The Theory of Implementation of Socially Optimal Decisions in Economics Luis C Corchón THE THEORY OF IMPLEMENTATION OF SOCIALLY OPTIMAL DECISIONS IN ECONOMICS This page intentionally left blank The Theory of Implementation of Socially Optimal Decisions in Economics Luis C Corch6n Professor of Economics Universidad de Alieante Alicante First published in Great Britain 1996 by MACMILLAN PRESS LTD Houndmills, Basingstoke, Hampshire RG21 6XS and London Companies and representatives throughout the world A catalogue record for this book is available from the British Library ISBN 0-333-65794-2 First published in the United States of America 1996 by ST MARTIN'S PRESS, INC., Scholarly and Reference Division, 175 Fifth Avenue, New York, N.Y 10010 ISBN 312-15953-6 Library of Congress Cataloging-in-Publication Data Corchon, Luis C The theory of implementation of socially optimal decisions in economics / Luis C Corch6n p em Includes bibliographical references and index ISBN 0-312-15953-6 Decision making Welfare economics I Title HD30.23.C668 1996 330.15'56-dc20 96-7685 CIP © Luis C Corchon 1996 All rights reserved No reproduction, copy or transmission of this publication may be made without written permission No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London WI P 9HE Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages 10 05 04 03 02 01 00 99 Printed in Great Britain "by The Ipswich Book Company Ltd Ipswich, Suffolk 98 97 96 To my wife, Maria del Mar This page intentionally left blank Contents Introduction xi Economies with Public Goods 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Introduction Efficiency and public goods The Core and public goods Lindahl's equilibrium Kaneko's ratio equilibrium Mas-Colell and Silvestre's cost-share equilibrium A criticism of the notions of equilibrium with an auctioneer Exercises References 1 11 15 17 20 23 Resource Allocation Mechanisms 25 2.1 Introduction An example of resource allocation: the case of Adam and Eve 2.3 Description of the economic environment 2.4 Social objectives and incentives 2.5 Mechanisms Strategic aspects 2.6 2.7 The problem of design 2.8 Summary 2.9 Exercises 2.10 References 25 Dominant Strategies and Direct Mechanisms 39 3.1 3.2 39 39 2.2 3.3 3.4 3.5 Introduction The revelation principle The impossibility of truthful implementation in economic environments (I) The impossibility of truthful implementation in economic environments (II) The manipulation of the initial endowments Vll 26 28 29 30 31 33 35 36 37 44 47 56 Contents viii 3.6 3.7 3.8 Conclusion Exercises References Implementation in Nash Equilibrium (I): General Results 4.1 4.2 Introduction Characterization of social choice correspondences implementable in Nash equilibrium 4.3· Implementation in Nash equilibrium in economic environments 4.4 Implementation when the feasible set is unknown and credible implementation 4.5 Exercises 4.6 References Appendix: the King Solomon problem Implementation in Nash Equilibrium (II): Applications 5.1 5.2 5.3 5.4 5.5 5.6 Introduction Implementing the Lindahl and the Walras correspondences by means of abstract mechanisms Doubly implementing the ratio and the Walras correspondences by means of market mechanisms Implementation of solutions to the problem of fair division Exercises References 58 59 62 65 65 66 73 78 80 83 85 89 89 90 96 104 108 110 Refining Nash Implementation 113 6.1 Introduction 6.2 Subgame perfect implementation 6.3 Implementation in undominated Nash equilibrium 6.4 Virtual implementation 6.5 Exercises 6.6 References Appendix I: the control of externalities Appendix II: double implementation in Nash and strict Nash equilibria 113 115 118 122 126 128 130 134 Contents Bayesian Implementation 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Introduction Resource allocation under uncertainty Games of incomplete information Necessary and sufficient conditions for Bayesian implementation Ex-post efficient allocations and incomplete information Exercises References IX 137 137 137 140 141 149 153 155 Notes and References 156 Index 159 Bayesian Implementation 151 Proof We will only sketch the proof: First, we present the implementing mechanism which is an adaptation of the canonical mechanism introduced in the proof of Proposition 2, Chapter Let S, = T, X A X A X IN Recall that as a result of assumption 1, t i reveals not only the preferences, information, etc of player i, but also the preferences and information of every other agent on the island where i lives The outcome function is similar to that described in Proposition 2, Chapter (but in this case we are dealing with a social choice function) Rule If the message profile, say s, implies that types are consistent, i.e on each island, each agent reveals the saffle preferences, information, etc for the people living there than any other agent on this island, g(s) = f¥(t) where g; is the social choice function to be implemented and t is the announced state Rule Suppose that the message profile, say s, implies a unique dissident, say i, on one island only Let t( -i) (resp t/ -i)) be the state (resp the type of i) implied by the types announced by everybody except i Let Si = (t i , a; b; n.) Then g(s) = at if a i E L, (@P(t( -i)), ttf-i)) and b, = f¥(t( -i)) And g(s) = f¥(t( -i)) otherwise Rule In any other case, an integer game If i sends the higher integer, a i is implemented Ties are broken by an exogenous rule Consider the truth-telling strategy o(t) = (t i, a, a, 0) where a is an arbitrary allocation It is clear that if n - agents follow this strategy, no 'deviation from the remaining agent is going to make any difference in any state It is clear that this strategy yields g;( ) Consider now a profile of strategies that are a Bayesian equilibrium, say, a* First notice that no dissidence is possible Indeed if there were a single dissident at state t', all her fellow islanders would announce an integer greater than any integer in 0"* Moreover, everybody outside the island would the same, justin case Other cases of non-consistent announcements are dealt with similarly Suppose now that at some announced state, say i, @P(t) is such that for some agent, say i, the utility function implied by i is not a Maskin-monotonic transformation of her true utility function But then, iii preferred to @P(i) according to i's true preferences such that Ii E L/3P(t( -i)), ttf-i)) Thus, by announcing Si = (ti,iii' g;(i), 0) with t, ::f= t( - i) she is better off since for every t i= i her dissidence does not count and for t = i her utility improves Notice how the following fact is crucial: in a Bayesian equilibrium, since, say, agent i is supposed to know the strategies taken by other 152 Implementation ofSocially Optimal Decisions agents In particular she knows if they are going to lie in some state of the world This goes beyond what could be considered reasonable Again we see that implementation theory takes game theory to its logical conclusions, exposing its limitations An implication of Proposition is that, given our assumption about islands, the fact that information is or is not complete makes no essential difference from the point of view of implementation This result can be understood as a generalization of Proposition in Chapter The cost of this generalization is the common knowledge assumption on islands Other assumptions can be dispensed with at the cost of some complications Thus, social choice correspondences can be implemented and the assumption that the environment is that of exchange economies can be generalized by introducing no veto power (see Definition 2, Chapter 4) Moreover the implementation is Robust in the sense that the same outcome is implemented with different equilibrium concepts and for every possible prior (see Corch6n and Ortufio-Ortfn (1991) and Exercise 7.18).2 A further generalization has been made by Yamato (1994) by considering more general information structures and showing that Nash and Robust implementation coincide Another implication of Proposition is the following The theory of Nash implementation has been criticized because it requires exact information Therefore a slight error in the evaluation of other people's characteristics might entail alternatives far away from those selected by the SCC, especially for a discontinuous outcome function The above mechanism however only requires exact complete information on a, usually small, subset of agents Thus, the above mechanism can be regarded as introducing some robustness to the information side of Nash implementation This is especially important in General Equilibrium models where the assumption that all agents have complete information is particularly unconvincing Bayesian Implementation 7.6 153 EXERCISES 7.1 Show that if type sets are finite, then any Bayesian game without consistent beliefs is equivalent to a Bayesian game with consistent beliefs (see Myerson, 'Bayesian Equilibrium and Incentive Compatibility: an Introduction', op cit., p 240) 7.2 Define two concepts of Pareto efficiency for allocation rules, namely exante efficiency and interim efficiency Is there any logical relationship between them (see Mas-Colell, Whinston and Green, Microeconomic Theory, Mimeo, Harvard University, chap 23, section F)? 7.3 Show that under some conditions Bayesian incentive compatibility constraints can be replaced with no loss of generality by the requirement that to report the truth about her own characteristic is a dominant strategy (see Mookherjee and Reichelstein, 'Dominant Strategy Implementation of Bayesian Incentive Compatible Allocation Rules', Journal of Economic Theory, 1992, vol 56, pp 379-99) 7.4 Show that by choosing posterior probabilities adequately, a mechanism is Bayesian incentive-compatible if and only if for any agent, to tell the truth about her own characteristic is a dominant strategy (see Ledyard, 'Incentive Compatibility and Incomplete Information', Journal of Economic Theory, 1978, vol 18, pp 171-89) Compare with Exercise 7.3 above 7.5 Suppose that utility functions are quasi-linear and that types are statistically independent Show that in an economy with a public and a private good there is an ex-post efficient social choice function (that is, one choosing an efficient project and satisfying budget balance) that satisfies Bayesian incentive compatibility This mechanism is called the Expected Externality Mechanism and is derived from D' Aspremont, Gerard-Varet and Arrow (see Mas-Colell, Whinston and Green, Microeconomic Theory, op cit., chapter 23, section D and example 23.AA.l) 7.6 Show by means of an example that when there are two players, there is no Bayesian incentive compatible mechanism that is ex-post efficient and individually rational (see Myerson, 'Bayesian Equilibrium and Incentive Compatibility: an Introduction', op cit., pp 251-2) Use the revelation principle to show that a social choice set that picks up ex-post efficient and individually rational allocations cannot be implemented In Bayesian equilibrium (this is a special case of the so-called Myerson-Satterthwaite Theorem, see MasColell, Whinston and Green, Microeconomic Theory,op cit., chapter 23, section E for a good exposition) 7.7 Write in detail the individual rationality (or participation) constraints corresponding to the cases of ex-ante, interim and ex-post preferences Give scenarios under which each of these constraints are relevant (see Mas-Colell, Whinston and Green, Microeconomic Theory, op cit., chapter 23, section E) 154 Implementation of Socially Optimal Decisions 7.8 Produce an example where the choice set that picks up Rational Expectations Walrasian allocations does not satisfy the Bayesian incentive compatibility constraints (see Blume and Easily (1990), 'Implementation of Walrasian Expectations Equilibria', Journal of Economic Theory, vol 51, pp 207-27) 7.9 Present an example in which an untruthful Bayesian equilibrium Pareto dominates the truthful Bayesian equilibrium from the point of view of those agents with non-trivial information (see Postlewaite and Schmeidler, op cit., p 17) 7.10 Define an augmented revelation mechanism and give an intuitive explanation of why it works (see Mookherjee and Reichelstein (1990), 'Implementation via Augmented Revelation Mechanisms', Review of Economic Studies, vol 57, pp 453-75) Show that in the case of Exercise above there is an augmented revelation mechanism that implements f i n Bayesian equilibrium (see Postlewaite and Schmeidler, op cit., p ) 7.11 Show by means of an example that Arrow-Debreu allocation rules not always satisfy Bayesian monotonicity and thus they are not Bayesian implementable (see Palfrey and Srivastava, 'On Bayesian Implementable Allocations', Review of Economic Studies, 1987, vol 54, p 202) 7.12 Show by means of an example that ex-ante efficient allocation rules not always satisfy Bayesian monotonicity and thus they are not Bayesian implementable (see Palfrey and Srivastava, 1987, op cit., example 1) 7.13 Show by means of an example that interim efficient allocation rules not always satisfy Bayesian monotonicity and thus they are not Bayesian implementable (see Palfrey and Srivastava, 1987, op cit., example 2) 7.14 Show by means of an example that ex-post efficient allocation rules not always satisfy Bayesian monotonicity and thus they are not Bayesian implementable (see Palfrey and Srivastava, 1987, op cit., example 3) 7.15 Show that rational expectations equilibrium (REE) in the framework of an exchange economy (with an interiority condition) satisfies Bayesian monotonicity (see Palfrey and Srivastava, 1987, op cit., Theorem 2) 7.16 Show that the interim envy-free correspondence satisfies Bayesian monotonicity (see Palfrey and Srivastava, 1987, op cit., Theorem 3) 7.17 Show by means of an example that in a general environment there are social choice sets satisfying Bayesian incentive compatibility, Bayesian monotonicity and a no-veto hypothesis that cannot be implemented in Bayesian equilibrium (see Jackson, 1991, p 471, Example 1) 7.18 Define an uniform Nash equilibrium Show that Proposition holds with this equilibrium Concept (Hint: truth-telling is obviously an equilibrium So in order to prove the theorem prove that the set of uniform Nash Bayesian Implementation 155 equilibria of a mechanism is contained in the set of Bayesian equilibria of this mechanism See Corchon-Ortufio-Ortfn, op.cit.) 7.7 REFERENCES The fundamental contribution to the understanding of games with incomplete information is: Harsanyi (1967/8), 'Games with Incomplete Information Played by Bayesian Players', Management Science, 14: pp 159-82, 320-34, 486- 502 The revelation principle in a Bayesian setting was proved (among others) by R Myerson (1979), 'Incentive Compatibility and the Bargaining Problem', Econometrica, 47; pp 61-74; and M Harris and R Townsend (1981), 'Resource Allocation with Asymmetric Information', Econometrica, 49, pp 33- 64 The last paper discusses several concepts of efficiency On this matter see also J.D Ledyard (1987), 'Incentive Compatibility' ,entry in J Eatwell, M Milgate and P Newman (eds), The New Palgrave Dictionary (Macmillan) The following survey presents a lucid discussion of the revelation principle: R.B Myerson (1985), 'Bayesian Equilibrium and Incentive Compatibility: an Introduction' in L Hurwicz, D Schmeidler and H Sonnenschein (eds), Social Goals and Social Organization, chapter On the theory of Bayesian implementation the reader may consult the following papers: A Postlewaite and D Schmeidler (1986), 'Implementation in Differential Information Economies', Journal of Economic Theory, 39, pp 1433; T Palfrey and S Srivastava (1989), 'Implementation with Incomplete Information in Exchange Economies', Econometrica, 57, pp 115-34; and M Jackson (1991), 'Bayesian Implementation', Econometrica, 59, pp 461-77 A good exposition of Bayesian implementation can be found in T Palfrey and S Srivastava (1991), Bayesian Implementation, Fundamentals of Pure and Applied Economics (New York: Harwood Academic Publishers); and T Palfrey (1992), 'Implementation in Bayesian Equilibrium: the Multiple Equilibrium Problem in Mechanism Design' in J.1 Laffont (ed.), Advances in Economic Theory (vol 1), VI World Congress of the Econometric Society (Cambridge University Press) Section 7.5 is based, almost entirely, on L Corch6n and I Ortufio-Ortfn (1991), 'Robust Implementation under Alternative Information Structures', May Institute of Mathematical Economics, Working Paper, University of Bielefeld, Germany Economic Design Vol 1, no 2, 1995 The main result obtained in the above paper has been generalized by: T Yamato (1994), 'Equivalence of Nash Implementability and Robust Implementability with Incomplete Information', Social Choice and Welfare, Vol 11(4) pp 289-303 Notes and References Introduction The work of Farquharson (1957) and Moulin (1981) should also be mentioned here as important forerunners of the Moore and Repullo approach Economies with Public Goods The usual convention in general equilibrium is just the opposite, that is, that private good outputs are positive and private good inputs are negative Notice that this convention does not apply to public goods since they are outputs and positive (public goods used as inputs could be considered at the cost of some complications) Resource Allocation Mechanism An alternative interpretation of Nash equilibrium is that it is the limit of a dynamic process of adjustment in which the players (endowed with an incomplete knowledge of the game) vary their strategies by means of some adjustment rule which is more or less rational Such an interpretation (which coincides with Cournot's presentation of his ideas on market equilibrium) is without a doubt more attractive than the one where complete information is assumed However, unfortunately it is not easy to formalize if we assume that the players are minimally rational Dominant Strategies and Direct Mechanism The Gibbard-Satterthwaite theorem does not assume that the social choice function must select Pareto efficient or individually rational allocations, but that its range must be of cardinality greater than 2 Notice that the actual price paid by the highest bidder is the same in the three kind of auctions (English, Dutch and Vickrey's) Thus it is not clear what the practical advantage of the latter is Moreover, the Vickrey auction has untruthful equilibria, i.e the second bidder is indifferent among any announcement about her reservation price This may be the reason of why, in practice, if a Vickrey auction is performed, objects are sometimes sold at very low prices Implementation in Nash Equilibrium (I): General Results In Section 4.4 we will consider the case where the feasible set is variable As we noticed in the previous chapter, if the dimension of both the domain of economies and the allocation space were finite, it would be pos- 156 Notes and References 157 sible to replace the announcement of a profile of preferences and an allocation for the announcement of two natural numbers This assumption can be relaxed to Al s -i)' being star-shaped with center in (Wi' 0), that is if (Xi' y) E A,( s _,), then any (x', y') = a/x;, y) + (1 - a) (Wi' 0) for some a; E (0, 1) also belongs to Als-) In fact in Hurwicz's version such an assumption is made which allows the differentibility to be relaxed to continuity and strict quasiconcavity to quasiconcavity Furthermore, Hurwicz assumes that is implementable The paper of Glazer and Ma in Games and Economic Behavior, 1989 and the survey of John Moore (1991) popularized the story among implementation theorist The latter credits the idea to A Rubinstein Implementation in Nash Equilibrium (II): Applications If the space of economies includes economies with several Lindahl equilibria, no social welfare function can be implemented in Nash equilibrium since the Lindahl correspondence is contained in any Nash implementable social choice correspondence (see exercise 27 in the previous chapter) The author has to confess that he has changed his mind on this matter over the years and that four years ago he was closer to the critics of Maskintype mechanisms that he is now See e.g the introduction of Corch6n and Wilkie (1990) I should add that when considering matters of strategy domination, I find the above criticisms much more compelling (see Jackson 1992» An analogy might be useful here Only I % of genetic information in human cells is actually used (see R Dawkins, The Blind Watchmaker, p 116, (Norton, New York) The other information may be there to be used just in case some unforeseeable event happens An Appendix to Chapter studies the implementation of the Ratio and the Lindahl correspondences in strict Nash equilibrium by means of the same mechanisms explained in this chapter The Walker mechanism belongs to a class of mechanisms called 'Tweed Ring' The story behind this name is that Tweed was the New York Mayor at the end of the previous century The famous cartoonist Thomas Nest pictured him and his friends in a circle Faced with the question 'Who stole the people's money?' each of them pointed their finger at the person next to him (See the entry corresponding on Caricature and Cartoon in the Encyclopaedia Britannica.) An alternative definition of monotonicity using MT( ) is the following