Perfect vs Imperfect Information and Bayesian Games

Perfect information means every information set contains a single node; each player knows exactly what has happened at every point in the game, as in chess. Imperfect information arises when some information sets contain multiple nodes, so a player cannot distinguish among several possible histories—examples include poker and simultaneous‑move games.

Complete information requires that all relevant facts are common knowledge before play begins; every player knows the payoff functions, the set of actions, and the structure of the game. Incomplete information leaves some facts known only to certain players, creating asymmetries. Typical settings are the used‑car market (where sellers know vehicle condition better than buyers), health‑insurance contracts (where insurers lack full medical histories), auctions (where bidders have private valuations), and bargaining situations (where political constraints or private facts differ across parties).

The Harsanyi Breakthrough

John Harsanyi’s 1967‑1968 insight transformed the analysis of games with incomplete information. He proposed modeling private information as a “move by nature” that randomly assigns each participant a type before the strategic interaction starts. The type captures the player’s entire mental state, including beliefs about the underlying state of nature and about other players’ types. This construction turns an incomplete‑information situation into a standard game with a well‑specified prior distribution, albeit with interpretational challenges about the pre‑deal stage.

Formalizing Bayesian Games

A Bayesian game is specified by five components:

• Players (i = 1 \dots n)
• Action sets (A_i) for each player
• State of nature (\theta \in \Theta) that determines fundamental payoffs
• Type spaces (T_i), where each type (t_i) encodes the player’s beliefs about (\theta) and about other players’ types
• Prior distribution (P) over the joint space (\Theta \times T_1 \times \dots \times T_n)

A strategy (s_i) is a function that maps a player’s realized type (t_i) to an action in (A_i); in other words, it is a complete contingent plan. The collection of all players’ strategies forms a strategy profile (s = (s_1, \dots, s_n)).

Equilibrium in Bayesian Games

Bayesian Nash Equilibrium (BNE) extends the Nash concept to settings with private information. For each player (i) and each possible type (t_i), the equilibrium strategy (s_i^*(t_i)) must solve

[ s_i^(t_i) \in \arg\max_{a_i \in A_i} \; \mathbb{E}{\theta, t{-i}\mid t_i}!\big[\,u_i(a_i, s_{-i}^(t_{-i}), \theta)\,\big], ]

where the expectation uses Bayes’ rule to update beliefs:

[ P(\theta, t_{-i}\mid t_i)=\frac{P(\theta, t_i, t_{-i})}{P(t_i)}. ]

Thus each type’s action maximizes expected utility given the player’s updated beliefs about the state of nature and opponents’ types. Consistency requires that, in equilibrium, beliefs about others’ strategies are correct. Moreover, a Bayesian Nash Equilibrium coincides with a Nash equilibrium of the ex‑ante strategic form game defined by the expected utilities before types are realized.

Illustrative Example

Consider a two‑person group project where each member chooses an effort level (e_i \in [0,1]). The payoff function (u_i = e_1 e_2 - e_i^2) may be adjusted by a state (\theta) that reflects external conditions. If a signal about (\theta) is observed with accuracy (q) (where (1/2 \le q \le 1)), each player forms beliefs about the other’s effort and the underlying state, then selects an effort that maximizes expected payoff given those beliefs. This simple setup demonstrates how types, priors, and Bayes’ rule interact in a Bayesian game.