Bayesian Games

A Bayesian game is a strategic interaction in which players have private information about their own characteristics — their 'type' — and must choose actions based on their beliefs about the types of others. The concept was introduced by John Harsanyi in 1967–68 as the natural setting for applying equilibrium analysis to situations of incomplete information, using what is now called the Harsanyi transformation to convert uncertainty about payoffs into uncertainty about a fictional move by Nature.

In a Bayesian game, each player's strategy is a function from types to actions: 'if I am this type, I will choose that action.' The solution concept is Bayesian Nash equilibrium: each player's type-contingent strategy maximizes expected utility, given her beliefs about other players' types and their strategies. The framework underlies modern mechanism design, auction theory, and signaling games — any domain where private information shapes strategic behavior.

The framework rests on the common prior assumption — that all players share the same probability distribution over types before receiving their private signals. This assumption is pragmatically necessary (without it, equilibrium analysis becomes intractable) but conceptually contentious. Do real negotiators share the same prior about each other's reservation prices? The common prior is not a psychological claim but a modeling convention — the price of making strategic uncertainty mathematically tractable.

Bayesian games are often presented as the general case of strategic interaction, with complete-information games as a special case where all types are identical. This is backwards. Complete-information games are the degenerate limit; Bayesian games are the normal case. Every real market, negotiation, and conflict is a game of incomplete information. The fact that game theory spent its first half-century analyzing the degenerate case says less about the world than about the mathematical convenience of assuming away uncertainty. Harsanyi's contribution was not to generalize game theory but to restore it to reality.