Indian Institute of Science Bangalore , India July 2012 Bayesian Mechanisms

Recall that we mentioned two possible routes to get around the Gibbard–Satterthwaite Impossibility Theorem. The first was to focus on restricted environments like the quasilinear environment, and the second one was to weaken the implementation concept and look for an SCF which is ex-post efficient, nondictatorial, and Bayesian incentive compatible. In this section, our objective is to explore the second route. Throughout this section, we will once again be working with the quasilinear environment. As we saw earlier, the quasilinear environments have a nice property that every social choice function in these environments is nondictatorial. Therefore, while working within a quasilinear environment, we do not have to worry about the nondictatorial part of the social choice function. We can just investigate whether there exists any SCF in quasilinear environment, which is both ex-post efficient and BIC, or equivalently, which has three properties — AE, BB, and BIC. Recall that in the previous section, we have already addressed the question whether there exists any SCF in quasilinear environments that is AE, BB, and DSIC, and we found that no function satisfies all these three properties. On the contrary, in this section, we will show that a wide range of SCFs in quasilinear environments satisfy three properties — AE, BB, and BIC.