Incomplete-Information Games in Large Populations with Anonymity

The paper provides theoretical foundations for models of strategic interdependence under uncertainty that have a continuum of agents and a decomposition of uncertainty into a macro component and an agent-specific micro component, with a law of large numbers for the latter. This macro-micro decomposition of uncertainty is implied by a condition of exchangeability of agents' types, which holds at the level of the prior if and only if it also holds at the level of agents' beliefs, i.e., posteriors. Under an additional condition of anonymity in payoffs, agents' behaviours are fully determined by their macro beliefs about the cross-section distribution of types and other macro variables and about the cross-section distribution of other agents' strategies. Any probability distribution over cross-section distributions of types and other macro variables is compatible with a fully specified belief system, but not every macro belief function is compatible with a common prior. The paper gives necessary and sufficient conditions for compatibility of a macro belief function with a common prior.