Bayesian Games with Baire Class 1 Payoff Functions

This paper considers Bayesian games where a player's payoff function is a Baire class 1 function. A Baire class 1 function (Baire (1905) is defined as a pointwise limit of a sequence of continuous functions and include semicontinuous functions, step functions, functions of bounded variations, and monotone functions. Bayesian games with Baire class 1 payoff's generalize the canonical model of Milgrom and Weber (1985) and encompass a large class of auction games and mechanism design problems. A Bayesian game with Baire class 1 payoff's has a Bayesian equilibrium via continuous approximations when the better reply security condition of Reny (1999) is satisfied. This continuous approximation approach can be regarded as a generalization of the path-following method to games with discontinuous payoff's. Using the continuous approximation method, we characterize the optimal auction mechanism with heterogeneous objects and multidimensional types with continuous distributions by unifying Myerson (1981), Mussa and Rosen (1978), and Rochet and Chone (1998). We illustrate the method by solving an example where the buyers private values are uniformly distributed on a square.