representing bayesian games without a common prior
Game-theoretic analyses of multi-agent systems typically assume that all agents have full knowledge of everyone's possible moves, information sets and utilities for each outcome. Bayesian games relax this assumption by allowing agents to have different "types," representing different beliefs about the game being played, and to have uncertainty over other agents' types. However, applications of Bayesian games almost universally assume that all agents share a common prior distribution over everyone's type. We argue, in concord with certain economists, that such games fail to accurately represent many situations. However, when the common prior assumption is abandoned, several modeling challenges arise, one of which is the emergence of complex belief hierarchies. In these cases it is necessary to specify which parts of other agents' beliefs are relevant to an agent's decision-making (or need be known by that agent). We address this issue by suggesting a concise way of representing Bayesian games with uncommon priors. Our representation centers around the concept of a block , which groups agents' view of (a) the game being played and (b) their posterior beliefs. This allows us to construct the belief graph, a graphical structure that allows agents' knowledge of other agents' beliefs to be carefully specified. Furthermore, when agents' views of the world are represented by extensive form games, our block structure places useful semantic constraints on the extensive form trees. Our representation can be used to naturally represent games with rich belief structures and interesting predicted behavior.
bayesian games common prior assumption belief hierarchy game theory formalism knowledge requirements equilibrium