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 FACULTY WORKING 
 PAPER NO. 89-1527 
 
 Mechanisms with No Regret: 
 Welfare Economics and 
 Information Reconsidered 
 
 FEB 2 3 1989 
 
 UK 
 
 "■UNJiS 
 
 
 Bhaskar Chakravorti 
 
 WORKING PAPER ON THE POLITICAL ECONOMY OF INSTITUTIONS 
 NO. 25 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 89-1527 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana- Champaign 
 
 January 1989 
 
 Mechanisms with No Regret: 
 Welfare Economics and Information Reconsidered 
 
 Bhaskar Chakravorti, Assistant Professor 
 Department of Economics 
 
 I have benefitted from comments and discussions in seminars at the 
 University of Illinois at Urbana -Champaign, Penn State University and 
 the Game Theory Conference in honor of Lloyd Shapley in Columbus, Ohio, 
 where this paper was presented. I thank the participants. 
 Conversations with Lanny Arvan were especially helpful. The usual 
 disclaimer applies. 
 
MECHANISMS WITH NO REGRET: 
 
 Welfare Economics and Information Reconsidered 
 
 by 
 
 Bhaskar Chakravorti 
 
 Department of Economics 
 
 University of Illinois at Urbana-Champaign 
 
 Champaign, IL 61820 
 
 February 1988 
 Revised: December 1988 
 
 Abstract 
 
 This paper achieves four objectives relating to the design of efficient 
 mechanisms in economies with asymmetric information. First, we show that 
 it is impossible to design an individually rational and efficient mechanism 
 using Bayesian equilibrium as the solution concept. Under complete 
 
 information, however, such mechanisms can be designed. Second, we attempt 
 to bridge this gap between complete information economies and an important 
 sub-class of asymmetric information economies by exploring the 
 possibilities with "mechanisms with no regret" that leak information. A 
 complete characterization is given of "No regret-implementation" using such 
 mechanisms. These mechanisms are played in four stages including a "regret 
 phase" and yield a refinement of Green and Laffont's (1987) posterior 
 implementability concept. Third, it is shown that though such mechanisms 
 cannot implement interim individually rational and efficient performance 
 standards, they can implement ex post individually rational and efficient 
 standards. Finally, it is shown that even under asymmetric information 
 such mechanisms implement any Nash implementable performance standard, such 
 as the core or the Walrasian correspondence. 
 
 Key Words: Implementation, Bayesian equilibrium with no regret, individual 
 rationality-efficiency, asymmetric information. JEL Code: 026, 
 
 I have benefited from comments and discussions in seminars at the 
 University of Illinois at Urbana-Champaign, Penn State University and the 
 Game Theory Conference in honor of Lloyd Shapley in Columbus, OH., where 
 this paper was presented. I thank the participants. Conversations with 
 Lanny Arvan were especially helpful. The usual disclaimer applies. 
 
1. Introduction 
 
 In economies with asymmetric information, a performance standard is a 
 non-empty set of socially "desirable" state-contingent allocations. Given 
 the incompleteness of information about the realized state of the world, a 
 mechanism is required to implement a given performance standard. An 
 implementing mechanism is a game of incomplete information with a non-empty 
 set of equilibrium outcomes that is contained within the standard. {Full 
 implementation requires coincidence of the two sets.) Most economies 
 aspire towards the two basic objectives of individual rationality and 
 efficiency. This paper reports a fundamental difficulty with any attempt to 
 design mechanisms for implementing individually rational-efficient 
 performance standards, and then proposes conditions under which there 
 exists a solution to this crucial problem. In light of the difficulties 
 that were just alluded to, this is the only result that demonstrates the 
 possibility of designing individually rational-efficient mechanisms in a 
 wide class of asymmetric information economies. Moreover, we show that 
 despite the information asymmetry it is possible to implement any standard 
 that is Nash-implementable (i.e. with complete information), such as the 
 core or the Walrasian correspondence. 
 
 By the Revelation Principle (Myerson (1979), Dasgupta, Hammond and 
 Maskin (1979), Harris and Townsend (1981), etc.), any performance standard 
 that has a non-empty intersection with the set of Bayesian equilibrium 
 outcomes of a game must satisfy an incentive comaptibility (or 
 self-selection) condition. We know of no satisfactory performance standard 
 that meets this condition. Thus, if we are interested in questions of 
 mechanism design in a wide class of environments, the conclusions are 
 
negative right away. 
 
 To surmount this initial hurdle, Palfrey and Srivastava (1987a) invoke 
 Postlewaite and Schmeidler's (1986) restriction on the informational 
 structure — non-exclusivity of information (NEI) — and then ask whether 
 performance standards that were fully implementable under complete 
 information can be fully implemented when information is asymmetric. The 
 NEI restriction essentially requires that if all agents except one pool 
 their private information they can deduce the information of the remaining 
 agent. It ensures incentive compatibility but is clearly restrictive. 
 
 Unfortunately, some restriction of this nature is necessary to evaluate the 
 implementability of natural economic performance standards. The striking 
 message of Palfrey and Srivastava is: despite the restriction and despite 
 the fact that incentive compatibility is satisfied, full Bayesian 
 implementation of either the interim or the ex post efficiency standard (as 
 defined in Holmstrom and Myerson (1983)) is impossible. This is in sharp 
 contrast to the positive results obtained under complete information. 
 
 Palfrey and Srivastava's (1987a) negative results provide the starting 
 point for this paper. We extend their arguments and show that the 
 situation is even more grim than is implied by their results. Define an 
 efficient Bayesian mechanism as a game such that all of its Bayesian 
 equilibrium outcomes are efficient (in either an interim or ex post sense). 
 However disappointing the Palfrey and Srivastava conclusions may be, they 
 do not imply that we can never hope to design an efficient mechanism since 
 there still exists the possibility of Bayesian-implementing (as opposed to 
 fully Bayesian-implementing) the efficiency standard or one of its subsets. 
 
 The first part of this paper argues that, indeed, even under NEI, we 
 cannot hope to find an efficient Bayesian mechanism if we impose an 
 
additional restriction of individual rationality on the standard. The 
 
 latter rules out uninteresting outcomes that assign all resources to one 
 individual. Such outcomes are efficient and trivially implementable. To 
 summarize, both the interim and the ex post individual 
 rationality-efficiency standard (and their respective subsets) are 
 non-implementable in Bayesian equilibria. An additional implication of 
 
 this part of the paper is: when one is interested in checking for 
 impossibility results, "implementation" (as opposed to "full 
 
 implementation") is the notion that one should focus on; a standard that is 
 not fully implementable may be implementable whereas one that is not 
 implementable can never be fully implementable. 
 
 These impossibility results indicate the cost, in terms of social 
 welfare, that is imposed on society by the presence of information 
 asymmetry. Under complete information, the literature on 
 
 Nash-implementation, (Maskin (1977), Hurwicz, Maskin and Postlewaite 
 (1984), etc.) has shown that individually rational-efficient mechanisms do 
 indeed exist. Can this gap be filled by a mechanism that not only 
 allocates resources but also "leaks" information? 
 
 The next step in the agenda is to tackle the problem of non-existence 
 of individually rational-efficient Bayesian mechanisms by addressing the 
 question posed in the previous paragraph. We take advantage of the fact 
 that the mechanism designer has the flexibility to not only choose the game 
 but also how it is played. Following the lead of Green and Laffont (1987), 
 we study the case of mechanisms with a "regret" phase. Green and Laffont 
 do not provide any specific structure to the changes in the rules of the 
 game. In this paper, we adopt a particular sequence of moves that allows 
 for a regret phase in a simultaneous-move game. This sequence is a slight 
 
modification of the structure that is implicit in Green and Laf font's 
 model. The resulting equilibrium that we obtain is a refinement of Bayesian 
 equilibria and of posterior implementable equilibria (due to Green and 
 Laffont). 
 
 The structure that we employ evolves in four stages and is informally 
 described with the help of an example in the following paragraphs. 
 
 Example 1: Consider a game with two players and three states of the 
 world: "Rain", "Shine" and "Clouds". Player 1 chooses T or B and is 
 completely uninformed. Her prior beliefs are that there is an equal chance 
 of any one of the states occurring. Player 2 chooses L, M or R and is 
 completely informed. Their payoffs are given in Figure 1. This is the 
 original game. The pure strategy Bayesian equilibria of the original game 
 are given in Figure 2. A game designer can introduce a regret phase in 
 rules of play of this game. 
 
 [Insert Figures 1 and 2 here.] 
 
 The game with a regret phase is played in four stages. In the first 
 stage, the players and the game designer calculate the set of Bayesian 
 equilibria in the original game. This constitutes the maximal set of 
 possible outcomes. Next, Nature selects a state of the world. In the 
 second stage, the players submit to the designer the actions they intend to 
 take in every equilibrium. These submissions are not commitments and are 
 made public knowledge by the designer. In the third stage, the agents are 
 given the opportunity to entertain the desire to unilaterally deviate from 
 one or more of the Bayesian equilibria. In the final stage, (as is usually 
 assumed in game theory) by some unbiased method a particular Bayesian 
 equilibrium, say s, is selected from the set comprising equilibria from 
 which no player wants to unilaterally deviate. s becomes common knowledge 
 
among the players and the designer. The game designer distributes the 
 outcome/payoffs to the players by checking the actions that each player had 
 submitted for s and using the payoff matrices in Figure 1. 
 
 It can be observed that this structure is different from the one that 
 underlies Green and Laffont's notion of posterior implementable equilibria. 
 In their case, the unbiased selection of the equilibrium is done before the 
 second stage. Thus, they would require that in the second stage the 
 players submit the actions that they intend to take in the particular 
 equilibrium that is chosen. 
 
 To return to the example, a Bayesian equilibrium will survive the 
 third stage if and only if the information conveyed by the second stage 
 public submissions is such that neither player has an incentive to 
 unilaterally deviate in stage three from the action he/she had submitted 
 for that equilibrium in the previous stage. Such an equilibrium is 
 
 referred to as a Bayesian equilibrium with no regret. 
 
 Since equilibria are pairs of strategies that are common knowledge 
 functions, Player 1 could conceivably acquire some information if Player 
 2's action were associated with a unique state in some Bayesian 
 equilibrium. However, not all Bayesian equilibria convey information in 
 this manner. 
 
 Let the set of Bayesian equilibria with no regret be denoted E(D. 
 Every Bayesian equilibrium with no regret must be immune to information 
 revealed by any other Bayesian equilibrium with no regret. This follows 
 from the fact that in stage four a Bayesian equilibrium with no regret will 
 be chosen to determine the final outcome. Given that in stage two the 
 players do not know which one will be chosen, they will not have an 
 incentive to unilaterally make arbitrary submissions for any equilibrium in 
 
E(D. This rules out unilateral manipulation on the part of the players by 
 the submission of messages that could mislead others. An equilibrium that 
 is not in E(D does not convey information since the players may submit 
 messages corresponding to such an equilibrium to mislead others about the 
 state of the world knowing that such equilibria shall never be used to 
 determine the final outcome. 
 
 The calculation of the set of Bayesian equilibria with no regret may 
 appear somewhat circular. We shall show how it is determined for the 
 
 example at hand. 
 
 * 
 Observe that the functions s', s" and s carry information if the 
 
 state is "Clouds". The equilibria s' and s remain self-enforcing even if 
 
 the information were revealed to Player 1 in the "Clouds" state by s', s" 
 
 * * 
 
 or s . This gives us a sufficiency condition: s' and s are Bayesian 
 
 equilibria with no regret. Are there any others? A necessary condition is 
 
 that a Bayesian equilibrium with no regret must not be destroyed by 
 
 information conveyed by another Bayesian equilibrium with no regret. It 
 
 can be checked that in s and s" Player 1 has an incentive to deviate in the 
 
 "Clouds" state given the information conveyed by s' and s . Thus, s' and 
 
 s constitute the set of Bayesian equilibria with no regret, E(D. 
 
 We conclude this example with two observations. First, note that the 
 
 * 
 Bayesian equilibria with no regret (e.g. s ) are not necessarily complete 
 
 information Nash equilibria of the game. Second, if we had employed Green 
 
 * 
 and Laffont's criterion, s, s' and s would have survived and s" would have 
 
 been eliminated. In sum, our criterion of regret requires greater 
 
 robustness from an equilibrium. While this is a strength from the 
 
 viewpoint of stability, it makes it more difficult to guarantee existence 
 
 of such equilibria. In this paper, we shall, however, prove the existence 
 
of the set of Bayesian equilibria with no regret for any mechanism that we 
 use for our implementation result. 
 
 The positive results in the paper make use of mechanisms with a regret 
 phase as outlined in the example above. A complete characterization is 
 provided for performance standards that are No-regret-( N R)-implementable 
 (i.e. we use Bayesian equilibrium with no regret as the appropriate 
 equilibrium concept). The characterization results are used to prove the 
 following arguments. The interim individual rationality-efficiency 
 
 standard cannot be NR-implemented. However, the ex post individual 
 rationality-efficiency standard is NR-implementable. In general, any 
 
 standard that is Nash-implementable under complete information is also 
 NR-implementable under asymmetric information. The sub-class of 
 
 environments in which we demonstrate these results satisfy the NEI 
 condition. Thus, we are successful in bridging the gap (in terms of 
 efficient mechanism design) between NEI environments and complete 
 information environments. 
 
 As mentioned earlier, NEI is clearly restrictive. However, given the 
 constraints imposed by the Revelation Principle, some condition "close" to 
 NEI is also necessary to obtain any positive result (Blume and Easley 
 (1987) prove the necessity of NEI for implementability of the Rational 
 Expectations Equilibrium standard). Moreover, NEI is satisfied in a large 
 variety of environments involving informational asymmetry. A few of these 
 are listed below: 
 
 (i) In large enough economies the private information of any single 
 individual can be fairly accurately predicted by pooling the information 
 observed by the rest of the population. 
 
 (ii) Individuals can be one of several "types". The proportions of 
 
each type in the population is known. 
 
 (iii) Information may depend on physical possession of some commodity 
 whose different varieties and total quantity is known. Each individual's 
 private holding is not common knowledge. 
 
 (iv) There are at least two fully informed individuals in the economy, 
 e.g. in certain markets with several sellers and buyers, all sellers are 
 presumed to be informed and all buyers are uninformed. 
 
 (v) Each individual has at least one other person who can verify 
 his/her information, e.g. close relatives, witnesses, etc. 
 
 (vi) Information is held by coalitions of individuals with the minimum 
 size of each coalition being two, e.g. family units with common preferences 
 over goods, preferences are determined by the school one attends, 
 collaborators who know each other's information, etc. 
 
 In sum, given that most individuals exist within certain social 
 institutions, truly exclusive relevant information may indeed be a rarity. 
 The success of real-world information collection mechanisms such as courts, 
 inquiry commissions, etc., which rely on gathering enough individuals to be 
 able to deduce the private information held by one individual, indicates 
 that NEI environments constitute a significant class. Thus, for a vast 
 number of realistic applications, a positive result that relies on the NEI 
 condition represents a significant improvement over impossibility results 
 or positive results under complete information. 
 
 Juxtaposed with related literature, our findings have several 
 interesting implications: (i) Green and Laffont's (1987) conclusions 
 regarding the application of games with a regret phase to the problem of 
 mechanism design were largely negative. We employ a stronger definition of 
 implementability and equilibrium and show that such games do, indeed, have 
 
important applications towards deriving positive results. (ii) The most 
 
 widely studied concept of implementation is Nash-implementation whose 
 origins lie in the classic work of Maskin (1977). Maskin's 
 
 characterization of Nash-implementable standards (also see Saijo (1988)) 
 has been criticized on the grounds that it applies only to complete 
 information settings. Our results show that the class of standards that 
 Maskin had identified as being implementable can be implemented (by 
 mechanisms with no regret) even when information is asymmetric. (iv) 
 Finally, we find that appropriate refinements of Bayesian equilibria 
 broadens the scope mechanism design — a fact discovered by Palfrey and 
 Srivastava (1987b) for economies with private values and by Moore and 
 Repullo (1988) for economies with complete information. Our results hold 
 for more general economies including those with common values. In 
 
 addition, the refinement is generated by changes made by the mechanism 
 designer and we do not rely on agents employing a specific refinement 
 criterion — observe that the Palfrey and Srivastava (1987b) mechanism is 
 based on elimination of dominated strategies and not successive elimination 
 of dominated strategies. The latter criterion would be more appealing from 
 a game-theoretic standpoint. 
 
 The following section presents the basic model that we shall use — an 
 exchange economy with privately informed agents. Section 3 introduces 
 Bayesian equilibria with no regret. Sections 4, 5 and 6 present results on 
 Bayesian-implementation and NR-implementation respectively and their 
 welfare inplications. The final section discusses possible extensions. 
 
 2. Preliminaries 
 
 10 
 
An asymmetric information economy, e, is a triple {L, N, 8}. L is a 
 set of goods, N is a set of agents and 6 is a set of states of the world. 
 All of these sets are non-empty and finite and the cardinalities of L and N 
 are given by I and n, respectively. & is the domain of all asymmetric 
 information economies. In the definitions that follow, we focus on a given 
 e e &. An explicit reference to e is dropped to minimize notational 
 burden. 
 
 Let e = {L, N, 9} be given. Every agent i e N is completely 
 
 * I 
 
 characterized by a list (u , w , IT , q ), where u : R x 6 -» R is agent i's 
 
 l 1 1 M I + 
 
 von Neumann-Morgenstern utility function; q (* 0) e R is agent i's 
 
 initial endowment of goods; IT is agent i's natural information partition 
 
 * 
 of 9 and q : 9 -> (0, 1) is agent i's prior probability distribution on 9. 
 
 Each constituent of this list is assumed to be given exogenously, and is 
 
 common knowledge in the sense of Aumann (1976). Let the function I : 8 -> 
 
 i 
 
 IT be defined by I°{6) = ie' € 9: there exists n € TT such that 6, Q' € 
 i i ii 
 
 it h The latter is agent i's natural information set in state B. By 
 "natural" information we refer to the information structure that the agent 
 is exogenously endowed with. This distinguishes it from the information 
 that can be acquired endogenously. In the sequel, let TT = x TT and Q = 
 Y u) . In addition, unless specified otherwise, x = [x ) and x = 
 
 M€N 1 r 1 1€N -i 
 
 (xr) 
 
 J j€N\(l> 
 
 A = iz e R : Y z < Q} is the set of feasible allocations. A 
 + ^1<=N 1 - 
 
 state-contingent allocation is a random variable /: 9 -» A. F is the domain 
 of such functions. A performance standard <p is a non-empty subset of F 
 such that for all 8 € 9 and all / € <p, /(8) * 0.. $ is the class of all 
 performance standards. 
 
 11 
 
Let P(X) denote the set of non-empty subsets of X. Agent i's 
 
 posterior probability distribution is the function q : x P(0) -> [0, 1] 
 
 defined by Bayes' Law, i.e. for all 9 e 6 and for all $ e P(8), 
 
 * 
 q (0) if € ?; 
 
 q (0. ? ) = 
 i 
 
 V e » *, (8 '> 
 
 0, otherwise. 
 
 /Igent i's expected utility from f e F, given } e PCI (Q)) is £ q (8', 
 
 1 0'e?' 
 
 ?)u (/ O'), 0'), and is written more compactly as EU (/ 9). Agent i's 
 ^-expected lower contour set at f is given by EL (f 9) = {g € F: Et/ C/ 
 9) > EU(g I m 
 
 The domain under consideration, S is defined as the collection of all 
 economies e ={L, N, Q} € g, that satisfy the following: 
 
 [Al] {strict monotonicity of preferences) Vi € N, V0 € 0, u (., .) is 
 
 t 
 strictly increasing in z 6 R , and 
 
 [A2] (non-exclusivity of information) Vi € N, V0 € 0, n I (0) = 
 
 ' J€N\(1> j 
 
 <e>. 
 
 [A3] | J\T | > 2. 
 
 A mechanism is a game T = {JV, M, £}, where, given that M is agent i's 
 message (or action) space, M = x M ; and £: M -» i4 is an outcome function. 
 Agent i's strategy is a random variable s : Q ^ M such that s. is 
 IT -measurable. Let S be the domain of such functions. Let S = X S . 
 
 11 1€N 1 
 
 3. Bayesian Equilibria with No Regret 
 
 The fundamental solution concept for games with asymmetric information 
 is that of Bayesian equilibrium due to Harsanyi (1967). This concept is 
 
 12 
 
defined as follows: 
 
 s € S is a Bayesian equilibrium of T = {N, M, £,) if Vi e N, V0 e 0, 
 
 Vs' € S , 
 i 1 
 
 £«(s\ s ) € EL (£os I I°(Q)). 
 1-1 1 ' 1 
 
 Let £°(r) denote the set o/ Bayesian equilibria of r and £°(D = {£°s e F: 
 s € E (D, T = {N, M, £». 
 
 We shall use the following structure to formally specify the changes 
 in the rules of play introduced by a regret phase in a game. The 
 construction given below shall be used throughout the paper. Alternative 
 constructions are, of course, possible. The eventual objective of this 
 paper is to provide one method of construction that a mechanism designer 
 can employ. 
 
 An original game T played with a regret phase is denoted T . 
 
 Suppose T is such that E°(D * . Let K = {k : £°(D -> M defined 
 ^ ill 
 
 by: 39 € such that Vs € E°(D, s (0) = k (s)K Let I (0, s) = {0' € 
 
 ill 
 
 J (0): s (0') = s (0)} be the information set revealed by s in 0. 
 l -l -I 
 
 T is played in four stages. 
 STAGE 1: 
 
 The agents and the mechanism designer compute E (D. 
 BETWEEN STAGES 1 AND 2: 
 
 Nature selects e 8. 
 STAGE 2: 
 
 Each agent i submits k € K to the designer. The designer makes 
 k public knowledge. 
 STAGE 3: 
 
 Each agent i is given the opportunity to entertain a desire to 
 deviate from any »c(s) for s e E°(D. Let the set of Bayesian equilibria 
 
 13 
 
that have not induced a desire for unilateral deviation by any i e N be 
 denoted £(D. 
 
 STAGE 4: 
 
 By some unbiased method, s € £(D is selected and is common 
 knowledge among the agents and the designer. The designer chooses the 
 outcome £(k(s)). 
 
 £(D is the set of Bayesian equilibria with no regret. Let E (D = 
 
 P 
 
 {£°s € F: s e £(r), r = iN, M, £}}. A game with a regret phase, T , for 
 which E(D * is a mechanism with no regret. 
 
 For the purposes of this paper, the following conditions shall 
 suffice for determining E(D. (A) is a sufficient condition for £(D * 
 and (B) is a necessary condition. 
 
 Condition (A): If 3s e E°(X; such that Vs' € E°(T;, Vi e JV, Vs" € S , 
 
 1 i 
 
 ^o( s , ' ( S ) e el^os | r ce, s';;, 
 
 then s € E(D. 
 
 Condition (B): If E(D * 0, then Vs, s' e E(D, Vi € A/, Vs" € S , 
 
 €o(s", s ) € ELC^os I ire, s'U 
 l -i i ^ ' i 
 
 Condition (A) provides a method for determining at least one Bayesian 
 equilibrium with no regret. If there exists an equilibrium in E (D that 
 is immune to information revealed by all the Bayesian equilibria, it must 
 be a Bayesian equilibrium with no regret. Condition (B) provides an 
 internal consistency condition for the set E(D. Every Bayesian 
 
 equilibrium with no regret must be immune to information revealed by any 
 other Bayesian equilibrium with no regret. This follows from the fact that 
 in stage four a Bayesian equilibrium with no regret will be chosen to 
 determine the final outcome. Given that in stage two the agents do not 
 
 14 
 
know which one will be chosen, they will not have an incentive to 
 unilaterally make arbitrary submissions for any equilibrium in £(D. This 
 rules out unilateral manipulation on the part of the agents by the 
 submission of messages that could mislead other agents. An equilibrium 
 that is not in £(D does not convey information since agents may submit 
 messages corresponding to such an equilibrium to mislead others about the 
 state of the world knowing that such equilibria shall never be used to 
 determine the final outcome. 
 
 4. Bayesian-Implementation and Welfare Implications 
 
 The classical approach to welfare economics has been to identify the 
 subset of allocations that are Pareto-efficient within the set of all 
 physically and technologically feasible allocations. Among these efficient 
 allocations, attention is generally focused on ones that are individually 
 rational. However, in asymmetric information economies, these welfare 
 evaluations must also take account of informational constraints — an 
 uninformed social planner cannot identify the set of individually 
 rational-efficient allocations in the absence of complete information about 
 the agents' preferences. The notion of an individually rational-efficient 
 allocation would vary depending on the extent of insurance that the 
 allocation provides each agent. Individual rationality and 
 
 Pareto-efficiency are thus extended to take account of different levels of 
 insurance and the appropriate notion depends on the timing of the welfare 
 analysis (see Holmstrom and Myerson (1983) for a detailed discussion). The 
 two primary concepts of efficiency are: 
 
 15 
 
Interim-efficiency: A state-contingent allocation / is 
 
 interim-efficient if there is no g e F such that Vi e N, V9 e 6, / € £L (g 
 
 i 
 
 | r°C8)) and / € int(EL (g | f°(9)) for some i e A/ and some 9 e 8. 
 
 Ex post-efficiency: A state-contingent allocation / is ex 
 post-efficient if there is no g € F such that Vi € N, V9 e 0, / e EL (g 
 
 i ' 
 
 {9}) and / e int(EL (g | {9}) for some i € N and some 9 e 8. 
 
 Given w e E defined by w(9) = u for all 9 € 8, let T l = (f e F: f is 
 interim efficient and Vi e N, V9 € 8, w € EL C/ | I°C9;;} and P e h {/ <= F: 
 / is ex post efficient and Vi 6 N, V9 € 8, w € EL C/ {9}} denote, 
 
 respectively, the sets of interim individually rational-efficient and ex 
 post individually rational-efficient performance standards. 
 
 Once a social planner decides on the appropriate notion of efficiency, 
 the question of implementing an efficient performance standard arises. 
 Once again, the informational asymmetry poses a constraint. A naive 
 mechanism in which each agent is asked to report his/her private 
 information to the planner will generally not ensure truth-telling as the 
 unique equilibrium strategy profile. Thus, we need to be guaranteed the 
 existence of a mechanism that implements the given standard. This is 
 defined (for the case where Bayesian equilibrium is the solution concept) 
 as follows: 
 
 A performance standard <p is Bayesian-implementable by T in e = {L, N, 
 Q} if o * E°(D S <p. 
 
 F r 
 
 A performance standard <p is Bayesian implementable (in a global sense) 
 if Ve € 8, 3T such that <p is Bayesian-implementable by T in e. 
 Remark: Full Bayesian-implementation of <p by V requires that E°(D = <p, 
 whereas weak Bayesian-implementation of (p by T requires that <p r\ E°(D *■ o. 
 
 Next, we present a condition that is necessary and, in economies in 8, 
 
 16 
 
sufficient for a performance standard to be Bayesian-implementable. Before 
 we present the condition, some additional definitions are needed. We shall 
 use the approach of Postlewaite and Schmeidler (1986) and Palfrey and 
 Srivastava (1987) to define a "collection of compatible manipulation 
 operators". 
 
 A collection of compatible manipulation operators for TT (CCMO), 
 denoted a = (a ) , is defined by 
 
 i i€N 
 
 (i) Vi € N, a : n -» II , 
 l l f 
 
 (ii) Vir € II, {fl ir * 0} » <H a Or ) * 0>. 
 
 1 ' 1€N 1 " i€N 1 1 
 
 Let 8 : 8 -» 8 be the deception induced by a and defined by 6 (8) = 
 H a (I (0)). By assumption A2, 8 is a well-defined function. 
 
 A performance standard <p satisfies Property M if the following is 
 true: 
 
 3<p'€ * such that <p' £ <p and V/ € F, VCCMO's a, 
 if (i) / e <p* and 
 
 (ii) vi e n, v g € f, ve € 8, ve' - e a (e), (g € EL(f | i°(e'))) => 
 fgoe a e ELff»e* I Ae)j;, 
 
 then /°8 € <p'. 
 
 Remark: <p satisfies Bayesian monotonicity if the Property M is modified as 
 
 follows: V/ e F, VCCMO's a, if (i)' and (ii)' imply /°8 a e <p, where (i)' / 
 
 € <p and (ii)' is the same as (ii) in the definition above. 
 
 Fact 1: There exists <p <= <t> such that <p satisfies Property M and violates 
 
 Bayesian monotonicity. 
 
 To check that this true, simply choose <p such that it is the union of 
 a Bayesian monotonic set <p' e * and a set <p" € $ which is not Bayesian 
 monotonic. For examples of such sets, see Palfrey and Srivastava (1987a). 
 Fact 2: Let e = {L, N, Q} be an economy in 8. A performance standard </> is 
 
 17 
 
fully Bayesian-implementable by a game in e if and only if it satisfies 
 
 Bayesian monotonicity. 
 
 Proof: See Palfrey and Srivastava (1985). ■ 
 
 The following proposition provides a parallel characterization of 
 Bayesian-implementability. 
 
 Proposition 1: Let e = {L, N, Q} be an economy in 8. A performance 
 standard <p is Bayesian-implementable by a game in e if and only if it 
 satisfies Property M. 
 
 Proof: (p is Bayesian-implementable if and only if there exists <p' e $ such 
 that <p' Q <p and <p' is fully Bayesian-implementable. Given Fact 2, the 
 proposition follows from the definitions. ■ 
 
 In conjunction with the facts given above, the proposition has an 
 interesting implication: a performance standard may be 
 
 Bayesian-implementable even though it violates Bayesian monotonicity. 
 
 i e 
 
 Palfrey and Srivastava' s (1987a) examples suggest that the T and T 
 sets* are not fully Bayesian-implementable. However, disappointing as this 
 may be, we are interested in a more crucial question: can we ever hope to 
 design a mechanism so that all of its Bayesian equilibrium outcomes are 
 individually rational-efficient (in either an interim or an ex post sense)? 
 In other words, is it possible to Bayesian-implement (i.e. to fully 
 Bayesian-implement some subset of) T or T in the domain of economies that 
 Palfrey and Srivastava have examined? It must be noted that their result 
 does not answer this latter question. The following results confirm that 
 the answer to the question is, indeed, negative. 
 
 The theorems are proved by way of counterexamples. We shall present 
 simple ones using linear economies. The negative results do not depend on 
 
 18 
 
the assumption of linearity and hold for other utility specifications as 
 well. 
 
 Theorem 1: If <p Q T , then <p is not Bayesian-implementable. 
 
 Proof: Consider the following example: 
 
 Example 2: We define e = {L, N, 9> as follows. Let L = {X, Y) with the 
 
 quantities of the two goods being denoted by the corresponding lower case 
 
 letters. Let N = {1, 2, 3, 4} and let 9 = {a, b, c). IT = TT = {(a), (b), 
 
 1 2 
 
 (c)> and TT = IT = {(a), (b, c)>, i.e. agents 1 and 2 are fully informed 
 and agents 3 and 4 cannot distinguish between states b and c when one of 
 them occurs. An allocation z is written as (x , y ) u> = ((0, 1), (0, 
 
 1 1 16N 
 
 1), (1, 0), (1, 0)). Each state is equally likely, i.e. q (a) = q (b) = 
 
 i i 
 
 * 1 I 
 
 q (c) = - for all i € N. u : \R x 8 -» IR is given as follows: 
 
 Vi € {1, 2), V9 € 9, a (z , 9) = 
 
 x + l.ly , 
 i J r 
 
 u (z . e) = 
 
 3 3 
 
 ( *3 + y 3 ] ' 
 
 if 9 = a 
 
 0.25U + y ) if 9 = b 
 
 3 3 
 
 0.75(x + y ) if 9 = c. 
 
 3 J 3 
 
 u „ (z „' 8) = 
 
 4 4 
 
 4 4 
 
 if G = a 
 
 0.75(x + y ) if 9 = b 
 
 4 4 
 
 0.25(x + y ) if 9 = c. 
 
 4 4 
 
 It can be checked that T * 0. Consider the following state-contingent 
 
 * 
 allocation, denoted / : 
 
 19 
 
z = 
 1 
 
 z = 
 
 2 
 
 z = 
 3 
 
 Z = 
 
 4 
 
 a 
 
 
 b 
 
 
 c 
 
 (0, 1) 
 
 
 (0, 1) 
 
 
 (0, 1) 
 
 (0, 1) 
 
 
 (0. 1) 
 
 
 (0, 1) 
 
 (1, 0) 
 
 
 (0, 0) 
 
 
 (5,0) 
 
 (1, 0) 
 
 
 (2, 0) 
 
 
 (i 0) 
 
 3 
 
 * 
 f 6 
 
 p\ 
 
 Also check 
 
 that 
 
 assumptions A1-A2 are 
 
 It can be checked that / e T 
 satisfied. 
 
 Choose any <p € $ such that <p £ ^P . We shall show that the hypothesis 
 
 of Property M is satisfied. Consider the function a : IT -» IT for all i e 
 
 r J ill 
 
 N defined by a (ti ) = {a) for all i e N and all ti € TT . Thus, a is a CCMO 
 11 11 
 
 and for all 9 e 8, a (0) = a. Pick / € </> and g 6 F. Write /(a) as (x, y) 
 and g(a) as (x\ y'). By construction, the utility functions of agents 1 
 and 2 are state-independent. Therefore, given that they are completely 
 informed, the hypothesis of Property M is trivially satisfied for these two 
 agents. The following relationships imply that the hypothesis of Property 
 M is met for the remaining agents: 
 
 x + y fc x' + y' >* 0.5[0.25(x + y ) + 0.75(x + y )] £ 0.5[0.25(x' + 
 
 3333 33 33 3 
 
 y') + 0.75(x' + y')]. 
 
 3 3 ^3 
 
 x + y > x' + y' => 0.5[0.75(x + y ) + 0.25(x + y )] > 0.5(0. 75(x' + 
 
 4444 44 4^4 4 
 
 y') + 0.25(x' + y')]. 
 
 4 44 
 
 For P to satisfy Property M, we must have f°Q e <p. foQ recommends the 
 following allocation in states b and c for agents 3 and 4: 
 
 b c 
 
 Agent 3 (x^ yj (x^ yj 
 
 Agent 4 (x , y ) (x , y ) 
 
 4 4 4 4 
 
 By interim individual rationality of /, (x , y ) * (0, 0). By the 
 
 construction of this example and given that / e <p Q P , x > and x > 0. 
 
 3 4 
 
 20 
 
Choose c such that min{x , x ) > c > 0. Consider an alternative rule h € F 
 
 3 4 
 
 defined by 
 
 h(a) = /(a), 
 
 Vi e {1, 2}, V0 e (b, c), h(8) = /(a), 
 
 i l 
 
 h (b) = (x -e, y ), 
 
 3 3 3 
 
 h (b) = (x +e, y ), 
 
 4 4 4 
 
 h (c) = (x +-, y ), 
 
 3 3 3 3 
 
 h (c) = (x --, y ). 
 
 4 4 3 4 
 
 Thus, El/ (h <b, c}) = 0.5[0.25(x - e + y ) + 0.75(x + - + y )] = 
 
 3 ' 3 ^3 3 3 J 3 
 
 0.5[0.25(x + y ) + 0.75(x + y )] = EU (/°9 a | <b, c». Trivially, for i 
 e {1, 2} and all e 8, EU (h | {6}) = EU {foQ a | {en. Also, trivially, 
 for i € {3, 4}, El/ (h | {a}) = £(/ (/oG a | {a}). However, EU (h | {b, c» = 
 0.5[0.75(x + e + y ) + 0.25(x - - + y )] > 0.5[0.25(x + y ) + 0.75(x + 
 
 4 • / 4 434 4" , 4 4 
 
 y )] = EU (/oG a I {b, c». Thus, /oe a <i P l . 
 
 4 4 ' 
 
 For any <p S T with / € <p, we find that /<>0 g <p. Thus, tP violates 
 Property M. By Proposition 1, it is not Bayesian-implementable in e. ■ 
 
 Theorem 2: If <p Q P , then <p is not Bayesian-implementable. 
 
 Proof: Consider the following example. 
 
 Example 3: Consider an economy e' which retains all the specifications of 
 
 e from Example 2 above with one exception: agent 3's utility function is 
 
 altered as follows: 
 
 u (z , 9) = 
 
 3 3 
 
 (* + y ), 
 
 3 J 3 
 
 if 9 = a 
 
 (x + 1.5y ) if 9 = b 
 
 3 3 
 
 (x + 0.5y ) if 9 = c. 
 
 3 3 
 
 Given these specifications, T * 0. Choose a : 17 -> IT as in the previous 
 proof, i.e. for all i e N, for all n e TT , a (ir ) = {a). Thus, a is a 
 
 111! 
 
 CCMO and for all 9 e 9, 9 a (9) = a. Choose any <p € * such that <p Q T* and 
 
 21 
 
pick / € <p. Next, we establish that the hypothesis of Property M is met. 
 From the relationships derived in the proof of Theorem 1, we know that the 
 relevant relationships for agents 1, 2 and 4 hold. To check that the 
 relevant relationship holds for agent 3, consider the following 
 observations: 
 
 Pick / € <p and g <= F. Write /(a) as (x\ y), and g{a) as (x\ y'). 
 Thus, we have 
 
 x + y > x' + y' => 0.5[(x: + 1.5y ) + (x + 0.5y )] £ 0.5[(x' + 
 
 3 J 3 3 J 3 3 3 3 ^ 3 3 
 
 1.5y') + W + 0.5y')]. 
 
 ^3 3 3 
 
 Thus, the hypothesis of Property M is satisfied. For T to satisfy 
 
 oc oc 
 
 Property M, we must have f°0 e <p. f°B recommends the following 
 
 allocation in states a and b for agents 1 and 3: 
 
 Agent 1 (x , yj (x^, y) 
 
 Agent 3 {x^ yj {x^, yj 
 
 By ex post individual rationality of /, (x , y ) * (0, 0). By the 
 
 construction of this example and given that / e <p Q T e , x > and y > 0. 
 
 Choose c such that min{— x , y } > e > 0. Consider an alternative rule h e 
 
 n 3 M 
 
 F defined by: 
 
 h (b) = (x +l.le, y -e) 
 l 11 
 
 h (b) = (x -1.1c, y +e) 
 
 3 3 J 3 
 
 Vi € {1, 3>, V9 € {a, c>, h (9) = /(a) 
 Vi € {2, 4}, h = / oe a . 
 Thus, EC/ (h {b» = x - l.le + 1.5c + 1.5y = x + 0.4c + 1.5y > x + 
 
 3 ' 3 y 3 3 ^3 3 
 
 l.5y 3 = EI/ (/«e a | {b» and £1/ (h | {b» = x + l.le + l.ly - 1.1c = x + 
 
 l.ly = EU (/oe a I <b>). For all i € N and all 9 e 6, /<>e a € EL (h I {9}). 
 li 1 i ' 
 
 Thus, foQ <£ <p Q 7> e . Given that this holds for all <p Q T e , P e does not 
 
 22 
 
satisfy Property M. By Proposition 1, it is not Bayesian-implementable in 
 
 5. No Regret Implementation 
 
 In this section, we discuss the implementability properties of 
 mechanisms with no regret and provide necessary and sufficient conditions 
 for this method of implementation. 
 
 A performance standard <p is No regret-implementable ( N R-implementable ) 
 by r R in e = {L, N, e} if * E (D Q <p. 
 
 F 
 
 A performance standard <p is N R-implementable (in a global sense) if Ve 
 
 e &, 3T such that <p is NR-implementable by T in e. 
 
 p 
 Remark: <p is fully NR-implementable by V if £ (D = <p. 
 
 Next, we define some properties that shall be used to characterize 
 NR-implementable standards. 
 
 A performance standard cp satisfies Property Ml with respect to T ={N, 
 M, £} if the following is true: 
 
 3(p' € * such that <p' Q <p and V/ e F, VCCMO's a, 
 if (i) / 6 <p' 
 
 (ii) Vi e N, Vg € F, V9 e 9, V9' = Q a (6), {Vs e E(D, g e EL(f \ 
 I°(0')) () EL(f I I(Q\ s))} =* {Vs' € S, go9 a e EL(f°Q a \ I°(G)) f) 
 EUf°6 a J I(Q, s'))}, 
 
 then /o0 € <p\ 
 
 A performance standard <p satisfies Property M2 with respect to T if V/ 
 € F, VCCMO's a, (i)' and (ii)' imply f°0 € <p, where (i)' / e <p and (ii)' 
 is the same as (ii) in the definition above. 
 
 23 
 
Theorem 3: Let e = {L, N, Q} be an economy in 8, and let <p be a performance 
 
 p 
 standard. If <p is NR-implementable by T in e, then <p satisfies Property 
 
 Ml with respect to T. 
 
 Proof: By definition of NR-implementation, for some game T = {N, M, *;}, 
 
 there exists (p' Q <p such that cp' is fully NR-implemented by T . Thus, for 
 
 all / e <p' there exists s € E{T) that / = £>°s. By Condition (B) the 
 
 following holds for all i € N, for all 9 e 8, for all s' e S , for all s" <= 
 
 E(D: 
 
 ^o( S ', s ) € el (/ I i°(e)) n eu/ I J( . s "» (i: 
 
 Choose a CCMO a. Next, suppose that for all i € N, for all 9 € 8, for all 
 
 9' = 9%), for all s" € £(D, for all g & EL(f \ I°(e')) fl EL (/ I 
 
 * 
 
 7 (9', s")), the following holds for all s € S: 
 
 go9 a € el (foe* | I°(e)) f) ELifoe" | i (e, s*); (2) 
 
 Given (1) and (2), for all i e N, for all 9 e 8, for all s' e S , for all 
 
 * 
 s e S, the following holds: 
 
 €•($;, s^jce" € ELCfoe* | i°(9)) H el (foe* \ lie, s*)) (3) 
 
 By definition of a, s »8 is IT -measurable for all i € N. By Condition 
 (A), (3) implies that s°9 a € E(D and /°9 a € E (D. By definition of full 
 NR-implementation of <p\ E (D £ ^'. Thus, /°9 e <p' and <p satisfies 
 Property Ml. ■ 
 
 Theorem 4: Let e = {L, N, Q} be an economy in 8 and let <p be a performance 
 standard. There exists T such that if <p satisfies Property M2 with respect 
 to T, then <p is NR-implementable by V in e. 
 
 Proof: The proof of this statement derives from the following lemmata. 
 Lemma 1: There exist T and s € E(T) such that for all i € N, for all 9 € 8, 
 
 24 
 
/ (q, s) = {e}. 
 
 i 
 
 Lemma 2: There exists T such that (i) T satisfies Lemma 1 and (ii) if <p 
 satisfies Property M2 with respect to T, then E (V) Q tp. 
 
 The proofs of these results are given in the appendix. ■ 
 
 Corollary to Theorem 4: Let e = {L, N, &} be an economy in 6, and let <p be 
 a performance standard. If <p is fully N R-implementable by T in e, then <p 
 satisfies Property M2 with respect to T. Conversely, there exists T such 
 that if <p satisfies Property M2 with respect to T, then <p is fully 
 N R-implementable by T in e. 
 
 6. No Regret Implementation and Welfare Implications 
 
 This section explores whether or not the individual 
 rationality-efficiency performance standards are implementable by no regret 
 mechanisms. We have bad news and good news. The bad news is that the 
 interim standard is not NR-implementable. The good news is that the ex 
 post standard is. Often we are not interested in implementing the entire 
 ex post individual rationality-efficiency set. Instead, we would like to 
 find out which subsets of this set are implementable. This brings us to 
 the most significant result: any performance standard that is 
 Nash-implementable is NR-implementable. Hence, extremely important 
 
 economic performance standards such as the core or the Walrasian 
 correspondence are implementable by mechanisms with no regret even in 
 asymmetric information economies. 
 
 25 
 
Theorem 5: "P is not NR-implementable. 
 
 Proof: Suppose that the theorem were not true, i.e. for all e € 8, there 
 
 1 R 
 
 exists a game T such that T is NR-implementable by T in e. Consider the 
 economy e defined in Example 2 (see proof of Theorem 1 above). By 
 definition of NR-implementation, for some V = {N, M, £,) in e, there exists 
 <p Q T such that for all f € <p, there exists s 6 E(D with ^s = /. In 
 addition, check that in the economy e in Example 2, the hypotheses of 
 Property Ml are satisfied with respect to T. Choose a CCMO a as in Example 
 2. Pick f € (p and g € F. Write /(a) as (x, y) and g(a) as (x\ y'). By 
 construction, the utility functions of agents 1 and 2 are 
 state-independent. Therefore, given that they are completely informed, the 
 hypothesis of Property Ml is trivially satisfied for these two agents. The 
 following relationships imply that the hypothesis of Property Ml is met for 
 the remaining agents: 
 
 x + y £ x' + y' => 0.5[0.25(x + y ) + 0.75(x + y )] > 0.5[0.25(x' + 
 
 3333 33 33 3 
 
 y') + 0.75(x' + y')]. 
 
 3 3 3 
 
 x + y > x' + y' =» 0.5[0.75(x + y ) + 0.25(x + y )] > 0.5[0.75(x' + 
 
 4444 4" / 4 4 y 4 4 
 
 y') + 0.25(x' + y')] and 
 
 4 44 
 
 x + y > x' + y' =* 0.25(x + y ) £ 0.25(x' + y') 
 
 3 y 3 3 y 3 3 J 3 3 J 3 
 
 x + y > x' + y' => 0.75(x + y ) £ 0.75(x' + y') 
 
 3 ^3 3 y 3 3 3 3 "^3 
 
 x + y > x' + y' => 0.75(x + y ) > 0.75(x' + y') 
 
 4 y 4 4 y 4 4 4 4 J 4 
 
 x + y >: x' + y' =» 0.25(x + y ) £ 0.25(x' + y') 
 
 4 4 4^4 4"'4 4" , 4 
 
 Given these observations, the hypothesis of Property Ml is met for any 
 game T. The rest of the proof of Theorem 5 follows the proof of Theorem 1 
 and it is established that f°B 4 <p. Thus, T does not satisfy Property Ml 
 with respect to r. Given Theorem 3 and the assumption that *P is 
 NR-implementable by T , we have a contradiction. ■ 
 
 26 
 
Theorem 6: T is NR-implementable. 
 
 This result follows from Lemma 3 and a more general result given in 
 Theorem 7. The lemma is proved in the appendix. We shall use the 
 following definition: 
 
 A performance standard <p satisfies Property M if the following is 
 true: 
 
 3<p'e $ such that <p' £ <p and V/ e F, VCCMO's a, 
 if (i) / € <p' and 
 
 (ii) vi € n, v g € f, ve € e, ve' = e a (e), (g e EL(f | {e'>;; => 
 {go9 a e ELffoQ* | <e>;>, 
 
 then f°Q e <p . 
 
 e * 
 
 Lemma 3: T satisfies Property M . 
 
 Theorem 7: Let <p be a performance standard. If <p is Bayesian-implementable 
 ( Nash-implementable) in economies with complete information (i.e. for all i 
 € N and all 6 e 9, I (6) = {6}), then <p is NR-implementable. 
 Proof: By Proposition 1, if <p is Bayesian-implementable in economies with 
 
 complete information, it must satisfy Property M . By definition, if <p 
 
 * 
 satisfies Property M , then there exists <p' Q <p which satisfies Property M2 
 
 with respect to any game T such that for all i € N and all € 6, there 
 
 exists s e £(D with I (6, s) = {Q}. By Lemma 1 and Lemma 2 and the 
 
 Corollary to Theorem 4, <p' is fully NR-implementable. Thus, <p is 
 
 NR-implementable. ■ 
 
 27 
 
7. Extensions 
 
 In many economic problems of interest, initial endowments are not 
 specified (for example, when resources are owned collectively) or a social 
 planner may wish to efficiently allocate resources in an "equitable" 
 manner. In such situations, we may want to replace the individual 
 
 rationality condition with some equity requirement, such as freedom from 
 envy (Foley (1967). The results reported in this paper, both and positive 
 ones, hold for the corresponding problem of implementing the interim and ex 
 post envy-free-efficient performance standards. 
 
 To summarize, we have shown that it is impossible to design 
 "interesting" efficient mechanisms under asymmetric information using 
 Bayesian equilibrium as the solution concept. Yet under complete 
 
 information such mechanisms are possible. By employing mechanisms with no 
 regret, similar to ones introduced by Green and Laffont (1987), we have a 
 solution to the problem. Such mechanisms provide channels for endogenous 
 leakage of information. In general, any success that has been achieved by 
 Nash-implementation theory for complete information economies can be 
 mimicked in situations where information is non-exclusive and incomplete. 
 
 28 
 
References 
 
 AUMANN, R. (1976): "Agreeing to Disagree," Annals of Statistics, 4, 
 
 1236-1239. 
 
 BLUME, L. AND D. EASLEY (1987): "Implementation of Expectations 
 
 Equilibria," mimeo Cornell University. 
 
 DASGUPTA, P., P. HAMMNOND AND E. MASKIN (1979): "The Implementation of 
 
 Social Choice Rules: Some General Results on Incentive Compatibility," 
 
 Review of Economic Studies, 46, 185-216. 
 
 FOLEY, D. (1967): "Resource Allocation and the Public Sector," Yale 
 
 Economic Essays, 7, 45-98. 
 
 GREEN, J. AND J. -J. LAFFONT (1987): "Posterior Implementability in a 
 
 Two-Person Decision Problem," Econometrica, 55, 69-94. 
 
 HARRIS, M. AND R. TOWNSEND (1981): "Resource Allocation under Asymmetric 
 
 Information," Econometrica, 49, 231-259. 
 
 HARSANYI, J. (1967): " Games with Incomplete Information Played by 
 
 'Bayesian' Players," Management Science, 14, 159-189, 320-334, 486-502. 
 
 HOLMSTROM, B. AND R. MYERSON (1983): "Efficient and Durable Decision Rules 
 
 with Incomplete Information," Econometrica, 51, 1799-1820. 
 
 MASKIN, E. (1977): "Nash Equilibrium and Welfare Optimality," mimeo, 
 
 M.I.T.. 
 
 MOORE, J. AND R. REPULLO (1988): "Subgame Perfect Implementation," 
 
 Econometrica (forthcoming). 
 
 MYERSON, R. (1979): "Incentive Compatibility and the Bargaining Problem," 
 
 Econometrica, 47, 61-74. 
 
 PALFREY, T. AND S. SRIVASTAVA (1985): "Implementation and Incomplete 
 
 Information," mimeo, Carnegie-Mellon University. 
 
 29 
 
PALFREY, T. AND S. SRIVASTAVA (1987a): "On Bayesian Implementable 
 
 Allocations," Review of Economic Studies, 54, 193-208. 
 
 PALFREY, T. AND S. SRIVASTAVA (1987b): "Mechanism Design with Incomplete 
 
 Information: A Solution to the Implementation Problem," mimeo California 
 
 Institute of Technology and Carnegie-Mellon University. 
 
 POSTLEWAITE, A. AND D. SCHMIEDLER (1986): "Implementation in Differential 
 
 Information Economies," Journal of Economic Theory, 39, 14-33. 
 
 SAIJO, T. (1988): "Strategy Space Reduction in Maskin's Theorem: Sufficient 
 
 Conditions for Nash-implementation," Econometrica, (forthcoming). 
 
 30 
 
Appendix 
 
 In this appendix, we shall prove the lemmata presented in the main 
 body of the paper. For this purpose, we shall devise an algorithm which 
 generates a game for every performance standard. Let §" denote the 
 algorithm and let S*(<p) denote the game that is generated when a particular 
 <p is applied to the algorithm given below. For all <p € $, ^§{<p) - iN, M, £} 
 is defined as follows: 
 
 (I) Vi € N, M = <m = (n (i), f(i), S(i)) € II x F X R + >. 
 
 Definition: Vi <= N, m satisfies Property y\i if the following conditions 
 hold: 
 
 (i) H tt (j) * 0. 
 
 1 I j€N\(i> j J 
 
 (ii) 3/ e ? such that Vj e JV\{i>, /(j) = /. 
 
 (iii) Vj e W\U>, 5(j) = 0. 
 
 * * 
 
 Definition: Vm € M, 9 (m) is defined such that f\ n (i) = {8 (m)}. 
 
 1 ' 1€N i 
 
 Definition: Vi 6 N, Vm € M , (m ) is defined such that tt ( j) 
 
 -1 -1 1 -1 ' ' j€N\(i> j J 
 
 = {9(m )>. 
 l -l 
 
 Definition: Vm € M, Kim) = ii e W: Vj € JVMO, 5(i) 2: 6{j)>. 
 
 (II) £: M -> A is given by the schematic diagram in Figure 3. 
 
 At 
 
Proof of Lemma h_ Choose / e <p. We shall show that there exists s e 
 
 E{£(<p)) such that £o S = / and for all i e JV and all 8 € 9, I (9, s) = ie). 
 
 Construct a strategy list s e S as follows: for all i <= N, for all 9 € 9, 
 
 s(e) = (JO), /, 0). It may be checked that for all i e A/, for all € 
 i l 
 
 0, s (9) satisfies Property y|£. By construction, Case 1 applies and ^»s 
 
 = /• 
 
 Consider a unilateral deviation to s' e S by agent i. We write s' = 
 
 l i i 
 
 (s* , s' , s' ). Note that by assumption A2, for all 9 € 9, 9 (s (9)) = 
 11 12 13 J 1-1 
 
 9. For all 9 € 0, there are two possibilities: 
 
 (a) s' (9) = /, in which case either Case 1 or Case 2 applies and 
 
 12 
 
 €(s'(e), s (9)) e </(e), oh 
 ^i-i 
 
 (b) s' (9) * /. Case 3 applies and e,{s'{d), s (9)) e </'(i)(e), 0}. 
 
 12 i -1 
 
 By strict monotonicity of preferences, u (/, 9) ^ u (0, 9) for all 9 € 9. 
 By construction for all j e N, s (9) = I (9). In conjunction with 
 assumption A2, this implies that for all 9 e 9, for all 9' € J°(8)M9), 
 
 there exists j € N\{i) such that s {6) * s (9'). Thus, for all i € N, 
 
 ji ji 
 
 for all 9 e 9, J (9, s) = {9>. If possibility (b) occurs, by Case 3, we 
 
 have £©(s\ s ) € ELif {9}) which implies that for all s" € S, %°{s\ 
 
 s) € EL(/ | I°(9)) f) ELC/ | 7(9, s")). By Condition (A), s e 
 
 E(^(<p)). Given ^«s = / € (p, we have £($*(<p)) * 0. ■ 
 
 Proof of Lemma 2i By Lemma 1, EV§{(p)) * 0. We need to show that E (&(v>)) 
 
 c <p. 
 
 Step L_ The proof of this lemma makes use of the following result: 
 
 Lemma 2.1: Let s € E(&(<p)). For all 9 € 9, s(6) satisfies Case 1. 
 
 Proof of Lemma 2.1: We shall establish that there cannot be a state 9 e 9 
 
 Ku 
 
such that s(0) = in (i), f(i), Sii)) satisfies either of the cases 2, 3 
 
 1 1€N 
 
 or 4. We write s as (s , s , s ) where the components of this list 
 
 1 il 12 13 
 
 denote the functions induced by restricting the range of s to II , F and IR 
 respectively. 
 
 Suppose that for some 9 € 0, s(0) satisfies one of the following: Case 
 2, Case 3 or Case 4. A contradiction will be established. Consider a 
 unilateral deviation by agent i e N to s' e S such that s' = is , s , 
 
 11 1 11 12 
 
 s' ). s" is such that for all J € WY(i>, for all 0' e 6, s. (0') < 
 
 13 13 J3 
 
 s' (6'). Let m = s(e) and m* = (s'(0), s (e)). We shall show that 
 
 13 1-1 
 
 there exists i € N such that the following hold: 
 
 € (m') > ^(m) (*) 
 
 V8' 6 8, ^(sjO*), s (9*)) ^ ^(s(9*)) (**) 
 
 There are two possibilities: 
 
 (i) Possibility 1: There exists j e N such that K{m) = {j}. 
 Therefore, for all i € N\{j), m does not satisfy Property y\i. Choose i 
 € N\{j}. By definition, Property y|£ is not met even if i deviates to s'_. 
 By the outcome rule associated with Case 4A, we have £, (m') = Q. Since 
 |AT\{j}| > 1, there exists i € N\{j} such that £ (m) * Q. Thus, (*) holds. 
 
 (ii) Possibility 2: There does not exist any J 6 N such that Kim) = 
 {j}. In this case, by construction, £(m) = 0. By the outcome rules 
 associated with Cases 2A, 3A and 4A, and given that for all 0' e 8, for all 
 / € <p, fie') * 0, there exists i 6 N such that £im') > 0. Thus, (*) 
 holds. 
 
 To check that (**) is true for the agent i for whom (*) holds, choose 
 0" e 8 with 0" * 0. There are, again, two possibilities: 
 
 (a) Possibility 1: There exists j e N such that Kisie")) = (J}. The 
 arguments given in (i) above apply. Thus, for all k e N\{j), £ is'ie"), 
 
 k k 
 
 Kb 
 
s (9")) = fl. Also, £(s'(9"), s (9")) = C(s(8")). Thus, (**) holds for 
 -k J -J 
 
 i. 
 
 (b) Possibility 2: There does not exist j € N such that K(s(9")) = 
 {j}. By construction, for some / € <p, £,(s(9")) € {/, OK By the outcome 
 rules associated with Cases 2A, 3A and 4A, we conclude that £(s'(9"), 
 s (9")) 6 {/. Q). Thus, (*•) holds for i. 
 
 Given that (*) and (**) are true, and given strict monotonicity of 
 preferences, we conclude that if s(0) does not satisfy Case 1, for some 9 e 
 8, then s 4. EC§(<p)). This contradicts the assumption that s € £(&(</?)). ■ 
 
 Step 2i Choose s <= E(£(<p)). By Lemma 2.1, for all 9 € 9, s(Q) must 
 satisfy Case 1. We write s as (s , s , s ). Thus, for all 9 € 0, 
 
 ' l 11 12 13 
 
 * 
 
 Os (9) * and 9 (s(9)) is well-defined. By IT -measurability of s, 
 1€N 11 1 i 
 
 (s ) defines a CCMO and we shall write it as a, where for all i 6 N, 
 
 11 1€N 
 
 * 
 
 for all 9 € 9, a (1(e)) = s (9). Observe that for all 9 € 9, 9 (s(9)) = 
 i i u 
 
 ct * 
 
 9 (9). By construction, for all 9 € 9, there exists / e <p such that 
 
 * 
 s (9) = / . This implies that there exists f e <p such that for all 9 € 9, 
 
 £(s(9)) = /(9 (s(9)), i.e. £<> s = /"°e a . We need to show that /°9 a € <p. 
 
 We shall first show that for all i e W, for all 9 € 9, for all 9" = 
 
 9 a (9), if g e EL if | 1(9', s )) for all s* € E(£(?)), then g°9 a e 
 
 EL (/°9 a | I°(e)) () EL (/o9 a | I (9, s)) for all s € S. In the case where 
 
 f = g, this is trivially true. To show that this is true even when f * g, 
 
 choose i e N and g e F such that g * f and for all 9 € 9 and for all 9' = 
 
 9 a (9), for all s € EVSdp)), g € EL(f \ 7(9', s )). By Lemma 1, there 
 
 exists s € E(£(<p)) such that for all 9 € 9, I (9, s ) = {9}. Thus, g e 
 
 ELY/ | 7(9', s )) for all s € E(£(p)) implies that g e EL (/ | <9*>). 
 
 Suppose i unilaterally deviates to s' e S where for all 9 6 9, s'(9) = 
 
 M 
 
(s (9), g, 5'U)). S'U) is such that for all J € N\{i), for all 9 e 0, 
 5'(i) > s (0). By construction, for all 9 € 9, 6 (s (9)) = 9 (s(9)). 
 
 J3 1-1 
 
 Thus, for all 9 € 0, (s'(9), s (9)) satisfies Case 3A and £((s'(9), 
 
 i -l i 
 
 * a 
 
 s (9)) = g(6 (s (9))) = g(Q (s(9)) = g(9 (9)). Observe that, by 
 
 -l ° i -l 
 
 * 
 Condition (B), s € £(S*(<p)) implies that for all 9 e Q, for all s e 
 
 EVS{<p)), g°e a € EL{foQ a | i°{q)) f) £U/°9 a | I (G, s*)). By Lemma 1, 
 
 there exists s € £(§"(<?)) such that 7 (9, s ) = {9} for all 9 e 0. Thus, 
 
 g°e a g £L (/" <>9 a | {9}) for all 9 € 8 and we conclude that for all s e S, 
 
 goe a e EHfoe" | r°(e)) fl EUf°e a | r(e, s)). 
 
 Given that the last conclusion holds for all i 6 W, by the fact that <p 
 satisfies Property M2 with respect to §"(<?), we have /°9 e <p. ■ 
 
 Proof of Lemma 3: Choose / e T and a CCMO a. Suppose that for all i e A/, 
 for all 9 6 0, for all 9' = 9 a (9), if /' € EL (f \ O')), then /'°g a e 
 EUf°e a | {9». 
 
 Next, suppose that / <>9 g <p . We shall prove that this yields a 
 contradiction. f°Q 4 T implies either one or both of the following 
 possibilities: 
 
 Possibility A: f°6 is not individually rational, i.e. given w € F defined 
 by w(9) = w for all 9 € 0, there exists j € N and 9 € such that w £ 
 
 EL(f°e a | <e». 
 
 Possibility B: f°G is not ex post efficient, i.e. there exists g, h 6 F 
 
 OL OL * * 
 
 such that h = g°e with h & EL (/°9 I <9 }) for some k e N and some 9 e 
 
 k ' 
 
 a 
 
 and /o9 € EL (h {9'» for all i e W and all 9' e 0. 
 
 By assumption, for all i 6 Af, for all 9 e 0, for all 9" = 9 (9), for 
 all /' € F if /* € EL(f | {9"}), then /'o9 a e EL(f°e a | <6». This 
 implies that for w, g, 9, 9 J and k given above, Possibility A implies ( + ) 
 
 kS 
 
and Possibility B implies (++) and (+++), where 
 
 w t el if | {e">) for e" = e a (e) (+) 
 
 g * EL (f I <e"» for e" = e a (e ) (++) 
 
 k ' 
 
 and for all i € N, for all 9' e 6, 
 
 / € ELig | {e'» (+++) 
 
 ( + ) contradicts the fact that / € T implies ex post individual rationality 
 
 of /. (++) and (+++) contradict the fact that / € T implies ex post 
 efficiency of /. Thus Possibilities A and B cannot occur and we conclude 
 that /°e a € P e . m 
 
 ) 
 
 h(o 
 
Rain 
 
 L M 
 
 R 
 
 Shine 
 
 M R 
 
 T 
 
 10, 10 
 
 10, 5 
 
 0, 
 
 B 
 
 5, 5 
 
 5, 10 
 
 0, 
 
 T 
 
 10, 10 
 
 5, 5 
 
 10, 
 
 B 
 
 5, 5 
 
 20, 1C 
 
 0, 
 
 Clouds 
 
 M R 
 
 T 
 
 5, 10 
 
 5, 10 
 
 10, 
 
 B 
 
 10, 10 
 
 5, 5 
 
 5, 10 
 
 Figure 1 
 
s (Rain) = 
 
 1 
 
 T; 
 
 s (Shine) = 
 
 1 
 
 ' T; 
 
 s (Clouds) 
 
 1 
 
 = T; 
 
 s'(Rain) = 
 l 
 
 T: 
 
 s' (Shine) = 
 l 
 
 ■ T; 
 
 s' (Clouds) 
 l 
 
 = 7"; 
 
 s"(Rain) = 
 i 
 
 B; 
 
 s"(Shine) = 
 
 i 
 
 ■■ B; 
 
 s"(Clouds) 
 
 i 
 
 = B; 
 
 • 
 
 
 s (Rain) = 
 l 
 
 B; 
 
 * 
 
 
 s (Shine) = 
 
 -- B; 
 
 s (Rain) = L 
 
 2 
 
 s (Shine) = L 
 
 2 
 
 s (Clouds) = L 
 
 2 
 
 s'(Rain) = L 
 
 2 
 
 s' (Shine) = L 
 
 2 
 
 s' (Clouds) = M 
 
 2 
 
 s"(Rain) = M 
 
 2 
 
 s"( Shine) = M 
 
 2 
 
 s"(Clouds) = R 
 
 2 
 
 s (Rain) = M 
 
 2 
 * 
 
 s (Shine) = M 
 2 
 
 s (Clouds) = B; 
 
 s (Clouds) = L 
 
 2 
 
 Figure 2 
 
 F2. 
 
Let m = (n (i), f(i) 5(i)) 
 
 I 1€N 
 
 Case 1: 
 
 * 
 Vi 6 W, (i) 39 € 6 such that 9 (m) = <9>, (ii) 3/ € <p such that /(i) = f, 
 
 and (iii) 5(i) = 0. 
 
 £(m) = /(9) 
 
 Case 2: 
 
 (i) 3/ € <p such that Vj € N, f{j) = /, (ii) 3i 6 JV such that m . satisfies 
 
 -i 
 
 Property y\i and (iii) the conditions of Case 1 are not all met. 
 
 Case 2A 
 
 Kim) = ii) 
 
 £(m) = /(e (m )) 
 1-1 
 
 Case 2B 
 
 Otherwise 
 
 £(m) = 
 
 Figure 3 
 
 F!> 
 
Case 3: 
 
 31 € N such that (i) Vj e N\{i}, f(i) * f(j) and (ii) m_ satisfies 
 Property y I i. 
 
 Case 3A 
 (i) Kim) = {i} 
 (ii) /(i) € EL(/(J)| {9(m )}), j * L 
 
 €(m) = /(i)(9(m )) 
 1 -i 
 
 Case 3B 
 
 Otherwise 
 
 1 
 
 €(m) = 
 
 Case 4: 
 
 Otherwise 
 
 Case 4A: 
 
 3i € N with K(m) = <i} 
 
 Case 4B 
 
 Otherwise 
 
 (€ (m). C (m)) = (£2, 0) 
 
 i -i 
 
 €(m) = 
 
 Figure 3 (Contd.) 
 
 F4- 
 
} 
 
 \ 
 
HECKMAN U 
 BINDERY INC. | I 
 
 
 JUN95 
 
 b~ i r pi™? N MANCHESTER. 
 Bml-io-n— < 1ND1ANA 46962 
 V J