Generalised Fictitious Play for a Continuum of Anonymous Players

Recently, efficient approximation algorithms for finding Nash equilibria have been developed for the interesting class of anonymous games, where a player’s utility does not depend on the identity of its opponents. In this paper, we tackle the problem of computing equilibria in such games with continuous player types, extending the framework to encompass settings with imperfect information. In particular, given the existence result for pure Bayes-Nash equilibiria in these games, we generalise the fictitious play algorithm by developing a novel procedure for finding a best response strategy, which is specifically designed to deal with continuous and, therefore, infinite type spaces. We then combine the best response computation with the general fictitious play structure to obtain an equilibrium. To illustrate the power of this approach, we apply our algorithm to the domain of simultaneous auctions with continuous private values and discrete bids, in which the algorithm shows quick convergence.