets the content from CP and is required to sub- license the content to SP 2 . The format of the sub-licensing contract matters. If the sub-licensing contract is a fixed fee contract, it does not generate any change in pricing or investment from a non- exclusive contract. The cost of purchasing the copyright is sunken, and SP 2 will still compete with SP 1 on prices and investment. The only difference would be that knowing the best strategy of SP 2 , SP 1 can extract all the profit of SP 2 by a fixed fee. So compared with a non-exclusive contract, fixed fee sub-licensing is profitable for SP 1 . Content provider is indifferent between (1) non-exclusive 20 Before an exclusive contract, some consumers prefer pirated music to the platform with a low price. Suppose consumers always prefer low prices over high prices. These consumers prefer previous pirated music to the platform with high prices under exclusive contracts by transitivity. In other words, they are made worse off by switching to a streaming platform. 85 contract and (2) fixed-fee sub-licensing contract, because under (2) it gets p s 1 (1,0)p s 1 (0,1). Since the fixed fee sub-licensing contract does not generate extra surplus of the market than non- exclusive contract, I havep s 1 (1,0)+p s 2 (1,0)=p 1 (1,1)+p 2 (1,1). f S, f ix 1 =p s 1 (1,0)p s 1 (0,1)=p 1 (1,1)p s 1 (0,1)+p 2 (1,1)p s 2 (1,0)= f NE 1 + f NE 2 Under a fixed fee sub-licensing contract, the content provider gets f S, f ix 1 , which is exactly the same as she got in the non-exclusive contract f NE 1 + f NE 2 . Corollary 15. Compared with a non-exclusive contract, the content provider is indifferent if the sub-licensing contract is a fixed fee contract. The total profit of the music industry does not change, and the licensed platform splits more surplus out of it. 3.6.4.2 Revenue Sharing Sub-licensing Contract As we have shown in Proposition 13, revenue sharing contracts can potentially (i) alleviate the concern about the insufficient incentive for anti-piracy investment and (ii) keep SP 2 in the market and strengthen competition. Actually, the sub-licensing contract in the Chinese music market consists of a fixed fee up to a threshold and revenue sharing after that. Besides risk sharing, the revenue sharing rate could be understood as an instrument to maximize joint profit, and the fixed fee part may represent the bargaining power of the two platforms. We use a simple linear revenue sharing with a revenue sharing rate r. Our model for revenue sharing sub-licensing contracts includes three periods. In the first period, A revenue sharing rate is exogenously given, and fixed fees are decided to ensure a fair split. 21 In the second period, two SPs decide their investment levels, and in the third period, subscription prices are determined. 21 For example, F 2 = p 2 D 2 as 2 2 +as 1 s 2 1r if SP 2 has no bargaining power. 86 First, consider the third stage. Platforms set prices to maximize their own profit, taking the level of service and contract form as given. Then back to the second stage, we solve for the investment levels. For simplicity, we assume v 1 = 0 that the two platforms are symmetric. p 1 = p 1 D 1 (p 1 ,s 1 , p 2 ,s 2 )+ r(p 2 D 2 (p 1 ,s 1 , p 2 ,s 2 ) F 2 )+ F 2 as 1 (s 1 + s 2 ) p 2 =(1 r)(p 2 D 2 (p 1 ,s 1 , p 2 ,s 2 ) F 2 ) as 2 (s 1 + s 2 ) Corollary 16. If the two platforms are symmetric, the total investment level increases with the revenue sharing rate. Corollary 17. If the two platforms are symmetric, a sub-licensing contract will result in a higher total profit for the market than a non-exclusive contract. Combining Corollary 17 and Lemma 5, we can conclude that under the assumption of symme- try, the content provider will prefer a sub-licensing contract to a non-exclusive contract. Corollary 16 indicates that a higher revenue sharing rate will induce a higher total investment level, just as Proposition 13 exhibits. To further discuss the impact of contract format on social surplus, we did a simple simulation, and the analysis is put into the appendix. 3.7 Discussion This paper has discussed the effects and incentives of exclusive and sub-licensing contracts in the digital streaming music market. We present the idea that the pros and cons of the exclusive or sub-licensing contract to society should be evaluated more carefully by the government. We show that anti-piracy investments are usually insufficient. The exclusive or sub-licensing contract can alleviate the free-riding issue, inducing a higher investment level at the cost of higher subscrip- tion fees. We provide discussions about the splitting of surplus between the content provider and 87 streaming platforms and show that under certain conditions, sub-licensing or exclusive contracts are preferred by content providers. The music streaming market has other interesting features that are worth exploring. The com- petition between TME, NetEase, and Ali Music is probably multi-dimensional. In this paper, We mainly focus on the music market, whereas in practice, these tech giants compete in gaming, E- commerce, and many more fields. Competitive cross-subsidization may arise because these firms have different comparative advantages (Chen and Rey, 2013). Another interesting phenomenon is that, although TME decides to sub-license 99% of its content to NetEase, TME is criticized for keeping the 1% core music content to itself. 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Consider experts observe a noisy signal T = S+e, where e has cumulative distribution H() and probability density function h(). We then consider the case whene goes to 0. Suppose all bidders use cutoff strategies. Individual bidders’ maximum bidding function is b I (v i ) and experts’ maximum bidding function is b E (T). 1. Experts Suppose there is only one expert. His expected payoff, R E , from bidding up to maximum bid A is Z [a,b] n+1 1 fmaxfb I (v i )g<Ag (S maxfb I (v i )g) Õ i f(v i )g(S)h(T S) Õ i dv i dS (H.1) The first-order condition yields ¶R E ¶A = Pr(fmaxfb I (v i )g= Ag) Z b a (S A)g(S)h(T S)dS= 0 (H.2) Therefore, we must have b E (T)= A= E(SjT), which means an expert bids his best guess of the common value, given his signal. Note that if experts’ signals are noisy, E(SjT)= E(TejT)= 94 T E(ejT). Sincee and T are positively correlated, the expert’s bid will shrink towards the true common value S. Whene goes to zero, the experts have a precise signal, b E (T)= E(TejT)= T . Thus he will bid his signal. The logic is the same for multiple experts with precise signals. Experts will bid their signal, b E (T)= T . One could also understand this result from a dominant strategy point of view. It is a weakly dominant strategy for experts to stay up to their signals. 2. Individual bidders Now suppose individual bidders use a cutoff strategy. That is to say, individual bidder i will submit a bid until B(v i ): The optimal cutoff strategy B(v i ) is obtained by maximizing her expected payoff, assuming that all individual bidders follow the same cutoff maximum bidding fuction. Her payoff from the cutoff strategy is: 22 R i = Z [a,b] n [¥,¥] 1 fB>maxb I (v i ),B>b E (S+e)g (rv i +(1r)S max(b(v i ),b E (S+e))) Õ j6=i f(v j )g(s)h(e) Õ j6=i dv j dsde Since we are in an open auction, the maximum bid B only affects bidder i’s winning probability, and should not affect her payment. Thus her payment is always max(b(v i ),b E (S+e)). Hence when we take the first-order condition, we only need to differentiate the indicator function. When taking first-order condition with respect to B, we are considering two margins where B= maxb I (v i ) or B= b E (S+e). This allows us to replace max(b(v i ),b E (S+e)) by B. Let p(v i ,S,B)=rv i +(1r)S B be the bidder’s payoff from winning. The first order condition is: 22 It is possible that when an individual bidder observes a current standing bid max(b(v i ),b E (S+e)), he will revise his belief about the common value to be higher than the lower bounda. Thus, we may have a different lower bound for the integral of S. For tractability, we ignore the belief updating of the lower bound at this moment and will leave this issue for our future study. 95 ¶R i ¶B = Z [a,b][¥,¥] 1 fB>b E (S+e)g p(v i ,S,B)g(S)h(e) (n 1) f(b 1 I (b))F n2 (b 1 I (b))dsde + Z [a,b] n 1 fB>maxb I (v i )g p(v i ,S,B)g(S)h(b 1 E (B) S) Õ j6=i f(v j ) Õ j6=i dv j ds Now let e go to zero. When experts’ signals are precise, then b E (S+e) = S. Hence we can simplify the first order condition to ¶R i ¶B =(n 1) f(b 1 I (B))F n2 (b 1 I (B)) Z B a (rv i +(1r)S B)g(S)dS (H.3) + F n1 (b 1 I (B))r(v i B)g(B)= 0 Because we condition on B= S for the second line, the payoffp(v i ,S,B)=rv i +(1r)S B= r(v i B), which does not depend on v j or S, for all j6= i. Since b I (v i )= B, we could replace b 1 I (B) by v i . Lastly, we divide both sides by(n 1) f(b 1 I (B))F n2 (b 1 I (B))G(B). We have Z B a [rv i +(1r)S B] g(S) G(B) dS+ F(v i ) (n 1) f(v i ) g(B) G(B) r(v i B)= 0 Rearrange the equation. The cutoff maximum bid B must satisfy: B=rv i +(1r)E(SjS< B)+ F(v i ) (n 1) f(v i ) g(B) G(B) r(v i B) (H.4) where b I (v i )= B Note that whenr = 1, the second term vanishes, and one could verify that B= v i is the unique solution. So we are back to the IPV case where bidders bid their private values. 96 H.2 Corollary 1 . Proof. Denote b I (v i ) as B. Rearranging the equation, we have rv i +(1r)E(SjS< B)+ F(v i ) (n 1) f(v i ) g(B) G(B) r(v i B) B= 0 (H.5) The left-hand side is actually the first-order condition of R i with respect to B. Because F() is log-concave, we know the left-hand side is increasing with v i . We then verify that the left-hand side is decreasing with B. First, note that the third term F(v i ) (n1) f(v i ) g(B) G(B) r(v i B) is decreasing with B, because v i B> 0 and g(B)/G(B) is decreasing due to log-concavity. It suffices to show(1r)E(SjS< B) B is decreasing in B. First, we do integral by parts: (1r)E(SjS< B) B =(1r) Z B a Sg(S) G(B) dS B (H.6) =rB(1r) Z B a G(S) G(B) dS (H.7) Then we differentiate the expression with respect to B: FOC[B]:1+(1r) g(B) R B a G(S)dS G 2 (b) Let H(B)= R B a G(S)dS. Since G() is log-concave, H() is also log-concave. The log-concavity of H() suggests that ( H h ) 0 > 0, which is equivalent to Hh 0 h 2 < 1. Expressing H with G, we have 97 g(B) R B a G(S)dS G 2 (b) < 1. Therefore, we have FOC[B] <r < 0. The left-hand side of the original equation is decreasing in B. By the implicit function theorem, B is strictly increasing in v i . H.3 Proof of Corollary 2. Proof. Use proof by contradiction. Denote b I (v i ) as B. Suppose we have B v i . Then we have B<rv i +(1r)E(SjS< B)<rv i +(1r)B (H.8) which is equivalent to B< v i , a contradiction. H.4 Proof of Corollary 3. Proof. Assume r = 0. We again prove by contradiction. Individual bidders’ bidding function b I (v i ) must satisfy b I (v i )= Z b I (v i ) a Sg(S) G(b I (v i )) dS (H.9) Suppose b I (v i )>a b I (v i )< Z b I (v i ) a b I (v i )g(S) G(b I (v i )) dS (H.10) = b I (v i )G(b I (v i ))/G(b I (v i )) (H.11) = b I (v i ) (H.12) 98 which is a contradiction. In conclusion, we have b I (v i )=a. The case for r = 1 is discussed in the main text. H.5 Proof of Corollary 4. Proof. The equation to solve B now becomes B=rv i +(1r) B 2 + rv 2 i (n 1)gB rv i (n 1)g (H.13) Multiply both sides by B v 2 i . Denote K= B v i and rearrange the equation: 1+r 2 K 2 r(1 1 (n 1)g )K r (n 1)g (H.14) Solving this quadratic function, we have b I (v i )= K(r,g)v i , where K(r,g)= r 1+r (1 1 (n1)g )+ r 1+r q (1 1 (n1)g ) 2 + 1+r r 2 (n1)g . 23 One can easily verify that K(r,g) is increasing in r and decreasing ing. H.6 Proof of Proposition 2. Proof. When an individual bidder wins the auction, her expected payment is 23 Another root, K = r 1+r (1 1 (n1)g ) r 1+r q (1 1 (n1)g ) 2 + 1+r r 2 (n1)g , is omitted because K has to be non- negative. 99 Z v i a bd ˆ F n1 (b)G(b) (H.15) Integrating the expected payment over the support, we have R 1 = Z ˆ b ˆ a Z v I ˆ a bd ˆ F n1 (b)G(b)d ˆ F(v I ) (H.16) Using integral by parts, we have R 1 = Z ˆ b ˆ a bd ˆ F n1 (b)G(b) Z ˆ b ˆ a ˆ F(b)d ˆ F n1 (b)G(b) (H.17) = Z ˆ b ˆ a b(1 ˆ F(b))d ˆ F n1 (b)G(b) (H.18) We perform integration by parts again: R 1 = Z ˆ b ˆ a ˆ F n1 (b)G(b)(1 ˆ F(b) b f(b))db (H.19) = Z ˆ b ˆ a ˆ F n1 (v)G(v) ˆ f(v)(v 1 ˆ F(v) ˆ f(v) )dv (H.20) In the last step, we replace b with v. Similarly, for the expert, his expected payment is Z v E a bd ˆ F n (b) (H.21) Integrate the expected payment over the support[a,b]: 100 R 2 = Z b a bd ˆ F(b)d f(v E ) (H.22) Doing the same integration by parts twice, as we did for individual bidders, one can get R 2 = Z b a ˆ F n (v)g(v)(v 1 G(v) g(v) )dv (H.23) Lastly, the total revenue of the auction is the sum of n individual bidders and 1 expert: R E = n Z ˆ b ˆ a ˆ F n1 (v)G(v) ˆ f(v)(v 1 ˆ F(v) ˆ f(v) )dv+ Z b a ˆ F n (v)g(v)(v 1 G(v) g(v) )dv (H.24) H.7 Proof of Proposition 3. Proof. The proof is similar to the proof of Proposition 2. Even though there are many experts, when calculating the revenue, we could think of the auction as having only one expert who al- ways bids his value S. The only difference to the proof of Proposition 2 is that when an expert wins the auction, his expected payment is R v E a v E d ˆ F n (b) instead of R v E a bd ˆ F n (b), due to Bertrand competition. 101 Table 3.1: log(Final Bids) (1) (2) (3) (4) % of Experts -0.379 -0.372 -0.817 -0.841 (0.0349) (0.0351) (0.0692) (0.0643) Total Bidder # 0.000481 0.000331 0.0000558 0.000271 (0.000242) (0.000251) (0.000260) (0.000279) Mileage -0.00713 -0.00645 -0.00581 -0.00566 (0.00149) (0.00148) (0.00149) (0.00149) Year -0.208 -0.210 -0.208 -0.206 (0.00359) (0.00361) (0.00361) (0.00360) Displacement 0.632 0.626 0.631 0.630 (0.0217) (0.0213) (0.0213) (0.0216) Length of Description 0.0000947 0.000140 0.000120 (0.0000451) (0.0000473) (0.0000473) Apparent Repair -0.0628 -0.0982 -0.0744 (0.0286) (0.0300) (0.0301) No Collision 0.0533 0.0168 0.0180 (0.0362) (0.0364) (0.0376) Engine issue -0.0665 -0.0636 -0.0731 (0.0223) (0.0228) (0.0232) Scratched -0.0700 -0.0593 -0.0645 (0.0193) (0.0197) (0.0200) Observations 24519 24519 24519 23940 R 2 0.735 0.737 0.729 0.728 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of individual bidders’ final bids. c. (3)’s instrument: percentage of experts in other auctions on the same date. d. (4)’s instrument: percentage of experts in the auctions held on the closest date. e. Bidders who deposit more money for 3 cars are considered experts. f. Fixed effects include car model fixed effect and month fixed effect. 102 Table 3.2: log(Auction Revenue) (1) (2) (3) (4) % of Experts -0.347 -0.345 -0.754 -0.672 (0.0280) (0.0281) (0.0545) (0.0722) Total Bidder # 0.00418 0.00424 0.00385 0.00456 (0.000255) (0.000273) (0.000280) (0.000341) Mileage -0.00609 -0.00587 -0.00514 -0.00515 (0.00140) (0.00140) (0.00143) (0.00145) Year -0.212 -0.213 -0.207 -0.210 (0.00327) (0.00327) (0.00336) (0.00357) Displacement 0.674 0.666 0.657 0.659 (0.0224) (0.0223) (0.0225) (0.0226) Text: Apparent Repair -0.131 -0.167 -0.150 (0.0290) (0.0308) (0.0319) Length of Description 0.000185 0.000162 0.000181 (0.0000404) (0.0000424) (0.0000421) Text: No Collision 0.0662 -0.00756 0.0298 (0.0483) (0.0493) (0.0508) Text: Engine Issue -0.0243 -0.0468 -0.0426 (0.0200) (0.0212) (0.0212) Text: Scratched -0.0326 -0.0305 -0.0341 (0.0169) (0.0172) (0.0173) Observations 4473 4473 4472 4386 R 2 0.821 0.823 0.811 0.815 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of auction price. c. (3)’s instrument: percentage of experts in other auctions on the same date. d. (4)’s instrument: percentage of experts in the auctions held on last date. e. Fixed effects include car model fixed effect and month fixed effect. 103 Table 3.3: log(Final Bids): Potential Bidders (1) (2) (3) (4) Potential % of Experts -0.581 -0.611 -0.838 -1.001 (0.0522) (0.0541) (0.0695) (0.0740) Total Bidder # 0.000291 0.000144 -0.0000108 0.000129 (0.000239) (0.000247) (0.000249) (0.000266) Mileage -0.00747 -0.00680 -0.00673 -0.00657 (0.00150) (0.00149) (0.00149) (0.00149) Year -0.209 -0.211 -0.211 -0.209 (0.00355) (0.00356) (0.00353) (0.00350) Displacement 0.632 0.627 0.629 0.630 (0.0217) (0.0213) (0.0213) (0.0215) Length of Description 0.000167 0.000208 0.000217 (0.0000467) (0.0000479) (0.0000489) Apparent Repair -0.0784 -0.0952 -0.0801 (0.0289) (0.0293) (0.0293) No Collision 0.0357 0.0179 0.0107 (0.0372) (0.0374) (0.0385) Engine issue -0.0920 -0.101 -0.118 (0.0222) (0.0223) (0.0227) Scratched -0.0598 -0.0527 -0.0535 (0.0191) (0.0191) (0.0195) Observations 24519 24519 24519 23940 R 2 0.735 0.737 0.737 0.735 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of individual bidders’ final bids. c. (3)’s instrument: percentage of experts in other auctions on the same date. d. (4)’s instrument: percentage of experts in the auctions held on the closest date. e. Bidders who deposit more money for 3 cars are considered experts. f. Fixed effects include car model fixed effect and month fixed effect. 104 Table 3.4: log(Final Bids): Compare Experts and Individuals (1) (2) (3) (4) Individuals Experts Individuals Experts % of Experts -0.372 -0.254 -0.841 -0.626 (0.0351) (0.0344) (0.0643) (0.0676) Length of Description 0.0000947 0.000290 0.000120 0.000275 (0.0000451) (0.0000453) (0.0000473) (0.0000465) Apparent Repair -0.0628 -0.0227 -0.0744 -0.0437 (0.0286) (0.0386) (0.0301) (0.0408) No Collision 0.0533 0.0113 0.0180 -0.0422 (0.0362) (0.0657) (0.0376) (0.0678) Engine issue -0.0665 -0.0306 -0.0731 -0.0577 (0.0223) (0.0231) (0.0232) (0.0237) Scratched -0.0700 -0.0121 -0.0645 -0.0168 (0.0193) (0.0181) (0.0200) (0.0185) Total Bidder # 0.000331 0.000706 0.000271 0.000273 (0.000251) (0.000286) (0.000279) (0.000319) Mileage -0.00645 -0.00211 -0.00566 -0.00160 (0.00148) (0.00178) (0.00149) (0.00183) Year -0.210 -0.230 -0.206 -0.225 (0.00361) (0.00365) (0.00360) (0.00370) Displacement 0.626 0.659 0.630 0.648 (0.0213) (0.0239) (0.0216) (0.0243) Observations 24519 22141 23940 22005 R 2 0.737 0.754 0.728 0.747 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of the final bids. c. (3)’s instrument: percentage of experts in other auctions on the same date. d. (4)’s instrument: percentage of experts in the auctions held on the closest date. e. Bidders who deposit more money for 3 cars are considered experts. f. Fixed effects include car model fixed effect and month fixed effect. 105 Table 3.5: log(Final Bids) with a new control: Department With Power (1) (2) (3) (4) % of Dealer -0.31 -0.31 -0.75 -0.74 (0.036) (0.036) (0.076) (0.068) Total Bidder # 0.00026 0.00017 -0.00013 0.000077 (0.00024) (0.00025) (0.00026) (0.00027) Mileage -0.0076 -0.0070 -0.0063 -0.0061 (0.0015) (0.0015) (0.0015) (0.0015) Year -0.21 -0.21 -0.21 -0.21 (0.0036) (0.0036) (0.0036) (0.0036) Displacement 0.63 0.62 0.63 0.63 (0.022) (0.021) (0.022) (0.022) Department With Power 0.14 0.13 0.076 0.079 (0.019) (0.020) (0.021) (0.021) Length of Description 0.000091 0.00012 0.00010 (0.000045) (0.000047) (0.000047) Text: Apparent Repair -0.068 -0.098 -0.071 (0.028) (0.029) (0.029) Text: No Collision 0.075 0.032 0.033 (0.037) (0.037) (0.038) Text: Engine Issue -0.058 -0.060 -0.070 (0.022) (0.022) (0.023) Text: Scratched -0.049 -0.051 -0.056 (0.020) (0.021) (0.021) Observations 24523 24523 24523 23946 R 2 0.74 0.74 0.73 0.73 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of individual bidders’ final bids. c. (3)’s instrument: percentage of dealers in other auctions on the same date. d. (4)’s instrument: percentage of dealers in the auctions held on last date. e. Fixed effects include car model fixed effect and month fixed effect. 106 Table 3.6: log(Final Bids): Dealer Deposits 5 Cars (1) (2) (3) (4) % of Dealer -0.431 -0.424 -1.392 -1.217 (0.0394) (0.0394) (0.107) (0.0825) Total Bidder # 0.000567 0.000471 -0.0000731 0.000288 (0.000235) (0.000248) (0.000267) (0.000279) Mileage -0.00598 -0.00524 -0.00447 -0.00433 (0.00146) (0.00146) (0.00149) (0.00149) Year -0.210 -0.213 -0.206 -0.206 (0.00355) (0.00356) (0.00361) (0.00361) Displacement 0.633 0.628 0.615 0.617 (0.0212) (0.0207) (0.0209) (0.0210) Length of Description 0.000114 0.000113 0.0000980 (0.0000432) (0.0000466) (0.0000456) Apparent Repair -0.0514 -0.111 -0.0793 (0.0286) (0.0312) (0.0305) No Collision 0.0490 -0.0249 -0.00608 (0.0384) (0.0390) (0.0399) Engine Issue -0.0564 -0.0676 -0.0737 (0.0214) (0.0229) (0.0227) Scratched -0.0670 -0.0607 -0.0681 (0.0187) (0.0196) (0.0196) Observations 32744 32744 32739 32095 R 2 0.740 0.742 0.717 0.725 Fixed effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of individual bidders’ final bids. c. (3)’s instrument: percentage of dealers in other auctions on the same date. d. (4)’s instrument: percentage of dealers in the auctions held on the closest date. e. Bidders who deposit more money for 5 cars are considered dealers. f. Fixed effects include car model fixed effect and month fixed effect. 107 Table 3.7: log(Final Bids): Dealer Deposits 4 Cars (1) (2) (3) (4) % of Dealer -0.387 -0.378 -1.027 -0.968 (0.0362) (0.0362) (0.0872) (0.0712) Total Bidder # 0.000434 0.000337 -0.000145 0.000115 (0.000233) (0.000245) (0.000257) (0.000267) Mileage -0.00646 -0.00577 -0.00510 -0.00504 (0.00150) (0.00149) (0.00151) (0.00152) Year -0.210 -0.213 -0.210 -0.208 (0.00356) (0.00357) (0.00357) (0.00357) Displacement 0.634 0.629 0.624 0.625 (0.0215) (0.0211) (0.0214) (0.0215) Length of Description 0.000120 0.000154 0.000132 (0.0000440) (0.0000460) (0.0000459) Apparent Repair -0.0490 -0.0909 -0.0626 (0.0285) (0.0301) (0.0301) No Collision 0.0420 -0.0130 -0.00509 (0.0381) (0.0388) (0.0398) Engine issue -0.0520 -0.0484 -0.0567 (0.0219) (0.0227) (0.0229) Scratched -0.0703 -0.0590 -0.0655 (0.0191) (0.0195) (0.0197) Observations 29340 29340 29335 28710 R 2 0.736 0.737 0.723 0.726 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of individual bidders’ final bids. c. (3)’s instrument: percentage of dealers in other auctions on the same date. d. (4)’s instrument: percentage of dealers in the auctions held on the closest date. e. Bidders who deposit more money for 4 cars are considered dealers. f. Fixed effects include car model fixed effect and month fixed effect. 108 Table 3.8: log(Final Bids): Dealer Deposits 2 Cars (1) (2) (3) (4) % of Dealer -0.342 -0.339 -0.683 -0.820 (0.0385) (0.0390) (0.0795) (0.0712) Total Bidder # 0.000262 0.0000743 -0.000164 -0.0000388 (0.000246) (0.000253) (0.000257) (0.000270) Mileage -0.00652 -0.00573 -0.00554 -0.00548 (0.00154) (0.00153) (0.00156) (0.00156) Year -0.207 -0.210 -0.209 -0.207 (0.00392) (0.00391) (0.00400) (0.00395) Displacement 0.635 0.628 0.644 0.651 (0.0228) (0.0223) (0.0229) (0.0233) Length of Description 0.0000994 0.000170 0.000178 (0.0000525) (0.0000554) (0.0000564) Apparent Repair -0.0720 -0.0962 -0.0812 (0.0302) (0.0314) (0.0321) No Collision 0.0564 0.0376 0.0331 (0.0365) (0.0370) (0.0383) Engine issue -0.0661 -0.0582 -0.0669 (0.0257) (0.0263) (0.0270) Scratched -0.0809 -0.0749 -0.0773 (0.0208) (0.0212) (0.0218) Observations 16556 16556 16556 16096 R 2 0.730 0.732 0.727 0.723 Fixed Effect Yes Yes Yes Yes Regression OLS OLS 2SLS 2SLS a. Standard errors are clustered by auction. b. Dependent variable: logarithm of individual bidders’ final bids. c. (3)’s instrument: percentage of dealers in other auctions on the same date. d. (4)’s instrument: percentage of dealers in the auctions held on the closest date. e. Bidders who deposit more money for 2 cars are considered dealers. f. Fixed effects include car model fixed effect and month fixed effect. 109 I Appendix of Chapter 2 I.1 Proof of Proposition 4 Proof. Consider the choice of user i. Denote the sum of other users’ tips as X i . If X i Z, then there will always be sufficient funds for the lottery no matter what x i is. If instead X i < Z, since choosing x i = ZX i would give user i zero payoff, and the right derivative evaluated at this point is ¶A ¶x + i x i =ZX i =1+ X i Z +ah 0 (0)> 0 (I.1) user i would always prefer x i > Z X i . Either way, user i would choose an x i such that the lottery expense can be covered by total tips, and thus the utility to be maximized would be x i + x i x i + X i Z+ h(a(x i + X i Z)) (I.2) The optimal x i can be determined by FOC 1+ X i (x i + X i ) 2 Z+ah 0 (a(x i + X i Z))= 0 due to that SOC 2 X i (x i + X i ) 3 Z+a 2 h 00 (a(x i + X i Z))< 0 always holds. 110 Denote T =å n i=1 x i Z as the excess of total tips over the prize Z. Summing over the FOC for all n users, we know that in equilibrium: n+ n 1 T+ Z Z+ nah 0 (aT)= 0 (I.3) or equivalently (1 1 n ) Z T+ Z = 1ah 0 (aT) (I.4) Note that LHS is decreasing in T , while RHS is strictly increasing in T since h 00 ()< 0. There- fore, for any n, there exists an unique and finite T(n) > 0 satisfying the equilibrium condition if both 1 1 n > 1ah 0 (0)( h 0 (0)> 3 2a 1 an and lim T!¥ [1ah 0 (aT)]> 0, lim Y!¥ h 0 (Y)< 1 a hold. Given T(n), user i’s optimal tips would be determined by 1+ T(n)+ Z x i (T(n)+ Z) 2 Z+ah 0 (aT(n))= 0, x i = T(n)+ Z n i.e., the equilibrium is symmetric. I.2 Proof of Corollary 6 Proof. (i) In the proof of 4, we have shown that T(n) is positive and finite in equilibrium. There- fore, R(n)=(1a)T(n) is also positive and finite. 111 To show that R(n) increases in n, we only need to show that T(n) increases in n. Rewrite the equilibrium condition as (1 1 n )Z=[1ah 0 (aT)](T+ Z) and hence T 0 (n)= 1 n 2 Z 1ah 0 (aT(n))a 2 h 00 (aT(n))(T(n)+ Z) > 0. (ii) As we have shown, user i can always guarantee a positive payoff by choosing x i = 0 when X i > Z and x i > Z X i when X i Z. Therefore, in equilibrium, each user’s payoff A(n)= 1 n T(n)+ h(aT(n)) must be positive. Taking the derivative of A(n) with respect to n A 0 (n)= 1 n 2 T(n)+[ 1 n +ah 0 (aT(n))]T 0 (n) Rewrite the equilibrium condition as [ 1 n +ah 0 (aT)]Z=[1ah 0 (aT)]T we shall see that 1 n +ah 0 (aT(n))=[1ah 0 (aT(n))] T(n) Z =(1 1 n ) T(n) T(n)+ Z 0 which implies A 0 (n)> 0. Proof of Corollary 7 112 Proof. To show that R(n) is strictly concave, we only need to show that T(n) is strictly concave. Following the equilibrium condition (1 1 n )Z=[1ah 0 (aT)](T+ Z) since T + Z is strictly increasing and convex in T , and 1ah 0 (aT) is strictly increasing and convex in T if h 0 () is concave, the RHS is strictly increasing and strictly convex in T . Moreover, the LHS is strictly increasing and strictly concave in n, which implies that T(n) and thus R(n) must be strictly concave. Now let’s turn to the effective price R(n) n . Taking derivatives ( R(n) n ) 0 = nR 0 (n) R(n) n 2 we shall see that the effective price increases in n if and only if x(n) D = nR 0 (n) R(n) 0 When R(n) is concave, we know that x 0 (n)= R 0 (n)+ nR 00 (n) R 0 (n)= nR 00 (n)< 0 Ifx(1)< 0, thenx(n)< 0 for all n 1, in which case we take ˆ n= 1. Ifx(1) 0, we need to examine if lim n!¥ x(n)< 0. To show this, recall first that T(n) increases in n. Moreover, let T solve Z T+ Z = 1ah 0 (aT) 113 and thus T(n) T for any n, which implies that lim n!¥ T 0 (n)= 0= lim n!¥ R 0 (n) Now, decomposex(n) as x(n)= nR 0 (n) R(n)=(n 1)R 0 (n)[R(n) R(1)]+ R 0 (n) R(1) = Z n 1 [R 0 (n) R 0 (t)]dt+ R 0 (n) R(1) The integral term is always strictly negative due to that R(n) is strictly concave, and we know that R(1) is positive. Since R 0 (n) approaches 0 as n!¥, it must be that lim n!¥ x(n)< 0 and thus there exists a unique ˆ n 1 such thatx(ˆ n)= 0 as we desire. I.3 Proof of Corollary 8 Proof. Recall that A 0 (n)= 1 n 2 T(n)+[ 1 n +ah 0 (aT(n))]T 0 (n) = 1 n 2 T(n)+[1ah 0 (aT(n))] T(n) Z 1 n 2 Z 1ah 0 (aT(n))a 2 h 00 (aT(n))(T(n)+ Z) = 1 n 2 T(n)[1+ 1ah 0 (aT(n)) 1ah 0 (aT(n))a 2 h 00 (aT(n))(T(n)+ Z) ] 2 n 2 T(n) 114 We show next that T(n) n 2 decreases in n, which is equivalent to p T(n) n decreases in n. Note that our analysis on T(n) only applies for n 1, and thus we need to extend p T(n) to the interval[0,1] by connecting (0,0) and (1, p T(1)). A sufficient condition is thus the concatenated function is overall strictly concave for n 0. When h 0 () is concave, we have shown that T(n) is strictly concave for n 1, and so is p T(n). We are left to show that concavity is not violated by the kink at n= 1: q T(1) ¶ ¶n q T(n)j n=1 = T 0 (1) 2 p T(1) , 2T(1) T 0 (1) Since T(1) is determined by 1ah 0 (aT(1))= 0 we know that T 0 (1)= Z a 2 h 00 (aT(1))(T(1)+ Z) Letf(T)= 1ah 0 (aT), and f 0 (T)=a 2 h 00 (aT)> 0,f 00 (T)=a 3 h 000 (aT) 0 It follows that f(T(1))f(0)= Z T(1) 0 f 0 (t)dt Z T(1) 0 f 0 (T(1))dt=f 0 (T(1))T(1) ,ah 0 (0) 1a 2 h 00 (aT(1))T(1) and thus 2T(1) T 0 (1), Z2a 2 h 00 (aT(1))T(1)(T(1)+ Z) ( Z 2(ah 0 (0) 1)(T(1)+ Z) ,(3 2ah 0 (0))Z 2(ah 0 (0) 1)T(1) 115 Since h 0 (0)> 3 2a , LHS is negative while RHS is positive, and the inequality must hold. To sum up, T(n) n 2 decreases in n, and thus A 0 (n)< 2t if T(1)<t, or equivalently 1ah 0 (at)> 0, h 0 (at)< 1 a . I.4 Proof of Proposition 5 Proof. Denote g i (n)= N exp( B(q i )+A(n i ) b ) å K j=1 exp( B(q j )+A(n j ) b ) , i= 1,2,:::,K and g(n) the vector function by stacking g i (n). Hence any participation equilibrium corresponds to a fixed point of the equation n= g(n) Since g is a continuous function mapping from[1,N] K to itself, the existence of a fixed point and thus a participation equilibrium is guaranteed by Brouwer’s fixed point theorem. As for uniqueness, consider the L ¥ norm:jjxjj ¥ = max i2N jx ki j. The following lemma from Tan and Zhou (2020) would be used in the proof later. Lemma 7. For any continuously differentiable function f(x 1 ,:::,x m ) on a convex domainQ R m , and for any x=(x 1 ,:::,x m ) and y=(y 1 ,:::,y m ), the following holds jf(x)f(y)j(max i jx i y i j)(sup z2Q m å j=1 j ¶f(z) ¶x j j) 116 Applying the lemma and we have jA(n i ) A(n 0 i )jjn i n 0 i j(sup n jA 0 (n)j) ) max i jA(n i ) A i (n 0 i )j max i jn i n 0 i j(sup n jA 0 (n)j) and jg i (n) g i (n 0 )j 1 b (max j jA(n j ) A(n 0 j )j)( sup a2R K N e a i (2å j6=i e a j ) (å K j=1 e a j ) 2 ) Denote B e = N b sup a2R K e a i (2å j6=i e a j ) (å K j=1 e a j ) 2 B f = sup n jA 0 (n)j and thus jjg(n) g(n 0 )jj ¥ = max i jg i (n) g i (n 0 )j B e (max j jA(n j ) A(n 0 j )j) B e B f max j jn j n 0 j j= B e B f jjn n 0 jj ¥ If B e B f < 1, by contraction mapping theorem, there exists a unique fixed point for g, i.e., the participation equilibrium is unique. We now turn to determining the values of B e . Denote m = e a i n = å j6=i e a j and thus B e = N b sup (m,n)2R 2 + 2mn (m+n) 2 = N 2b 117 Therefore, the condition for contraction mapping is reduced to B f = sup n jA 0 (n)j< 1 B e = 2b N or equivalently, for any n, A 0 (n)< 2b N . The second part of the proposition is obtained by applying Corollary 8 I.5 Proof of Corollary 9 Proof. When the quality profile is symmetric, we can immediately check that a symmetric partici- pation equilibrium where n i = N K is indeed a fixed point for the equation n= g(n), and thus it must be the unique participation equilibrium. When the quality profile is semi-symmetric, without loss of generality, suppose platform 1 chooses a quality level q 1 while all other platforms choose the same quality level q. It’s sufficient to show that, for n 1 satisfying lnn 1 ln(N n 1 )+ ln(K 1)= 1 b [B(q 1 )+ A(n 1 ) B(q) A( N n 1 K 1 )] the semi-symmetric participation profile(n 1 , Nn 1 K1 ,:::, Nn 1 K1 ) is indeed an equilibrium, which fur- ther reduces to showing the existence and uniqueness of such n 1 . Rearrange the terms such that n 1 only appears on LHS: b lnn 1 b ln(N n 1 )+b ln(K 1) A(n 1 )+ A( N n 1 K 1 )= B(q 1 ) B(q) 118 The derivative of LHS with respect to n 1 would be b( 1 n 1 + 1 N n 1 ) A 0 (n 1 ) 1 K 1 A 0 ( N n 1 K 1 )> 4b N K K 1 2b N 0 The monotonicity, along with the fact that LHS approaches negative (positive) infinity as n 1 goes to 0(N), guarantees the existence and uniqueness of n 1 and thus the corollary. I.6 Proof of Proposition 6 Proof. Following the setting in the proof of 9, it’s sufficient to show that platform 1’s profit function is quasi-concave in its quality choice q 1 , holding fixed the other platform’s quality choice q. The FOC is given by R 0 (n 1 ) ¶n 1 ¶q 1 c= 0 and from the proof of 9 ¶n 1 ¶q 1 = B 0 (q 1 ) b( 1 n 1 + 1 Nn 1 ) A 0 (n 1 ) 1 K1 A 0 ( Nn 1 K1 ) > 0 Plug in the expression and divide by B 0 (q 1 ) > 0, a sufficient condition for the quasi-concavity would thus be ¶ ¶n 1 R 0 (n 1 ) b( 1 n 1 + 1 Nn 1 ) A 0 (n 1 ) 1 K1 A 0 ( Nn 1 K1 ) ! ¶n 1 ¶q 1 ¶ ¶q 1 c B 0 (q) < 0 which holds under our assumptions. To derive the equilibrium quality investment level, let q 1 = q= q e ,n 1 = N n 1 K 1 = N K = n e 119 in the FOC: R 0 (n e ) B 0 (q e ) b K K1 1 n e K K1 A 0 (n e ) c= 0 , B 0 (q e )= cK K 1 b n e A 0 (n e ) n e R 0 (n e ) The existence and uniqueness of such q e are guaranteed by that B(q) is strictly concave and follows the Inada conditions. I.7 Proof of Corollary 10 Proof. (i) is straight forward from equation 2.8. (ii) The right-hand side can be written as cK K1 b nR‘(n) cK K1 A 0 (n) R 0 (n) , which is decreasing with n under our assumption. By concavity of B(q), we have q increases with n. Proof. When N goes to infinity, the LHS of inequality 2.13 also goes to infinity since its denomi- nator goes to 0, while its numerator is positive and finite. Whenb goes to infinity, the LHS of inequality 2.13 goes to infinity since its numerator goes to infinity. In fact we can derive a sufficient condition asb > (K1)(1a) (K2)a (h 0 ) 1 ( 1 a ). I.8 Proof of Lemma 1 Proof. (i):A(n)= 1 n R ¯ v v 1F(v) f(v) dF n (v)= R ¯ v v F n1 (v)dv R ¯ v v F n (v)dv, and we use integral by parts to get A(n)=d(n)d(n1)= d(n)d(n1) n(n1) =d 0 (n). The last part is a first-order approximation. 120 (ii): A 0 (n) = d 00 (n). So A 0 (n) < 0 due to the concavity of the expectation of largest order statistics. (iii): Directly from equation (2.15). (iv)) Differentiate R(n)=d(n) nA(n) with respect to n, and plug ind 0 (n)= A(n). I.9 Proof of Proposition 10 Proof. According to equation (2.20), we have ¶R 1 (q 1 ,q 2 ) ¶q 1 q ¶R 1 (q 1 ,q 2 ) ¶q 1 q > 0 ¶R 1 ¶q 1 (q 1 ,q 2 ) ¶R 1 ¶q 1 (q 1 ,q 2 )+ ¶R 1 ¶q 1 (q 1 ,q 2 ) ¶R 1 ¶q 1 (q 1 ,q 2 )> 0 ¶ 2 R 1 ¶q 2 1 ( ¯ q 1 , ¯ q 2 )(q 1 q 1 )+ ¶ 2 R 1 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 )(q 2 q 2 )> 0 (I.5) Similarly, for platform 2, we must have ¶ 2 R 2 ¶q 2 2 ( ¯ q 1 , ¯ q 2 )(q 2 q 2 )+ ¶ 2 R 2 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 )(q 1 q 1 )> 0 (I.6) Note that ¶R 1 ¶q 1 (q 1 ,q 2 ) and ¶ 2 R 2 ¶q 2 2 ( ¯ q 1 , ¯ q 2 ) are negative, while ¶ 2 R 2 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 ) is assumed to be positive. For both inequalities to hold, we must have q 1 and q 2 move in the same direction (i.e., it is impossible to have one firm raising its quality while the other platform reducing its quality.) We prove that both platforms can’t raise their quality after the merger. We do a proof by contradiction. Suppose q 1 q 1 =Dq 1 > 0 and q 2 q 2 =Dq 2 > 0. 121 ¶ 2 R 1 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 )Dq 2 > ¶ 2 R 1 ¶q 2 1 ( ¯ q 1 , ¯ q 2 )Dq 1 (I.7) ¶ 2 R 2 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 )Dq 2 > ¶ 2 R 2 ¶q 2 2 ( ¯ q 1 , ¯ q 2 )Dq 1 (I.8) Deviding the two inequality, we have: ¶ 2 R 1 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 ) ¶ 2 R 2 ¶q 1 ¶q 2 ( ˜ q 1 , ˜ q 2 )> ¶ 2 R 1 ¶q 2 1 ( ¯ q 1 , ¯ q 2 ) ¶ 2 R 2 ¶q 2 2 ( ¯ q 1 , ¯ q 2 ) This contradicts Assumption 4(ii). Therefore, under Assumption 3(i), 4(i) and 4(ii), the equi- librium quality level is lower after the merger. J Appendices of Chapter 3 J.1 Proof of Lemma 3 We discuss the existence of equilibria. Our assumption 1 restricts that the demand functions are continuous. Assume price and investment is taken from compact intervals (for example, p2[0,m] and s2[0,m], where m is sufficiently large). By the Weierstrass theorem, the maximizer set is non- empty, and by Berge’s maximum theorem, the optimal prices and investments are continuous in the other parameters in the profit functions. We could also show that our profit function is concave and the best response is unique (thus convex valued): p 1 = p 1 D 1 (p 1 , p 2 ,s 1 + s 2 ) g(s 1 + s 2 ) Under assumption 1, we have ¶ 2 P 1 ¶ p 2 1 = 2 ¶D 1 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 2 1 < 0 and ¶ 2 P 1 ¶s 2 1 = p 1 ¶ 2 D 1 ¶s 2 1 g 00 (s 1 +s 2 )< 0. These conditions ensure that each best response function has a unique solution 24 . 24 Assume the Hessian matrix is negative semi-definite. 122 Since the best response function is non-empty, convex-valued, continuous (thus upper hemi- continuous), by Kakutani’s fixed point theorem, there exists a fixed point of the best response function, which constitutes an equilibrium of the model. J.2 Proof of Proposition 11 Proof. There are many moving parts. We prove the proposition step by step: Note that p 1 = p 1 D 1 (p 1 , p 2 ,s 1 + s 2 ) s 1 g(s 1 + s 2 ) p 2 = p 2 D 2 (p 1 , p 2 ,s 1 + s 2 ) s 2 g(s 1 + s 2 ) And if the two platforms merge, the profit function is: P=p 1 +p 2 = p 1 D 1 (p 1 , p 2 ,s 1 + s 2 )+ p 2 D 2 (p 1 , p 2 ,s 1 + s 2 )(s 1 + s 2 )g(s 1 + s 2 ) Suppose the profit functions are quasi-linear and we have the following first order conditions: ¶p i ¶s i = p i ¶D i ¶s i g s i ¶g ¶s i (J.1) ¶P ¶s i = p i ¶D i ¶s i g s i ¶g ¶s i + p j ¶D j ¶s i s j ¶g ¶s i (J.2) ¶p i ¶ p i = D i + p i ¶D i ¶ p i (J.3) ¶P ¶ p i = D i + p i ¶D i ¶ p i + p j ¶D j ¶ p i (J.4) 123 Step 1: First we show that, holding price fixed, the merger induces a higher total investment for the two platforms. We call the competition case as NE (Nash equilibrium) and the integrated case as I (Integration). Evaluated at the Nash equilibrium investment level: ¶p 1 ¶s 1 (s NE 1 ,s NE 2 )= ¶P ¶s 1 (s NE 1 + s NE 2 )(p 2 ¶D 2 ¶s 1 (s NE 1 + s NE 2 ) s 2 ¶g ¶s 1 (s NE 1 + s NE 2 )) = ¶P ¶s 1 (s NE 1 + s NE 2 ) g(s NE 1 + s NE 2 ) By(1) = 0 Since g(s NE 1 + s NE 2 )> 0, we have ¶P ¶s 1 (s I 1 + s I 2 )= 0< ¶P ¶s 1 (s NE 1 + s NE 2 ). By concavity ofP, we know s I 1 + s I 2 > s NE 1 + s NE 2 if price is fixed. Step 2: Then we show that, holding investment level fixed, the integration of the two platforms will induce higher prices. Similarly, evaluate at the Nash equilibrium level: ¶p 1 ¶ p 1 (p NE 1 , p NE 2 )= ¶P ¶ p 1 (p NE 1 , p NE 2 ) p 2 ¶D 2 ¶ p 1 (p NE 1 , p NE 2 )= 0 Since ¶D 2 ¶ p 1 (p NE 1 , p NE 2 )> 0, we have ¶P ¶ p 1 (p I 1 , p I 2 )= 0< ¶P ¶ p 1 (p NE 1 , p NE 2 ). By concavity ofP and symmetry, we know p I 1 > p NE 1 and p I 2 > p NE 2 if total investment is fixed. Step 3: In the first two steps, we showed that moving from competition to integration, total investments or prices tend to increase. In this step we discuss the complementarity between price and investment levels and conclude that the integrated platform will price higher and spend more on investment. First we discuss in an integrated platform, how does an increase in the prices affect total invest- ment. We do this by total differentiate first order condition ¶P ¶s = 0 with respect to p i , and solve for ¶s ¶ p i . The result is (take ¶s ¶ p 1 as an example): 124 ¶s ¶ p i = ¶D 1 ¶s p 1 ¶ 2 D 1 ¶s¶ p 1 p 2 ¶ 2 D 2 ¶s¶ p 1 p 1 ¶ 2 D 1 ¶s 2 + p 2 ¶ 2 D 2 ¶s 2 2g 0 (s) sg 00 (s) > 0 So with prices increasing, the integrated platform has more incentive to make investments. Step 4: Then, we analyze how does an increase in total investments affects pricing. To do this, we total differentiate the first-order conditions ¶P ¶ p 1 = 0, ¶P ¶ p 2 = 0 with respect to total investment level s. The result is slightly more complicated. We get the following system of equations: 2 6 4 2 ¶D 1 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 2 1 + p 2 ¶ 2 D 2 ¶ p 2 1 ¶D 1 ¶ p 2 + ¶D 2 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 1 ¶ p 2 + p 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 ¶D 1 ¶ p 2 + ¶D 2 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 1 ¶ p 2 + p 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 2 ¶D 2 ¶ p 2 + p 1 ¶ 2 D 1 ¶ p 2 2 + p 2 ¶ 2 D 2 ¶ p 2 2 3 7 5 2 6 4 ¶ p 1 ¶s ¶ p 2 ¶s 3 7 5 = 2 6 4 ¶D 1 ¶s p 1 ¶ 2 D 1 ¶s¶ p 1 p 2 ¶ 2 D 2 ¶ p 1 ¶s ¶D 2 ¶s p 1 ¶ 2 D 1 ¶s¶ p 2 p 2 ¶ 2 D 2 ¶ p 2 ¶s 3 7 5 < 2 6 4 0 0 3 7 5 (J.5) By applying Cramer’s rule, we know the necessary and sufficient condition for ¶ p 1 ¶s > 0 and ¶ p 2 ¶s > 0 is the determinant of the left giant matrix to be positive. The determinant is: (2 ¶D 1 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 2 1 + p 2 ¶ 2 D 2 ¶ p 2 1 )(2 ¶D 2 ¶ p 2 + p 1 ¶ 2 D 1 ¶ p 2 2 + p 2 ¶ 2 D 2 ¶ p 2 2 ) (J.6) ( ¶D 1 ¶ p 2 + ¶D 2 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 1 ¶ p 2 + p 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 ) 2 The first term is a product of two SOCs so the product is positive. equation (J.6) is positive if j ¶D i ¶ p i j>j ¶D i ¶ p j j,j ¶D i ¶ p i j>j ¶D j ¶ p i j andj ¶ 2 D i ¶ p 2 i j>j ¶ 2 D i ¶ p i ¶ p j j8i, j=f1,2g . Special cases: Linear and Logit demand system We can show that these conditions are valid under classic linear demand or the Logit demand system. The case for linear demand is simple since the cross-derivative for linear demand is 0. We provide a simple analysis of Logit demand. 125 Consider a Logit demand function D i = e v i p i e v 0 s +e v 1 p 1+e v 2 p 2 With a little bit algebra, we havej ¶D i ¶ p i jj ¶D i ¶ p j j =j ¶D i ¶ p i jj ¶D j ¶ p i j = (1 D i D j )D i > 0 and j ¶ 2 D i ¶ p 2 i jj ¶ 2 D i ¶D j ¶D i j= D i (1 D i D j )j2D i 1j> 0. So the two assumptions are satisfied. J.3 Proof of proposition 12 and Corollary 12 Taking the first order conditions with respect to s 1 and s 2 and total differentiate both sides with r, we have 2 6 4 p 1 ¶ 2 D 1 ¶s 2 1 + rp 2 ¶ 2 D 2 ¶s 2 1 2g 0 (s) s 1 g 00 (s) p 1 ¶ 2 D 1 ¶s 1 ¶s 2 + rp 2 ¶ 2 D 2 ¶s 1 ¶s 2 g 0 (s) s 1 g 00 (s) (1 r)p 2 ¶ 2 D 2 ¶s 1 ¶s 2 g 0 (s) s 2 g 00 (s) (1 r)p 2 ¶ 2 D 2 ¶s 2 2 2g 0 (s) s 2 g 00 (s) 3 7 5 2 6 4 ¶s 1 ¶r ¶s 2 ¶r 3 7 5 = 2 6 4 p 2 ¶D 2 ¶s 1 p 2 ¶D 2 ¶s 2 3 7 5 Applying Cramer’s rule, and notice that due to the additive property of s 1 and s 2 in demand function, ¶ 2 D i ¶s 2 i = ¶ 2 D i ¶s i ¶s j = ¶ 2 D i ¶s 2 j and ¶D i ¶s i = ¶D i ¶s j . Let A be the big matrix on the left. It’s determinant jAj= 3g 0 (s) 2 + g 0 (s)g 00 (s)(s 1 + s 2 ) g 0 (s)(p 1 ¶ 2 D 1 ¶s 2 + p 2 ¶ 2 D 2 ¶s 2 ) is positive under our assumption. Note that if g 0 (s)= 0 the determinant is 0 and we can’t apply Cramer’s rule. We have the two derivatives as: 2 6 4 ¶s 1 ¶r ¶s 2 ¶r 3 7 5 = 2 6 6 4 p 2 2 ¶D 2 ¶s ¶ 2 D 2 ¶s 2 p 1 p 2 ¶D 2 ¶s ¶ 2 D 1 ¶s 2 +[3g 0 (s)+(s 1 +s 2 )g 00 (s)]p 2 ¶D 2 ¶s jAj p 2 2 ¶D 2 ¶s ¶ 2 D 2 ¶s 2 +p 1 p 2 ¶D 2 ¶s ¶ 2 D 1 ¶s 2 [3g 0 (s)+(s 1 +s 2 )g 00 (s)]p 2 ¶D 2 ¶s jAj 3 7 7 5 From our assumption, we can derive that ¶s 1 ¶r > 0 and ¶s 2 ¶r < 0. What is interesting is that ¶s 1 +¶s 2 ¶r = 0. That is to say, absent price changes, the revenue sharing among platforms is irrelevant to a change in the investment level. Note that the irrelevant result depends on the assumptions that the first-order condition is suffi- cient, and both platforms choose an interior solution of investments. However, as r varies, we are 126 likely to get a corner solution. 25 Without loss of generality, assume SP 2 is in the corner that s 2 = 0. Therefore, we have s 1 = argmax s 1 p 1 D 1 (s 1 )+ rp 2 D 2 (s 1 ) g(s 1 ) From the FOC, we get ¶s 1 ¶r = p 2 ¶D 2 ¶s 1 p 1 ¶ 2 D 1 ¶s 2 1 + rp 2 ¶ 2 D 2 2 ¶s 2 1 g 00 (s 1 ) > 0 So if we reach a corner solution, our irrelevant result does not hold, and the total investment increases with revenue sharing rate. J.4 More details about the proof of Proposition 13 J.4.1 Price increases with r Assumption 7. ¶ 2 D i ¶ p i ¶ p j 0 Proof of price increases with r under assumption 5,7 We fix investment s i . Taking FOC of each firm’s profit function with respect to price p i , and total differentiating with respect to r, we get the following system of equations: 2 6 4 2 ¶D 1 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 2 1 + rp 2 ¶ 2 D 2 ¶ p 2 1 ¶D 2 ¶ p 2 + p 1 ¶ 2 D 1 ¶ p 1 ¶ p 2 + r ¶D 2 ¶ p 1 + rp 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 ¶D 2 ¶ p 1 + p 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 2 ¶D 2 ¶ p 2 + p 2 ¶ 2 D 2 ¶ p 2 2 3 7 5 2 6 4 ¶ p 1 ¶r ¶ p 2 ¶r 3 7 5 = 2 6 4 p 2 ¶D 2 ¶ p 1 0 3 7 5 Apply Cramer’s rule. The denominator is positive thanks to assumption 5, and the numerator is positive by combining assumption 7 to assumption 5. As a result, we have ¶ p 1 ¶r > 0 and ¶ p 2 ¶r > 0. Proof of price increases with r under Logit demand 25 Note that f(s)= s also results in a corner solution following the same discussion. 127 Logit demand does not necessarily satisfy assumption 7. Consider a Logit demand function D i = e v i p i e v 0 s +e v 1 p 1+e v 2 p 2 . with s= s 1 + s 2 . First order conditions of both platforms yields 1+ p 1 (D 1 1)+ rp 2 D 2 = 0 (J.7) 1+ p 2 D 2 p 2 = 0 (J.8) After taking the total derivative with respect to r, and substituting FOC into the derivatives, we get 2 6 4 1 p 1 D 1 D 2 p 2 D 1 D 2 1 3 7 5 2 6 4 ¶ p 1 ¶r 2 ¶ p 2 ¶r 2 3 7 5 = 2 6 4 p 2 D 2 0 3 7 5 (J.9) Let B denote the left matrix. Then jBj= 1 p 1 p 2 D 2 1 D 2 2 Plug in p 2 = 1 1D 2 and p 1 = 1 1D 1 (1+ r D 2 1D 2 ) jBj= 1 D 2 1 D 2 2 (1 D 1 )(1 D 2 ) (1+ r D 2 1 D 2 ) Since D 1 < 1 D 2 and D 2 < 1 D 1 > 1 D 1 D 2 (1+ r D 2 1 D 2 ) > 1 D 2 (D 1 + D 2 ) > 0 128 SincejBj> 0 2 6 4 ¶ p 1 ¶r 2 ¶ p 2 ¶r 2 3 7 5 = 2 6 4 p 2 D 2 jBj p 2 2 D 2 2 D 1 jBj 3 7 5 (J.10) Both are positive. J.4.2 Price increases with total investment s Proof of price increases with r under assumption 5,7 We fix the revenue-sharing rate r. After taking FOC of each firm’s profit function with respect to price p i , and doing a total differentiate with respect to total investment s, we get the following system of equations: 2 6 4 2 ¶D 1 ¶ p 1 + p 1 ¶ 2 D 1 ¶ p 2 1 + rp 2 ¶ 2 D 2 ¶ p 2 1 ¶D 2 ¶ p 2 + p 1 ¶ 2 D 1 ¶ p 1 ¶ p 2 + r ¶D 2 ¶ p 1 + rp 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 ¶D 2 ¶ p 1 + p 2 ¶ 2 D 2 ¶ p 1 ¶ p 2 2 ¶D 2 ¶ p 2 + p 2 ¶ 2 D 2 ¶ p 2 2 3 7 5 2 6 4 ¶ p 1 ¶s ¶ p 2 ¶s 3 7 5 = 2 6 4 ¶D 1 ¶s p 1 ¶ 2 D 1 ¶ p 1 ¶s rp 2 ¶ 2 D 2 ¶ p 1 ¶s ¶D 2 ¶s p 2 ¶ 2 D 2 ¶ p 2 ¶s 3 7 5 (J.11) Similar to previous discussions, we apply Cramer’s rule. The denominator is positive thanks to assumption 1, and the numerator is positive by combining assumption 7 to assumption 5. As a result, we have ¶ p 1 ¶s > 0 and ¶ p 2 ¶s > 0. Proof of price increases with s under Logit demand Using the same demand function and first order condition, buy take total derivative with respect to s. Using similar algebra, 129 2 6 4 1 p 1 D 1 D 2 p 2 D 1 D 2 1 3 7 5 2 6 4 ¶ p 1 ¶s ¶ p 2 ¶s 3 7 5 = 2 6 4 (1 p 1 )D 0 0 3 7 5 (J.12) Applying Cramer’s rule: 2 6 4 ¶ p 1 ¶s ¶ p 2 ¶s 3 7 5 = 2 6 4 (p 1 1)D 0 jBj (p 1 1)p 2 D 0 D 1 D 2 jBj 3 7 5 (J.13) By first order condition (28), we know p 1 1= p 1 D 1 + rp 2 D 2 > 0, and we showedjBj> 0. So it follows that ¶ p 1 ¶s > 0 and ¶ p 2 ¶s > 0. J.5 Extension of lemma 5 Suppose we don’t assume commitment. So CP can not force exclusively licensed SP to sub-license. Whether SP 1 will sub-license solely depends on the profitability of sub-licensing. CP anticipates SP 1 ’s choice, and charge p s 1 (1,0)p s 1 (0,1) if anticipate sub-licensing, and p 1 (1,0)p 1 (0,1) if not. Because CP can only charge a fixed fee, we know SP 1 will choose sub-licensing ifp s (1,0)> p 1 (1,0). In equilibrium, SP 1 ’s total profit is p s 1 (1,0)(p s 1 (1,0)p s 1 (0,1))= p s 1 (0,1). the li- censed platform is going to get its disagreement point anyway. Case 1: Supposep s (1,0)>p 1 (1,0), CP expect SP 1 to choose sub-licensing. Then CP is going to choose exclusive if p 1 (1,1)p s 1 (0,1)+p 2 (1,1)p s 2 (1,0)>p s 1 (1,0)p s 1 (0,1) or equivalently 130 P ( 1,1)+P 2 (1,1)>P s (1,0) Case 2: Supposep s (1,0)p 1 (1,0)p 1 (0,1) or equivalently P ( 1,1)+P 2 (1,1)>P(1,0) So in both cases, CP will choose the form of contract that generates the highest total profit for the market. Lemma 4 is still robust. J.6 More details about the Salop circle demand system Salop demand system for non-exclusive or sub-licensing contract The locations can be found as x 1 = 1 6 + v 1 p 1 +c+s 1 +s 2 2t , x 2 = 5 6 + p 1 p 2 v 1 2t and x 3 = 1 2 + p 2 c(s 1 +s 2 ) 2t . Hence the demands for each platform and pirate music are D 1 = x 1 + 1 x 2 = 1 3 + 2v 1 + c 3 + s 1 + s 2 + p 2 2p 1 2t , D 2 = x 2 x 3 = 1 3 + v 1 + p 1 + s 1 + s 2 + c 3 2p 2 2t , D 3 = x 3 x 1 = 1 3 + v 1 + p 1 + p 2 2[c 3 + s 1 + s 2 ] 2t Salop demand system for exclusive contract 131 In the model, the locations of the platforms do not change, because consumers taste should not change due to exclusive contract. The only difference is that the market share of SP 2 is split by platforms SP 1 and SP 3 and the marginal contribution s 2 is driven to zero. In this case, x EX 1 = 1 6 + v 1 p 1 +c+s 1 2t and x EX 2 = 2 3 + v 1 +p 1 cs 1 2t . The demand of SP 1 is D EX 1 = 1 x EX 2 + x EX 1 = 1 2 + v 1 p 1 +c+s 1 t . Results for Salop circle demand system Result 1. (Integrated) If the two streaming platforms are fully integrated, the optimal investment level is s I 1 + s I 2 = 6c+3v 1 +4t 6(4at1) . The optimal prices are p I 1 = 3(v 1 +c)+2t 6 + s I 1 +s I 2 2 . p I 2 = 3c+2t 6 + s I 1 +s I 2 2 . Result 2. (Non-exclusive) Under non-exclusive contract, the investments in equilibrium are s NE 1 = 2(10at 2 6v 1 +15atc+48atv 1 ) 15at(27at4) , s NE 2 = 2(10at 2 +6v 1 +15atc33atv 1 ) 15at(9at2) . The prices in equilibrium are p NE 1 = 3(10at 2 2v 1 +15atc+21atv 1 ) 5at(27at4) , p NE 2 = 3(10at 2 +2v 1 +15atc6atv 1 ) 5at(27at4) . Result 3. (Exclusive) The investment level under exclusive contract is s EX 1 = 2c+2v 1 +t 2(4at1) . The price of the exclusively licensed platform is p EX 1 = 2ats EX 1 Result 4. (Revenue-sharing) Under sub-licensing contract with r = 1, the prices are p SUB 1 = 7(3v 1 +24a(v 1 +c)t+16at 2 ) 4(98at23) , p SUB 2 = 280at(3c+2t)+v 1 (171336at) 24(98at23) . The investment level is s SUB 1 = 276c+129v 1 +184t 12(98at23) and s SUB 1 = 0. Proof of corollary 13 Proof. Observe the total investment level s I 1 + s I 2 = 6c+3v 1 +4t 6(4at1) and s NE 1 + s NE 2 = 2(6c+3v 1 +4t) 3(27at6) . Then it is straightforward to see s I 1 + s I 2 s NE 1 s NE 2 = 11at(6c+3v 1 +4t) 6(4at1)(27at4) > 0. For the second part of corollary, simply take the difference: s NE 1 s NE 2 = 2v 1 5at > 0 Proof of corollary 14 Proof. (1)Evaluate the difference in investment: s EX 1 s NE 1 s NE 2 = (19at 2)v 1 + 66atc+ 4t 6(4at 1)(27at 4) > 0 132 is positive under assumption 6. The difference increases with v 1 . (2) Take the difference of subscription fee under of SP 1 under exclusive and non-exclusive contract: p EX 1 p NE 1 = (18a 2 t 2 + 47at 6)v 1 +(18at+ 1)5atc+ 5at 2 (3at+ 2) 5(4at 1)(27at 4) is positive under assumption 6. (3) We take the derivative ofP EX P NE with respect to v 1 , and verify that the numerator is positive under assumption 6. Proof of corollary 16 Proof. Two cases emerge from sub-licensing. The exists an r such that s 2 at r is driven down to 0. When r < r , both platforms invest, while s 1 increases with r and s 2 decreases with r, just like our intuition in section 4. When r passed a threshold, SP 2 is no longer interested in anti-piracy investment, and s 2 remains at 0. However, SP 1 still benefit from the a higher revenue sharing rate, and thus have incentives to increase s 1 . Mathematically, assuming symmetric platform v 1 = 0. If r< r : ¶s 1 + s 2 ¶r = 5at(15 r)(25 19r)(2t+ 3c) (3a(15 r) 2 t+ 3r 2 + 5r 100) 2 > 0 If r> r ¶s 1 + s 2 ¶r = 10at(15 r)(155 9r)(2t+ 3c) 3(2a(15 r) 2 t+ 3r 2 45r 50) 2 > 0 Proof of Corollary 17 Proof. Suppose r< r , as we defined in Corollary 5. We can show that the derivative ¶P ¶r = 5a 2 t(15 r)(25 19r)(3c+ 2t) 2 (3r 2 + 5r 100+ 9a(15 r) 2 t) 6(3r 2 + 5r 100+ 3a(15 r) 2 t) 3 133 which can be shown to be positive under assumption 6 that a> 1/4t. Suppose r> r ¶P ¶r = 10a 2 t(15 r)(3c+ 2t) 2 (3(4750 r(5525+ r(610+ 19r)))+ 2at(15 r) 2 (25 19r)) 9(50 45r+ 3r 2 + 2a(15 r) 2 t) 3 By assumption 6, the denominator is positive. Taking the derivative of the numerator with respect to r, we verify that this derivative is always negative. So it suffices to check whether the numerator is positive under r= 1. It turns out that when r= 1. Then the numerator at r= 1 is 140a 2 t(3c+ 2t) 2 (2352at 552)> 0. J.7 Simulations on social welfare In this section, we use simulation to illustrate the previous results. From the previous discussion, we presented that, under non-exclusive contracts, platforms compete on prices but prefer to free- ride on investment, which means insufficient incentive for investment. Exclusive contracts alone can induce a higher level of investment, but that also comes with higher prices. Exclusive contracts alone also hurt consumers by foreclosing the rival platforms. Consumers who used to be subscribed to SP 2 have to pay higher transportation costs for switching to SP 1 or SP 3 . Sub-licensing is a method to compromise between the two cases, which also results in a higher total investment than the one under the non-exclusive contract. We did a few simulations to show how social welfare changes with market dominance(v 1 ) for different contracts. The producer sur- plus and consumer surplus are 26 : PS= p 1 D 1 + p 2 D 2 a(s 1 + s 2 ) 2 CS= v+(v 1 p 1 )D 1 p 2 D 2 cD 3 t( Z x 1 0 xdx+ Z 1/3x 1 0 xdx 26 The consumer surplus for exclusive contract alone is a little bit different in the transportation cost. 134 + Z x 3 1/3 0 xdx+ Z 2/3x 3 0 xdx+ Z x 2 2/3 0 xdx+ Z 1x 2 0 xdx) The producer surplus is the joint profit of the industry, which is split by two platforms and the content provider. Consumer surplus is the summation of utilities of all consumers, which includes valuation of music (positive), subscription fees, searching cost, and transportation cost (all negative). The simulation results are demonstrated in Figure 5. 27 27 I fix v= 5, a= 1, t = 4, c= 1, and vary v 1 from 0 to 4. I set t to be relatively large to make boundary conditions satisfied. This also represents that consumers have relatively strong preferences over platforms. All the boundary conditions are verified. 135 Figure 3.5: Comparative Statics about the Dominance of SP 1 Anti-piracy investments protect content, thus incentivizing musicians to compose better music. So in the long-run anti-piracy investment could be good for society. In the short run, anti-piracy has mixed effects. On the one hand, Anti-piracy forces consumers who prefer pirate music to choose a legal platform, which decreases their utility. The investment itself is also costly in that going to court is both expensive and time-consuming. On the other hand, the anti-piracy investment raises 136 the platforms’ revenues and forces consumers to listen to music with better quality. In this sense, the anti-piracy investment could enhance welfare, depending on which force is stronger. Our simulations only consider short-run social welfare. We observe that when market dom- inance is not very strong in the short run, a non-exclusive contract is the best way for society. Besides, a revenue-sharing contract results in similar social welfare as the non-exclusive contract. When market dominance is mild, the sub-licensing contract induces higher social welfare than ex- clusive contract alone, which indicates that a little bit of collusion could be good for society. When market dominance is very strong, an exclusive contract alone will lead to a level of social surplus even higher than a non-exclusive contract. In terms of total investment level, both exclusive and sub-licensing will induce a relatively high investment level than the non-exclusive contract. If there is no market dominance, the sub- licensing will induce a higher investment level. In contrast, with strong market dominance, the exclusively licensed platform will invest more than the total investment under sub-licensing. If the exclusively licensed platform does not have strong dominance, it prefers a sub-licensing contract to an exclusive contract. However, if the licensed platform is of strong market dominance, it prefers to exclude the other platform from the market. This is an interesting result that replicates the findings of classic literature. Economists believe that a dominant platform prefers incompati- bility to compatibility, while a small platform prefers compatibility (see Casadesus-Masanell and Ruiz-Aliseda (2008) and Adner, Jianqing Chen, and F. Zhu (2016)). In the model, piracy is a com- mon enemy to SP 1 and SP 2 . Whether SP 1 will frenemy SP 2 or not depends on how good SP 2 is. If SP 2 has a relatively low quality, SP 1 may want to exclude SP 2 rather than cooperate. The results may help explain why the copyright war was initially exclusive but later evolved to sub-licensing after the government intervened: at least sub-licensing is better than non-exclusive for SP 1 . Lastly, exclusivity or sub-licensing induces a higher price compared to the non-exclusive case. The price is slightly lower under sub-licensing than exclusivity, possibly because sub-licensing brings back competition on prices. 137
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Three essays on industrial organization
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anti-piracy investments
bidder asymmetry
exclusive contract
quality competition
sub-licensing
the winner's blessing
the winner's curse
user externalities
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