Mechanisms with Evidence: Commitment and Robustness1 Elchanan Ben-Porath 2 Eddie Dekel 3 Barton L. Lipman4 First Draft January 2017 1 We thank the National Science Foundation, grant SES–0820333 (Dekel), and the US–Israel Binational Science Foundation for support for this research. 2 Department of Economics and Center for Rationality, Hebrew University. Email: [email protected] 3 Economics Department, Northwestern University, and School of Economics, Tel Aviv University. Email: [email protected] 4 Department of Economics, Boston University. Email: [email protected] Abstract We show that in a class of I–agent mechanism design problems with evidence, commitment is unnecessary, randomization has no value, and robust incentive compatibility has no cost. In particular, for each agent i, we construct a simple disclosure game between the principal and agent i where the equilibrium strategies of the agents in these disclosure games give their equilibrium strategies in the game corresponding to the mechanism but where the principal is not committed to his response. In this equilibrium, the principal obtains the same payoff as in the optimal mechanism with commitment. As an application, we show that certain costly verification models can be characterized using equilibrium analysis of an associated model of evidence. 1 Introduction We show that in a class of I–agent mechanism design problems with evidence, randomization has no value for the principal and robust incentive compatibility has no cost. Also, commitment is unnecessary in the sense that there is an equilibrium of the game when the principal is not committed to the mechanism with the same outcome as in the optimal mechanism with commitment. We also show that this equilibrium can be computed from a collection of I auxiliary games, where the ith game is a simple disclosure game between agent i and the principal. As an application, we show that certain mechanism design problems with costly verification instead of evidence can be solved via an associated evidence model.1 To understand the class of mechanism design problems we consider, consider the following examples. Example 1. The simple allocation problem. The principal has a single unit of an indivisible good which he can allocate to one of I agents. Each agent has a type which affects the value to the principal of allocating the good to that agent. Each agent prefers getting the good to not getting it, regardless of her type. Types are independent across agents and monetary transfers are not possible. Each agent may have concrete evidence which proves to the principal some facts about her type. For example, the principal may be a dean with one job slot to allocate to a department in the College. Each department wants the slot and each has private information regarding the characteristics of the person the department would likely hire with the slot, information that is relevant to the value to the dean of assigning the slot to the department. Alternatively, the principal may be a state government which needs to choose a city in which to locate a public hospital. The state wants to place the hospital where it will be most efficiently utilized, but each city wants the hospital and has private information on local needs. The state could ask the city to bear the cost of the hospital, but that would imply diverting the city’s funds from other projects that the government considers important. Extensions: More complex allocation problems. A broader class of allocation problems will also fit in our framework. For example, consider again the example of a dean given above, but suppose the dean has several job slots to allocate where each department can have at most one and there are fewer slots than departments. A related problem is the allocation of a budget across divisions by the head of a firm. Suppose the organization has a fixed amount of money to allocate and that the value produced by a division is a function of its budget and its privately known productivity. Alternatively, consider a task allocation problem where the principal is a manager who must choose an 1 In a model with costly verification, the agents do not have evidence to present but the principal can learn the true type of an agent at a cost. 1 employee to carry out a particular job. Suppose none of the employees wants to do the task and each has private information about how well he would do it. Finally, we could consider a task that some employees might want to do and others would not want to do, where both the employee’s ability and desire to do the job are private information. Example 2. The public goods problem. The principal has to choose whether or not to provide a public good which affects the utility of I agents. If the principal provides the good, the cost must be evenly divided among the agents. Each agent has a type which determines her willingness to pay for the good. If the willingness to pay exceeds her share of the cost, she wants the good to be provided and otherwise prefers that it not be provided. Types are independent across agents and monetary transfers other than the cost sharing are not possible. Each agent may have evidence which enables her to prove some facts to the principal about the value of the public good to her. The principal wishes to maximize the sum of the agents’ utilities. For example, the principal may be a government agency deciding whether or not to build a hospital in a particular city and the agents may be residents of that city who will be taxed to pay for the hospital if it is built. Then an agent might show documentation of a health condition or past emergency room visits to prove to the principal that she has a high value for a nearby hospital. Alternatively, the principal can maximize a weighted sum of the agents’ utilities plus a utility of her own for the public good. We will show that optimal mechanisms for these examples share several significant features. First, commitment is not necessary. In other words, if the principal is not committed to the mechanism, there is still an equilibrium of the game with the same outcome as in the optimal mechanism. Second, the optimal mechanism is deterministic — the principal does not need to randomize. Third, the optimal mechanism is not just incentive compatible but is also what we will call robustly incentive compatible. We define this precisely later, but for now simply note that it is a strengthening of dominant strategy incentive compatibility. Thus the robustness of dominant strategy incentive compatibility comes at no cost to the principal. The robustness of the mechanism in turn implies similar robustness properties of the equilibrium which achieves the same outcome.2 One useful implication of this result is that we can compute optimal mechanisms by considering equilibria of the game without commitment. In particular, we give a relatively simple characterization of an optimal equilibrium for the principal which does not rely on much information regarding the principal’s preferences or the structure of the set of actions. More specifically, we construct a collection of I auxiliary games, one for each agent, where the game for agent i is a simple disclosure game between agent 2 This does not mean that the “truth telling” strategies used in the mechanism are also used in the game without commitment. In general, agents may be mixing over reports and evidence in the equilibrium of the game. 2 i and the principal. The equilibrium of the game without commitment between the I agents and the principal which has the same outcome as the optimal mechanism can be constructed by assigning to agent i her equilibrium strategy in her auxiliary game. This makes determining the optimal mechanism straightforward in some cases. To illustrate, we consider optimal mechanisms when the evidence technology is the one originally proposed by Dye (1985). In Dye’s model, each agent has some probability of having evidence that would enable her to exactly prove her true type and otherwise has no evidence at all. When we apply this approach to the simple allocation problem described in Example 1 above or to the public good problem of Example 2, we find optimal mechanisms reminiscient of optimal mechanisms in a different context, namely, under costly verification. We discuss this connection to Ben-Porath, Dekel, and Lipman (2014) and to Erlanson and Kleiner (2015) in Section 5 where we show that a class of costly verification models can be solved using our results for evidence models. The paper is organized as follows. Section 2 presents the formal model. In Section 2.5, we state the main results sketched above, including the characterization of the best equilibrium for the principal. The proof of this theorem is sketched in Section 4. In Section 3, we specialize to the Dye (1985) evidence structure and provide a characterization of optimal mechanisms in this setting. We then use this characterization to give optimal mechanisms for a variety of more specific settings including the simple allocation problem and the public goods problem. We also show that under some conditions, optimal mechanisms for costly verification instead of evidence can be solved using the optimal mechanisms for Dye evidence. We offer concluding remarks in Section 5. Proofs not contained in the text are in the Appendix. Related literature. Our work is related to the literature on mechanism design with evidence. The first paper on this topic was Green and Laffont (1986). We make use of results in Bull and Watson (2007) and Deneckere and Severinov (2008).3 A particularly relevant subset of this literature is a set of papers on one–agent mechanism design problems which show that, under certain conditions, the principal does not need commitment to obtain the same outcome as under the optimal mechanism. This was first shown by Glazer and Rubinstein (2004, 2006) and extended by Sher (2011) and by Hart, Kremer, and Perry (2016, forthcoming). We discuss these papers in more detail in Section 5. Also, our result showing that commitment is not necessary can be thought of as a characterization of equilibria in games with evidence. Hence our work is also related to the literature on communication games with evidence. The first papers on this topic are Grossman (1981) and Milgrom (1981). Our work makes particular use of Dye (1985) and Jung and Kwon (1988). Finally, the papers most closely related to our application to 3 Other papers which are less directly related include Ben-Porath and Lipman (2012), Kartik and Tercieux (2012), and Sher and Vohra (2015). 3 costly verification models are Ben-Porath, Dekel, and Lipman (2014) and Erlanson and Kleiner (2015). 2 Model and Results The set of agents is I = {1, . . . , I} where I ≥ 1. The principal has a finite set of actions A and can randomize over these. For example, in the simple allocation problem, we have A = I where a = i means that the good is allocated to i. More generally, a can be interpreted as an allocation of money (where money is finitely divisible) as well as other goods, public or private. Each agent i has private information in the form of a type ti where types are distributed independently across agents. The finite set of types of i is denoted Ti and the (full support) prior is denoted ρi . 2.1 Preferences Given action a by the principal and type profile t, agent i’s utility is ūi (a, ti ), independent of t−i . We need significantly more structure on the agents’ utility functions than this “private values” assumption, as we discuss in detail below. The principal’s utility is v(a, t) = u0 (a) + X ūi (a, ti )v̄i (ti ). i For notational convenience, we define v̄0 (t0 ) = 1 so that we can write this as with the convention that the sum runs from i = 0 to I. P i ūi (a, ti )v̄i (ti ) There are two ways to interpret the principal’s utility function. The most obvious is a social welfare interpretation where the principal maximizes a weighted sum of the agent’s utilities and v̄i (ti ) determines how much he “cares” about agent i’s utility. On the other hand, this utility function does not require the principal to care about the agents at all. A different interpretation is to think of v̄i (ti ) as measuring the extent to which the principal’s interests are aligned with those of agent i. That is, a high value of v̄i (ti ) doesn’t mean that the principal likes agent i but means that the principal likes what agent i likes.4 Of course, one can also interpret the model as assuming both motivations for the principal. 4 For example, consider the simple allocation problem where the principal is the head of an organization who needs to choose one of the agents to promote. Assume that every agent wishes to be promoted, so i’s utility is 1 if he is promoted and 0 otherwise. So in this context, it is natural to assume that v̄i (ti ) measures the ability of type ti , not how much the principal cares about ti ’s utility. 4 Another important issue for interpretation is that we cannot entirely separate assumptions about the principal’s utility function and the agents’ utility functions. For example, suppose v̄i (ti ) > 0 for all ti and all i. Then consider changing agent i’s utility function from ūi (a, ti ) to ûi (a, ti ) = ūi (a, ti )v̄i (ti ) and changing the principal’s utility P function to i ûi (a, ti ). Because ûi (a, ti ) is a positive affine transformation of ūi (a, ti ), we haven’t changed best responses for the agents. Clearly, the principal’s preferences have not changed since this is simply a different way of writing the same function. Hence we cannot separate vi (ti ) into the part that comes from how the principal evaluates i’s utility and how agent i evaluates outcomes. Also, note that we allow v̄i (ti ) to be zero or negative. Thus the principal’s interests can be in conflict with those of some or all agents in a way which depends on the agents’ types. Turning to the details of our assumptions on the agents’ utility functions, we go beyond the type–independent preferences that the literature has assumed, but require that the type dependence takes a particularly simple, multiplicatively separable, form. Specifically, we say that ūi (a, ti ) satisfies simple type dependence if there exist functions ui : A → R and βi : Ti → R such that ūi (a, ti ) = ui (a)βi (ti ) where βi (ti ) 6= 0 for all ti ∈ Ti .5 After explaining how these preferences capture the examples above, we provide a renormalization that gives a more useful form for analyzing the model and which also helps show how this model generalizes the cases considered in the literature. To show that simple type dependence accommodates all the examples discussed in the introduction, we illustrate with two examples. First, consider the simple allocation problem, Example 1. Let A = {1, . . . , I} where a = i means the principal allocates the good to agent i. Since every agent desires the good regardless of ti , we let βi (ti ) = 1 for all i and let ui (i) = 1 and ui (j) = 0 for all j 6= i. Finally, let u0 (a) ≡ 0. Then we can interpret v̄i (ti ) as the value to the principal of allocating the good to agent i when his type is ti . As another example, consider the public goods problem, Example 2. Let A = {0, 1}, where 1 corresponds to providing the good and 0 to not providing it. Let βi (ti ) be the value of the public good to type ti net of i’s share of the cost of provision. Letting ui (a) = a, then the utility of agent i is ūi (a, ti ) = ui (a)βi (ti ). If we take the utility of the principal to be the sum of the utilities of the agents, then letting v̄i (ti ) = 1 for every ti P and every i, the utility of the principal is v(a, t) = i ūi (a, ti )v̄i (ti ). 5 If βi (ti ) = 0 for some ti , then that type is indifferent over all actions by the principal and so will always truthfully reveal. Hence we may as well disregard such types. 5 We now renormalize the agents’ utility functions. Let ui (a, ti ) = ūi (a, ti ) . |βi (ti )| Clearly, ui (·, ti ) represents the same preferences over ∆(A) as ūi (·, ti ) for every ti and hence the model is strategically equivalent if we use ui for i’s utility function. Note that ® ui (a, ti ) = ui (a), if ti ∈ Ti+ ; −ui (a), if ti ∈ Ti− where Ti+ = {ti ∈ Ti | βi (ti ) > 0} and Ti− = Ti \ Ti+ . We refer to Ti+ as the positive types of i and Ti− as the negative types of agent i. Thus all types have indifference curves over ∆(A) defined by ui (a), though types may differ in terms of the direction of increase in utility. Also, types can differ in terms of preference intensity as measured by βi (ti ). This intensity factor does not have implications for i’s preferences over ∆(A) but does for the principal’s in the sense that a change in βi (ti ), all else equal, changes the principal’s preferences over ∆(A) conditional on ti .6,7 With this rewriting of the agents’ utilities, we can rewrite the principal’s utility as X X v(a, t) = ui (a)βi (ti )v̄i (ti ) = ui (a)vi (ti ), i i where vi (ti ) = βi (ti )v̄i (ti ) (with β0 (t0 ) defined to be 1). We will typically write the utility functions in this form henceforth. While the assumption of simple type dependence is restrictive in general, it obviously has type–indepdendent preferences as a special case. If we set Ti− = ∅, we have the assumption used in most of the literature on mechanism design with evidence and, in 6 It may seem odd to have a part of the agent’s utility function which is irrelevant to her preferences. We can think of βi as measuring the intensity of i’s preferences over A relative to some other actions which are not under the principal’s control and do not affect the principal’s choices. In other words, if agent i’s utility function is ui (a)βi (ti ) + gi (wi ) where wi is a bundle of private goods chosen by the agent, then βi is relevant to i’s preferences overall, but not for i’s preferences with respect to those choices which are included in the model. For example, in the public goods problem discussed above, our interpretation of βi (ti ) as the monetary value of the public good to ti minus her share of the costs implicitly treats wi as money and assumes gi (wi ) = wi . 7 This does not require us to assume that the principal “cares” about the intensity of the agents’ preferences. If βi (ti ) > βi (t0i ), we could have v̄i (ti ) < v̄i (t0i ) to an extent which offsets this, leaving the principal’s preferences conditional on ti the same as his preferences conditional on t0i . On the other hand, this formulation allows the principal to respond to differences in intensities. The public goods problem discussed above is one where the intensity of agent preferences naturally matters to the principal. For another example, consider the simple allocation problem described above where the principal is a utilitarian. Then the principal wishes to allocate the good to the agent whose preference for the good is the “most intense.” 6 particular, the papers on the value of commitment. More broadly, in many settings, the agent has only two type–independent indifference curves over A and in this case, simple type–dependence is without loss of generality. For example, if the principal has only two actions, then, obviously, there can only be two indifference curves (at most). The type–independent version of this setting is the case originally considered by Glazer and Rubinstein (2004, 2006). Similarly, consider a type–dependent version of the simple allocation problem where each agent cares only about whether she receives the good or not, but some types prefer to get the good and others prefer not to.8 Here the principal has as many actions as there are agents (more if she can keep the good), but each agent has only two indifference curves over A. In this case, there are only two (nontrivial) preferences over ∆(A), so this formulation is not restrictive in that context. 2.2 Evidence Each agent may have evidence which would prove some claims about herself. To model T evidence, we assume that for every i, there is a function Ei : Ti → 22 i . In other words, Ei (ti ) is a collection of subsets of Ti , interpreted as the set of events that ti can prove. The idea is that if ei ∈ Ei (ti ), then type ti has some set of documents or other tangible evidence which she can present to the principal which demonstrates conclusively that her type is in the set ei ⊂ Ti . For example, if agent i presents a house deed with her name on it, it proves that she is one of the types who owns a house. We require the following properties. First, proof is true. Formally, ei ∈ Ei (ti ) implies ti ∈ ei . Second, proof is consistent in the sense that si ∈ ei ∈ E(ti ) implies ei ∈ Ei (si ). In other words, if there is a piece of evidence that some type can present which does not rule out si , then it must be true that si could present that evidence. Clearly, if si could not present it, the evidence actually refutes the possibility of si . Putting these two properties together, we have ti ∈ ei if and only if ei ∈ Ei (ti ). The last property we assume is not necessary for the model to be internally consistent but is a convenient simplifying assumption used in much of the literature. This property was introduced as the full reports condition by Lipman and Seppi (1995), but is more commonly referred to as normality, following Bull and Watson (2007). The condition says that there is one event that ti can present which summarizes all the evidence she has available. Intuitively, this condition means that there are no time or other restrictions on the evidence an agent can present, so that she can present everything she has. Formally, the statement is that for every ti , we have \ ei ∈ Ei (ti ). ei ∈Ei (ti ) 8 For example, if the “good” is a task assignment as discussed in the extensions of Example 1 in the introduction, this formulation is natural. 7 That is, the event proved by showing all of ti ’s evidence is itself an event that ti can prove. Henceforth, we denote this maximally informative event by \ Mi (ti ) = ei . ei ∈Ei (ti ) We sometimes refer to ti presenting Mi (ti ) as presenting maximal evidence. 2.3 Mechanisms Before formally defining a mechanism, we note that given our assumptions, it is without loss of generality to focus on mechanisms where the agents simultaneously make cheap talk reports of types and present evidence and where each agent truthfully reveals her type and presents maximal evidence. This version of the Revelation Principle has been shown by, among others, Bull and Watson (2007) and Deneckere and Severinov (2008). Formally, Q let Ei = ∪ti ∈Ti Ei (ti ) and E = i Ei . A mechanism is then a function P : T × E → ∆(A). For notational brevity, given a mechanism P , ti ∈ Ti , (si , ei ) ∈ Ti × Ei (ti ), and (t−i , e−i ) ∈ T−i × E−i , let ũi (si , ei , t−i , e−i | ti , P ) = X P (a | si , ei , t−i , e−i )ui (a, ti ) a and ûi (si , ei | ti , P ) = Et−i ũi (si , ei , t−i , M−i (t−i ) | ti , P ). In words, ũi (si , ei , t−i , e−i | ti , P ) is agent i’s expected utility under mechanism P when her type is ti but she reports si , presents evidence ei , and expects all other agents to claim types t−i and report evidence e−i . Then ûi (si , ei | ti , P ) is i’s expected utility from reporting (si , ei ) when her type is ti and she expects the other agents to report their types truthfully and to provide maximal evidence. A mechanism P is incentive compatible if for every agent i, ûi (ti , Mi (ti ) | ti , P ) ≥ ûi (si , ei | ti , P ), for all si , ti ∈ Ti and all ei ∈ Ei (ti ). In words, just as stated above, the agent finds it optimal to report her type truthfully and present maximal evidence given that every other agent does the same. The principal’s expected payoff from an incentive compatible mechanism P is X Et P (a | t, M (t))v(a, t). a Our main result is that if the agents’ preferences satisfy simple type dependence, then for the principal, commitment is not necessary, there is no cost to robust incentive compatibility, and randomization has no value. We now make this more precise. 8 Before defining our notion of robust incentive compatibility, we begin with more standard notions. A mechanism is ex post incentive compatible if for every agent i, ũi (ti , Mi (ti ), t−i , M−i (t−i ) | ti , P ) ≥ ũi (si , ei , t−i , M−i (t−i ) | ti , P ), for all si , ti ∈ Ti , all t−i ∈ T−i , and all ei ∈ Ei (ti ). In other words, a mechanism is ex post incentive compatible if each agent i has an incentive to report honestly and present maximal evidence even if she knows all the other agents’ types and that they are reporting truthfully. Say that a reporting strategy σj : Tj → Tj × Ej is feasible if whenever σj (tj ) = (sj , ej ), we have ej ∈ Ej (tj ). A mechanism is dominant strategy incentive compatible if for every agent i, Et−i ũi (ti , Mi (ti ), σ−i (t−i ) | ti , P ) ≥ Et−i ũi (si , ei , σ−i (t−i ) | ti , P ) for all si , ti ∈ Ti , all feasible σ−i : T−i → T−i × E−i , and all ei ∈ Ei (ti ). That is, a mechanism is dominant strategy incentive compatible if each agent i has an incentive to report honestly and present maximal evidence given any feasible strategies for her opponents. In mechanisms with evidence, neither of these notions of incentive compatibility implies the other. A mechanism could be ex post incentive compatible, but an agent might want to deviate if she knew another agent were going to report (si , ei ) where ei 6= Mi (si ). That is, an agent might want to deviate from truth telling and maximal evidence if she knew another agent was going to deviate from truth telling and maximal evidence in a detectable way. Similarly, a mechanism could be dominant strategy incentive compatible but an agent could wish to deviate if she knew the specific types of her opponents. The robustness notion we will use combines both the ex post and dominant strategy features of the above definitions. We say that a mechanism is robustly incentive compatible if for every agent i, ũi (ti , Mi (ti ), t−i , e−i | ti , P ) ≥ ũi (si , ei , t−i , e−i | ti , P ), for all si , ti ∈ Ti , all t−i ∈ T−i , all e−i ∈ E−i , and all ei ∈ Ei (ti ). In other words, even if i knew the exact type and evidence reports of all other agents, it would be optimal to report truthfully and provide maximal evidence regardless of what those reports are. As noted above, ex post incentive compatibility and dominant strategy incentive compatibility are not equivalent in mechanisms with evidence even with independent private values. Robust incentive compatibility implies both ex post incentive compatibility and dominant strategy incentive compatibility, but is not implied by either. We give an example in Appendix A to illustrate. If a mechanism is robustly incentive compatible, then it has several desirable properties. First, the mechanism does not rely on the principal knowing the beliefs of the 9 agents about each other’s types or strategies. Second, the outcome of the mechanism need not change if the agents report publicly and sequentially, rather than simultaneously, regardless of the order in which they report. Obviously, robust incentive compatibility implies incentive compatibility, but the converse is not true. Hence the best robustly incentive compatible mechanism for the principal yields her a weakly lower expected payoff than the best incentive compatible mechanism, typically strictly lower. Our result states assumptions under which there is no difference — that is, the best incentive compatible mechanism for the principal is robustly incentive compatible. We say a mechanism P is deterministic if for every (t, e) ∈ T ×E, P (t, e) is a degenerate distribution. In other words, for every report and presentation of evidence, whether or not it involves truth telling and maximal evidence, the principal chooses an a ∈ A without randomizing. Of course, randomization is an important feature of optimal mechanisms in some settings. We will show that under our assumptions, there is an optimal mechanism which is deterministic. 2.4 Games Finally, to state what it means that commitment is not necessary, we must define what the principal can accomplish in the absence of commitment. Without commitment, we assume that there is a game in which, just as in the revelation mechanism, agents simultaneously make type reports and present evidence, perhaps with randomization. The principal observes these choices and then chooses some allocation a, again perhaps with randomization. For clarity, we refer to this as the game without commitment. More formally, the set of strategies for agent i, Σi , is the set of functions σi : Ti → ∆(Ti × Ei ) such that σi (si , ei | ti ) > 0 implies ei ∈ Ei (ti ). That is, if agent i is type ti and puts positive probability on providing evidence ei , then this evidence must be feasible for ti in the sense that ei ∈ Ei (ti ).9 The principal’s set of feasible strategies, ΣP , is the set of functions σP : T × E → ∆(A). A belief by the principal is a function µ : T × E → ∆(T ) giving the principal’s beliefs about t as a function of the profile of reports and evidence presentation. For notational convenience, given σ−i ∈ Σ−i , σP ∈ ΣP , a ∈ A, and (si , ei ) ∈ Ti × Ei , let Qi (a | si , ei , σ−i , σP ) = Et−i X (s−i ,e−i ) σP (a | s, e) Y σj (sj , ej | tj ). j6=i This is the probability the principal chooses allocation a given that she uses strategy σP , agents other than i use strategies σj , j 6= i, and agent i reports si and presents evidence 9 We do not require ti to report truthfully and do not require his claim of a type to be consistent with the evidence he presents. That is, we could have σi (si , ei | ti ) > 0 even though si 6= ti and ei ∈ / Ei (si ). 10 ei . We study perfect Bayesian equilibria of this game. Our definition is the natural adaptation of Fudenberg and Tirole’s (1991) definition of perfect Bayesian equilibrium for games with observed actions and independent types to allow type–dependent sets of feasible actions. See Appendix B for details. The equilibria which will give the principal the same payoff as in the optimal mechanism will satisfy a certain robustness property that, for lack of a better phrase, we simply call robustness. Specifically, a perfect Bayesian equilibrium (σ, µ) is robust if for every i and every ti ∈ Ti , σi (si , ei | ti ) > 0 implies (si , ei ) ∈ arg max X s0i ∈Ti ,e0i ∈Ei (ti ) σP (a | s0i , e0i , s−i , e−i )ui (a, ti ), ∀(s−i , e−i ) ∈ T−i × E−i . a∈A In other words, σi (ti ) is optimal for ti regardless of the actions played by the other agents, given the strategy of the principal. Note that i’s strategy is robust with respect to the strategies of the other agents, but not with respect to the principal’s strategy. Given a perfect Bayesian equilibrium (σ, µ), the principal’s expected utility is Et X XY (s,e)∈T ×E a σi (si , ei | ti )σP (a | s, e)v(a, t). i We will show that there is a robust perfect Bayesian equilibrium of this game which gives the principal the same expected utility as the optimal mechanism. In this sense, the principal does not need the commitment assumed in characterizing the optimal mechanism. When we show that commitment is unnecessary, we will construct an equilibrium with the same outcome as in the optimal mechanism. The equilibrium construction is particularly simple in that it can be constructed from a set of I one–agent games which do not depend on A or preferences over A. Specifically, we define the auxiliary game for agent i as follows. This is a game with two players, the principal and agent i. Agent i has type set Ti . Type ti has action set Ti × Ei (ti ). The principal has action set X ⊆ R where X is the compact interval [minj mintj ∈Tj vj (tj ), maxj maxtj ∈Tj vj (tj )]. Agent i’s payoff as a function of ti and the principal’s choice of x is ® x, if ti ∈ Ti+ ; −x, otherwise. The principal’s utility in this situation is −(x − vi (ti ))2 . In other words, the artifical game is a persuasion game where positive types want the principal to believe that vi (ti ) is large and negative types want him to believe it is small. The structure of A and ui (a) play no role. As in the original game defined above, a strategy for agent i is a function σi : Ti → ∆(Ti × Ei ) with the property that σi (si , ei | ti ) > 0 implies ei ∈ Ei (ti ). We denote a strategy for the principal as Xi : Ti × Ei → X. 11 2.5 Results: Commitment, Determinism, and Robust Incentive Compatibility Our main results are stated in the following theorem. Theorem 1. If every ui exhibits simple type dependence, then there is an optimal incentive compatible mechanism for the principal which is deterministic and robustly incentive compatible. In addition, there is a robust perfect Bayesian equilibrium of the game without commitment with the same outcome as in this optimal mechanism. In this equilibrium, agent i’s strategy is also a perfect Bayesian equilibrium strategy in the auxiliary game for agent i. 3 3.1 Optimal Mechanisms with Dye Evidence Characterizing the Optimal Mechanism In light of Theorem 1, we can compute the outcomes of optimal mechanisms by identifying the best perfect Bayesian equilibrium for the principal. In particular, we can compute these equilibria by considering the auxiliary game for each agent i. In some cases, these equilibria are very easy to characterize. In this section, we illustrate by considering optimal mechanisms with a particular evidence structure introduced by Dye (1985) and studied extensively in both the economics and accounting literatures. After characterizing optimal mechanisms with Dye evidence, we show that these results can also be used to characterize optimal mechanisms in a different setting. Specifically, in certain models without evidence but where the principal can verify the type of an agent at a cost, we show that the optimal mechanism can be computed from the optimal mechanism for an associated Dye evidence model. We say that the model has Dye evidence if for every i, for all ti ∈ Ti , either Ei (ti ) = {Ti } or Ei (ti ) = {{ti }, Ti }. In other words, any given type either has no evidence in the sense that she can only prove the trivial event Ti or has access to perfect evidence and can then choose between proving nothing (i.e., proving Ti ) and proving exactly her type. Let Ti0 denote the set of ti ∈ Ti with Ei (ti ) = {Ti }. In what follows, we sometimes refer to types who have only trivial evidence as having no evidence and types with Ei (ti ) = {Ti , {ti }} as having evidence. A small complication in stating our results is that there is an essentially irrelevant but unavoidable multiplicity of equilibrium in our auxiliary games. To understand this, note that our auxiliary games differ in one respect from the usual persuasion games in the 12 literature in that agent i both presents evidence and makes a cheap talk claim regarding her type in the former. Of course, if these cheap talk claims convey information, we can always permute agent i’s use of these claims and the principal’s interpretation of them to obtain another equilibrium. There is also another form of multiplicity which is more standard in the literature on games with evidence. In some cases, we may have an equilibrium where the principal has the same beliefs about the agent whether she presents evidence e or evidence e0 . In these cases, we can construct an equilibrium where the agent presents evidence e and another where she presents evidence e0 . Note that in both of these cases, the principal’s beliefs about the agent along the equilibrium path are the same across these various equilibria. That is, if the agent is type t, the belief the principal will have about t is the same across these equilibria. With this issue in mind, we say that an equilibrium in the auxiliary game for agent i is essentially unique if all equilibria have the same outcome in this sense. To be precise, given equilibria (σi∗ , Xi∗ ) and (σ̂i∗ , X̂i∗ ) of the auxiliary game for i, we say these equilibria are essentially equivalent if for every x ∈ X and every ti ∈ Ti , we have σi∗ ({(si , ei ) ∈ Ti × Ei (ti ) | Xi∗ (si , ei ) = x} | ti ) = σ̂i∗ (si , ei ) ∈ Ti × Ei (ti ) | X̂i∗ (si , ei ) = x | ti . Ä¶ © ä If there is an equilibrium with the property that every other equilibrium is essentially equivalent to it, we say the equilibrium is essentially unique. The simplest case to consider with Dye evidence is where the utility functions are not type dependent at all. We say that the model exhibits type–independent utility if ui (a, ti ) is independent of ti for all i and a. In other words, Ti− = ∅, so ui (a, ti ) = ui (a) for all ti . The following results build on well–known characterizations of equilibria in evidence games using the Dye evidence structure. Theorem 2. In any model with Dye evidence, for every i, there exists a unique vi∗ such that î ó vi∗ = E vi (ti ) | ti ∈ Ti0 or vi (ti ) ≤ vi∗ . If Ti− = ∅, the essentially unique equilibrium in the auxiliary game for i is a pure strategy equilibrium where every type makes the same cheap–talk claim, say s∗i , and only types with evidence and with vi (ti ) > vi∗ present (nontrivial) evidence. That is, type ti sends (s∗i , e∗i (ti )) with probability 1 where e∗i (ti ) ® = Ti , if ti ∈ Ti0 or vi (ti ) ≤ vi∗ ; {ti }, otherwise. 13 To see the intuition, note first that cheap talk cannot be credible in this game since every type wants the principal to believe that vi is large. So if i has no evidence (i.e., can only prove the trivial event Ti ), then she has no ability to convey any information to the principal — she can only send an uninformative cheap talk message and prove nothing. If i can prove her type is ti , she wants to do so only if vi (ti ) is at least as large as what the principal would believe if she showed no evidence. Thus types with evidence but lower values of vi (ti ) will pool with the types who have no evidence, leading to an expectation of vi (ti ) equal to vi∗ . In this equilibrium, the principal’s expectation of vi (ti ) will be vi∗ given a type with no evidence or with vi (ti ) ≤ vi∗ and will equal the true value otherwise. More formally, let ® ∗ vi , if ti ∈ Ti0 or vi (ti ) ≤ vi∗ ; v̂i (ti ) = vi (ti ), otherwise. For every v̂ = (v̂1 , . . . , v̂I ), let p̂(· | v̂) denote any p ∈ ∆(A) maximizing " X # p(a) u0 (a) + X ui (a)v̂i . i a∈A The following is a corollary to Theorems 1 and 2. Corollary 1. In any model with type–independent utility and Dye evidence, there is an optimal mechanism P with P (· | t, M (t)) = p̂(· | v̂(t)). In other words, with or without commitment, the outcome selected by the principal when the profile of types is t is p̂(· | v̂(t)). We can use Corollary 1 to give simple characterizations of optimal mechanisms in many cases of interest. Example 3. The simple allocation problem (Example 1) with Dye evidence. In this case, p̂(i | t) > 0 iff v̂i (ti ) = maxj v̂j (tj ). That is, the good is given to one of the agents with the highest v̂j (tj ) or, equivalently, who is believed to have the highest vj . We can break indifferences in a particularly simple way and recast this characterization in the form of a favored–agent mechanism. More specifically, say that P is a favored–agent mechanism if there is a threshold v ∗ ∈ R and an agent i, the favored agent, such that the following holds. First, if no agent j 6= i proves that vj (tj ) > v ∗ , then i receives the good. Second, if some agent j 6= i does prove that vj (tj ) > v ∗ , then the good is given to the agent who proves the highest vj (tj ) (where this may be agent i). Then a favored–agent mechanism where the favored agent is any i satisfying vi∗ = maxj vj∗ and the threshold v ∗ is given by vi∗ is an optimal mechanism. To see this, fix any 14 t. By definition, v̂j (tj ) ≥ vj∗ for all j. Hence if vi∗ ≥ vj∗ for all j, then v̂i (ti ) ≥ vj∗ for all j. Hence for any j such that Ej (tj ) = {Tj } or vj (tj ) ≤ vj∗ , we have v̂i (ti ) ≥ vi∗ ≥ vj∗ = v̂j (tj ). So if every j 6= i satisfies this, it is optimal for the principal to give the good to i. Otherwise, it is optimal for him to give it to any agent who proves the highest value. As we discuss further below, this mechanism is reminiscent of the favored–agent mechanism discussed by Ben-Porath, Dekel, and Lipman (2014) for the allocation problem with costly verification. We now extend the simple allocation problem as discussed in the introduction. Example 4. The multi–unit allocation problem with Dye evidence. It is not hard to extend the above analysis to the case where the principal has multiple identical units of the good to allocate. Suppose he has K < I units and, for simplicity, assume he must allocate all of them. Suppose each agent can only have either 0 or 1 unit. Then the principal’s action can be thought of as selecting a subset of {1, . . . , I} of cardinality P K. The principal’s utility given the set Î is i∈Î vi (ti ). As before, agent i’s utility is 0 if she does not get a unit and 1 if she does. In this case, it is easy to see that the principal allocates units to the K agents with the highest values of v̂i (ti ) as computed above. It’s not difficult to show that this can be interpreted as a kind of recursive favored–agent mechanism.10 Example 5. Allocating a “bad.” Another setting of interest is where the principal has to choose one agent to carry out an unpleasant task (e.g., serve as department chair). It is easy to see that this problem is effectively identical to having I − 1 goods to allocate since not receiving the assignment is the same as receiving a good. Thus we can treat the principal’s set of feasible actions as the set of subsets of {1, . . . , I} of cardinality I − 1, interpreted as the set of agents who are not assigned the task. The one aspect of this example that may seem odd is that the principal’s utility if he assigns the task to P agent i is then j6=i vj (tj ). On the other hand, it is an innocuous renormalization of the P principal’s utility function to subtract the allocation–independent term j vj (tj ) from her utility. In this case, we see that the principal’s payoff to assigning the task to agent i is −vi (ti ), so vi (ti ) is naturally interpreted as ti ’s level of incompetence in carrying out the task. One can apply the analysis of the previous example for the special case of K = I − 1 to characterize the optimal mechanism for this example. While the case of type–independent utility with Dye evidence is particularly tractable, More specifically, we allocate the first unit to the agent with the highest value of vi∗ if no other agent proves a higher value and to the agent with the highest proven value otherwise. Once removing this agent and unit, we follow the same procedure for the second unit, and so on. It is easy to see that the agent with the highest value of vi∗ is the most favored agent in the sense that at least K agents must prove a value above her vi∗ for her to not get a unit. Similarly, the agent with the second–highest value of vi∗ is the second–most favored agent in the sense that at least K − 1 of the “lower ranked” agents must prove a value above her vi∗ for her not to get a unit, etc. 10 15 the case of simple type dependence is not much more difficult. To see the intuition, again consider the auxiliary game for i where some types wish to persuade the principal that vi (ti ) is large and other types want to convince him vi (ti ) is small. Suppose that when the agent doesn’t prove her type, she makes a cheap talk claim regarding whether her type is positive (i.e., she wants the principal to think vi (ti ) is large) or negative (i.e., the reverse). Let vi+ denote the principal’s belief about vi if i does not prove her type but says it is positive and let vi− be the analog for the case where i claims her type is negative. If vi+ > vi− , then every positive type without evidence prefers to truthfully report that her type is positive and every negative type without evidence will honestly reveal that her type is negative since this leads to the best possible belief from i’s point of view. If i is a positive type with evidence, she will want to prove her type only if vi (ti ) > vi+ , while a negative type with evidence will prove her type only if vi (ti ) < vi− . Hence for this to be an equilibrium, we must have î vi+ = E vi (ti ) | (ti ∈ Ti+ ∩ Ti0 ) or (ti ∈ Ti+ \ Ti0 and vi (ti ) ≤ vi+ ) ó and vi− = E vi (ti ) | (ti ∈ Ti− ∩ Ti0 ) or (ti ∈ Ti− \ Ti0 and vi (ti ) ≥ vi− ) . î ó Suppose this gives a unique value for vi+ and vi− . If these values do not satisfy vi+ ≥ vi− , then we can’t have an equilibrium of this kind since the positive types who don’t present evidence will prefer to act like the negative type and vice versa. In this case, we must pool all types. If we do have vi+ ≥ vi− , then these strategies do give an equilibrium. In the case where vi+ = vi− , the cheap talk does not convey any extra information. When vi+ > vi− , cheap talk is useful, but there is another equilibrium as well where cheap talk is disregarded or treated as “babbling,” as in all models with cheap talk. The following lemma provides the background for the equilibrium characterization. Lemma 1. In any model with Dye evidence, for every i, there exists a unique triple vi+ , vi− , and vi∗ such that î ó vi+ = E vi (ti ) | (ti ∈ Ti+ ∩ Ti0 ) or (ti ∈ Ti+ \ Ti0 and vi (ti ) ≤ vi+ ) , vi− = E vi (ti ) | (ti ∈ Ti− ∩ Ti0 ) or (ti ∈ Ti− \ Ti0 and vi (ti ) ≥ vi− ) , î ó and vi∗ = E vi (ti ) | (ti ∈ Ti0 ) or (ti ∈ Ti− \ Ti0 and vi (ti ) ≥ vi∗ ) or (ti ∈ Ti+ \ Ti0 and vi (ti ) ≤ vi∗ ) . î ó Using these cutoffs, we can characterize the equilibria of the auxiliary game for i. Theorem 3. If vi+ ≤ vi− , then there is an essentially unique equilibrium in the auxiliary game for i. In this pure strategy equilibrium, there is a fixed type ŝi such that ti reports (ŝi , e∗i (ti )) where e∗i (ti ) ® = Ti , if ti ∈ Ti0 or (ti ∈ Ti+ and vi (ti ) ≤ vi∗ ) or (ti ∈ Ti− and vi (ti ) ≥ vi∗ ); {ti }, otherwise. 16 If vi+ > vi− , there are two equilibria that are not essentially equivalent to one another and every other equilibrium is essentially equivalent to one of the two. The first is exactly the − same strategy profile as above. In this second equilibrium, there are types ŝ+ i and ŝi with + − k k k ŝi 6= ŝi such that ti ∈ Ti sends (ŝi , ei (ti )), k ∈ {−, +}, where e+ i (ti ) Ti , if ti ∈ Ti0 or vi (ti ) ≤ vi+ ; {ti }, otherwise, ® Ti , if ti ∈ Ti0 or vi (ti ) ≥ vi− ; {ti }, otherwise. = and e− i (ti ) ® = Thus if vi+ > vi− , then there are (essentially) two equilibria in the auxiliary game. As the result below will show, we can always compare these equilibria for the principal and the better one is the one which separates the positive and negative types. Thus this is the equilibrium that corresponds to the optimal mechanism. With this in mind, now define v̂i (ti ) as follows. If vi+ > vi− , we let vi+ , if ti ∈ Ti0 ∩ Ti+ or ti ∈ Ti+ \ Ti0 and vi (ti ) ≤ vi+ ; − if ti ∈ Ti0 ∩ Ti− or ti ∈ Ti− \ Ti0 and vi (ti ) ≥ vi− ; v̂i (ti ) = vi , vi (ti ), otherwise. If vi+ ≤ vi− , let vi (ti ), if (ti ∈ Ti+ \ Ti0 and vi (ti ) ≥ vi∗ ) or (ti ∈ Ti− \ Ti0 and vi (ti ) ≤ vi∗ ); vi∗ , otherwise. (1) For each t ∈ T , let p̂(· | v̂) denote any p ∈ ∆(A) maximizing ® v̂i (ti ) = " X # p(a) u0 (a) + X ui (a, ti )v̂i . i a∈A The following result is a corollary to Theorems 1 and 3. Corollary 2. In any model with simple type dependence and Dye evidence, there is an optimal mechanism P with P (· | t, M (t)) = p̂(· | v̂(t)). In other words, the outcome selected by the principal when the profile of types is t is p̂(· | v̂(t)). The only part of this result that does not follow immediately from Theorems 1 and 3 is the claim above that when vi+ > vi− , the equilibrium that is better for the principal is the one that separates the positive and negative types. This equilibrium is better since it provides more information for the principal. This is shown in Appendix F. 17 Example 8. The public–goods problem. As an application, consider the public goods model discussed in Section 1. For simplicity, we write out the optimal mechanism only for the case where vi+ > vi− for all i, but similar comments apply more generally. We know that in equilibrium, given a profile of types t, the principal’s expectation of vi will be given by v̂i (ti ) defined in equation (1) above. Then the principal will provide the P public good iff i v̂i (ti ) > 0. Just as the analysis of Example 3 above was reminiscent of Ben-Porath, Dekel, and Lipman’s (2014) analysis of allocation with costly verification, the optimal mechanism in this example is very reminiscent of the optimal mechanism under costly verification identified by Erlanson and Kleiner (2015). 3.2 Costly Verification In Ben-Porath, Dekel, and Lipman (2014) and Erlanson and Kleiner (2015), costly verification is modeled by assuming the principal can pay a cost ci to “check” or learn the realization of agent i’s type, ti . The agent cannot affect this verification process. By contrast, in the evidence model we consider here, the principal cannot acquire information about an agent without somehow inducing the agent to reveal it. Despite this difference, the optimal mechanisms look very similar. Ben-Porath, Dekel, and Lipman identify an optimal mechanism in the costly–verification version of the simple allocation problem which is very similar to the optimal mechanism here. In both cases, there is a favored agent and a threshold. In both cases, if no non–favored agent “reports” above the threshold, the favored agent receives the object regardless of his report. Here, “reporting above the threshold” means to prove a value of vi (ti ) above the threshold. In Ben-Porath, Dekel, and Lipman, it means to make a cheap talk report of a type such that the type minus the checking cost is above the threshold. In both cases, if some non–favored agent “reports” above the threshold, the good goes to the agent with the highest such report. In the costly verification model, this is after checking this type. Similarly, Erlanson and Kleiner consider the public goods model under costly verification. In their mechanism and in the optimal mechanism here, we compute “adjusted reports” for each agent i given ti . In both cases, the adjusted report for a positive type is max{vi+ , vi (ti )}, while the adjusted report for a negative type is min{vi− , vi (ti )} for certain cutoffs vi+ and vi− . Just as with the allocation problem, the difference between these two scenarios is that the report is proven in the evidence model and is a cheap talk claim of a type adjusted by the verification cost in the costly verification model. In both the allocation and public goods problems, these reports are adjusted by the verification cost and then summed to determine the optimal action by the principal. Again, this includes some checking in the costly verification model. 18 These parallels are special cases of a more general result that certain costly verification models can be rewritten as a Dye evidence model, so that the optimal mechanism can be computed directly from our results about optimal mechanisms with evidence. In the subsequent text, we explain this for the simple allocation problem. We give the more general result and explain the connection to Erlanson and Kleiner in Appendix G. So consider the following alternative model. Again, the principal has a single unit of an indivisible good to allocate, each agent prefers to receive the good, and vi (ti ) is the payoff to the principal to allocating the good to agent i. For simplicity, we assume here that vi (ti ) > 0 for all ti and all i and that no two types have the same value of vi (ti ). Instead of assuming that agents may have evidence to present, assume that the principal can pay a cost ci > 0 to learn the type of agent i, which we refer to as checking i. As shown in Ben-Porath, Dekel, and Lipman, we can characterize an optimal mechanism as specifying functions p : T → ∆({1, . . . , I}) and qi : T → [0, 1] where p(t) gives the probability distribution over which agent the principal gives the good to as a function of type reports t and qi (t) gives the probability that the principal checks i given reports t. The principal’s objective function then is " Et # X pi (t)vi (ti ) − qi (t)ci i where p(t) = (p1 (t), . . . , pI (t)). The incentive compatibility constraints are p̂i (ti ) ≥ p̂i (t0i ) − q̂i (t0i ), ∀ti , t0i ∈ Ti , ∀i where p̂i (ti ) = Et−i pi (t) and q̂i (ti ) = Et−i qi (t). To see this, note that if type ti reports truthfully, then whether he is checked or not, he will receive the good with expected probability p̂i (ti ). On the other hand, if he misreports and claims to be type t0i , he will be checked with expected probability q̂i (t0i ). In this case, the principal will learn that he has lied and will not give him the good. Thus his probability of receiving the good is the same as t0i ’s probability, minus the probability of being checked. It is not hard to show that the solution is monotonic in the sense that p̂i (ti ) ≥ p̂i (t0i ) if vi (ti ) ≥ vi (t0i ). For each i, let t0i be the type with the smallest value of vi (ti ). The monotonicity of the solution implies that if incentive compatibility holds for type t0i , then it holds for every other type of agent i. Hence we can rewrite the incentive compatibility constraints as q̂i (t0i ) ≥ p̂i (t0i ) − p̂i (t0i ), ∀t0i ∈ Ti , ∀i. It is easy to see that the optimal solution must set q̂i as small as possible since checking is costly. Hence q̂i (ti ) = p̂i (ti ) − p̂i (t0i ) for all ti . We can then rewrite the objective function as î ó X X Eti [p̂i (ti )vi (ti ) − q̂i (ti )ci ] = Eti p̂i (ti )(vi (ti ) − ci ) + p̂i (t0i )ci . i i 19 Thus we can solve the principal’s problem by choosing p to maximize the above subject to the constraint that p̂i (ti ) ≥ p̂i (t0i ) for all ti ∈ Ti and all i. Rewriting the objective function once more, we can write it as X Eti [p̂i (ti )ṽi (ti )] = Et i where # pi (t)ṽi (ti ) i ( ṽi (ti ) = " X vi (ti ) − ci , vi (t0i ) − ci + ci , ρi (t0i ) if ti 6= t0i if ti = t0i . (Recall that ρi is the principal’s prior over Ti .) Now consider the simple allocation problem with Dye evidence where the value to the principal of allocating the good to agent i is ṽi (ti ). Assume that Ei (t0i ) = {Ti } and Ei (ti ) = {{ti }, Ti } for all ti 6= t0i . In this case, the objective function is the same as the one above. The incentive compatibility constraint is simply that no type who can prove his type wishes to imitate the type who cannot. That is, p̂i (ti ) ≥ p̂i (t0i ), the same incentive compatibility constraint as in the costly verification model. Thus we can directly apply our characterization of optimal mechanisms with Dye evidence to derive the solution to this problem. It is straightforward to use this to give a characterization for the original costly verification model by “inverting” the ṽi ’s and writing the solution in terms of the original vi ’s. In particular, we obtain the optimal mechanism identified by Ben-Porath, Dekel, and Lipman. To see this, for each i, define the cutoffs ṽi∗ from the ṽi functions the same way we defined vi∗ from the vi functions. That is, ṽi∗ is the expectation of ṽi conditional on ti not having evidence (here being the type t0i ) or having ṽi (ti ) ≤ ṽi∗ . As shown above, the optimal mechanism for this allocation problem with evidence is to select a favored agent who has ṽi∗ ≥ ṽj∗ for all j 6= i and to set threshold ṽi∗ . This implies that it is optimal to give the good to i if ṽj (tj ) = vj (tj ) − cj ≤ ṽi∗ for all j 6= i and to give the good to that agent j who maximizes vj (tj ) − cj otherwise. This is exactly the mechanism discussed by Ben-Porath, Dekel, and Lipman. One can use this approach to characterize optimal mechanisms with costly verification for less simple allocation problems such as the extensions of Example 1 in Section 1 and for the model of Erlanson and Kleiner, as discussed in Appendix G. 4 Proof Sketch In this section, we sketch the proof of Theorem 1. For simplicity, we sketch the proof in the context of a special case, namely, the simple allocation problem. So assume for 20 this section that the principal has one unit of an indivisible good to allocate to some agent. All agents desire the good and the principal must give it to one of them. So A = {1, . . . , I} where a = i means that the principal allocates the good to agent i. The utility functions of the agents are type independent with ® ui (a) = 1, if a = i 0, otherwise. The payoff to the principal to allocating the good to agent i given type profile t is vi (ti ) which we assume is strictly positive for all i and all ti . One convenient simplification in the type independent case is that we can write a mechanism as a function only from type reports into choices by the principal, where it is understood that if i claims type ti , she also reports maximal evidence for ti , Mi (ti ). This works in the type independent case because if i claims type ti but does not show evidence Mi (ti ), the principal knows how to punish i — namely, he can give i the good with zero probability, the worst possible outcome for i. This will deter any such “obvious” deviation. (Of course, the mechanism must still deter the more subtle deviations to reporting some si 6= ti and providing evidence Mi (si ).) So for this proof sketch, we will write a mechanism as a function P : T → ∆(A). Fix an optimal mechanism P . Given this mechanism, we can construct the probability that any given type of any given agent receives the good. Let p̂i (ti ) = Et−i P (i | ti , t−i ). This is type ti ’s probability of being allocated the good in mechanism P . Partition each Ti according to equality under p̂i . In other words, for each α ∈ [0, 1], let Tiα = {ti ∈ Ti | p̂i (ti ) = α}. Of course, since Ti is finite, there are only finitely many values of α such that Tiα 6= ∅. Unless stated otherwise, any reference below to a Tiα set assumes that this set is nonempty. Let Ti denote the partition of Ti so defined and T the induced (product) partition of T . We refer to T as the mechanism partition. It is easy to see that incentive compatibility is equivalent to the statement that Mi (si ) ∈ Ei (ti ) implies p̂i (ti ) ≥ p̂i (si ). In other words, if ti can report si credibly in the sense that ti has available the maximal evidence of si , then the mechanism must give the good to ti at least as often as si . The first key observation is that without loss of generality, we can take the mechanism to be measurable with respect to the mechanism partition T . While this property may seem technical, it is the key property behind our results and is not generally true for models with more general type dependence than we allow in Theorem 1. 21 To see why this property holds in the simple allocation problem, suppose it is violated. In other words, suppose we have some pair of types si , s0i ∈ Ti such that p̂i (si ) = p̂i (s0i ) but P is not measurable with respect to {si , s0i } in the sense that there is some t−i ∈ T−i with P (· | si , t−i ) 6= P (· | s0i , t−i ). Consider the alternative mechanism P ∗ which is identical to P unless i’s report is either si or s0i . For either of these actions by i, P ∗ specifies the expected allocation generated by P . In other words, if q is the probability of type si conditional on {si , s0i }, then for every a ∈ A and every t−i ∈ T−i , we set P ∗ (a | si , t−i ) = P ∗ (a | s0i , t−i ) = qP (a | si , t−i ) + (1 − q)P (a | s0i , t−i ). It is easy to see that the incentives of agents j 6= i are completely unaffected. By the private–values assumption, the payoffs to these agents don’t depend on i’s type directly — they are only affected by i’s type through its effect on the outcome chosen by the principal. Since this change in the mechanism preserves the probability distribution over outcomes from the point of view of these agents, their incentives are unaffected. Also, the incentives of agent i are not affected. The payoff to i from reporting anything other than si or s0i are not changed. The expected payoff to i from reporting si was p̂i (si ) in the original mechanism, while the expected payoff from reporting s0i was p̂i (s0i ). In the new mechanism, we have “averaged” these two types together, so that in the new mechanism, the probability i receives the good if she reports si is now q p̂i (si )+(1−q)p̂i (s0i ). But since p̂i (si ) = p̂i (s0i ), this means that the probability that i receives the good if she reports si does not change and similarly for s0i . Hence the expected payoff to i from every action is the same under P and P ∗ , so P ∗ must be incentive compatible.11 To see that this change does not affect the principal’s payoff, recall that the principal’s utility function is X v(a, t) = uj (a)vj (tj ). j Under the original mechanism, the principal’s expected payoff is Et X a P (a | t) X uj (a)vj (tj ) = j X ñ Etj Et−j X ô P (a | t)uj (a) vj (tj ) a j = X Etj p̂j (tj )vj (tj ). j As noted above, for every j and every type tj ∈ Tj , the probability tj receives the good is unchanged in the new mechanism. Hence the expected payoff of the principal is 11 Note that this would be true more generally if all types have the same indifference curves. In this case, if si is indifferent between reporting si or lying and claiming to be type s0i , s0i would also be indifferent between these reports. Hence both types would have their payoffs unchanged if we replace the response to either report with the averaged response. This is the key way we use our assumption of simple type dependence. 22 unchanged as well. So without loss of generality, we can change from P to P ∗ . Repeating as needed, we construct an alternative optimal mechanism which is measurable with respect to T . To see why this property is critical, fix any event T̂ in the mechanism partition T and suppose that the principal learns only that the type profile t is contained in T̂ . Since the mechanism is measurable with respect to T , it specifies the same response for every t ∈ T̂ . Suppose, though, that this response is not sequentially rational for the principal in the sense that learning that t ∈ T̂ would lead him to strictly prefer some different response from what the mechanism specifies so that the commitment to the mechanism is crucial. If this is true, we can consider the following alternative mechanism. If t ∈ / T̂ , the new mechanism is the same as the original one. If t ∈ T̂ , then the new mechanism chooses the same allocation as the old one with probability 1 − ε and chooses the strictly better response for the principal with probability ε for some sufficiently small ε > 0. It is easy to see that this alternative mechanism must give the principal a strictly higher expected payoff than the original mechanism. Hence if the new mechanism is incentive compatible, we have a contradiction to the optimality of the original mechanism. To see that the new mechanism is incentive compatible for sufficiently small ε > 0, fix any ti and any si 6= ti with Mi (si ) ∈ Ei (ti ). Since the original mechanism is incentive compatible, we know that p̂i (ti ) ≥ p̂i (si ). Suppose in the original mechanism, we have p̂i (ti ) = p̂i (si ). Since the new mechanism is measurable with respect to T , we must have this same equality in the new mechanism. Hence in this case, ti has no incentive to claim to be si . So suppose in the original mechanism, we had p̂i (ti ) > p̂i (si ). Then if we choose ε sufficiently small, the interim probabilities must still satisfy this condition. Hence the new mechanism must be incentive compatible, a contradiction. This result is the key to Theorem 1. This argument shows that if the principal learns only the event of the mechanism partition containing t, then he wants to follow the mechanism — commitment would not be needed. To complete the proof, we need to show that we can construct an equilibrium of the game without commitment where this is all the useful information the principal receives. This will imply that the principal is sequentially rational, at least on the equilibrium path. As we explain below, we construct an equilibrium where the principal receives at least this information and does not receive more information that he can use. The result above also tells us a great deal about the structure of the optimal mechanism. In particular, for any event T̂i in the partition of Ti , let v̄i (T̂i ) = E[vi (ti ) | ti ∈ T̂i ]. Q Given any event T̂ = j T̂j , the optimal mechanism must give the good to some agent i ∈ arg maxj v̄j (T̂j ). Ignoring ties to keep the discussion simple, we see that this pins down the optimal mechanism given the partition T and that the optimal mechanism is 23 deterministic.12 In particular, we can take the optimal mechanism to be measurable with respect to these beliefs. This result also implies that the mechanism is robustly incentive compatible. Because the mechanism allocates the good to that agent i with the highest estimated vi , incentive compatibility implies that every agent i must maximize v̄i (T̂i ) by her honest report. Hence whatever i thinks the other agents are doing, she could not do better by misreporting. Intuitively, this is also why our auxiliary game construction works. In the mechanism, each agent i’s report is chosen to try to persuade the principal that vi is large. Strategically, then, we can analyze i’s incentives by studying a persuasion game with only agent i where this is i’s only objective. We conclude this proof sketch by explaining how we use the auxiliary games to construct equilibrium strategies which obtain the same outcome in the game without commitment as in the optimal mechanism. This construction has four steps. First, we consider equilibria in what we call the restricted auxiliary game for i. In this game, type ti is restricted to sending evidence she has which is maximal evidence for some si which is in the same event of the mechanism partition as ti . That is, if si , ti ∈ Tiα for some α, then in the restricted auxiliary game for i, ti can send evidence Mi (si ) if Mi (si ) ∈ Ei (ti ). Agent i’s objective in this game is simply to make the principal believe that vi (ti ) is as high as possible. In other words, we let the principal’s action in this game be the choice of a number x where his payoff is −(x − vi (ti ))2 and where agent i’s utility is x. Thus x is, in effect, the principal’s estimate of vi (ti ). In this game, the principal must learn at least that ti ∈ Tiα since, by construction, the only messages available to ti reveal that ti ∈ Tiα . Second, we show the claim mentioned above: that given this information, the principal cannot do better than to implement the outcome of the optimal mechanism. More specifically, for each i, fix an equilibrium of the restricted auxiliary game. For any ti ∈ Ti , the equilibrium strategy followed by ti in the restricted game for i determines the principal’s equilibrium expected value of vi which we denote vi∗ (ti ).13 For a profile of types t, let v ∗ (t) = (v1∗ (t1 ), . . . , vI∗ (tI )). Then the second step is to show that given type profile t, it is optimal for the principal to follow the allocation prescribed by the optimal mechanism given t when his expectation of the vi ’s is given by v ∗ (t). In this sense, the equilibrium does not give him information which he can use to improve on the 12 The result also makes clear that even when there are ties, we can change the mechanism to be deterministic without affecting the principal’s payoff. In the proof, we show how this can be done without affecting the agents’ incentives. 13 Agent i’s equilibrium strategy in the restricted game could be mixed, but optimality then requires that the principal’s belief in response to every pure strategy in the support must be the same. Hence the principal’s belief in response to the equilibrium strategy of any type is unambiguously defined. 24 mechanism. To see why this holds, suppose there is some t0 for which the principal strictly prefers a different allocation of the good given beliefs v ∗ (t0 ) than what the mechanism calls for. We derive a contradiction similarly to the way we showed the sequential rationality property Q above. That is, let T̂ = i T̂i be the event of the mechanism partition containing t0 . Consider the following alternative mechanism. If t ∈ / T̂ , the new mechanism is the same as the original one. If t ∈ T̂ , then the new mechanism chooses the same allocation as the old one with probability 1 − ε. With probability ε, the new mechanism chooses any optimal allocation given the beliefs v ∗ (t) which is measurable with respect to these beliefs. By assumption, this new mechanism yields the principal a strictly higher payoff for any ε > 0. Hence this mechanism cannot be incentive compatible. But the new mechanism is incentive compatible. Fix any si , ti ∈ Ti with Mi (si ) ∈ Ei (ti ). If si is a type that ti strictly prefers in the original mechanism not to imitate, then for ε sufficiently small, ti prefers not to imitate si in the new mechanism. Hence if there is a violation of incentive compatibility, si and ti must be in the same event of the mechanism partition and hence must both be in T̂i , the only event of the mechanism partition where the new mechanism differs from the old one. But if ti prefers imitating si , it must be because si gets the good strictly more often than ti . Given our construction, this must mean that vi∗ (si ) > vi∗ (ti ). But given that these beliefs come from an equilibrium of the restricted auxiliary game, this means that ti cannot play si ’s equilibrium strategy and hence Mi (si ) ∈ / Ei (ti ). Hence this does not give a violation of incentive compatibility. Before turning to the third step, we show a useful implication. Fix events Tiα and from the partition Ti where α > β. (Recall that Tiα is the set of ti who receive the good with probability α in the optimal mechanism.) Then vi∗ (ti ) ≥ vi∗ (t0i ) for all ti ∈ Tiα and t0i ∈ Tiβ . To see this, note that the argument above shows that the probability that vi∗ (ti ) = maxj vj∗ (tj ) must be at least α, while the probability that vi∗ (t0i ) > maxj6=i vj∗ (tj ) cannot exceed β. Since α > β, this requires vi∗ (ti ) ≥ vi∗ (t0i ). Tiβ The third step is to use this fact to show that by appropriately specifying beliefs in response to evidence which is not maximal for any type, we obtain an equilibrium of the unrestricted auxiliary game for i, where the payoffs are the same as in the restricted game but where ti can send any evidence she possesses. Specifically, in the unrestricted auxiliary game for i, if i presents evidence ei which is not maximal for any type, then the principal puts probability 1 on any type ti consistent with ei which minimizes vi (ti ). To see why this works, consider a deviation by type ti in the unrestricted auxiliary game to a message that was not available to her in the restricted game. By construction, this is either evidence which is not maximal for any type or is maximal evidence but for a type in a different event of the mechanism partition. First, consider the case of a deviation to evidence which is not maximal for any type. Since ti could present Mi (ti ) 25 in the restricted game, the belief in response to her equilibrium strategy in the restricted game must be at least as large as the belief in response to Mi (ti ). The belief in response to Mi (ti ) must be at least vi (si ) for any si which minimizes vi (si ) subject to Mi (ti ) ∈ Ei (si ). Since Mi (ti ) rules out as many si ’s as ti can rule out, this must be weakly larger than the belief in response to any evidence ti can send that is not maximal for any type. Hence ti would not deviate to such evidence. So consider a deviation to evidence which is maximal for some other type but where this type is in a different event of the mechanism partition. That is, if ti is in partition event Tiα , then the deviation is to Mi (si ) for si in some other partition event Tiβ for β 6= α. By incentive compatibility, the fact that ti can send si ’s maximal evidence implies β < α. Given the result above, then, the belief ti generates in equilibrium must be greater than or equal to the belief si generates, so ti has no incentive to deviate to si ’s equilibrium strategy. Hence ti has no incentive to deviate to any evidence si could feasibly send, including Mi (si ). In the fourth and final step, we construct an equilibrium of the game without commitment by taking the strategies of the agents and the beliefs of the principal to be the same as in the unrestricted auxiliary games and the strategy of the principal to be optimal given his beliefs and measurable with respect to them. Since the strategy of the principal is optimal given his beliefs and since his beliefs are consistent with the strategies of the agents, this is an equilibrium of the game without commitment as long as no agent wishes to deviate. But it is easy to see that an agent can gain only by deviating to a strategy which yields a larger belief about vi . Since the strategy of agent i comes from a game where this is precisely what she maximizes, no such deviation is possible. Finally, as shown in the second step, even though the principal obtains more information in the game than the partition T , following what the mechanism prescribes and ignoring this extra information is optimal for the principal. Hence the payoff of the principal in this equilibrium is exactly his payoff in the optimal mechanism. 5 Connection to the Literature In this section, we discuss in more detail how our results relate to the literature. As mentioned in Section 1, there are earlier results for the one–agent setting showing that commitment is not necessary to the principal. Our result extends these in several ways. First, we consider multiple agents. Second, because we have multiple agents, we can consider robust incentive compatibility — that is, the question of robustness with respect to agents’ beliefs about other agents, an issue absent in the one–agent setting. Third, our characterization of these equilibrium strategies is novel. 26 Even when we restrict our analysis to the case of I = 1 so that we also only have one agent, our results are not nested by the previous literature. Most significantly, all previous results in the literature assume that the one agent’s preferences are independent of her type, while we allow simple type–dependence. So to clarify the relationship to the literature, for the remainder of this discussion, we discuss the one–agent case, so t refers to the type of the single agent, T her set of types, and u her utility function. The first papers to show that a result of the form that commitment is not necessary in a mechanism design problem with evidence were Glazer and Rubinstein (2004, 2006). They used weaker assumptions on evidence as they did not assume normality, but they assumed that the principal only had two actions available and the agent’s preference was type–independent. By contrast, as noted, in the one–agent, two–action case, our assumption of simple type–dependence imposes no restrictions on the preferences of the agent. Sher (2011) generalizes the Glazer–Rubinstein result via a concavity assumption. He assumes type–independent utility for the agent and that the principal’s utility can be written as a concave function of the agent’s utility. In the one–agent version of our model, the principal’s utility function is v(a, t) = u0 (a) + v(t)u(a). Since this depends on a directly, not just through u(a), it is not nested by (nor does it nest) Sher’s assumptions, even in the type–independent version of our model. Hart, Kremer, and Perry (2016, forthcoming) give versions of the Glazer–Rubinstein result which, like our result, assume normality of evidence. Again, they assume type– independent utility for the agent. Hart, Kremer, and Perry (forthcoming) assume that the principal cannot randomize. In addition, they weaken Sher’s concavity assumption and instead assume that for each t ∈ T , the utility function of the principal over A can P be written as v(a, t) = ϕt (u(a)) where given any µ ∈ ∆(T ), t µ(t)ϕt is single–peaked or, equivalently, strictly quasi–concave. Because we allow the principal’s utility to depend on a directly, not only through u(a), our model violates this assumption, even in the type–independent version of our model. Also, we prove that the principal does not need to randomize, while Hart, Kremer, and Perry assume he cannot. Hart, Kremer, and Perry (2016) allow the principal to randomize. Their main assumption is called PUB or Principal’s Uniform Best. This states that if we fix any indifference curve for the agent, then there is a point on that indifference curve which is best for the principal independently of t. In the one–agent version of our model, we have v(a, t) = u0 (a) + u(a)v(t). Hence holding fixed the agent’s utility, for any t, the best P lottery over a is any p on the indifference curve which maximizes a p(a)u0 (a). Thus except for the type–dependence we allow, our assumptions are nested in their one–agent case. An additional contribution of Hart, Kremer, and Perry (forthcoming) is that they 27 identify a refinement of equilibrium in the disclosure game that corresponds to the principal’s best equilibrium. Our result showing that the principal’s best equilibrium in the game without commitment can be found using I separate one–agent disclosure games is analogous in that it also provides a means to understand or compute this equilibrium. We note that Hart, Kremer, and Perry’s approach is defined only for games with type–independent preferences (where we can restrict attention to messages that contain maximal evidence) and so does not appear to carry over to the more general case we consider. 28 Appendix A Example In this subsection, we give an example to show that in games with evidence, even with independent private values, the requirements of robust incentive compatibility, dominant strategy incentive compatibility, and ex post incentive compatibility differ. The reason that ex post incentive compatibility and robust incentive compatibility are not equivalent is that robust incentive compatibility requires truth–telling to be optimal even when other agents deviate from truth–telling with maximal evidence. In the absence of evidence, the fact that we have independent private values means that agent i is unaffected by whether the claims of other agents are true or not. Hence these two notions would be the same in that case. But with evidence, we can have reports by the other agents that would be impossible under truth–telling with maximal evidence. The reason that dominant strategy incentive compatibility is not the same as robust incentive compatibility is that dominant strategy incentive compatibility only requires that truth telling and maximal evidence be a best reply to any strategy function by the other agents. In the absence of evidence, the other agents could be playing constant strategies, implying that truth telling and maximal evidence must be a best reply to any reports by the other agents. In mechanisms with evidence, however, constant strategies may not be possible. To see both points in a simple example, suppose I = 2 and Ti = {αi , βi }, i = 1, 2. Suppose Ei (αi ) = {{αi }} and Ei (βi ) = {{βi }}, i = 1, 2. Suppose the principal has just two actions, denoted 0 and 1. Assume u1 (a) = a and u2 (a) = 0 for all a.14 Say that agent i reports consistently if she reports (αi , {αi }) or (βi , {βi }) and reports inconsistently otherwise. Note that all three versions of incentive compatibility say that consistent reports are optimal and differ only in the circumstances under which consistent reports are required to be optimal. Assume that the prior probability that t2 = β2 is strictly below 1/2. Consider the mechanism where the principal chooses a = 1 if one of the following is true. First, 1’s report is consistent and 2’s report (consistent or not) has evidence {α2 }. Second, both reports are consistent and 2’s evidence presentation is {β2 }. Third, both reports are inconsistent and 2’s evidence presentation is {β2 }. If the reports do not satisfy one of these three conditions, then the principal chooses a = 0. Because 2 is indifferent 14 It is not difficult to give a symmetric but more complex example where neither agent is completely indifferent. 29 between a = 0 and a = 1, the mechanism satisfies robust incentive compatibility for him. However, it is not robustly incentive compatible for 1. To see this, simply note that 1’s best response to a report by 2 of (α2 , {β2 }) is to be inconsistent. On the other hand, this mechanism is both ex post incentive compatible and dominant strategy incentive compatible for 1. To see that it is ex post incentive compatible, note that if 2 is consistent, then 1’s best response is always to be consistent, regardless of the type profile. To see that it is dominant strategy incentive compatible, note that for any feasible strategy for 2, 1’s expected payoff to any consistent report is at least the probability that t2 = α2 , while the payoff to any inconsistent report is at most the probability that t2 = β2 . Since the former strictly exceeds the latter, reporting consistently is a dominant strategy. B Equilibrium Definition Our definition of perfect Bayesian equilibrium is identical to that of Fudenberg and Tirole (1991) but adapted to allow type–dependent sets of feasible actions. We say that (σ1 , . . . , σI , σP , µ) is a perfect Bayesian equilibrium if the following conditions hold. First, for every i and every ti ∈ Ti , σi (si , ei | ti ) > 0 implies (si , ei ) ∈ arg max 0 s0i ∈Ti ,ei ∈Ei (ti ) Qi (a | s0i , e0i , σ−i , σP )ui (a, ti ). X a∈A Second, for every (s, e) ∈ T × E, σP (a | s, e) > 0 implies a ∈ arg max 0 a ∈A X µ(t | s, e)v(a0 , t). t∈T Third, for every (s, e), µ(· | s, e) respects independence across agents. That is, i’s report (si , ei ) only affects the principal’s beliefs about ti and his beliefs about ti and tj respect independence for all i 6= j. Formally, we have functions µi : Ti × Ei → ∆(Ti ) such that for all t ∈ T and all (s, e) ∈ T × E, µ(t | s, e) = Y µi (ti | si , ei ). i Fourth, for all (s, e), µ(· | s, e) respects feasibility. That is, the principal’s beliefs must put zero probability on any type which is infeasible given (s, e). Formally, for every ti ∈ Ti and (si , ei ) ∈ Ti × Ei , we have µi (ti | si , ei ) = 0 if ei ∈ / Ei (ti ). Finally, the principal’s beliefs are consistent with Bayes’ rule whenever possible in the sense that for every (si , ei ) ∈ Ti × Ei such that there exists ti with σi (si , ei | ti ) > 0, we 30 have σi (si , ei | ti )ρi (ti ) . 0 0 t0i ∈Ti σi (si , ei | ti )ρi (ti ) µi (ti | si , ei ) = P (Recall that ρi is the principal’s prior over ti .) C Proof of Theorem 1 Throughout the Appendix, we assume that each ui satisfies simple type dependence and consequently write the agents’ utility functions as ui (a), if ti ∈ Ti+ ; −ui (a), if ti ∈ Ti− , ® ui (ai , ti ) = where Ti+ ∪ Ti− = Ti . We write the principal’s utility function as v(a, t) = u0 (a) + X ui (a)vi (ti ) = I X i ui (a)vi (ti ), i=0 where we rewrite as explained in Section 2.1. For each i = 1, . . . , I, let Ri ≡ Ti × Ei . Given a mechanism P and ri ∈ Ri , let ûi (ri ; P ) = Et−i X P (a | ri , t−i , M−i (t−i ))ui (a). a Recall that the Revelation Principle for this class of problems says that we can restrict attention to equilibria where ti honestly states her type and provides maximal evidence. Hence ûi (ri ; P ) will be the expected utility of ti from report ri in the mechanism. Fix an optimal mechanism P . For each α ∈ R, let Riα = {ri ∈ Ri | ûi (ri ; P ) = α}. Finiteness of Ti implies that Ei is finite. Given this, “most” Riα will be empty. When we refer to one of these sets below, we often take as given that it is nonempty. Note that the nonempty Riα ’s form a partition of Ri . In what follows, we refer to this partition as the mechanism partition for i, denoted {Riα } and refer to the product partition of R formed Q Q by the cells i Riαi simply as the mechanism partition, denoted { i Riαi }. It will also be useful to define Tiα = {ti ∈ Ti | ûi (ti , Mi (ti ); P ) = α} = {ti ∈ Ti | (ti , Mi (ti )) ∈ Riα }. Note that we could have some values of α with ûi (ti , ei ; P ) = α only for ei 6= Mi (ti ) in which case Riα 6= ∅ but Tiα = ∅. 31 Lemma 2. Fix (si , ei ) ∈ Riα . For any ti ∈ Ti+ , if (ti , Mi (ti )) ∈ Riβ with α > β, then we have ei ∈ / Ei (ti ). For any ti ∈ Ti− , if (ti , Mi (ti )) ∈ Riβ with α < β, we have ei ∈ / Ei (ti ). Proof. Follows from incentive compatibility. Lemma 3. Without loss of generality, we can assume the optimal mechanism P has the property that for all i and all α, if Riα 6= ∅, then Tiα 6= ∅. Proof. Suppose the optimal mechanism does not have this property. Fix any Riα 6= ∅ such that Tiα = ∅. By the Revelation Principle, the ri ∈ Riα are not used in equilibrium since Tiα = ∅ implies they are all of the form (ti , ei ) where ei 6= Mi (ti ). Intuitively, then, we can change the outcome from these off equilibrium reports so that they remain off equilibrium without changing the principal’s payoff. More specifically, choose any β such that Tiβ 6= ∅ and there does not exist γ ∈ (α, β) ∪ (β, α) with Tiγ 6= ∅. It is easy to see that such a β must exist. First, there must be some β with Tiβ 6= ∅ since the nonempty Tiβ sets partition Ti . So we can simply choose the smallest β > α such that Tiβ 6= ∅ if such a β exists and the largest β < α with Tiβ 6= ∅ otherwise. Fix any (t̂i , êi ) ∈ Riβ and consider the mechanism P ∗ given by ∗ P (· | t, e) = ® P (· | t, e), if (ti , ei ) ∈ / Riα ; P (· | t̂i , êi , t−i , e−i ), otherwise. Note that we have only changed the mechanism for reports by i which are in Riα and hence are not of the form (ti , Mi (ti )). Hence the incentive compatibility of P for j 6= i implies incentive compatibility of P ∗ for j 6= i. Similarly, the principal’s payoff from P ∗ is the same as his payoff from P . To see that P ∗ is incentive compatible for i, fix any ti and any (si , ei ) such that ei ∈ Ei (ti ). Clearly, ûi (ti , Mi (ti ); P ∗ ) = ûi (ti , Mi (ti ); P ). If (si , ei ) ∈ / Riα , then ûi (si , ei ; P ∗ ) = ûi (si , ei ; P ), so the fact that ti prefers reporting (ti , Mi (ti )) to reporting (si , ei ) in P implies the same is true for P ∗ . So suppose (si , ei ) ∈ Riα . In this case, ûi (si , ei ; P ∗ ) = ûi (t̂i , êi ; P ) = β, while ûi (si , ei ; P ) = α. For concreteness, suppose β > α (the case where β < α is analogous). From the way we chose β, we cannot have α ≤ ûi (ti , Mi (ti ); P ) < β. So either ûi (ti , Mi (ti ); P ) < α = ûi (si , ei ; P ) < β = ûi (si , ei ; P ∗ ) or ûi (si , ei ; P ) = α < β = ûi (si , ei ; P ∗ ) ≤ ûi (ti , Mi (ti ); P ). 32 Recall that ei ∈ Ei (ti ) by assumption. Hence in the former case, incentive compatibility implies ti ∈ Ti− and therefore ti prefers reporting (ti , Mi (ti )) to reporting (si , ei ) in P ∗ . In the latter case, incentive compatibility implies ti ∈ Ti+ and therefore ti (weakly) prefers reporting (ti , Mi (ti )) to reporting (si , ei ) in P ∗ . Either way, P ∗ is incentive compatible and is also an optimal mechanism. By repeating this argument, we construct an optimal mechanism with the desired property. Lemma 4. Without loss of generality, we can take the mechanism P to be measurable with respect to the mechanism partition for each i, {Riα }, in the sense that if (si , ei ), (ti , e0i ) ∈ Riα , then P (· | si , ei , t−i , e−i ) = P (· | ti , e0i , t−i , e−i ) for all (t−i , e−i ) ∈ R−i . Q Hence we can take P to be measurable with respect to the mechanism partition { i Riαi } Q in the sense that P (· | s, e) = P (· | s0 , e0 ) if (s, e), (s0 , e0 ) ∈ i Riαi . Proof. Fix an optimal mechanism P which is not measurable in this sense. We construct an alternative mechanism which is measurable, is incentive compatible, and has the same payoff for the principal as P . Fix any i and any α such that Riα 6= ∅. By Lemma 3, Tiα 6= ∅. Define a mechanism P ∗ by P ∗ (· | si , ei , t−i , e−i ) = P (· | si , ei , t−i , e−i ) if (si , ei ) ∈ / Riα and otherwise P ∗ (a | si , ei , t−i , e−i ) = Eti (P (a | ti , Mi (ti ), t−i , e−i ) | (ti , Mi (ti )) ∈ Riα ), for all a ∈ A and all (t−i , e−i ) ∈ R−i . For any agent j 6= i, the expected payoff under the mechanism, both from honest reporting with maximal evidence and from any deviation, is unaffected. Hence we have incentive compatibility of P ∗ for all j 6= i from incentive compatibility of P . For agent i for (si , ei ) ∈ Riα , we have ∗ ûi (si , ei ; P ) = Et−i ñ X ô ∗ P (a | si , ei , t−i , M−i (t−i ))ui (a) a = Et−i ñ X ô Eti [P (a | ti , Mi (ti ), t−i , M−i (t−i ))) | (ti , Mi (ti )) ∈ Riα ]ui (a) a ñ = Eti Et−i Ç X å P (a | ti , Mi (ti ), t−i , M−i (t−i ))ui (a) | (ti , Mi (ti )) ∈ ô Riα ) a = Eti [α | (ti , Mi (ti )) ∈ Riα ] = α = ûi (si , ei ; P ). So every type of agent i receives the same expected payoff under the new mechanism for every report as she did in the original mechanism. Hence the incentive compatibility of P implies incentive compatibility of P ∗ . It is easy to see that P ∗ gives the principal the 33 same expected payoff as P . Hence P ∗ is an optimal mechanism as well. Iterating this construction, we construct an optimal mechanism which is measurable with respect to the mechanism partition. To see that this implies measurability with respect to the mechanism partition, apply the argument above iteratively over i. In light of Lemma 4, we henceforth assume P is measurable with respect to the mechanism partition. Given any partition R of T × E, we say that a mechanism P̃ (not necessarily the optimal mechanism) is sequentially rational given R if the following is true. First, P̃ is measurable with respect to R. Second, for every event E of the partition, for every (t, M (t)) ∈ E, P̃ (· | t, M (t)) is some p ∈ ∆(A) which maximizes X p(a)Et [v(a, t) | (t, M (t)) ∈ E] . a In other words, the mechanism is optimal for the principal given the information contained in the partition R. Lemma 5. P is sequentially rational given the mechanism partition. Proof. Suppose not. Fix any (t, M (t)) and let P (· | t, M (t)) = p̄. Let R̂ = the event of the mechanism partition containing (t, M (t)) and suppose X p̃(a)E[v(a, t) | (t, M (t)) ∈ R̂] > a X Q j α Rj j denote p̄(a)E[v(a, t) | (t, M (t)) ∈ R̂]. a We construct a new mechanism P ∗ as follows. For any (t, e) ∈ / R̂, P ∗ (· | t, e) = P (· | t, e). For (t, e) ∈ R̂, P ∗ (· | t, e) = (1 − ε)p̄ + εp̃ for some small ε > 0. Clearly, for any ε ∈ (0, 1), P ∗ yields a strictly higher payoff for the principal than P . We now show that that for ε sufficiently small, P ∗ is incentive compatible. To see this, fix any (ti , Mi (ti )) and any (si , ei ) with ei ∈ Ei (ti ). Suppose that under P , ti strictly preferred reporting (ti , Mi (ti )) to reporting (si , ei ). Then for ε sufficiently small, this must still be true. So suppose ti was indifferent between (ti , Mi (ti )) and (si , ei ). That is, ûi (ti , Mi (ti ); P ) = ûi (si , ei ; P ). But then by measurability of the new mechanism P ∗ with respect to the mechanism partition of the original mechanism P , ti must still be indifferent between these reports in the new mechanism, so it remains incentive compatible. This contradicts the optimality of P . 34 In what follows, for any α such that Tiα 6= ∅, let v̄i (α) = E[vi (ti ) | (ti , Mi (ti )) ∈ Riα ]. The following lemma will be useful. Lemma 6. Let U = {(ū0 , ū1 , . . . , ūI ) ∈ RI+1 | ∃p ∈ ∆(A) with X p(a)ui (a) = ūi , ∀i}. a Given any belief of the principal over each Ti , let v̂i denote the expectation of vi (ti ) under the belief over Ti and let v̂ = (1, v̂1 , . . . , v̂I ). Let U ∗ (v̂) denote the set of u ∈ U maximizing v̂ · u. Fix any i, v, and v 0 such that vi > vi0 and vj0 = vj for j 6= i. Fix any u ∈ U ∗ (v) and any u0 ∈ U ∗ (v 0 ). Then ui ≥ u0i . Proof. This result is standard, but we include a proof for completeness. Clearly, we must have v · u ≥ v · u0 v 0 · u0 ≥ v 0 · u implying (v − v 0 ) · (u − u0 ) ≥ 0. But this is (vi − vi0 )(ui − u0i ). Since vi > vi0 , we must have ui ≥ u0i . Lemma 7. For all α > β, we have v̄i (α) ≥ v̄i (β). In other words, (weakly) “more valuable” sets for the principal receive higher utilities. α Proof. Fix α > β. Since α > β, there must exist events Tj j for j 6= i such that ūi (α) ≡ X P (a | t, M (t))ui (a) > a X P (a | t0i , Mi (t0i ), t−i , M−i (t−i ))ui (a) ≡ ūi (β), a α where ti is an arbitrary element of Tiα , t0i is an arbitrary element of Tiβ , and tj ∈ Tj j Q α for each j 6= i. Let T̂−i = j6=i Tj j . Let v̄j = v̄j (αj ) for j 6= i, let v̄ α denote the vector (v̄i (α), v̄−i ), and define v̄ β analogously. For each j 6= i, let ūj (α) = X P (a | t, M (t))uj (a) a for any t ∈ Tiα × T̂−i and define ūj (β) analogously using any t ∈ Tiβ × T̂−i . Finally, let ūα denote the vector (ūi (α), ū−i (α)) and define ūβ analogously. From Lemma 5, we know that ūα maximizes v̄ α · u over u that can be generated by some p ∈ ∆(A). Similarly, ūβ maximizes v̄ β · u. Since ūαi > ūβi , Lemma 6 implies that v̄i (α) ≥ v̄i (β). 35 Lemma 8. Without loss of generality, we can take the mechanism P to be measurable with respect to v̄i in the sense that if v̄i (α) = v̄i (β) and (si , ei ) ∈ Riα , (s0i , e0i ) ∈ Riβ , then P (· | si , ei , t−i , e−i ) = P (· | s0i , e0i , t−i , e−i ) for all (t−i , e−i ) ∈ R−i . In other words, we can take the mechanism to have the property that α 6= β implies v̄i (α) 6= v̄i (β). Proof. Fix an optimal mechanism p which does not satisfy this property. Fix the relevant α and let A = {β | Riβ 6= ∅ and v̄i (β) = v̄i (α)}. By assumption, there exists at least one β 6= α with β ∈ A. By Lemma 7, we have that α0 > β 0 implies v̄i (α0 ) ≥ v̄i (β 0 ). Hence for any γ ∈ / A, either γ is strictly smaller than every β ∈ A or γ is strictly larger than every β ∈ A. Define a new mechanism P ∗ by setting P ∗ (· | si , ei , t−i , e−i ) = P (· | si , ei , t−i , e−i ) if (si , ei ) ∈ / ∪β∈A Riβ and otherwise, P ∗ (a | si , ei , t−i , e−i ) = Eti [P (a | ti , Mi (ti ), t−i , e−i ) | (ti , Mi (ti )) ∈ ∪β∈A Riβ ], for all a ∈ A and all (t−i , e−i ) ∈ R−i . We now show that P ∗ is incentive compatible and gives the principal the same payoff as P , establishing the claim. To see that P ∗ is incentive compatible, note that the interim payoff to tj for any feasible (sj , ej ) for j 6= i is unaffected by this change. Hence we have incentive compatibility for any j 6= i. So fix any ti and any (si , ei ) 6= (ti , Mi (ti )) with ei ∈ Ei (ti ). If neither (ti , Mi (ti )) nor (si , ei ) is contained in ∪β∈A Riβ , then the response to either report is unaffected, so incentive compatibility of P implies that ti prefers reporting (ti , Mi (ti )) to reporting (si , ei ). If both are contained in ∪β∈A Riβ , then the expected payoff under P ∗ is the same in response to either report, so this incentive compatibility constraint holds. So suppose (ti , Mi (ti )) ∈ ∪β∈A Riβ and (si , ei ) is not. Then (si , ei ) ∈ Riγ for some γ that is either below every β ∈ A or above every β ∈ A. If γ is below every β ∈ A, then ûi (ti , Mi (ti ); P ∗ ) > ûi (s, ei ; P ∗ ) and ûi (t, Mi (ti ); P ) > ûi (si , ei ; P ). The latter inequality and the incentive compatibility of P implies ti ∈ Ti+ , so that the former inequality implies ti prefers reporting (ti , Mi (ti )) to reporting (si , ei ). Similarly, If γ is above every β ∈ A, then both inequalities are strictly reversed, implying ti ∈ Ti− and that ti prefers reporting (ti , Mi (ti )) to reporting (si , ei ). A similar argument holds for the case where (si , ei ) ∈ ∪β∈A Riβ and (ti , Mi (ti )) is not. Hence P ∗ is incentive compatible. To see that P ∗ yields the same expected payoff to the principal as P , recall that by the Revelation Principle, the specification of P ∗ on messages other than those of the α form (t, M (t)) are irrelevant to the principal’s payoffs. Fix any t−i and let v̄j = v̄j (Rj j ) α for the αj with (tj , Mj (tj )) ∈ Rj j . For any (ti , Mi (ti )) ∈ Riβ , β ∈ A, let pβ = P (· | 36 ti , Mi (ti ), t−i , M−i (t−i )). By Lemma 5, P is sequentially rational for the principal. Hence for every β ∈ A, pβ maximizes X β p (a)[u i (a)v̄i (β) + a X uj (a)v̄j ]. j6=i Since v̄i (α) = v̄i (β) for all β ∈ A, the only way we can have pα 6= pβ is if the principal is indifferent between pα and pβ . Obviously, then, the fact that P ∗ differs from P in such situations has no payoff consequences. Hence P ∗ yields the principal the same payoff as P. In light of this result, we henceforth assume P is measurable with respect to v̄ in the sense defined above. Lemma 9. P is robustly incentive compatible. Proof. Suppose not. Then either there exists ti ∈ Ti+ , ei ∈ Ei (ti ), si ∈ Ti , t̄−i ∈ T−i , and ē−i ∈ E−i such that X P (a | ti , Mi (ti ), t̄−i , ē−i )ui (a) < a X P (a | si , ei , t̄−i , ē−i )ui (a) (2) a or some ti ∈ Ti− , ei ∈ Ei (ti ), si ∈ Ti , t̄−i ∈ T−i , and ē−i ∈ E−i with the opposite strict inequality. Since these cases are entirely symmetric, we consider only the former, so fix ti , si , ei , t̄−i , and ē−i satisfying equation (2). Assume (si , ei ) ∈ Riα and (ti , Mi (ti )) ∈ Riβ . By measurability with respect to the mechanism partition for i, we know that α 6= β. By Lemma 8, v̄i (α) 6= v̄i (β). By incentive compatibility and the fact that ti ∈ Ti+ , we have α < β and hence by Lemma 7, v̄i (α) < v̄i (β). As in the proof of Lemma 7, for each j, including j = i, let ūαj = X P (a | t0i , Mi (t0i ), t0−i , M−i (t0−i ))uj (a) a α for any (t0i , Mi (t0i ) ∈ Riα and any (t0−i , M−i (t0−i )) ∈ j6=i Rj j . (By Lemma 3, such t0i and t0−i must exist.) Similarly, define ūβj using some t0i with (t0i , Mi (t0i )) ∈ Riβ . Finally, let v̄ α denote the vector (v̄i (α), v̄−i ) and define v̄ β analogously. By Lemma 6, v̄i (β) > v̄i (α) implies ūβi ≥ ūαi . But ūβi is the left–hand side of equation (2) and ūαi is the right–hand side, a contradiction. Q Lemma 10. There exists an optimal mechanism P which is robustly incentive compatible and is deterministic in the sense that P (a | t, M (t)) ∈ {0, 1} for all a ∈ A and t ∈ T . Proof. Given an arbitrary mechanism P̃ , let Πi (P̃ ) denote the mechanism partition of Ri induced by P̃ . Given ri ∈ Ri , let πi (ri | P̃ ) denote the event of Πi (P̃ ) containing ri . By 37 Lemmas 4 and 8, we know that there exists an optimal mechanism P which is strictly measurable with respect to Πi (P ) for all i in the sense that if Eti [vi (ti ) | (ti , Mi (ti )) ∈ πi (ri | P )] = Eti [vi (ti ) | (ti , Mi (ti )) ∈ πi (ri0 | P )], then P (· | ri , r−i ) = P (· | ri0 , r−i ), ∀r−i . In other words, P is measurable with respect to the beliefs about the vi ’s induced by the mechanism partition. Let P denote the set of optimal mechanisms P̃ such that P̃ is strictly measurable with respect to each Πi (P̃ ) in this sense. Finally, let P denote any mechanism in P which is minimal in the sense that there is no P 0 ∈ P which is strictly measurable with respect to each Πi (P 0 ) and for which Πi (P 0 ) has weakly fewer elements than Πi (P ) for all i, strictly fewer for some i. By finiteness of T , such a P must exist. If P is deterministic, we are done, so suppose it is not. In light of Lemma 5, this can only occur when the principal is indifferent ex post. In other words, if P (a∗ | r) > 0 for some a∗ ∈ A and some r ∈ R, then X P (a | r)Et [v(a, t) | (t, M (t)) ∈ π(r | P )] = Et [v(a∗ , t) | (t, M (t)) ∈ π(r | P )]. a Hence there must exist a deterministic mechanism, say P ∗ , which is strictly measurable with respect to each Πi (P ) which yields the same expected payoff for the principal. We now show that P ∗ is incentive compatible and strictly measurable with respect to each Πi (P ∗ ). To show incentive compatibility, suppose P ∗ is not incentive compatible. Then either there exists ti ∈ Ti+ , si ∈ Ti , and ei ∈ Ei (ti ) such that ûi (ti , Mi (ti ); P ∗ ) < ûi (si , ei ; P ∗ ) or ti ∈ Ti− , si ∈ Ti , and ei ∈ Ei (ti ) with the reverse strict inequality. Because P ∗ is measurable with respect to each Πi (P ), this implies ûi (ti , Mi (ti ); P ) 6= ûi (si , ei ; P ). Hence incentive compatibility of P implies ûi (ti , Mi (ti ); P ) > ûi (si , ei ; P ) if ti ∈ Ti+ and the reverse strict inequality if ti ∈ Ti− . Consider the mechanism P λ ≡ λP + (1 − λ)P ∗ . For every λ ∈ [0, 1], this mechanism has the same payoff for the principal as P . Clearly, for λ sufficiently close to 1, P λ is incentive compatible. Let λ∗ be the smallest λ such that P λ is incentive compatible. It is easy to see that such a λ∗ exists and that it satisfies λ∗ ûi (ti , Mi (ti ); P ) + (1 − λ∗ )ûi (ti , Mi (ti ); P ∗ ) = λ∗ ûi (si , ei ; P ) + (1 − λ∗ )ûi (si , ei ; P ∗ ) for some ti , si ∈ Ti and ei ∈ Ei (ti ) such that πi (ti , Mi (ti ) | P ) 6= πi (si , ei | P ). Hence ∗ for every i, Πi (P λ ) either equals or coarsens Πi (P ), coarsening for some i. By the 38 ∗ same reasoning as Lemmas 4 and 8, there exists another mechanism P ∗∗ with Πi (P λ ) = Πi (P ∗∗ ) for every i which is strictly measurable with respect to each Πi (P ∗∗ ) and yields ∗ the principal the same expected payoff as P λ . This contradicts the minimality of P . Hence P ∗ is incentive compatible. To see that P ∗ is strictly measurable with respect to each Πi (P ∗ ), suppose it is not. By construction, this means that each Πi (P ∗ ) is either equal to or a coarsening of Πi (P ) and is a coarsening for some i. Again following the same reasoning as Lemmas 4 and 8, there exists another mechanism P ∗∗ with Πi (P ∗ ) = Πi (P ∗∗ ) for every i which is strictly measurable with respect to each Πi (P ∗ ) and yields the principal the same expected payoff as P ∗ . This again contradicts the minimality of P . Finally, the same reasoning as in the proof of Lemma 9 shows that P ∗ is robustly incentive compatible. We now construct an equilibrium for the game which yields the same payoff for the principal as P . In particular, the strategy for agent i in this game is the same as i’s strategy in an equilibrium of the auxiliary game for i. Recall that the auxiliary game for i is a two–player game between i and the principal. i has a set of types Ti where the prior over Ti is the same as in the mechanism design problem. If i is type ti , then her set of feasible actions is Zi (ti ) ≡ Ti × Ei (ti ). The principal’s set of feasible actions is X = [minj mintj ∈Tj vj (tj ), maxj maxtj ∈Tj vj (tj )]. The game is sequential. First, agent i learns her type ti ∈ Ti . Then she chooses an action zi ∈ Zi (ti ). Next, the principal observes this action and chooses x ∈ X. If i’s type is ti and the principal chooses action x, then the principal’s payoff is −(x − vi (ti ))2 , while i’s payoff is ® x, if ti ∈ Ti+ ; −x, otherwise. Denote a strategy for i in this game by σi (· | ti ), a function from Ti to ∆(Zi (ti )). Let the principal’s belief be denoted qi (· | si , ei ) where this is a function from Ri = Ti × Ei to ∆(Ti ). Finally, the principal’s action in response to (si , ei ) is denoted Xi : Ri → X. We construct the relevant equilibrium of the auxiliary game for i by first considering what we will call the restricted auxiliary game. In the restricted game, type ti cannot choose any action in Ri but can only choose actions in Riα where α is the unique α such that ti ∈ Tiα . Fix any i and any perfect Bayesian equilibrium (σi∗ , Xi∗ , qi∗ ) of the restricted auxiliary game for i.15 Obviously, sequential rationality implies that Xi∗ (si , ei ) is the expectation 15 To see that such an equilibrium must exist, consider the game where i is restricted to putting probability ε > 0 on each of her pure strategies. By standard results, this game has a Nash equilibrium. 39 of vi (ti ) given the belief qi∗ or vi (ti )qi∗ (ti | si , ei ). X ti ∈Ti Let X̂i∗ (ti ) denote the action chosen by the principal when i is type ti . That is, X̂i∗ : Ti → X and is given by X̂i∗ (ti ) = Xi∗ (si , ei ), for some (si , ei ) ∈ supp(σi∗ (· | ti )). Note that the principal’s optimal action is always pure and that ti is never indifferent between two distinct actions by the principal. Hence every message in the support of ti ’s mixed strategy must lead to the same response by the principal. Thus the definition above is unambiguous. Clearly, for this to be an equilibrium, it must be true that if ti ∈ Ti+ , X̂i∗ (ti ) = max α Xi∗ (si , ei ), (si ,ei )∈Zi (ti )∩Ri while for ti ∈ Ti− , X̂i∗ (ti ) = min (si ,ei )∈Zi (ti )∩Riα Xi∗ (si , ei ). By construction, in any equilibrium of the restricted auxiliary game for i, the principal learns at least which event of the mechanism partition for i that ti lies in. This is true because if ti ∈ Tiα , then ti can only send (si , ei ) ∈ Riα . Hence observing (si , ei ) reveals the relevant value of α. Since the optimal mechanism is measurable with respect to the mechanism partition, this means that the principal must have enough information to carry out the optimal mechanism if this is the information the agents reveal to him. On the other hand, the principal may learn more than just that ti ∈ Tiα in the equilibrium. The following lemma shows that this extra information, if any, cannot be useful for the principal. Lemma 11. For each i, fix any equilibrium of the restricted auxiliary game for i and Q any αi such that Tiαi 6= ∅. Then for every t ∈ i Tiαi , P (· | t, M (t)) ∈ arg max X p∈∆(A) a p(a) X ui (a)X̂i∗ (ti ). i In other words, given the belief formed by the principal in the equilibria at profile t, it is optimal for him to follow the optimal mechanism. As ε ↓ 0 (taking subsequences as needed), these strategies converge to a Nash equilibrium of the restricted auxiliary game by upper hemicontinuity of the Nash equilibrium correspondence. These strategies and the limiting beliefs for the principal must also be a perfect Bayesian equilibrium since the principal’s limiting strategy must be optimal given his limiting belief. 40 Proof. Suppose not. For any v̂ = (v̂1 , . . . , v̂I ), let p̃(· | v̂) denote any p(·) ∈ ∆(A) which maximizes X X p(a) ui (a)v̂i . a i Clearly, there exists p(· | t) with X p(a | t) a X ui (a)X̂i∗ (ti ) > X P (a | ti , Mi (ti )) a i X ui (a)X̂i∗ (ti ) (3) i if and only if this holds for p(· | t) = p̃(· | X̂ ∗ (t)) where X̂ ∗ (t) = (X̂1∗ (t1 ), . . . , X̂I∗ (tI )). Given any (si , ei ) ∈ Riαi , let ( v̂i (si , ei ) = X̂i∗ (si ), if ei = Mi (si ); ∗ Xi (si , ei ), otherwise. Given (s, e) ∈ i Riαi , let v̂(s, e) = (v̂1 (s1 , e1 ), . . . , v̂I (sI , eI )). Fix a small ε > 0 and define a new mechanism P ∗ by Q ® ∗ P (· | s, e) = εp̃(· | v̂(s, e)) + (1 − ε)P (· | s, e), if (si , ei ) ∈ Riαi , ∀i; P (· | s, e), otherwise. In other words, for those types t ∈ i Tiαi , we assign a convex combination of the p̃ that will be optimal for the principal given the belief they will induce in the restricted auxiliary games and the original mechanism, assuming they report honestly and provide maximal evidence. If they deviate from maximal evidence, we assign a convex combination of the p̃ optimal for the principal given the induced beliefs in the restricted auxiliary games given those deviations and the original mechanism. Finally, for all other type profiles, the mechanism is unchanged. Q We now show that P ∗ is incentive compatible. So fix some ti ∈ Ti and (si , ei ) such that ei ∈ Ei (ti ). If ti strictly prefers reporting (ti , Mi (ti )) to reporting (si , ei ) under P , then for ε sufficiently small, ti still has this strict preference. So suppose that ti is indifferent between reporting (ti , Mi (ti )) to reporting (si , ei ) under P , so (ti , Mi (ti )) and (si , ei ) are in the same event of the mechanism partition for i. Clearly, if that event is not Riαi , then P ∗ still treats these reports the same way, so ti is still indifferent. So assume (ti , Mi (ti )), (si , ei ) ∈ Riα . The only way ti would not be indifferent under P is if Xi∗ (si , ei ) 6= X̂i∗ (ti ). If ti ∈ Ti+ , we know that X̂i∗ (ti ) ≥ Xi∗ (si , ei ). By Lemma 6, this implies ∗ Et−i X p̃(a ∗ | X̂i∗ (ti ), X̂−i (t−i ))ui (a) | t−i ∈ a ≥ Et−i Y α Tj j j6=i X p̃(a ∗ | Xi∗ (si , ei ), X̂−i (t−i ))ui (a) | t−i ∈ a Y j6=i 41 α Tj j . Since P is incentive compatible, this implies ti prefers reporting maximal evidence to reporting (si , ei ) in P ∗ . A similar argument applies to ti ∈ Ti− . Hence P ∗ is incentive compatible. But then we have a contradiction. By hypothesis, P is the optimal incentive compatible mechanism, so the fact that P ∗ is also incentive compatible implies that it cannot yield the principal a strictly higher payoff than P . Lemma 12. Fix α > β such that Tiα 6= ∅ and Tiβ 6= ∅ and any equilibrium of the restricted auxiliary game for i. Then for every ti ∈ Tiα and t0i ∈ Tiβ , we have X̂i∗ (ti ) ≥ X̂i∗ (t0i ). Proof. Since α > β, there exists t̂−i ∈ T−i such that uαi ≡ X P (a | ti , Mi (ti ), t̂−i , M−i (t̂−i ))ui (a) > a X P (a | t0i , Mi (t0i ), t̂−i , M−i (t̂−i ))ui (a) ≡ uβi . a For each j 6= i, let uαj = X P (a | ti , Mi (ti ), t̂−i , M−i (t̂−i ))uj (a), a and define uβj analogously. By Lemma 11, we know that pα ≡ P (· | ti , Mi (ti ), t̂−i , M−i (t̂−i )) maximizes over p(·) ∈ ∆(A) X p(a) ui (a)X̂i∗ (ti ) + a X uj (a)X̂j∗ (t̂j ) j6=i and pβ defined analogously maximizes the analog for t0i . Hence by Lemma 6, uαi > uβi implies X̂i∗ (ti ) ≥ X̂i∗ (t0i ). We now show how we can modify an equilibrium of the restricted auxiliary game for i to construct an equilibrium of the unrestricted auxiliary game for i with the same equilibrium path. So fix any equilibrium of the restricted auxiliary game for i, say (σi∗ , Xi∗ , qi∗ ). We first show that in the unrestricted auxiliary game for i, agent i does not have a profitable deviation from these strategies to any (si , ei ) which has positive probability in equilibrium (i.e., with σi∗ (si , ei | ti ) > 0 for some ti ). Fix any ti ∈ Tiα . Since these strategies are an equilibrium of the restricted game, ti does not have a profitable deviation to any (si , ei ) ∈ Riα with ei ∈ Ei (ti ). So consider a deviation by ti to some (si , ei ) ∈ Riβ for β 6= α such that (si , ei ) has positive probability under the equilibrium strategies. By Lemma 12, we know that a deviation to any (si , ei ) ∈ Riβ , β 6= α, which has positive probability in equilibrium must at least weakly increase the principal’s belief if β > α and decrease it if β < α. If ti ∈ Ti+ , then Lemma 2 implies that for every (si , ei ) ∈ Riβ with β > α, we have ei ∈ / Ei (ti ). Hence ti cannot deviate 42 to an (si , ei ) which has positive probability in equilibrium and increases the principal’s belief. Similarly, a negative type cannot feasibly deviate to any (si , ei ) which has positive probability in equilibrium and decreases the principal’s belief. So we only need to ensure that there is no profitable deviation to an (si , ei ) which has zero probability in the restricted game. Fix any such (si , ei ) and suppose (si , ei ) ∈ Riα . Let Fi = {ti ∈ Ti | ei ∈ Ei (ti )}. Since we have an equilibrium of the restricted game, we know that min+ X̂i∗ (t̄i ) ≥ Xi∗ (si , ei ) ≥ max− X̂i∗ (t̄i ). t̄i ∈Fi ∩ Ti ∩ Tiα t̄i ∈Fi ∩ Ti ∩ Tiα That is, the worst off positive type who can send evidence ei prefers the belief she induces to deviating to (si , ei ) and analogously for the negative types. Let T̂i+ denote the set of ti ∈ Fi ∩ Ti+ with ti ∈ Tiβ for some β 6= α. By Lemma 2, we must have β > α for each ti ∈ T̂i+ . If Fi ∩ Ti+ ∩ Tiα 6= ∅, then by Lemma 12, we have X̂i∗ (ti ) ≥ min+ t̄i ∈Fi ∩ Ti ∩ Tiα X̂i∗ (t̄i ) ≥ Xi∗ (si , ei ), ∀ti ∈ T̂i+ , so no positive type can gain by deviating to (si , ei ). So assume Fi ∩ Ti+ ∩ Tiα = ∅. Fix t̂i ∈ T̂i+ such that X̂i∗ (t̂i ) ≤ X̂i∗ (ti ) for all ti ∈ T̂i+ . By the same reasoning as above, if X̂i∗ (t̂i ) ≥ Xi∗ (si , ei ), no positive type can gain by deviating to (si , ei ). So assume Xi∗ (si , ei ) > X̂i∗ (t̂i ). Suppose there is any t0i ∈ F such that X̂i∗ (t̂i ) ≥ vi (t0i ). If so, redefine the belief in response to (si , ei ) by setting it equal to a convex combination of the equilibrium belief from the restricted game and a degenerate distribution on t0i chosen to make the expectation of vi equal to X̂i∗ (t̂i ). By construction, then, no positive type will be able to gain by deviating to (si , ei ). For any ti ∈ Fi ∩ Ti− with ti ∈ Tiγ , γ 6= α, Lemma 2 implies γ < α. Hence by Lemma 12, X̂i∗ (ti ) ≤ X̂i∗ (t̂i ). Hence no negative type can gain by deviating to (si , ei ) either. Hence we can assume that there is no t0i ∈ Fi with X̂i∗ (t̂i ) ≥ vi (t0i ). That is, every type t0i who can send ei has vi (t0i ) > X̂i∗ (t̂i ). Clearly, if t̂i sends Mi (t̂i ), an option which must be feasible in the restricted auxiliary game for i, she must prove at least as much as ei . Hence she has a strategy available in the restricted game which must lead to a belief by the principal above X̂i∗ (t̂i ), a contradiction. Summarizing, we see that either no positive type can gain by sending (si , ei ) given the belief from the restricted auxiliary game for i this leads to or we can change that belief in such a way that no positive or negative type can gain. The symmetric argument for negative types then shows that by only changing off equilibrium beliefs, we can turn an equilibrium for the restricted auxiliary game for i into an equilibrium of the unrestricted game. 43 To complete the proof, we now use the equilibrium strategies of the auxiliary games to construct equilibrium strategies for the real game. The strategy for agent i is the same as her strategy in the equilibrium of the auxiliary game for i. Similarly, the principal’s belief about ti when he observes (si , ei ) is given by his belief in the auxiliary game for i. Given the principal’s beliefs, sequential rationality tells us what his action must be at any information set where he has a unique optimal choice given his beliefs. However, we need to specify his actions at information sets with multiple optimal choices. On the equilibrium path, we will specify his actions to follow the optimal mechanism, but information sets off the equilibrium path are more subtle. To construct the principal’s equilibrium strategy, we divide the possible (s, e) profiles he may observe into three sets. First, if (s, e) has positive probability under the equilibrium strategies of the agents, the principal chooses what the mechanism prescribes given the types. To be more specific, if (si , ei ) ∈ Riαi and σi∗ (si , ei | ti ) > 0 for some ti for each i, then the principal chooses P (· | t̂, M (t̂)) for any t̂ such that t̂i ∈ Tiαi for all i. Second, if (s, e) has the property that (si , ei ) has zero probability under the equilibrium strategies of the agents for at least two i, then the principal chooses any optimal p(·) given his beliefs. Obviously, the specification of the principal’s strategy on such histories does not affect equilibrium considerations for the agents. Third, consider any (s, e) such that (si , ei ) has zero probability under the equilibrium strategies for exactly one i. If Xi∗ (si , ei ) 6= X̂i∗ (ti ) for all ti , then we can take the principal’s response to (s, e) to be any optimal p(·) ∈ ∆(A) given his beliefs. If (si , ei ) ∈ Riα and there exists ti ∈ Tiα with Xi∗ (si , ei ) = X̂i∗ (ti ), then we can treat (si , ei ) the same way as any positive probability (s0i , e0i ) ∈ Riα as specified above. Next, suppose (si , ei ) ∈ Riα but the only ti ’s satisfying Xi∗ (si , ei ) = X̂i∗ (ti ) have ti ∈ / Tiα . If all such ti are in the same Tiβ , then we treat (si , ei ) the same way as any positive probability (s0i , e0i ) in Riβ . Finally, suppose (si , ei ) ∈ Riα , there is no ti ∈ Tiα with Xi∗ (si , ei ) = X̂i∗ (ti ), and there exists t̄ki ∈ Tiβk with Xi∗ (si , ei ) = X̂i∗ (t̄ki ), k = 1, 2, with β1 > β2 . By Lemma 12, β1 > β2 implies that every expectation of vi induced in equilibrium by a type in Tiβ1 must weakly exceed every expectation induced by a type in Tiβ2 . Hence it must be true that X̂i∗ (t̄1i ) = minti ∈T β1 X̂i∗ (t1 ) and X̂i∗ (t̄2i ) = maxti ∈T β1 X̂i∗ (ti ). If β2 > α, then take the i i principal’s response to (si , ei ) to be the same as his response to any (s0i , e0i ) ∈ Riβ2 which has positive probability. If α > β1 , then take the principal’s response to (si , ei ) to be the same as his response to any (s0i , e0i ) ∈ Riβ1 which has positive probability. Finally, if β1 > α > β2 , take the principal’s response to be a 50–50 mixture between his response given any (s0i , e0i ) ∈ Riβ1 on the equilibrium path and the response given any (s00i , e00i ) ∈ Riβ2 on the equilibrium path. (This case can only arise if X̂i∗ (t̄1i ) = X̂i (t̄2i ) = X̂i∗ (t0i ) for all t0i ∈ Tiα .) To see that these strategies form an equilibrium, first note that Lemma 11 implies 44 that the principal is choosing a best reply given his beliefs in response to every (s, e) which has positive probability in equilibrium. The construction above ensures that the principal is also sequentially rational in response to any (s, e) which has zero probability in equilibrium. Turning to the agents, consider any ti ∈ Ti and consider a deviation by ti to some (si , ei ) with σi∗ (si , ei | ti ) = 0. If Xi∗ (si , ei ) 6= X̂i∗ (ti ), then the fact that these strategies are an equilibrium of the auxiliary game for i implies that if ti ∈ Ti+ , we have X̂i∗ (ti ) > Xi∗ (si , ei ) and the reverse strict inequality if ti ∈ Ti− . By Lemma 6, this implies that ti is at least weakly worse off deviating to (si , ei ). So suppose Xi∗ (si , ei ) = X̂i∗ (ti ). That is, suppose the principal has the same belief about vi under the deviation as he would following equilibrium play by ti . For concreteness, assume ti ∈ Ti+ — an analogous argument covers the case where ti ∈ Ti− . Assume that ti ∈ Tiα . Since ei ∈ Ei (ti ) by hypothesis, Lemma 2 implies that (si , ei ) ∈ Riβ for β ≤ α. If there is some type t0i 6= ti who sends (si , ei ) with positive probability in equilibrium, then the outcome is the same as in the optimal mechanism given any t0i ∈ Tiβ while the outcome if ti follows the strategy from the equilibrium of the auxiliary game is the same in the optimal mechanism given ti . By incentive compatibility, ti does not gain by deviating to (si , ei ). So assume (si , ei ) is not sent with positive probability by any type in equilibrium. If β = α or if there is no γ 6= α with t0i ∈ Tiγ and Xi∗ (si , ei ) = X̂i∗ (t0i ), then (si , ei ) is treated the same way as any (s0i , e0i ) ∈ Riα which does have positive probability, so the outcome is the same as if ti followed the equilibrium strategy from the auxiliary game. Hence, again, he does not gain by deviating. Finally, suppose (si , ei ) has zero probability in equilibrium, α > β, and there is some γ 6= α with t0i ∈ Tiγ and Xi∗ (si , ei ) = X̂i∗ (t0i ). If γ > α, then, again, (si , ei ) is treated the same way as any (s0i , e0i ) ∈ Riα which has positive probability in the equilibrium of the auxiliary game, so the outcome is again the same as if ti followed the equilibrium strategy from the auxiliary game. Hence, again, she does not gain by deviating. If α > β > γ, the response to (si , ei ) is a 50–50 randomization between the way the principal would respond to positive probability (s0i , e0i ) ∈ Riα and the way he would respond to positive probability (s0i , e0i ) ∈ Riγ . This is strictly worse for ti than the response to ti ’s equilibrium strategy. Finally, if α > γ > β, the principal’s response is the same as his response to any positive probability (s0i , e0i ) ∈ Riγ , again worse for ti than following the equilibrium strategy. 45 D Proof of Theorem 2 We first show there exists vi∗ solving vi∗ = E[vi (ti ) | ti ∈ Ti0 or vi (ti ) ≤ vi∗ ]. (4) If Ti = Ti0 , then it is easy to see that vi∗ = E(vi (ti )) satisfies (4). On the other hand, if Ti0 = ∅, then vi∗ = minti ∈Ti vi (ti ) satisfies (4). In what follows, assume Ti0 6= ∅ and Ti0 6= Ti . n+1 n ). (If we Write Ti \ Ti0 as {t1i , . . . , tN i } where without loss of generality vi (ti ) < vi (ti 0 0 0 0 have ti , ti ∈ Ti \ Ti with ti 6= ti and vi (ti ) = vi (ti ), we can treat these two types as if they were one type for the purposes of this calculation.) For n = 1, . . . , N , let gin = E[vi (ti ) | ti ∈ Ti0 or ti = tki , for some k ≤ n] and let gi0 = E(vi (ti ) | ti ∈ Ti0 ). Suppose that there is no solution to equation (4). If gi0 ≤ vi (t1i ), then vi∗ = gi0 satisfies (4). Hence gi0 > vi (t1i ). But gi1 is a convex combination of vi (t1i ) and gi0 , with strictly positive weight on each term, so vi (t1i ) < gi1 < gi0 . Again, if gi1 ≤ vi (t2i ), then vi∗ = gi1 satisfies (4), so we must have gi1 > vi (t2i ), implying vi (t2i ) < gi2 < gi1 . Similar reasoning N gives gin−1 > gin > vi (tni ) for n = 1, . . . , N . In particular, giN > vi (tN i ). But gi = E[vi (ti )], ∗ N so this implies vi = gi solves equation (4), a contradiction. Hence a solution exists. To see that the solution is unique, suppose to the contrary that vi1 and vi2 both solve (4) where vi1 > vi2 . Let Tik = Ti0 ∪ {ti ∈ Ti \ Ti0 | vi (ti ) ≤ vik }, so vik = E[vi (ti ) | ti ∈ Tik ]. Since vi1 > vi2 , we have Ti2 ⊂ Ti1 and Ti1 \ Ti2 = {ti ∈ Ti \ Ti0 | vi2 < vi (ti ) ≤ vi1 }. Note that vi1 is a convex combination of vi2 and E[vi (ti ) | ti ∈ Ti1 \ Ti2 ]. But every ti ∈ Ti1 \ Ti2 has vi (ti ) ≤ vi1 , so we must have E[vi (ti ) | ti ∈ Ti1 \ Ti2 ] ≤ vi1 ≤ vi2 , contradicting vi1 > vi2 . To construct equilibrium strategies, first note that we must have x∗i (si , {ti }) = vi (ti ). That is, if ti proves her type, the principal must infer correctly. Thus we only need to 46 determine the principal’s beliefs in response to reports of the form (si , Ti ) where the agent proves nothing. It is easy to see that if Ti0 = ∅, then vi∗ = minti ∈Ti vi (ti ) and that the essentially unique equilibrium has every type proving her type. This is the usual unraveling argument. Any type t0i with vi (t0i ) = maxti ∈Ti vi (ti ) must strictly prefer proving her type to pooling with lower types and so must prove her type. But then any type with the next highest possible value of vi (ti ) cannot pool with higher types and so must prove her type, etc. So for the rest of this proof, assume Ti0 6= ∅. Clearly, we cannot have (si , Ti ) and (s0i , Ti ), both with positive probability in equilibrium with x∗i (si , Ti ) 6= x∗i (s0i , Ti ). Since all types are positive, every type strictly prefers whichever of these reports yields the larger x in response. Hence we may as well fix some s∗i and suppose that the only (si , Ti ) sent with positive probability in equilibrium is (s∗i , Ti ) where x∗i (s∗i , Ti ) ≥ x∗i (si , Ti ) for all si ∈ Ti . Let ṽi = x∗i (s∗i , Ti ). From the above, we know that types ti ∈ Ti0 send report (s∗i , Ti ). Any type ti ∈ / Ti0 can send either (s∗i , Ti ) and obtain response ṽi or can send some (si , {ti }) and receive response vi (ti ). Hence ti chooses the former only if ṽi ≥ vi (ti ). Ignoring indifference for a moment, we see that this implies that ṽi must be the vi∗ defined in equation (4). To address indifference, note that v ∗ is not changed if we add or remove from the set of types sending this message a type with vi (ti ) = vi∗ . Hence we have the same outcome regardless. E Proof of Lemma 1 and Theorem 3 For Lemma 1, the existence and uniqueness of vi+ follows from Theorem 2 taking the set of types to be Ti+ . For vi− , note that Theorem 2 applied to the function −vi (ti ) and types Ti− implies that there is a unique vi− satisfying −vi− = E[−vi (ti ) | ti ∈ Ti0 ∩ Ti− or (ti ∈ Ti− \ Ti0 and − vi (ti ) ≤ −vi− )] which can be rewritten as the definition of vi− . Next, we show that there exists vi∗ solving vi∗ =E vi (ti ) | (ti ∈ Ti0 ) or (ti ∈ Ti− \ Ti0 and vi (ti ) ≥ vi∗ ) î or (ti ∈ Ti+ \ Ti0 and vi (ti ) ≤ vi∗ ) . ó (5) Let the function of vi∗ on the right–hand side be denoted gi (vi∗ ). So we seek to prove that there exists vi∗ solving vi∗ = gi (vi∗ ). 47 As with Theorem 2, the proof is by contradiction. So suppose there is no vi∗ solving / Ti0 . Without loss of this equation. Let vi1 , . . . , viN denote the values of vi (ti ) for ti ∈ generality, assume vik < vik+1 for k = 1, . . . , N − 1. First, note that for vi∗ ≤ vi1 , we have gi (vi∗ ) = E[vi (ti ) | ti ∈ Ti0 ∪ Ti− ]. So the assumption that there is no vi∗ with vi∗ = gi (vi∗ ) implies E[vi (ti ) | ti ∈ Ti0 ∪ Ti− ] > vi1 as otherwise we can set vi∗ = E[vi (ti ) | ti ∈ Ti0 ∪ Ti− ] and obtain a solution to equation (5). Clearly, the function gi (vi∗ ) is constant in vi∗ for vi∗ ∈ (vik , vik+1 ). Hence if gi (vik ) ∈ we have a solution to equation (5) in this interval. (vik , vik+1 ), Also, if gi (vik ) > vik+1 , then gi (vik+1 ) > vik+1 . To see this, first suppose that vik+1 ∈ vi (Ti− ). In this case, gi (vik ) is a convex combination of gi (vik+1 ) and vik+1 . Since gi (vik ) > vik+1 by assumption, we must have gi (vik+1 ) ≥ gi (vik ) > vik+1 , implying the claim. Alternatively, suppose vik+1 ∈ vi (Ti+ ). In this case, gi (vik+1 ) is a convex combination of gi (vik ) and vik+1 . Since gi (vik ) > vik+1 , this implies gi (vik+1 ) > vik+1 . As shown above, we start with gi (vi1 ) > vi1 , so by induction, we have gi (viN ) > viN . But gi (vi∗ ) = E[vi (ti ) | ti ∈ Ti0 ∪ Ti+ ] for all vi∗ ≥ viN . So there exists vi∗ > viN solving (5), a contradiction. To show uniqueness, suppose to the contrary that vi1 and vi2 are both solutions to equation (5) where vi1 > vi2 . Let Tik+ = {ti ∈ Ti+ \ Ti0 | vi (ti ) ≤ vik }, k = 1, 2 and Tik− = {ti ∈ Ti− \ Ti0 | vi (ti ) ≥ vik }, k = 1, 2. Clearly, since vi1 > vi2 , we have Ti2+ ⊆ Ti1+ and Ti1− ⊆ Ti2− . But vik = E[vi (ti ) | ti ∈ Ti0 ∪ Tik+ ∪ Tik− ]. Let ṽi = E[vi (ti ) | ti ∈ Ti0 ∪ Ti2+ ∪ Ti1− ]. Then vi1 is a convex combination of ṽi and E[vi (ti ) | ti ∈ Ti1+ \ Ti2+ ], while vi2 is a convex combination of ṽi and E[vi (ti ) | ti ∈ Ti2− \ Ti1− ]. It is easy to see that vi2 ≤ E[vi (ti ) | ti ∈ Ti1+ \ Ti2+ ] ≤ vi1 since vi2 ≤ vi (ti ) ≤ vi1 for all ti ∈ Ti1+ \ Ti2+ . Similarly, vi2 ≤ E[vi (ti ) | ti ∈ Ti2− \ Ti1− ] ≤ vi1 . 48 Since vi1 is a convex combination of ṽi and something smaller than vi1 , we must have ṽi ≥ vi1 . But since vi2 is a convex combination of ṽi and something larger than vi2 , we must have vi2 ≥ ṽi . Hence vi1 ≤ ṽi ≤ vi2 , contradicting vi1 > vi2 . Turning to Theorem 3, we construct equilibrium strategies as follows. First, note that if x∗i (si , Ti ) > x∗i (s0i , Ti ), then no positive type will send report (s0i , Ti ) and no negative type will send (si , Ti ). Hence there are, at most, two distinct values of x∗i (si , Ti ) observed on the equilibrium path. Let ṽi+ = maxsi ∈Ti x∗i (si , Ti ) and ṽi− = minsi ∈Ti x∗i (si , Ti ). For the moment, assume ṽi+ > ṽi− . Then it is easy to see that every positive type ti ∈ Ti0 sends a report generating ṽi+ as does every positive type ti ∈ / Ti0 with vi (ti ) ≤ ṽi+ . Similarly, every negative type ti ∈ Ti0 or not in Ti0 but with vi (ti ) ≥ ṽi− sends some report generating ṽi− . All other types ti send a report of the form (si , {ti }). Given this, it is clear that ṽi+ must equal vi+ and ṽi− must equal vi− . This is an equilibrium iff vi+ ≥ vi− . Note that if vi+ = vi− , then the expectation of vi given the set of types sending either report must also be the same value. Thus in this case, we have vi− = vi+ = vi∗ . Regardless of the relationship between vi− and vi+ , there is also an equilibrium where the principal’s beliefs ignore the type report and condition only on the evidence. Letting ṽi denote the principal’s expected value of vi condition on the evidence report ei = Ti , we see that positive types with vi (ti ) > ṽi will prove their types as will negative types with vi (ti ) < ṽi . Hence ṽi must satisfy equation (5), so ṽi = vi∗ . F Proof of Corollary 2 When vi+ ≤ vi− , there is only one equilibrium in the auxiliary game for i, so the claim follows. When vi+ > vi− , however, there are (essentially) two equilibria. In one, type reports are used to separate positive types from negative types. All positive types with evidence and vi (ti ) > vi+ prove their types, as do all negative types with evidence and vi (ti ) < vi− . All other positive types send one type report and evidence ei = Ti , while all other negative types send another type report and the same evidence. In what follows, we refer to this equilibrium as the cheap–talk equilibrium as it uses the “cheap talk” of type reports to help separate. In the other equilibrium, the principal’s beliefs depend only on the evidence presented, so type reports are irrelevant. All positive types with evidence and vi (ti ) > vi∗ prove their types as do all negative types with evidence and vi (ti ) < vi∗ . All other types report some fixed type report and evidence ei = Ti . We refer 49 to this equilibrium as the non–talk equilibrium.16 Since there are two equilibria in the auxiliary game for i in this case, we need to determine which strategies for i are used in the equilibrium of the game which has the same outcome as the optimal mechanism. Clearly, if the principal is better off under one set of strategies than the other, then these must be the strategies used since the equilibrium corresponding to the optimal mechanism must be the best possible equilibrium for the principal. We now show that the principal’s payoff is always at least weakly larger in the cheap– talk equilibrium, completing the proof of Corollary 2. First, we show that vi+ > vi− implies vi+ ≥ vi∗ ≥ vi− with at least one strict inequality. To see this, suppose to the contrary that vi∗ > vi+ > vi− . Define the following sets of types: T̂i− = {ti ∈ Ti− | ti ∈ Ti0 or vi (ti ) ≥ vi− } T̂i+ = {ti ∈ Ti+ | ti ∈ Ti0 or vi (ti ) ≤ vi+ } T̂i∗− = {ti ∈ Ti− | ti ∈ Ti0 or vi (ti ) ≥ vi∗ } T̂i∗+ = {ti ∈ Ti+ | ti ∈ Ti0 or vi (ti ) ≤ vi∗ } In other words, the types in T̂i− are the negative types who “pool” together in the cheap–talk equilibrium, while T̂i+ is the set of positive types who pool together in this equilibrium. Similarly, T̂i∗− and T̂i∗+ are, respectively, the set of negative and positive types who all pool together in the non–talk equilibrium. By definition, vi− = E[vi (ti ) | ti ∈ T̂i− ] vi+ = E[vi (ti ) | ti ∈ T̂i+ ] vi∗ = E[vi (ti ) | ti ∈ T̂i∗− ∪ T̂i∗+ ] Hence vi∗ is a convex combination of E[vi (ti ) | ti ∈ T̂i∗− ] and E[vi (ti ) | ti ∈ T̂i∗+ ]. Since vi− < vi∗ , we see that T̂i∗− ⊆ T̂i− . Note that if ti ∈ T̂i− but ti ∈ / T̂i∗− , then vi− ≤ vi (ti ) < vi∗ . Hence vi− = E[vi (ti ) | ti ∈ T̂i− ] is a convex combination of E[vi (ti ) | ti ∈ T̂i∗− ] and the expectation of vi (ti ) for a set of types all with vi (ti ) > vi− . Hence vi∗ > vi− = E[vi (ti ) | ti ∈ Ti− ] > E[vi (ti ) | ti ∈ T̂i∗− ]. Similarly, vi+ < vi∗ implies that T̂i+ ⊆ T̂i∗+ . Since the types in T̂i∗+ \ T̂i+ all satisfy vi+ ≤ vi (ti ) < vi∗ , we see that E[vi (ti ) | ti ∈ T̂i∗+ ] is a convex combination of vi+ = E[vi (ti ) | 16 We refer to this as a non–talk equilibrium rather than as a babbling equilibrium since, unlike in the usual babbling equilibria in the literature, the use of evidence does enable some communication. 50 ti ∈ T̂i+ ] < vi∗ and an expectation of vi (ti ) for a set of types with vi (ti ) < vi∗ . Hence E[vi (ti ) | ti ∈ T̂i∗+ ] < vi∗ . But then we have vi∗ is a convex combination of two terms which are strictly smaller than vi∗ , a contradiction. A similar argument rules out the possibility that vi+ > vi− > vi∗ . Consider the game between the agents and the principal. We know that there is a robust PBE with the same outcome as in the optimal mechanism. We know i’s strategy in this equilibrium must either be the one she uses in the cheap–talk equilibrium or the one she uses in the non–talk equilibrium. Fix the strategies of all agents other than i. We know these strategies are defined from the auxiliary games for these agents, independently of which strategy i uses or the principal’s response to i. Thus we can simply determine which strategy by i leads to a higher payoff for the principal. Note that the principal’s payoff for a fixed a is linear in his expectation of vi . Hence his maximized payoff is convex in his expectation of vi . We now show that the distribution of beliefs for the principal in the cheap–talk equilibrium is a mean–preserving spread of the distribution in the non–talk equilibrium, completing the proof. To be precise, let (σi1 , x1i ) denote the cheap–talk equilibrium strategies and (σi2 , x2i ) the non–talk equilibrium strategies from the auxiliary game for i. For k = 1, 2, define probability distributions B k over R by B k (v̂i ) = ρi Ä¶ ti ∈ Ti | Xik (ti ) = v̂i ©ä . (Recall that Xik (ti ) = xki (si , ei ) for any (si , ei ) with σik (si , ei | ti ) > 0 and that ρi is the prior over Ti .) The law of iterated expectations implies X v̂i B k (v̂i ) = E[vi (ti )], k = 1, 2. v̂i ∈supp(B k ) Hence the two distributions have the same mean. Consider any v̂i < vi− . Since vi− ≤ vi∗ , for k = 1 or k = 2, we have Xik (ti ) = v̂i if and only if there is a negative type with evidence who has vi (ti ) = v̂i . Similarly, since vi∗ ≤ vi+ , for any v̂i > vi+ , we have Xik (ti ) = v̂i iff there is a positive type with evidence who has vi (ti ) = v̂i . Hence B 1 (v̂i ) = B 2 (v̂i ) for any v̂i ∈ / [vi− , vi+ ]. Also, we have B 1 (v̂i ) = 0 for all v̂i ∈ (vi− , vi+ ). Any type with vi in this range either (1) is positive and chooses to induce belief vi+ or (2) is negative and chooses to induce belief vi− . Under B 2 , however, many of the types generating beliefs concentrated at vi− or vi+ in the cheap–talk equilibrium instead generate beliefs in (vi− , vi+ ). In particular, types without evidence or types with evidence they prefer not to show induce the belief vi∗ , a positive type with evidence who has vi (ti ) ∈ (vi∗ , vi+ ) generate the belief vi (ti ), and similarly for negative types. Hence B 1 is a mean–preserving spread of B 2 . 51 G Costly Verification In this section, we show that for a particular class of costly verification models with simple type dependence, the optimal mechanism can be computed using our results for optimal mechanisms with Dye evidence. To be specific, we continue to let A denote the finite set of actions available to the principal, Ti the finite set of types of agent i with the same distributional assumptions as in the text, and continue to assume that agent i’s utility function can be written as ® ui (a, ti ) = ui (a), if ti ∈ Ti+ ; −ui (a), if ti ∈ Ti− , and that the principal’s utility function can be written as v(a, t) = I X ui (a)vi (ti ). i=0 We add three further assumptions on preferences. First, we assume that each agent has exactly two indifference curves in A.17 That is, for each agent i, we can partition A into nonempty18 sets A0i and A1i where ® ui (a) = 0, if a ∈ A0i ; 1, if a ∈ A1i . For example, this assumption holds in the allocation example and most of the related problems discussed in Example 1 of Section 1 as well as the public goods problem discussed in Example 2. It also holds in the public goods problem discussed in Erlanson and Kleiner (2015) (after renormalizing their statement of the payoffs). Second, we assume that for all i, either Ti− = ∅ or vi (ti ) > vi (t0i ) for all ti ∈ Ti+ and t0i ∈ Ti− . In other words, either we have type–independent preferences or every positive type has a higher vi than every negative type. This assumpion on the comparison of positive and negative types is made by Erlanson and Kleiner. For the costly verification model, the agents do not have evidence to present. Instead, the principal can check agent i at a cost ci > 0. “Checking” agent i means that the principal learns agent i’s type ti . We will show that the optimal mechanism for this problem can be computed by an appropriate “translation” of a related mechanism design problem with Dye evidence instead of costly verification. 17 This also includes “agent 0” — that is, this also applies to the utility function u0 (a). Alternatively, we can simply assume that u0 is identically zero. 18 If either set is empty for i 6= 0, then the agent is indifferent over all choices by the principal and incentive compatibility is trivially satisfied. Hence we can disregard any such agent. 52 Note that our assumptions imply that if vi (ti ) = vi (t0i ), then either both are positive types or both are negative. Since agents do not have evidence, this means that ti and t0i are identical and there is no need for the model (or the principal) to distinguish them. Hence without loss of generality, we assume that if ti 6= t0i , then vi (ti ) 6= vi (t0i ). k+1 k i Consequently, we can write the type set for i as Ti = {t0i , . . . , tK ) i } where vi (ti ) < vi (ti for k = 0, . . . , Ki − 1. It is not hard to prove a simple analog of the Revelation Principle for this class of problems, namely, that it is without loss of generality to focus on mechanisms with the following structure. First, all agents simultaneously make cheap talk reports of types to the principal. The mechanism specifies a probability distribution over which agents to check and what a ∈ A to choose as a function of the reports. Each agent will have an incentive to report his type honestly, so when the principal checks an agent, he finds that the report was truthful. Off the equilibrium path, if the principal finds that an agent has lied, the principal chooses any action which is worst for that agent. (Since the agents all expect the other agents to report honestly, the specification of the mechanism for histories where multiple agents are found to have lied is irrelevant.) Hence we can write a mechanism as a function P : T → ∆(2I ×A) where P (Q, a | t) is the probability that the principal checks the agents in the set Q ⊆ I and chooses action a ∈ A when the type reports are t and the result of the checking verifies that the reports were honest. The expected payoff of the principal from such a mechanism is Ñ X Et P (Q, a | t) é v(a, t) − (Q,a)∈2I ×A X ci . i∈Q Let p(a | t) = X P (Q, a | t) Q⊆I qi (t) = X X P (Q, a | t). a∈A Q⊆I|i∈Q Then we can rewrite the principal’s expected payoff as " # X Et p(a | t)v(a, t) − P i qi (t)ci . i a∈A Using the fact that v(a, t) = X ui (a)vi (ti ), we can rewrite this as " Et # X i vi (ti ) X p(a | t)ui (a) − X i a∈A Let pi (t) = X a∈A1i 53 p(a | t). qi (t)ci . In other words, pi (t) is the probability that the principal selects an action a such that ui (a) = 1 given type profile t. Then the principal’s expected payoff is " X Et # (pi (t)vi (ti ) − qi (t)ci ) = X i Eti [p̂i (ti )vi (ti ) − q̂i (ti )ci ], i where p̂i (ti ) = Et−i pi (t) and q̂i (ti ) = Et−i qi (ti , t−i ). If agent i of type ti reports truthfully, his expected utility in mechanism P is X Et−i P (Q, a | t)ui (a) (Q,a)∈2I ×A if ti ∈ Ti+ and this times −1 otherwise. So the expected payoff to a positive type from reporting truthfully is p̂i (ti ), while the expected payoff to a negative type is −p̂i (ti ). If agent i is type ti but reports t0i 6= ti , then he may be caught lying. In this case, as noted above, the principal will choose an action which minimizes his payoff. So if ti ∈ Ti+ , his payoff will be 0 if he is caught lying, while if ti ∈ Ti− , it will be −1. Hence for a positive type, the expected payoff to the deviation is Et−i P (Q, a | t0i , t−i )ui (a) X (Q,a)∈2I ×A|i∈Q / = Et−i P (Q, a | t0i , t−i )ui (a) − X (Q,a)∈2I ×A (Q,a)∈2I ×A|i∈Q = p̂i (t0i ) − Et−i P (Q, a | t0i , t−i )ui (a) X X P (Q, a | t0i , t−i ) . (Q,a)∈2I ×A|i∈Q,a∈A1i We will simplify this expression further below. If a negative type is caught reporting falsely, the principal chooses an action setting ui (a) = 1 so that the agent’s payoff is −1. Hence the expected payoff to a negative type 54 ti from claiming to be t0i 6= ti is Et−i P (Q, a | t0i , t−i )(−ui (a)) − X P (Q, a | t0i , t−i ) X (Q,a)∈2I ×A|i∈Q (Q,a)∈2I ×A|i∈Q / = Et−i − P (Q, a | t0i , t−i )ui (a) − X X (Q,a)∈2I ×A|i∈Q (Q,a)∈2I ×A = −p̂i (t0i ) − Et−i P (Q, a | t0i , t−i )(1 − ui (a)) P (Q, a | t0i , t−i ) . X (Q,a)∈2I ×A|i∈Q,a∈A0i Summarizing, the incentive compatibility constraint for agent i is that for all positive types ti ∈ Ti+ , we have p̂i (ti ) ≥ p̂i (t0i ) − Et−i X P (Q, a | t0i , t−i ) , ∀t0i 6= ti (6) (Q,a)∈2I ×A|i∈Q,a∈A1i and for all negative types ti ∈ Ti− , we have p̂i (ti ) ≤ p̂i (t0i ) + Et−i X P (Q, a | t0i , t−i ) , ∀t0i 6= ti . (7) (Q,a)∈2I ×A|i∈Q,a∈A0i Note that the right–hand side of each incentive compatibility constraint is independent of ti . Hence (6) holds for all positive types ti iff it holds for the positive type with the smallest p̂i (ti ) and (7) holds for all negative types ti iff it holds for that negative type with the largest p̂i (ti ). It is not hard to show that the optimal mechanism must be monotonic in the sense that p̂i (tki ) ≤ p̂i (tk+1 ) for k = 0, . . . , Ki − 1. To see this, recall that vi (tki ) < vi (tk+1 ), i i so the principal is better off with higher values of pi associated with higher values of ti . So suppose we have an incentive compatible mechanism with p̂i (tki ) > p̂i (tk+1 ) for some i k and i. Consider the mechanism which reverses the roles of these types — i.e., assigns the outcome (Q, a) to (tki , t−i ) that it would have assigned to (tk+1 , t−i ) and vice versa.19 i Then this altered mechanism is also incentive compatible and yields the principal a higher expected payoff. By assumption, for every i, either Ti− = ∅ or vi (ti ) > vi (t0i ) for all ti ∈ Ti+ , t0i ∈ Ti− . Hence if there are Ji negative types(where Ji can be zero), the negative types i are t0i , . . . , tJi i −1 and the positive types are tJi i , . . . , tK i . Thus the positive type with the 19 To be precise, this implicitly assumes the two types have the same prior probability. If not, we can reverse the role of one of the types and “part of” the other. 55 lowest p̂i (ti ) is tJi i , while the negative type with the highest p̂i (ti ) is tiJi −1 and we have p̂i (tJi 1 −1 ) ≤ p̂i (tJi i ). So we can write the incentive compatibility constraints (6) and (7) as p̂i (tJi i ) ≥ p̂i (t0i ) − Et−i X P (Q, a | t0i , t−i ) , ∀t0i 6= ti (8) (Q,a)∈2I ×A|i∈Q,a∈A1i and p̂i (tJi i −1 ) ≤ p̂i (t0i ) + Et−i X P (Q, a | t0i , t−i ) , ∀t0i 6= ti . (9) (Q,a)∈2I ×A|i∈Q,a∈A0i The following lemma simplifies the incentive compatibility constraints. Lemma 13. In any optimal mechanism, we have P (Q, a | ti , t−i ) = 0, ∀t−i if ti ∈ Ti+ , i ∈ Q, and a ∈ A0i and P (Q, a | ti , t−i ) = 0, ∀t−i if ti ∈ Ti− , i ∈ Q, and a ∈ A1i . Consequently, we can rewrite the incentive compatibility constraints (8) and (9) as p̂i (tJi i ) ≥ p̂i (ti ) − q̂i (ti ), ∀ti ∈ Ti+ (10) p̂i (tJi i −1 ) ≤ p̂i (ti ) + q̂i (ti ), ∀ti ∈ Ti− . (11) and Proof. First, we show that we only require (8) for t0i ∈ Ti+ and (9) for t0i ∈ Ti− . Specifically, we show that monotonicity of p̂i implies that (8) holds for all t0i ∈ Ti− and (9) holds for all t0i ∈ Ti+ . To see this, fix any t0i ∈ Ti− . By assumption, vi (t0i ) ≤ vi (tJi i ), so monotonicity implies p̂i (tJi i ) ≥ p̂i (t0i ). Since p̂i (t0i ) is weakly larger than the right–hand side of (8), this implies (8) holds. A similar argument shows that monotonicity implies (9) for any t0i ∈ Ti+ . Next, suppose, contrary to the statement of the lemma, that we have an optimal mechanism P with the property that P (Q, a | ti , t−i ) > 0 for some t−i ∈ T−i , ti ∈ Ti+ , i ∈ Q, and a ∈ A0i . In other words, there is a positive probability that the principal checks some positive type and then chooses an action giving that agent a payoff of zero. Construct a new mechanism P ∗ as follows. For any (Q0 , a0 ) 6= (Q, a) or t0 6= t, let P ∗ (Q0 , a0 | t0 ) = P (Q0 , a0 | t0 ). Let P ∗ (Q, a | t) = 0 and let P ∗ (Q \ {i}, a | t) = P (Q, a | t) + P (Q \ {i}, a | t). In other words, if i is checked but gets a zero payoff at (Q, a), we shift this probability to (Q \ {i}, a), where i does not get checked but still gets the same zero payoff. It is easy to see that the incentive compatibility constraints for any agent j 6= i are unaffected. Since ti is a positive type, the only incentive compatibility 56 constraint for i that is potentially affected is (8) at t0i = ti or where ti = tJi i . But since we have only changed the checking probability and not the marginal probabilities over actions a ∈ A, p̂i (ti ) is unaffected by this change in the mechanism. Similarly, the second term on the right–hand side of (8) for t0i = ti only involves actions in A1i , so this term also is unaffected. Hence P ∗ is incentive compatible. Finally, since the probability over A is unchanged but the principal checks less often, his payoff must be strictly larger, a contradiction. So suppose we have an optimal mechanism P with the property that P (Q, a | ti , t−i ) > 0 for some t−i ∈ T−i , ti ∈ Ti− , i ∈ Q, and a ∈ A1i , contrary to the statement of the lemma. That is, we have a strictly positive probability that the principal checks some negative type and then chooses an action giving that agent a payoff of −1. Construct a new mechanism P ∗ exactly as in the previous case. That is, for any (Q0 , a0 ) 6= (Q, a) or t0 6= t, we do not change the mechanism, so P ∗ (Q0 , a0 | t0 ) = P (Q0 , a0 | t0 ). Again, we let P ∗ (Q, a | t) = 0 and let P ∗ (Q \ {i}, a | t) = P (Q, a | t) + P (Q \ {i}, a | t). Again, it is easy to see that the incentive compatibility constraints for any agent j 6= i are unaffected. Since ti is a negative type, the only incentive compatibility constraint for i that is potentially affected is (9) at t0i = ti or where ti = tiJi −1 . But since we have only changed the checking probability and not the probabilities over actions, p̂i (ti ) is unaffected by this change in the mechanism. Analogously to the previous case, the second term on the right–hand side of (9) only involves actions in A0i , so this term also is unaffected. Hence P ∗ is incentive compatible. Finally, just as before, the probability over A is unchanged but the checking probabilities are lower, making the principal strictly better off, a contradiction. To conclude, consider equation (8) for t0i in light of the above. Since P (Q, a | t0i , t−i ) = 0 if a ∈ A0i , we see that X (Q,a)∈2I ×A|i∈Q,a∈A1i P (Q, a | t0i , t−i ) = X P (Q, a | t0i , t−i ) = qi (t0i , t−i ). (Q,a)∈2I ×A|i∈Q Hence we can rewrite (8) as p̂i (tJi i ) ≥ p̂i (t0i ) − q̂i (t0i ) for all t0i ∈ Ti+ . A similar argument applied to (9) completes the proof. In light of Lemma 13, we can directly compute q̂i (t0i ) for all t0i . Since q̂i is costly for the principal, we see that the inequalities in equations (10) and (11) must hold with equality, so ( p̂i (ti ) − p̂i (tJi i ), if ti ∈ Ti+ ; q̂i (ti ) = p̂i (tiJi −1 ) − p̂i (ti ), if ti ∈ Ti− . 57 We can use this to substitute into the objective function for q̂i to rewrite it as X Eti [p̂i (ti )vi (ti ) − q̂i (ti )ci ] = i ñ i −1 X JX i + Ki X ρi (tki )[p̂i (tki )(vi (tki ) + ci ) − p̂i (tiJi −1 )ci ] (12) k=0 ρi (tki )[p̂i (tki )(vi (tki ) − ci ) + p̂i (tJi i )ci ] ô . k=Ji The only remaining incentive constraints are that p̂i (ti ) ≤ p̂i (tiJi −1 ) ≤ p̂i (tJi i ) for all negative types ti and p̂i (tJi i −1 ) ≤ p̂i (tJi i ) ≤ p̂i (ti ) for all positive types ti . Now consider a different mechanism design problem, this one with evidence instead of costly verification. We have the same set of types as in the problem above and the i same ui functions. As above, types t0i , . . . , tiJi −1 are negative and types tJi i , . . . , tK are i positive. The principal’s objective function is now X ui (a)ṽi (ti ) i where vi (ti ) − ci , vi (ti ) + ci , ṽi (ti ) = vi (tJi i ) − ci + ciJ , ρi (ti i ) ci vi (tJi i −1 ) + ci − Ji −1 , ρi (ti ) if ti ∈ Ti+ and ti 6= tJi i ; if ti ∈ Ti− and ti = 6 tiJi −1 ; if ti = tJi i ; if ti = tiJi −1 . It is easy to see that this specification of ṽi makes the principal’s objective function in this problem the same as the expression in equation (12). We specify the evidence structure as follows. For any ti other than tiJi −1 or tJi i , we have Ei (ti ) = {{ti }, Ti }. Also, Ei (tiJi −1 ) = Ei (tJi i ) = {Ti }. The incentive compatibility constraints for this problem, then, are the following. First, since types tiJi −1 and tJi i can each claim to be the other and send the other’s (trivial) maximal evidence, each must weakly prefer her own allocation. Since tiJi −1 is a negative type and tJi i is positive, this implies p̂i (tJi i −1 ) ≤ p̂i (tJi i ). This implies that any other negative type prefers imitating tJi i −1 to imitating tJi i , while any positive type has the opposite preference. Hence the only other incentive compatibility constraints are p̂i (ti ) ≤ p̂i (tiJi −1 ) for any negative type ti and p̂i (ti ) ≥ p̂i (tJi i ) for any positive type ti , exactly the same constraints as in the costly verification model. Hence we can apply our results on optimal mechanisms with Dye evidence to compute the optimal mechanism for the evidence model as a function of ṽi . We can then substitute in terms of vi to rewrite in terms of the original costly verification model. It is straightforward to show that doing so for the case considered in Ben-Porath, Dekel, and Lipman (2014) or for the case considered in Erlanson and Kleiner (2015) yields the optimal mechanism identified there. 58 Because the assumptions used here also cover several of the variations of Example 1 discussed in Section 1, we can also use this approach and the characterization given in Examples 4 and 5 of Section 3.1 to characterize optimal mechanisms with costly verification for the case where the principal allocates multiple identical goods or the case where he allocates a “bad.” While this connection between costly verification mechanisms and Dye evidence mechanisms is likely to hold for some more general assumptions than we have used, some assumption beyond simple type dependence is necessary for it. 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