Profitable Bayesian implementation

In mechanism design theory, a designer would like to implement a desired social choice function which specifies her favorite outcome for each possible profile of all agents' types. Since agents' types are modelled as their private information, what the designer can do is to construct a mechanism and choose an outcome after observing a specific profile of agents' strategies. Traditionally, the designer has no way to adjust agents' types and hence may be in a dilemma in the sense that even if she is not satisfied with some outcome with low profit, she has to announce it because she must obey the mechanism designed by herself. In this paper, we generalize the mechanism design theory to a case where the designer can adjust the type distribution of agents, and propose a novel notion, \\emph{i.e.}, profitable Bayesian implementation. After defining a series of notions such as adjusted types, optimal adjustment cost and profitable Bayesian implementability, we propose that the revelation principle does not hold in this generalized case. Finally, we construct an auction example to show that the designer can obtain an expected profit greater than the maximum profit that she can obtain in the traditional optimal auction, at the same time each agent's \\emph{ex ante} expected profit is also increased.