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Manipulating consumer opinion on social media using trolls and influencers

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Manipulating consumer opinion on social media using trolls and influencers

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Content Manipulating Consumer Opinion on Social Media Using Trolls and Influencers by Lei (Amy) Pei 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 (BUSINESS ADMINISTRATION) May 2020 Copyright2020 Lei(Amy)Pei ACKNOWLEDGEMENTS First and foremost, I would like to thank my committee chair, advisor, and mentor, Dina Mayzlin, for her expertise, guidance, generosity, and patience throughout my academic journey at USC. Without her help, this dissertation would not have been possible. I am tremendously grateful to havecommitteemembersLanLuo,AnthonyDukes,OdilonCamara,andJoeNunes,fortheirkind support,feedback,andencouragement. To my advisor and all committee members, your sharp insights and passion for research have deeply inspired me. Thank you for serving on my committee and teaching me invaluable lessons abouthowtoconductandpresentresearch. To my parents, thank you for loving me, believing in me, and being there for me every step alongtheway. To all my friends and fellow students in the Marshall Ph.D. program, your hard work and determination constantly inspire me. I am grateful to have spent six years with such wonderful colleagues. ii TableofContents ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi ChapterI: DoCurationAlgorithmsAmplifytheEffectofTrollsonUsers? . . . . . . . . . . 1 1.1 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 ModelSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Curation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ChapterII:InfluencingtheInfluencers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Extension: PartialDisclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Appendix A:ProofsforChapterI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Appendix B:ProofsforChapterII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 iii ListofTables 2.1 VariableDefinitionsforProposition2.3.4.1 . . . . . . . . . . . . . . . . . . . . . . . 54 iv ListofFigures 1.1 TheNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 ExampleofaTrollInfiltratingaUser’sNetwork . . . . . . . . . . . . . . . . . . . . . 8 1.3 TimelineoftheGame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 NoTroll,Troll (UnlimitedBandwidth) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 ’sExpectedUtilityUnderCurationvsNoCuration . . . . . . . . . . . . . . . . . . 28 2.1 TimelineoftheGame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Thepersuasivenessofthepositivereview,j%¹jG0ºd 0 j . . . . . . . . . . . . . . 47 2.3 EquilibriumFirmProfitwithAffiliationversusProfitunderIndependentInfluencer . . 55 2.4 EquilibriumLevelofAffiliation (0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5 Consumer’sPosteriorBeliefinEquilibrium,%¹jG0 º . . . . . . . . . . . . . . . . 57 2.6 Consumer’sValueofInformationinEquilibrium . . . . . . . . . . . . . . . . . . . . 59 2.7 ThePaymenttotheInfluencerinEquilibrium,(¹0 º . . . . . . . . . . . . . . . . . . 60 v Abstract Social media is an important source of information for consumers. In particular, there are two important ways in which consumers acquire information on social media: 1) online influencers, and 2) consumers’ social connections on platforms such as Facebook. Despite its popularity, information on social media can be manipulated by interested parties. The first chapter considers the effectiveness and welfare implications of curation algorithms in the presence of “trolls" (fake accounts that promote a certain agenda). One important unintended consequence of curation that has not been considered by either the existing literature or the industry is the amplification of the effectoftrollsonusers. Iftheplatformhaspreciseknowledgeabouttheuser’spreferences,curation improvesuserwelfareeveninthepresenceoftrollsbyaggregatinginformationinanefficientway. However, if the platform has imprecise knowledge about the user’s preferences, curation may decreaseuserwelfarerelativetonocurationbyshowingtheusercontentfromtrustedsourcesthat is biased toward the troll’s agenda. Without an effective means to eliminate trolls or the ability to acquirepreciseuserinformation,socialmediaplatformsmayconsiderturningoffcurationinorder to protect users against the bias of trolls. The second chapter shows that sponsored content by influencers is effective in driving sales even if the affiliation between the firm and the influencer is fullydisclosedtotheconsumers. Thefirm’soptimalaffiliationdecisiondependsontheconsumer’s awareness of the product and prior belief on product fit. When the consumer’s prior belief is low, the firm partially affiliates with the influencer to preserve the persuasiveness of the review. In contrast, when the consumer’s prior belief is high, the firm fully affiliates with the influencer to maximize awareness and to prevent a negative review. Prior research has primarily focused on the firm’scovertmanipulationofconsumeropinionthroughfakereviews,whichbenefitsthefirmatthe expense of the consumer. However, under certain conditions, the firm’s involvement in the review process can be Pareto-improving. Furthermore, the consumer may benefit from partial disclosure comparedtofulldisclosureifhispriorbeliefislow. vi Chapter 1 DoCurationAlgorithmsAmplifytheEffectofTrollsonUsers? Inthespringof2019,LosAngelesCountyconcurrentlyexperiencedoutbreaksofmeaslesandper- tussis,bothchildhooddiseaseswithexistingeffectivevaccines. Measles,inparticular,wasthought to have been eliminated in 2000.1 Many health experts believed that the outbreaks were linked to thelowimmunizationratesamongchildren. Infact,insomeofLA’swealthiestneighborhoods,the immunizationratesofchildrenin2014wereaslowasthoseinChadandSouthSudan.2 Localand state health officials blamed the low vaccination rates on misinformation about vaccines online. Interestingly,someoftheaccountsthatspread“anti-vax"misinformationonTwitterwerefoundto belinkedtoRussia’sInternetResearchAgency.3 In March 2019, Tesla CEO Elon Musk announced that there would be “a massive wave of deliveries"ofnewTeslacarsthroughoutEurope,China,andNorthAmerica4. However,agrowing community of Tesla skeptics on Twitter was determined to convince the public that the claim was false. One member of the community, Machine Planet, tweeted that he flew a plane over Tesla’s storage lots and saw more than 100 empty car-carrier trailers sitting idle rather than being loaded for deliveries. Like most members of this community, Machine Planet ended his tweet with the Twitter hashtag $TslaQ, Tesla’s stock symbol followed by the letter Q indicating that a company is in bankruptcy. Some suspected that the $TslaQ members were in fact Tesla short-sellers who hopedthatthecompany’sstockpricewouldcrash.5 Both the Russian accounts spreading anti-vaccine information, and the $TslaQ members at- 1https://www.nytimes.com/2019/04/23/us/measles-outbreak-los-angeles-county.html; https://www.latimes.com/local/california/la-me-ln-whooping-cough-harvard-20190226-story.html 2https://www.hollywoodreporter.com/features/los-angeles-vaccination-rates/ 3A recent study finds that Russian trolls have been polarizing the vaccination debate by amplifying both pro- and anti-vaxcontentonTwitter(Broniatowskietal., 2018) 4https://www.businessinsider.com/elon-musk-tesla-workers-car-deliveries-should-primary-priority-leaked-email- 2019-3 5https://www.latimes.com/business/autos/la-fi-hy-tesla-short-sellers-musk-20190408-story.html 1 tempted to change others’ beliefs on social media in a way that was consistent with their agenda. Inbothcases,theindividualsinvolveddidsoanonymouslyorunderassumedidentities. Usingthe popularInternetjargon,theyare“trolls."Trollsmanipulateconsumersbyexposingthemtobiased ($TslaQ members post only anti-Tesla content) or false (vaccines are toxic) information. Existing researchhasfocusedonthedirecteffectoftrollsonconsumers. Thatis,whetherconsumers’beliefs andactionswereaffectedduetotheirdirectexposuretothetrolls’posts. Forexample,trollscreated public Facebook pages under the guise of local social activists in order to mobilize unsuspecting users.6 These strategies appeared to be effective: trolls amplified ideological polarization among U.S.votersbytweetingandre-tweetingpoliticallydivisivecontentin2016(Stewartetal.,2018). Curation algorithms are meant to solve the basic information-processing problem that users face on social media. Even though consumers face a high volume of information produced by many social connections, they can only consume a small subset of this information due to time constraints;consumershave“limitedbandwidth."Toaddressthis,manyplatformschoosetocurate content in order to expose each user to a personalized subset of posts that is more informative and engaging compared to a random subset of posts that the user would otherwise sample. We definecurationastheselectionofcontentaccordingtospecificcriteria,ataskthatisautomatedby curationalgorithms. Onepossibleunintendedconsequenceofthistypeofalgorithmisthecreation offilter-bubbles: theconcernthatplatformsonlyexposeuserstopoststhatconformtotheirbeliefs (Lazer, 2015). For example, the platform may mostly show right-leaning posts to users on the right side of the political spectrum. We focus on a different unintended consequence of curation algorithms, onethathasnotbeenstudiedbefore: thevulnerabilityofcurationalgorithmstotrolls. We use an analytical model to investigate when and how curation algorithms are vulnerable to the trolls’ influence. We consider a simple network with one focal user and her three network connections: twofriendsandoneacquaintance,whomaybeatrollthatinfiltratedtheuser’snetwork under an assumed identity. We also assume that the user prefers to see a friend’s post. There are 6Forexamples,seehttps://www.nytimes.com/2018/08/02/technology/facebook-fake-accounts.html, and https://www.nytimes.com/2018/08/14/technology/facebook-disinformation-black-elevation.html 2 two states of the world: the measles vaccine is either harmful or beneficial. The user’s legitimate connections send posts that are informative of the state of the world, while the troll always sends a post that supports his agenda. We assume that the user has limited bandwidth and is able to viewonlyonepost,whiletheplatformobservesthefullsetofpostsaswellasthestrengthofeach connection. Undernocuration,theusersamplesonepostatrandomandobservestheidentityofitssender. Incontrast,undercuration,theuserisshownonepostspecificallyselectedforherbythealgorithm. We refer to the process of boiling down all the information to a single post as “information compression."Sincetheplatformcannotperfectlydistinguishtrollsfromgenuineusers,thetroll’s post may make a certain type of content appear more popular among the user’s connections. For example, suppose that the users’ friends are split between pro-vax and anti-vax content, and the trollpostsonlyanti-vaxcontent. Herethealgorithmchoosestodisplaytheanti-vaxpostbyafriend since it reflects the majority sentiment. The user who sees the algorithm-curated anti-vax post infers that this represents the majority opinion. However, the composition of the majority matters forherdecisionsincetheacquaintance’spostislessreliable. Heretheusercannotfullyrecoverthe fullsetofpostspriortocompressionsinceshedoesnotobserveeitherthepostsortheidentitiesof theothertwosenders. We show that if the platform has perfect knowledge about the user, curation achieves the first- best optimal compression. In this case, the platform is able to show a post that matches the user’s optimal action under full information. On the other hand, if the platform does not have perfect information, we show that the first best-optimal compression cannot be achieved. In this case, the algorithmsometimesshowstheuseranincorrectpost,i.e.,onethatresultsinanincorrectdecision. Since the user is unable to unpack the curated post, she still chooses to follow it as long as the probability of this error is not too high. We show that the user may be worse off under curation than under no curation where the user is less likely to be misled by the troll. Hence, the curation algorithm may inadvertently enable the troll to affect the user’s action. In this sense, it actually amplifiestheeffectofthetrolls. 3 We contribute substantively to the literature by demonstrating an unintended consequence of curationthathasnotbeenconsideredinpriorresearch,whichprimarilyfocusedonwhethercuration algorithms led to filter bubbles in the absence of trolls or the direct effect of trolls. Our model illustrates the crucial trade-off that social media platforms face when curating content to users: while curation increases the amount of information received by the user through compression, it also amplifies the effect of trolls by making it harder for the user to directly discount the trolls’ content. Inrecentyears,socialmediaplatformscameunderincreasingscrutinyfromthepublicas wellasregulatorsforfacilitatingmisinformationandpoliticalpolarization. Ourresultssuggestthat successfulcurationinthepresenceoftrollsrequirestheplatformtoacquireverypreciseinformation abouttheuser’spreferences. Intheabsenceofsuchinformation,theplatformmayneedtomodify theuseofcurationalgorithms. The rest of the paper is organized as follows. In Section 1.1, we discuss the related literature. In Section 1.2, we describe our model setup. In Section 1.3, we present our model of curation and analyzeitseffectonuserwelfare. Lastly,weconcludeinSection1.4. 1.1 LiteratureReview Our research is related to several literature streams: curation algorithms, consumer learning in social networks, misinformation, and mechanism design. In this section, we review the prior empiricalandtheoreticalfindingsandexplainhowourworkrelatestotheliterature. First,thischapterisrelatedtoresearchoncurationalgorithms,whichmainlyaddressestwomain questions: 1) do curation algorithms reduce the diversity of information received by individuals? And 2) how do curation algorithms affect the network structure and users’ content posting and sharingbehavior? Forthefirstquestion,priorpapersfindthatcurationalgorithmsthatrecommend content often lead to filter bubbles, a situation where users only receive information and opinions that conform to and reinforce their own beliefs (Bakshy et al., 2015; Flaxman et al., 2016; T. T. Nguyen et al., 2014). On the other hand, algorithms that recommend products, or recommender systems, increase the diversity of consumers’ product choices (Hosanagar et al., 2014; D. Lee 4 & Hosanagar, 2019). For the second question, prior papers find that curation algorithms that recommend both content and connections make users more likely to connect with others and also exertgreatereffortinproducingcontent(Berman&Katona,n.d.;Iyer&Katona,2016). Inaddition, users tend to connect with others who have similar posting behavior (Bhattacharya et al., 2019). Users are also more likely to share content posted by others who are similar, such as those with commonfollowersandfolloweesontheplatform(Pengetal.,2018). Second, this chapter is related to the literature on how information on social media affects individuals’ decisions. For example, Mousavi and Gu (2019) finds that information on Twitter affectsU.S.Congressrepresentatives’votingdecisions. Morecommonly,theinformationinsocial networks influences consumers’ decisions to adopt new products (Ameri et al., 2019; J. Zhang et al., 2015; Y. Zhang & Godes, 2018). These papers show that learningfrom social ties generally improves consumers’ decision quality. However, the prior literature did not consider the effect of trolls or misinformation on learning in social networks, which is the focus of this chapter. Below wediscussrelatedresearchonthistopic. Interestedpartiesoftenseektoinfluenceconsumers’opinionsthroughmanipulatinginformation onlineorspreadingmisinformation. Forexample,firmscreatefakereviews(Dellarocas,2006;S.Y. Lee et al., 2018; Mayzlin, 2006; Mayzlin et al., 2014); anti-vax groups promote fake stories about vaccines(Broniatowskietal.,2018;Chiou&Tucker,2018);andtheInternetResearchAgency(the infamous Russian “troll farm” operation) spread fake news about political candidates (Allcott & Gentzkow,2017;Guessetal.,2018;Linvill&Warren,2018;Schiff,2018). Priorliteraturehasalsostudiedhowmisinformationisspreadonsocialmediaplatforms. Shao etal.(2018) showsthatonTwitter,trollsspreadmisinformationbyamplifyingexistingcontentand targetinginfluentialusersthroughrepliesandmentions. Spangheretal.(2018) demonstratesthat on Facebook, trolls induce authentic engagement with fake public pages by maintaining contact withlegitimateusersandpushingthemtojointhosepages. Vosoughietal.(2018) findsthatfake news becomes viral because legitimate users share it. In contrast, Shao et al. (2018) finds that 5 bots amplify the reach of false news by re-sharing links posted initially by humans. In the same vein, Spangher et al. (2018) finds empirical evidence that trolls have promoted “a diverse array of left-leaning, right-leaning, and apolitical content." Acemoglu et al. (2010) proposes a game- theoretical model of the spread of misinformation, which occurs because there exist individual stubborn agents who change others’ beliefs without changing their own beliefs. While the above papers demonstrate how trolls operate on social media, the effect of trolls under curation is not modeledorconsidered. Priorworkonmisinformationalsoproposesseveralmethodstocurtailandcontrolmisinforma- tion on social media. One method of containing the spread of misinformation is to target a subset ofhighlyinfluentialusersanddissuadethemfrombelievinganongoingmisinformationcampaign, thereby minimizing the number of users who end up believing the misinformation (Budak et al., 2011; N. P. Nguyen et al., 2012). This idea is similar to immunization: “immunizing" a critical group of users against misinformation in order to prevent an epidemic. Another method is to implementan“inspection"typeofmechanismthatselectivelyverifiescertainposts(Papanastasiou, 2020;Tambuscioetal.,2015). CandoganandDrakopoulos(2017) consideramechanismdesign problem of a platform that needs to balance user engagement against misinformation, where the platformhasprivateknowledgeoftheaccuracyofitscontent. Inourmodel,theplatformdoesnot have private knowledge about the identity of trolls; however, it faces a similar trade-off between amplifying trolls and increasing the value of information when designing the curation algorithm. Inaddition,westudytheproblemofmisinformationundercuration. Wearethefirsttosuggestthat theplatformmayreducetheinfluenceoftrollsbyabandoningcuration. Finally,optimalalgorithmdesignisanalogoustomechanismdesign,wheretheplatformisthe social planner. From this perspective, the curation algorithm we model here is analogous to the Myersonian mediator (Myerson, 1986), an incentive-compatible agent who collects information from everyone else and then makes a recommendation. In the case of curation, the platform observes all the signals from the focal user’s social ties and shows the user only one signal. In ordertomakethecorrectrecommendationtoeachuser,curationrequiresperfectknowledgeofthe 6 environment and preferences of the users. The critique of the Myersonian mechanism (Myerson, 1981) is that it requires common knowledge of players’ preferences (Bergemann & Morris, 2005; Wilson, 1985). If this assumption is incorrect, then fine-tuning the mechanism can actually cause more harm than good. This critique is parallel to ourfinding that when the platform has imperfect informationabouttheuser,curationcanactuallylowertheuser’swelfareratherthanimprove it. 1.2 ModelSetup Weconsiderasimplesocialnetworkconsistingofafocaluser,,andherthreesocialcontacts: and . We refer to as the receiver and to her contacts as the senders. and are ’s friends, whereas is her acquaintance. For example, may know both and in real life, but is someone with whom she only interacts online. There are two possible states of the world: f!'g. For instance, the two states correspond to whether or not vaccines are actually harmful or beneficial. has a prior belief about the true state of the world: %A¹'º= d 0 ¡ 1 2 . In other words, sheisbiasedtowardsstate',whichinourexampleimpliesthatbelievesthatthemeaslesvaccine ismorelikelytobebeneficialtoherchild. Figure1.1: TheNetwork A B C D Friend Friend Acquaintance The receiver Eachoneofthesenders,,,and,postsanindependentandnoisysignalaboutthetruestate 7 of the world,B 9 2f;Ag, where 92fg. For example, and may post information that the measles vaccine is safe and effective, whereas may claim that the vaccine is toxic. may receive the full set or the subset of all the available signals. We denote the set of signals received by as<. We assume that the signals sent by ’s friends match the true state of the world with probabilityU¡ d 0 . Thatis,%¹B 9 =;j!º=%¹B 9 =Aj'º=U,where 92fg;notethatwemake the simplifying assumption that the friends’ signals are equally informative. Given<, updates herpriorbeliefaboutthestateoftheworld,denotedby\=%¹'j<º. Trolls infiltrate users’ networks by posing as legitimate mutual friends or acquaintances.7 For example,atrollfirststealstheidentityofalegitimateuserbyduplicatinghisFacebookprofile. From this spoofed profile, the troll then sends out friend requests to the user’s legitimate connections, who may accept the friend requests since they appear to come from a mutual friend. Figure 1.2 illustratesanactualFacebookuser’sexperienceofacceptingafriendrequestfromsomeonewhom he believed was an acquaintance. However, this user later suspected that the “acquaintance" may, infact,beatrollsinceheexhibitedtroll-likebehavioronline. Figure1.2: ExampleofaTrollInfiltratingaUser’sNetwork We allow for the possibility that the acquaintance, , could in fact be a troll that infiltrated 7The “check-in" function on Facebook allows users to record the places they visited and the events they attended on their profiles. This information often can be viewed publicly, allowing trolls to create bogus identities that target theseusers. 8 ’s network: %¹trollº = C, where 0 C 1. To simplify, we assume that there exists one left troll who always sends;. Hence, the left troll’s signal is uncorrelated with the state of the world: %¹B =;jtroll!º=%¹B =;jtroll'º= 1. TheparameterC measurestheleveloftrollinfiltration. For example, ifC = 1, trolls have fully infiltrated the platform, such that any social contact who is not a friend is for sure a troll. IfC = 0, there are no trolls on the platform, and all social contacts arelegitimateusers. TheprobabilitythatB matchesthetruestateoftheworldisgivenasfollows: %¹B =;j!º=%¹nottrollº%¹B =;j!nottrollº¸%¹trollº%¹B =;j!trollº =¹1CºU¸C (1.1) Similarly, %¹B =;j'º=¹1Cº¹1Uº¸C (1.2) %¹B =Aj!º=¹1Cº¹1Uº (1.3) %¹B =Aj'º=¹1CºU (1.4) Weassumethat’stotalutilityisthesumofthesocialutility((*)theinformationutility(*). The socialutilitycapturestheideathatthereceiverpreferstointeractwithsomesendersoverotherson socialmedia. Theinformationutilityisthestandardutilityfunction,whichdependsontheplayer’s actionandthestateoftheworld. *=(*¸* (1.5) Weassumethatderivessocialutility((*)fromreceivingsignalssentbyfriends(ratherthan acquaintances). Thesocialutilityisassumedasfollows: (*= 8 > > > > < > > > > : X if 92fg; 0 otherwise (1.6) We also assume that obtains information utility (*) of 1 if she takes an action that matches the true state of the world. For instance, if the underlying state of the world is that vaccines are 9 beneficial, a parent who vaccinates her child receives a positive utility. Similarly, if vaccines were harmful,a parentwouldbenefitfromnotvaccinatingherchild. *= 8 > > > > < > > > > : 1 if¹0 ! andstate!º or¹0 ' andstate'º 0 otherwise (1.7) chooses an action that maximizes her expected utility, given her belief on the state of the world. In particular, she takes action0 ' if\ 1 2 and0 ! if\ 1 2 . Since ’s prior is biased toward state ',shechooses0 ' bydefaultintheabsenceofanysignals. The timing of the game is as follows (see Figure 1.3). AtC= 1, Nature chooses the state of the world,whichisunknowntoanyplayer. AtC= 2,,,and eachsendsanindependentsignal. At C = 3, receives the set (or a subset) of the three available signals. In particular, we consider two cases. Inthecaseofunlimitedbandwidth,seesallthreesignals: <=fB B B g. Incontrast,in the case of limited bandwidth, can only receive one signal: <=fB 9 g, where 92fg. At C= 4, updatesherbeliefandtakesanaction. Herinformationutilityisafunctionoftherealized stateoftheworldandheraction. Figure1.3: TimelineoftheGame B, C, and D each sends an independent signal, . Nature chooses the state of the world, ∈ { , }. A observes = { } , , . A observes = { ൟ , ∈ { , , }. Unlimited bandwidth Limited bandwidth A updates her belief and takes an action ( ). t=1 t=2 t=3 t=4 10 1.2.1 UnlimitedBandwidth In this section, we consider the full information case where has unlimited bandwidth. That is, observes all three signals at no cost. The timing of the game is illustrated on the left branch inFigure1.3. We derive ’s optimal action and expected utility given the full set of signals, which serves as a benchmark for our analysis and welfare comparisons for the later sections of the chap- ter. The presence of the left troll adds noise to B but does not affect the precision of B or B . This implies that the precision of the signals differs depending on the identities of the senders, which are observed by the receiver. There are a total of eight different realizations of<: ff;;;gf;;Agf;A;gf;AAgfA;;gfA;AgfAA;gfAAAgg. We can show that if and send the same signal (e.g.,f;;Ag), ’s signal does not affect ’s optimal decision. Given that the friends’ signals are equally informative, ’s signal at most “cancels out" either ’s or ’s signal. The remaining signal from a friend is strong enough to change ’s belief since we assume thatU ¡ d 0 . This implies that whenever the friends ( and) areinagreement,’sdecisionwillbeinthesamedirectionastheirsignals. On the other hand, if and disagree (e.g.,f;A;g), their signals cancel each other out, and ’s belief effectively depends on’s signal alone. The presence of the troll weakens’s signal, butaslongastheprobabilitythat isatrollissmall,aweak; signalfrom maystillsufficiently change ’s belief and affect her optimal decision. Consider the scenario< =f;A;g. ’s belief followingthissignalrealizationis %¹'jf;A;gº\jf;A;g= %A¹B =;j'º%A¹'º %A¹B =;j'º%A¹'º¸%A¹B =;j!º%A¹!º (1.8) Substitutingequation(1.1)and(1.2)intoequation(1.8),wehave \jf;A;g= »¹1Cº¹1Uº¸C¼d 0 »¹1Cº¹1Uº¸C¼d 0 ¸»¹1CºU¸C¼¹1d 0 º (1.9) optimallychoosesaction0 ! if\¹f;A;gº 1 2 . RewritingthisinequalityintermsofC,weobtain C Ud 0 U¸d 0 1 ¯ C. In other words, optimally chooses an action in the direction ofB ifC ¯ C. 11 Note that ¯ C is a decreasing function ofd 0 . This is because the more extreme’s prior is, the more preciseB needs to be to act as a tie-breaker.8 On the other hand, Equation (1.9) also implies that \jf;A;g 1 2 ifC ¡ ¯ C. In this case, optimally chooses 0 ' since B is too noisy to affect ’s decision. Lemma1.2.1.1summarizesA’soptimaldecisionrulegivenunlimited bandwidth,herexpected utility given her decision, and the total effect of the troll on her welfare. If C ¯ C, optimally chooses 0 ! whenever the number of ; signals exceeds the number of A signals and chooses 0 ' otherwise. That is, follows the “majority" rule, where her optimal decision is in the direction of the majority of the signals. In contrast, ifC ¡ ¯ C, optimally chooses0 ! if and only if both and send;;otherwise,shechooses0 ' . Lemma 1.2.1.1. Consider the case of unlimited bandwidth, where ¯ C Ud 0 U¸d 0 1 , 0 ¯ C 1, ’s optimaldecisionruleis 0= 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ifC ¯ C 8 > > > > < > > > > : 0 ! if #; ¡ #A 0 ' if #A ¡ #; ifC ¡ ¯ C 8 > > > > < > > > > : 0 ! ifB =B =; 0 ' otherwise (1.10) whichgivesherthefollowingexpectedutility * Troll = 8 > > > > < > > > > : U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º¸ 2X ifC ¯ C U 2 ¸ 2U¹1Uºd 0 ¸ 2X ifC ¡ ¯ C Thedifferencein’sexpectedutility intheabsencevspresenceofthelefttroll: NoTroll,Troll * NoTroll * Troll = min»2U¹1Uº¹Ud 0 º 2CU¹1Uº¹U¸d 0 1º¼ Proof. SeetheAppendixA. 8NotethattheassumptionU¡ d 0 ¡ 1 2 implies that 0 ¯ C 1. 12 The presence of the left troll introduces noise to the information received by and lowers her expected utility. NoTroll,Troll is the difference between ’s expected utility in the absence of the trollandherexpectedutilityinthepresenceofthetroll. Notethatsince’ssocialutilityisidentical in both cases, NoTroll,Troll consists of only the difference in ’s expected information utility. We refer to this as (the magnitude of) ’s information loss due to the presence of the troll. Depicted in Figure 1.4, the information loss is increasing inC up until ¯ C and then becomes constant. WhenC issmall,’soptimaldecisiondependsonallthreesignals: thetwostrongsignalsfromthefriends and one weak signal from the acquaintance. The presence of the troll causes a direct information loss to by lowering the precision ofB , which is increasing inC. In contrast, whenC is large,B no longer affects’s optimal decision, and takes0 ! if and only if both and send;. In other words, ’s optimal decision effectively depends on only the two strong signals: B and B . The presenceofthelefttrolleffectivelydestroysthemarginalvalueofB . 1.2.2 InformationLossInthePresenceoftheTroll The total effect of U on ’s information loss is ambiguous because there are two opposing effects as U increases. First, a higher U increases the precision of B , which leads to a higher marginalvalueofB andhencealargerlossofinformationinthepresenceofthetroll. However,as U becomessufficientlyhigh, and aremorelikelytobecorrectaboutthestateoftheworldand agree with each other, which in turn decreases the marginal value ofB . Therefore, the potential loss of information is also smaller. Shown in Panel b) of Figure 1.4, the change in the information lossasafunctionofU isnonlinearandexhibitsaninverse-Ushape. To illustrate how the precision of the signal affects ’s information loss in the presence of the troll,weusetheTeslaexampleintroducedearlier. SupposeisaninvestorinTesla’sstockandtries to decide whether she should short it. Her friends work in the electric car industry and are likely to have reliable information about the company. If the acquaintance is also an industry insider, his post will provide additional information to . However, if the acquaintance is in fact a troll, his post would contain no actual information about Tesla. In this sense, receives a lower expected 13 information utility due to the presence of the troll. On the other hand, if the friends are Tesla employees who have accurate information, then the additional information from the acquaintance is less important from the user’s perspective. In this case, the anti-Telsa troll portending to be the acquaintanceislesslikelytoinfluencetheuser’sdecision. Figure1.4: NoTroll,Troll (UnlimitedBandwidth) a) NoTroll,Troll asafunctionofC b) NoTroll,Troll asafunctionofU t - 0 1 t 0.06 Δ 1 0.6 0.7 0.82 0.9 α 0. 0.06 0.04 Δ d= 06 1.2.3 LimitedBandwidth Intheprevioussection,weshow’soptimaldecisionandexpectedutilityunderfullinformation, wheresheobservesthefullsetofsignals. However,inreallife,atypicaluserisunabletoreadand process all posts that are available to her. Time and information processing constraints imply that theuserisonlyabletoreadasubsetoftheposts. Weoperationalizethisconstraintbylimiting’s bandwidth. In our model, limited bandwidth means that can only read and process one signal insteadofthree,receivingimperfectinformation. Thetimingofthegameisillustratedastheright branchinFigure1.3. ThetimelineisthesameasintheunlimitedbandwidthcaseexceptforC= 3, where < consists of only one signal rather than three. Without curation, draws one signal at randomwithprobability 1 3 andupdatesherbeliefbasedontheobservedsignal. As discussed in Section 1.2.1, eitherB orB alone changes ’s belief and optimal decision. For example, ’s belief followingB = ; is less than 1 2 , and therefore her optimal decision is0 ! . On the other hand,B affects’s decision if and only ifC ¯ C. The following Lemma summarizes 14 ’soptimaldecisionrulegivenlimited bandwidth,herexpectedutilitygivenherdecision,andthe totaleffectofthetrollonherwelfare. Lemma 1.2.3.1. Consider the case of limited bandwidth, where ¯ C Ud 0 U¸d 0 1 , 0 ¯ C 1. ’s optimaldecisionruleis: 0= 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ifC ¯ C 8 > > > > < > > > > : 0 ! ifB 9 =; 0 ' ifB 9 =A ifC ¡ ¯ C 8 > > > > < > > > > : 0 ! ifB =; orB =; 0 ' otherwise Theuser’sexpectedutilityis: * Troll = 8 > > > > < > > > > : U 1 3 C¹U¸d 0 1º¸ 2 3 X ifC ¯ C U 1 3 ¹Ud 0 º¸ 2 3 X ifC ¡ ¯ C (1.11) Thedifferencein’sexpectedutility intheabsencevspresenceofthelefttrollis: NoTroll,Troll =* NoTroll * Troll = min 1 3 C¹U¸d 0 1º 1 3 ¹Ud 0 º Proof. SeetheAppendixA. Under limited bandwidth, may take an incorrect action because she receives incomplete information. Forexample,shewouldoptimallytakeaction0 ! basedonadrawofB =;. However, ifthefullsetofthesignalsisfB B B g=f;AAg,thenthecorrectactionunderfullinformation should be0 ' . In addition, receives a social utility of only 2 3 X since she draws a friend’s signal two-thirdsofthetime. Hence,haslowerwelfareherecomparedtotheunlimitedbandwidthcase. ’sinformationloss(measuredby NoTroll,Troll )dependsonthespecificsignalshedraws. Since the troll only affects B , experiences no information loss if she draws B or B . In this sense, the troll’s signal “crowds out" the friends’ signals. Note that’s information loss is nonlinear and decreasing for high values ofU in the unlimited bandwidth case because observing two additional 15 signals reduces the marginal value ofB whenU is high. Since observes only one signal here, the marginal value ofB is always increasing inU. Therefore, ’s information loss under limited bandwidth is linear and non-decreasing inU. WhenC is small, the information loss is increasing inC since a higherC implies a noisierB . WhenC is large, the information loss becomes constant sinceB is toonoisyandlongeraffects’sdecision. 1.3 Curation Before we turn to curation, we need to define a few terms and make a few assumptions. First, weassumethattheplatformhasnobandwidthconstraints. Thatis,theplatformobservesallsignals andisabletoprocessthematnocost. Assumption1.3.0.1. Theplatformobservesallsignals. Second, we assume that both the platform and the user are aware of the presence of the left troll. In addition, the level of troll infiltration, measured byC in our model, is public knowledge. This assumption reflects the widespread awareness of trolls on social media following a series of highlypublicizedinvestigationsintoRussia’smeddlinginUSpolitics.9 Assumption1.3.0.2. TheprobabilityC thattheacquaintanceisalefttrollispublicknowledge. We also assume that the platform’s objective function is centered on the user. In reality, platforms maximize profit from advertising, for example, rather than the user’s expected utility. However, in our stylized model, there is no conflict between maximizing the user’s utility and the firm’s profit. We essentially assume that the firm’s profit is driven by the users’ satisfaction with theplatformthroughretention.10 Assumption 1.3.0.3. The objective function of the curation algorithm is to maximize the user’s expectedutilitygivenherbandwidthconstraint. 9SchiffReport2018: https://intelligence.house.gov/social-media-content/default.aspx 10Facebookstatedthatthecuratedcontentonitsplatformistoinformandentertainuserswhileprioritizingcontent fromfriendsandfamily(Mosseri,2016). 16 Finally, we can turn to the curation algorithm itself. First, note that the algorithm’s objective is to select a subset of signals that maximize ’s expected utility given ’s bandwidth constraint. Therefore, the curation algorithm is simply a mapping from the full set of signals to a subset of signals: (! ˜ (, where ( =fB B B g and ˜ ( (. We refer to this mapping as “information compression." Definition 1.3.0.1. Let( be a set of binary signals, and let ˜ ( be a set of binary signals. The function 5 :(! ˜ (isacompressionif ¡ . Furthermore,itisthefirstbestoptimalcompression iftheuser’s expected information utilityisthesamegiven ˜ ( and(. Sinceweassumethathasthebandwidthtoreadonlyonesignal,thecompressedsetcontains onlyonebinarysignal,B : ,where:2fg. WerefertoB : asthecurated signal,andwerefer to the curation algorithm as a mapping from( toB : . Note that each binary signal contains 1 bit of data (which takes the value of either; orA, and analogously, 0 or 1), and the full set of signals contains3bitsofdata. Bychoosingonesignaltoshowtotheuser,thealgorithmcompresses3bits ofdatainto1bit. Thealgorithmachievesthefirst-bestoptimalcompressionunderlimitedbandwidthifitenables the user to take the optimal action given the observed set of signals within the user’s bandwidth. Thealgorithmcandosobyreplicating’soptimaldecisionintheunlimitedbandwidthcase,which requires that 1) the algorithm is able to perform the mapping from( to0 (the platform can infer whattheuserwouldchoosetodounderfullinformation),and2)thereexistsaone-to-onemapping between0 andB : (theinstructionsabouttheoptimalactioncanbeencodedintothecuratedsignal). Notethatthelatteristheoreticallypossiblesinceboth0 andB : arebinary. However,achievingthe first-best optimal compression alone is not a sufficient condition to achieve optimal curation since theconsumeralsocaresaboutsocialutility,whichisafunctionoftheidentityofthesender. Definition1.3.0.2. Anoptimalalgorithmmaximizestheuser’sexpectedutilitygiventhebandwidth constraint. 17 Finally, here we consider an equilibrium where the consumer chooses to follow the curated signalonlyifitincreasesherexpectedinformationutility. Thisrequirestheconsumertoknowhow thealgorithmworks. Assumption 1.3.0.4. The user knows the mapping function between the full set of signals and the curatedsignal,S!B : . 1.3.1 ThePlatformHasPerfectUserInformation Inthissection,weexplorehowtheperformanceofthealgorithmisaffectedbythepresenceofthe trollwhentheplatformhasperfectinformationabouttheuser’spreference,whichisoperationalized asherpriorbeliefaboutthestateoftheworld. Thisrepresentsalltheinformationshehasgathered independently both on the platform and offline. For example, the information that the user is exposed to on the platform up to any point in time can be observed through the type of post she hasinteractedwith(e.g.clicking,liking,commentingon,orsharing)aswellasthefrequencyofthe interactions. However,theusermayalsoobtaininformationfromothersoffline,whichtheplatform does not observe. Therefore, even if the user’s activity on the platform is perfectly observed, and a large amount of data is collected on the user (disregarding a potential violation of privacy), the platform’sinformationontheuserisstillnotperfect. Inthissection,wefirstexploreahypothetical case where the platform has perfect knowledge about the user’s prior. The results in this section serve as a benchmark for the welfare comparisons between when the platform has perfect user informationandwhenitdoesnot. Definition 1.3.1.1. The platform has perfect user information if it observesfB B B g (which includes the identity of each sender and the strength of the connection between the sender and the user),d 0 ,andC. We show that the following algorithm maximizes ’s expected total utility by 1) maximizing ’sexpectedinformationutilitythroughoptimalcompression,and2)maximizing’ssocialutility byprioritizingafriend’ssignal. 18 Lemma1.3.1.1. Thefollowingalgorithm,denotedby! " ¹Cº :S!B : ,isoptimal. • IfC ¯ C: B : = 8 > > > > < > > > > : B iffB B B g2ff;;;gf;;Agf;A;gfA;Agg B iffB B B g2ffAAAg¹AA;º¹A;;º¹;AAºg • IfC ¡ ¯ C: B : = 8 > > > > < > > > > : B iffB B B g2ff;;;gf;;AgfA;AgfA;;gg B iffB B B g2ffAAAgfAA;gf;AAgf;A;gg Proof. Seetheappendix. The algorithm works as follows. If both and send;, thenB =; is selected as the curated signal. Similarly, if both and sendA, thenB =A is selected as the curated signal. If and send opposite signals, then the curated signal depends on the level ofC. IfC ¯ C, the majority rule applies. That is, an; from a friend is selected if the number of; signals exceeds the number ofA signals. Similarly, anA signal from a friend is selected if the numberA signals exceed the number of ; signals. In other words, the algorithm selects the friend’s signal that matches the majority opinion. Ontheotherhand,ifC ¡ ¯ C andthefriends’opinionsaresplit,thealgorithmalwaysselects theA signal from the friend who sent it. In this case, the friend’s signal that matches ’s prior belief is selected. For example, iffB B B g=f;A;g, then the algorithm selectsB =A as the curated signal. Essentially, the algorithm’s decision regarding whether the acquaintance’s signal breaksthetiebetween and dependsonthecutoff ¯ C. Underthecurationalgorithm,theuser() whoobservesthecuratedsignalB : =; infersthatthiswasthemajoritysignalifC ¯ C orinfersthat both friends sent; ifC ¡ ¯ C. Similarly, the user who observes the curated signalB : =A infers that either this was the majority signal (in the case ofC ¯ C) or infers that at least one friend sentA (in 19 thecasewhereC ¡ ¯ C). Giventhis,’sposteriorbelieffollowingthecuratedsignalis: \¹B : =;º= %A¹B : =;j'º%A¹'º %A¹B : =;j'º%A¹'º¸%A¹B : =;j!º%A¹!º (1.12) \¹B : =Aº= %A¹B : =Aj'º%A¹'º %A¹B : =Aj'º%A¹'º¸%A¹B : =Aj!º%A¹!º (1.13) IfC ¯ C,then %A¹B : =;j'º=%A¹#; ¡ #Aj'º=¹1Uº 2 ¸ 2U¹1Uº 2 ¸ 2CU 2 ¹1Uº (1.14) %A¹B : =;j!º=%A¹#; ¡ #Aj!º=U 2 ¸ 2U 2 ¹1Uº¸ 2CU¹1Uº 2 (1.15) IfC ¡ ¯ C,then %A¹B : =;j'º=%A¹#; ¡ #Aj'º=¹1Uº 2 (1.16) %A¹B : =;j!º=%A¹#; ¡ #Aj!º=U 2 (1.17) takes action 0 ! if her posterior belief given the curated signal is less than 1 2 , or %A¹B : = ;j'º 1 2 . Otherwise, she takes action 0 ' . Re-writing this inequality in terms of C, we obtain C U¹1Uº¹2U 1º¸¹Ud 0 º 2U¹1Uº¹U¸d 0 1º ˜ C. This upper bound onC implies that would take an action in the same direction as the curated signal in equilibrium, which we refer to as "following" the curatedsignalaslongasC islessthanorequalto ˜ C.11 Note that ˜ C and ¯ C have different interpretations. In the unlimited bandwidth and limited band- width case, ¯ C is the cutoff above which discardsB . Under curation, ˜ C is the cutoff above which discards the curated signal. Since the platform knows ’s prior, it can derive ’s cutoff for discardingB , ¯ C,anduseittodeterminehowthesignalsshouldbeaggregated. Inotherwords,here theplatform’scutofffordiscardingB isalso ¯ C,coincidingwiththeuser’s. Since ˜ C isalwaysgreater than ¯ C undertheoptimalcurationalgorithm,theconditionC ˜ C isalwayssatisfied. Thisimpliesthat under the optimal curation algorithm, always follows the curated signal. By Definition 1.3.0.1, the curation algorithm achieves the first-best optimal compression of the signals since takes the 11ThereisanupperboundonU forwhich ˜ C 1: U ¹1d 0 º¸ q 13d 0 ¸3d 2 0 2d 0 1 ¯ U. AsU! 1, ˜ C!1. SinceC2»0 1¼, ˜ C!1impliesthatasU becomesveryhigh, always follows the curated signal. 20 sameactionunderthecurationalgorithmasshewouldintheunlimitedbandwidthcase. Moreover, the algorithm maximizes ’s social utility by always showing her a friend’s signal that matches the optimal action, which guarantees that she receives a social utility ofX. Hence, by Definition 1.3.0.2, this is an optimal curation algorithm given ’s bandwidth constraint. Note, however, that theoptimalcurationalgorithmisnotunique. Claim1.3.1.1. Theoptimalalgorithmisnotunique. To show this, consider an alternative algorithm ! "2 ¹Cº (presented in Appendix A) that is identicalto! " ¹Cº exceptforthefollowingchange: inthecasewherebothfriends’signalsarein the same direction (e.g.fB B B g=f;;;g), the algorithm shows eitherB =; orB =; with equalprobability. makesthesameoptionaldecisionunderthisalternativealgorithmandreceives the same expected utility as under ! " ¹Cº. The results under optimal curation as summarized below. Lemma1.3.1.2. Considerthecaseoftheuserwithlimitedbandwidthconstraintandtheplatform with perfect user information. The platform can construct an (optimal) curation algorithm that achievesthefirstbestoptimalcompression,where’sequilibriumdecisionruleis: 0= 8 > > > > < > > > > : 0 ! ifB : =; 0 ' ifB : =A (1.18) whichgives thefollowingexpectedutility * curation A = 8 > > > > < > > > > : U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º¸X ifC ¯ C U 2 ¸ 2U¹1Uºd 0 ¸X ifC ¡ ¯ C The difference between the user’s expected utility under the first best optimal curation and under 21 nocuration, DA0C8>=! ,is DA08C>=! =* curation * ! = 8 > > > > < > > > > : U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1ºU¸ 1 3 C¹U¸d 0 1º¸ 1 3 X 0 ifC ¯ C U 2 ¸ 2U¹1Uºd 0 U¸ 1 3 ¹Ud 0 º¸ 1 3 X 0 ifC ¡ ¯ C Proof. SeetheAppendixA. Undernocuration,thetroll’ssignal“crowdsout"thesignalsof’sfriendssinceobservingB implies that does not observeB orB . Under curation, on the other hand, all signals are taken into account by the algorithm, and a friend’s signal is always selected. By observing the curated signal,isabletomakeaninferenceabouttherealizationofthefullsetofsignalsandtakethesame action as she would under full information. Therefore, here obtains the same level of expected information utility as in the unlimited bandwidth case. Note that due to her bandwidth constraint, receives a social utility of X under the curation algorithm instead of 2X in the hypothetical case of unlimited bandwidth. Nonetheless, still enjoys a higher social utility compared to the limited bandwidth case without curation, where she only receives an expected social utility of 2 3 X. Therefore,’sgaininexpectedsocialutilityundertheoptimalcurationalgorithmis 1 3 X. The extent of the informational gain due to the curation algorithm depends on the precision of the signals. In a “split case" where and send opposite signals, new information only comes throughtheacquaintance. WhenU isrelativelylow(around’sprior),theprobabilityofasplit case is high. Therefore, the benefit of compression with respect to gaining additional information is small. On the other hand, whenU is high, and are more likely to send the same signal that is also likely to be correct about the state of the world. In this case, compression is very useful. By aggregating the signals across ’s connections, the curation algorithm achieves more precise information and minimizes the effect of the potential troll’s signal. AsU increases, the benefit of compressionalsoincreases. However,whenU isveryhigh(closeto 1),thebenefitofcompression diminishes since any one signal drawn at random is already highly informative. The marginal 22 informationvalueofadditionalsignalsbecomesrelativelyinsignificant. Therefore,theinformation gain under curation first increases for intermediate values ofU and then decreases for very high valuesofU,exhibitinganinverseU-shape. Insummary, theoptimal curationalgorithm solves’sbandwidth problemby determiningthe optimal action given ’s prior and the full set of signals and then encoding the instruction for the optimalactionintoacuratedsignal. Inequilibrium, followsthecuratedsignalandbecomesjust as informed with respect to her action as under full information. Therefore, is always better off under the optimal curation algorithm with perfect user information relative to no curation, even in thepresenceoftrolls. Proposition1.3.1.1. Optimalcurationincreasestheuser’sexpectedutilityinthepresenceoftrolls iftheuserhaslimitedbandwidth,andtheplatformhasperfectuserinformation. Proof. SeeAppendixA. Despite its benefit, the optimal curation algorithm comes with one important caveat. That is, does not observe the full set of signals nor the identities of all the senders, even though this informationallowstoproperlydiscountthetroll’snoiseanddeterminethecorrectactiontotake. Withoutobservingsuchinformation, reliesonthecurationalgorithmtomaketherightdecision for her. We have shown that since the curation algorithm is optimized specifically for ’s prior, it acts as a perfect agent for and never gives an incorrect instruction, i.e. one that is inconsistent with ’s optimal decision under full information. However, the algorithm’s judgment may not be reliable if it is not optimized for ’s true prior. This problem arises when the platform does not haveperfectinformationon,acaseweexploreinthenextsection. 1.3.2 ThePlatformHasImperfect UserInformation In the preceding section, we show that curation with perfect user information always increases the welfare of the user even in the presence of the troll. However, this result relies on the strong assumption that the platform’s information on the user is perfect, which is unlikely to hold in 23 reality. The platform may obtain some information on the user by observing her online activity, but such information is incomplete since her offline activity, which may also affect her prior, is unobserved. Furthermore, a new user would have little online activity that the platform could observe. Therefore, it is nearly impossible for even the most sophisticated curation algorithms to acquire precise knowledge about the user. In this section, we relax the assumption of perfect user information. Inparticular,weassumethattheplatformdoesnotknow’sprior,and isawareof thisuncertaintyonthepartoftheplatform. Assumption1.3.2.1. TheplatformhasimperfectuserinformationifitobservesfB B B g(which includes the identity of each sender and the strength of the connection between the sender and the user)andC,butdoesnotobserved 0 . Suppose that the platform assumes a prior of 1 2 12. Consider the following algorithm, ! " , basedonthisassumedprior: B : = 8 > > > > < > > > > : B iffB B B g2ff;;;gf;;Agf;A;gfA;Agg B iffB B B g2ffAAAg¹AA;º¹A;;º¹;AAºg (1.19) Note that this leads to a mapping from( toB : that is simply the“majority rule" regardless ofC (for simplicity,weomitC from! " ¹Cº): B : = 8 > > > > < > > > > : ; if #; ¡ #A A if #A ¡ #; (1.20) The above algorithm is optimal from the platform’s perspective. For a user with a neutral prior, even a very noisy signal would move her belief to the left or right of 1 2 . This implies that such a hypothetical user would follow the majority opinion of her connections even if B is very noisy. Therefore,theoptimalalgorithmalwaysshowsafriend’ssignalthatisconsistentwiththemajority opinion. Under this curation algorithm, ’s posterior belief following the curated signal is given 12We provide an example of a distribution of priors for which ! " is the optimal algorithm, or in other words whentheplatform’sassumptionofaneutralprior is correct, in the Technical Appendix 24 by Equation (1.12) and (1.13) in the previous section. takes action0 ! if\¹B : =;º 1 2 and0 ' otherwise. Re-writing this inequality in terms ofC, we obtain the same cutoff ˜ C as in Section 1.3.1 withrespecttowhether followsthecuratedsignal. DifferentfromthepreviousSection,herethe algorithmdoesnothaveacutoffonC sinceitneverdiscardsB .13 Although the current algorithm is optimal with respect to the assumed neutral prior, it is imperfect since it does not achieve the first-best optimal compression. In particular, it would sometimes make a mistake from ’s perspective. Consider the casefB B B g =f;A;g and C ¡ ¯ C. The correct curated signal given ’s true prior and the full set of signals should beB =A. However, the algorithm based on a neutral prior would instead select B = ;, representing the majority opinion. Nonetheless, the information encoded in the curated signal can still be useful to . For example, by observing B = ;, can infer that the majority opinion is; and at least one of her friends sent;. Alternatively, can reject this information and choose to take her default action. She faces a trade-off between accepting this information with a margin of error and not accepting any information at all. In equilibrium, chooses to follow the curated signal as long as doingsoincreases herexpectedinformationutilitycompared torejectingthisinformation. Thisis truewhen theprobabilitythealgorithm’serrorissufficientlylow,orC ˜ C. The following Lemma summarizes ’s equilibrium decision rule and expected total utility underthecurationalgorithmwhere’spriorisassumedtobe 1 2 ,aswellastheeffectofthetrollon herwelfare. Lemma 1.3.2.1. Consider the case of the user with limited bandwidth constraint and the plat- form with imperfect user information, where ˜ C = U¹1Uº¹2U 1º¸¹Ud 0 º 2U¹1Uº¹U¸d 0 1º , 0 C 1. ’s 13In the extreme case whereC = 1, the user with a neutral prior would be indifferent between taking action0 ! and 0 ' . Weassumethatinthiscasethealgorithmstill shows the majority signal. 25 equilibriumdecisionruleis: 0= 8 > > > > > > > > < > > > > > > > > : ifC ˜ C 8 > > > > < > > > > : 0 ! ifB : =; 0 ' ifB : =A ifC ¡ ˜ C 0 ' (1.21) Theuser’sexpectedutilityundercurationis: * curation A = 8 > > > > < > > > > : U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º¸X ifC ˜ C d 0 ¸X ifC ¡ ˜ C (1.22) Thedifferencebetweentheuser’sexpectedutilityundercurationandundernocuration, DA0C8>=! , is DA0C8>=! = 8 > > > > < > > > > : U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1ºU¸ 1 3 C¹U¸d 0 1º¸ 1 3 X ifC ˜ C d 0 U¸ 1 3 ¹Ud 0 º¸ 1 3 X ifC ¡ ˜ C (1.23) Proof. SeeAppendixA. The algorithm imperfectly “bundles" the troll’s signal with the friends’ signals; that is, the algorithm groups different realizations of the full set of signals together for which the same instruction is given. For example, here the algorithm groups all cases where the ; signal is the majorityandinstructstheusertotakeaction0 ! . ThisisthecorrectbundlingifC ¯ C,butnotifC ¡ ¯ C. InthecasewhereCisbigenoughsuchthatthebundlingisincorrectandyetsmallenoughthattheuser still follows the algorithm’s instruction, may be misled by the algorithm’s incorrect instruction. There is a positive probability of this occurring since ¯ C C ˜ C is a non-empty set. Moreover, when the probability of the algorithm’s error is sufficiently high (C ¡ ˜ C), ultimately disregards the curated signal and receives no improvement in her expected information utility relative to no curation. Therefore, may not always be better off under curation. We summarize our results on ’swelfareundercurationwithimperfectuserinformationinthefollowingproposition. 26 Proposition 1.3.2.1. Consider the case where the user has limited bandwidth and the plat- form has imperfect user information, where ˜ C = U¹1Uº¹2U 1º¸¹Ud 0 º 2U¹1Uº¹U¸d 0 1º , and ˆ C = ˜ C 1 3 2¹Ud 0 ºX 2U¹1Uº¹U¸d 0 1º 1. IfX 2¹Ud 0 º, then the social value of being connected to friends is so high that the user alwaysobtainshigherexpected utilityundercurationthanundernocuration. 2. If X 2¹Ud 0 º, then the user obtains higher expected utility under curation than under no curation whenC ˆ C, and receives lower expected utility under curation than under no curationwhen ˆ C C ˜ C. Proof. SeeAppendixA. The central idea of Proposition 1.3.2.1 is that the user could be worse off under the curation algorithm with unknown prior, even if it is optimal (from the platform’s perspective). Panel a) of Figure1.5showsthat’sexpectedutilityundercurationislowerthanherexpectedutilityunderno curation whenC is large. There are two effects. First, follows the curated signal and sometimes takes an incorrect action whenC is in an intermediate range, or ˆ C C ˜ C. Second, for a largeC (C ¡ ˜ C), no longer follows the curated signal and receives no new information under curation. In this case, a randomly drawn signal may provide more information. However, if the social utility from receiving the friends’ signals is sufficiently large, is still better off under curation since the higher social utility can offset the lack of information utility. This trade-off is given by the expression X 2¹Ud 0 º. The difference between U and d 0 measures the change in ’s belief when she is exposed to new information (the signals). The bigger is the change in ’s belief, the biggeristheinformationvaluerelativetothesocialvalue. Forexample,intheextremecasewhere U= d 0 , there is no change in ’s belief after receiving a signal. Panel b) of Figure 1.5 shows that when the social value is sufficiently high, orX ¡ 2¹Ud 0 º, ’s expected utility under curation is higherthanthatundernocuration. 27 Figure1.5: ’sExpectedUtilityUnderCurationvsNoCuration EU Curation A EU LB A t ˜ 0 1 t  0 1 t 0 1 t 0.95 0.8 0.83 EU Curation A a) δ =0.05 EU Curation A EU LB A t ˜ 0 1 t  0 1 t 0 1 t 1.05 0.9 EU Curation A b) δ =0.15 d 0 = 075U= 08 Theadvantageofcurationisimproving’swelfarethroughincreasingbothherexpectedinfor- mationutilityandexpectedsocialutilitywhilesatisfyingherbandwidthconstraint. Toaccomplish this,thecurationalgorithmcompressesthefullsetofsignalsintoonesignalthatinstructstheuser which optimal action to take. In addition, the curation algorithm always shows a friend’s signal in ordertomaximizeherexpectedsocialutility. Thecaveatofsuchasystemisthatdoesnotobserve nor can she reverse the compression to recover the original set of signals. More importantly, she does not observe the troll’s signal, which prevents her from properly discounting it. For example, observingB =; onlytellsthatthemajorityopinionis;,but’soptimalactiondiffersdepending on the composition of the majority. Not knowing this information creates additional uncertainty that does not exist under no curation, where randomly draws a signal and always observes the identityofthesender. Usingtheexampleweintroducedatthebeginningofthepaper,if seesan anti-vaxpostfromanacquaintancewithwhomsheonlyinteractsonline,shecandiscounthispost basedonhowlikelyheisatroll. Under no curation, the troll can only affect when she drawsB andC ¯ C. Under curation, however, the troll can affect whenever her friends’ opinions are divided andC ˜ C, in which case the curated signal is biased toward the troll’s agenda. In our modelC is assumed to be exogenous 28 and the troll cannot change the level ofC. We will discuss more about this in the next paragraph. Curationamplifiestheeffectofthetrollbyallowinghimtoinfluence anonymously(inthesense that doesnotobservehissignal)andindirectly(throughherfriends). Ourmodelshowsthatitis easier for the troll to push his agenda when the opinions among legitimate users are divided. This theoretical result is consistent with the empirical observations that Russian trolls focus on highly divisiveissuessuchasrace,abortion, andgunownershiprights.14 To summarize, curation improves ’s welfare if 1) the platform has perfect information about herpriorbelief,and2)theleveloftrollnoiseontheplatformissufficientlysmall. Thisimpliesthat the platform could improve the performance of the curation algorithm by either acquiring perfect user information or lowering the level of troll infiltration (C). Acquiring perfect user information may be unfeasible since even if the platform is able to collect all user data online, it still does not observetheusers’activitiesandinteractionsoffline. Moreover,attemptingtocollectalargeamount of information on each user may violate privacy regulations and also alienate users who wish to protecttheir ownprivacy. Therefore, amorefeasible waytoimprove theefficiencyof thecuration algorithmmaybetoreduceC. Inourmodel,C isassumedtobeexogenous. Inreality,social media platforms have the technology to detect and remove troll accounts. However, this technology can only curtail the level of troll activity and does not eliminate trolls because they can re-join the platformundernewassumedidentitiesorusesophisticatedmethodstoevadedetection(atleastfor alongenoughtimetohaveanimpactonlegitimateusers). Therefore,C canbeendogenizedwhere theplatformimplementsacostlytechnologytoloweritslevel. 1.4 Conclusion Curationsolvestheuser’sproblemoflimitedbandwidthbycompressingallinformationintoa curated signal. We show that in the presence of trolls, the effectiveness of the curation algorithm dependsontheplatform’sknowledgeabouttheuser. Whentheplatformknowstheuser’sprior,the curationalgorithmachievesfirst-bestoptimalcompressionandalwayscorrectlybundlesthesignals. 14https://www.nytimes.com/2017/10/02/technology/facebook-russia-ads-.html 29 That is, the algorithm correctly reconstructs the mapping from the full set of signals to the user’s optimal action, replicating her optimal decision rule under full information. In equilibrium, the user follows the curated signal and obtains the same information utility as under full information. Moreover,thecurationalgorithmimprovestheuser’ssocialutilitybyalwaysshowingherafriend’s signal. However, if the platform does not know the user’s prior, the curation algorithm cannot achieve first-best optimal curation. In this case, the algorithm always bundles cases that have the same majorityopinion. Theuserrealizesthisandinfersthattheinformationcuratedtoherrepresentsthe majority opinion among her connections. However, she does not observe which senders make up themajority,whichiscrucialfordeterminingtheoptimalaction. Ifthemajorityconsistsofoneonly friend and the acquaintance, following the majority opinion may not always be the optimal action when there is a high probability that the acquaintance is a troll in disguise. Hence, the bundling basedonthemajorityopinionissometimesincorrectfromtheuser’sperspective. Moreover,since the user is unable to directly observe the original set of signals, she cannot distinguish different cases of majority. Instead, she must choose between taking the algorithm’s bundling with a grain ofsaltorignoringitandgivinguppotentiallyusefulinformation. In equilibrium, the user would follow the curated signal as long as the probability of the algorithm’smis-bundlingorerrorisnottoohigh. Sincethetroll’ssignalisnevershowntotheuser underthecurationalgorithm,itinasense“launders"theidentityofthetrollbyhidinghissignalin thebackgroundandyetallowingittoaffectwhatcontentisshowntotheuser. Asaresult,curation with imperfect user information sometimes misleads the user, making her worse off compared to no curation. The presence of the troll, in general, lowers user welfare regardless of the use of curation, but the troll has an additional effect on the user when her friends’ opinions are divided, a feature unique to curation. We show that information compression as a solution to the user’s bandwidthproblemisno“freelunch": itincreasestheamountofinformationreceivedbytheuser attheexpenseofamplifyingtheeffectoftrolls. Whilethischapteristhefirststepinunderstanding howcurationalgorithmsperforminthepresenceoftrolls,real-lifesocialmediaplatformsaremuch 30 more complex environments than the simple network we model. Building on our model, there are severalinterestingfuturedirectionsofresearchtobeexplored. 1.4.1 FutureResearch Our parsimonious analytical model has several limitations. First, the current model assumes thatsignalshavethesameprecision,andtheplatformperfectlyobservestheprecisionofthesignals. This is unlikely to hold in reality and more complex networks. For example, some friends may have extreme beliefs and tend to send one type of signal. We also assume that all signals are independent,buttheycouldbeinfactcorrelatedamongfriendssincetheyoftencross-shareposts. This is more likely to be the case when the user and the senders have many mutual friends. Since friends’ signals are correlated, simply aggregating the signals produces little new information. In ourmodel,theplatformimplicitlyputsequalweightsonallsignalssincetheyareindependentand have the same precision. However, in the case of correlated signals and different signal precision, the platform needs to weight the signals differently. One direction for future research is to study howtoconstructtheoptimalalgorithmgivenamorecomplexinformationstructure. Second, we assume that the platform has information about all the signals sent by different users. In addition to the information from the user’s social connections, the curation algorithms employed by real-life social media platforms also take into account the information beyond the user’s immediate network. On Facebook, the user may see popular content from public pages. However, this type of algorithm may be more susceptible to the influence of trolls who not only lurk in the networks of numerous users but also create public pages and produce content that gains popularity among authentic users. Facebook recently uncovered a large number of public Facebook pages that appeared to belong to social activists, but they were linked to the Russian Intelligence Agency15. Another direction of research is to study how the presence of trolls affects theperformanceofalgorithmsthatincorporatepublicinformation. Furthermore, we assume that all information is common except for the user’s prior. In reality, 15https://newsroom.fb.com/news/2018/11/last-weeks-takedowns 31 theremaybemorediscrepanciesbetweenwhattheuserknowsabouttheplatformandviceversa. For one,theplatformandtheusercouldhavedifferentperceptionsofthesignalprecision. Forexample, suppose that the platform observes that a user always shares anti-vax content with his connections and determines that his signal is not very informative. However, the individual connections of this user may only see his post once and do not observe a holistic picture of his behavior on the platform. Therefore, this user’s post may appear more informative to his connections than to the platform. In addition, the user may not be fully aware of how the algorithm works. That is, the user may mistakenly believe that the curated algorithm is optimized for her prior. Hence, the third directionoffutureresearchistointroduceahigherdegreeofasymmetricinformation. Inaddition,trollsarestrategicagentswhorespondtoboththecurationalgorithmaswellasthe platform’ssecuritymechanismdesignedtopolicethem. Ontheotherhand,wehaveshownthatthe platformcanincreasetheefficiencyofthealgorithmbyloweringtheleveloftrollnoise. Therefore, the fourth direction of future research is to endogenize the presence of trolls and model them as a player in the game. Our model can also be generalized to more complex scenarios and networks. Amodelthatincorporatesmanyoftherealisticfeaturesmaynolongerbemathematicallytractable orhavecloseformsolutions. Instead,simulationsareappropriateforanalyzingcomplexnetworks. Lastbutnotleast,wecanalsotestthepredictionsofourmodelonanempiricalnetworktakenfrom TwitterorFacebook. 32 Chapter 2 InfluencingtheInfluencers 2.1 Introduction Social media influencers are category enthusiasts (for example, fashion, beauty, fitness, auto, photography,technology)whogainfollowersduetothehighqualityofthecontentthattheyproduce and share. Given the importance of influencers’ recommendations on consumers’ decisions, firms oftenexpendresourcesonmanagingtheinfluencers,apracticeknownas“influencermarketing."1 In particular, we focus on a certain aspect of influencer marketing: the firm’s management of the influencers’ product reviews. We use examples from the beauty category, which is particularly active in terms of influencer activity. In our examples, we focus on mid-level influencers with a large enough following for review management to make sense, but not so large such that the firm wouldtailortheproductdesigntofittheinfluencer’staste.2 Towhatextentaresocialmediainfluencerssimilartotheopinionleaderswhohavebeenstudied beginningwith KatzandLazarsfeld(1955)? Clearly,influencersareopinionleadersinthatothers seektheiradvice,andtheyinfluencealargenumberoftheirpeers. Whatisnovelisthatsocialmedia influencers publicly post their content, and hence their influence extends beyond their immediate socialnetwork. Thisvisibilityalsoenablesfirmstotargettheminordertomanagetheirreviews. Therearetwotypesofinfluencerreviews—affiliatedandindependent,dependingonwhether there is a relationship between the firm and the influencer. As an example of an affiliated review, consideravideopraisingtheL’OccitaneAlmondShowerOilbyCarolineHirons,abeautyinfluencer with 220k YouTube subscribers.3 We also observe independent influencer reviews of beauty 1It is hard to estimate the exact size of the market, but one industry study estimates that it is around $2 billion (Mediakix,2017)acrossallcategories. 2The size of the subscriber base of the influencers we use is in the ten to hundred thousand range. In contrast, for the top 10 beauty influencers based on the rankings in Forbes (2017) , the subscriber base ranged from 2.6 to 11 millionaccordingtoYouTubedataobtainedon Nov 27, 2018. 3Thedisclosurestates“ThispostissponsoredbyL’Occitane."https://www.youtube.com/watch?v=A9KpEEAhe5w 33 products. For example, consider two independent reviews (one positive and one negative) of the BeccaAqua LuminousPerfecting Foundation. Mallory Cornelison,a beautyinfluencer with 103k YouTubesubscribers,praisedtheproductfordeliveringjusttherightamountofnaturalglowtoher skin.4 Incontrast,BriannaStanko,abeautyinfluencerwith178kYouTubesubscribers,complained thatthefoundationdidnotdeliverenoughglowtoherskin.5 What is the trade-off that the firm faces when choosing whether and how closely to affiliate withtheinfluencer? SinceFTCregulationsrequireinfluencerstopubliclydiscloseanyrelationship with brands,6 a positive recommendation from an independent influencer is more credible than a positive recommendation from an influencer who has a close relationship with the firm. However, firmsmaybenefitfromincreasedcontrolovertheinfluencer’smessagethatcomeswithaffiliation. For instance, had Becca affiliated with Stanko, it may have prevented a negative review of its foundation. We ask the following research questions. First, when does the firm want to affiliate with the influencer versus allow her to remain independent? Second, what is the optimal level of affiliation? Third, since affiliated reviews are often biased in favor of the firm, what is the impact of affiliation on consumer welfare? Finally, to what extent do different disclosure regimes (the amountofdetailthattheinfluencerdisclosesaboutherrelationshipwiththefirm)affectbothprofit andconsumerwelfare? Weproposeamodelwiththreeplayers: thefirm,theinfluencerandthefollower(theconsumer). Initially,noneoftheplayersobservethefitoftheproducttothefollower. Theinfluencercanacquire asignalontheproductfitinordertowriteaproductreview,whichmaybeeitherpositiveornegative, dependingonthesignal. Thereviewprovidesinformationtotheconsumerintwoways: 1)itraises awareness of the product, and 2) it changes the consumer’s beliefs about the product fit, which we refer to as the “persuasiveness" of the review. Both of these factors determine the consumer’s value of information from the review, which impacts both the influencer’s and the consumer’s utility. The firm may try to manipulate the review process by offering the influencer payment in 4https://www.youtube.com/watch?v=XfDfe1V3SAE 5https://www.youtube.com/watch?v=j2JAYiCRuEY 6https://www.ftc.gov/tips-advice/business-center/guidance/ftcs-endorsement-guides-what-people-are-asking 34 exchange for affiliation, where an increase in the affiliation level increases the probability that the influencer posts a positive review. The affiliation contract must satisfy the influencer’s incentive compatibility constraint since she can choose to remain independent. Since the influencer must disclosetheprecisenatureofheraffiliationwiththefirm,ahigherlevelofaffiliationdecreasesthe persuasivenessofthepositivereview. Therefore,thefirmfacesatrade-offbetweentheprobability andthepersuasivenessofthepositivereview. We show that the firm’s optimal decision to affiliate with the influencer depends on the con- sumer’s prior belief and awareness. In equilibrium, the firm’s payment to the influencer makes her exactly indifferent between accepting the positively-biased affiliation contract and remaining independent. When the consumer’s prior belief on the product fit is relatively low, an aware con- sumer does not buy based on the prior, but a positive review may convince him to buy. Hence, in order to induce a purchase, the positive review needs to be persuasive enough. Here the firm introducessomepositivebiasintothereviewprocessbutdoesnotfullyaffiliatewiththeinfluencer. In equilibrium, affiliation is set such that the consumer is just indifferent between buying or not buyingtheproductfollowingapositivereview. Sincetheconsumereitherdoesnotbuyorisindif- ferent between no purchase and purchase, the consumer’s value of information from the affiliated review is exactly zero: the firm destroys the review’s value of information. In contrast, when the consumer’s prior belief is relatively high, an aware consumer’s default action is to purchase. Hence, the persuasiveness of a positive review is irrelevant from the firm’s perspective. On the otherhand,thefirmbenefitsfrompreventinganegativereviewfrombeingpostedsinceitdissuades the consumer from buying. Therefore, here the firm chooses to fully affiliate with the influencer to both maximize awareness and to prevent a negative review. While the firm destroys the value of information from persuasiveness, it does not destroy the value of information from awareness. In fact, the consumer strictly prefers for the firm to be involved in the case where the independent influencerchoosesnottopostareview,andtheprioronproductfitishigh. While we assume full disclosure in the base model, in practice it is often difficult for the influencer to disclose the precise terms of her affiliation contract, especially if the terms of the 35 contractarecomplex. Forexample,whilemanyinfluencersdisclosethattheyreceivefreeproducts frombrands,itisuncleartowhatextenttheyskewtheirreviewsoftheseproducts. Inanextension to the model, we consider the case where the influencer is able to only credibly communicate whether or not her review is affiliated or independent but not the exact level of affiliation. In this scenario, we find that the firm prefers to let the influencer remain independent if the prior is low. That is, partial disclosure forces the firm not to corrupt the influencer’s review, which benefits the consumer. Hence, a less stringent disclosure regime, in this case, benefits the consumer and hurts thefirm. This chapter contributes to the literature on the firm’s management of social interactions. In particular, Mayzlin (2006) and Dellarocas (2006) model the surreptitious manipulation of word ofmouthbythefirm. Incontrast,weexaminehowthefirmcanmanipulatetheinfluencer’sreviews through publicly-disclosed affiliation. Substantively, the current study is also closely-related to Mitchell (2017) , who models the influencer’s trade-off between offering advice and posting ads. One important difference between ours and the other paper is that while we explicitly model the interaction between the firm and the influencer, in Mitchell (2017) the advertising market is exogenously given. Finally, the firm’s manipulation of the review process is a form of Bayesian persuasion. In particular, in Kamenica and Gentzkow (2011) , the sender (in our case, the firm) manipulates the information structure (or the review process) to maximize own payoff. Our theoretical contribution is in modeling the information structure as the outcome of the interaction between three players: the firm, the influencer, and the consumer. Hence we endogenize the cost tothefirmofmanipulatingtheinformationstructureofthegame. The structure of this chapter is the following. In Section 2.2, we discuss the related literature. In Section 2.3, we analyze our main model, where the influencer must fully reveal the amount of affiliation. In Section 2.4, we analyze an extension where only partial disclosure is possible. We concludeinSection2.5. 36 2.2 LiteratureReview Our work relates to two broad literature streams: 1) the empirical and the theory literature on word of mouth and the firm’s management of social interactions, and 2) the theory literature on communication and persuasion. We discuss how our work relates to these two streams in turn below. Asocialmediainfluencerisaconsumerwhoisalsoanopinionleaderinaspecificproductcat- egory. Inthissense,theinfluencer’srecommendationsarerelatedtoconsumerwordofmouth. The existing literature demonstrates that word of mouth causally impacts purchase decisions (Cheva- lier & Mayzlin, 2006; Chintagunta et al., 2010). Most studies show that positive word of mouth increasessales,whilenegativewordofmouthdecreasessales(Rosarioetal.,2016). Giventheimportanceofwordofmouthonproductadoption,somefirmstrytomanagewordof mouth or, more broadly, social interactions (see Godes et al. (2005) ). One way to manage social interactionsistoencourageexistingconsumerstoengageinwordofmouth. Thequestionofwhich consumersthefirmshouldtargettoseedtheconversationsinasocialnetworkhasgeneratedmuch debate in the literature. For example, Watts and Dodds (2007) suggests that easily-influenced consumersshouldbetargeted,while Hinzetal.(2011) showsthatthefirmbenefitsfromtargeting the better-connected consumers. In addition, empirically the effect of opinion leadership on the firm’s sales has been shown to depend on a number of factors, such as the sender’s loyalty to the product (Godes & Mayzlin, 2009), whether the stage is trial or repeat (Iyengar et al., 2011), and the type of relationship between the sender and receiver (Chen et al., 2017). Several recent empirical papers have shown that firms can manage word of mouth by directly responding to consumers’ reviews online (Chevalier et al., 2018; Ma et al., 2015; Proserpio & Zervas, 2017; Wang&Chaudhry,2018). Inadditiontoencouragingconsumerstoengageinwordofmouth,thefirmcansurreptitiously manipulateonlinewordofmouththrougheithermanufacturingpositivewordofmouthforitsown product or negative word of mouth for the rival, “promotional chat” (Dellarocas, 2006; Mayzlin, 37 2006; Mayzlin et al., 2014). In these papers, consumers infer the amount of fake word of mouth in equilibrium and are more skeptical of online recommendations. Despite this, word of mouth affects sales as long as there is enough real word of mouth, and the costs of faking word of mouth are not too low. Our research is closely related to this literature since we also address the question ofthefirm’smanipulationofonlinerecommendations. Intermsofresults,inboththesepapersand our work, the firm’s involvement renders recommendations less persuasive. One major distinction between our work and this literature is that in our model, the firm’s manipulation of reviews by influencersispubliclydisclosed,andhencetheconsumersareabletodirectlyobservewhetherand how closely the influencer is affiliated with the firm. Other distinctions arise from the specifics of the social media influencer context. By definition, there are fewer social media influencers, and their popularity is observable. Hence, it is feasible for the firm to affiliate with the sender in order to prevent a negative review. In contrast, it seems less feasible for the firm to prevent the spread of negative word of mouth through affiliation. Hence, our model is not equivalent to promotional chatwithdisclosure. The result that the firm’s involvement negatively affects the persuasiveness of the message is consistent with the consumer behavior literature that shows that consumers tend to be skeptical toward messages that are designed to persuade them. See, for example, Campbell and Kirmani (2000) and Campbell et al. (2013) ). In fact, there is empirical evidence for this effect in the social media influencer context. For example, Carr and Hayes (2014) , and Hwang and Jeong (2016) show that consumers perceive product reviews by a blogger who discloses affiliation with the firm to be less credible, and Evans et al. (2017) show that Instagram posts by influencers are lesspersuasivewhenthepostsaredisclosedtobeaffiliated. Another very relevant literature stream includes the theory papers on communication and persuasion. Startingwith CrawfordandSobel(1982),priorpapershavestudiedthecommunication game between informed senders and uninformed receivers who have misaligned preferences (see Sobel (2013) for a review). A related stream of literature examines the relationship between the expertandthereceiver(see OttavianiandSorensen,2006b(2006a,2006b) forexample). Inthese 38 papers, there is uncertainty about the extent of the sender’s expertise, which is revealed through the accuracy of her recommendations. Even though our research relates to these papers in that we alsostudyacommunicationgamebetweenaninformedsender(theinfluencer)andtheuninformed receiver(theconsumer),thereareimportantdifferencesinfocus. Whilethepapersabovefocuson the amount of information that is transmitted between the sender and the receiver in the presence of either misaligned incentives or the desire to signal one’s type, we focus on whether and how a third party, the firm, can strategically manipulate the communication between the sender and the receiver. Fromthemodelingperspective,ourworkisclosesttotheBayesianpersuasionortheinformation design literature (see Bergemann and Morris (2016) for a literature review). Kamenica and Gentzkow(2011) showsthatifthesender’spayoffisconvexinthereceiver’sbeliefs,thesendercan manipulatetheinformationstructure(thedesignoftheexperiment)inordertoincreaseherpayoff. This result holds even if the receiver is informed about the information structure. Our research fits into this framework in the following ways. First, since the consumer’s decision to buy or not to buy is discrete, the firm’s profit is convex in the consumer’s beliefs. Second, the strength of theaffiliationchangestheinformationstructureofthegame—theinformativenessofthereviews. Third, similar to the standard result in Bayesian Persuasion, the firm benefits from manipulating theinformationstructure,whichisaccomplishedinourmodelthroughaffiliation. However,thereareseveralkeydistinctionsbetweenourworkandmostoftheBayesianpersua- sion literature. In Kamenica and Gentzkow (2011) , the sender chooses the information structure that benefits him the most, but the authors abstract away from the actual process of changing the signals. Incontrast,weexplicitlymodeltherelationshipbetweenaffiliationandtheresultinginfor- mation structure. In addition, we model the cost of changing the information structure. That is, in our model any change in the information structure is the result of a contract between two different strategicplayers: thefirmandtheinfluencer. Finally,duetothespecificsoftheinfluencercontext, we model features such as awareness, cost of information acquisition, and different disclosure regimes. 39 Finally, two papers are substantively related to ours since they both study the issue of biased influencers. Mitchell (2017) models the trade-off that the influencer faces between showing ads (and possibly losing followers) or providing good advice (and forgoing revenue) in order to maintain the relationship with her followers. The paper finds a “reap and sow" cycle where the influenceralternatesbetweengoodadviceandadsintherepeatedgame. Mitchell(2017) focuses on the dynamics of reputation-building by the influencer and does not model the incentives of the firm,i.e.,theadvertisingmarketisexogenoustothemodel. Incontrast,wefocusontheactionsof thefirmandtheresultinginferencebytheconsumer. Hamami(2018) considersathree-playergame(theseller,thereviewingfirm,andtheconsumer) where the reviewing firm can sign a contract with the seller to inflate her reviews. A critical assumptionin Hamami(2018) isthatthereviewingfirm’smessagedrivesthepricethattheseller charges, which implies that the seller’s payoff is linear in the consumer’s belief. This violates the convexity requirement of Bayesian Persuasion, which implies that there is no benefit to inflating reviews if they are purely informational. Instead, the author focuses on the benefit of review inflationduetonetworkeffects. 2.3 Model Westartwithabriefoverviewofthemodel. Therearethreeplayers: therisk-neutralinfluencer (“she”), her follower (the risk-neutral consumer, “he"), and the monopolist firm that wants to sell its product at price ? to the consumer. We assume that the consumer “follows”7 the influencer whose taste matches his own. The influencer’s post may make the consumer aware of the product and provide information on product fit. The firm may enter into a relationship with the influencer in order to increase awareness of the product and/or increase the probability of a favorable rec- ommendation. Assuming that the influencer cares about the information value of her content, she mustbecompensatedforbiasingherreview. Inthebasemodel,weassumethattheinfluencermust disclosethefullextentofheraffiliationwiththefirmtotheconsumer. Theconsumerwhoobserves 7Thatis,theconsumerconsumesthecontent posted by the influencer. 40 the influencer’s review forms an inference on the product fit and makes a purchase decision. We lookforanequilibriuminthefirm’sandtheinfluencer’sactionsaswellasintheconsumer’sbelief andaction. Note that in our model, both the product price and the product design (the fit of the product to the influencer’s preference) are exogenous to the game that we analyze. These assumptions not only simplify our analysis but also reflect our observations of the relative impact of many influencersonthemarketplace. Thatis,wewishtomodelthefirm’sinteractionswiththemid-level influencer whose following is big enough to warrant the firm’s attempt to manipulate her review through affiliation, but not so significant that the firm wishes to create or price the product based onherpreferences. Alltheexamplesofinfluencerswehaveusedfitthisdefinition. Thisrepresents the boundary condition for the applicability of our model. That is, if the size of the influencer’s followingisasignificantportionofthemarket,ourmodeldoesnotapply. Whilewedosometimes observefirmscollaboratingwithverytopinfluencerstocreateproducts,8thisisadifferentproblem from what we study in this chapter. In the cases of collaboration, the influencer is the co-creator andendorseroftheproduct;therefore,manipulationoftheinfluencer’sreviewisnolongeranissue. Next, we describe the set-up of the game in more detail. Please also see Figure 1.3. AtC = 1, Nature chooses the fit between the product and the consumer. We also assume that the influencer and the consumer have identical preferences. Let denote the product fit,2fg, where standsfor“goodfit",and standsfor“badfit."TheconsumerreceivesutilityE ifhepurchasesthe product that fits his taste and 0 otherwise: *¹º =E and*¹º = 0. The consumer’s prior belief on the product fit is%¹º = d 0 , whered 0 2»0 1¼. At the start of the game, neither the firm, the influencer,ortheconsumerobserve. Tomodelimperfectawareness,weassumethatwithprobability_ theconsumerisawareofthe product and hence forms beliefd 0 on its fit. With probability 1_, he is not aware of the product and does not include it in his consideration set. If he does become aware of the product, his prior 8For example, Lancome collaborated with Camila Coelho, a beauty influencer with 3.4 million subscribers on YouTubeand7.6millionfollowersonInstagram, to create the “L’Absolu Rouge x Camila Coelho" lipstick. 41 Figure2.1: TimelineoftheGame Nature chooses the product fit. The firm makes an offer to the influencer. The influencer posts a review of the product and reveals her affiliation. The influencer posts an independent review. The influencer invests c to receive a private signal on the product fit. The consumer observes the influencer’s post and affiliation and makes a purchase decision. t=1 t=2 t=5 t=3 t=4 The influencer does not invest to receive a private signal. The consumer is aware of the product and makes a purchase decision based on her prior. The consumer observes an independent post and makes a purchase decision. The firm makes no offer to the influencer. The consumer is unaware of the product and does not purchase. λ 1 − λ Firm makes affiliation decision. The influencer invests c to receive a private signal. The influencer does not post a review. The influencer is affiliated. The influencer is independent Offer is accepted Offer is rejected onthefitisd 0 .9 AtC= 2,thefirmmaymakeatake-it-or-leave-itofferofpaymenttotheinfluencerinexchange for an increased likelihood of a positive product review. The influencer can accept or reject the offer. We assume that the information sets of the influencer and the firm are identical at the time the contract is signed.10 Let\2f0 1g, where\ = 1 denotes affiliation, and\ = 0 independence. Thefirm’sofferspecifiestheextenttowhichtheinfluenceragreestoskewherreviews, 00 1, in exchange for the payment (¹\0º to the influencer —¹\0(¹\0ºº, where (¹10º 0 and (¹0 0º = 0. We define the nature of the bias that the firm introduces, 0, when we describe the review process atC = 4. If the influencer accepts the offer, \ = 1, and she is obligated to post a 9Conceptually this is equivalent to a model where the firm with a certain reputation (which determines the prior belief d 0 ) has a new product, and the consumer is aware of this product with probability _. When the consumer becomesawareoftheproduct,hisbeliefonthe product fit is driven by the firm’s reputation. 10As we discussed before, there is a large number of mid-level influencers, and the taste of any one of them is idiosyncratic. Thismakesithardforthefirmtopredictwhethertheproductwillfitthetasteofanyoneofthemapriori. This assumption rules out a different set of signaling equilibria where the nature of the affiliation contract conveys additionalinformationonthelikelihoodofproduct fit. 42 review. Anindependentinfluencer,(\= 0),isundernosuchobligation. AtC= 3,theindependentinfluencerwhoeitherturneddownthefirm’sofferorwasnotoffered a contract atC = 2 can choose to invest cost2 to receive a private signalB on the product fit. We assumethatreceivingBisarequirementforpostingareview,whichimpliesthattheinfluencerwho isaffiliatedwiththefirmiscontractuallyobligatedtoobtainB atacost2. Thesignalisinformation that the influencer obtains by trying out the product. For example, the influencer could purchase andsampleafoundationortryitoutatthestore. Shecouldeitherlikeordisliketheresult. Thatis, B2f61g,where6 standsfor“observedgoodfit,"and1 standsfor“observedbadfit."Weassume that the signal is inherently noisy due to external factors — the influencer may not like the result due to poor lighting at the store, for example. That is, %¹6j º = %¹1j º = W, andW ¡ 05, where W is observable. A higher W implies less noise in the signal, which also translates to less noiseinthereviewprocess,whichwedescribeindetailinC= 4. Weassumethattheinfluencerderivesutilityfromthevalueofinformation(+$)ofherreview to the consumer, where 0 U 1 measures the weight that the influencer puts on+$ relative to(. We discuss the concept of the value of information in detail in Section 2.3.2. The desire to deliver high value of information may be due to altruism on the part of the influencer or simply due to the desire to maintain a high reputation, which yields more followers in the long term. The influencerwhoinvests2 toreceiveB obtainsthefollowingexpectedutility, * infl =U+$¹\0º¸(¹\0º2 (2.1) We refer to the payment to the affiliated influencer as(¹0º instead of(¹10º to simplify notation. Weonlyconsidercontractswithpositivepaymentsfromthefirmtotheinfluencer,(¹0º ¡ 0.11 AtC = 4, the affiliated influencer is contractually obligated to post a review on product fit and must reveal her affiliation with the firm. The independent influencer who received the signalB at C= 3canalsopostareview. Thereviewisbinary: <=fGBg,whereGdenotesapositivemessage (positivereview)andB denotesanegativemessage(negativereview). Wedenotetheabsenceofa 11Theresultsarequalitativelythesameifwe include(¹0º= 0. 43 review by< =;. We assume that conditional on having the signal, the act of posting a review is costless. The independent influencer posts the review if her expected utility from posting exceeds heroutsideoption,* 0 . Forsimplicity,weset* 0 = 0. Theindependentinfluencerpostsareviewif * infl jB=U+$¹0 0ºjB 0. As we show in Claim B.0.6.1 in the Appendix B,+$¹0 0ºjB 0, whichimpliesthattheindependentinfluencerwhoreceivesB alwayspostsareview. Therefore,the decisiontoinvestinB isequivalenttothedecisiontoinvestinpostingareview. We assume that the independent influencer’s review reflects the signal that she received. That is,iftheinfluencerlikedtheproduct,shepostsapositivereview,andiftheinfluencerdidnotlikethe product,shepostsanegativereview. Inotherwords,%¹Gj6º= 1,and%¹Bj 1º= 1. Thisimplies thatfortheindependentinfluencer,%¹Gjº=%¹Bj º=W and%¹Gj º=%¹Bjº= 1W. Affiliation in our model increases the probability that the influencer posts a positive review and decreases the probability that she posts a negative review. As before, the affiliated influencer always posts a positive review if she likes the product: %¹Gj 60º = %¹Gj 6º = 1. However, the influencer who is affiliated at level 0 with the firm posts a positive review even if she does not like the product with probability %¹Gj 10º = 0. Hence, the probability that the influencer posts a positive review is%¹Gj0º = %¹6º¸0%¹1º, where0%¹1º is the amount of distortion due to affiliation. If 0 = 0, there is no positive distortion. In contrast, if 0 = 1, the influencer is contractuallyobligatedtoalwayssend<=G. We assume that the contract between the firm and the influencer is legally binding: neither partycandeviatefromthecontractafteritissigned. Whileinaone-shotgametheinfluencermay beabletodeviatefromthecontractwithoutdetection,overtimetheinfluencerwhohopestowork with the firm must deliver the agreed-upon0. Moreover, we do find examples of firms refusing paymenttoinfluencersbasedontheirunhappinesswiththevalenceofthecontentposted.12 Inthebasemodelwealsoassumefulldisclosure: theaffiliatedinfluencermustdisclosewhether or not she is affiliated and the extent of the affiliation,¹\0º. The fact that\ must be disclosed is 12Seehttps://www.youtube.com/watch?v=2Jo1QWURQ_U. 44 very much in line with existing FTC regulations, “... if there’s a connection between an endorser andthemarketerthatconsumerswouldnotexpectanditwouldaffecthowconsumersevaluatethe endorsement,thatconnectionshouldbedisclosed."13Whethertheexactlevelofaffiliation,0,either mustbeorcanbedisclosedislessclear. ThesameFTCguidestates,“Whatmattersiswhetherthe information would have an effect on the weight readers would give your review. So whether you got $100 or $1,000 you could simply say you were ‘paid.’ " Given this, the full disclosure model can be considered more of an idealized limit case. In Section 2.4, we consider a variant of the modelwithpartialdisclosure,wheretheinfluencerdisclosesonly\ andnot0. AtC = 5, there are three possible cases. In the case of the affiliated influencer, the consumer observes¹<\= 10º. Inthecaseoftheindependentinfluencerwhopostsareview,theconsumer observes¹<\ = 0º. In both these cases, the consumer updates his belief and makes a purchase decisionbasedonthatbelief. Inthecaseoftheindependentinfluencerwhodoesnotpostareview, the consumer observes < =;. If the consumer is aware of the product, he makes the purchase decisionbasedonhispriorbelief. If,ontheotherhand,heisnotawareoftheproduct,hedoesnot buytheproduct. Theconsumer’srealizedpayoffifhepurchasestheproductisE? iftheproduct isagoodfitand? iftheproductisabadfit. 2.3.1 ConsumerInferenceandPurchaseDecisionatC= 5 The equilibrium concept we use is Perfect Bayesian Nash Equillibrium (PBNE). We solve for theequilibriumusingbackwardinduction. Let’s examine the consumer inference on product fit atC = 5. If the consumer sees no review, (<=;),withprobability 1_ heisnotawareoftheproductandisnotabletopurchaseit. Forall othercases,theconsumer’sinformationsetisk=¹<\0\ 0 " º. Here<isthereviewposted by the influencer,¹\0º is the affiliation disclosed in the review (where0= 0 for the independent influencer and 0 0 1 for the affiliated influencer), and¹\ 0 º is the affiliation expected in equilibrium. Finally, " is the set of possible messages that the consumer expects to observe in 13https://www.ftc.gov/tips-advice/business-center/guidance/ftcs-endorsement-guides-what-people-are- asking#disclose 45 equilibrium. That is, if the consumer expects to observe a review in equilibrium, " =fGBg, and if no review is expected," =;. The observed actions are in equilibrium if¹\0º =¹\ 0 º and<2 " ,andarenotinequilibriumotherwise. We first describe the belief formation on the equilibrium path:¹\0º =¹\ 0 º and<2 " . Note that, since there is no cost of posting content in our model, the independent influencer who obtains B always posts a review at C = 4; the absence of a review does not signal any private information on the part of the influencer. In addition, the contract is signed at C = 2 before any private information is revealed, after which the review process is set into motion. Hence, the presence or the absence of affiliation does not signal any private information on product fit. Thisimpliesthatallthepayoff-relevantinformationinequilibriumisconveyedthroughthereview processandthedisclosedaffiliation. First, suppose that the aware consumer does not see a review of the product, (< =;). The consumer purchases the product if and only if %¹ºE ? 0. That is, he buys the product if his prior is high enough: d 0 ¡ ? E :. Second, suppose the consumer observes a review <. Sinceallactionsareinequilibrium,theonlypayoff-relevantinformationiscontainedinthesubset k 0 =¹<0º. The consumer updates his belief about the product fit to%¹jk 0 º and purchases the product if and only if * = %¹jk 0 ºE ? 0 or %¹jk 0 º :. The posterior belief on the equilibriumpathcanbederivedusingBayes’rule. %¹jG0º= %¹Gj0º%¹º %¹Gj0º%¹º¸%¹Gj 0º%¹º (2.2) = »W¸0¹1Wº¼d 0 »W¸0¹1Wº¼d 0 ¸»¹1Wº¸0W¼¹1d 0 º (2.3) %¹jB0º= %¹Bj0º%¹º %¹Bj0º%¹º¸%¹Bj 0º%¹º (2.4) = ¹1Wºd 0 ¹1Wºd 0 ¸W¹1d 0 º (2.5) 46 wheretheprobabilityofapositivereviewgiventruefitandaffiliationis %¹Gj0º=%¹Gj60º%¹6jº¸%¹Gj 10º%¹1jº=W¸0¹1Wº (2.6) %¹Gj 0º=%¹Gj60º%¹6j º¸%¹Gj 10º%¹1j º=¹1Wº¸0W (2.7) There are a few observations that we can make about Equation 2.3 and 2.5. First, note that %¹jB0ºisnotafunctionof0sinceareviewcanonlyhaveapositivebiasbyassumption. Second, %¹jG0º d 0 %¹jB0º. Finally, we define the persuasiveness of review< as the change in the consumer’s belief following the review:j%¹j<0ºd 0 j. In our model the persuasiveness of G is monotonically decreasing in0 since m%¹jG0º m0 0 (see Figure 2.2). (Note that an increase in %¹j<0º implies an increase in the persuasiveness of< and vice versa). That is, an increase in affiliationmakestheconsumermoreskepticalofG. Figure2.2: Thepersuasivenessofthepositivereview,j%¹jG0ºd 0 j 0.1 0.5 1 a 0 0.1 0.21 |P(G| ,a)- ρ 0 | W= 07d 0 = 045 Finally, we need to specify the consumer’s out-of-equilibrium beliefs when observed actions do not match what the consumer expects to occur in equilibrium. Since neither the firm nor the influencer has any private information when the contract is signed, and since we assume that the contract is legally binding, we do not have a classic signaling game. Hence, we are not able to use themachineryofrefinementstorestrictthebeliefs. Instead,wemakethefollowingassumptions, 47 Assumption2.3.1.1. Weassumethefollowingout-of-equilibriumbeliefsfortheawareconsumer: 1. Iftheconsumerexpectstoobserveareviewinequilibriumbutnoreviewisposted,%¹jkº= d 0 . 2. If¹\0º<¹\ 0 º,theconsumerupdateshisbeliefonproductfitusingobserved0 and<. Theassumptionsareconsistentwithequilibriumbehaviorinthatnoadditionalprivateinforma- tion is revealed by the absence of a review, and the consumer uses the observable affiliation level, 0,inhisinference. Given Assumption 2.3.1.1 along with the inference made in equilibrium, we can focus on k 0 =¹<0º toderivetheconsumerinferenceandoptimalpurchasedecision. Lemma 2.3.1.1. Consider the aware consumer’s posterior beliefs and actions at C = 5, given k 0 =¹<0º. Letd¹0º :¹1W¹10ºº ¹1:º¹W¸0¹1Wºº¸:¹1W¹10ºº and ¯ d :W ¹1:º¹1Wº¸:W . 1. Ifd 0 2»0d¹0ºº,theconsumer neverbuystheproduct. 2. Ifd 0 2»d¹0º:º,theconsumeronlybuystheproductifheseesapositivereview,<=G. 3. If d 0 2 »: ¯ dº, the consumer buys the product if he sees a positive review or no review, <=fG;g,buthedoesnotbuyifheseesanegativereview,<=B. 4. Ifd 0 2» ¯ d 1¼,theconsumeralwaysbuystheproduct. Proof. SeeAppendixB. The consumer’s optimal purchase decision depends on whether his posterior belief is greater than or equal to :. First, consider the case where the consumer never buys (region 1 in Lemma 2.3.1.1). Heretheconsumerdoesnotbuytheproductevenifheseesapositivereview: %¹jG0º :. We can restate the condition in terms of the prior: ifd 0 2»0d¹0ºº, the consumer never buys. 48 Second, consider the other extreme case where the consumer always buys (region 4 in Lemma 2.3.1.1). Here, the consumer buys the product even after a negative review: %¹jBº :. Expressed intermsoftheprior,ifd 0 2» ¯ d 1¼,theconsumeralwaysbuys. Finally let’s turn to the intermediate region, where %¹jBº : %¹jG0º. Here the posterior following the positive message is above :, and the posterior following the negative message is below :. Hence, the consumer buys following a positive review and does not buy following a negative review: he follows the influencer’s recommendation. The decision following noreviewdependsonwhethertheprior,d 0 ,isbeloworabove:. Ifd 0 : %¹jG0º (region2 in Lemma 2.3.1.1), the consumer buys the product only if he sees a positive review. On the other hand, if %¹jBº : d 0 (region 3 in Lemma 2.3.1.1), the aware consumer buys the product eitherfollowingapositivereviewornoreview. As 0 increases, the consumer’s belief does not change as much following a positive review. Thisimpliesthattheregionwheretheconsumerneverbuys(region1)grows,andtheregionwhere the consumer buys only following a positive recommendation (region 2) shrinks. As we can see fromLemma2.3.1.1,d¹0º separatesthesetworegions. Consistentwithourintuition,wecanshow thatd¹0º isstrictlyincreasingin0. Inthelimit,as0 approaches1,d¹0º=:: region2disappears. Inthiscase,theconsumerdoesnotbuytheproductifhispriorisbelow:. 2.3.2 TheValueofInformation The value of information delivered by the influencer’s review, which we denote by+$¹\0º, is the difference between the consumer’s expected utility after he observes the review and the consumer’s expected utility in the absence of a review. Note that since we assume risk neutrality, his expected utility is just the expected payoff. We denote the value of information from an independentinfluencer’sreviewby+$¹0 0º. +$¹0 0º=* con j<2fGB;g* con j<=; (2.8) 49 Sincetheaffiliatedinfluencermustpostareview,wehave +$¹10º=* con j0<2fGBg* con j<=; (2.9) Theinfluencer’sreviewsimpacttheconsumer’sexpectedutilitybychanginghisactions,eitherdue tothechangeinhisbeliefsorthechangeinhisawareness. Let’sfirstconsiderhowtheinfluencer’s reviews can change the consumer’s actions through affecting his beliefs, i.e., the persuasiveness of the review system. As discussed in Lemma 2.3.1.1, if the consumer’s prior is in region 2, he does not buy based on his prior but buys after seeingG. Therefore, in this region, exposure to the influencer’s review provides information value to the consumer by enabling him to purchase the product. On the other hand, if the consumer’s prior is in region 3, he buys based on the prior or after seeingG, but does not buy after seeingB. Therefore, the exposure to the influencer’s reviews provides information valueto the consumer by preventing him from buyinga product that is likely to be a bad fit. We can also show that in regions 2 and 3, the value of information is strictly decreasing in0, which is a direct implication of the fact that an increase in0 decreases the persuasiveness of reviews. In contrast, if the consumer’s prior is in region 1 or 4, his actions are unchanged by the influencer’s reviews; the persuasiveness of the review does not affect the value ofinformation. Reviews may also deliver value by informing the unaware consumer of the existence of the product. Awareness can result in a change of actions (from no purchase to purchase) if the consumer’spriorisabove:,whichisthecaseforregions3or4inLemma 2311. Anincreasein awareness (an increase in_) implies an increase in consumer’s expected utility in the absence of influencer’sreviews. Hence,anincreasein_ leadstoalowervalueofinformation. Wesummarize ourfindingsinLemma2.3.2.1. Lemma2.3.2.1. Thevalueofinformationoftheinfluencer’sreviewisasfollows, 1. Ifd 0 2»0d¹0ºº,+$¹\0º= 0 2. Ifd 0 2»d¹0º:º,+$¹\0º=%¹Gj0º h %¹jG0ºE? i 0and m+$¹\0º m0 0. 50 3. Ifd 0 2»: ¯ dº,+$¹\0º=%¹Gj0º h %¹jG0ºE? i _ h %¹ºE? i ¡ 0and m+$¹\0º m0 0. 4. Ifd 0 2» ¯ d 1¼,+$¹\0º=¹1_º h %¹ºE? i ¡ 0 Proof. SeeAppendixB 2.3.3 TheInfluencer’sDecisionsatC= 2 3 4 The influencer who is affiliated with the firm is contractually obligated to acquire the signal B on product fit at a cost 2 atC = 3 and to post a review atC = 4. We analyze the independent influencer’s decisiontoinvestininformationacquisitionusingbackwardinduction. Lemma 2.3.3.1. The independent influencer’s expected utility is maxf0U+$¹0 0º2g. Below wedescribewhensheacquiresinformationatC= 3andpostsareviewatC= 4(whichwesimplify to“postsareview"), 1. Ifd 0 2»0d¹0ºº,theinfluencerdoesnotpostareview. 2. Ifd 0 2»d¹0º:º,theinfluencerpostsareviewifU%¹Gº h %¹jGºE? i 2 3. Ifd 0 2»: ¯ dº,theinfluencerpostsareviewifU %¹Gº h %¹jGºE? i _ h %¹ºE? i 2 4. Ifd 0 2» ¯ d 1¼,theinfluencerpostsareviewifU¹1_º h %¹ºE? i 2 Proof. SeeAppendixB. Finally, atC = 2, the influencer who has been offered (¹0º in return for affiliation 0 accepts the offer if her expected utility under affiliation weakly exceeds her expected utility if she remains independent: * infl =(¹0º¸U+$¹10º2 maxf0U+$¹0 0º2g 51 Adding2 tobothsidesandmovingterms,thisbecomes (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 (2.10) 2.3.4 Firm’sOptimalAffiliationDecisionatC= 2 Wefinallycometothecentralquestionofthischapter: whendoesthefirmoptimallychooseto allow the influencer to remain independent versus affiliate with the influencer atC= 2? We denote by \ thefirm’sprofitgivenitsaffiliationchoice,where\2f0 1g. Wesolveforthefirm’sdecision toaffiliatewiththeinfluencer,\ ,thelevelofaffiliation,0 ,andthecorrespondingpayment tothe influencer,(¹0 º,inequilibrium. First, we derive the firm’s profit if the influencer is independent, 0 . The profit depends on whethertheinfluencerpostsareviewandtheconsumer’sbeliefsfollowingthereview. 0 = 8 > > > > < > > > > : %¹Gº 1f%¹jGº :g?¸%¹Bº 1f%¹jBº :g? ifU+$¹0 0º2 _ 1fd 0 :g? >C4AF8B4 (2.11) Next,wederive 1 ¹0 º,thehighestprofitthatthefirmcanattainunderaffiliation. Todoso,we define the affiliated firm’s problem with respect to the optimal level of affiliation and the payment to the influencer.14 Note that the firm’s problem is constrained by the fact that the influencer must (weakly)preferthecontracttobeingindependentasspecifiedbyEquation(2.10). 1 ¹0 º=max 0(¹0º %¹Gj0º 1f%¹jG0º :g?¸%¹Bj0º 1f%¹jB0º :g?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 (2.12) The equilibrium strategies and beliefs are summarized in Proposition 2.3.4.1, and all variable definitionsareinTable2.1. 14WeshowintheproofofProposition2.3.4.1thatasub-optimalaffiliationlevelcannotbeinequilibriumsincethe firmwoulddeviateto0 . 52 Proposition 2.3.4.1. The following describes the PBNE equilibria of the three-player game. The equilibriumisuniqueincases1-3. 1. Supposed 0 2»0d¹0ºº,thereisnoaffiliation,noindependentreviewandnopurchase. 2. Supposed 0 2»d¹0º:º, (a) If 2 ¯ 1 , the firm pays (¹ ¯ 0º ¡ 0 to affiliate at 0 = ¯ 0, 0 ¯ 0 1, %¹jB ¯ 0º %¹jG ¯ 0º=:. HeretheconsumerbuysifG anddoesnotbuyifB. Finally,+$¹1 ¯ 0º= 0. (b) Otherwise,thereisnoaffiliation,noindependentreview,andnopurchase. 3. Supposed 0 2»: ¯ dº, (a) If2 maxfU+$¹0º ¯ 2 g,0 = 1,(¹1º ¡ 0,%¹jB 1º : %¹jG 1º=d 0 . HeretheconsumerbuysifG anddoesnotbuyifB. Finally,+$¹1 1º=¹1_º¹d 0 E ?º. (b) Otherwise,thereisnoaffiliation,noindependentreview,andtheawareconsumerbuys. 4. Supposed 0 2» ¯ d 1¼, (a) If 2 U+$¹0 0º, there is no affiliation, and the influencer posts an independent review. IfU+$¹0 0º 2 ¯ 2 , the firm affiliates at any0 2»0 1¼, and( ¡ 0. In bothcases,theconsumeralwaysbuysand+$ =¹1_º¹d 0 E?º. (b) Otherwise, there is no affiliation, no review is posted, and only the aware consumer buys. Proof. SeeAppendixB. The basic idea behind the proof to Proposition 2.3.4.1 is the following. Note that if the consumer’s prior is very low (region 1), even a positive signal from the independent influencer is 53 Table2.1: VariableDefinitionsforProposition2.3.4.1 d¹0º= :¹1Wº ¹1:ºW¸:¹1Wº , ¯ d= :W ¹1:º¹1Wº¸:W , ¯ 0= W¹1:ºd 0 :¹1d 0 º¹1Wº W:¹1d 0 º¹1Wº¹1:ºd 0 ¯ 1 h d 0 ¹W¸ ¯ 0¹1Wºº¸¹1d 0 º¹1W¸ ¯ 0Wº i ? ¯ 2 =¹1_º h ¹1Uº?¸Ud 0 E i (¹ ¯ 0º= maxfU+$¹0º2g,(¹1º= maxfU+$¹0º2gU+$¹1º, ( = maxf02U+$¹0ºg notpersuasiveenoughtoconvincetheconsumertopurchase. Hencereviewsplaynoroleinregion 1. Forillustrationpurposes,assumethatU+$¹0 0º2andd 0 2»d¹0º:º. Thatis,theindependent influencer posts a review, and the consumer buys if and only if the independent influencer posts G. Next, we need to derive the optimal affiliation level as defined by Equation (2.12). Given our assumptionon2,theconstraintin(2.12)becomes(¹0ºU»+$¹0 0º+$¹10º¼. Theconstraint mustbebindingorthefirmcanlower(¹0ºtoincreaseitsprofit: (¹0º=U»+$¹0 0º+$¹10º¼. In other words, the influencer must be compensated for the decrease in the consumer’s value of information due to the bias that results from affiliation. The expression for (¹0º can be substituted into the firm’s objective function, which becomes 1 ¹0º = %¹Gj0º 1f%¹jG0º :g?¸U»+$¹10º+$¹0 0º¼. Weshowthat 1 ¹0ºisincreasingin0aslongas%¹jG0º :. Hence, the firm optimally sets0 to be as high as possible, up to the point where the consumer still choosestobuytheproductafterseeingapositivereview: %¹jG0 º=:. Whathappensiftheindependentinfluencerdoesnotchoosetopostareview—U+$¹0 0º 2? Here the cost of acquiring information is too high relative to the benefit, U+$¹0 0º, from the perspectiveoftheinfluencer. Ifthefirmchoosestoaffiliate,theconstraintin(2.12)becomes(¹0º= 2U+$¹10º. Hence, the firm subsidizes the influencer’s investment to acquire information. The objective function becomes 1 ¹0º = %¹Gj0º 1f%¹jG0º :g?¸U+$¹10º2. As before,thisisincreasingin0,andhencethefirmincreases0 aslongastheconsumerisstillwilling to buy afterG. Since the influencer must be compensated for2 in order to review, a very high2 implies that affiliation is too expensive. In this case, the firm prefers to let the influencer remain independent. Notethatthisalsoimpliesthatthefirmpreferssomeformofaffiliation(asopposedto 54 independence) as long as2 is not too high. As we can see in Proposition 2.3.4.1, affiliation occurs in all regions where the cost is low enough (regions 20, 30, and 40) and does not occur when the costishigh(regions 21, 31,and 41). However,case 41 isuniquebecauseaffiliationdoesnotoccur eitherifthecostistoolow. Wewilldiscussthisindetailbelow. We graph the firm’s profit under optimal affiliation (solid line) and the firm’s profit under no affiliation (dashed line) as a function of the prior in Figure 2.3. There are two ways in which the firm potentially benefits from affiliating with the influencer — the solid line is above the dashed line. The first is to skew the reviews in favor of the firm. The second is to subsidize information acquisition by the influencer, where none would occur without the firm’s involvement. This is the case ifU+$¹0 0º 2 ¯ 1 in region 20 where no sale occurs without affiliation — the dashed line is zero. This is also the case ifU+$¹0 0º 2 ¯ 2 in regions 30 and 40 where only the awareconsumerbuysbasedontheprior—thedashedlineisflatintheprior. Figure2.3: EquilibriumFirmProfitwithAffiliationversusProfitunderIndependentInfluencer Firmprofitwith optimal affiliation Firmprofitwithoutaffiliation Region 1 Region 2a Region 3a Region 4a ρ k ρ 1 ρ 0 c Profit ?= 3E= 15U= 08W= 07_= 082= 015 Since Proposition 2.3.4.1 is somewhat dense, we break our discussion into three (related) topics: 1) optimal affiliation, consumer beliefs and the value of information in equilibrium, 2) the 55 equilibriumpaymenttotheinfluencer, and3)consumerwelfare. Optimalaffiliation,consumerbeliefs,andthevalueofinformationinequilibrium The optimal affiliation, 0 , depends on the consumer’s default action. We graphically illustrate 0 in Figure 2.4 and%¹jG0 º in Figure 2.5. Following the discussion above, we focus on the region of the parameter space where 2 is relatively low (regions 20, 30 and 40). If the prior is below the cutoff : (region 20), the consumer’s default action is not to purchase the product, and theconsumerneedstobepersuadedbytheinfluencertobuy. Byaffiliatingwiththeinfluencer,the firm can increase the probability that the influencer posts a positive review,G, even as doing so decreasesthepersuasivenessofthereview,%¹jG0ºorj%¹jG0ºd 0 j. Thefirm’soptimallevel ofaffiliationissuchthattheconsumerisexactlyindifferentbetweenpurchasingandnotpurchasing afterseeingapositivereview: %¹jG ¯ 0º=:. Heretheinfluencer’sreviewsarepartiallydistorted upwards, but there is still information inG. We show in the Proof for Proposition 2.3.4.1 that ¯ 0 is increasingintheprior,d 0 . Thismakesintuitivesense. Anincreasein ¯ 0resultsinalessinformative positive review. Asd 0 increases, a less informativeG signal is needed to increase the consumer’s belieffromd 0 to:. Hence,anincreaseind 0 resultsinahigher ¯ 0. Figure2.4: EquilibriumLevelofAffiliation(0 ) Region 1 Region 2a Region 3a Region 4a ρ k ρ 1 ρ 0 1 a * a.c ≤ αVOI(0,0) Region 1 Region 2a Region 3a Region 4a ρ k ρ 1 ρ 0 1 a * b. αVOI(0,0) <c ≤Min{C 1 ,C 2 } a a E= 3?= 15U= 08W= 07_= 08 56 Figure2.5: Consumer’sPosteriorBeliefinEquilibrium,%¹jG0 º Region 1 Region 2a Region 3a Region 4a ρ k ρ 1 ρ 0 1 k P(G| ) P(G| ,a * ) E= 3?= 15U= 08W= 07_= 08 Ifthepriorisabovethecutoff:,theconsumer’sdefaultactionistopurchasetheproduct. Ifthe priorisinregion 30,theinfluencercanonlyruinasalewithanegativereviewfromtheperspective of the firm. Hence here the firm prefers to fully affiliate with the influencer: 0 = 1, ensuring that the influencer always posts a positive review: %¹jG 1º = d 0 . The value of affiliation to the firm hereistopreventanypotentialnegativereviewsandtoincreaseawareness. Let’s consider the case where the prior is very high (region 40): the aware consumer buys even after a negative review. Hence, affiliation is potentially valuable to the firm only in terms of awareness and not persuasiveness. There are two cases here. Panel a of Figure 2.4 illustrates the firstcasewherethecostofacquiringthesignalisrelativelylow,2U+$¹0 0º. Theindependent influencer always posts a review, resulting in perfect awareness. The firm chooses not to affiliate withtheinfluencersinceawarenessisnotanissuehere. PanelbofFigure2.4illustratesthesecond casewherethecostisintheintermediaterange,U+$¹0 0º 2 ¯ 2 . Theindependentinfluencer does not post a review on her own. Therefore, the firm prefers to affiliate to increase awareness. Since at this high prior level persuasiveness does not matter, the firm is indifferent between any 57 affiliation level 0 0 1. Here d 0 %¹jG0 º %¹jGº. In summary, the firm only chooses to affiliate in this region if there is imperfect awareness or the influencer does not choose to independently post a review. Let’s consider the case where2 is relatively low,2U+$¹0 0º, and where the prior on product fit is relatively high: regions 3 and 4. In both of these cases, the influencerispositivelypredisposedtowardsthefirm: sheislikelytoreceivesignalB=6. However, thefirmchoosestoonlyaffiliateinregion3butnotinregion4. Itaffiliatesinregion3topreventa potentialnegativereviewbytheinfluencer. Inregion4,itdoesnotaffiliatebecauseheretheaware consumer always buys, and the independent influencer always posts a review, which guarantees awareness. Hence, affiliation does not occur in the region where the influencer is most positively predisposedtowardsthefirm–theoptimalaffiliationisnotmonotonicintheconsumer’sprior.15 Finally, we consider the value of information that the consumer derives from the influencer’s reviewinequilibrium. Theconsumerobtainsvaluealongtwopotentialdimensions: 1)thereview’s ability to change the consumer’s actions through changing beliefs — its persuasiveness, or 2) the review’s ability to make the consumer aware of the product — the increase in awareness. The incentives between the firm and the consumer are aligned when it comes to consumer awareness: both the firm and the consumer benefit from an increase in awareness as long as the prior is high enough. This,however,isnotthecasewithrespecttopersuasiveness: whiletheconsumerwantsto figureoutthetruefit,thefirm’sobjectiveistomanipulatethereviewsystemsuchthattheconsumer is more likely to buy. We find that in equilibrium, the firm destroys the consumer’s value from the persuasiveness of the influencer’s reviews by manipulating the level of affiliation,0. The only remaining value is through awareness only. In Panel 1 in Figure 2.6, we illustrate the value of information,+$¹10 º, that the consumer derives from the influencer’s reviews in equilibrium for _ 1. The value of information in region 20 is+$¹1 ¯ 0º = %¹Gj ¯ 0º»%¹jG ¯ 0ºE ?¼ = %¹Gj ¯ 0º»:E?¼= 0: thefirmcompletelydestroystheconsumer’svalueofinformation. Notethat in this region, the only relevant dimension is persuasiveness since an aware consumer does not choosetopurchase. Incontrast,ford 0 : (regions 30and 40),+$¹10 º=¹1_º¹d 0 E?º ¡ 0. 15Wethankareviewerforpointingthisout. 58 Here all the value is derived through awareness only. If awareness is not an issue,_= 1, the value of information is zero for all d 0 (see Panel3 in Figure 2.6). Again this is due to the fact that the firmdestroysallthevalueofinformationfrompersuasiveness. Figure2.6: Consumer’sValueofInformationinEquilibrium ρ k ρ 1 ρ 0 VOI(0, 0) a. VOI(0,0), λ<1 ρ k ρ 1 ρ 0 VOI(1,a * ) b. VOI(1, a * ), λ<1 ρ k ρ 1 ρ 0 VOI(0, 0) c. VOI(0,0), λ=1 ρ k ρ 1 ρ 0 VOI(1,a * ) d. VOI(1, a * ), λ=1 Region 2a Region 1 Region 3a Region 4a Region 2a Region 1 Region 3a Region 4a Region 2a Region 1 Region 3a Region 4a Region 2a Region 1 Region 3a Region 4a E= 3?= 15U= 08W= 07_= 08 PaymenttotheInfluencerandtheFirm’sProfitinEquilibrium Let’s turn to the equilibrium compensation that the firm pays the influencer,(¹0 º. As we discuss before, the firm optimally sets(¹0 º = maxfU+$¹0 0º2gU+$¹10 º. Suppose that2 = 0. This implies that the independent influencer chooses to post a review since U+$¹0 0º 0. It also implies that (¹0 º = U»+$¹0 0º+$¹10 º¼: the firm compensates the influencer for the destruction of the value of information of her signal through affiliation. That is, the compensation is proportional to the difference in the value of information of the “clean” signal (reviews unadulterated by the firm’s affiliation) and the value received from the signal under 59 equilibrium affiliation. Graphically, this is the difference in the curve pictured in Panel 0 and the curve in Panel 1 in Figure 2.6, which results in the curve in Panel 0 in Figure 2.7. As we increase2,atsomepointtheindependentinfluencernolongerchoosestopostindependentlysince U+$¹0 0º 2. Hence, (¹0 º = 2U+$¹10 º: the firm must compensate the influencer for the cost of acquiring information minus the utility that the influencer obtains from the value of information of the review under affiliation. The firm essentially needs to subsidize review production where none otherwise would take place. As we see in Panels 1-3 in Figure 2.7, an increase in 2 results in kinks in the compensation curve. Generally speaking, (¹0 º is weakly increasing in d 0 up until the cutoff: and weakly decreasing in d 0 after:, with the compensation peaking at the cutoff: — the point of maximum uncertainty at which the value of information is thegreatest. Aswediscussedbefore,if2 becomestoolarge,thefirmprefersnottoaffiliate. Figure2.7: ThePaymenttotheInfluencerinEquilibrium,(¹0 º ρ k ρ 1 ρ 0 c S(a * ) a.c =0 ρ k ρ 1 ρ 0 c S(a * ) b.c =0.1 ρ k ρ 1 ρ 0 c S(a * ) c.c =0.15 ρ k ρ 1 ρ 0 c S(a * ) d.c=0.2 Region 2a Region 1 Region 3a Region 4a Region 2a Region 1 Region 3a Region 4a Region 2a Region 1 Region 3a Region 4a Region 2a Region 1 Region 3a Region 4a E= 3?= 15U= 08W= 07_= 08 60 TheEffectoftheFirm’sInvolvementonConsumerWelfare Thefirm’sinvolvementcandecrease(andsometimescompletelydestroy)thevalueofinformationin reviews. Isitalwaysthecasethatthefirm’sinvolvementhasanegativeeffectonconsumerwelfare? The answer depends on whether the firm only destroys the persuasiveness of the independent influencer’srevieworalsoenablesthepostingofareviewwhenonewouldnototherwisebeposted bytheindependentinfluencer. Inthecasewheretheindependentinfluencerchoosestopostareview, the firm’s involvement results in a transfer of surplus from the consumer to the firm. However, if thefirmenablesthepostingofareview,andthevalueofinformationinequilibriumispositivedue totheincreaseinawareness,theinvolvementofthefirmmayactuallyhelptheconsumer. Notethat U measurestheextenttowhichtheinfluencerinternalizestheconsumer’svalueofinformation. As Udecreases,U+$¹0 0º 2ismorelikelytohold. Thatis,inthecasewheretheinfluencervalues theconsumer’svalueofinformationless,thefirm’sinvolvementismorelikelytobehelpfultothe consumer. Corollary 2.3.4.1. If d 0 : andU+$¹0 0º 2 ¯ 2 , the firm’s affiliation results in a review where none would be posted by an independent influencer. Here the consumer strictly prefers for the firm to be involved since+$¹10 º =¹1_º¹d 0 E ?º ¡ 0, the firm is strictly better off since it makes non-zero profit, and the influencer is indifferent between the firm’s involvement or independence. Hencethefirm’sinvolvementresultsinaParetoimprovement. Proof. SeeAppendixB. Corollary 2.3.4.1 implies that it is sometimes socially efficient for the firm to bear the cost of information acquisition since surplus is transferred from the consumer to the firm as a result of thefirm’smanipulationofreviews. Inparticular,thefirm’soptimalaffiliationdecisiondetermines whether a review is posted and to what extent the review is biased, internalizing the costs of both acquiringanddestroyinginformation. 61 2.4 Extension: PartialDisclosure In this section, we consider an alternative disclosure regime — partial disclosure. That is, we considerthecasewheretheinfluenceronlydiscloseswhetherthereisanyaffiliationwiththefirm, \, but does not disclose the precise nature of the affiliation,0. Hence the consumer only observes \2f0 1g andinfersthelevelofaffiliation,whichwedenoteby ˜ 0. Several empirical observations motivate this model extension. First, as we discussed before, it is not clear that FTC’s current regulatory requirements are very rigorous. Second, we observe imperfect compliance with existing FTC disclosure regulations. For example, in April 2017, the FTCsentoutletterstomorethan90influencersandbrands,remindingthemthatthey“shouldclearly and conspicuously disclose their relationships to brands when promoting or endorsing products through social media" (FTC, 2017). Third, it is not clear whether it is feasible to disclose the exactnatureoftherelationshipbetweentheinfluencerandthefirm,especiallyonalow-bandwidth mediumsuchasInstagram. Let’s consider the firm’s optimal level of affiliation, 0 , in an equilibrium where the firm affiliates with the influencer, \ = 1. Here0 is observable to the firm and the influencer but not to the consumer whose belief and actions are based on ˜ 0. We calculate+$ from the influencer’s perspectivesincesheneedstoagreetothefirm’scontract. Theaffiliatedfirm’sproblembecomes 1 ¹0 ˜ 0º=max 0(¹0º %¹Gj0º 1f%¹jG ˜ 0º :g?¸%¹Bj0º 1f%¹jB ˜ 0º :g?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 (2.13) Wedefinetheout-of-equilibrium beliefsbelow. Assumption2.4.0.1. Weassumethefollowingout-of-equilibriumbeliefsfortheawareconsumer: 1. Iftheconsumerexpectstoobserveareviewinequilibriumbutnoreviewisposted,%¹jkº= d 0 . 62 2. Iftheconsumerexpectsnoaffiliationinequilibriumbutobservesaffiliationortheconsumer expectsfullaffiliation, ˜ 0= 1,butobserves<=B,thentheconsumerupdateshisbeliefusing ˜ 0= 1n. The first assumption is the same as before. The second assumption implies skepticism — the consumer expects a high amount of affiliation when he sees deviations from equilibrium. Finally, wearereadytopresenttheequilibriumresults. Proposition2.4.0.1. ConsiderthePBNEequilibriumunderpartialdisclosure. TheresultingPBNE isidenticaltothefulldisclosurePBNEincases 1, 3,and 4inProposition1. Supposethatd 0 2»d¹0º:º (case 2 inProposition2.3.4.1), 1. If+$¹0 0º 2, there is no affiliation, the independent influencer posts a review, and the consumerbuystheproductifandonlyif<=G. 2. If+$¹0 0º 2,thereisnoaffiliation,noreview,andnopurchase. Proof. SeeAppendixB. First, note that the majority of the results are exactly the same as before. Hence most of our results go through under a less rigorous disclosure regime. We do see one important difference – whereasthefirmaffiliatesatanintermediatelevelunderfulldisclosureinregion 20,thefirmdoes not affiliate for d 0 : under partial disclosure. What is the intuition for this result? Consider a candidateequilibriumwithaffiliationinthisregion. AsweshowintheproofofProposition2.4.0.1, theconsumerrationallyinfersthatthefirmwouldchoosetoaffiliatewiththeinfluencerfully. Since 0 is not observable to the consumer, the firm cannot credibly commit to limit its affiliation level. Hence, the firm prefers not to affiliate at all in order to maintain the persuasiveness ofG in the regionwheretheconsumermustbepersuadedinordertobuy. Note that the consumer actually benefits from partial disclosure. Under full disclosure, the equilibriumaffiliationlevelis0 = ¯ 0suchthattheconsumer’svalueofinformationfromthereview 63 isexactlyzero. Incontrast,underpartialdisclosure,thefirmdoesnotaffiliatewiththeinfluencerat all. Sincethefirmdoesnotbiasthereviewprocess,theconsumer’svalueofinformationisstrictly higherunderpartialdisclosure. Thefirm,ontheotherhand,preferstohaveafulldisclosureregime since this yields a higher profit. Full disclosure acts as a commitment device to allow the firm to haveanintermediatelevelofaffiliationasopposedtobeingeitherfullyaffiliatedornotaffiliatedat all. As we saw before, under full disclosure the firm subsidizes the review process in the case where d 0 2»d¹0º:º and+$¹0 0º 2 ¯ 1 . In this case, no review is posted under partial disclosure, but an affiliated review is posted under full disclosure. Of course, since, in either case, the consumer’s value of information is zero, this makes no difference to consumer welfare. Here thefirmagainbenefitsfromfulldisclosure. 2.5 Conclusion Wemodelthethree-playergamebetweenthefirm,theinfluencer,andtheconsumer. Ourmodel makes predictions on when and how closely the firm affiliates with the influencer, and the effect thatthisaffiliationhasonallthreeplayers. We show that affiliating with the influencer potentially benefits the firm in two different ways, 1) it enables the posting of a review that raises awareness of the product, and 2) it increases the probability of a positive review. The affiliation decision depends on the cost of information acquisition, the consumer’s prior, and the disclosure regime. Under full disclosure, the firm affiliates at an intermediate level if the prior is low, and affiliates fully if the prior is high. Here the firm’s involvement completely destroys the value of information from the persuasiveness of reviews. However, the value of information from the increase in awareness is positive if the prior is high. In fact, the consumer benefits from the firm’s involvement if the prior is high, and the cost of information acquisition is in the intermediate range. In this region, the firm subsidizes the acquisition of information for the influencer. We also find that the consumer may benefit from a partialdisclosureregimecomparedtoafulldisclosureregimeifthepriorislow. 64 We have several important theoretical contributions above and beyond what is known through theBayesianPersuasionliteratureineconomics. Onecontributionofourresearchistoendogenize the cost of changing the information structure that is costlessly accomplished in the earlier paper by adding the third player, the influencer. In addition, the influencer context, the modeling of awareness, the addition of the cost of information acquisition, and the comparison of the two differentdisclosureregimes,areallnew. Inthissense,ourmodelisquitedifferentfrom Kamenica andGentzkow(2011). From a policy perspective, our results provide insights into how different disclosure regimes affect consumer welfare. For example, we show that more disclosure does not always benefit the consumer. In fact, partial disclosure may benefit the consumer by forcing the firm to allow the influencertoremainindependent. Wealsoshowthatsometimestheconsumermaybenefitfromthe presence of the firm, especially if the influencer does not fully internalize the consumer’s benefit fromthevalueofinformation. We show that affiliation allows the firm to silence the influencer by paying her to prevent the possibility of a negative review. This result applies to a variety of settings. For example, in the politicalrealm,anoppositionleadermaybeaskedtojointheadministrationinordertopreventany potentialcriticism. The current research has several limitations that present opportunities for future research. Our model applies to mid-level influencers only; therefore, other firm decisions such as pricing and product design are exogenous to our model. However, this assumption may no longer hold for top-level influencers who have millions or even tens of millions of followers. The present model alsoassumesthatneitherthefirmnortheinfluencerhasanyprivateinformationabouttheproduct fit before a contract is signed. Future research can relax this assumption and explore how the equilibrium outcome would change when either party has private information. 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Marketing Science, 37(3), 425– 444.https://doi.org/10.1287/mksc.2017.1076 74 75 Appendix A ProofsforChapterI A.0.1 ProofofLemma1.2.1.1 Thereceiver’sposteriorbeliefsarethefollowing: %A¹'j;;;º= %A¹;;;j'º%A¹'º %A¹;;;j'º%A¹'º¸%A¹;;;j!º%A¹!º = ¹1Uº 2 »¹1Cº¹1Uº¸C¼d 0 ¹1Uº 2 »¹1Cº¹1Uº¸C¼d 0 ¸U 2 »¹1CºU¸C¼¹1d 0 º %A¹'jAAAº= %A¹AAAj'º%A¹'º %A¹AAAj'º%A¹'º¸%A¹AAAj!º%A¹!º = U 3 d 0 U 3 d 0 ¸¹1Uº 3 ¹1d 0 º %A¹'jB =;B =;B =Aº= %A¹B =;B =;B =Aj'º%A¹'º %A¹B =;B =;B =Aj'º%A¹'º¸%A¹B =;B =;B =Aj!º%A¹!º = ¹1Uºd 0 ¹1Uºd 0 ¸U¹1d 0 º %A¹'jB =AB =AB =;º= %A¹B =AB =AB =;j'º%A¹'º %A¹B =AB =AB =;j'º%A¹'º¸%A¹B =AB =AB =;j!º%A¹!º = U 2 »¹1Cº¹1Uº¸C¼d 0 U 2 »¹1Cº¹1Uº¸C¼d 0 ¸¹1Uº 2 »¹1CºU¸C¼¹1d 0 º %A¹'jB =;B =AB =;º= %A¹B =;B =AB =;j'º%A¹'º %A¹B =;B =AB =;j'º%A¹'º¸%A¹B =;B =AB =;j!º%A¹!º = »¹1Cº¹1Uº¸C¼d 0 »¹1Cº¹1Uº¸C¼d 0 ¸»¹1CºU¸C¼¹1d 0 º %A¹'jB =;B =AB =Aº= %A¹B =;B =AB =Aj'º%A¹'º %A¹B =;B =AB =Aj'º%A¹'º¸%A¹B =;B =AB =Aj!º%A¹!º = Ud 0 Ud 0 ¸¹1Uº¹1d 0 º 76 »¹1Uº 2 ¸ 2U¹1Uº 2 ¸ 2CU 2 ¹1Uº¼d 0 »¹1Uº 2 ¸ 2U¹1Uº 2 ¸ 2CU 2 ¹1Uº¼d 0 ¸»U 2 ¸ 2U 2 ¹1Uº¸ 2CU¹1Uº 2 ¼¹1d 0 º = 1 2 »¹1Uº 2 ¸ 2U¹1Uº 2 ¸ 2CU 2 ¹1Uº¼d 0 =»U 2 ¸ 2U 2 ¹1Uº¸ 2CU¹1Uº 2 ¼¹1d 0 º 2CU¹1Uº¹U¸d 0 1º=U¹1Uº¹2U 1º¸¹Ud 0 º C= U¹1Uº¹2U 1º¸¹Ud 0 º 2U¹1Uº¹U¸d 0 1º WhenC ¯ C,theprobabilitythatthereceivertakesaction0 ! giventhetruestateis! is %¹#; ¡ #Aj!º=%A¹;;Aj!º¸%A¹;;;j!º¸%A¹;A;j!º¸%A¹A;;j!º =U 2 ¸ 2U¹1Uº»¹1CºU¸C¼ = 3U 2 2U 3 ¸ 2CU¹1Uº 2 =U 2 ¸ 2U 2 ¹1Uº¸ 2CU¹1Uº 2 Theprobabilitythatshetakestakesaction0 ' giventhetruestateis' is %¹#A ¡ #;j'º=%A¹AAAj'º¸%A¹AA;j'º¸%A¹A;Aj'º¸%A¹;AAj'º =U 2 ¸ 2U 2 ¹1Uº¹1Cº = 3U 2 2U 3 2CU 2 ¹1Uº Thereceiver’sexpectedutilityinthepresenceoftrollsis: * ULBC¯ C A =%¹#; ¡ #Aj!º%¹!º¸%¹#A ¡ #;j'º%¹'º¸ 2X =»3U 2 2U 3 ¸ 2CU¹1Uº 2 ¼¹1d 0 º¸»3U 2 2U 3 2CU 2 ¹1Uº¼d 0 ¸ 2X = 3U 2 2U 3 2CU¹1Uº¹U¸d 0 1º¸ 2X =U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º¸ 2X WhenC ¡ ¯ C,theprobabilitythatthereceivertakesaction0 ! whenthetruestateis! is %A¹0 ! j!º=%A¹;;j!º=U 2 77 Theprobabilitythatshetakesaction0 ' whenthetruestateis' is: %A¹0 ' j'º= 1%A¹;;j'º= 1¹1Uº 2 ’sexpectedutilityinthepresenceof trollsis: * ULB A =%'¹0 ! j!º%A¹!º¸%A¹0 ' j'º%A¹'º¸ 2X =U 2 ¹1d 0 º¸»1¹1Uº 2 ¼d 0 ¸ 2X =U 2 ¸ 2U¹1Uºd 0 ¸ 2X A.0.2 ProofofLemma1.2.3.1 ’sposteriorbeliefsgivenB 9 9 = are: %A¹'jB 9 =;º= 8 > > > > < > > > > : ¹1Uºd 0 ¹1Uºd 0 ¸U¹1d 0 º if 9 = »¹1Cº¹1Uº¸C¼d 0 »¹1Cº¹1Uº¸C¼d 0 ¸»¹1CºU¸C¼¹1d 0 º if 9 = %A¹'jB 9 =Aº= Ud 0 Ud 0 ¸¹1Uº¹1d 0 º IfC ¯ C,theprobabilitythatthereceivertakesaction0 ! whenthetruestateis! is %A¹0 ! j!º= Õ 9= 1 3 %A¹B 9 =;j!º=U¸ 1 3 C¹1Uº Theprobabilitythatshetakesaction0 ' whenthetruestateis' is %A¹0 ' j'º= Õ 92fg 1 3 %A¹B 9 =Aj'º=U 1 3 CU Thereceiver’sexpectedutilityis * LBC¯ C =%A¹0 ! j!º%A¹!º¸%A¹0 ' j'º%A¹'º¸ Õ 92fg %A¹B 9 ºX =U 1 3 C¹U¸d 0 1º¸ 2 3 X IfC ¡ ¯ C,theprobabilitythatthereceivertakesaction0 ! whenthetruestateis! is %A¹0 ! j!º= 1 3 %A¹B =;j!º¸ 1 3 %A¹B =;j!º = 2 3 U 78 Theprobabilitythatshetakesaction0 ' whenthetruestateis' is %A¹0 ' j'º= 1 3 %A¹B =Aj'º¸ 1 3 %A¹B =Aj'º¸ 1 3 %A¹B º = 2 3 U¸ 1 3 ’sexpectedutilityis: * LBC¡¯ C A =%A¹0 ! j!º%A¹!º¸%A¹0 ' j'º%A¹'º¸ 2 3 X = 2 3 U¹1d 0 º¸ 2 3 U¸ 1 3 d 0 ¸ 2 3 X = 2 3 U¸ 1 3 d 0 ¸ 2 3 X Theexpectedwelfarelossfromthetroll,!>BB ! iscalculatedasfollows: * LBC=0 * LBC ¯ C =U¸ 2 3 X U 1 3 C¹U¸d 0 1º¸ 2 3 X = 1 3 C¹U¸d 0 1º * LBC=0 * LBC¡¯ C =U¸ 2 3 2 3 U¸ 1 3 d 0 ¸ 2 3 X = 1 3 ¹Ud 0 º A.0.3 ProofofLemma1.3.1.1 Proof. ! " ¹Cº :(!B : leadstothefollowingmapping: B : = 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ifC ¯ C 8 > > > > < > > > > : ; if #; ¡ #A A if #A ¡ #; ifC ¡ ¯ C 8 > > > > < > > > > : ; ifB =B =; A ifotherwise (A.1) ThereexistsauniquemappingbetweenB : and0,where 0= 8 > > > > < > > > > : 0 ! ifB : =; 0 ' ifB : =A (A.2) 79 RecallinLemma1.2.1.1,’soptimaldecisionrulegiventhefullsetofsignals,(!0,is 0= 8 > > > > > > > > > > > > < > > > > > > > > > > > > : ifC ¯ C 8 > > > > < > > > > : 0 ! if #; ¡ #A 0 ' if #A ¡ #; ifC ¡ ¯ C 8 > > > > < > > > > : 0 ! ifB =B =; 0 ' otherwise (A.3) Replace 0 with B : , mapping (A.1) and (A.3) are identical. Therefore ! " ¹Cº replicates ’s optimaldecisionrule. ! " ¹Cº achievesoptimalcompressionand’sexpectedinformationutility ismaximizedunder! " ¹Cº. The social value is also maximized under ! 1 . Given the limited bandwidth constraint, the maximum possible social value that could be obtained by the receiver isX. Since B 2fB B g under ! 1 , is guaranteed to receive a social value of X. The above algorithm is optimal becausebecauseitmaximizesboththeinformationutilityandsocialvaluetothereceivergiven’s bandwidth constraint. However, the optimal algorithm is not unique. For example, consider the case whereC ¯ A andB =;B =;B =A. The algorithm could sendB with probability ? and sendB withprobability 1?. Thisdoesnotchangethetotalexpectedutilityreceivedby. A.0.4 ! "2 The algorithm below is also replicates ’s optimal decision rule in the unlimited bandwidth case. 80 • IfC ¯ C: B : = 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : B orB wequalprob iffB B B g2ff;;;gf;;Agg B iffB B B g2ff;A;gg B iffB B B g2ffA;;gg B orB wequalprob iffB B B g2ffAAAgfAA;gg B iffB B B g2ffA;Agg B iffB B B g2ff;AAgg • IfC ¡ ¯ C: B : = 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : B orB wequalprob iffB B B g2ff;;;gf;;Agg B iffB B B g2ff;A;gg B iffB B B g2ffA;;gg B orB wequalprob iffB B B g2ffAAAgfAA;gg B iffB B B g2ffA;Agg B iffB B B g2ff;AAgg A.0.5 ProofofLemma1.3.1.2 Proof. followsthecuratedsignalinequilibriumsince\¹B : =;j'º 1 2 and\¹B : =Aj'º ¡ 1 2 . 1. C ¯ C. If followsthecuratedsignal,herexpectedutilityis * DA0C8>= =%A¹0 ! j!º%A¹!º¸%A¹0 ' j'º%A¹'º =%A¹B : j!º%A¹!º¸%A¹B : j'º%A¹'º =%A¹#; ¡ #Aj!º%A¹!º¸%A¹#A ¡ #;j'º%A¹'º =U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º¸X 81 * LB =U 1 3 C¹U¸d 0 1º¸ 2 3 X * DA0C8>= * ! =U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1ºU¸ 1 3 C¹U¸d 0 1º¸ 1 3 X 2. C ¡ ¯ C, * DA0C8>= =%A¹0 ! j!º%A¹!º¸%A¹0 ' j'º%A¹'º =%A¹B : j!º%A¹!º¸%A¹B : j'º%A¹'º =%A¹B =B =;j!º%A¹!º¸»1%A¹B =B =;j'º¼%A¹'º =U 2 ¹1d 0 º¸U¹2Uºd 0 =U 2 ¸ 2Ud 0 2U 2 d 0 =U 2 ¸ 2U¹1Uºd 0 ¸X * ! = 2 3 U¸ 1 3 d 0 =U 1 3 ¹Ud 0 º¸ 2 3 X * DA0C8>= * ! =U 2 ¸ 2U¹1Uºd 0 ¸ 1 3 ¹Ud 0 º¸ 1 3 X 0 IfC ¯ C,thedifferencein’sexpectedutilityis Curation,LB =U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1ºU¸ 1 3 C¹U¸d 0 1º¸ 1 3 X =U¹1Uº¹2U 1ºC¹U¸d 0 1º¹2U¹1Uº 1 3 º¸ 1 3 X Since this Curation,LB is increasing inC and in this caseC is at most ¯ C, we substiteC = ¯ C into the equationandobtain U¹1Uº¹2U 1º¹Ud 0 º¹2U¹1Uº 1 3 º¸ 1 3 X =U¹1Uº¹2U 1º 2U¹1Uº¹Ud 0 º¸ 1 3 ¹Ud 0 º =U¹1Uº2UU¹1Uº¹Ud 0 º2U¹1Uº¸ 1 3 ¹Ud 0 º = 2U¹1Uº¹UU¸d 0 ºU¹1Uº¸ 1 3 ¹Ud 0 º =U¹1Uº¹2d 0 1º¸ 1 2 ¹1Uº 0 82 A.0.6 ProofofProposition1.3.1.1 Proof. ThisfollowsdirectlyfromLemma1.3.1.2since’sexpectedutilityundercurationisalways greaterthanorequaltoherexpectedutilityundernocuration. A.0.7 ProofofLemma1.3.2.1 Proof. WhenC ˜ C, followsthecuratedsignalsince\¹B : =;j'º 1 2 . Herexpectedutility is: * curationC˜ C receiver =%A¹B : =;j!º%A¹!º¸%A¹B : =Aj'º%A¹'º¸X =»U 2 ¸ 2U 2 ¹1Uº¸ 2CU 2 ¹1Uº¼¹1d 0 º¸»U 2 ¸ 2U 2 ¹1Uº 2CU 2 ¹1Uº¼d 0 ¸X = 3U 2 2U 3 2CU¹1Uº¹U¸d 0 1º¸X =U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º¸X When C ¡ ˜ C, does not follow the curated signal and instead takes her default action since \¹B : =;j'º ¡ 1 2 . Herexpectedutilityis * curationC¡˜ C receiver =%A¹B : =;j'º%A¹'º¸%A¹B : =Aj'º%A¹'º¸X =»%A¹B : =;j'º¸%A¹B : =Aj'º¼%A¹'º¸X =%A¹'º¸X =d 0 ¸X Theexpectedinformationutilitylossfromtrollsis: • IfC ˜ C,* curation,notrolls * curation,trolls = 2CU¹1Uº¹U¸d 0 1º • ifC ¡ ˜ C. * curation,notrolls * curation,trolls =U 2 ¸ 2U 2 ¹1Uºd 0 Therefore,!>BB curation = min 2CU¹1Uº¹U¸d 0 1º U 2 ¸ 2U 2 ¹1Uºd 0 A.0.8 ProofofProposition1.3.2.1 First, note that ¯ C ˜ C. This implies that has a higher tolerance of trolls under curation than undernocuration. 83 Proof. Case1: C ¯ C. ’sexpectedtotalutilityundercurationwithimperfectknowledgeis * curation =U 2 ¸ 2U 2 ¹1Uº 2CU¹1Uº¹U¸d 0 1º which is equal to’s expected total utility under curation with perfect knowledge. By Proposition 1.3.1.1,’s expectedutilityishigherundercurationthanundernocuration. Case2: When ¯ C C ˜ C. * curation = 3U 2 2U 3 2CU¹1Uº¹U¸d 0 1º¸X * LB¯ CC receiver = 2 3 U¸ 1 3 d 0 ¸ 2 3 X =U 1 3 ¹Ud 0 º¸ 2 3 X Aisindifferentbetweencurationandnocurationwhen* curation,unknownC˜ C receiver =* LB¯ CC receiver ,orwhen 3U 2 2U 3 2CU¹1Uº¹U¸d 0 1º¸X=U 1 3 ¹Ud 0 º¸ 2 3 U 2 ¸ 2U 2 ¹1UºU¸ 1 3 ¹Ud 0 º¸X 2 3 X= 2CU¹1Uº¹U¸d 0 1º U 2 ¸ 2U 2 ¹1UºU¸¹Ud 0 º¹Ud 0 º¸ 1 3 ¹Ud 0 º¸ 1 3 X= 2CU¹1Uº¹U¸d 0 1º U¹1Uº¹2U 1º¸¹Ud 0 º 2U¹1Uº¹U¸d 0 1º 1 3 2¹Ud 0 ºX 2U¹1Uº¹U¸d 0 1º =C ˆ C ˜ C 1 3 2¹Ud 0 ºX 2U¹1Uº¹U¸d 0 1º =C Notethat ˜ C ¡ ¯ C. IfX 2¹Ud 0 º,then ¯ C ˆ C ˜ C. Since* curation isdecreasinginC,thereceiver obtains higher expected utility under curation than under no curation whenC ˆ C, and gets lower expectedutilityundercurationthanundernocurationwhenC ¡ ˆ C. IfX ¡ 2¹Ud 0 º, then ¯ C ˜ C ˆ C, the receiver gets higher total expected utility under curation comparedtonocurationuptoC= ˜ C. 84 Case3: C ¡ ˜ C. * curation,unknownC¡˜ C receiver =d 0 ¸X * LBC¡˜ C receiver = 2 3 U¸ 1 3 d 0 ¸ 2 3 X * curation,unknownC¡˜ C receiver * LBC¡˜ C receiver = 1 3 X 2 3 ¹Ud 0 º ThereceiverreceiveshigherexpectedutilityundercurationthannocurationwhenX¡ 2¹Ud 0 º. ThereceiverreceiveslowerexpectedutilityundercurationthannocurationwhenX 2¹Ud 0 º. 85 Appendix B ProofsforChapterII B.0.1 ProofofLemma2.3.1.1 1. First,considerthecasewheretheconsumer’spriorisverylowsuchthathisposteriorfollowing a positive review is still below :. This implies that 0 %¹jG0º :. Given the fact that m%¹jG0º d 0 ¡ 0 and m%¹jG0º 0 0, we can restate the condition in terms of the prior: 0 d 0 d¹0º,whered¹0º isderivedasfollows: %¹jG0º= »W¸0¹1Wº¼d 0 »W¸0¹1Wº¼d 0 ¸»¹1Wº¸0W¼¹1d 0 º : (B.1) d 0 :»1W¹10º¼ ¹1:º»W¸0¹1Wº¼¸:»1W¹10º¼ d¹0º (B.2) Note that the lower bound d¹0º is a function of 0, as it is derived from %¹jG0º. This lower bound exists for any02»0 1¼ such that ifd 0 2»0d¹0ºº (region 1), inequality (B.1) holdsandtheconsumerneverbuys. 2. Second, consider the case where the consumer’s prior is slightly below :, but his posterior followingapositivereviewisabove:. Thisimpliesthatd 0 : %¹jG0º,whichcanbe re-written as d¹0º d 0 :. Therefore, if d 0 2»d¹0º:º (region 2), the aware consumer buys ifandonlyifheseesapositivereview. 3. Third, consider the case where the consumer’s prior is slightly above :, but his posterior following a negative review is below:. This implies that%¹jBº : d 0 , which can be re-writtenas: d 0 ¯ d,where ¯ d isderivedasfollows: %¹jBº= ¹1Wºd 0 ¹1Wºd 0 ¸W¹1d 0 º : (B.3) d 0 :W ¹1:º¹1Wº¸:W ¯ d (B.4) 86 Notethattheupperbound ¯ disnotafunctionof0,asitisderivedfrom%¹jBº. Thereexists an upper bound ¯ d such that ifd 0 2»: ¯ dº (region 3), the aware consumer buys if he sees no revieworapositivereview,but doesnotbuyifheseesanegativereview. 4. Lastly, consider the case where the consumer’s prior is very high such that his posterior following a negative review is still above:. This implies that: %¹jBº 1, which can bere-writtenas ¯ d d 0 1. Therefore,ifd 0 2» ¯ d 1¼ (region4),theawareconsumeralways buys. B.0.2 ProofofLemma2.3.2.1 Weuse resultsinLemma2.3.1.1inthisproof. 1. Whenthepriorisinregion1,theconsumerneverbuys. Inotherwords,theconsumer’saction does not change and his expected utility remains 0 whether the influencer posts a review or not,and+$¹\ 0º= 0. 2. Whenthepriorisinregion2,theconsumerbuysifandonlyif<=G. Intheabsenceofthe influencer’s review, the aware consumer does not buy and receives 0 expected utility. The influencer posts a positive review with probability %¹Gj0º, following which the consumer buys and receives an expected utility of%¹Gj0º h %¹jG0ºE? i . Therefore,+$¹\0º = %¹Gj0º h %¹jG0ºE? i . Note that here the consumer’s information value is a function of 0. We can show that+$¹\0º is decreasing in0. Using the fact that%¹Gj0º%¹jG0º = %¹º%¹Gj0º,wecanre-write+$¹\0º as+$¹\0º=%¹º%¹Gj0ºE%¹Gj0º?. Sinceinthisregion%¹jBº :,wehavethat ¹1Wºd 0 ¹1Wºd 0 ¸W¹1d 0 º ? E or¹1Wºd 0 E h ¹1Wºd 0 ¸W¹1d 0 º i ?. Thisinturnimpliesthat m+$¹\0º m0 =¹1Wºd 0 E»¹1Wºd 0 ¸ W¹1d 0 º¼? 0. 3. The aware consumer buys if < =fG;g. In the absence of the influencer’s review, with probability_ the consumer is aware of the product and buys , receiving an expected utility 87 of _ h %¹ºE ? i . If the influencer posts a review, with probability %¹Gj0º she posts a positive review, following which the consumer buys and receives an expected utility of %¹Gj0º h %¹jG0ºE? i . Therefore,+$¹\0º=%¹Gj0º h %¹jG0ºE? i _ h %¹ºE? i . 4. Theawareconsumeralwaysbuyswhenhispriorisinregion4irrespectiveoftheinfluencer’s review. In the absence of the influencer’s review, with probability_ the consumer is aware of the product and buys, receiving an expected utility of_ h %¹ºE ? i . If the influencer postsareview,theconsumerbuyswithcertainty,receivinganexpectedutilityof%¹d 0 ºE?. Therefore,+$¹\0º=¹1_º h %¹ºE? i . B.0.3 ProofofLemma2.3.3.1 We use the results in Lemma 2.3.2.1 in this proof. The four regions of priors are the same as before. However, since the independent influencer’s decision to post reviews is independent of0, thelowerboundhereisd¹0º. 1. Whentheconsumer’spriorisinregion1,theconsumerderives 0informationvaluefromthe influencer’sreview: U+$¹0 0º2 0. Theindependentinfluencerdoesnotpostareview. 2. Whentheconsumer’spriorisin region2,+$¹0 0º=%¹Gº h %¹jGºE? i . Theindepen- dentinfluencerpostsareviewifU%¹Gº h %¹jGºE? i 2 3. Whentheconsumer’spriorisinregion3,+$¹0 0º=%¹Gº h %¹jGºE? i _ h %¹ºE? i . TheindependentinfluencerpostsareviewifU %¹Gº h %¹jGººE? i _ h %¹ºE? i 2. 4. When the consumer’s prior is in region 4, +$¹0 0º = ¹1_º h %¹ºE ? i 2. The independentinfluencerpostsareviewifU¹1_º h %¹ºE? i 2. B.0.4 ProofofProposition2.3.4.1 WeuseLemma2.3.1.1with0= 0todeterminetheconsumerbehaviorineachregion. Similarly, we use Lemma 2.3.2.1 to obtain the value of information and Lemma 2.3.3.1 to determine the independentinfluencer’sdecisiontopost. 88 Case1 d 0 2»0d¹0ºº. Here the consumer’sprior is so lowsuch that no messagewill convince him tobuy. Thefirmisnotabletomakeanyprofitandisnotwillingtopayforaffiliation. Based onLemma2.3.3.1,theindependentinfluencerdoesnotpostareview. Case2 d 0 2»d¹0º:º. Here the consumer buys only if he observesG. Let’s consider a candidate equilibriumwithaffiliation,¹\ = 10 º,wherewederive0 asfollows. Asthefirmincreases 0 from 0, it increases%¹Gj0º but decreases%¹jG0º. The consumer buys the product as long as%¹jG0º = »W¸0¹1Wº¼d 0 »W¸0¹1Wº¼d 0 ¸»¹1Wº¸0W¼¹1d 0 º :. With some algebra, thisreducesto 0 W¹1:ºd 0 :¹1d 0 º¹1Wº W:¹1d 0 º¹1Wº¹1:ºd 0 ¯ 0 Wecanshowthat 0 ¯ 0 1. a)Toshowthat ¯ 0 1,weneedtoshowthatW¹1:ºd 0 :¹1d 0 º¹1Wº W:¹1d 0 º ¹1Wº¹1:ºd 0 . After some algebraic simplification, this reduces to d 0 :, which of courseistrueinthisregionbydefinition. b) To show that ¯ 0 0, we can show that the denominator of ¯ 0 is strictly greater than zero forW 1 2 andd 0 :. The numerator of ¯ 0 is strictly increasing ind 0 and is equal to zero at d¹0º. Hence,thenumeratorof ¯ 0 isweaklypositiveinthisregion,whichimpliesthat ¯ 0 0. Finally, m ¯ 0 md 0 = 2W1 »W:¹1d 0 º¹1Wº¹1:ºd 0 ¼ 2 ¡ 0 forW ¡ 1 2 , and m ¯ 0 mW = : 2 ¹1d 0 º 2 ¹1:º 2 d 2 0 »W:¹1d 0 º¹1Wº¹1:ºd 0 ¼ 2 ¡ 0 ford 0 :. Taking into account the fact the condition0 ¯ 0 in order for the consumer to buy afterG, theaffiliatedfirm’sproblembecomes, 1 ¹0 º=max 0(¹0º %¹Gj0º?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 ¹1º 00 ¯ 0 ¹2º Clearly, constraint (1) must be binding or the firm can simply lower (¹0º which would increase its profit. (¹0º = maxfU+$¹0 0º2gU+$¹10º. Using Lemma 2.3.2.1, 89 we have+$¹10º = %¹Gj0º h %¹jG0ºE ? i and+$¹0 0º = %¹Gº h %¹jGºE ? i . Substitutingtheexpressionfor(¹0º backintothefirm’sprofit,weobtain 1 ¹0º=%¹Gj0º?¸U+$¹10º maxfU+$¹0 0º2g =%¹Gj0º?¸U%¹Gj0º h %¹jG0ºE? i maxfU+$¹0 0º2g Next we use the fact that%¹Gj0º%¹jG0º = %¹º%¹Gj0º (Bayes’ Rule) to rewrite the firm’sprofit, 1 ¹0º=%¹Gj0º?¸U%¹º%¹Gj0ºEU%¹Gj0º? maxfU+$¹0 0º2g =¹1Uº%¹Gj0º?¸U%¹º%¹Gj0ºE maxfU+$¹0 0º2g Intheexpressionabove,onlythefirsttwotermsarefunctionsof0. Infact,both¹1Uº%¹Gj0º and U%¹Gj0º are strictly increasing functions of 0. Hence, constraint (2) is also bind- ing: 0 = ¯ 0. Note that at 0 = ¯ 0, %¹jG ¯ 0º = :. That is, at 0 = ¯ 0, the consumer is just indifferent between buying or not buying the product. This implies that+$¹1 ¯ 0º = %¹Gj ¯ 0º h %¹jG ¯ 0ºE? i =%¹Gj ¯ 0º»:E?¼= 0 (¹ ¯ 0º= maxfU+$¹0 0º2gU+$¹1 ¯ 0º= maxfU+$¹0 0º2g¡ 0,and 1 ¹ ¯ 0º=%¹Gj ¯ 0º maxfU+$¹0 0º2g. Next, let’s turn to uniqueness. First, is there another possible equilibrium with0 < ¯ 0? Per Assumption1,underadeviationtheconsumermakesinferenceusingtheobservedaffiliation level,0. Since the firm’s profit is (uniquely) maximized at0 = ¯ 0, the firm has no incentive todeviateto0< ¯ 0. Hence,this istheuniqueequilibriumwithaffiliation. Case2a • Case 2ai: U+$¹0º 2: the independent influencer posts a review and 0 = %¹Gº?. Theaffiliatedfirm’smaximizedprofitis 1 ¹ ¯ 0º=¹1Uº%¹Gj ¯ 0º?¸U%¹º%¹Gj0ºE U+$¹0 0º. Second,wouldthefirmwanttodeviatetonoaffiliation,\ = 0? 1 ¹ ¯ 0º 0 = ?¹1Uº h ¯ 0¹1Wºd 0 ¸ ¯ 0W¹1d 0 º i ¸Ud 0 ¯ 0¹1WºE 0 90 We can see that the above expression is always greater than or equal to 0 – the firm does not want to deviate to \ = 0. Therefore, we have shown that the equilibrium ¹\ = 1 0 = ¯ 0º is unique in this region. The firm’s equilibrium profit is 1 ¹ ¯ 0º = ? ¯ 0¹1Uº h ¹1Wºd 0 ¸W¹1d 0 º i ¸Ud 0 ¯ 0¹1WºE¸ h Wd 0 ¸¹1Wº¹1d 0 º i ?. • Case 2aii: Suppose thatU+$¹0 0º 2: the independent influencer does not post a reviewand 0 = 0. Thefirm’smaximalprofitwithaffiliationis 1 ¹ ¯ 0º=%¹Gj ¯ 0º?2. The firm does not deviate to no affiliation if 1 ¹ ¯ 0º 0 = h d 0 ¹W¸ ¯ 0¹1Wºº¸¹1 d 0 º¹1W¸ ¯ 0Wº i ?2 0. We define, ¯ 1 h d 0 ¹W¸ ¯ 0¹1Wºº¸¹1d 0 º¹1W¸ ¯ 0Wº i ?. Therefore, there is a unique equilibrium with affiliation at0 = ¯ 0 ifU+$¹0 0º 2 ¯ 1 , where the firm’s profitis 1 ¹ ¯ 0º= ¯ 1 2. Case2b In this region2 ¡ ¯ 1 . We can show that the unique equilibrium is one where the firm does notaffiliatewiththeinfluencer,andtheindependentinlfuencerdoesnotpostareview. First,notethat+$¹0 0º=d 0 WE h d 0 W¸¹1d 0 º¹1Wº i ?,whichimpliesthatU+$¹0 0º ¯ 1 . Hence,theindependentinfluencerdoesnotpostareviewsinceU+$¹0 0º ¯ 1 2. Notethatthefirm’soptimalprofitfromdeviationtoaffiliationis 1 ¹ ¯ 0º= ¯ 1 2 0. Hence, thefirmdoesnotwanttodeviatetoaffiliation. Case3 d 0 2»: ¯ dº. Here the consumer buys if he observesG or no review and does not buy if he observesB. Consider a candidate equilibrium with affiliation,\ = 1, where0 is derived below. Since%¹jG ¯ 0º : and%¹jB ¯ 0º :,wecanre-writetheaffiliatedfirm’sproblem as, 1 ¹0 º=max 0(¹0º %A¹Gj0º?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 ¹1º 0 1 ¹2º 0 0 ¹3º 91 As before, constraint (1) must be binding or the firm can lower (¹0º in order to increase its profit: (¹0º = maxfU+$¹0 0º2gU+$¹10º. Using Lemma 2.3.2.1, we have +$¹10º=%¹Gj0º h %A¹jG0ºE? i _ h %¹ºE? i and+$¹0 0º=%¹Gº h %¹jGºE ? i _ h %¹ºE? i . Substitutingfor(¹0º intothefirm’sobjectivefunctionandusingBayes’ Rule,wehave 1 ¹0º=%¹Gj0º?¸U%¹Gj0º h %¹jG0ºE? i U_ h %¹ºE? i maxfU+$¹0 0º2g =¹1Uº%¹Gj0º?¸U%¹º%¹Gj0ºEU_ h %¹ºE? i maxfU+$¹0 0º2g Since 1 is strictly increasing in 0, constraint (2) is also binding: 0 = 1. (Since this is the unique optimum, the firm does not want to deviate to any0 < 1). Note that at0 = 1, the influencer’s review has no information above and beyond raising awareness. That is, the review does not change the consumer’s prior, d 0 . This also implies that+$¹1 1º = ¹1_º h %¹ºE? i . Wecanshowthat(¹1º= maxfU+$¹0 0º2gU+$¹1 1ºU+$¹0 0ºU+$¹1 1º ¡ 0: +$¹0 0º+$¹1 1º=d 0 E¹W 1º¸? h d 0 ¸W 2d 0 W i ¡ 0 The last inequality comes from the fact that in this region the consumer does not buy if he sees a negative review. That is, %¹Bº h %¹jBºE ? i = %¹BjºE%¹Bº? 0, which impliesthatd 0 E¹W 1º¸? h d 0 ¸W 2d 0 W i ¡ 0. Case3a • Case3ai: U+$¹0 0º2: theindependentinfluencerpostsareviewand 0 =%¹Gº?. Thefirm’soptimalprofitwithaffiliationis 1 ¹1º=¹1Uº%¹Gj1º?¸U%¹º%¹Gj 1ºEU_ h %¹ºE? i U+$¹0 0º = h 1U¸U%¹Gº i ?¸U%¹º h 1%¹Gjº i E 92 Thefirmprefersnottodeviatetonoaffiliationif 1 ¹1º 0 = h 1U¸U%¹Gº i ?¸U%¹º h 1%¹Gjº i E%¹Gº? = ?¹1Uº»1%¹Gº¼¸U%¹º»1%¹Gjº¼E 0 Thefirm’sprofitis 1 ¹1º= h 1U¸U¹Wd 0 ¸¹1Wº¹1d 0 ºº i ?¸U¹1Wºd 0 E. • Case 3aii: Suppose thatU+$¹0 0º 2: the independent influencer does not post a review. Here 0 =_? since the aware consumer buys if he sees no review. The firm’s profitwithaffiliationis: 1 ¹1º=¹1Uº%¹Gj1º?¸U%¹º%¹Gj 1ºEU_ h %¹ºE? i 2 =¹1Uº?¸U%¹ºEU_ h %¹ºE? i 2 The firm prefers not to deviate to no affiliation if 1 ¹0 º 0 =¹1_º h ¹1Uº?¸ Ud 0 E i 2 0. Wedefine ¯ 2 =¹1_º h ¹1Uº?¸Ud 0 E i . Therefore,thereisauniqueequilibriumwith affiliation,where 1 ¹0 º=¹1U¸U_º?¸U¹1_ºd 0 E2,ifU+$¹0º 2 ¯ 2 . Case 3b As before, we can show that there is a unique equilibrium with no affiliation if 2¡ maxfU+$¹0 0º ¯ 2 g. Heretheindependentinfluencerdoesnotpostareview. Case4 d 0 2» ¯ d 1¼. In this region the consumer always buys regardless of the valence of review. Therefore, the level of affiliation does not affect the decision to buy. However, the review maystill bevaluabletothe firm ifitincreases awareness(_ 1). Inaddition,using Lemma 2, we can see that the consumer’s value of information is also not affected by the value of0. Thatis,+$¹10º=+$¹0 0º. Hencethepayment( isnotafunctionof0. Giventhis,the firm’soptimalaffiliationproblembecomes, 1 ¹0 º=max 0(¹0º ?( s.t. (¸U+$¹0 0º maxfU+$¹0 0º2g 0 ¹1º 93 As before, constraint (1) is binding. (= maxfU+$¹0 0º2gU+$¹0 0º. Using Lemma 2.3.2.1, we have+$¹0 0º =¹1_º h %¹ºE ? i . Hence, ( = maxfU+$¹0 0º2g U+$¹0 0º= maxf02U+$¹0 0ºg 0. Case4a • Case4ai: U+$¹0 0º2: theindependentinfluencerpostsareviewand 0 = ?. The paymenttotheinfluenceris(= 0. Sinceweonlyconsider(¹0º ¡ 0,anycontractwith affiliationwillbemorecostly(andyieldthesamerevenue)tothefirm. Hence,thefirm strictlyprefersnoaffiliationand(= 0: thisistheuniqueequilibriumhere. • Case4aii: U+$¹0 0º 2: theindependentinfluencerdoesnotpostareviewand 0 = _?. Substituting( intothefirm’sobjectivefunction,thefirm’sprofitwithaffiliation(at anylevel0 )becomes 1 ¹0 º= ?¸U+$¹0 0º2= ?¸U¹1_º h %¹ºE? i 2. Note that if 1 ¹0 º 0 =¹1_º h ¹1Uº?¸Ud 0 E i 2 0, or2¹1_º h ¹1 Uº?¸Ud 0 E i ¯ 2 , the firm weakly prefers affiliation to no affiliation, which rules out an equilibrium with no affiliation. The firm is indifferent between any affiliation level 0 : theequilibriumlevelofaffiliationisnotunique. Therefore, if U+$¹0 0º 2 ¯ 2 , there is a unique equilibrium where the firm affiliateswiththeinfluenceratany0 2»0 1¼. Thefirm’sequilibriumprofitis 1 ¹0 º= h 1U¹1_º i ?¸U¹1_ºd 0 E2,andthefirm’sequilibriumpaymenttotheinfluencer is(=2U¹1_º h %¹ºE? i ¡ 0. Case 4b If2 ¡ ¯ 2 , there is a unique equilibrium with no affiliation. Here the independent influencer does not post a review. The aware consumer buys. The firm’s profit without affiliationis 0 =_?. B.0.5 ProofofCorollary2.3.4.1 SinceU+$¹0 0º 2, the independent influencer does not post a review — the receiver does notexpendresourcesacquiringtheinformation,andtheconsumerdoesnotreceiveanyvaluefrom 94 receiving it. We also show in the proof for Proposition 2.3.4.1 that the firm’s optimal affiliation contract is such that the influencer is indifferent between posting and not posting a review. Based on the proof for Proposition 2.3.4.1, the cases that satisfyU+$¹0 0º 2 are case 288, case 388, andcase 488. 1. Case 2ii. Per Proposition 2.3.4.1, here affiliation occurs if 2 ¯ 1 , and+$¹10 º = 0. Hencetheconsumerisjustindifferentbetweenfirminvolvementornofirminvolvement. 2. Case3iiandCase4ii. Hereaffiliationoccursif2 ¯ 2 . Here+$¹10 º=¹1_º¹d 0 E?º ¡ 0. Hence,theconsumerstrictlyprefersfirminvolvement. B.0.6 TheIndependentInfluencerNeverWithholdsaSignal. Claim B.0.6.1. The information value from the independent influencer’s review, conditional onB, isalwaysnon-negative:+$¹0 0ºj6 0and+$¹0 0ºj1 0. Since+$¹0 0º= 0ifthereis no reviewposted,thisimpliesthattheindependentinfluencerwhoreceivesB alwayspostsareview. Proof. The independent influencer posts< =G if she receivesB = 6 and< =B if she receives B=1. WeusetheresultsinLemma2.3.1.1andLemma2.3.2.1with0= 0inthisproof. 1. d 0 2»0d¹0ºº. If the influencer posts a review,+$¹0 0ºj6 =+$¹0 0ºj1 = 0, since the consumerdoesnotbuyinthisregion. 2. d 0 2»d¹0º:º. Suppose that the influencer receivesB =6. The consumer’s utility with no review is 0, since he does not buy based on his prior. If the influencer posts < =G, the consumer buys, and his expected utility is %¹jGºE ? ¡ 0. Therefore, +$¹0 0ºj6 = %¹jGºE? ¡ 0. SupposetheinfluencerreceivesB=1. Ifsheposts<=B,theconsumer doesnotbuytheproduct,andhisexpectedutilityis 0. Therefore,+$¹0 0ºj1= 0. 3. d 0 2»: ¯ dº. SupposethattheinfluencerreceivesB=6. Conditionalonthisinformation,the consumer’sexpectedutility(fromtheinfluencer’sperspective)withnoreviewis_ h %¹j6ºE 95 ? i ¡ 0sincehebuystheproductbasedonhisprioralone. Iftheinfluencerdoespost<=G, the consumer’s utility is %¹jGºE ? ¡ 0. Therefore,+$¹0 0ºj6 = h %¹jGºE ? i _ h %¹j6ºE? i ¡ 0. However, since we assume that the influencer postsG if she receives B=6,%¹j6º=%¹jGº. Hence,+$¹0 0ºj6=¹1_º h %¹jGºE? i ¡ 0. Suppose that the influencer receivesB=1. Conditional on this information, the consumer’s expected utility (from the influencer’s perspective) with no review is_»%¹jBºE?¼ 0. Iftheinfluencerdoespost<=B,theconsumerwouldnotbuytheproductandhaveautility of 0. Therefore,+$¹0 0ºj1= 0_»%¹jBºE?¼=_»?%¹jBºE¼ ¡ 0. 4. d 0 2» ¯ d 1¼. The consumer always buys regardless of<. Suppose B = 6. Conditional on this information, the consumer’s expected utility (from the influencer’s perspective) with no review is _ h %¹j6ºE ? i _ h %¹jGºE ? i ¡ 0. If the influencer posts < =G, the consumer’sexpectedutilityis%¹jGºE?. Therefore,+$¹0 0ºj6=¹1_º h %¹jGºE ? i ¡ 0. Similarly, suppose B = 1. The consumer’s expected utility (from the influencer’s perspective) with no review is _ h %¹j1ºE ? i ¡ 0. If the influencer posts < =B, the consumer would still buy the product, and his expected utility is%¹jBºE?. We assume that the independent influencer posts < =B if she receives B = 1, %¹j1º = %¹jBº. Therefore,+$¹0 0ºj1=¹1_º h %¹jBºE? i ¡ 0. B.0.7 ProofofProposition2.4.0.1 To prove the Proposition, we need to re-do all the Lemmas from Section 2.3.1 to Section 2.3.4 inthenewset-upwheretheconsumerdoesnotdirectlyobserve0. Lemma B.0.7.1. Consider the aware consumer’s posterior beliefs and actions at C = 5, given k 0 =¹<0º. Letd¹ ˜ 0º :¹1W¹1 ˜ 0ºº ¹1:º¹W¸ ˜ 0¹1Wºº¸:¹1W¹1 ˜ 0ºº and ¯ d :W ¹1:º¹1Wº¸:W . 1. Ifd 0 2»0d¹ ˜ 0ºº,theconsumerneverbuystheproduct. 96 2. Ifd 0 2»d¹ ˜ 0º:º,theconsumeronlybuystheproductifheseesapositivereview,<=G. 3. If d 0 2 »: ¯ dº, the consumer buys the product if he sees a positive review or no review, <=fG;g,buthedoesnotbuyifheseesanegativereview,<=B. 4. Ifd 0 2» ¯ d 1¼,theconsumeralwaysbuytheproduct. Proof. TheproofisidenticaltotheproofofLemma1inthemaintext,wherewesubstitute ˜ 0 for0 sincetheconsumerdoesnotobserve0. Lemma B.0.7.2. The value of information of the influencer’s review from the perspective of the influencerisasfollows, 1. Ifd 0 2»0d¹ ˜ 0ºº,+$¹\0º= 0 2. Ifd 0 2»d¹ ˜ 0º:º,+$¹\0º=%¹Gj0º h %¹jG0ºE? i 0and m+$¹10º m0 0. 3. Ifd 0 2»: ¯ dº,+$¹\0º=%¹Gj0º h %¹jG0ºE? i _ h %¹ºE? i ¡ 0and m+$¹10º m0 0. 4. Ifd 0 2» ¯ d 1¼,+$¹\0º=¹1_º h %¹ºE? i ¡ 0 Proof. Theconsumer’sdecisionruleonwhethertobuyornottobuyfollowing<isafunctionof ˜ 0. That is why the regions are a function of ˜ 0. (See Lemma 1 in the Technical Appendix). However, the expected utility of the consumer is calculated here from the perspective of the influencer who observesthetrue0. Thatiswhy0isbeingusedinthecalculation. Wecouldalsocalculatethe+$ fromtheperspectiveoftheconsumer,andinthiscasewewoulduse ˜ 0. However,sincethecontract is between the firm and the influencer, the relevant+$ is from the perspective of the influencer. TherestoftheproofisidenticaltotheproofofLemma2inthemainbodyofthetext. Lemma B.0.7.3. The independent influencer’s expected utility is maxf0U+$¹0 0º2g. Below wedescribewhensheacquiresinformationatC= 3andpostsareviewatC= 4(whichwesimplify to“postsareview"), 97 1. Ifd 0 2»0d¹0ºº,theinfluencerdoesnotpostareview. 2. Ifd 0 2»d¹0º:º,theinfluencerpostsareviewifU%¹Gº h %¹jGºE? i 2 3. Ifd 0 2»: ¯ dº,theinfluencerpostsareviewifU %¹Gº h %¹jGºE? i _ h %¹ºE? i 2 4. Ifd 0 2» ¯ d 1¼,theinfluencerpostsareviewifU¹1_º h %¹ºE? i 2 Proof. Sincethisinvolvestheindependentinfluencer,theproofisexactlythesameastheproofof Lemma3inthemainbodyofthetext. WenowredotheproofofProposition1withafewchangesaccountingforpartialobservability of0. Thefirm’sproblemis, 1 ¹0 º=max 0(¹0º %¹Gj0º 1f%¹jG ˜ 0º :g?¸%¹Bj0º 1f%¹jB ˜ 0º :g?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 (B.5) Notethat,aswearguedinLemma2,+$ isafunctionof0sincetheinfluencerobserves0. The consumerinferenceintheobjectivefunctionisafunctionof ˜ 0. Theposteriorbeliefsaredefinedas follows, %¹jG ˜ 0º= »W¸ ˜ 0¹1Wº¼d 0 »W¸ ˜ 0¹1Wº¼d 0 ¸»¹1Wº¸ ˜ 0W¼¹1d 0 º (B.6) %¹jB ˜ 0º= ¹1Wºd 0 ¹1Wºd 0 ¸W¹1d 0 º (B.7) Thefirm’saffiliationdecisiondependsontherelativemagnitudesof 0 and 1 ¹0 º, \ = 8 > > > > < > > > > : 1 if 1 ¹0 º 0 0 >C4AF8B4 98 We use Lemma 2.3.1.1 with0 = 0 to determine the consumer behavior in each region. Similarly, we use Lemma 2.3.2.1 to obtain the value of information and Lemma 2.3.3.1 to determine the independentinfluencer’sdecisiontopost. Case1 d 0 2»0d¹0ºº. Here the consumer’sprior is so lowsuch that no messagewill convince him tobuy. Thefirmisnotabletomakeanyprofitandisnotwillingtopayforaffiliation. Based onLemma2.3.3.1,theindependentinfluencerdoesnotpostareview. Case2 d 0 2»d¹0º:º. Let’s first consider a potential equilibrium with affiliation, \ = 1. What will be the equilibrium level of affiliation,0 ? In this region, the consumer buys only if he observesG. As the firm increases0, it increases %¹Gj0º, but %¹jG ˜ 0º does not change since0 is not observed by the consumer. The consumer buys the product if%¹jG ˜ 0º :. Hence,thefirm’smaximizationproblembecomes, max 0 1 = 1f%¹jG ˜ 0º :g%¹Gj0º?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 ¹1º 00 1 ¹2º Theconsumerbuystheproductaslongas ˜ 0 W¹1:ºd 0 :¹1d 0 º¹1Wº W:¹1d 0 º¹1Wº¹1:ºd 0 ¯ 0 Wehaveshownbeforethat 0 ¯ 0 1. Wecanre-writethefirm’sproblemasfollows: max 0 1 =%¹Gj0º?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 ¹1º ˜ 0 ¯ 0 ¹2º 0 0 ¹3º Notethatthisisidenticaltotheproblemthatwehadunderfulldisclosure,exceptforconstraint 2. But again, an increase in0 does not change ˜ 0. As before, constraint 1 is binding, and the 99 expressionfor(¹0º canbesubstitutedbackintotheobjectivefunction. Wecanshowthat 1 =¹1Uº%¹Gj0º?¸U%¹º%¹Gj0ºE maxfU+$¹0 0º2g Intheexpressionabove,onlythefirsttwotermsarefunctionsof0. Bothtermsareincreasing functions of0. Hence,0 = 1 — the firm fully affiliates with the influencer. That is, if we had a candidate equilibrium with affiliation (\ = 1), the consumer would infer that ˜ 0 = 1. But this results in a contradiction since for this equilibrium to hold, it must be the case that ˜ 0 ¯ 0 1. Hence,therecannotexistanequilibriumwithaffiliationinthisregion. Let’s next consider an equilibrium with no affiliation, \ = 0. Here the firm makes 0 = %¹Gº? if the independent influencer posts a review —+¹0 0º 0. If the firm deviates to \= 1,ourassumptionontheout-of-equilibriumbeliefsimpliesthat ˜ 0= 1n. Hence,there wouldbenopurchasesince ˜ 0 ¡ ¯ 0. Thisimpliesthatthethefirmhasnoincentivetodeviate. Case3 d 0 2»: ¯ dº. Here the consumer buys if he sees no review or a positive review, but he does not buy if he sees a negative review. Let’s consider a candidate equilibrium with affiliation, \ = 1. Thefirm’saffiliationproblemis, max 0 1 =%A¹Gj0º?(¹0º s.t. (¹0º¸U+$¹10º maxfU+$¹0 0º2g 0 ¹1º ˜ 0 1 ¹2º 0 0 ¹3º Again, we can show that 0 = 1 and hence ˜ 0 = 1. Note that at ˜ 0 = 1, the signal has no informationaboveandbeyondawareness. Thatis,thesignaldoesnotchangetheconsumer’s prior belief,d 0 . This also implies that+$¹1 1º=¹1_º h %¹ºE? i . We can show that for2 maxfU+$¹0 0º ¯ 2 g, the firm has no incentive to deviate to\= 0 since that yields strictlylessprofitforthefirm. 100 Next, let’s consider another candidate equilibrium where \ = 0. We can show that this cannot be an equilibrium for2 maxfU+$¹0 0º ¯ 2 g since the firm would make strictly moreprofitbydeviatingto\ = 1,wherebyassumption ˜ 0= 1n. Wecanalsoshowthatfor2¡ maxfU+$¹0 0º ¯ 2 g,noaffiliationisinequilibrium. Case4 d 0 2» ¯ d 1¼. Inthisregiontheconsumeralwaysbuysregardlessthevalenceofreview. Hence, affiliation is only important for the purpose of awareness. The rest of the proof is almost identicaltotheproofofProposition1,case4. Theonlydifferenceisintheout-of-equilibrium beliefs,buttheresultsremainthesame. 101

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Asset Metadata

Creator Pei, Lei Amy (author)

Core Title Manipulating consumer opinion on social media using trolls and influencers

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 04/08/2020

Defense Date 03/10/2020

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

Tag curation algorithm,game theory,influencer marketing,information design,Marketing,misinformation,OAI-PMH Harvest,online reviews,recommendations,social media, trolls,word of mouth

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Mayzlin, Dina (committee chair), Camara, Odilon (committee member), Dukes, Anthony (committee member), Luo, Lan (committee member), Nunes, Joseph (committee member)

Creator Email amyleipei@gmail.com,leipei@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-280460

Unique identifier UC11674884

Identifier etd-PeiLeiAmy-8250.pdf (filename),usctheses-c89-280460 (legacy record id)

Legacy Identifier etd-PeiLeiAmy-8250.pdf

Dmrecord 280460

Document Type Dissertation

Rights Pei, Lei Amy

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA