of Economic Design An extensive form solution to the adverse selection problem in principal/multi-agent environments * John Duggan

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1 Rev. EcoD. Design (1998) of Economic Design Springer-Verlag 1998 An extensive form solution to the adverse selection problem in principal/multi-agent environments * John Duggan Department of Political Science and Department of Economics. University of Rochester, Rochester, NY 14627, USA ( dugg@iroi.cc.rochesteredu) Received: 30 April 1997 / Accepted: 16 September 1997 Abstract When agents have quasi-linear preferences, every incentive compatible social choice function can be implemented by a simple extensive form mechanism, even if agents are allowed to use mixed strategies. The second stage of the mechanism, which is used to elicit the agents' true preferences, is not reached in equilibrium; it gives agents strict dominant strategies, so equilibrium outcomes are not sensitive to agents' beliefs off the equilibrium path. This solves the multiple equilibrium problem of a principal facing several agents: the mechanism implements any solution to the principal's second best maximization problem. The specification of incentive compatibility constraints in the principal's problem presupposes a precise knowledge of the agents' beliefs. However, the above mechanism can be modified to implement the principal's second best (to within arbitrarily small perturbations of transfers), regardless of the agents' conditional beliefs. JEL classification: D78, D82 Key words: Principal-agent, incentive compatible, implementation, extensive form, incomplete information 1 Introduction It is standard in the theory of incentives to analyze the principal-agent problem by invoking the revelation principle; the performance of an arbitrary mechanism can be duplicated by the truthful equilibrium of an incentive compatible direct * This paper was presented at the 1995 Canadian Economic Theory meetings in Montreal; at the 1995 conference. New Directions in the Theory of Markets and Games, in honor of Robert Aumann; at the 1995 Winter Meetings of the Econometric Society in San Francisco; and at Alabama, Caltech. Duke, Ohio State, and Wisconsin. 1 thank participants there; and Jim Bergin for his comments; and Tom Palfrey for discussions of belief revelation. Funding from SSHRCC is gratefully acknowledged.

2 168 J. Duggaa mechanism. In this paper, I consider two difficulties with the standard approach in settings of pure adverse selection. First, it is known that the optimal incentive compatible mechanism may admit non-truthfial Bayesian equilihria giving the principal a payoff below second best. This problem is especially important when two or more agents are involved, for some non-truthful equilibria may Pareto dominate truth from the point of view of the agents. To account for this possibility, it would seem that an "implementation" constraint, in addition to incentive compatibility, must be imposed on the principal. However, I construct an extensive form mechanism that achieves the principal's second best in equilibrium and, under a minimal sequential rationality hypothesis, precludes unwanted outcomes. Thus, the principal's implementation constraint is effectively slack. A second difficulty less widely recognized than multiple equilibria, but nonetheless important in applications of incentive theory, is that the formulation of the principal's second best problem (and, therefore, its solution) presumes exact knowledge of the agents' beliefs. One approach in the absence of this knowledge would be to strengthen the incentive compatibility constraints to demand that truthful reporting be a dominant strategy for the agents, but this would likely result in an expected payoff to the principal below second best. I propose a method for eliciting the agents' conditional beliefs and modifying the above mechanism to deliver a payoff to the principal arbitrarily close to second best, whatever the conditional beliefs of the agents happen to be. Thus, the principal's welfare can be sustained in the presence of uncertainty about the beliefs of the agents. 1.1 The multiple equilibrium problem Several papers have addressed the multiple equilibrium problem in moral hazard settings with two agents. Mookerherjee [24] offers an example of the multiple equilibrium problem in a model of pure moral hazard. Ma [17] adopts Mookerherjee's model and establishes conditions under which the principal's desired actions are implementable in perfect Bayesian equilibrium. Demski and Sappington [13] address the issue of multiple equilibria in a model of moral hazard and adverse selection. They propose to strengthen the incentive compatibility constraint for one agent, requiring that truthful reporting be a dominant strategy, but this generally leads to an expected payoff to the principal below second best. Giving agents non-type messages. Ma et al. [18] achieve the principal's second best outcomes with the unique Bayesian equilibrium of an "indirect" mechanism. Arya and Glover [5] construct a mechanism that approximates the principal's second best in a model of pure moral hazard with two agents operating independent production technologies, and Arya et al. [6] analyze the case in which the technologies are correlated.' ' A nice feature of Arya and Glover [5] and Arya et al. [6] is that their arguments rely only on the iterative deletioa of strictly dominated strategies.

3 Extensive form solution 169 The multiple equilibrium problem in pure adverse selection settings is analyzed in the more abstract framework of Bayesian implementation, which extends Maskin's [ 19, 20] work on Nash implementation to incomplete information environments: each agent is assumed to have a set of possible types, with typecontingent preferences and beliefs about other agents' types. Here, the outcome desired by the principal as a function of agents' realized types is given by a social choice function. The principal's second best problem is then to find, subject to the familiar incentive compatibility constraint, the social choice function delivering the highest expected payoff. The solution to this problem may be achieved, in a sense, by the corresponding direct mechanism; ask the agents for their types and choose the outcome determined by the social choice function on the basis of those reports. The incentive compatibility constraint ensures that tmthful reporting is a Bayesian equilibrium of this mechanism, but, as noted above, there may be others. The idea of implementation is to augment the message spaces of agents with non-type messages in a way that eliminates unwanted equilibria yet leaves at least one equilibrium supporting the outcomes desired by the principal. In other words, the problem is to implement the principal's social choice function in Bayesian equilibrium. Important work in Bayesian implementation includes Postlewaite and Schmeidler [32]. Palfrey and Srivastava [27. 29]. and Mookherjee and Reichelstein [25]. Jackson [15] extends the results of the preceding literature, showing that the implementable social choice functions necessarily satisfy incentive compatibility and a condition called Bayesian monotonicity. In economic environments, the conjunction of these conditions is also sufficient for Bayesian implementation.^ Thus, we can formulate the implementation constraint (to be appended to the incentive compatibility constraint) as follows: the principal's choice must be restricted to the class of Bayesian monotonic social choice functions. If this new constraint is binding, the result is again an expected payoff to the principal below second best.' Much of the literature on implementation can be seen as an effort to circumvent the monotonicity constraints arising in complete and incomplete information environments. There are two main approaches to this problem: equilibrium refinements and virtual implementation.* Both approaches have been successful, but this paper focuses on the first. Assuming agents have complete infonnation. Palfrey and Srivastava [30] show that nearly every social choice function is implementable in undominated Nash equilibrium, and Moore and Repullo [26] and Abreu and Sen [3] apply the sub-game perfection refinement to extensive form mechanisms with permissive results. The lesson of these papers is that refinements simplify the task of eliminating unwanted equilibria and. therefore, often ^ Sec Palfrey and Srivastava [31] for a thorough survey of tbe literature. Duggan [11] extends Jackson's results to implementation problems with uncountable sets of types. ^ See Palfrey and Srivastava [27] and Matsushima [22] for results on the restrictiveness of Bayesian moootonicity. * The virtual (or "approximate") implementation approach is pursued in complete informadoo environments by Matsushima [1] and Abreu and Sen [4], For incomplete information environments, see Abreu and Matsushima [2], Arya et al. [7], and Duggan [12].

4 170 J. Duggan increase the scope for implementation. This logic would seem to apply equally well to incomplete information environments. Extensive form refinements have only infrequently been employed while allowing for incomplete information. Bergin and Sen [9] offer necessary conditions and a sufficient condition, called "posterior reversal," for implementability with extensive form mechanisms. Brusco [10] offers a characterization of perfect Bayesian implementability (with acceptable belief assessments), and Baliga [8] gives results for implementation in sequential equilibrium. Though extensive form implementation under incomplete information is exceedingly complex, these authors maintain a high level of generality with respect to the set of outcomes and the preferences of agents. The cost of this generality is that the sufficiency results of these papers rely on highly complex mechanisms, and they are sensitive to the conventions adopted regarding agents' beliefs off the equilibrium path. In addition, (as in much of the implementation literature) these mechanisms may admit unwanted mixed strategy equilibria. In this paper, I exploit the relatively simple structure of principal/multi-agent environments, in which agents have quasi-linear preferences over a finite set of public decisions and a transferable private good. I assume that these preferences are completely determined by each agent's own type (a condition called private values), and that they extend in linear fashion to lotteries over public decisions. I focus on social choice functions that are value-measurable, in the sense that chosen outcomes cannot change unless at least one agent's preferences do also.^ The first result of the paper is that, assuming there are at least three agents, every incentive compatible, value-meastirable social choice function is implementable by a simple extensive form mechanism.^ The implementing mechanism is similar to one constructed by Moore and Repullo [26] for complete information environments where agents have quasi-linear preferences: it consists of only two stages, the second of which is reached only out of equilibrium and is used to elicit agents' true preferences. Agents have strict dominant strategies in the second stage, so the equilibrium outcomes of the mechanism are not sensitive to conventions regarding beliefs off the equilibrium path. Another attractive feature of the mechanism is that it produces the appropriate equilibrium outcomes even when agents are allowed to use mixed strategies. I formalize the notion of robustness with respect to beliefs off the equilibrium path by defining the concept of implementation in sequentially rational strategies. The definition involves two polar equilibrium concepts for extensive form games with incomplete infonnation, both imposing sequential rationality along the path of play but differing in their requirements off the equilibrium path. The first, weak sequential equilibrium, requires only that agents play strictly dominant strategies off the path of play, if they are available. (Updating is therefore not an issue.) The second, strong sequential equilibrium, requires that agents actually have strictly ' When any two of an agent's types determine distinct preferences, every social choice function is value-measurable. This condition is called value-distinguished types. * In a similar spirit. Palfrey and Srivastava [28] prove that the implementation constraint ceases to be binding when tbe strategic form refinement of undominated Bayesian equilibrium is used.

5 Extensive form solution 171 dominant strategies off the path of play. A mechanism implements a social choice function in sequentially rational strategies if it simultaneously implements the social choice function in weak and strong sequential equilibrium. Thus, the concept of implementation in sequentially rational strategies is extremely demanding: it insures that the principal's social choice function is supported in a strong sense (by a strong sequential equilibrium) and that unwanted equilibria are excluded in a strong sense (there are no undesirable weak sequential equilibria). Preferences are elicited in the second stage with the use of a "preference revelation device" that gives the agents strict dominant strategies to reveal their preferences tnithfully. A similar technique is used in Duggan [12]. where I show that incentive compatibility is necessary and sufficient for virtual implementation in Bayesian equilibrium. Because that paper employs the concept of Bayesian equilibrium, the advantages of preference revelation only accrue if the equilibrium outcomes of the mechanism place an arbitrarily small but positive weight on the outcomes of the revelation device. Thus, the results in that paper are for virtual implementation. Using extensive form refinements, the preference revelation device can be used effectively in a second stage that is reached with zero probability in equilibrium. Consequently, this paper provides exact implementation results. 1.2 Global implementation The specification of the incentive compatibility constraint requires exact knowledge of the agents' beliefs, which is likely to be unavailable in practice. I relax this requirement, though I maintain the standard assumption that the beliefs of the agents are common knowledge among the agents themselves. One approach to the principal/multi-agent problem under these conditions would be to impose dominant strategy incentive compatibility constraints on the principal's optimization problem, but this would most likely result in an expected payoff to the principal below second best (calculated for the true beliefs of the agents). This is perhaps to be expected if the principal does not know the agents' beliefs. However, since beliefs are assumed to be common knowledge among the agents, we may suppose they play Bayesian equilibria corresponding to their true beliefs. If the principal's objectives are flexible, in the sense that an outcome desired given one specification of the agents' beliefs is not necessarily desired given another (as is true of the principal's second best), then it may be possible to construct a mechanism that implements those objectives, regardless of the beliefs of the agents. I refer to this as global implementation. To capture the dependence of the principal's desired outcome on both agents' types and their beliefs, I define the notion of a generalized social choice function and extend the concepts of incentive compatibility and implementation to this framework. The second result of the paper is that, assuming there are at least three agents and types are value-distinguished, every incentive compatible generalized social choice function is globally implementable in sequentially rational

6 ITS J. Duggan Strategies (to within arbitrarily small perturbations of transfers). Since the principal's second best, formulated as a generalized social choice function, is incentive compatible, the principal's welfare is essentially unaffected by ignorance of the agents' beliefs. To prove this result. I construct a "belief revelation device" and incorporate it into the extensive form mechanism discussed above.^ In every weak sequential equilibrium the agents report their types truthfully, and the belief revelation device then gives each type of each agent an unambiguous incentive to report his conditional beliefs truthfully. Relying on the common knowledge assumption, we can then use this information to choose the outcome dictated by the generalized social choice function. J.3 Outline of paper In Section 2,1 set forth the framework of the paper and offer a somewhat informal definition of sequentially rational implementation. In Section 3, I present the preference revelation device, and, in Section 4,1 use it to construct a mechanism to implement an arbitrary incentive compatible, value-measurable social choice function. In Section 5, I present the belief revelation device, and, in Section 6. I use it to implement an arbitrary incentive compatible generalized social choice function. In Section 7, I consider several ways in which these results can be extended. The assumption that there are at least three agents is maintained throughout the paper, but the first theorem extends to the case of two agents if a budgetbalancing requirement is relaxed. The mechanism constructed in Section 4 violates the best response property, but if mixed strategies are not at issue then it may easily be modified to guarantee the existence of best responses. The paper focuses on social choice functions, but when there are three or more agents the results of the paper can be extended to social choice sets (see Jackson [15]). The issue of bounded transfers is also handled easily. Though the mechanism is tailored to environments with a transferable good and quasi-linear preferences, the ideas behind its construction are more general. For example, I indicate how it may be modified for use in private good exchange economies. It can also be used when agents have infinite sets of types. Appendix A contains of a precise treatment of extensive fcnn mechanisms in incomplete information environments and a precise formulation of implementation in sequentially rational strategies. ' Although the modified mechanism is extensive in form, the principles of belief revelation can also be applied to the standard Bayesian implementation problem confronting a hypothetical social planner: the planner's objectives may be represented by a generalized social choice function and implemented up to small transfers using the same techniques.

7 Extensive form solution ' '73 2 Preliminaries Let W = {I,...,n} denote the set of agents, indexed by i and ;. Let Y = {y\y-^yk} denote a finite set of public decisions with K > 2, and let Z = A(Y) = (z G R+lIlf=i2fc = 1} denote the set of probability distributions on public decisions. Assume that n > 3 and that there is a transferable private good, with allocations denoted f = (ti,...,t ) G R". Let 7 denote the set of allocations t that art feasible, in the sense that ^-^^ r, = 0. Let X = Z x T be the set of outcomes, with generic element x = {z, t). Let 0i denote the set of i's types, with generic element ^,. Let G = Xi^f^Oi, with generic element 0 = (9i,...,6n)- Assume that each Bi is finite. Each i has prior beliefs Pi on 0, which are assumed to be common knowledge and to be diffuse, i.e., for all ( and all 6, pi{9) > 0. These priors induce conditional beliefs Pi(9^i\9i) calculated from p, using Bayes rule. Agent /'s payoff from public decision yk and transfer t, contingent on 9, is Ui(yk,9i)-^- ti. (This functional form implicitly assumes private values.) These payoffs are extended to outcomes {z,t) according to expected utility. That is, i's payoff from iz,t) is.k=\ The agent's preferences contingent on 9i are more conveniently represented by the utility vector «,(^,) = (".(yi, ^i),, WiCy/f, 9i)). Then i's payoff from (z, r) is simply u,(5;)-2+f/. The type-contingent preferences of each agent are normalized so that the components of each u,(^,) sum to zero,^ and it is assumed that for all I e N and all 5, e Gi, H,(^,) ^ 0. In words, no agent is ever indifferent between every pair of public decisions. Let uid) = (u (^i). -."n(^n)) denote the utility matrix contingent on 9, and let u = max^,^ "i(^i)!- A social choice function (SCF) is a mapping 0 : 0 -> X, which can be written as (^, r), where ^ : G -> Z and r : 0 -^ T. It is incentive compatible if, for all I en and all Si,9'- ^Gi, Thus, <f> is incentive compatible if, when all types of every agent report truthfully, no type of any agent has an incentive to lie. A SCF 4> is value-measurable if. for all 9,9' G, u(9) = u{9') implies <ji{9) = (f>(6'). In words, the outcomes chosen at two type profiles can differ only if the preferences of at least one agent change as well. This is weaker ' This is an "intercept" aormalizadon, achieved by adding a constant to each type's utility function. A "scale" normalization is already implicit in setting the coefficient of li in the agent's payoff equal to one.

8 174 J. Duggan than the assumption that types are value-distinguished, in the sense that, for all 6,6' e 0, u{6) = u{d') implies 9 = 9'. In environments for which types are value-distinguished, every social choice function is value-measurable. Extensive form mechanisms, described in detail in Appendix A.I, induce extensive form games of incomplete information. Many equilibrium concepts, such as perfect Bayesian equilibrium (see Fudenberg and Tirole [14]) and sequential equilibrium (see Kreps and Wilson [16]), require sequential rationality and differ only with respect to their requirements of agents' beliefs off the equilibrium path. Let PB denote the set of (possibly mixed) perfect Bayesian equilibria. Given 9 and a mixed strategy profile a, a probability distribution, denoted {J.{.\a., 6), on X is determined. A mechanism implements a SCF 0 in perfect Bayesian equilibrium if PB^ 0 and, for all <j PB and all 5 e 0, {<l>i9)} = supp^(.j(7,9). Alternatively, for all BeG, A similar definition holds for sequential equilibrium. A more effective mechanism would support the outcomes of (p with pure sequential equilibria, denoted by SEQ, while excluding all other outcomes from the support of mixed perfect Bayesian equilibria. A mechanism simultaneously implements (j) in perfect Bayesian and pure sequential equilibrium if it implements 4> in both equilibrium concepts. Since SEQCPB, this holds exactly when, for all [j snpp{x{.\a,6)c{m}q [J A more demanding concept of simultaneous implementation is defined in Appendix A.2 using two polar equilibrium concepts. The first, weak sequential equilibrium, requires sequential rationality along the path of play and, off the play path, it requires only that agents use strict dominant strategies when they are available. Letting WSEQ denote the set of mixed weak sequential equilibria, it is clear that PBCWSEQ. The second, strong sequential equilibrium, requires sequential rationality and, at every node off the path of play, agents must have Strictly dominant strategies. Letting SSEQ denote the set of pure strong sequential equilibria, clearly SSEQCSEQ. A mechanism implements 0 in sequentially rational strategies if it simultaneously implements it in weak and pure strong sequential equilibrium. Implementation in sequentially rational strategies implies not only simultaneous implementation in perfect Bayesian and sequential equilibrium, but in any variant of these equilibrium concepts that preserves the idea of sequential rationality. No conventions about beliefs off the equilibrium path are needed.

9 Exttnsive form solutioa A preference revelation device In the environments described in Section 2, it is possible to elicit any agent's preferences, conditional on a given type, as a strict dominant strategy.' I construct a preference revelation device in two steps. In the first step I will attempt to elicit an agetit's type-contingent preferences without using transfers, but it will only be possible to elicit the direction (not the norm) of the agent's utility vector. Have agent i announce a vector 0' R'^ with components summing to zero. Based on this announcement, a public decision Ci^') is determined as follows. Let z = (l/k,...,l/k), and let ^(t)') = z+v'/lk. Note that, in order to ensure that ^'(0') is indeed a probability distribution, we must bound the norm of I'S reports. Require, therefore, that \\v'\\ < 1. Assuming no transfers are made then, conditional on type 0,, i chooses 0' to maximize ^*(v') Ui{9i) subject to Z]jfc»i %-^ ^"*^ 11^'II = ^- Equivalently, i solves and < 1. Since we assume 0. ^^ unique solution to this problem is v* = Geometrically, whenat = 3, Z is simply the face of the unit simplex in R-', and type ^,'s indifference curves are parallel straight lines with gradient u,(^,)- As depicted in Fig. 1, ^' allows the agent to pick out lotteries within a ball around I. Since u,(^/) ^ 0 by assumption, the utility maximizing point in this ball is the furthest from z, indicating the direction of u,(5,)- Thus, ^* elicits the relative values of public decisions to i, but not the monetary intensity of those evaluations. For this, transfers are needed. Fig. 1. Rather than allowing i to choose freely any point in the ball around f, let the distance from z increase with a number m' [0,u] announced by the agent See Dus aii [12] for a more general coostniction.

10 IM j J. Duggan in addition to v'. The agent will be required to pay to move further from z. Fix 0 < a < l/2ku, and define m'd' = z+am'd Regardless of m', the optimal v' remains v* = «,(^i)/ «,(^,). Thus, I's problem reduces to or max m'] u,(^,) j -. m'e[0,i7] 2 The first order necessary condition for a solution immediately yields m" = ]H,(5,) ]. Thus, (^*,r") gives each of agent i's types a strict dominant strategy to reveal the direction {v') and norm (m*) of its utility vector, and the agent's utility vector can then be calculated as m*v*. Moreover, since a may be taken arbitrarily close to zero and m' is bounded above by H, the mechanism may use arbitrarily small transfers. 4 Imptementing incentive compatible SCF's Every incentive compatible, value-measurable SCF may be implemented in sequentially rational strategies by a two-stage mechanism informally described as follows. In the first stage, agents report their types. In addition, each agent reports a non-negative rational number (positive reports are interpreted as objections on the grounds that some other agent's type is being misrepresented with positive probability), an agent (the accused), and a public decision. If no objections are registered, the mechanism terminates with the outcome given by the SCF on the basis of the agents' reported types. If exactly one agent objects, the mechanism proceeds to a second stage, where the preferences of the accused agent are elicited using the preference revelation device constructed in Section 3. If more than one agent objects, a "rational number game" (similar to an integer game or modulo game) is triggered. The mechanism is constructed so that objections do not occur in equilibrium. If the mechanism proceeds to the second stage, the outcome is determined by the preference revelation device (as far as the accused agent is concerned), so the agent's strict dominant strategy is to report his preferences truthfully. The transfer to the accuser, however, will depend on the preferences revealed by the accused. (This transfer will not affect the incentives of the accused.) If the accused reports the same preferences as in the first stage of the mechanism (indicating that the initial report was truthful), the accuser pays a small fine that decreases to zero with the accuser's reported rational number. Respecting the budget-balancing requirement, this fine is distributed across the uninvolved

11 Extemive form solution 177 agents (there is at least one, since n > 3). Otherwise (the preferences of the accused were misrepresented in the first stage), the accuser is rewarded with a transfer from the uninvolved agents. This reward may be arbitrarily small but must be fixed independently of the accuser's reported rational number. It will be proved that, in a mixed weak sequential equilibrium, the strategy of every type ^, of agent i places positive probability only on type reports fi, such that UiiOi) = Ui0i). Roughly, the argument runs as follows. If some agent j were to report, with positive probability, a type 9j such that Uj{9j) ^ Uj{9j), then any other agent i can object by reporting a positive rational number and accusing;, sending the mechanism to the second stage and receiving a reward with positive probability. By choosing his rational number small enough, / can ensure that this expected reward outweighs any expected penalties. Thus, the behavior conjectured for j cannot occur in equilibrium. Since the SCF to be implemented is assumed to be value-measurable, the equilibrium outcomes of the mechanism must then coincide with those of the SCF. Furthermore, it is readily established that there is at least one pure, strong sequential equilibrium, and the mechanism therefore implements the SCF in sequentially rational strategies. The mechanism, denoted T, is defined formally as follows. Take an arbitrary social choice function ^ = (^,r), fix 0 < c < 1, let r = max^,; \ri(6)\, and parameterize T* so that it is bounded in norm by c and r. By choosing c appropriately, transfers to or fi^om any agent in excess of J may be restricted to be arbitrarily small. Stage I In the first stage, agents simultaneously announce types, different agents, public decisions, and rational numbers in the interval [0,c]. Denote I's reportby (^'J',Jt',r') e, x W \ {J} X {1,...,A:} X (QPI [0,C]), and let ^ = (^',..., ^). There are three cases of importance. Case l.j: For all / N,? ' = 0. In this case, let the outcome of the mechanism be 4>i9). Case 1.2: There exists i N such that r'' > 0 and, for ally 4" ' ''^ = 0- In this case, the mechanism proceeds to the second stage. Case 1.3: All other reports. If i reports the lowest positive rational number (with ties going to the lowest indexed agent) then the outcome of the mechanism is j f i, where e e Z places probability one on the ^'th public decision. Stage 2 Let I denote the accusing agent andy ~j' the accused. Agenty announces v' and rh^. There are two cases. Case 2.1: If m-'v' ^ Uj{ ') then the outcome is

12 m S. Duggan The Mechanism, F: Stage 1 Each i announces every t reports f* = 0 exactly one i reports f'>0 all other cases "rational number game" Stage 2 f plays the preference revelation device. reporting the truth j' was reporting falsely i pays a fine (decreasing to zero in f') J receives reward c Fig. 2. Z = Case 2.2: If ihw = Uj{&) the outcome is z = {\- U = Ti0)-f'2u The extensive form is depicted in Fig. 2. Using this construction, the first result of the paper can now be stated.

13 Extensive rorm solution 179 Theorem \. If <f> is incentive compatible and value-measurable then F implements (f> in sequentially rational strategies. Proof Take any tr ewseq and axiy 9 e G. If Cases 1.2 or 1.3 occur with positive probability then there exist J and r' such that, with positive probability, i reports r' and r' is the lowest positive number reported. Then there is some j ^ 1 whose strategy puts positive probability on zero or a number greater than r' (or J > i and; reports r' with positive probability). Consider the deviation forj transfering that probability to any rational number r,q < r < r', and announcing his best public decision. With positive probability, r and r' will both be reported and r will be the lowest reported positive number, so that Case 1.3 obtains, andy's payoff will increase by at least c in this event. (This is true eveti if;'s report moves the mechanism from the second stage.) This increment is independent of r. In the event Case 1.2 is realized,_/'s payoff may be lower as a result of this deviation, but this decrement goes to zero with r. In the event Case 1.3 is realized and r is not the lowest reported positive number, j's payoff may be lower from deviating, but the probability of this event goes to zero with r. Thus, by choosing r close enough to zero.y's expected payoff is higher, contradicting the supposition that cr GWSEQ, Therefore, Case 1.1 holds with probability one at a and 9, and the support of ^(.{cr, 6) is concentrated on the set of outcomes 4>i.9) such that 9 is reported with positive probability. Suppose that, with positive probability, some i reports ^' such that u,c^,) / Ui{9'). Consider the deviation for; ^ i reporting r > 0 and accusing i. This takes the mechanism to the second stage, where, since a WSEQ, i reports m'v' = Ui{9i). If m'o' ^ Ui{9') then j is rewarded by a payment of c. This increment is fixed independently of r. If m'v' = Ui(9'),j''s payoff may be lower from deviating, but this decrement goes to zero with r. By choosing r close enough to zero, j's expected payoff is higher, a contradiction. Therefore, the support of fi(.\ct, 9) is concentrated on outcomes (f>i&) such that 9 is reported with positive probability and u{9) = u(9). By value-measurability, it is concentrated on 4>(9). Finally, consider the strategy profile a' that has each type 9i of agent i report (^j,i + 1,1,0) in the first stage (where n + 1 = I) and report truthfully in the second. Agents are following strict dominant strategies in the second stage. By incentive compatibility and the construction of punishments in the second stage, no agent can gain by deviating in the first stage. (This is true even if an agent moves the mechanism to the second stage, where ^* may yield public decisions preferable to ^.) Therefore, a* GSSEQ and, for all 9 G, the support of p.{.\(t'',9) is concentrated on <f>i9). D 5 A belief revelation device When agents have quasi-linear utility functions, it is possible to put in place incentives for the agents to truthfully reveal their conditional beliefs. Have agent i announce p' A(G^i)< and have all other agents announce their types 9~'.

14 tw J. Duggan Based on these announcements, i is paid where the parameter b > 0 may be chosen arbitrarily small. Conditional on type 9j and assuming other agents announce their types truthfully, / chooses p' to maximize = ^ E Note that the second equality above relies on E E p\q'-ifpiib-i%)= E which follows because, for all 9'_j e B-i, p'(9'_^)^ appears on the lefthand side once for each 9-i =f 6'_^, where it is multiplied by p,{9-i\9i). For each 5_,, the unique solution to max ^l^p,c0_,[fl;) - ^ \^Piifi-i\Q{). Thus, when other agents announce their types truthfully, f, gives i a strict best response to report the agent's true beliefs about ^_, conditional on any type ^,. Since b may be taken arbitrarily close to zero, the mechanism may use arbitrarily small transfers. ti Global implementation Without the beliefs of the agents, the class of incentive compatible SCF's is not well-defined. A SCF, ^, that is incentive compatible for one system of conditional beliefs may not be incentive compatible for another. It is, therefore, generally impossible to construct a mechanism that achieves ^ for every possible specification of conditional beliefs. If the principal's objectives are flexible (i.e., the desirability of an outcome is allowed to depend on the beliefs of the agents), however, it may be possible to design a mechanism that implements them for whichever beliefs happen to obtain. This is the idea of global implementation. In fact, the solution to the principal's second best problem does depend on the

15 Extensive form soludon 181 agent's beliefs (through the incentive compatibility constraint), and Theorem 2 shows that it is possible to globally implement these objectives to within arbitrarily small perturbations of transfers. Let Bi denote the set of mappings /?, : 0, -^ A {G-i), with the interpretation that Pt(9i) represents the conditional beliefs of agent i at type ^,. Let B = Xi^ffBi. An element of B, denoted /? = (/?!,...,^n), is a conditional belief structure specifying conditional beliefs for the agents. It is important to note that these concepts are meant to formalize uncertainty on the part of the principal, rather than the agents they have common knowledge of each other's priors and, therefore, the implied conditional belief structure. Therefore, I simply use belief structures /? 6 S to index the environments possible from the principal's perspective and, following the work on Nash implementation, I analyze the properties of a fixed mechanism under variable belief structures. The problem can be equivalently formulated in Bayesian terms, if so desired.^*' The concept of feasibility is weakened in this section to allow for arbitrarily small leakages of the transferable good. For e > 0, let 7* = {/ e R" - e < E,e/v '' ^ 0}' ^"^ ^^^^^ X' = Z X ^^ A SCF 0 : 0 ^ XMs budget^balanced if, for all ^ O, 4>{9) e X. The objectives of the principal are now formalized as a generalized social choice function (GSCF), which is a mapping ^ : B xb ^ K^. Given a conditional belief structure /3, <f (/?,.) is a SCF in the usual sense. Write ^0 _ (^^^r^) for <P(/3,.). A mechanism globally implements ^ in sequentially rational strategies if, for all /? 6 S, it implements <j)^ in sequentially rational strategies. Given a ball 5 C R" containing zero, a mechanism implements a SCF 4) = (^, r) in sequentially rational strategies at P e B to within S if, for all 6 e 0, \J supp/iz(. cr,^) C {((6)} C J and where fj.z{-\<^,6) and ^T(-W,9) are the marginals of }J.(.\(T,9) on Z and T, respectively. Note the explicit dependence of WSEQ and SSEQ on /?. As with implementation in sequentially rational strategies, the weak sequential equilibria must, for each 9, place probability one on the lottery ^(9) over public decisions. Morever, ^(6) must be supported by at least one pure strong sequential equilibrium. Unlike implementation in sequentially rational strategies, transfers may lie anywhere within the ball S around T{6). A GSCF 0 is globally implementable in sequentially rational strategies up to small transfers if, for every open ball 5 C R" containing zero, there exists "^ In the framework of Mcrtens and Zamir [23], I am assuming ihat ihcrc is a commod knowledge partition of the space of universal type profiles such that (i) corresponding lo each ^ S is an element, d, of the partition, (ii) profiles ^ ^ can be matched with profiles fl in a way that preserves preferences, and (iti) conditional beliefs over & are given hy 0.

16 182 I. Ehiggan a mechanism that, for every P B, implements ^ in sequentially rational strategies at P to within S. Assuming value-distinguished types, a slight modification of P establishes that every incentive compatible and budget-balanced GSCF is globally implementable in sequentially rational strategies up to small transfers. The modified mechanism has each agent report a conditional belief structure, along with a type, an agent, a non-negative rational number, and a public decision. If no agents object by reporting positive rational numbers and the agents agree on their reported beliefs (with at most one dissenter), then the outcome of the mechanism is given by the GSCF on the basis of the beliefs and types reported by agents, with extra transfers determined by the belief revelation device. The belief revelation device can be parameterized so that these extra transfers lie within any given ball around the desired transfers. The mechanism is designed so that no dissension occurs in equilibrium. Objections are handled as in F, and the same arguments as in the proof of Theorem 1 establish that, in equilibrium, the agents report their types truthfully. The belief revelation device then ensures that agents report their own conditional beliefs truthfully. Thus, the outcomes of the mechanism are given by the GSCF on the basis of truthful reports. Though the equilibrium outcomes of the mechanism place the desired probabilities on public decisions, the belief revelation device calls for agents to make arbitrarily small payments in excess of the transfers called for by the GSCF. It is for this reason that the GSCF is required to be budget-balanced. (This could be weakened slightly to require only that ^ be interior-valued in the sense defined in Subsection 7.4.) A close look at the mechanism constructed in the proof of Theorem 2 reveals that an agent Ts excess payment cannot simply be spread across the remaining agents, for that amount depends on the reported types of the other agents through f^, defined in Section 5. Theorem 2. Fix e > 0 and assume value-distinguished types. If 4>^ is incentive compatible and budget-balanced for all P ^ B then ^ is globally implementable in sequentially rational strategies up to small transfers. Proof Fixing S, a modification of P establishes the desired result. An agent / now reports ( 'J'J',k',r') Bi x B x N \ {i} x {I,...,K} x (Qn[O.c]), where c > 0 may be chosen arbitrarily small. Whereas before there were three relevant cases, there are now four. Case 1.1: (no partial consensus) There do not exist i N and p Q B such that, for all; ^t,^ =/3. Case J.2: Partial consensus and, for all i e N, r' = 0. Case 1.3: Partial consensus and there exists i e N such that r^ > 0 and, for all j^irr^^o. Case 1.4: All other reports. In Cases 1.2, L3, and 1.4, there exist i e N a.tid 0 e B such that, for all j ^ i, pi = p. For these cases, the outcomes of the mechanism are determined as in P for the SCF 0^, with the exception that an amount fi(0i{9'),6~') - b

17 I Extensive form solution 183 is added to each agent i's transfer, where b > O'ls the bound from Section 5 on the extra transfers to the agents. It follows that the extra payment to each agent is negative. To ensure feasibility, b must be set so that b < e/2n. To ensure that equilibrium transfers lie within the ball S around the desired transfers, it must be that 2b^/n is less than the diameter of S. Thus, agent i's own reported conditional beliefs are used with the reported types of other agents to determine an extra transfer according to the belief revelation device. In Case 1.1, if i reports the lowest positive rational number (with ties going to the lowest indexed agent) then the outcome of the mechanism is y = e*' Now take any p B,(T GWSEQ(/3), and (9 e 0. Clearly, Case 1.1 must hold with zero probability, so there exist i and /3 such that every type of every agent j ^ i places probability one on $. Furthermore, the arguments in the proof of Theorem 1 show that Case 1.2 must hold with probability one, so the support of /j(.]cr,5) is concentrated on the set of outcomes ^0,9) such that 9 is reported with positive probability. The arguments in the proof of Theorem 1 also show that, for all i and all 9' reported with positive probability, Ui(9') = u,(9i). Under the assumption of value-distinguished types, this implies that each agent i reports 9' = 9i with probability one. Because of the additional transfers from the belief revelation device, it is a strict best response for / to report/?' with /th component satisfyingp^(^,) = /3,(5). Therefore, 4 = /?, supp^z(. a,e) C {^^(0)}, and supp^h.k,^) C T^{9) + S. It is easy to see that the strategy profile defined in the proof of Theorem 1, modified so that every type of every agent reports his conditional beliefs truthfully, is a pure strong sequential equilibrium. 7 Extensions 7.1 Two agents The mechanism f, constructed in Section 4, relies on the existence of at least three agents only to balance payments, ensuring feasible outcomes. If the concept of feasibility is weakened to X = K x {r e R" J^t^N '' ^ ^l' Theorem 1 holds for n = 2 with slight modifications to F. In Case 2, the total payment to / and j may be positive, so that a lump sum (independent oij's reported preferences) must be deducted from y's payment. 7.2 The best response property As it is currently defined, the mechanism F fails to give agents best reponses for some strategy profiles. This may occur, in particular, when an agent wishes

18 184 J. Duggan to report a positive rational number, but one as small as possible. This feature is needed when agents are allowed to use mixed strategies, because an agent may be reporting falsely with arbitrarily small probability, and any other agent's expected reward to objecting may be correspondingly small. If agents use only pure strategies and each agent's set of types is finite (as we have assumed), the probability that an agent misrepresents his type has a positive lower bound, as does every other agent's expected reward for objecting. The outcomes of r in Case 2, as it is defined in Section 4, depend on the accusing agent's reported rational number (denoted r'). If the probability that an agent's reward for objecting has a positive lower bound, the number r' in the outcomes specified in Case 2 can be replaced by a fixed parameter r (0,1) that, if chosen low enough, preserves the incentive properties of the mechanism. The modified mechanism guarantees the existence of best responses for the agents. 7.3 Social choice sets The focus of Section 4 is on SCF's 4> : Q ^ X. Thus, for each 9 exactly one outcome is selected. A more general formulation would define ^ as a social choice correspondence., selecting a non-empty set of outcomes for each 6. In incomplete information environments, even more generality is possible. A social choice set, denoted F, is a collection of social choice functions. A social choice set F is incentive compatible if each 0 G F is incentive compatible. Given a mechanism and strategy profile a, let g(a{9)) = supp/.i(. a, 9). If this set is a singleton for all 9, we can view g o cr as a SCF itself. An extensive fonm mechanism implements F in sequentially rational strategies if, for all 6 eo, {g o a\a e WSEQ} CFC{goa\(TG SSEQ}. Assuming n > 3, F can be modified to implement any incentive compatible social choice set using standard techniques." 7.4 Bounded transfers The construction of F relies on the ability to reward an agent with a payment in excess of 7 = maxj^g r,(5) by an arbitrarily small amount. If the set T of feasible allocations is bounded, perhaps T = {t e R" J^i^N ^' = ^ ^"^ r, > -1 for all i}, this may not be possible. In this example, the highest feasible payment to an agent is n I. If Ti{9) = n 1 for some i and 9, it would not be possible to reward i with an additional payment. A similar problem arises with respect to penalties for agents. Write t > t' when, for all i, r, > f/. A SCF <^ is interior-valued if, for all 9, there exist t,t' & T such that f > r($) > /'. Because the rewards and penalties These social choice sets musi also satisfy closure (see Jackson [15]). When priors are diffuse, as they are in all but the last two subsections of this paper, every social choice set satisfies closure.

19 Extensive form solutioa 185 used by F can be made arbitrarily small, the assumptions of Theorem 1 can be strengthened so that the theorem is true even when T is bounded: if (f) is incentive compatible, value-measurable, and interior-valued then f, parameterized appropriately to ensure feasible outcomes, implements <p in sequentially rational strategies. 7.5 Private good exchange economies In a private good exchange economy, each agent i consumes a bundle xi 6 Rt of L commodities. An allocation x = (xi,...,xn) is feasible if Yli^N^' = u, where w is the aggregate endowment of commodities. Let Y be the set of feasible allocations, and let X be the set of probability distributions over feasible allocations. Agent i's preferences are given by a utility function Ui : Y x 9, -> R that is constant in x_, and monotonic and continuous in Xj. Preferences over probability distributions are given by expected utility. The key features here and in the principal/multi-agent environments of Section 2 are continuity of preferences and the possibility of independently rewarding and punishing agents by transfers of a private good. For simplicity, the focus of this subsection will be SCF's ^ that are nonstochastic, in the sense that each (pib) places probability one on a single allocation. Thus, (p(9) may be interpreted as an allocation rather than a probability distribution. Write 0 = {(f>i,...,(!> ), where 4>i(9) is the commodity bundle allocated to agent i at type profile 9. Furthermore, assume n > 3. (This is not necessary if resources can be feasibly destroyed.) Transfers of commodities are bounded in exchange economies, but this is not a problem for non-stochastic SCF's 0 such that, for all i and all 9, <f>i{9) > 0. Altering slightly the usage in the previous subsection, such an SCF is interior-valued. A potentially more serious problem is the existence of a preference revelation device in private good exchange economies. It is proved in Duggan [12], however, that such mechanisms exist in a wide range of environments, including private good exchange economies. For an agent j, let x.*{u^) denote the probability distribution on allocations fory determined by the revelation device when j reports utility function u^ : K > R. Now consider an incentive compatible, value-measurable, interior-valued, non-stochastic SCF (j). The arguments of Section 4 show 0 is implementable in sequentially rational strategies by F, modified as follows. In the first stage, each agent i now reports {9'J',r'). Three cases are distinguished, just as in Section 4. In Case 1, the outcome of the mechanism is (f>(9). In Case 2, the mechanism proceeds to a second stage. Letting i denote the accusing agent and j = j' the accused, j will report a utility function «'. If u^ ^ Uj(.\9^) then, with probability 1 r',j receives (t>j{9) while i receives the leftover endowment; and with probability r',;'s allocation is x^iw) while i receives the leftover endowment. If H-' = Uj{.\^) then, with probability 1 r', the outcome is ^{9)\ and with probability r',y's allocation is Xj'(u^) and the leftover endowment is allocated to

20 186 J, Duggan any agent other than i and j. In Case 3, the agent reporting the lowest positive rational number (with ties going to the lowest indexed agent) receives the entire endowment Clearly, in a weak sequential equilibrium. Case 1 must hold with probability one. The mechanism is constructed so tbat, following an objection by a single agent, the accused agent has a strict dominant strategy to report truthfully in the second stage. Were any agent to report, with positive probability, a type with the wrong preferences, another agent could deviate profitably by reporting a low enough positive number. Thus, every weak sequential equilibrium produces outcomes consistent with <f>. The strategy profile that has agents report their types truthfully along with zero is a pure strong sequential equilibrium. Thus, this modification of F implements ^ in sequentially rational strategies. 7.6 Non-diffuse priors The assumption of diffuse priors can be weakened substantially, at the cost of some additional complications. Instead, assume merely that agents' priors have common support: for all i, j, and 9, pi{9) = 0 if and only if pj{9) = 0. Let ^ denote the set of type profiles with positive probability according to the agents' priors, and, for each i, let ^,- = {9, G Oi\pi({9i} x _,) > 0} denote the set of i 's types with positive marginal probability. The definition of incentive compatibility must be adapted to the weaker assumption of common support. A SCF (^ = (^,T) would be incentive compatible if, for all i e JV, al! ${ G ^f, and The quantifiers implicit in the definitions of weak and strong sequential equilibrium in Appendix A.2 should be understood to range over 0, for each i, and the definition of implementation in sequentially rational strategies must also be modified to hold for all 9 e&. With these adjustments, F still implements every incentive compatible SCF in sequentially rational strategies. Consider any weak sequential equilibrium, take any 9 & and suppose that Cases 2 or 3 hold with positive probability, so some agent i is announcing a positive number with positive probability. We can, in fact, assume that i's number is the lowest reported at 9 with positive probability. A contradiction arises if, for some; ^ i, there is a type with positive marginal probability that can gain from deviating. By the assumption of common support, 9j is just such a type. Therefore, Case 1 must hold with probability one. The rest of the arguments in Section 4 proceed similarly.

21 Extensive Form solution Infinite sets of types ' Few issues are raised when agents are allowed to have countable sets of types, as long as agents' priors have common support. When the 0, are uncountable, however, technical difficulties make a general treatment of extensive form games more challenging. The difficulties are fewer for the mechanism F, since it consists ofonly two stages. Moreover, Theorem 8 of Duggan [12] establishes the existence of a preference revelation device in uncountable environments: it is required only that there exist a countable set of outcomes over which no type of any agent is entirely indifferent and that each Ui(.\9i) be continuous. Thus, updating of prior beliefs is not an issue In the second stage. After supplying the problem with some needed structure, I argue that F implements every incentive compatible, value-measurable SCF, even when the agent's type sets are uncountable. I will suppose that agents are restricted to pure strategies for the remainder of this discussion. A cr-algebra on 0, must be explicitly specified and O equipped with the product (7-algebra, on which each pi is defined. The condition of common support is extended to such environments as follows: for all i and j and all measurable A C 0, pi(a) > 0 if and only if pj(a) > 0. Call this condition mutual absolute continuity. The definition of incentive compatibility can then be written to hold for each i and p,-a.e. type 9i, as long as (i) the SCF is measurable with respect to the product a-algebra on &, and («) each agent is restricted to measurable type-reporting strategies. Now consider an incentive compatible, value-measurable SCF that is measurable with respect to the product a-algebra on G. Consider a (measurable) weak sequential equilibrium a and suppose that, with positive probability, there is some agent who reports a positive rational number. (This is unambiguous, since we assume mutual absolute continuity.) Then some number r must, with positive probability, be the lowest reported (recall that Q n [0,c] is countable), and it follows that some agent i must submit this lowest report with positive probability. Then some j / / must, with positive probability, report zero or a positive number higher than r. That is, the set {9j G 9j\<7j{9j) reports 0 or r^ > r} has positive marginal probability according to pj. Consider the deviation a' identical to (Tj except that each type in this set reports / ', 0 < r' < r, so that, with positive probability, j now reports the lowest positive rational number. This raises 7's payoff by an increment that is fixed as r' goes to zero. By picking r' low enough, ;' s expected payoff is higher, a contradiction. Therefore, no agent reports a positive number with positive probability. The other arguments in the proof of Theorem I go through much as above. Once the notational apparatus is in place, the greatest difficulty presented by uncountable type sets is the same as that created by mixed strategies: the probability that an agent reports a positive number or falsely reports a type may be arbitrarily

22 188 J. Duggan small. Because F is constructed to address mixed strategies, uncountable sets of types are easily handled. Appendix AJ Extensive form mechanisms An extensive form mechanism consists of a game tree, an information structure, a labeling of nodes in the game tree, and an outcome function. Formally, a game tree is a set Q of nodes ordered by a precedence relation -< and an initial node ^0-^^ The relation -; is used to denote immediate precedence, and >- denotes immediate succession. Let S{q) = {q'\g -< q'} denote the immediate successors of ^. A path from qi E Q is a sequence (.q\,q2,- ) such that for al! i,qi ^ gj+j. If a node q' lies on a path from q, the path connects q to q'. The initial node is characterized by the property that it is connected to every non-initial node. (Thus, it is unique.) A node is terminal if it precedes no other nodes. An information structure consists of an equivalence relation '^ on Q and a mapping i : Q ^ N assigning agents to nodes. Let [q] denote the equivalence class of q according to ~, which will be interpreted as the infonnation set containing q. Each non-initial node q is labeled a{q), interpreted as the action necessary to reach q from its predecessor. Let A{q) = {oc{q')\q -< q'} denote the set of actions available at q, and let A = \}g^^^a{q). An outcome function g maps terminal nodes q to outcomes g{q) X. It is required that 1. If.? -- q' then i{q) = t(q'). 2. Ifq^ q', q -< q'\ and a{q') = a{q") then q' = q". 3. If q-^ q' ihtn A{q) = A{q'). 4. All paths are finite. ' 5. A{q) is countable. ' The last assumption is made for reasons of technical expedience. Together with a utility function u, and prior beliefs p,- for each agent /, an extensive form mechanism induces an extensive form game of incomplete information. A behavioral mixed strategy for i is a mapping a^ : L~^{i)xGi > A{A) such that (a) q ^ q' implies o-,(. g,^,) = (7i{.\q',9i), and (b) s\ipp<7i{.\q,9i) C A{q). Let Ei denote the set of i's strategies, and let S = Xi^i^Ej, with elements a = (ctj,...,itrt), referred to as strategy profiles. Given q E Q and 9 e 0, a strategy profile a induces a probability distribution, denoted fj.*i.\q,(j,9), on nodes q' connected to q. Let q = gi, q' = qi, and let (9I,-..,9L) be the path connecting q io q'. Then This induces a probability distribution, denoted /i(. ^, a, 6), on outcomes, defined by '^ -< is a precedence relation if it is transitive and q' -< q and q" -< q imply q' = q".

23 Extensive form solution 189 for alljc. Let^(. a-, ^) = fi(.\qq,a,b). A pure strategy profiler is a strategy profile such that, for all i, all q and all 9^, jsuppcr,(. g,^,) = 1. Agent i's beliefs at any point in the game are represented by an assessment Pi : t ~'(0 X Gi -* AiQ X 0^i) such that (a) q ^ q' implies ^.( i^,^,) = Pi{.\q'A), and (b) supp/'.c-k.^i) C [q] x a_/. Thus, i's beliefs at node q conditional on 9i are P,(k,^.)- Let P = (?i,...,/* ) denote a profile of assessments. Given q e Q,(T E, and 5, 0,, an assessment for i determines a probability distribution, denoted X'{.\q,a,9i,Pi), on Q x G_i. Let This also determines a probability distribution, denoted A(.ji?,(7,^,,F,), on outcomes, defined by for all X. A.2 Sequentially rational strategies Consider the following two notions of equilibrium. A strategy profile cr is a weak sequential equilibrium if there exists an assessment profile P such that {cr,p) satisfies (i) For alii e/v.al!^, 6>,, and all <? & L-^{i)with X'i[q]xG-i\qo,(T,9i,pi) > 0. 0-,- 6 arg max V[u,(^,) z+fi]a(x ^,(ff-,ct-,)>^/,^i), wbere Pi(.\q,9i) is defined on subsets of [q] x 0_i by P(\a B^- -^*(-I^O'^'^'-P') (ii) For alii N.all^, e0/,andall^ t-'(i) with A* 0, (Ti{a\qA) = 1 if. for all a', all p!, all q' S{q) n a-'(a), and all Thus, weak sequential equilibrium is weak indeed. Condition (i) requires sequential rationality along the path of play, where agents update using Bayes rule, and condition (ii) requires that, off the path of play, if an action is a strict dominant strategy for an agent then he will play it. In particular, every perfect Bayesian equilibrium is a weak sequential equilibrium. Given an extensive form

24 190 J. Duggan mechanism and utility functions and prior beliefs for agents, let WSEQ denote the weak sequential equilibrium strategy profiles. At the opposite extreme are the strong sequential equilibria. These are the strategy profiles a for which there exist P such that (a, F) satisfies (i), (ii), and (iii) For all I G/V, all 5, e e,, a n d a l l ^ e i~\i)with X'{[q]xe^i\qQ,(T,ei,pi) = 0, if ai{a\q,9i) > 0 then, for all a\ all P/, all q' Siq) n a-\a), and all Condition (iii) requires that, off the path of play, agents must have strictly dominant strategies. A strong sequential equilibrium presumes very little about the sequential rationality of agents and is therefore quite strong. In particular, every strong sequential equilibrium is a sequential equilibrium. Let SSEQ denote the pure strong sequential equilibrium strategy profiles. An extensive form mechanism implements a SCF (f> in sequentially rational strategies if, for all 9 0., [j supp/x(. (j,e) C {0(0)} C U snpptsi.\a,9). Since PBCWSFQ and SSFQCSEQ, implementation in sequentially rational strategies implies simultaneous implementation in perfect Bayesian and pure sequential equilibrium. References 1. Abreu, D,, Matsushima, H. (1992a) Virtual implementation in iteratively undominated strategics: complete infonnation. Econometrica 60; Abreu, D., Matsushima, H. (1992b} Virtual implementation in iteratively undominated strategies: incomplete information. Mimeo 3. Abreu. D., Sen. A. (1990) Sub-game perfect implementation: a necessary and almost sufficieni condition. J- Econ. Theory 50: Abreu, D., Sen, A. (1991) Virtual implementation in Nash equilibrium. Econometrica 59: Arya, A., Glover, J. (1995) A simple forecasting mechanism for moral hazard settings, J. Econ. Theory 66: Arya, A., Glover, J., Hughes, J. (1997) Implementing coordinated team play. J. Econ. Theory 74: Arya, A., GloverJ., Young, R. (1995) Virtual implementation in separable Bayesian environments using simple mechanisms. Games Econ. Behavior 9: Baliga, S. (1997) Implementation in economic environments with incomplete information: the use of multi-stage games. Mimeo, Northwestern University 9. Bergin, J., Sen, A. (1995) Extensive form implementation in incomplete information ecoqomic environments. Mimeo, Queen's University 10, Brusco, S. (1995) Perfect Bayesian implementation. Econ. Theory 5: , Duggan, J, (1994) Bayesian implementation. Ph.D. Thesis, California Institute of Technology 12, Duggan, J. (1997) Virtual Bayesian implementation. Econometrica 65:

25 Exteosive form solution Demski, J., Sappiogton, D. (1984) Optimal incentive contracts with multiple agents. J. Econ. Theory 33: Fudenberg, D.. Tirole, J, (1991) Perfect Bayesian and sequential equilibrium. J. Econ. Theory 53; Jackson, M. (1991) Bayesian implementation, Econometrica 59: 461^ Kreps, D, Wilson, R. (1982) Sequential equilibrium- Econometrica 50: Ma, C, (1988) Unique implementation of incentive contracts with many agents. Rev. Econ. Stud. 55: Ma, C, Moore, J., Turabull, S. (1988) Stopping agents from "cheating." J, Econ. Theory 46: Masldn, E. (1977) Nash equilibrium and welfare optimality. Mimco, Northwestern University. 20. Maskin. E. (1988) The theory of implemenution in Nash equilibrium- (n: Hurwicz, L. et ai. (eds.) Social goals and social Organization: essays in honor of Elisha A. Pazner. Cambridge University Press. Cambridge, pp Matsushima, H. (1988) A new approach to the implementation problem. J. Econ. Theory 45: 12& Matsushima. H. (1993) Bayesian monotonicity with side payments, J- Econ, Theory 59: Mertens, J.-F,. Zamir. S. (1985) Formulation of Bayesian analysis for games with incomplete information. Int. J. Game Theory 14: Mookherjee, D, (1984) Optimal incentive schemes with many agents. Rev. Econ. Stud, 51: Mookherjee, D., Reichelstein, S. (1990) Implementation via augmented revelation mechanisms. Rev, Econ, Stud. 57: Moore, J., Repullo, R, (1988) Sub-game perfect implementation, Econometrica 56: Palfrey, T,, Srivastava, S. (1987) On Bayesian implementahlc allocations. Rev. Econ. Stud. 54: Palfrey. T,. Srivastava, S- (1989a) Mechanism design with incomplete information: a solution to the implementation problem, J, Polit. Econ, 97: Palfrey, T,, Srivastava, S. (1989b) ImplemenUtion with incomplete infonnation in exchange ccooomies. Econometrica 57: Palfrey, T,. Srivastava, S. (1991) Nash implementation using undominated strategies. Econometrica 59: Palfrey, T., Srivasuva, S. (1993) Bayesian implementation. (Fundamentals of Pure and Applied Economics 53) Harwood, Langhome 32- Postlewaite, A,. Schmeidler, D, (1986) Implementation in di^erential infonnation economies. J. Econ. Theory 39: 14-33

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