Connection between Graphical Potential Games and Markov Random Fields with an Extension to Bayesian Networks

Abstract

A probabilistic graphical model is a graphical representation of a joint probability distribution, in which the conditional independencies among random variables are specified via an underlying graph. We connect the probabilistic graphical models to some special types of games. Graphical potential games are the intersection of potential games and graphical games. They have characteristics from both of these two classes of games, namely, potential functions of potential games and graphical structure of graphical games. We review that there is a bijection between the normalized graphical potential games and the corresponding Markov Random Fields. We use a similar method to study the structure of Bayesian networks and define two types of games on directed graphs whose nodes are players. One is the directed graphical game, which is defined based on the assumption that the utility of player i only depends on the parent of i in the graph. The other one is the Bayesian-factorable potential game. The potential function of the game gives rise to the probability distribution, which can be factorized as in a Bayesian network. We explore the connections between such games and Bayesian networks.