Markov chain Monte Carlo for incomplete information discrete games

Abstract

This paper outlines a Bayesian approach to estimating discrete games of incomplete information. The MCMC routine proposed features two changes to the traditional Metropolis-Hastings algorithm to facilitate the estimation of games. First, we propose a new approach to sample equilibrium probabilities using a probabilistic equilibrium selection rule that allows for the evaluation of the parameter posterior. Second, we propose a di¤erential evolution based MCMC sampler which is capable of handling the unwieldy posterior that only has support on the equilibrium manifold. We also present two applications to demonstrate the feasibility of our proposed methodology.

Efficient computation of discrete games: Estimating the effect of Apple on market structure

“…For the estimation of the model parameters, this study adopts a Bayesian approach because it provides a unified methodology for inference and decision (Rossi et al., 2005). The Bayesian approach can properly reflect parameter uncertainty, including the equilibrium selection when evaluating the desirability of a firm's location choice (Misra, 2013; Narayanan, 2013). Despite its advantages, the Bayesian approach is computationally more challenging than many frequentist methods (Betancourt et al., 2017).…”

“…Second, the framework can capture players’ strategic effects—both negative (competitive) and positive (complementary) effects of their own and rivals’ actions. Thus, researchers have used discrete games to examine players’ strategic behavior that affects market dynamics (Arcidiacono et al., 2020; Berry, 1992; Bresnahan & Reiss, 1990, 1991; Ciliberto et al., 2021; Ciliberto & Tamer, 2009; Mazzeo, 2002; Misra, 2013; Narayanan, 2013; Seim, 2006; Zhu et al., 2009). Generally, there are two types of discrete games: those of complete information where players have full knowledge regarding their competitors and those of incomplete information where players possess private information not known to their rivals.…”

Multiplicity of equilibria and information structures in empirical games: challenges and prospects

“…Besanko, Doraszelski, Lu & Satterthwaite (2010a, 2010b, Kryukov (2010, 2012) and Besanko, Doraszelski & Kryukov (2013). ** See Narayanan (2013) and Misra (2013) for Bayesian approaches to estimate complete or incomplete information games, respectively, that mix over equilibrium. Aguirregabiria and Magesan (2012) study identification of dynamic games when players' beliefs about rivals' actions may be biased.…”

Modeling marketplace behavior

“…Imai et al (2009) propose a Bayesian estimator for models of forwardlooking behavior that overcomes difficulties in the evaluation of high dimensional value functions. Misra (2011) proposes use of data augmentation to condition on error realizations in models of discrete games with incomplete information. Narayanan (2011) proposes a Bayesian estimator for estimating models of complete games.…”