Computing pure Bayesian-Nash equilibria in games with finite actions and continuous types

We extend the well-known fictitious play (FP) algorithm to compute pure-strategy Bayesian-Nash equilibria in private-value games of incomplete information with finite actions and con-tinuous types (G-FACTs). We prove that, if the frequency distribution of actions (fictitiousplay beliefs) converges, then there exists a pure-strategy equilibrium strategy that is con-sistent with it. We furthermore develop an algorithm to convert the converged distributionof actions into an equilibrium strategy for a wide class of games where utility functions arelinear in type. This algorithm can also be used to compute pure -Nash equilibria whendistributions are not fully converged. We then apply our algorithm to find equilibria inan important and previously unsolved game: simultaneous sealed-bid, second-price auctionswhere various types of items (e.g., substitutes or complements) are sold. Finally, we providean analytical characterization of equilibria in games with linear utilities. Specifically, weshow how equilibria can be found by solving a system of polynomial equations. For a specialcase of simultaneous auctions, we also solve the equations confirming the results obtainednumerically.