The return function: A new computable perspective on Bayesian–Nash equilibria

In this paper, we suggest a new approach called the return function to deal with the determination of Bayesian–Nash equilibria in games of incomplete information. Whereas in the traditional approach players reply to each others’ strategies, here each player replies to his own return function. In short, given a player’s choice of action and the other players’ strategies, the return function of that given player is the probability distribution of the outcome. Interestingly, we show that the dynamics of best-reply strategies, which are hard to compute in practice, are mapped to an observable and easier-to-compute dynamics of return functions. We propose a new algorithm for computing Bayesian–Nash equilibria, and illustrate its implementation on a cake-cutting problem. Finally, we prove the convergence of the dynamics of return functions to the Bayesian–Nash equilibrium under fairly general topological assumptions.