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A Bayesian game models situations where players have private information (types). Each player has a type from a set T_i, a payoff function u_i(a, t), and beliefs over others' types. A Bayesian Nash equilibrium is a strategy for each type such that each type's strategy maximizes expected payoff given beliefs. This framework unifies signaling, screening, and mechanism design.
The game theory you encountered in introductory microeconomics assumed that all players know everything relevant — the payoffs, the strategies available, and crucially, who they are playing against. This is the complete information assumption, and it is often unrealistic. In most real strategic situations, you face uncertainty about the other party: Does the seller know something about this car that I don't? How much does my rival bidder value this contract? Is my opponent in negotiation a tough type or a soft type? Bayesian game theory is the framework for analyzing exactly these situations.
The central innovation is the concept of a type. A type is a bundle of private information that a player holds and others do not observe. In an auction, your type is your valuation for the item. In an insurance market, your type is your risk level. In a negotiation, your type might be your reservation price. Each player knows their own type but only has probabilistic beliefs about others' types — typically modeled as a common prior distribution that everyone agrees on (the Harsanyi assumption). When you play, you cannot condition on information you don't have; you must form expectations.
A strategy in a Bayesian game is therefore richer than in a standard game. It is a function from types to actions: "if I am type t, I will take action a(t)." A high-value bidder bids differently from a low-value bidder, so the equilibrium specifies a rule for every possible type, not a single action. A Bayesian Nash equilibrium is a profile of such strategies — one for each player — such that no type of any player can gain by deviating. Each type is best-responding given its beliefs about others' type distributions and the strategies those other types are playing.
Consider a first-price sealed-bid auction as a clean example. You must submit a bid without seeing anyone else's. Your type is your private value v. You know that others' values are drawn from some distribution (say, uniform on [0, 100]). The equilibrium bidding strategy — derivable from the expected payoff calculation — has you bid less than your true value (bid-shading), because winning at a price equal to your value earns zero profit. The precise amount to shade depends on the number of bidders and the distribution. This is a Bayesian Nash equilibrium: given that everyone follows the same bidding function, no single bidder can improve their expected payoff by deviating.
This framework is foundational for everything that follows: mechanism design asks how to structure rules to achieve outcomes when participants have private types; signaling models players actively trying to reveal or conceal their types; screening models the uninformed party trying to sort types through contract design. All of these are applications of the Bayesian game structure.