Harnessing beliefs to optimally disclose contestants’ types

Serena, Marco Article — Published Version Harnessing beliefs to optimally disclose contestants’ types Economic Theory Provided in Cooperation with: Springer Nature Suggested Citation: Serena, Marco (2021) : Harnessing beliefs to optimally disclose contestants’ types, Economic Theory, ISSN 1432-0479, Springer, Berlin, Heidelberg, Vol. 74, Iss. 3, pp. 763-792, https://doi.org/10.1007/s00199-021-01378-1 This Version is available at: https://hdl.handle.net/10419/286757 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Economic Theory (2022) 74:763–792 https://doi.org/10.1007/s00199-021-01378-1 RESEARCH ARTICLE Harnessing beliefs to optimally disclose contestants’ types Marco Serena1 Received: 2 October 2017 / Accepted: 9 July 2021 / Published online: 23 July 2021 © The Author(s) 2021, corrected publication 2021 Abstract A contestant’s effort depends on her knowledge of her rival’s type. This knowledge is often limited in real-life contests. We propose a model where the principal of a contest has commitment power to verifiably disclose contestants’ types. We investigate the optimal disclosure policy to stimulate contestants’ efforts. Full disclosure stimulates more (less) effort than full concealment if high-types are more (less) likely than lowtypes. However, regardless of the likelihood of types, the optimal policy is that of contingent disclosure; it is optimal to commit to disclosing if both contestants are high types and concealing otherwise. Keywords Contests ·Strategic complements ·Strategic substitutes ·Information JEL Classification C72 ·D82 This article is based on the first chapter of my Ph.D. dissertation at Universidad Carlos III, Madrid. Previous versions of this paper circulated under the name “Information in contests” in 2013–2015, and “Harnessing beliefs to stimulate efforts” in 2016–2018. The supervision of Luis Corchón is greatly appreciated. I am pleased to acknowledge useful comments by a Co-Editor, three reviewers, and Carmen Beviá, Guillermo Caruana, Robert Edwards, Tore Ellingsen, Christian Ewerhart, Natalia Fabra, Dawei Fang, Qiang Fu, Aart Gerritsen, Andrea Guariso, Angel Hernando-Veciana, René Kirkegaard, Harald Lang, Jingfeng Lu, Thomas Mariotti, Ricardo Martinez, Diego Moreno, Ron Siegel, Giancarlo Spagnolo, and Huseyin Yildirim. I would like to thank participants at the 2013 GSE Summer Forum (Pompeu Fabra), the 2013 Young Researchers MOVE Workshop on Contests and Tournaments (UAB), the First Microeconomics Graduate Workshop (CEMFI), the 2014 Economics PhD Conference (Warwick), the 25th International Conference on Game Theory (Stony Brook), the 2014 MACCI Enter Workshop (Mannheim), the 2014 European Winter Meeting of the Econometric Society (Carlos III), the 2015 ECARES Summer School (ULB), the 2015 SAET Conference (Cambridge), and the 2016 CBESS Conference on Contests (UEA). I acknowledge financial support from ECO2011-25330/ECON. All errors are my own. BMarco Serena [email protected] 1Max Planck Institute for Tax Law and Public Finance, Munich, Germany 123 764 M. Serena 1 Introduction Running relative performance contests is a popular way to stimulate workers’ efforts. It is estimated that roughly between one third (Loew 2015) and two thirds (The Wall Street Journal 2012) of firms use some such form of internal ranking. A manager can provide some information to the competing workers about each others’ types, and competing workers do not a priori have this information on their rivals unless informed by the manager. Workers’ types can be, for instance, their past performance, their inherent ability or even their identity itself. If the manager discloses types before the proclamation of the winner of the contest (i.e., the worker who obtains the promotion, the tenure, or the bonus), then such a disclosure affects workers’ efforts, and consequently the firm’s profit. Analogous information disclosure problems arise in other settings. For instance, the organizer of a sports tournament might disclose information about players that would otherwise be too costly or impossible to retrieve by the players themselves; in chess tournaments, players’ Elo ranking is sometimes displayed on the pairing-list board, and in the NBA fine-tuned data collected by advanced tracking cameras are disclosed online. Similarly, scholars compete for grants by submitting research projects, and they presumably do not know who they are competing against unless the grant-awarding entity publicly discloses the list of participants. The awareness of a rival’s type affects a contestant’s effort. For example, an average researcher aware of being shortlisted to compete for a grant against a leading scholar might give up hope and exert little effort. In the present paper we explore the effort-maximizing disclosure policy. Our model has an informed principal (manager, sports tournament organizer, grantawarding entity) who maximizes the aggregate effort of two contestants (workers, players, scholars) who compete in a Tullock contest with discriminatory parameter r∈(0,1](Tullock 1980). First, the principal commits to a costless and verifiable disclosure policy about contestants’ types (modeled as marginal costs of efforts). Then, nature assigns a type to every contestant. Contestants know their own type, but their knowledge of their rival’s type depends on the principal’s disclosure policy.1Our model assumes that there are two contestants. This assumption admittedly stylizes the applications we described, but it allows us to abstract from contestants’ endogenous participation decisions, and thus to isolate the effect of the disclosure policy on contestants’ efforts. Each contestant is of high-type with probability p∈[0,1], and low-type with the complementary probability. Types are drawn independently and p is common knowledge. We find that the probability of high-types pdrives the optimal disclosure policy. We provide an intuitive explanation in what follows. We describe two building blocks of our results. First, around the equilibrium of our setup the effort of the high- (low-) type and that of her rival are strategic complements 1The principal commits to a disclosure policy ex-ante, that is, before observing types. This captures the idea that the principal chooses and commits to the rules of the contest, among which the disclosure policy, which are implemented every time the contest is run. This argument is similar to the one in Rayo and Segal (2010), where the precommitment power of the sender is a way to build long-run reputation in the eyes of the sequence of “short-run” receivers. In such a repeated game, losses of reputation would jeopardize the sender’s long-run profits. Moreoever, in “Appendix B.1” we show that the principal is better off committing to her disclosure policy ex-ante , rather than ex-post, if she could choose. 123 Harnessing beliefs to optimally disclose contestants’ types 765 (substitutes). Second, the effort of a contestant, regardless of her type, is maximum when she is up against a contestant of the same type. We call this situation an even contest, and its effort the “stimulated effort,” in contrast to an uneven contest and the “discouraged effort” where types differ.2 With these two building blocks in mind, consider the effect of the probability of high-types pon the optimal disclosure policy. Trivially, in the extreme case of p∈ {0,1}, any attempt to conceal types in order to affect efforts is fruitless since contestants know p, and thus disclosure policies are all outcome-equivalent. We analyze the case of a high (low), but non-extreme, value of pinCaseI(II)below. Case I (high p) Consider a high-type. Under disclosure, a high-type will exert the stimulated effort with high probability (p) and the discouraged effort with low probability (1 −p). Under concealment, a high-type believes she is in an even (uneven) contest with high (low) probability, and knows that her most likely rival (a high-type) has the same beliefs. Thus, the resulting level of effort of the high-type under concealment is close to the effort that the principal expects from the high-type under disclosure, which in fact is the high-type’s stimulated effort with high probability (p) and the high-type’s discouraged effort with low probability (1 −p). Consider a low-type. Under disclosure, a low-type will exert the stimulated effort with low probability and the discouraged effort with high probability. Under concealment, a low-type believes she is most likely up against a high-type who believes she is most likely up against another high-type, and so on. Thus, a low-type under concealment, despite knowing she is most likely up against a high-type, exerts significantly less effort than in a high-vs-low contest under disclosure, because she knows that her (most likely) high-type rival believes that the contest is (most likely) even, rather than uneven. This discouragement of a low-type is obviously only present under concealment. For brevity, we call this discouragement caused by concealment when pis high the negative p-Effect (hereafter, “-pE”), which is stronger the more pis far from 1/2 and close to 1. Case II (low p) In a nutshell, everything said for Case I carries over by just swapping highand lowtype, and changing “less ” and “discouragement ”into“more” and “encouragement.” In fact, the key difference is that now the p-Effect is positive. For brevity, we call this encouragement caused by concealment when pis low the positive p-Effect (hereafter, “+pE”), which is stronger the more pis far from 1/2 and close to 0. The present paper makes the case that the p-Effects drive the optimal disclosure policy in contests. In fact, we find that in a standard Tullock contest aggregate effort is maximized by disclosure when p∈(1/2,1)in order to avoid the—pE, and by concealment when p∈(0,1/2)in order to make the most out of the +pE.3Concealment and disclosure yield the same aggregate effort in the remaining cases; namely, 2When two contestants are of the same type, they maximize the same payoff function, and this typically yields the maximum individual effort in standard two-player Tullock contests (see, e.g., Baik 1994;Nti 1999). 3Recall that +pEand-pE are present only under concealment. 123 766 M. Serena the symmetric prior p=1/2, because +pE and -pE cancel out, and the degenerate prior p∈{0,1}. These are the results of Sect. 4. In Sect. 5, we consider a principal who can precommit to disclosing or concealing contestants’ types contingently on the realization of types. We show that the optimal information disclosure, regardless of p, is to commit to disclosing contestants’ types if contestants are both high-types and concealing all other realizations of types. Such an optimal contingent disclosure policy thus improves upon the full disclosure and full concealment policies considered in Sect. 4. The intuition behind this result relies again on +pE and -pE, and it will be given in Sect. 5, after presenting the result. After a discussion of the related literature in Sect. 2, the main body of the paper (Sects. 3–6) includes what is needed to understand the intuition behind the optimal disclosure policies, and everything else is moved to the appendices. In particular, Sect. 3 spells out the contest model and the disclosure game. Sections 4and 5investigate the optimal disclosure policy for a principal who maximizes aggregate effort: in Sect. 4 the principal can only commit to either fully disclosing or fully concealing contestants’ types, and in Sect. 5the principal can commit to disclosing contestants’ types contingently on the realization of types. Section 6discusses the results. Appendix A contains the proofs and Appendix B extensions. 2 Related literature The most closely related literature is that on how information affects efforts in Tullock contests (e.g., Hurley and Shogren 1998a; Hurely and Shogren 1998b; Denter et al. 2011; Epstein and Mealem 2013; Heijnen and Schoonbeek 2016; Zhang and Zhou 2016; Chen et al. 2017,2018b). These papers—in contrast to ours—assume that the type of one contestant is common knowledge and that of the other is private information. In other words, the private information the principal may disclose is one-sided among the two contestants. We believe that a disclosure policy equally capable of disclosing each contestant’s type is a sensible assumption in the applications discussed in the Introduction. From a more technical perspective, the prevailing assumption of one-sided private information contest shuts down the higher-order reasoning behind the p-Effects which drives the results of the present paper. Perhaps one reason for the popularity of the one-dimensional private information approach is its tractability, in that it admits a closed-form solution for equilibrium efforts, contrary to our context.4 In fact, in the words of Zhang and Zhou (2016), “It is natural to ask what happens if both contestants possess private information. Several technical challenges emerge accordingly, [...] a characterization of the equilibrium is usually not obtainable.” We sidestep the lack of a closed-form solution by showing that, in Tullock contests, equilibrium efforts always satisfy a property that depends on contestants’ beliefs about the rival’s type (see (6) in Appendix A), and that this property suffices to fully characterize the optimal disclosure policy. Furthermore, this novel property happens to generalize a well-known wisdom of complete information Tullock contests—namely the ratio 4Even if r=1in(1), a contest does not admit a closed-form solution for equilibrium efforts in our fully-private information setting unless p∈{0,1/2,1}. The case p=1/2 is analyzed in Malueg and Yates (2004). 123 Harnessing beliefs to optimally disclose contestants’ types 767 of types equals the ratio of efforts—to Tullock contests with a variety of contestants’ beliefs about the rival’s type. The literature has proposed ways that sidestep the lack of a closed-form solution for the equilibrium efforts of private information Tullock contests: among others, a modified contest success function (see Wasser 2013), a binary effort space (see Dubey 2013), or numerical simulations (see Hurley and Shogren 1998a; Wasser 2013). In particular, Wasser (2013) compared full disclosure and full concealment through numerical simulations, while in Theorem 1 we formally prove his numerical result under binary distribution of types; however, not having an analytical result impeded him from explaining the reason why the probability of highand low-types is what drives the optimal disclosure policy. The key role of the probability of highand lowtypes is one of our main contributions, and we show how it extends to some non-binary distributions (see, Appendix B.3). The present paper is, to the best of my knowledge, the first to study type-contingent information disclosure in contests with two-sided asymmetric information. Following the present paper, Chen et al. (2018a) study the joint optimal design of timing and information disclosure in Tullock contests, and Lu et al. (2018) study the optimal information disclosure in all-pay auctions, where an arbitrarily higher effort secures victory. 5In particular, Lu et al. (2018) extend the present analysis to all-pay auctions. Their work considerably contributes to the literature in that they provide a full characterization of the equilibrium bidding function in all-pay auctions under a regime of contingent information disclosure, where Siegel’s (2014) approach is not applicable. In contrast to our Tullock success function, the all-pay auction yields payoffs which are not quasiconcave, hence leading to a nondegenerate mixed strategy equilibrium even in the complete information case. This generates structural differences between the two settings which make the results hardly comparable. For instance, in Lu et al.’s (2018) equilibrium participation depends on beliefs and distribution of types. Furthermore, the elegant characterization of the p-Effects is possible thanks to the strategic complementarity (substitutability) of the high- (low-) type’s effort around the equilibrium, which we find in our setup with Tullock contest success function with exponent r≤1.6 Other branches of the literature are related to the present paper. First, the literature on the disclosure of contestants’ dynamic performance (e.g., Aoyagi 2010; Goltsman and Mukherjee 2011) or of the number of contestants (e.g., Myerson and Wärneryd 2006; Lim and Matros 2009;Fuetal.2011). Second, the literature on contestants’ incentive to acquire information (e.g., Yildirim 2005; Denter et al. 2011). In our model it is the principal, rather than the contestants, who has control over the information that contestants acquire. Third, the literature on contestants’ incentive to disclose their private information. While players have no incentive to do so in all-pay auctions (e.g., Kovenock et al. 2015), in Tullock contests players may want to disclose their information (Wu and Zheng 2017). Fourth, and finally, the Bayesian persuasion literature 5An earlier version of the present paper is Serena (2016), which Lu et al. (2018) cite. The earliest version is Serena (2014); see the reference in Vázquez Sedano (2015). 6Beside Lu et al.’s (2018) extension of our results to all-pay auctions, the comparison between expected aggregate effort under full disclosure or full concealment is in Fu et al. (2014) and Kovenock et al. (2015), who find that full concealment dominates full disclosure. 123 768 M. Serena (see Kamenica and Gentzkow 2011, henceforth KG), where a sender commits to a disclosure policy on a stochastic state of the world before its realization. In this class of models, it is not a novelty to find optimality of contingent information disclosure, like in our setting; in particular, it is optimal for a sender (i.e., a prosecutor in KG, and the principal here) to disclose to a receiver (i.e., a judge in KG, and the set of two contestants here) only the realization which is most favorable to the sender (i.e., the defendant being guilty in KG, and both contestants being high-types here) and to pool the other signals in a strategic way (i.e., randomizing between innocent and guilty signals in KG, and concealing here). The disclosure policy space is more general under the Bayesian persuasion approach than in our setting. Nevertheless, our disclosure policy space is simpler to implement, in that it involves no stochastic messages, and thus it requires a weaker commitment condition than Bayesian persuasion. Furthermore, our disclosure policy space is not affected by the technical challenges of applying Bayesian persuasion to analyze the disclosure of both contestants’ types. (Three such technical challenges are discussed by Zhang and Zhou 2016; Section 4). 3 A model of a contest The contest technology. Two contestants, indexed by i=1,2, compete for a prize of value V>0 by exerting effort ei≥0. Each contestant has a probability of winning a prize equal to7 pi(ei,ej)=er i er i+er j if ei+ej>0 1 2if ei+ej=0 (1) with i,j=1,2,j= i, and 0 <r≤1. When at least one player exerts strictly positive effort, (1) uniquely satisfies a set of appealing axioms (see Skaperdas 1996). The contestants’ payoff. Contestants are risk-neutral. The cost of effort is linear, and contestant iis of type θi, which determines her marginal cost of effort. In particular, the payoff of a contestant of type θiwhen she exerts effort eiand her rival exerts effort ejis pi(ei,ej)V−ei θi . Contestant’s type θiis an independent draw from the commonly-known prior,8 θi=h l with probability p with probability 1 −p(2) with p∈[0,1]and h>l>0. Thus, being a high-type rather than a low-type brings about a lower marginal cost of effort. 7In Appendix B.4 we briefly discuss a more general contest technology. 8In Appendix B.3 we discuss the case of a continuum of types. 123 Harnessing beliefs to optimally disclose contestants’ types 769 Table 1 Possible realizations of contestants’ types, with corresponding probability and aggregate equilibrium effort under P=Dand P=C Realizations of {θ1,θ 2}Prob. Aggregate effort if DAggregate effort if C {h,h}p22ehh 2eh {l,l}(1−p)22ell 2el {h,l}or{l,h}2p(1−p)ehl +elh eh+el The timing of the game. First, before types are realized, the principal commits to a verifiable and costless disclosure policy P, which is observed by the contestants. Then, types are realized, each contestant ilearns θiand may or may not be informed by the principal of θj, according to P. Finally, contestants simultaneously choose efforts. The principal and the disclosure policy. The principal maximizes the expected aggregate effort when choosing the disclosure policy P.9In Sect. 4, the principal chooses between two extreme disclosure policies on types: disclosure P=D(that is, contestants are informed of their rival’s type), and concealment P=C(that is, contestants are not informed and contestants’ posterior belief about their rival’s types equals the prior (2)). In Sect. 5we enlarge the space of P’s: the principal commits to disclosing or concealing contestants’ types contingently on type realizations.10 Each Pinduces a system of beliefs for the contestants, and each system of beliefs induces an expected level of aggregate effort. We discuss the Pwhich maximizes aggregate effort in the main text, while intermediary results on how systems of beliefs affect aggregate effort appear in “Appendix A”. Equilibrium. Since contestants are ex-ante symmetric, it is natural to focus on typesymmetric equilibria; that is, contestants of the same type follow the same equilibrium strategy. The existence of a unique equilibrium has already been proved in our setting (see Einy et al. 2015, and Ewerhart and Quartieri 2019). Our results in the main text are valid for both interior and corner equilibria; formally, we assume interior equilibrium throughout the paper, and at the end of “Appendix A” we build on our findings to explain why considering corner equilibria does not affect our results. 4 Optimal disclosure policy: disclose or conceal? If the principal chooses between committing to full disclosure (P=D) or full concealment (P=C) only, which one yields the greatest expected aggregate effort? When committing to a P, the principal does not know whether the contest will be between two high-types, two low-types, or a high-type and a low-type. Table 1provides a summary of the possible realizations of types, the corresponding probabilities, and the aggregate equilibrium effort (i.e., the principal’s payoff). The notation is as follows: the first subindex in eis the type of the contestant exerting effort, and the 9In Appendix B.2 we discuss alternative objective functions for the principal. 10 In Appendix B.1 we discuss the optimal disclosure policy if the principal does not have commitment power, so that the disclosure policy is de facto chosen ex-post. 123 770 M. Serena Fig. 1 High-type’s equilibrium efforts as functions of passuming r=1, V=1, l=1andh=2. The lines ehh and ehl are her two efforts under D. The dashed line disclosure-ehis thus the principal’s expectation of a high-type’s effort under D. The line concealment-ehis the effort of a high-type under C. The segment called +pE is the positive p-Effect, which is depicted at p→0 second subindex is the type of her rival, in case she is told (that is, under D). Thus, for instance, ehl is the equilibrium effort of a high competing against a low (under D), and ehis the equilibrium effort of a high who does not know her rival (under C). If the contest is even—{h,h}or{l,l}—, then Dmaximizes aggregate effort because it prevents contestants from thinking that they are competing in an uneven contest (i.e., 2ehh ≥2ehand 2ell ≥2el). If instead the contest is uneven—{h,l}or{l,h}—, then C maximizes aggregate effort because it prevents discouragement. Therefore, finding the optimal Pboils down to the ex-ante trade-off between the benefits of Dif the contest is even and the benefits of Cif the contest is uneven. In what follows we gradually build the intuition that will eventually lead to Theorem 1using the graphical support of Figs. 1and 2. In Fig. 1we focus on the effort of a high-type.11 The horizontal lines are her two possible equilibrium efforts under disclosure—against another high-type or against a low-type—, where pdoes not affect efforts and ehh >ehl because evenness of types stimulate efforts. We call disclosure-ehthe effort that the principal expects from the high-type under disclosure, which equals pehh +(1−p)ehl. Thus, the disclosure-eh is a straight line in pgoing from ehl to ehh, as depicted. We call concealment-ehthe equilibrium effort of a high-type under concealment. Ranking concealment-ehand disclosure-eh(and concealment-eland disclosure-elin Fig. 2) unveils the optimal disclosure policy. The concealment-ehis increasing in pbecause pincreases the probability of an even contest. When p→0, the high-type believes she is up against 11 Since a closed-form solution for equilibrium efforts does not exist, figures are created using numerical simulations on the system of FOCs (see (4)and(5) in “Appendix A”) with r=1, V=1, l=1andh=2. This parametrization does not affect the qualitative features of those figures. 123 Harnessing beliefs to optimally disclose contestants’ types 777 The FOCs are necessary and sufficient to characterize the best reply, which is continuously differentiable and bounded.16 To lighten the notation we omit the dependencies of ehand elon p,h,l,rand V. A type-symmetric equilibrium is a pair (eh,el)satisfying (4) and (5). Finally, isolate the second addends of the left-hand sides in (4) and (5), consider their ratio, (1−p)rer−1 her l prer−1 ler h = 1 h−pr 4ehV 1 l−(1−p)r 4elV ⇐⇒ (1−p)el peh = 1 h−pr 4ehV 1 l−(1−p)r 4elV cross-multiply, (1−p)el l−(1−p)2r 4V=peh h−p2r 4V ⇐⇒ 4 rV (1−p)el l=4 rV peh h+(1−2p), and finally use the expressions of ehh and ell in (3), in order to derive the following key relation between equilibrium efforts under Dand under C, el ell (1−p)=eh ehh p+(1−2p). (6) Using the above preliminaries (i.e., (3)–(6)), we are ready to proceed with the three main steps needed to prove Theorem 1. Step 1. We show that πD−C=0iffp∈{0,0.5,1}. First, we analyze when πD−C takes the value 0. πD−C=p2[2ehh −2eh]+2p(1−p)[ehl +elh −eh−el]+(1−p)2[2ell −2el]=0, i.e., p2ehh +(1−p)2ell +p(1−p)[ehl +elh]−peh−(1−p)el=0.(7) If p=0orp=1, by el=ell and eh=ehh respectively, (7) holds and πD−C=0. Thus, from now on we can focus on p∈(0,1). 16 See Yildirim (2005) and Morgan and Várdy (2007). 123 778 M. Serena Substitution of (3) and (6)into(7) yields p2rh 4V+(1−p)2rl 4V+p(1−p)r(h+l)hrlr (hr+lr)2V−ph+l heh−1−2p 4rlV =0, pr h+l 4V+(1−p)r(h+l)hrlr (hr+lr)2V=h+l heh, pehh +(1−p)ehl =eh.(8) Hence, (8) is a condition for the indifference between Dand Cwritten in terms of the efforts exerted by the high-type only. This condition coincides with equality of concealment-ehand disclosure-eh(see Fig. 1). With a similar procedure used to find (8)—that is, by substituting (6)intoehrather than into el–we can obtain the value of elfor which the administrator is indifferent between Dand C, which symmetrically to (8)is (1−p)ell +pelh =el.(9) This condition coincides with equality of concealment-eland disclosure-el(see Fig. 2). We plug (8) and (9)into(4) and see if any p∈(0,1)solves the resulting equation— thus, yielding πD−C=0. 17 First, use (3) to rewrite the indifference conditions (8) and (9)foreland ehas eh=rh p(hr+lr)2+4(1−p)hrlr 4(hr+lr)2V,(10) el=rl(1−p)(hr+lr)2+4phrlr 4(hr+lr)2V.(11) These efforts are those that, if exerted under full concealment, lead to indifference between Cand D.Now,using(4) and (5), we check whether these effort levels are reached for some parameter values. Hence, we rewrite (4)as pr h 4V+(1−p)rher her l (er h+er l)2V=eh.(12) Plugging (10) into the right-hand side of (12), and after simple simplifications, we obtain the following er her l (er h+er l)2=hrlr (hr+lr)2.(13) 17 Remark: the fact that (8)and(9) are sufficient for πD−C=0 could have already been noticed in (7), but we needed to use (6) to show that they are also necessary for πD−C=0. 123 Harnessing beliefs to optimally disclose contestants’ types 779 Finally, we plug (10) and (11) where we defined J=p(hr+lr)2+4(1−p)hrlrand K=(1−p)(hr+lr)2+4phrlrinto (13), and obtain hrlrJrKr (hrJr+lrKr)2=hrlr (hr+lr)2 ⇐⇒ l2rKr(Jr−Kr)=h2rJr(Jr−Kr), (14) and the unique solution of (14)isJ=K, which is equivalent to p(hr+lr)2+4(1−p)hrlr=(1−p)(hr+lr)2+4phrlr ⇐⇒ 4(1−2p)hrlr=(1−2p)(hr+lr)2, whose unique solution is p=0.5. Similar algebra shows that (10), (11) and p=0.5 satisfy (5)—besides satisfying (4) as proved. Hence, we proved that there are only three values of pfor which πD−C=0: 0, 0.5, and 1. Step 2. We write the system (4) and (5) as a unique equation in terms of ehand parameters only, and then we make use of the implicit function theorem to evaluate the derivative of πD−Cin p=0.5, and prove that it is strictly positive. That is, ∂πD−C ∂pp=0.5 >0. Remember that efforts under Dare not functions of p, unlike the efforts under C.We omitted this detail so far in the notation, and we now write it when it would otherwise yield confusion as we need to differentiate with respect to p. To simplify πD−Cwe use the same steps used to move from (7)to(8), where we simplified a p, and get ∂ ∂pp2ehh +p(1−p)ehl −peh(p)p=0.5 >0, 2pehh +ehl −2pehl −eh(p)−p∂eh(p) ∂pp=0.5 >0, ehh −eh(0.5)>1 2∂eh(p) ∂pp=0.5.(15) When p=0.5, we know from Step 1 that πD−C=0, and hence from (8), eh(0.5)= ehh+ehl 2. Therefore, (15) is equivalent to ehh −ehl >∂eh(p) ∂pp=0.5 (16) 123 780 M. Serena The left-hand side of (16) is known by (3). The right-hand side is trickier. First, isolate elin (6) and use ehh and ell from (3) to obtain: el=4pleh+(1−2p)rlhV 4(1−p)h.(17) Use (17)into(4), and obtain f(eh,p)≡pr 4eh V+4rhrlrr(1−p)r+1er−1 h[h(1−2p)V+4peh]r [4rhr(1−p)rer h+(4pleh+hl(1−2p)V)r]2−1 h=0. The defined f(eh,p)is an equation in pand ehonly, and hence by the implicit function theorem ∂eh(p) ∂pp=0.5 =− ∂f(eh,p) ∂pp=0.5 ∂f(eh,p) ∂ehp=0.5 .(18) We will eventually plug (18)into(16) to conclude the proof of Step 2. We start with the denominator of (18): ∂f(eh,p) ∂ehp=0.5 =∂f(eh,0.5) ∂eh =∂ ∂ehr 8eh V+hrlrr 2(hr+lr)2eh Vp=0.5 =−r e2 hh2r+l2r+6hrlr 8(hr+lr)2Vp=0.5 (19) Note that when p=0.5 equilibrium effort is eh=rhh2r+l2r+6hrlr 8(hr+lr)2V. This is easy to verify, since p=0.5 corresponds to the contest of Malueg and Yates (2004)—see their expression (20). We use this expression into (19) and obtain ∂f(eh,p) ∂ehp=0.5 =− 1 hehp=0.5 . 123 Harnessing beliefs to optimally disclose contestants’ types 781 Hence, expression (18) reads ∂eh(p) ∂pp=0.5 =− ∂f(eh,p) ∂pp=0.5 ∂f(eh,p) ∂ehp=0.5 =heh ∂f(eh,p) ∂pp=0.5 =rh 4V+4rhr+1lrrer h ∂ ∂p (1−p)r+1[h(1−2p)V+4peh]r [4rhr(1−p)rer h+(4pleh+hl(1−2p)V)r]2p=0.5 =rh 4V+4rhr+1lrrer h ∂ ∂pa(p)b(p) [c(p)]2p=0.5 (20) where we defined a(p)=(1−p)r+1, b(p)=[h(1−2p)V+4peh]r, c(p)=4rhr(1−p)rer h+(4pleh+hl(1−2p)V)r. Hence, ∂ ∂pa(p)b(p) [c(p)]2p=0.5 =a(p)b(p)+a(p)b(p) [c(p)]2p=0.5 −2a(p)b(p)c(p) [c(p)]3p=0.5 . (21) From the definitions of the functions a,band ccompute their values and their derivatives when p=0.5. a(0.5)=1 2r+1a(0.5)=− r+1 2r b(0.5)=2rer hb(0.5)=2r+1rer−1 h[eh−hV/2] c(0.5)=2rer h(hr+lr)c(0.5)=−2r+1hrrer h+2r+1lrrer−1 h(eh−hV/2) 123 782 M. Serena Plug these results into (21) to write (20) in the following way18 ∂eh(p) ∂pp=0.5 =rh 4V+ −4rhr+1lrrer her h(r+1)−rer−1 h(eh−hV/2) 2rer h(hr+lr)2 + rer h2r+1−hrer h+lrer−1 h(eh−hV/2) 2rer h(hr+lr)3⎤ ⎦ =rh 4V+4hr+1lrrlr−hr (hr+lr)(hr+lr)2+4hrlrV −hr+1lrrlr(2r+1)+hr(1−2r) (hr+lr)3V, where we used the fact that eh=rhh2r+l2r+6hrlr 8(hr+lr)2Vwhen p=0.5. Therefore, we can finally evaluate expression (16). rh 4−rh hrlr (hr+lr)2>rh 4+4hr+1lrrlr−hr (hr+lr)(hr+lr)2+4hrlr+ −hr+1lrrlr(2r+1)+hr(1−2r) (hr+lr)3 ⇐⇒ 2rlr−hr (hr+lr)2>4lr−hr (hr+lr)2+4hrlr ⇐⇒ 2(hr+lr)2>r(hr+lr)2+4hrlr. By r≤1, it suffices to show that 2(hr+lr)2>(hr+lr)2+4hrlr ⇐⇒ (hr−lr)2>0, and the result follows. Step 3. The continuity of ehand elin pdirectly follows from the Maximum Theorem applied to contestants’ payoff, which is continuous and strictly concave in own effort. The continuity of πD−Cin pfollows from the continuity of ehand el. Proof of Theorem 2and Proposition 3 If p∈{0,1}, the principal is trivially indifferent between all disclosure policies. The ranking between π{D,D,D}and π{C,C,C}is known from Theorem 1. Hence, in what follows, we focus on p∈(0,1)and on all rankings except the one between π{D,D,D}and π{C,C,C}. 18 For the sake of brevity, we use ehrather than eh|p=1 2 . 123 Harnessing beliefs to optimally disclose contestants’ types 783 First, we spell out some Preliminaries, including in Lemma 5two key monotonicities of efforts in beliefs. Second, building on Preliminaries and Lemma 5, we prove Theorem 2in three steps. Denoting by πPthe expected sum of efforts under disclosure policy P, we prove that π{D,D,D}>π {C,C,D}(Step 1), that π{D,C,C}>π {D,D,D} (Step 2), and that π{D,C,C}>π {C,C,C}(Step 3). Finally, note that in all the remaining policies (i.e., {C,D,D},{D,C,D},{D,D,C},{C,D,C}) contestants perfectly infer types, and thus they are outcome-equivalent to P={D,D,D}. Theorem 2thus follows from the three steps. Finally, in order to prove Proposition 3, notice the Steps 1–2 imply π{D,C,C}> π{D,D,D}>π {C,C,D}, and Step 3 implies that {C,C,C}is not the best policy. In the additional Step 4 below we prove that {C,C,C}is not the worst policy either. By Theorem 1the full ranking of Corollary 4then follows. Preliminaries First, we derive two key equilibrium properties, (22)and (23), which hold for the two new disclosure policies, {C,C,D}and {D,C,C}, by plugging the appropriate p. In particular, for policy {C,C,D}we plug p=1in(5), as a low-type observing C under policy P={C,C,D}is sure she is up against a high-type, and following the same steps used to derive (6) from (4) and (5) we obtain 4leh−4(1−p)hel=prhlV ,(22) and similarly, for policy {D,C,C},weplugp=0in(4), as a high-type observing C under policy P={D,C,C}is sure she is up against a low-type, and following the same steps used to derive (6) from (4) and (5) we obtain 4hel−4lpeh=(1−p)rhlV.(23) The second preliminary result is the following key lemma. Lemma 5 The equilibrium effort of the low-type decreases in the high-type’s belief of being in an even contest. The equilibrium effort of the high-type increases in the low-type’s belief of being in an even contest. Proof of Lemma 5For the sake of the proof of this lemma only, we denote by ph(pl) the belief of a high (low) type of being in an even contest; that is, of being against another high (low) type. Hence, we rewrite the system of FOCs (4) and (5)as phA+(1−ph)B=1 h plC+(1−pl)D=1 l ,(24) where we define A≡r 4eh V,B≡rer−1 her l er h+er l2V,C≡r 4el V,D≡rer−1 ler h er h+er l2V. We prove the statements of the lemma building on five intermediary results. 123 784 M. Serena First, we show that eh>el.(25) Assume by contradiction that eh≤el. Since 1 h<1 l,(24) implies phA+(1−ph)B<plC+(1−pl)D.(26) Also, it is routine to show that A≥B,A≥C,C≥Dand B≥D. Thus, A≥ max{B,C}≥min{B,C}≥D.IfB≥C, then a contradiction is immediately reached by A≥B≥C≥Dand (26 ). If instead C>B, then using the definitions of B and Cand cross-multiplying, we obtain er h+er l2>4er+1 ler−1 h, which requires er h+er l2>4e2r l, which in turn can be written as er h+3er ler h−er l>0. This contradicts eh≤el. Second, we show that ∂[phA+(1−ph)B] ∂eh ≤0≤∂[phA+(1−ph)B] ∂el . (27) Result (27) has an easy economic interpretation; a high type’s marginal payoff from higher effort is decreasing in eh(as the payoff function is concave), and is increasing in elfor el<eh(encouragement effect). In fact, Adecreases in eh.Bdecreases in ehbecause its numerator decreases in ehand its denominator increases in eh.Ais constant in el.Bincreases in elbecause ∂B ∂el =rer−1 h rer−1 ler h+er l−2re2r−1 l er h+er l3V =r2er−1 her−1 l er h−er l er h+er l3V>0, where the last inequality follows from (25). Third, as can be proved with similar, thus omitted, steps as for (27), ∂[plC+(1−pl)D] ∂eh ≤0,∂[plC+(1−pl)D] ∂el ≤0.(28) Result (28) has an analogous economic interpretation as the one provided for (27). Fourth, we show that efforts are smaller than their complete-information levels in an even contest, or formally, eh≤rhV 4and el≤rlV 4.(29) Consider the first equation of (24). If ph=1, eh=rhV/4. Consider now the effect of lowering ph(<1). Then, by A≥B, the convex combination phA+(1−ph)B 123 Harnessing beliefs to optimally disclose contestants’ types 785 decreases in ph. To keep it equal to the constant 1/h, then ehmust decrease because both Aand Bare decreasing functions of eh, as proved above. Hence, eh≤rhV/4. The proof of el≤rlV/4 is analogous. Fifth, following the same steps used to derive (6), but for the general beliefs phand plas defined in this proof, we generalize ( 6)to 4el rlV −pl 1−pl = 4eh rhV −ph 1−ph ,(30) and recall that both eland ehdepend on (ph,pl). The first statement of Lemma 5can be written as ∂el/∂ ph<0. Assume by contradiction that ∂el/∂ ph≥0. The left-hand side of (30) increases in el, and thus by ∂el/∂ ph≥0 it also increases in ph. Since the left-hand side of (30) increases in ph, also the right-hand side of (30) has to increase in ph. However, x−ph 1−phdecreases in phwhenever x≤1 (which holds by ( 29)), hence the only way to have the righthand side of (30) increasing in phis that ∂eh/∂ ph≥0. By (28), ∂el/∂ ph≥0 and ∂eh/∂ ph≥0 lead to a contradiction. Therefore, ∂el/∂ ph<0, proving the first statement of Lemma 5. The second statement of Lemma 5can be written as ∂eh/∂ pl>0. First, as we just proved, ∂el/∂ ph<0, and hence by (28) and plC+(1−pl)D=1/l,itmust be that ∂eh/∂ ph>0. Second, by (27) and phA+(1−ph)B=1/hwe obtain ∂eh/∂ pl>0⇐⇒ ∂el/∂ pl>0. Therefore, the last step to finish the proof of the second statement of Lemma 5is to assume that ∂eh/∂ pl≤0 and ∂el/∂ pl≤0 and obtain a contradiction. For brevity, we denote these two conditions e l≤0 and e h≤0 in what follows. We differentiate the second equation of (24) with respect to pl(for the differential of (1−ph)Bwe apply the formula in (21)) E   rer−1 l+plr(r−1)er−2 le ler l F   −plr2e2r−2 le l 4e2r l V+ + G   −rer−1 ler h H   +(1−pl)r(r−1)er−2 ler he l+rer−1 lrer−1 he h er h+er l2V+ I   − 2(1−pl)rer−1 ler hrer−1 he h+rer−1 le l er h+er l3V=0. We prove that E+F+G+H+I>0 in order to achieve a contradiction and thus end the proof. Term Fis trivially positive.19 Term E+Gis positive since it can be 19 Recall that we are under assumptions e l≤0ande h≤0. 123 786 M. Serena written as E+G=re2r−1 ler h−er l2+plr(r−1)e2r−2 le ler h+er l2 4e2r ler h+er l2>0. And similarly, H+I=(1−pl)r(r−1)er−2 ler he l+rer−1 lrer−1 he her h+er l−2rer−1 ler hrer−1 he h+rer−1 le l er h+er l3 ≥(1−pl)rer−1 lrer−1 he her h+er l−2rer−1 lre2r−1 he h er h+er l3 =(1−pl)rer−1 lrer−1 he h er l−er h er h+er l3≥0, where the last inequality holds true by (25) and e h≤0. Therefore, e l≤0 and e h≤0 lead to a contradiction and the result follows.  Using (22), (23) and Lemma 5, we are finally ready to proceed with the four main steps needed to prove Theorem 2and Proposition 3. The proofs of the four steps are alike. First, we simplify the difference in the expected sum of efforts under the two disclosure policies using (6), (22) and (23). Second, we use Lemma 5to conclude the proofs. Step 1.π{D,D,D}>π {C,C,D}. In this step we denote by ehand elthe efforts under Cand policy {C,C,D}; that is, it is commonly known that the high-type does not know her rival’s type and the low-type knows her rival’s type. The claim is equivalent to p2(2ehh −2eh)+2p(1−p)(ehl +elh −eh−el)>0 pehh +(1−p)(ehl +elh)−(1−p)el−eh>0 prh 4V+(1−p)rhrlr(h+l) (hr+lr)2V−(1−p)el−eh>0, whereweused(3) in the last step. Now, use (22) to eliminate eh, and obtain prh 4V+(1−p)rhrlr(h+l) (hr+lr)2V−(1−p)el−(1−p)h lel−prh 4V>0 rhrlr(h+l) (hr+lr)2V−h+l lel>0 rhrlr+1 (hr+lr)2V>el elh >el,(31) where the last step is implied by (3). 123