# Bayesian Implementation

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## Abstract

In this chapter we will be concerned with the theory of resource allocation under uncertainty. We will assume that uncertainty only affects the preferences of the members of the society but not their endowments or the productive capabilities that determine the set of feasible allocations. In section 7.2 we will present the main concepts under which the theory will be built later on, namely the notions of a type and a state. We will see how the concept of an allocation has to be redefined in our new setting. Section 7.3 reviews the main game-theoretical concepts that will be used in this chapter. We will see that the notion of strategy must also be revised. With this new notion in hand, we define our main equilibrium concept in this chapter, namely that of a Bayesian equilibrium. We also define what we mean by Bayesian implementation. Section 7.4 studies necessary and sufficient conditions for Bayesian implementation. We will see that a form of the revelation principle holds in our framework: Bayesian implementation implies that truthful revelation is a Bayesian equilibrium (Proposition 1). The latter property is known as Bayesian incentive compatibility. Proposition 2 states that a form of monotonicity, known as Bayesian monotonicity, is also necessary for Bayesian implementation. Proposition 3 shows that, in the case of exchange economies, Bayesian incentive compatibility and Bayesian monotonicity are also sufficient for Bayesian implementation. The last section studies a case in which Bayesian monotonicity can be replaced by Maskin monotonicity because of the special structure of information (Proposition 4). This implies that full information (constrained) Walrasian allocations are Bayesian-implementable.

## Keywords

Nash Equilibrium Social Choice Incentive Compatibility Social Choice Function Incentive Compatibility Constraint## Preview

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## References

- The fundamental contribution to the understanding of games with incomplete information is: J. Harsanyi (1967/8), ‘Games with Incomplete Information Played by Bayesian Players’,
*Management Science*, 14: pp. 159–82, 320–34, 486502.CrossRefGoogle Scholar - The revelation principle in a Bayesian setting was proved (among others) by R. Myerson (1979), ‘Incentive Compatibility and the Bargaining Problem’,
*Econometrica*, 47, pp. 61–74CrossRefGoogle Scholar - M. Harris and R. Townsend (1981), ‘Resource Allocation with Asymmetric Information’,
*Econometrica*, 49, pp. 33–64.CrossRefGoogle Scholar - The last paper discusses several concepts of efficiency. On this matter see also J.O. Ledyard (1987), ‘Incentive Compatibility’, entry in J. Eatwell, M. Milgate and P. Newman (eds),
*The New Palgrave Dictionary*(Macmillan).Google Scholar - 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 8.Google Scholar - On the theory of Bayesian implementation the reader may consult the following papers: A. Postlewaite and D. Schmeidler (1986), ‘Implementation in Differential Information Economies’,
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*Econometrica*, 59, pp. 461–77.CrossRefGoogle Scholar - 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)Google Scholar - T. Palfrey (1992), ‘Implementation in Bayesian Equilibrium: the Multiple Equilibrium Problem in Mechanism Design’ in J.J. Laffont (ed.),
*Advances in Economic Theory*(vol. 1), VI World Congress of the Econometric Society (Cambridge University Press).Google Scholar - Section 7.5 is based, almost entirely, on L. Corchbn and I. Ortuno-Ortin (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.Google Scholar - 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’,
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