A Sufficient And Necessary Condition Of 2-person Bayesian Implementation

For a given environment that consists of a set of agents,a set of feasible allocations and agents' information sets,the theory of mechanism design is concerned with the problem of incentives.When the social choice rule has choosed a desired results,whether and how we can design a mechanism that make the personal interests equal to the desired goal of designer.According to the information is complete or incomplete,the theory of mechanism design can be divided into two aspects.The first aspect is Nash imlementation under the environment with complete information. And the second aspect is Bayesian implementation under the environment with incomplete information.This paper focuses on Bayesian implementation with incomplete information.There are a lot of papers on Bayesian implementation,but most of these papers have assumed that there are at least three agents,with the exception of Mookherjee and Reichelstein(1990),Dutta and Sen(1994).In the literature of Dutta and Sen(1994),they have shown the theory of 2-person Bayesian implementation in general environments with incomplete information. They defined two classical conditions(condition A and condition A' in Chapter 4),which are sufficient condition and necessary condition respectively,and laid theoretical foundation for the theory of 2-person Bayesian implementation.This paper will quote these two conditions and connect with the third condion(condition B in Chapter 4) to extend the results of Dutta and Sen to consider 2-person Bayesian implementation in pure exchange economic environments.It will establish a model of 2-person Bayesian mechanism under above environments and give a sufficient and necessary condition of social choice rule that can be Bayesian implemented when the individual preferences and initial endowments are both incomplete information.It will show that,in pure exchange economic environments with two individuals,and utility functions U~i:R_+~ι×T→R_+ are stictly increasing in consumption for each type t∈T,a social choice set F is Bayesian implementable if and only if condition A and condition B or condition A' and condition B are satisfied.