A Folk Theorem for Contract Games with Multiple Principals and Agents

Transcription

1 A Folk Theorem for Contract Games with Multiple Principals and Agents Siyang Xiong March 27, 2013 Abstract We fully characterize the set of deteristic equilibrium allocations in a competing contract game with multiple principals and agents. Compared to similar folk theorems in the literature, our main theorem relaxes an indispensable assumption on equilibrium existence in Yamashita [8], and does not require a particular communication protocol as required in Peters and Troncoso-Valverde [6]. I thank Eddie Dekel, Songying Fang, Mike Peters, Balázs Szentes, Stephen Wolff, Takuro Yamashita and anonymous referees for helpful comments. I thank the National Science Foundation (grant SES ) for financial support. All remaining errors are my own. Department of Economics, Rice University, xiong@rice.edu 1

2 1 Introduction In the presence of asymmetric information, economic players may write contracts to elicit other players information, in order to make optimal decisions. Most of the large literature on contract theory focuses on the classical principal-agent (i.e., one principal and one or multiple agents) model and the common-agency (i.e., one agent and multiple principals) model. 1 However, both multiple principals and multiple agents are usually involved in many economic situations. For example, there are many competing insurance companies that offer different policies, and there are many consumers trying to find the best insurance plans. To understand such situations, we study a two-stage multi-principal-multi-agent game in this paper. In this contract game, the payoffs of the players 2 depend on both the states and the actions. States are privately observed by the agents, while actions are chosen by the principals. In stage 1, the principals offer contracts, which specify their committed actions contingent on the messages reported by the agents. In stage 2, the agents report messages and the contracts are executed. Our objective is to fully characterize the set of deteristic equilibrium allocations 3 in this contract game. Our main result (Theorem 1) says that the equilibrium allocations are fully characterized by two incentive compatibility (IC) conditions for the agents and the principals. truthful reporting by the agents forms a Bayes Nash equilibrium (I) agents: ; in the direct mechanisms defined by the allocation (II) principals: every principal is endowed with a -max- value (as defined in (5)); under the allocation, every principal achieves utility larger than the -max- value. Condition (I) must hold in every equilibrium; the intuition is the same as the revelation principle in the classical principal-agent model. However, the revelation principle 1 See Bolton and Dewatripont [1] and Martimort [4] for surveys on the principal-agent model and the comon-agency model, respectively. 2 Throughout the paper, we use she and he to refer to a principal and an agent, respectively. We use players to refer to both principals and agents. 3 We discuss stochastic equilibrium allocations in Section

3 ignores IC for the principals, so we need condition (II). The -max- value in condition (II) has the same role as the -max value in a usual 2-player normal-form game given other principals and agents strategies, principal j can always find a contract that guarantees her -max- value. As a result, in any equilibrium, all principals must achieve utility larger than the -max- values, i.e., condition (II) must hold in every equilibrium. Conversely, for any allocation satisfying conditions (I) and (II), we construct an equilibrium to implement the allocation. Specifically, for any principal k, principals k can find punishing contracts c k k under which principal k achieves utility less than her max- value. 4 Then, the intuition of the equilibrium construction is clear: in stage 1, each principal offers a set of contracts, which includes the equilibrium-allocation contract and the punishing contracts; in stage 2, the agents choose the equilibrium-allocation contract on the equilibrium path and they choose the punishing contracts c k k whenever principal k deviates from the equilibrium path. Yamashita [8] s theorem shares a similar intuition. However, his proof requires the agents play pure strategies in the subgames in stage 2, which poses a technical problem a pure-strategy equilibrium does not exist in many games. To eliate this problem, Yamashita assumes the equilibrium existence for all the possible subgames in stage 2. Equivalently, any contracts that induce no pure-strategy Nash equilibrium in stage 2 are automatically excluded from the principals choice sets in stage 1. This limits the scope of Yamashita s theorem, as illustrated by Example 1 in Section 2. In this example, market information is observed only by two agents who have diametrically opposed preferences. Thus, any contract offered by the principal defines a zero-sum game for the agents. Since pure-strategy Nash equilibria exist only in degenerate zero-sum games, no valid contract can extract the market information. As a result, the unique equilibrium allocation is inefficient. In this paper, we allow for mixed strategies in stage 2. Under some usual technical conditions (i.e., compact message spaces and continuous contracts), a mixed-strategy Nash equilibrium always exists in stage 2 (Glicksberg s Theorem, see [2]). Hence, we do not need the equilibrium existence assumption in [8]. Equivalently, we do not place any restrictions on the set of contracts offered by the principals in stage 1. As a result, our 4 We adopt the standard notation of letting k denote all members of a group other than individual k. 3

4 folk theorem expands the scope of Yamashita s theorem. For instance, if we allow for mixed-strategy equilibria in Example 1, the principals can find a contract to elicit market information. Consequently, an efficient equilibrium allocation exists. It is worth noting that each principal in our contract game is allowed to offer only a single contract, while we let each principal offer a set of contracts, as suggested by the equilibrium construction above. One methodological contribution of this paper is to introduce a way to embed such a set of contracts into one single valid contract. One caveat of our equilibrium characterization is that it is complicated, because the -max- values are hard to compute. Peters and Troncoso-Valverde [6] provide a much simpler characterization, but they require a particular complicated protocol for communication among the players. 5 We discuss these issues in Sections 5.4 and 5.5. The remainder of the paper is organized as follows. An illustrating example is provided in Section 2. The model is defined in Section 3. We present the main result in Section 4. Section 5 concludes with discussions. Technical proofs can be found in Appendix A. 2 Pure-strategy implementation versus mixed-strategy implementation: an example We use the following simple example to illustrate the major difference between our setup and that in Yamashita [8]: mixed strategies are allowed in our setup, while only pure strategies are valid in [8]. Example 1 There are two agents i 1,i 2, and one principal j. 6 The state is θ 0, 1 Θ, which is uniformly distributed. Suppose i 1 and i 2 observe θ, but j does not. Principal j has to choose an action a 1, 0, 1 A. The players are expected-utility maximizers, with Bernoulli utility 5 The complicated communication protocol is rarely observed in most economic situations. In such cases, our folk theorem applies. 6 For simplicity, we consider only two agents and one principal. The example remains valid if we add additional payoff-irrelevant agents and principals (i.e., these agents do not observe the states, these principals do not take payoff-relevant actions, and their payoffs are always constant). 4

5 functions defined as follows: u j a, θ 0, if a 0; 2, if a 1 θ; 3, if a 1 and θ 0; u i1 a, θ a; u i2 a, θ a. Following Yamashita [8], a valid contract for principal j is a function c j : M i1 M i2 A, where M i1 and M i2 are some exogenously given message spaces for i 1 and i 2, respectively. Upon receiving the contract, the agents send their messages and then the contract is executed. Pure-strategy implementation As in [8], the agents are required to play pure strategies, and we assume the existence of pure-strategy equilibria in the subgames defined by all the valid contracts of the principal. Equivalently, any contract that induces no pure-strategy Nash equilibria in stage 2 is not available to j. For every a A, let z a denote the constant allocation, in which a is taken at both states by principal j. It is easy to draw the following observations. (1) Only constant allocations can be implemented by pure-strategy equilibria. 7 (2) Principal j strictly prefers z a0 to z a1 or z a1. By 1 and 2, z a0 is the unique equilibrium allocation, under which the players achieve a payoff of 0. We show below that z a0 is not ex-ante pareto efficient. 7 Suppose otherwise. I.e., a at θ 0 and a a at θ 1 is implemented by some contract c j and some pure-strategy Nash equilibrium m 1, m 2 at θ 0 and m1, 2 m at θ 1. Suppose cj m 1, m2 a. Without loss of generality, suppose a i1 a and a i2 a. By the IC of i 2 at state θ 0, we have a i2 a. Hence, we have a i2 a i2 a. Furthermore, by the IC of i 1 at state θ 1, we have a i1 a, which implies a i2 a, contradicting a i2 a i2 a. 5

6 Mixed-strategy implementation Consider the contract ζ j for principal j defined as follows: 8 Suppose we allow the agents to play mixed strategies. ζ j : m 2 m 2 m 2 m 1 a 0 a 1 a 1 m 1 a 0 a 1 a 1 The following strategy profile of the agents forms a mixed-strategy Nash equilibrium in the game defined by ζ j. agent i 1 : at both states, report m 1 and m 1 with equal probability (i.e., 1 2 ); agent i 2 : at state θ 0, report m 2 with probability 1 at state θ 1, report m 2 and m 2 with equal probability (i.e., 1 2 ). Under this equilibrium, the agents fully reveal the true states, and principal j achieves the maximal ex-ante payoff , while both agents get an ex-ante payoff of 0. Therefore, this equilibrium allocation is pareto efficient, and it pareto doates z a0. 3 Setup 3.1 The contract game There are two sets of players: the principals 1,..., J and the agents 1,..., I. Each agent i privately observes his type θ i Θ i, and each principal j has to take an action y j j. Let Θ Θ i and j. The Bernoulli utility functions for i j principal j and agent i are denoted by v j : Θ R and u i : Θ R, respectively. We assume 1, 2, Θ,,, and θ Θ is distributed according to a common prior p Θ. Let ji denote the set of messages that agent i can send to principal j. Throughout this paper, we assume ji j i 0, 1 R for every j, i. 8 The contract ζ j is not valid for prinicipal j in [8], because no pure-strategy Nash equilibrium exists in the game defined by ζ j. 6

7 A strategy of each principal j is a valid contract, i.e., a continuous function c j : ji Y j. 9 Let j denote j s strategy set and j. A strategy of i j each agent i is a (type, contract)-contingent signalling scheme, i.e., a function S i : Θ i ji. 10 Let i denote i s strategy set and i. That is, in this j i contract game, the principals first choose contracts c c j in stage 1. Upon j observing c and θ i, each agent i sends (possibly stochastic) messages to the principals in stage 2, and then the contracts are implemented. I.e., this is a 2-stage dynamic game. Each c chosen by the principals in stage 1 defines a subgame for the agents in stage 2. In every such subgame, each agent i chooses a behavior strategy s i : Θ i ji in j the set sub i Θi ji. Define sub i sub. j i We use S i and s i to denote generic elements in i and i sub, respectively. Given c, we use S i c i sub to denote the behavior strategy induced by S i and c, i.e., S i cθ i S i θ i, c. Let φ s, θ ji denote the independent 11 distribution over message j, i profiles induced by any s, θ sub Θ, i.e., φ s, θ E i i i s i θ i E i, measurable E i i ji. i j Let ψ c, m denote the independent distribution over action profiles induced by 9 We require the codomain of c j be Y j rather than Yj, so that c j can be made continuous and Glicksberg s Theorem can be applied later. 10 We implicitly embed mixed strategies of the agents into our model, i.e., a pure strategy for agent i is a function from Θ i to j ji, and a mixed strategy is a function from Θ i to 11 Given each θ Θ, every agent i takes his strategies s i θ i the distribution φ s, θ induced by s, θ is a distribution over i i I. ji. j ji independently. As a result, j ji which is independent across j 7

8 cj any c, m j, mji j, i ji, i.e., j, i yj mji yj ψ c, m j c j i, yj j j. Given c, s, θ, Γ c, s, θ, y defined below calculates the probability that the action profile y is taken by the principals: Γ c, s, θ, y m j, i ji ψ c, my d φ s, θ. Define V j : sub R and U i : sub Θ i R as follows, where, given c, s, V j c, s and U i c, s, θ i denote the expected payoffs of principal j and agent i who observes θ i, respectively: U i c, s, θ i V j c, s θθ θ i Θ i v j y, θ Γ c, s, θ, y y v j y, θ i, θ i Γ c, s, θ i, θ i, y y p θ ; p θ i, θ i p θ i, θ. i θ i Θ i In the subgame defined by any c, the continuity of c j and the compactness of ji ensure that a (mixed-strategy) Nash equilibrium (NE) always exists by j j Glicksberg s Theorem (see [2]). Let c sub denote the set of Nash equilibria in the subgame defined by c, i.e., c s sub : U i c, s i, s i, θ i U i c, s i, s i, θ i, i, θ, s Θ sub. Hence, c for any c. Throughout the paper, for every c, fix some e c e i c i c sub. (1) We adopt the solution concept of subgame perfect Nash equilibrium (SPNE) 12 for this 2-stage contract game. Specifically, it is defined as follows. 12 Throughout the paper, we use SPNE and NE to represent the equilibrium in the 2-stage contract game and the subgame in stage 2, respectively. 8

9 cj Definition 1 c, S j, S i i is a SPNE if V j cj, c j, S c j, c j Vj c j, c j, S c j, c j, j, c, (2) and S i c i c, c. 3.2 Transformation of a mixed strategy through a homeomorphism Let A and B denote two subsets of the message spaces such that A is homeomorphic to B. Fix any homeomorphism ξ : A B. In our proofs below, we need to transform a mixed strategy in A (i.e., α A) to a mixed strategy in B (i.e., β B) via the homeomorphism ξ. Specifically, for any α A, define ξ α Bas follows: ξ α E α a A : ξ a E, measurable E B. That is, ξ renames the elements in A to the homeomorphic elements in B, and ξ α is just α, respecting this renag. 4 Main result 4.1 Equilibrium allocations A (deteristic) allocation is a function z : Θ ; let z j : Θ j denote the jth projection of z. As in [8], we say z is incentive compatible iff zj u i θ i, θ i j, θ i, θ i p θ i, θ i (3) θ i Θ i θ i Θ i u i zj θji, θ i j, θ i, θ i We say an allocation z is induced by c, s sub iff Note that if z is induced by c, s, then p θ i, θ i, i, θ i, θ ji Θ j i Θ i. z θ y Γ c, s, θ, y 1, θ, y Θ Y. V j c, s v j z θ, θ p θ, j. (4) θθ 9

10 Let Z IC denote the set of incentive compatible allocations. Let Z denote the set of allocations induced by SPNEs. 13 It is easy to show Z Z IC. 14 Define v j inf c j j sup c j j sc V j note that v j is well-defined by Lemma 3 in Appendix A.1. However, c j j j : sup c j j c j j sc V j sup c j j cj, c j, s, j ; (5) cj, c j, s may or may not exist. Define sc V j cj, c j, s is well-defined. The following is our main result. Theorem 1 Z z Z IC : v j z θ, θ p θ v j for every j ; θθ v j z θ, θ p θ v j for every j /. θθ Consider the following thought experiment: We ask each agent i to report his type θ i, and require the principals carry out their actions according to z θ i i, i.e., j takes the action z j θ. Theorem 1 says that we can implement z by a SPNE in the contract game defined in Section 3.1 iff in the thought experiment, no agent finds it profitable to deviate from the truthful report and every principal j achieves at least an expected payoff of v j. 15 The only if (i.e., ) direction of the proof of Theorem 1, which is similar to that in Yamashita [8], can be found in Appendix A.2. We focus on the if (i.e., ) direction in Section I.e., Z allocation z : there exists a SPNE c, S such that z is induced by c, S c. 14 The intuition is just the revelation principle. See also the proof of Lemma 1 in Yamashita [8, p ]. 15 Different from [8], no principal in can achieve the payoff v j in any SPNE, due to the technical difficulty associated with the infinite-message mechanisms that we consider. 10

11 4.2 The proof of Theorem 1: the if direction Throughout this subsection, we fix any z Z IC such that v j z θ, θ p θ v j for θθ every j and v j z θ, θ p θ v j for every j /. We will construct a SPNE θθ c, S such that z is induced by c, S c. The following two lemmas will be used later, and their proofs can be found in Appendix A.3 and A.4. Lemma 1 There exists c, s sub such that s c and z is induced by c, s. Lemma 2 For any k, there exists c k k c k j find s c k, c k k c k, c k k satisfying jk k such that for any c k k, we can V k c k, c k k c, s k, c k k v k z θ, θ p θ. (6) θθ Lemma 1 says that z is induced by c, s and s is a NE in the subgame defined by c. Lemma 2 says that for any principal k, principals k have the punishing contracts c k k and the agents have the punishing NE s c k, c k k to deter principal k from deviating from the allocation z. It is worth noting that the principals punishing contracts c k k does not depend on the deviating strategy (i.e., c k c k ) chosen by principal k, while the agents punishing NE s c k, c k k does A sketch of the proof This contract game has two stages. In stage 1, the principals choose c, and in stage 2, the agents choose s. Ignoring the IC of the principals, Lemma 1 says that z can be implemented by c, s, and the IC of the agents is satisfied, i.e., s c. To ensure the IC of the principals, we modify the rules of the game, hypothetically. Suppose there is an extra stage between the two stages, which we call stage 1.5. In stage 1.5, every principal j observes all the contracts chosen in stage 1, and then she revises 11

12 her contract according to the following protocol 16 : revise to c k j if principal k j is the unique principal who deviates from the strategy profile c defined in Lemma 1; do not revise otherwise. Lastly, based on the final contracts offered in stage 1.5, the agents choose s in stage 2. Given the extra stage and the described protocol, the following strategy profile forms a SPNE which implements z. A) Equilibrium Path: in stage 1, the principals offer c; in stage 1.5, no revision occurs; in stage 2, the agents choose s; B) Off-Equilibrium Path - case i): in stage 1, the principals offer c k, c k c; in stage 1.5, principals k revise their contracts to c k k and principal k does not revise her contract; in stage 2, the agents choose s c k, c k k ; C) Off-Equilibrium Path - other cases: in stage 1, the principals offer c c and two or more principals deviate from c; in stage 1.5, no revision occurs; in stage 2, the agents choose e c c fixed in (1). Lemma 1 implies that z is implemented on the equilibrium path. Furthermore, the IC of the principals is satisfied: if principal k deviates unilaterally from c in stage 1, principals k would choose the punishing contracts c k k in stage 1.5 and the agents would play the punishing NE s c k, c k k in stage 2. By (6) in Lemma 2, principal k does not find it profitable to deviate from the equilibrium path in stage 1. The problem for the scheme above is that stage 1.5 does not exist in our original contract game. In particular, the principals cannot observe who deviates in stage 1, even though they have the punishing contracts to deter deviation. Instead, it is the agents (in stage 2) who can observe the deviating principal. Furthermore, in stage 2, even if the agents inform the principals about the deviating one in stage 1, the principals do not have a chance to revise their contracts after stage 1. Hence, it should be the agents that revise the contracts for the principals in stage 2. To achieve this without stage 1.5, roughly, in stage 1 on the equilibrium path, we let each principal j offer a set of contracts including c j and c k j for every k j 16 In stage 1.5, the principals follow the protocol mechanically, i.e., they do not behave strategically. 12

13 and delegate the choice of contracts to the agents in stage 2: on the equilibrium path in which no principal deviates, the agents choose c j for the principals and play s to j implement z ; on the off-equilibrium path in which principal k deviates unilaterally to c k, the agents choose c k k c for principals k and play s k, c k k to punish principal k. Still, we are facing one problem. The set of contracts described above is not a valid contract in the contract game defined in Section 3.1. We embed the set of contracts into one single valid contract in the next subsection The embedding and the extension To define each c j : ji j, we first break each ji into several parts, and i then define c j on these parts separately. For any E ji, we use c j E to denote c j i with the restricted domain E. 17 Fix any a 0, a 1,..., a 2, a 2 1 such that i 0 a 0 a 1... a 2 a Recall ji 0, 1. Define k k ji a 2k, a 2k1 for every j, i and every k 0, 1, 2,...,. That is, we fix 1 disjoint closed intervals in, and the agents use messages in these disjoint intervals to tell the principals about who deviates from the equilibrium path in stage 1: m ji 0 ji means no one deviates in stage 1, and m ji k ji (with k 0) means principal k deviates in stage 1. For every k 0, 1, 2,...,, k is homeomorphic to ; let h k : k be a homeomorphism. We use h k 1 to denote the inverse function of h k. Furthermore, let ID k : k k be the identity function, i.e., ID k m m for every m k. Since k is closed, by the Tietze Extension Theorem (see [5, p.219]), there exists a continuous function g k : k such that g k m ID k m m for every m k. That is, g k, while preserving the definition of ID k on k, extends the domain continuously to. We describe the embedding in three cases. 17 I.e., c j E : E j satisfies c j E m c j m, m E. 13

14 A) The equilibrium path: the principals offer c in stage 1. Recall s c in Lemma 1, i.e., for every i, θ Θ, we have U i c, s i, s i, θ i U i c, s i, s i, θ i, s i For every j, define c j 0 : ji i i c mji j c i j h 0 m ji i 0 ji j as follows: Θi ji. (7) j, m ji i 0 ji. i I.e., on the equilibrium path, the agents reports messages in 0 ji. Upon receiving the messages, the principals translate them to messages in ji via the homeomorphism h 0, and then implement c. follows: For every i, consider the homeomorphism H 0 : ji 0 ji defined as j j H 0 mji j For every θ i Θ i, note that s i θ i h 0 1 mji, m ji j j j ji and H 0 s i θ i j ji. 0 ji j. That is, each agent i encodes the messages via h 0 1 and transforms the strategy si to H 0 s i, while each principal j decodes messages via h 0 and transforms the strategy H 0 s i back to s i before executing the contract c j. As implied by (7), for every i, θ Θ, we have Θi U i c, H 0 s i, H 0 s i, θ i U i c, si, H0 s i, θ i, si ji 0. (8) j On the equilibrium path, if we hypothetically require agents report messages in 0 ji,(8) would imply that H 0 s is a NE in the subgame defined by c. However, agent i may deviate to report messages outside 0 ji. To make H0 s remain a NE without the hypothetical requirement, we need to extend c j 0 to c ji j as follows: ji 0 i j, i i mji c j i ji 0 j, i i m ji, m j, i i c j 0 ji i ji 0 j, i g 0 m ji, i I.e., any unilateral deviation to a message outside 0 ji is first translated to a message inside 0 ji via g0. As implied by (8), H 0 s remains a NE in the subgame defined by c j. 14.

15 B) The off-equilibrium path with unilateral deviation In stage 1, suppose principal k offers c k c k and principals k offer c k. By Lemma 2, s s c k, c k k c k, c k k, i.e., for every i, θ Θ, we have U i c k, c k k, s i, s i, θ i U i c k, c k k, si, s i, θ i, si For every j k, define c j i c mji j c k j i k ji : k ji j as follows: i h k m ji i, m ji i k ji. i Θi ji. (9) j I.e., the agents reports messages in k ji, and principals k translate them to messages in ji via h k before implementing c k k. For every i, consider the homeomorphism H k : ji j ki k ji jk defined as follows: H k m ki, mji m ki, jk For every θ i Θ i, note that s i θ i h k 1 mji jk, m ki, mji jk ji and H k s i θ i j ki k ji. jk ki k ji jk That is, each agent i encodes the messages to principals k via h k 1 and transforms the strategy s i to H k s i, while principals k decode messages via h k and transform the strategy H k s i back to s i before executing the contract c k k. As implied by (9), for every i, θ Θ, we have ck U i, c k, H k s i, H k s i, θ i s i ck U i, c k, s i, H k s i, θ i, (10) Θi ki ji k. jk If we hypothetically require agents report messages in k ji to each principal j k, (10) would imply that H k s is a NE in the subgame defined by c k, c k. However, agent i may deviate to report messages outside k ji. To make Hk s remain a NE without the hypothetical requirement, we need to extend c j k to c ji j i c j i mji c ji k i j j, i i m ji, m j, i i 15 k ji ji k j, i i as follows: ji k j, i g k m ji i.,.

16 I.e., any unilateral deviation to a message outside k ji is first translated to a message inside k ji via gk. As implied by (10), H k s remains a NE in the subgame defined by ck, c k. C) The other off-equilibrium paths In the above, we have defined each c j on the restricted domain ji k j, i. Since the restricted domain is closed and c j is kj i continuous on it, by the Tietze Extension Theorem, there exists a continuous function c j : ji j such that i c j m c j m kj which completes the definition of c j. kj i i m, ji k j, i ji k j, i, The definition of S Finally, we define S as follows: S c H 0 s; S c k, c k H k s c k, c k k, k, c k k c k ; where e c c is fixed for every c in (1). S c e c otherwise; This completes the definition of c, S. Clearly, c, S is a SPNE as argued above, and z is induced by c, S c. 5 Discussions We conclude the paper with several discussions. 16

17 5.1 Finite messages versus infinite messages Unlike Yamashita [8], we have to consider infinite-message spaces. In our proof of the if direction of Theorem 1, a message m ji sent from agent i to principal j plays two roles: i) it recommends a contract c j for principal j, and ii) it reports i s message when c j is adopted. That is, we have to establish a surjective function from the message set M ji to L M ji, where L denotes the set of potential contracts recommended by i. Clearly, such a surjective function exists only if M ji is an infinite set. Nevertheless, it is without loss of generality to focus on infinite-message mechanisms. Let Q j denote the set of all finite-message mechanisms for principal j, i.e., j q j : W ji ii j Wji, j, i. Although j j, every element in j can be represented by some element in j : For any allocation z that is induced by some q Π jj j and s q, there exist c Π jj j and s c such that z is induced by c and s. 18 Thus, any allocation, which is implementable via finite-message mechanisms, can be implemented via infinite-message mechanisms. 5.2 Stochastic equilibrium allocations For notational ease, we focus on deteristic equilibrium allocations throughout the paper. With some modification, our result also applies to stochastic equilibrium allocations. A stochastic allocation is a function β : Θ. We say a stochastic allocation β is independent if β θ is an independent distribution over Y for every θ; otherwise, we say that β is correlated. Recall that we use a direct mechanism on the equilibrium path to implement a potential equilibrium allocation. However, an implicit assumption of our model is that the principals take actions independently for each profile of messages received. Consequently, a stochastic allocation can be implemented on the equilibrium path if and only if it is independent. Hence, Theorem 1 remains true for independent stochastic equilibrium allocations. 18 The proof is similar to the proof of Lemma 1 and is omitted. 17

18 For correlated stochastic equilibrium allocations, our characterization is true if we endow the principals with the correlation-generating device used in Kalai, Kalai, Lehrer and Samet [3] and Peters and Troncoso-Valverde [6], so that, for each profile of message received, the principals are induced by the correlation device to take correlated actions. 5.3 Szentes critique Following the setup of Yamashita [8], Szentes [7] constructs a complete-information example, in which a principal cannot even achieve her -max utility. The problem, according to [7], is that the principals delegate their actions to the agents in [8]. Szentes then modifies Yamashita s model to a no-delegation model, and proves a folk theorem under the complete-information setup. 19 In Yamashita s setup, the principals are restricted to choose deteristic contracts. If we allow for random contracts, every principal in Szentes example is able to achieve her -max value. Hence, our model, which allows for random contracts, is immune to the critique raised by Szentes [7]. 5.4 A caveat and a re-interpretation A caveat of our theorem is that v j defined in (5) is difficult to compute, which seems to make Theorem 1 hard to use. However, the following result shows that we can apply Theorem 1 without knowing v j. Theorem 2 For any z Z, we have z Z IC : v j z θ, θ p θ v j z θ, θ p θ, j θθ θθ Z. Theorem 2 says that any IC allocation, which doates another SPNE allocation regarding the principals payoffs, can be implemented by a SPNE. Though Theorem 2 is immediately implied by Theorem 1, the arguments in Section 4.2 provide a constructive 19 It remains an open question to extend Szentes folk theorem to incomplete-information setups. 18

19 way to prove Theorem 2 directly. Suppose z Z is implemented by a SPNE c, S. Then, c k can be used as punishing contracts for principals k to deter principal k from deviating from the equilibrium path for z, i.e., v k z θ, θ p θ v k z θ, θ p θ V k c, S c V k c k, c k, S c k, c k, c k k, θθ θθ where the last inequality follows from the fact that c, S is a SPNE. Therefore, with c k k replaced by c k, the argument in Section 4.2 proves Theorem 2. We use the following example to illustrate the idea. Example 2 (A Public Good Project) Multiple principals decide whether to participate in a publicgood project and how much to invest if they participate. The project can be completed if and only if every principal participates. Furthermore, multiple agents privately observe the states which deteres the payoffs of the project to all the players. Clearly, it is a SPNE that no principal participates, and all players get 0 in this equilibrium. Furthermore, by Theorem 2, any IC allocation, in which the principals gets non-negative payoffs, can be implemented by a SPNE Comparison to Peters and Troncoso-Valverde [6] We differ from Yamashita [8] only on the strategy spaces: we allow for mixed strategies, but [8] does not. We differ from Peters and Troncoso-Valverde[6] on three respects. First, as in Yamashita [8], we follow the classical setup in contract theory, in which imal communication is required: the principals first offer the contracts, i.e., functions from (exogenously given) message spaces to action spaces; the agents then send their messages, and the contracts are executed. However, Peters and Troncoso-Valverde [6] requires a particular procedure of two-stage communication among all players. Due to this difference, our equilibrium characterization is more complicated than that in [6]. Hence, an alternative way to interpret [6] is that it finds a communication protocol which induces a simple and elegant characterization of equilibrium allocations. 20 To implement such an allocation, the principals use the direct mechanism on the equilibrium path; if any principal unilaterally deviates, the agents recommed non-participation for the other principals. 19

20 Second, we adopt the solution concept of SPNE, while Bayesian Nash equilibrium is adopted in [6]. That is, different from [6], we require the agents equilibrium strategy profile remains a continuation equilibrium in the subgame defined by any contract profile offered by the principals. 21 Due to this difference, we define v j as a -max- value, while [6] defines v j as a -max value, where the extra takes care of all potential equilibria in subgames. Finally, in Peters and Troncoso-Valverde [6], every player is both a principal and an agent. Through the two-stage communication procedure, all players observe anyone who deviates from the equilibrium path, and they can revise their equilibrium contracts to punishing contracts so as to punish the deviator. However, in Yamashita [8] and this paper, though the principals are endowed with punishing contracts, they have to offer their contracts before observing the deviating principal, and they cannot revise their contracts later. Furthermore, the agents, who observe the deviating principal, are not endowed with punishing contracts. Hence, it is the agents who tell the principals how to revise their equilibrium contracts to punishing contracts. Thus, different from [6], we overcome a conceptual and technical obstacle: the transmission of information regarding the deviating principal from the agents (who observe the deviator but are not endowed with punishing contracts ) to the principals (who do not observe the deviator but are endowed with punishing contracts ). A Appendix A.1 Lemma 3 Lemma 3 V j c, s is well defined for every j, c. sc Proof. Fix any j, c. Suppose inf V j c, s α. Then, for any positive integer n, there exists s n csuch that V j c, s n α sc n 1. Since sub is compact, s n has a convergent subsequence. With abuse of notation, let s n denote this convergent subsequence, i.e., s n s for some s sub. 21 See more discussion in Troncoso-Valverde [6, Section 7]. 20

21 Θi Note that ji is endowed with weak topology and i sub ji j j is endowed with product topology. Furthermore, for any j, i, θ i, both V j c, s and U i c, s, θ i are continuous on S sub. 22 Consequently, the NE is upper hemi-continuous, i.e., s n c for all n and s n s imply s c. Hence, V j c, s inf sc V j c, s α. (11) Furthermore, V j c, s n α n 1 for all n and sn s imply V j c, s lim V j c, s n lim α 1 α. (12) n n n (11) and (12) imply V j c, s α inf V j c, s. Finally, s cimplies that V j c, s sc sc is well defined. A.2 Proof of Theorem 1: the only if direction Fix any allocation z Z, i.e., there exists a SPNE c, S such that z is induced by c, S c. For notational ease, let s denote S c. Hence, by (4), we have We now show V j c, s v j z θ, θ p θ, j. θθ v j z θ, θ p θ v j if j ; θθ v j z θ, θ p θ v j if j /. θθ Suppose otherwise. I.e., there are two cases: 1) there exists j such that v j z θ, θ θθ p θ v j ; 2) there exists j such that v j z θ, θ p θ v j. θθ In case 1), V j c, s v j z θ, θ p θ θθ v j inf c j j sup c j j sc V j cj, c j, s sup c j j sc j,c j V j cj, c j, s. 22 This is immediately implied by the definition of weak topology and the fact that c is continuous. 21

22 In case 2), V j c, s v j z θ, θ p θ θθ v j inf c j j sup c j j sc V j cj, c j, s sup where the last inequality follows from j /, i.e., arg c j j does not exist. Therefore, in both cases, we have V j c, s sup c j j which implies that there exists c j j s.t. V j c, s sc j,c j V j s c j,c j V j c j, c j, S c j j sc j,c j V j sup c j j cj, c j, s, V j c j, c j, s c j, c j. I.e., V j c j, c j, S c j, c j V j c j, c j, S c j, c j, contradicting the fact that c, S is an equilibrium (i.e., (2)). sc V j cj, c j, s, cj, c j, s A.3 Proof of Lemma 1 Lemma 1 There exists c, s sub such that s c and z is induced by c, s. Proof. Recall z Z IC. Suppose every principal j offers the direct mechanism z j (where z z j ). Since j z is incentive compatible (see (3)), truthful reporting by the agents is a NE and z is implemented. That is, (3) is equivalent to zj u i θ i, θ i j, θ i, θ i p θ i, θ i (13) θ i Θ i zj u i θji, θ i θ ji j Θ i j, θ i, θ i p θ i, θ i dρ, ρ Θ i. θ i Θ i 22

23 However, z j is not a valid contract, i.e., it is not a continuous function from ji i to j. We thus need to transform every z j to a valid contract. First, based on z j : Θ i j, consider the continuous function Γ z j i j defined as follows: : Θ i i Γ z j µi i yj θ θ Θ: z j θ i y j µ i i Θ i, y j j. i µ i θ i, That is, under the direct mechanism z j, the agents report deteristic types, while Γ z j is the stochastic direct mechanism induced by z j, in which the agents report stochastic types. Clearly, truthful reporting remains a NE under Γ z j. Second, for any a, we use δ a to denote the Dirac measure on the set a. For every i, fix any injective function i : Θ i ji. We use i θ i to represent θ i, i.e., by reporting i θ i ji, agent i informs principal j of i s type θ i. Consider the continuous function Λ j : i θ i i : θ Θ Λ j i θ i i δ θi Θ i, i i, θ Θ. Since i θ i i : θ Θ is a closed subset of ji, by the Tietze Extension Theorem, there exists a continuous i function Λ j : ji i Θ i, s.t. Λ j i θ i i Λ j i i θ i i, θ Θ. That is, i θ i ji is used by agent i to inform principal j about i s type θ i, and any other message in ji reported by i would suggest a stochastic type in Θ i. Third, define c, s as follows: c j Γ z j Λ j, j ; 23

24 s i θ i δ i θ i, i, θ Θ. j I.e., c j is the generalized stochastic direct mechanism, and s i is the truthful reporting strategy. By (13), s c and z is induced by c, s. A.4 Proof of Lemma 2 Lemma 2 For any k, there exists c k k c k j k such that for any c k k, we can jk find s c k, c k k c k, c k k satisfying V k c k, c k k c, s k, c k k v k z θ, θ p θ. (14) θθ Proof. First, for any k, we will show below that there exists c k k k such that sup c k sc k k,ck k V k c k, ck k, s v k z θ, θ p θ. (15) θθ Second, fix any k and c k k described above. For any c k k, pick some s arg sc k,c k k V k c k, c k k, s. Hence, V k c k, c k k, s c k, c k k Therefore, (14) is implied by (16) and (15). sup c k k sc k,c k k V k sup c k k sc k,ck k V k c k, c k k, s c k, ck k, s. c k, c k k Finally, we prove (15) by considering two cases: k and k /. For any k, pick c k k arg sup c k k sc k,c k V k c k, c k, s. Then, sc k,ck k V k c k k c k, ck k, s sup c k k c k k sc k,c k V k v k v k z θ, θ p θ, θθ 24 c k, c k, s (16)

25 i.e., (15) holds. For any k /, since inf c k k sup c k k sc k,c k V k by the definition of "inf," we have c k k k s.t. sup c k k i.e., (15) holds. sc k, ck k V k c k, c k, s v k v k z θ, θ p θ, θθ c k, ck k, s v k z θ, θ p θ, θθ References [1] Bolton, P. and M. Dewatripont (2004): "Contract Theory," Cambridge, MA: The MIT Press. [2] Fudenberg, D. and J. Tirole (1991): "Game Theory," Cambridge, MA: The MIT Press. [3] Kalai, A.T., E. Kalai, E. Lehrer and D. Samat (2010): "A Commitment Folk Theorem," Games and Economic Behavior 69, [4] Martimort, D. (2006): "Multi-Contracting Mechanism Design," in Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Vol. I, ed. by R. Blundell, W. K. Newey, and T. Persson. Cambridge, MA: Cambridge University Press, [5] Munkres, J. (2000): "Topology," Upper Saddle River, NJ: Prentice-Hall. [6] Peters, M. and C. Troncoso-Valverde (2013): "A Folk Theorem for Competing Mechanisms" Journal of Economic Theory, forthcog. [7] Szentes, B. (2009): "A Note on Mechanism Games with Multiple Principals and Three or More Agents by T. Yamashita," manuscript, University of Chicago. [8] Yamashita, T. (2010): "Mechanism Games with Multiple Principals and Three or More Agents," Econometrica 78,