Information Revelation in Competing Mechanism Games

Andrea Attar∗ Eloisa Campioni† Gwenaël Piaser‡

November 22, 2011

Abstract We consider multiple-principal multiple-agent games of incomplete information. In this context, we identify a class of direct and incentive compatible mechanisms: each principal privately recommends to each agent to reveal her private information to the other principals, and each agent behaves truthfully. We show that there is a rationale in restricting attention to this class of mechanisms: If all principals use direct incentive compatible mechanisms that induce agents to behave truthfully and obediently in every continuation equilibrium, there are no incentives to unilaterally deviate towards more sophisticated mechanisms. We develop examples to show that private recommendations are a key element of our construction, and that the restriction to direct incentive compatible mechanisms is not sufﬁcient to provide a complete characterization of all pure strategy equilibria.

Key words: Incomplete information, competing mechanisms, information revelation. JEL Classiﬁcation: D82.

∗University of Rome II, Tor Vergata and Toulouse School of Economics (IDEI-PWRI); [email protected]. †University of Rome II, Tor Vergata; [email protected]. ‡IPAG Business School, Paris; [email protected]. 1 Introduction

This paper contributes to the analysis of competing mechanism games with incomplete information. Specifically, we consider a scenario in which several principals simultaneously design incentive contracts in the presence of several agents who are privately informed about their char- acteristics. This class of strategic interactions has provided a natural framework to represent the relationship between competition and incentives under incomplete information. The situation in which only one agent is considered (common agency) has become a refer- ence set-up to formalize competition in financial markets (Attar, Mariotti, and Salanie, 2011; Biais, Martimort, and Rochet, 2000), conflicts between regulators (Calzolari, 2004; Laffont and Pouyet, 2004), and to revisit classical issues in oligopoly theory (d’Aspremont and Dos Santos Ferreira, 2010; Martimort and Stole, 2009; Stole, 1995). Building on the seminal con- tribution of McAfee (1993), multiple-principal multiple-agent models have been developed to represent markets in which buyers can choose between different auctions, as it is the case in many online tradings (Ellison, Fudenberg, and Möbius, 2004; Peters and Severinov, 1997). Ver- tical restraints contexts and competition among hierarchies under asymmetric information have also been modeled as games with several principals and agents.1 Despite the increased economic relevance of competing mechanism models, there is still little consensus on their general predictive power. The recent contributions of Yamashita (2010), Peters and Szentes (2011), and Peters and Troncoso-Valverde (2011) provide different instances of a Folk Theorem: if no restriction is put on the set of available mechanisms, then a very large number of allocations can be supported as Perfect Bayesian equilibria of a competing mechanism game in which players play a pure strategy.2 One natural way to overcome such in- determinacy has been to restrict attention to some specific classes of mechanisms. Following the traditional approaches to oligopolistic screening, most economic models of competing mechanisms restricted attention to a class of simple communication mechanisms. That is, principals commit to message-contingent decisions which induce agents to truthfully reveal their private information:

[...] the literature on competing mechanisms [McAfee (1993), Peters and Severinov (1997)] has restricted principals to direct mechanisms in which agents report only private information about their preferences, (see Epstein and Peters, 1999, p.121).

1See Gal-Or (1997) for a survey of the early contributions in this area and Martimort and Piccolo (2010) for a recent application. 2These contributions often postulate the presence of at least two agents; Martimort and Stole (2003) suggest a similar result in a class of common agency models of complete information.

1 It is now an established result that such a restriction involves a loss of generality even in single agent environments.3 One should however notice that economic applications of common agency have identified a strong rationale for making use of these mechanisms. At any pure strategy equilibrium, every single competitor behaves as a monopolist facing a single agent whose reservation utility is endogenous. The optimal behavior of this monopolist can therefore be characterized through type-revelation mechanisms that induce the agent to behave truthfully. Given this property, and provided that the agent’s indirect utility function satisfies some regularity conditions, several features of equilibrium allocations can be derived.4 In the same spirit, although on a more theoretical ground, Theorem 2 in Peters (2003) establishes that pure strategy equilibria of common agency games in which principals offer truth-telling mechanisms are robust to unilateral deviations towards any arbitrary indirect mechanism. That is, the corresponding outcomes will be supported in a pure strategy equilibrium of the game in which principals can make use of general communication mechanisms. The present work develops a similar analysis in the context of multiple-principal multiple- agent models of incomplete information. The explicit consideration of several agents introduces additional strategic effects. To illustrate them, observe that, from the viewpoint of a single principal, the messages that agents send to his rivals can be seen as hidden actions. We interpret this idea in the light of Myerson (1982) construction. Given the profile of mechanisms proposed by his opponents, a single principal is facing the same situation as if he was interacting with several agents who have private information and take some non-contractible actions, i.e. the messages they send to the other principals. It is hence natural to ask whether he could gain by introducing some uncertainty over his decisions, in such a way to induce the agents to correlate on their message choices. One should however appreciate that, in our multiple-principal multiple-agent framework, agents simultaneously send messages to principals. This constitutes a major differ- ence with the Myerson (1982) setting, in which the agents’ non contractible actions (efforts) are taken after having sent messages to principals and having received recommendations from them. To evaluate the role of private recommendations, we therefore allow principals to privately communicate with agents before receiving the messages from them. In such a context, we say that a mechanism is direct if the messages that can be sent by any single agent and the recommendations she is allowed to receive can only consist of her private information. In addition, we say

3The existence of equilibrium outcomes that can be supported through arbitrary (indirect) communication mechanisms but not through direct revelation ones has been acknowledged as a failure of the revelation principle in common agency games (Martimort and Stole, 2002; Peters, 2001). 4This methodology has been intensively used in most applications of common agency models with incomplete information to Finance (Biais, Martimort, and Rochet, 2000; Khalil, Martimort, and Parigi, 2007), Public Economics and Regulation Theory (Bond and Gresik, 1996; Calzolari, 2004; Laffont and Pouyet, 2004; Olsen and Osmundsen, 2001), and to Political Economy (Martimort and Semenov, 2008). A general exposition of the methodology is provided in Martimort and Stole (2003) and Martimort and Stole (2009).

2 that a direct mechanism is incentive compatible from the viewpoint of any principal j if, given the behavior of his competitors, it induces each agent to truthfully reveal her type to principal j and to obey his recommendations. Direct and incentive compatible mechanisms have an obvious appeal, they are easy to characterize. We show that, in competing mechanism games of incomplete information, there is a rationale to restrict principals to make use of such mechanisms. That is, every pure strategy equilibrium of the direct mechanism game in which principals use incentive compatible mechanisms and agents behave in a truthful and obedient way in every continuation equilibrium survives unilateral deviations towards more sophisticated communication schemes. The result therefore extends Theorem 2 of Peters (2003) to games with several agents. In addition, it provides a sufficient condition to establish that a strategy profile constitutes an equilibrium: one can indeed restrict attention to the unilateral deviations of a single principal which induce agents to behave truthfully and honestly with him. Although incentive compatible mechanisms play a major role in our construction, one cannot safely restrict attention to such mechanisms when determining the optimal behavior of a single principal for a given profile of mechanisms offered by his rivals. We stress this point with a simple example analyzing the situation in which principal one faces a competitor who offers an indirect mechanism involving some private recommendations. From the viewpoint of principal one, the recommendations sent by his rival constitute an additional source of private information for agents. He therefore has an incentive to use a sophisticated communication mechanism to extract this additional information. More generally, we investigate whether, despite their central role in our analysis, incentive compatible mechanisms may constitute a general device to characterize equilibria in competing mechanism games of incomplete information. We provide a negative answer to this question by developing a second example which shows that there exist equilibrium outcomes of a game in which principals use arbitrary communication mechanisms that cannot be reproduced by restricting them to use direct and incentive compatible mechanisms. The example hence suggests a failure of the revelation principle in this setting.5

The mechanisms we consider allow a principal to correlate agents’ behaviors by introducing some uncertainty over his actions. We argue that the possibility to send private recommendations is a key element to generate such uncertainty. We support this view by developing an example of a two-principal two-agent game of incomplete information in which the equilibrium outcomes supported by simple type-revelation mechanisms, which do not allow for any recommendation from principals to agents, fail to survive if a single principal deviates towards an indirect mech-

5Relative to other examples of the failure of the revelation principle in competing mechanism games (see for example Martimort and Stole (2002), Peters (2001), and Peck (1997)) we do explicitly consider private recommendations.

3 anism. This indeed suggests that the restriction to type-revelation and truthful mechanisms, usually postulated in the applied literature, may be seriously problematic.

The remaining of the paper is organized as follows. Section 2 introduces the general model. Section 3 discusses the robustness of equilibria supported by direct and incentive compatible mechanisms. Section 4 shows that the equilibrium outcomes sustain by simple type-revelation mechanisms may not survive against unilateral deviations towards more sophisticated mechanisms. Section 5 concludes.

2 The Model

We refer to a scenario in which several principals (indexed by j ∈ J = {1,...,J}) contract with several agents (indexed by i ∈ I = {1,...,I}). Each agent has private information about her preferences. The information available to agent i is represented by a type ωi ∈ Ωi. We denote i i ω ∈ Ω = ×i∈I Ω a state of the world and we take the sets Ω to be ﬁnite. Principals have common beliefs on the probability distribution of ω, which we denote F. Each principal j can take an action y j ∈ Yj. The sets Yj are all compact and Borel measurable. i Principal j’s preferences are represented by the function v j : Y × Ω → R+, and u : Y ×

Ω → R+ represents the agent i’s preferences, with Y = × j∈J Yj. For any given array y˜ =

(y˜1,y˜2,...,y˜J) ∈ ∆(Y), the state contingent utilities of agent i and principal j are:

Z Z Z Z Z Z i i u˜ (y˜,ω) = ... u (y,ω) dy˜1dy˜2...dy˜J and v˜j(y˜,ω) = ... vi(y,ω) dy˜1dy˜2...dy˜J Y1 Y2 YJ Y1 Y2 YJ

Final allocations are determined by the multilateral contracts that principals exchange with agents. Every single principal has the full control of the communication structure with the only constraint that he cannot make his decisions contingent on the contracts proposed by his rivals. Speciﬁcally, we consider a situation where principals can both send and receive messages from agents. Along the lines of Myerson (1982), we refer to the messages sent by principals as to recommendations. Principals’ decisions depend on the messages they receive from agents and on the recommendations they send. The messages sent by each agent, in turn, depend on her initial information and on the private recommendations she receives from any of the principals. i More formally, we take R j to be the set of recommendations that can be sent by principal j i to agent i, and we denote M j the set of messages that can be used by agent i to communicate i i with principal j. Both R j and M j are ﬁnite sets for every i and j. We typically write R j =

4 i i 6 ×i∈I R j, M j = ×i∈I M j, M = × j∈J M j and R = × j∈J R j. A mechanism for principal j is the pair γ j = (χ j,φ j), where χ j ∈ ∆(R j) is a probability distribution over the set of recommendations

R j and φ j : R j × M j → ∆(Yj) is a mapping that associates a (stochastic) action to messages and recommendations. We denote Γ j the set of all such mechanisms, and we write Γ = × j∈J Γ j. We consider the following extensive form game. First, principals offer mechanisms and simultaneously send private recommendations to agents. Having observed the offered mechanisms, her private recommendations and her own type, each agent sends back a message to each of the principals. In a next step, any principal takes the (eventually stochastic) action prescribed by his mechanism, given the recommendations he sent and the messages he received. Observe that, from the point of view of each single principal j, the messages sent by each agent i to his − j opponents can be interpreted as hidden actions. In such a competing mechanism setting, however, communication and moral hazard cannot be separated. When agent i communicates her private information to principal j, she simultaneously chooses her hidden actions. This pre- vents a single principal from making recommendations contingent on the messages he receives from agents. Here, principals send recommendations before receiving any message, relying on the ex-ante distribution function F. A private recommendation sent by principal j to agent i may hence prescribe one array of messages to be sent to all principals for every agent i’s type. We denote GΓ the multiple-principal multiple-agent game relative to the set of mechanisms Γ. In this game, a (pure) strategy for principal j is a mechanism γ ∈ Γ . A strategy for agent i is ¡ ¢ j j the measurable mapping λi : Γ × Ri × Ωi → ∆ Mi , with Mi = × Mi , and Ri = × Ri . For ¡ ¢j∈J j j∈J j notational convenience, for every agent i we denote λi mi;γ,ri,ωi the probability of sending the array of messages mi when her type is ωi, when she has received the array of recommendation ri having the principals played the mechanisms γ. ¡ ¢ Every proﬁle of agents’ strategies λ = λ1,λ2,...,λI induces a probability distribution over principals’ decisions. Given the array of mechanisms γ = (γ1,γ2,...,γJ), the state of the world λ ω and the realization of the recommendations r = (r1,r2,...,rJ) ∈ R, we take ψ j ∈ ∆(Yj) to be the probability distribution over principal j’s actions induced by the strategy proﬁle λ, and we let ψλ be the joint distribution over all principals. The expected payoffs to each agent i of type ωi who received the array of recommendations ri can be written as:

Z Z Z ¡ ¯ ¢ Ui(λ,γ,ω,ri) = u˜i(y˜,ω) F(ω−i|ωi) dψλ(γ,ri,r−i,ωi,ω−i) dω−i dP r−i ¯ri , R−i Ω−i Im(γ)

6 Observe that we treat the message spaces available to every single principal as given;³ an´ alternative route to deﬁne a mechanism would be to let each principal j choosing the array of message spaces Mi , as it is done in j i∈I Peters (2001). This alternative formulation can be introduced in the model without affecting our results.

5 ¡ ¯ ¢ where P r−i ¯ri is the posterior probability distribution over recommendations jointly used by principals. The payoff to principal j is: Z Z Z λ Vj (λ,γ,ω) = v˜j (y˜,ω) F (ω) dψ (γ,r,ω) dω dP(r). r∈R Ω Im(γ) ¡ ¢ We typically say that the strategy proﬁle λ = λ1,λ2,...,λI constitutes a continuation equilibrium relative to Γ if for every i ∈ I and for every γ ∈ Γ:

i ¡ i¢ i ¡ i0 −i ¡ i0 −i¢ ¡ i0 −i¢ ¡ i0 −i¢ i¢ U λ,γ,ω,r ≥ U λ ,λ ,γ1 λ ,λ ,γ2 λ ,λ ,...,γJ λ ,λ ,ω,r .

for all λi0. We take Perfect Bayesian Equilibrium (PBE) to be the relevant solution concept.

In the following, we will also consider situations in which principals are restricted to offer “direct” and “incentive compatible” mechanisms. A mechanism available to principal j is said to h¡ ¢ i iJ−1 i i i i |Ω | be direct if one has M j = Ω and R j = Ω for all i and j. That is, in a direct mechanism both the messages and the recommendations used by each agent and by each principal can only D D consist of a type. Let Γ j be the set of such direct mechanisms for principal j and G be the multiple-principal multiple-agent game in which principals are restricted to make use of direct mechanisms. A direct mechanism is incentive compatible from the point of view of principal j if, given the mechanisms offered by the other principals, it induces a continuation equilibrium in which:

• each agent i of type k sends to principal jˆ, with jˆ6= j, the message she was recommended by principal j,

• each agent i of type k reports her true type to principal j.

This notion of incentive compatibility therefore parallels that originally formulated by My- erson (1982), in terms of agents being truthful and obedient, provided that one takes as given the behavior of J − 1 principals. In a ﬁnal step, we consider the restricted class of mechanisms in which agents only communicate their own type, but principals do not privately communicate with agents. That is, i i i M j = Ω and R j is a singleton for every i and j. For each principal j, we hence denote “type- revelation" mechanism the mapping γˆ j : Ω → ∆(Yj) and let Γˆ j be the set of type-revelation mechanisms available to principal j. For a given proﬁle of mechanisms γ− j ∈ Γ− j, we then say that γˆ j ∈ Γˆ j ⊆ Γ j is a type-revelation and truthful (i.e. incentive compatible) mechanism if the array (γˆ j,γ− j) induces a continuation equilibrium in which agents truthfully reveal their type to

6 principal j. As we already remarked in the Introduction, most economic applications of competing mechanism games with incomplete information restrict principals to offer type-revelation and truthful mechanisms, which motivates our interest in them.

3 Robust Equilibria

It is now an established finding that the equilibrium outcomes of multiple-principal multiple- agent games are typically indeterminate when no restriction is put on the set of mechanisms available to principals.7 This suggests that, for a theory of competing mechanisms to have some predictive power, it should restrict attention to specific classes of mechanism, in such a way to guarantee that the model is tractable enough. In this section, we argue that direct incentive compatible mechanisms may serve this role in games with several agents. In particular, we show that these mechanisms satisfy a basic requirement of any economic model of competition. That is, the corresponding equilibrium outcomes survive when a single principal is allowed to deviate towards arbitrary communication mechanisms. This robustness issue has already been formalized in the competing mechanism literature. Specifically, Peters (2003) showed that any pure strategy equilibrium relative to simple type- revelation mechanisms is weakly robust when several principals deal with only one agent (common agency). That is, following any unilateral deviation towards a complex mechanism, there always exists a continuation equilibrium that punishes a deviating principal. In games with several agents, attention has been typically restricted to a different notion of equilibrium robustness, i.e. to strongly robust equilibria.8 In contexts where the agents’ types are publicly known, Han (2007) and Attar, Campioni, Piaser, and Rajan (2010) have identified simple mechanisms that satisfy the following property: any pure strategy strongly robust equilibrium relative to such simple mechanisms remains a pure strategy strongly robust equilibrium in any competing mechanism game where complex communication mechanisms are available.9 We show in the present section that direct incentive compatible mechanisms, as previously defined, serve the same role in multiple-principal multiple-agent games of incomplete information. Specifically, we restrict attention to those equilibria in which agents behave truthfully and obediently on the equilibrium path, and each principal exploits the power of incentive compati-

7See, among others, Yamashita (2010), Peters and Szentes (2011), and Peters and Troncoso-Valverde (2011). 8An equilibrium of any competing mechanism game is said to be strongly robust relative to the set of available mechanisms if each principal’s equilibrium payoff is no less than his payoffs in any of the continuation equilibria both on and off the equilibrium path following his unilateral deviations. 9In particular, Han (2007) considers a situation where agents’ actions are fully contractible, and Attar, Campioni, Piaser, and Rajan (2010) extend the analysis to moral hazard contexts.

7 bility out of equilibrium. That is, following any single principal’s deviation, each player believes that the agents will coordinate on the continuation equilibrium in which every agent truthfully reports her private information to deviating principal and obeys his recommendations. The following proposition formally identiﬁes the role of incentive compatibility in characterizing pure strategy equilibria of the game GD. ³ ´ Proposition 1 Consider the game GD. Let the strategy proﬁle γ˜,λ˜ be such that:

(i) In the continuation equilibrium λ˜ (γ˜,ω), each agent truthfully reveals her type to each of the principals and obeys the recommendations she receives, i.e. the mechanism γ˜ is incentive compatible.

(ii) For each principal j, one has

¡ ¢ ³ ´ 0 0 0 ˜ Vj γ˜ j ,γ˜− j,λ (γ˜ j ,γ˜− j,ω),ω ≤ Vj γ˜,λ,ω ³ ´ 0 D 0 0 for every γ˜ j ∈ Γ j , with λ γ˜ j ,γ˜− j,ω being the continuation equilibrium in which agents are truthful and obedient to principal j. ³ ´ Then, γ˜,λ˜ is an equilibrium of GD. ³ ´ 00 00 ˜00 Proof. Suppose not. There exists a direct mechanism γ˜ j = χ˜ j ,φ j such that ³ ¡ ¢ ´ ³ ´ 00 ˜ 00 ˜ Vj γ˜ j ,γ˜− j,λ γ˜ j ,γ˜− j,ω ,ω > Vj γ˜,λ,ω . (1)

We now show that if (1) holds, one can always construct an equivalent incentive compatible mechanism for principal j which violates (ii). ³ ´ ˜ i i 00 i i We ﬁrst take λ− j m− j;γ˜ j ,γ˜− j,r j,ω to be the probability that agent i sends the messages ¡ ¢ mi ∈ Ωi J−1 to principals − j in the continuation equilibrium induced by the mechanisms ³ − j ´ 00 i i γ˜ ,γ˜− j , given her type ω and given the private recommendation r she receives from princi- j ³ j ´ 10 00 ˜ i i 00 i i pal j. In particular, following the deviation to γ˜ j , the strategy λ− j m− j;γ˜ j ,γ˜− j,r j,ω can be used to describe the messages sent to the non-deviating principals − j for every recommendation h¡ ¢ i iJ−1 i i i |Ω | r j ∈ R j = Ω privately received by each agent i = 1,2,...,I. We can hence construct the vector of recommendations to agent i:

10Because we have assumed ﬁnite communication sets, there exists at least one such a continuation equilibrium

8 Ã ! Z ¡ ¢ ¡ ¢ 0i ˜ i i 00 i i 00 i i χ˜ j = λ− j m− j;γ˜ j ,γ˜− j,r j,ω χ˜ j r j dr j , (2) ri ∈Ri j j ωi∈Ωi ³ ´ 00 i i in which χ˜ j r j is the (marginal) probability that the private recommendation r j is sent by h¡ ¢ i iJ−1 00 0i i |Ω | principal j to agent i in the deviating mechanism χ˜ j . The vector χ˜ j ∈ ∆ Ω reproduces the probability distributions over the³ messages´ sent by each agent i to principals other than ˜ 00 j in the continuation equilibrium λ γ˜ ,γ˜− j when agents obey to the recommendations that j ³ ´ 0 0i principal j makes. Finally, we let χ˜ j = χ˜ j be the collection of private recommendations i∈I sent to all agents.

Consider now the mapping φ j : R j × Ω → ∆(Yj) such that: Z ˜00 ˜ 00 φ j (r j,ω) = φ j (m j,r j)λ j(m j;γ˜ j ,γ˜− j,r j,ω)dm j, (3) m j∈Ω ³ ´ ˜ 00 in which λ j m j;γ˜ j ,γ˜− j,r j,ω is the probability distribution over the messages sent by agents to principal j in the continuation equilibrium induced by the deviation. For any given array of recommendations r j ∈ R j, and given the mechanisms γ˜− j, the type-contingent mapping φ j ˜00 induces the same probability distribution over principal j’s actions as that generated by φ j in the continuation equilibrium λ˜ when agents truthfully report their private information to principal j. To construct an incentive compatible mechanism inducing the same probability distribution as ˜00 φ j , we must incorporate the requirement of obediency to principal j’s recommendations, r j. ˜00 This is indeed not immediate since, following the deviation to φ j , it may happen that different recommendations of principal j induce agents to send the same messages to principals − j in the continuation equilibrium. We address the issue as follows. First, take any array of messages m− j and let ( ) · ¸J−1 ¡ ¢ i ¡ ¢ i |Ω | ˜ 00 r j (m− j) = r j ∈ Ω : λ− j m− j;γ˜ j ,γ˜− j,r j,ω > 0 be the set of private recommendations in the support of the deviating mechanism that induces ˜ agents to play m− j with a strictly positive probability in λ− j. Second, consider the mapping

R 00 φ (r j,ω) χ˜ (r j)dr j ˜0 r j∈r j(m− j) j j φ j (m− j,ω) = R 00 . (4) χ˜ (r j) dr j r j∈r j(m− j) j

9 ³ ³ ´´ 00 00 i with χ˜ j (r j) = χ˜ j r j being the joint probability over all agents of sending the array i∈I h¡ ¢ i iJ−1 i |Ω | of recommendations r j. For each array of messages m− j ∈ Ω sent with a strictly ˜ 00 ˜0 positive probability in the continuation equilibrium λ(γ˜ j ,γ˜− j), the mapping φ j(.) induces the same distribution over principal j’s actions as φ j(.) as long as agents obey the corresponding recommendations of principal j. Observe that, by sending the recommended array of messages m− j, agents induce principal j to select a random action taking into account the uncertainty 00 associated to χ˜ j (r j). This exactly reproduces the uncertainty in principal j’s behavior that can be generated by non incentive compatible mechanisms that associate different recommendations to the same message. ³ ´ 0 0 ˜0 To summarize: given the mechanisms γ˜− j, the mechanism γ˜ j = χ˜ j,φ j is incentive compatible from the point of view of principal j. That is, there exists a continuation equilibrium 0 induced by the array (γ˜ j ,γ˜− j) in which agents behave in a truthful and obedient way to principal 0 D j. Given such behaviors, γ˜ j constitutes a proﬁtable deviation for principal j in the game G , which contradicts condition (ii).

The intuition for Proposition 1 can be provided along the lines of the canonical Myerson (1982) construction. When contemplating a unilateral deviation, a single principal is facing a moral hazard problem: he cannot contract on the messages that agents send to his opponents. In such a context, any payoff attainable by this deviating principal could also be obtained by recommending agents to truthfully reveal their type to each of the non deviating principals. In the corresponding continuation equilibrium, it is always optimal for every agent to follow this recommendation as long as all other agents do so. Importantly, the result holds true although our extensive form slightly differs from the Myerson’s one. In particular, the deviating principal sends his recommendations before, and not after, having received the agents’ messages. One should also observe that, starting from Myerson, we analyze a situation in which principals compete through mechanisms. In this context, we argue that the notion of incentive compatibility is also relevant when a single principal contemplates a unilateral deviation. Proposition 1 shows that, when analyzing such unilateral deviations, it is sufﬁcient to consider those towards incentive compatible mechanisms. Any proﬁle of incentive compatible mechanisms that survives these deviations constitutes an equilibrium in GD. The same reasoning can indeed be used to characterize a set of equilibrium mechanisms in any game GΓ. The result is established in the following: ³ ´ Corollary 1 Let γ˜,λ˜ be a pure strategy PBE in the game GD satisfying conditions (i) and (ii) of Proposition 1. Then, the corresponding equilibrium outcome can be supported as a pure strategy PBE in any competing mechanism game GΓ.

10 Γ Proof. Consider any indirect mechanism game G , in which Γ = (Γ1,...,ΓJ) is the set of mechanisms available to principals. As in the proofs³ of Theorem´ 1 and 2 in Peters (2003), we have to extend the continuation equilibrium λ˜ = λ˜ 1,λ˜ 2,...,λ˜ I in the game GD to all proﬁle of mechanisms in GΓ so to support the original equilibrium. For every γ ∈ Γ and³ω ∈ Ω, given the array of private´ recommendations ¡ ¢ i ¡ ¢ −i ¡ ¢ ri,r−i received by agents, we take λ = λ γ,ri,ω ,λ γ,r−i,ω to be the continuation equilibrium on such extended set. In particular, let

i λ (γ,ω) = λ˜ i (γ,ω) ∀i,∀ω, and ∀γ ∈ ΓD ⊆ Γ (5)

that is, each type of every agent sends exactly the same array of messages in the "enlarged" game in which all communication mechanisms are feasible. For all γ ∈ Γ \ ΓD, we let λ(γ,r,ω) be any continuation equilibrium.³ ´ Finally, we set γ = χ (r ),φ (r ,m ) = γ˜ for every principal j , for every r ∈ R and j ¡ ¢ j j j j j j j j m j ∈ M j , and let γ = γ . j j∈J ³ ´ ³ ´ It is clear that the profile γ,λ in the game GΓ induces the same distribution as γ˜,λ˜ in D the game G over the decisions of the principals and of the agents. ³ ´ 0 0 0 D Suppose now that principal j deviates towards the mechanism γ j = χ j,φ j 6∈ Γ j , such that: ³ ¡ ¢ ´ ³ ´ 0 0 ˜ Vj γ j ,γ˜− j,λ γ j ,γ˜− j,r j,ω ,ω > Vj γ˜,λ,ω (6) ³ ´ 0 i −i in which λ γ ,γ˜− j,r ,r ,ω is a continuation equilibrium induced by such a deviation, j ³ j j ´ with the property that ri ,r−i is any array of private recommendations sent by principal j to ³ ´ j j i −i 0 agents such that r j,r j ∈ supp χ j. The same reasoning developed in the proof of Proposition 1 can now be used to show that if (6) holds, one can always construct an equivalent incentive compatible mechanism for³ principal´ j which violates (ii). Specifically, one can always construct 0 0 ˜0 D a direct mechanism γ˜ j = χ˜ j,φ j ∈ Γ j as it is done in equation (2) − (4), but integrating over i i 0 the given set of messages M j and R j. If the mechanism γ j constitutes a profitable deviation Γ D for principal j in the game G , then, in the game G , one can construct a direct³ and incentive´ 0 ˜ compatible mechanism γ˜ j which violates (ii). It follows from Proposition 1 that γ˜,λ cannot be a PBE in GD, which constitutes a contradiction. The common logic underlying Proposition 1 and Corollary 1 can be grasped by observing that, in each candidate equilibrium of GD considered in Proposition 1, every principal is simply recommending to agents to truthfully reveal their types to his opponents. Following a unilateral deviation of principal j to some arbitrary mechanism, the "effective" recommendations he sends

11 constitute a source of uncertainty for agents. Proposition 1 and Corollary 1 essentially show that it is always possible for the deviating principal to generate such an uncertainty by means of a direct incentive compatible mechanisms, provided that agents truthfully reveal their initial private information and obey his recommendations in the corresponding continuation equilibrium. Given these results, it seems natural to ask whether the best reply of a single principal j − j to any array of mechanisms γ− j ∈ Γ can be characterized by incentive compatible mechanisms.11 Unfortunately, this intuition would not be correct: the optimal behavior of a single principal cannot be always reproduced by incentive compatible mechanisms if his rivals use more sophisticated communication schemes. We stress this point in the following example.

Example 1

Let J = 2 and I = 1. Denote the principals as P1 and P2; the allocation spaces are Y1 = {y11,y12} for P1, and Y2 = {y21,y22} for P2. There is complete information about the agent’s type, i.e.

Ω = {ω}. Communication is modeled as follows. P1 receives a message m1 ∈ M1 = {m11,m12} from the single agent, and he cannot send any recommendation, that is R1 = {r1}. P2 does not receive any message from the agent, i.e. M2 = {m2}, but he is allowed to send to the agent a private recommendation r2 ∈ R2 = {r21,r22}. The payoffs are represented in the table below in which the ﬁrst payoff is that of P1, who chooses the row, the second payoff is that of P2, who chooses the column, and the last payoff is that of the agent:

y21 y22

y11 (1,2,1)(0,2,0)

y12 (0,2,0)(1,2,1)

µ ¶ 1 1 Suppose now that P2 proposes the (indirect) mechanism γ = (χ ,φ ) with χ = , , 2 2 2 2 2 2 and φ2 (r2i) = y2i, for i = {1,2}. In the mechanism γ2, P2 chooses y21 (y22) with probability one whenever he recommends r21 (r22). We argue that it is a best reply for P1 to offer the (indirect) mechanism

( y11 if he receives the message m11 γ1(m1) = y12 if he receives the message m12

11We thank one of the referees for pointing us in this direction.

12 Indeed, given the mechanisms (γ1,γ2), one can check that it is optimal for the agent to send m11 to P1 when she receives r21 from P2, and to send him m12 to P1 when she receives r22 from

P2. By proposing the mechanism γ1, P1 can therefore get his maximal payoff of 1.

Consider now the situation in which, when determining his best reply to γ2, P1 is restricted to use incentive compatible mechanisms. In our complete information setting, an incentive compatible mechanism is a (eventually stochastic) take-it or leave-it offer. Given the randomization chosen by P2, it is immediate to verify that there is no incentive compatible mechanism that yields P1 a payoff of 1.

The example emphasizes that incentive compatible mechanisms may not be ﬂexible enough to characterize the optimal behavior of P1 when P2 sends nontrivial recommendations. From the viewpoint of P1, each private recommendation sent by P2 generates an additional private information for the agent. In the example, P1 strictly gains by asking the agent to communicate this new private information.12

More generally, one can show that restricting attention to equilibrium outcomes of the game GD in which principals use incentive compatible mechanisms inducing agents to behave truthfully and obediently in every continuation equilibrium involves a loss of generality. That is, there exist equilibrium outcomes of a mechanism game that cannot be reproduced by looking at incentive compatible mechanisms. To illustrate this point, we develop a second example.

Example 2

Let J = 3 and I = 2. Denote the principals as P1, P2 and P3, and the agents as A1 and A2. The allocation spaces are Y1 = {y11,y12} for P1, Y2 = {y21,y22} for P2 and Y3 = {y31,y32,y33} for P3. The agents have no private information, i.e. Ω1 = Ω2 = {ω}.

12Of course, we could generalize the concept of direct mechanism in order to be always able to characterize best replies as incentive compatible mechanisms. Take our Example 1, if P1 could ask to the agent to report not only her type but also the recommendation she received from P2, then we could get a payoff of 1. Using Myerson’s¡ ¢ theorem Γ we could state a general result: take an arbitrary game G and an arbitrary array of mechanisms γ j,γ− j , if there 0 is some indirect mechanism γ j with a continuation equilibrium profitable for principal j, then there is an incentive compatible and obedient mechanism γ˜ j that is profitable for j with a continuation equilibrium in which agents are ˜ i i i truthful and obedient to j. To do this it would be enough to define for every agent i his type set as Ω = Ω ∪l6= j Rl and the Revelation Principle¡ would¢ give the result. Hence, whether j can profitably deviate to such a mechanism yields a sufficient condition for γ j,γ− j to be an equilibrium. However, generalizing direct mechanisms in this way would lead to an infinite regress problem à la Epstein and Peters (1999) making it difficult to characterize equilibria in the direct mechanism game. That is, if principal j asks to the agent what recommendations she received, then he will construct a mechanism that makes the allocation that he selects depend on the mechanisms that the other principals have offered. Each principal’s mechanism then would depend on whether other principals’ mechanisms depend on his mechanism, and so on. Proposition 1 and Corollary 1 do not suffer from this problem.

13 The payoffs of the game are represented in the three tables below where the ﬁrst payoff is that of P1, who chooses the row, the second payoff is that of P2, who chooses the column, the third payoff is that of P3 who chooses the table, and the last payoffs are those of A1 and A2.

y21 y22

y11 (2,1,1,1,1)(−10,−1,0,0,0)

y12 (−10,1,1,1)(−10,−1,0,0,0)

Table 1: Players’ payoffs if P3 chooses Y31

y21 y22

y11 (−10,−1,0,0,0)(−10,1,1,1,1)

y12 (2,−1,0,0,0)(2,1,1,1,1)

Table 2: Players’ payoffs if P3 chooses Y32

y21 y22

y11 (−1,−1,−1,−1,−1)(−1,−1,−1,−1,−1)

y12 (−1,−1,−1,−1,−1)(−1,−1,−1,−1,−1)

Table 3: Players’ payoffs if P3 chooses y33

P1 and P3 communicate with the agents by receiving messages. The sets of available mes- 1 2 1 2 1 2 1 2 1 2 sages are M1 = M1 = M3 = M3 = {ma;mb}, M2 = M2 = {m} and R1 = R1 = R3 = R3 = {r}, 1 2 R2 = R2 = {ra,rb}.

We consider the situation where P1 proposes the indirect mechanism:

( y11 if he receives the message ma from both A1 and A2 γ1 = y12 otherwise,

and P3 proposes the indirect mechanism:

  y31 if he receives the message ma from both A1 and A2, γ3 = y32 if he receives mb from A1 and A2,  y33 if he receives different messages from A1 and A2.

Finally, P2 proposes an indirect mechanisms which involves recommendations, denoted γ2.

This mechanism consists of the lottery mixing equally over ra and rb. In addition, the actions of

14 P2 are perfectly correlated with the recommendations he sends. If ra (rb) realizes, then y21 (y22) is chosen with probability one.

When they have received the recommendation ra, agents play the following continuation game, where (mk,ml) denotes the strategy “I send the message mk to P1 and the message ml to P3”:

(ma,ma)(ma,mb)(mb,ma)(mb,mb)

(ma,ma) (1,1)(−1,−1)(1,1)(−1,−1)

(ma,mb) (−1,−1)(0,0)(−1,−1)(0,0)

(mb,ma) (1,1)(−1,−1)(1,1)(−1,−1)

(mb,mb) (−1,1)(0,0)(0,0)(0,0)

Table 4: Agents’ payoffs in the continuation game when the recommendation is ra.

This continuation game has a pure strategy equilibrium in which both agents send the message ma to P3 and ma to P1. If the recommendation sent by P2 is rb, one can check that the corresponding continuation game over messages has a pure strategy equilibrium in which both agents send the message mb to P3 and P1.

Given the mechanisms γ1, γ2 and γ3, the payoff proﬁle induced by such continuation equilibria is (2,1,1,1,1). Since each of the principals achieves his maximal payoff, there are no proﬁtable unilateral deviations, which shows that these strategies form an equilibrium in our competing mechanism game where principals can communicate with agents through non de- generate messages. At equilibrium, a single principal uses these messages to get additional informations over the recommendations sent by his competitors. One should observe that, if we consider the game GD in which principals are restricted to make use of incentive compatible direct mechanisms, P1 cannot anymore achieve a payoff of 2 at equilibrium. When choosing an incentive compatible direct mechanism, P1 does not receive any message from agents, which makes impossible for him to correlate his actions with the recommendations sent by P2.

As it stands, the example emphasizes a failure of the revelation principle in competing mechanism settings. The set of equilibrium outcomes supported through arbitrary indirect mechanisms is larger than the set of outcomes supported through direct incentive compatible ones. The novel feature of the example is the explicit consideration of private recommendations.13 In this

13This indeed differentiates our Example 2 from standard examples documenting such a failure in single agent (Martimort and Stole (2002), Peters (2001)) and multiple-agent (Peck (1997)) settings.

15 respect, the result can be re-interpreted in the following way: if the notion of incentive compatible mechanism available to each principal j is extended to include the private recommendations that he can send to agents only concerning the messages to be sent to all − j principals, then restricting principals to mechanisms that induce all agents to behave honestly and truthfully in each continuation equilibrium involves a loss of generality. To restore the revelation principle, we could extend our notion of incentive compatible mechanisms by letting agents communicate not only their type but also the realizations of the recommendations sent by the other principals, but we would incur in the inﬁnite regress problem as explained by Epstein and Peters (1999) seriously limiting the applicability of our results.14

4 The Need for Recommendations

We have shown in the last section that each equilibrium outcome of the game GD in which agents behave in a truthful and obedient wayin every continuation equilibrium survives the introduction of more complicated communication schemes. At this point, one should observe that, although incentive compatibility plays a key role in our analysis, economic models of multiple-principal multiple-agent games with incomplete information have typically restricted attention to a sim- pler class of mechanisms. Starting from McAfee (1993), most contributions considered a setting in which principals commit to type-revelation mechanisms which induce agents to truthfully reveal their private information. In particular, agents do not receive any private recommendations on their message choices. It seems then natural to investigate whether one can further simplify our analysis by restricting attention to mechanisms where agents are only asked to reveal their type, with private recommendations playing no role. In the present section, we suggest that our analysis cannot be simpliﬁed in a straightforward way. To this extent, we develop a third example stressing the key role of private recommendations in guaranteeing that equilibria supported by direct and incentive compatible mechanisms are robust. Speciﬁcally, we show there exist equilibrium outcomes of the game in which principals are restricted to simple revelation-type mechanisms that cannot be supported at equilibrium when principals can arbitrarily select their mechanisms. We therefore emphasize that Proposition 1 does not apply to the game in which principals are restricted to type-revelation mechanisms and agents behave truthfully in every continuation equilibrium. At the same time, the example illustrates the logic underlying Proposition 1. Indeed, the payoff attainable by any single principal when he deviates to an arbitrary indirect mechanisms, when his rivals use type-revelation mechanisms, could have as well been achieved if he deviated to some incentive compatible mechanism, making a crucial use of private recommendations.

14See footnote 12 at page 13 for a discussion on this issue.

16 Example 3

Let I = J = 2. Denote the principals as P1 and P2, and the agents as A1 and A2. The allocation spaces are Y = {y ,y ,y ,y } for P1, and Y = {y ,y ,y ,y } for P2. Each agent can 1 11 12 13 14 © ª 2 21 22 23 24 be of two types, and Ω1 = Ω2 = ω1,ω2 . Agents’ types are perfectly correlated, thus the ©¡ ¢ ¡ ¢ª ¡ ¢ two states of the world are ω = ω1,ω1 , ω2,ω2 ; in addition, we take prob ω1,ω1 = ¡ ¢ prob ω2,ω2 = 0.5. The payoffs of the game are represented in the two tables below. In each table, the ﬁrst payoff is that of P1, who chooses the row, the second payoff is that of P2, who chooses the column, and the last two payoffs are those of A1 and A2, respectively. The ﬁrst ¡ ¢ table shows the payoffs corresponding to the state ω1,ω1 :

y21 y22 y23 y24

y11 (α,0,−1,−1)(α,0,0,0)(α,0,0,9)(16,1,15,10)

y12 (α,0,11,0)(α,0,−1,−1)(16 + ε,0,10,15)(α,0,0,0)

y13 (α,0,0,0)(16 + ε,0,10,15)(α,0,−1,−1)(α,0,11,0)

y14 (16,0,15,10)(α,0,0,9)(α,0,0,0)(a,0,−1,−1) ¡ ¢ Table 5: Players’ payoffs in state ω1,ω1 ¡ ¢ The second table shows the payoffs corresponding to the state ω2,ω2 :

y21 y22 y23 y24

y11 (α,0,−1,−1)(α,0,0,0)(α,0,0,9)(16,0,15,10)

y12 (α,0,11,0)(α,0,−1,−1)(16 + ε,0,10,15)(α,0,0,0)

y13 (α,0,0,0)(16 + ε,0,10,15)(α,0,−1,−1)(α,0,11,0)

y14 (16,1,15,10)(α,0,0,9)(α,0,0,0)(α,0,−1,−1) ¡ ¢ Table 6: Players’ payoffs in state ω2,ω2

Observe that only P2’s preferences are state-dependent. That is, incomplete information is neither affecting the agents’ behaviors nor the P1’s one. In addition, ε ∈ R+ is taken to be arbitrarily close to zero. We postulate that each agent always participates with both principals. This is essentially done to simplify the exposition, and it is in line with the approach followed in standard examples of ﬁnite competing mechanism games.15 For the purpose of the example, a type-revelation i mechanism proposed by principal j is hence any mapping γˆ j : × Ω → ∆(Yj). We develop our i∈I argument via a series of claims.

15See the examples proposed in Martimort and Stole (2002), Peters (2001) and Han (2007), among others.

17 Claim 1 Suppose both principals are restricted to offering type-revelation mechanisms. Then, if α is small enough, the proﬁle of payoffs (16,1,15,10) can be supported in an equilibrium in which principals play a pure strategy and agents truthfully reveal their type.

Proof. Consider the following behaviors. P1 selects the mechanism γˆ1:

 1  y11 if he receives the message ω from both A1 and A2  ¡ ¢  y if he receives ω1 from A1 and ω2 from A2 γˆ m1,m2 = 13 1 1 1  2 1  y12 if he receives ω from A1 and ω from A2  2 y14 if he receives the message ω from both A1 and A2

P2 selects the mechanism γˆ2:

 1  y24 if he receives the message ω from both A1 and A2  ¡ ¢  y if he receives ω1 from A1 and ω2 from A2 γˆ m1,m2 = 23 2 2 2  2 1  y22 if he receives ω from A1 and ω from A2  2 y21 if he receives the message ω from both A1 and A2

Given the mechanisms γˆ and γˆ , agents play a continuation game over messages. In this ex- 1 2 ¡ ¢ ample, the agents’ payoffs associated to this message game are independent of whether ω1,ω1 ¡ ¢ or ω2,ω2 realizes. They are represented in the following table where the ﬁrst element in each cell denotes the payoff to A1 who chooses the row, and the second element denotes the payoff to A2.

¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ω1,ω2 ω1,ω1 ω2,ω2 ω2,ω1 ¡ ¢ ω1,ω2 (−1,−1)(0,0)(0,0)(10,15) ¡ ¢ ω1,ω1 (0,9)(15,10)(−1,−1)(11,0) ¡ ¢ ω2,ω2 (11,0)(−1,−1)(15,10)(0,9) ¡ ¢ ω2,ω1 (10,15)(0,0)(0,0)(−1,−1)

Table 3: Agents’ payoffs in the continuation game induced by γˆ1 and γˆ2

One can check that there is a continuation equilibrium where both agents truthfully reveal their types to each of the principals: ¡ ¢ 1 1 1 1 • if ω ,ω realizes, then both A1 and A2 truthfully reveal ω to P1 and P2; that is: m1 = 1 1 2 2 1 m2 = ω and m1 = m2 = ω ;

18 ¡ ¢ 2 2 2 1 • if ω ,ω realizes, then both A1 and A2 truthfully reveal ω to P1 and P2; that is: m1 = 1 2 2 2 2 m2 = ω and m1 = m2 = ω .

These behaviors induce the proﬁle of payoffs (16,1,15,10). We now show that none of the principals can gain by deviating towards an alternative type-revelation mechanism. Since P2 is getting his maximal payoff of 1, one should only consider deviations of P1. Consider ﬁrst the case in which P1 deviates proposing a single take-it or leave-it offer to 0 0 agents, irrespective of the received messages. We take this deviation to be y1; if y1 is a determin- istic offer, four alternative continuation games can be induced. Following a deviation towards 0 y1 ∈ {y11,y12,y13,y14}, the agents strategically set the messages to be sent to P2 who sticks to his equilibrium mechanism γˆ2. The corresponding agents’ payoffs are represented in the four tables below.

0 0 y1 = y11 y1 = y12 2 2 2 1 2 1 m2 = ω m2 = ω ω ω 2 2 2 m1 = ω (−1,−1)(0,0) ω (11,0)(−1,−1) 2 1 1 m1 = ω (0,9)(15,10) ω (10,15)(0,0) 0 0 y1 = y13 y1 = y14 ω2 ω1 ω2 ω1 ω2 (0,0)(10,15) ω2 (15,10)(0,9) ω1 (−1,−1)(11,0) ω1 (0,0)(−1,−1)

We argue that there is no continuation game yielding a payoff strictly greater than 16 to P1. More precisely,

0 1. If y1 = y11, the agents’ continuation game admits only one Nash equilibrium: both A1 and A2 report ω1 to P2. The corresponding payoff to P1 is 16.

0 2. If y1 = y12, the agents’ continuation game admits only one Nash equilibrium: both A1 and A2 report ω2 to P2. The corresponding payoff to P1 is α.

0 3. If y1 = y13, the agents’ continuation game admits only one Nash equilibrium: both A1 and A2 report ω1 to P2. The corresponding P1 payoff is α.

0 4. If y1 = y14, the agents’ continuation game admits only one Nash equilibrium: both A1 and A2 report ω2 to P2. The corresponding payoff to P1 is 16.

19 Suppose now that P1 deviates toward a stochastic take-it or leave-it offer (i.e. a randomization over {y11,y12,y13,y14}), and consider the continuation game induced by such a stochastic offer. One can check that if agents coordinate on a pure strategy equilibrium, then α can always be chosen low enough to guarantee that P1 cannot achieve a payoff strictly greater than 16. The only way for P1 to gain is therefore to have agents playing a mixed strategy in the continuation game. Let δ1k, with k = 1,2,3,4 be the probability to play y1k in the randomization selected by P1. For a deviation to be profitable one should have (δ12 + δ13) > 0, otherwise P1 will not achieve a payoff greater than 16. Once again, it can be checked that, in any of such mixed strategy equilibria, the level of α can be fixed to be small enough to make the deviation unprofitable.16

Finally, we argue that there is no need to consider deviations to more sophisticated type- revelation and truthful mechanisms, which assign different actions to different profile of types declared by agents, for P1. Let assume that P1 plays the truthful mechanism γˆ0 , which is such ¡ ¢ ¡ ¢ 1 0 1 1 0 2 2 that γˆ1 ω ,ω = y and γˆ1 ω ,ω = y, where y and y are lotteries over the decision set Y1. Since γˆ0 is truthful and types are perfectly correlated, each agent sends the message ω to P1 1 ¡ ¢ ¡ ¢i when the state of the world is ωi,ωi , with i = 1,2. If the state of the world is ω1,ω1 , both agents anticipate that y will be implemented. They hence behave in the same way as if they were only proposed the take-it or leave-it offer y. Analogously, when the state of the world is ¡ ¢ ω2,ω2 , the corresponding agents’ behaviors could be reproduced by making them only the take-it or leave-it offer y. Since there are no profitable deviations through take-it or leave-it offers, and given that P1’s preferences are type-independent (i.e. he cannot benefit by making losses on some types and gains on others), no profitable deviation towards an arbitrary type- revelation mechanism is possible.

Claim 2 Suppose P2 keeps playing the same type-revelation mechanism γˆ2. Then, P1 can profitably deviate to an alternative mechanism attaining a payoff strictly higher than 16. Following this deviation, both agents will lie with a strictly positive probability.

Proof. Suppose that P1 proposes the mechanism:

 1  y11 if he receives the message ω from both A1 and A2  ¡ ¢  y if he receives ω1 from A1 and ω2 from A2 γˆ00 m1,m2 = 12 1 1 1  2 1  y13 if he receives ω from A1 and ω from A2  2 y14 if he receives the message ω from both A1 and A2

16 Observe that when the probability δ12 + δ13 approaches zero, agents will play a pure strategy equilibrium yielding a payoff strictly smaller than 16 to P1.

20 00 The new array of mechanisms {γˆ1,γˆ2} induces the following message game among agents: ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ω1,ω2 ω1,ω1 ω2,ω2 ω2,ω1 ¡ ¢ ω1,ω2 (−1,−1)(0,0)(11,0)(−1,−1) ¡ ¢ ω1,ω1 (0,9)(15,10)(10,15)(0,0) ¡ ¢ ω2,ω2 (0,0)(10,15)(15,10)(0,9) ¡ ¢ ω2,ω1 (−1,−1)(11,0)(0,0)(−1,−1)

0 Table 4: Agents’ payoffs in the continuation game induced by {γ˜1,γ˜2}. ¡ ¢ ¡ ¢ Observe that the strategies ω1,ω2 and ω2,ω1 are strictly dominated for both agents, hence we can eliminate them. In the reduced game, there is no equilibrium in pure strategies. The unique mixed-strategy Nash equilibrium prescribes that: ¡ ¢ ¡ ¢ • Agent 1 mixes between ω1,ω1 and ω2,ω2 with equal probabilities. ¡ ¢ ¡ ¢ • Agent 2 mixes between ω1,ω1 and ω2,ω2 with equal probabilities.

Simple computations show that these strategies give to principal P1 a payoff strictly greater than 16. With some strictly positive probability, the mechanism offered by P1 induces the agents to send contradicting information to the principals: one agent will send the message ω1, while the other sends the message ω2. Hence, the mechanism offered by P1 does not induce truthful information revelation.

The above reasoning can be interpreted in the light of Proposition 1. In particular, the prof- 00 itable deviation γˆ1 can be reproduced by P1 if he makes use of a direct incentive compatible mechanism. Given P2’s equilibrium mechanism, we now suppose that P1 has the possibility to send private recommendations to the agents over the messages that they send to P2. © ª 0 00 1 2 2 Then, P1 can offer the mechanism γ1 deﬁned as: ∀(ω ,ω ) ∈ ω ,ω ,  0 00  P1(ω1,ω1) = 1/4, φ(y11;ω1,ω1,ω ,ω ) = 1   P (ω ,ω ) = 1/4, φ(y ;ω ,ω ,ω0,ω00) = 1 γ = 1 1 2 12 1 2 1  0 00  P1(ω2,ω1) = 1/4, φ(y13,ω2,ω1,ω ,ω ) = 1  0 00 P1(ω2,ω2) = 1/4, φ(y14,ω2,ω2,ω ,ω ) = 1.

In the mechanisms γ1, P1 uses a very simple form of recommendations: he sends only one private signal to each of the agents, irrespective of their type.17 Private recommendations, however, are a key ingredient of γ1: if A1 receives the recommendation ω1 from P1, she believes

17 i To simplify notation, we denote ωi with i = 1,2 a vector of two elements, each equal to ω .

21 with probability 1/2 that A2 has received the recommendation ω1 and that P1 will take the action y11. With the same probability, she believes that A2 has received the recommendation ω2 1 and that P1 will take the action y12. Hence, it is optimal for her to report ω to P2. Following a similar reasoning, it can be checked that the direct mechanism γ1 is incentive compatible and that it guarantees to P1 a payoff strictly greater than 16.

The example stresses that if P1 introduces some uncertainty over the decisions he is effec- tively selecting, by inducing a mixed-strategy equilibrium in the agents’ continuation game, he can get a higher payoff. The peculiarity of the example is that this uncertainty cannot be mapped into a type-revelation mechanism; thus, given the mechanisms proposed by his opponents, a single principal can strictly gain through non-truthful mechanisms. The presence of several agents is crucial to get the result. In a single agent environment, provided that all principals are playing a pure strategy, uncertainty is fully revealed at equilibrium: the best reply to any array of mechanisms can always be formulated as a type-revelation mechanism.18 More generally, the example hence suggests that the standard methodology developed to characterize pure strategy equilibria in common agency games of incomplete information does not extend to multiple agent environments.19

5 Conclusion

This paper contributes to analyze multiple-principal multiple-agent games of incomplete information. We show that if each agent can simultaneously negotiate with several principals, limiting principals to use simple type-revelation and truthful mechanisms can be restrictive. In the presence of several agents, a principal tries to exploit correlation among agents’ behaviors in his own interest and can proﬁtably induce (untruthful) random behaviors. We therefore introduce a class of direct incentive compatible mechanisms that are robust against unilateral deviations towards more sophisticated communication schemes. These mechanisms have a clear analogy with those originally employed by Myerson (1982), and are therefore relatively tractable. Interpreting our results in the light of the existing literature on multiple-principal multiple- agent models, one is led to the conclusion that there is only little foundation for the restriction to type-revelation mechanisms which has been postulated in economic applications. A natural question to ask could therefore be: is there any institutional setting where this restriction is meaningful? In particular, it seems important to consider the situation where agents can at

18Theorem 2 in Peters (2003) formalizes this intuition. 19Most economic applications of common agency under incomplete information construct the the best response of each principal to any given proﬁle of his competitors’ mechanisms by looking at simple menu offers corresponding to type-revelation and truthful mechanisms. See Martimort and Stole (2009) for an exposition of this methodology.

22 most participate with one principal, which is taken as a shortcut for exclusive competition. This assumption has been typically introduced in most competing mechanism models of incomplete information.20 In such a scenario, one can indeed show that the best reply of each single principal to any proﬁle of mechanisms offered by his rivals can always be characterized through simple type-revelation mechanisms. As a consequence, any equilibrium outcome supported by type-revelation mechanisms survives if a single principal deviates towards a sophisticated communication scheme, since the uncertainty generated at any (mixed strategy) continuation equilibrium of the agents’ message game can always be reproduced by type-revelation mechanisms.21

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