Estimation of discrete games of complete information, which have been applied to a variety of contexts such as market entry, technology adoption and peer effects, is challenging due to the presence of multiple equilibria. In this paper, we take a Bayesian MCMC approach to this problem, specifying a prior over multiple equilibrium selection mechanisms reflecting the analysts uncertainty over them. We develop a sampler, using the reversible jump algorithm to generate draws from the posterior distribution of parameters across these equilibrium selection rules. The algorithm is flexible in that it can be used both in situations where the equilibrium selection rule is identified and when it is not, and accommodates heterogeneity in equilibrium selection. We explore the methodology using both simulated data and two empirical applications, one in the context of joint consumption, using a dataset of casino visit decisions by married couples, and the second in the context of market entry by competing chains in the retail stationery market. We demonstrate the importance of accounting for multiple equilibrium selection rules in these applications and show that taking an empirical approach to the issue, such as the one we have demonstrated, can be useful.
