Bayesian Estimation of Discrete Games of Complete Information

Discrete games of complete information have been used to analyze a variety of contexts such as market entry, technology adoption and peer effects. They are extensions of discrete choice models, where payoffs of each player are dependent on actions of other players, and each outcome is modeled as Nash equilibria of a game, where the players share common knowledge about the payoffs of all the players in the game. An issue with such games is that they typically have multiple equilibria, leading to the absence of a one-to-one mapping between parameters and outcomes. Theory typically has little to say about equilibrium selection in these games. Researchers have therefore had to make simplifying assumptions, either analyzing outcomes that do not have multiplicity, or making ad-hoc assumptions about equilibrium selection. Another approach has been to use a bounds approach to set identify rather than point identify the parameters. A third approach has been to empirically estimate the equilibrium selection rule. In this paper, we take a Bayesian MCMC approach to estimate the parameters of the payoff functions in such games. Instead of making ad-hoc assumptions on equilibrium selection, we specify a prior over the possible equilibria, reflecting the analyst's uncertainty about equilibrium selection and find posterior estimates for the parameters that accounts for this uncertainty. We develop a sampler using the reversible jump algorithm to navigate the parameter space corresponding to multiple equilibria and obtain posterior draws whose marginal distributions are potentially multi-modal. When the equilibria are not identified, it goes beyond the bounds approach by providing posterior distributions of parameters, which may be important given that there are likely regions of low density for the parameters within the bounds. When data allow us to identify the equilibrium, our approach generates posterior estimates of the probability of specific equilibria, jointly with the estimates for the parameters. Our approach can also be cast in a hierarchical framework, allowing not just for heterogeneity in parameters, but also in equilibrium selection. Thus, it complements and extends the existing literature on dealing with multiplicity in discrete games. We first demonstrate the methodology using simulated data, exploring the methodology in depth. We then present 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 show the importance of accounting for multiple equilibria in these application, and demonstrate how inferences can be distorted by making the typically used equilibrium selection assumptions. Our applications show that it is important for empirical researchers to take the issue of multiplicity of equilibria seriously, and that taking an empirical approach to the issue, such as the one we have demonstrated, can be very useful.