Bayesian Games and Bayesian Nash Equilibrium

Why Private Information Changes the Game

In a complete-information game, each player knows the payoff-relevant circumstances that determine everyone’s incentives. With private information, a player may know something—such as their cost, quality, or willingness to pay—that others do not observe. The player’s action can then depend on that information, while opponents choose their actions based on beliefs about it.

A Bayesian game represents this situation explicitly. It specifies the possible private information of each player, how likely the different type profiles are, and what each player earns for each combination of types and actions. A solution must describe not just what players do, but how their actions depend on their own types. That type-contingent description is essential when using a strategic model to interpret observed behavior.

Finite Bayesian Games: Types, Beliefs, and Strategies

Definition: A finite Bayesian game consists of a finite set of players $i=1,\ldots,N$; a finite type space $T_i$ and action space $A_i$ for each player; a payoff function $u_i(a_1,\ldots,a_N,t_1,\ldots,t_N)$ for each player; and a common prior probability distribution $\pi(t_1,\ldots,t_N)$ over type profiles. Each player observes their own type before choosing an action, but not necessarily the other players’ types. The game’s rules, payoffs, and prior are common knowledge.

A type is the player-specific information relevant to their payoff or decision. For example, a firm’s type could be its production cost. A type profile $t=(t_1,\ldots,t_N)$ records the types of all players. The prior $\pi$ describes uncertainty about that profile before players observe their own types; it need not make types independent.

After observing type $t_i$, player $i$ forms a belief about opponents’ types. For a type $t_i$ with positive prior probability, the conditional belief is

\mu_i(t_{-i}\mid t_i) = \frac{\pi(t_i,t_{-i})}{\sum_{\tilde t_{-i}}\pi(t_i,\tilde t_{-i})}.

Here $t_{-i}$ denotes the types of all players other than $i$. If types are independent, the conditional distribution of opponents’ types does not depend on $t_i$. If types are correlated, observing one’s own type can change beliefs about opponents.

A pure strategy specifies an action for every type a player might have. It is therefore a function $s_i:T_i\to A_i$, not just one action. Given opponents’ pure strategies, type $t_i$ evaluates an action by averaging over the opponents’ types according to their conditional beliefs:

U_i(a_i,s_{-i}\mid t_i) = \sum_{t_{-i}}\mu_i(t_{-i}\mid t_i) u_i\bigl(a_i,s_{-i}(t_{-i}),t_i,t_{-i}\bigr).

The expression $s_{-i}(t_{-i})$ is the opponents’ action profile prescribed by their strategies at type profile $t_{-i}$. A type’s best response maximizes this conditional expected payoff over available actions.

Definition: A pure-strategy Bayesian Nash equilibrium is a profile of strategies $s^*=(s_1^*,\ldots,s_N^*)$ such that, for every player $i$ and every type $t_i$ with positive probability, $$ s_i^*(t_i)\in\arg\max_{a_i\in A_i} U_i(a_i,s_{-i}^*\mid t_i). $$ Each type’s prescribed action must be optimal given that type’s beliefs and the other players’ type-contingent strategies.

For finite games, strategies can also randomize: a mixed strategy specifies a probability distribution over actions for each type. The equilibrium condition is then that each type’s strategy assigns positive probability only to actions maximizing its conditional expected payoff. The examples below use pure strategies so that the type-by-type incentives can be checked directly.

How Conditional Expected Payoffs Are Derived

The expected-payoff calculation follows from the definition of conditional expectation. For a fixed type $t_i$, each possible opponents’ type profile $t_{-i}$ leads to a payoff determined by the actions prescribed for that profile. Weight each payoff by the probability of that profile conditional on $t_i$, then add the weighted payoffs. To verify an equilibrium, compare this result across the player’s available actions separately for every type.

\begin{aligned} U_i(a_i,s_{-i}\mid t_i) &=\sum_{t_{-i}} P(T_{-i}=t_{-i}\mid T_i=t_i)\, u_i\bigl(a_i,s_{-i}(t_{-i}),t_i,t_{-i}\bigr)\\ &=\sum_{t_{-i}}\mu_i(t_{-i}\mid t_i)\, u_i\bigl(a_i,s_{-i}(t_{-i}),t_i,t_{-i}\bigr). \end{aligned}

The key conditioning is on the player’s own observed type. Using the unconditional prior in its place is justified only when the relevant types are independent. The following examples show how to perform the calculation and check the full equilibrium conditions.

Worked Example: A Private-Cost Firm and an Informed Decision

A firm has private type $H$ with probability $0.75$, or type $L$ with probability $0.25$. It chooses Invest ($I$) or Stay out ($O$). A second player, who has no private type, simultaneously chooses Support ($S$) or Decline ($D$). The firm’s payoffs are:

If it invests and receives support, its payoff is 2 when type $H$ and $-1$ when type $L$. If it invests and receives no support, its payoff is $-1$ for type $H$ and $-2$ for type $L$. Staying out gives the firm 0 for either type, regardless of the second player’s action. The second player receives 1 when choosing $S$ and the firm invests, $-2$ when choosing $S$ and the firm stays out, and 0 from choosing $D$.

Consider the strategy profile in which type $H$ invests, type $L$ stays out, and the second player supports. Given support, type $H$’s payoff from investing is 2, compared with 0 from staying out, so $H$ prefers $I$. Type $L$’s payoff from investing is $-1$, compared with 0 from staying out, so $L$ prefers $O$.

The second player believes the firm invests with probability $0.75$, because this happens exactly when the firm is type $H$. Its expected payoff from support is $0.75(1)+0.25(-2)=0.75-0.50=0.25$. Declining gives 0. Since $0.25>0$, support is optimal. All types’ prescribed actions are best responses, so the proposed profile is a pure-strategy Bayesian Nash equilibrium.

Type-Contingent Best Responses with Independent Types

Worked Example: A High Type Acts, a Low Type Waits

Two players each have type $H$ or $L$, independently, with probability $0.5$ for each type. Each chooses $A$ or $B$. A player choosing $A$ earns 2 if the other player also chooses $A$, and 0 from that part of the payoff otherwise; the player then pays a type-dependent cost. The costs are 0.5 for type $H$ and 1.5 for type $L$. Choosing $B$ gives a payoff of 0.

Consider the strategy for each player: choose $A$ if type $H$, and $B$ if type $L$. Under the other player’s strategy, $A$ is chosen exactly when that player is type $H$. Independence means that, conditional on either own type, the probability the opponent chooses $A$ is $0.5$.

For a type-$H$ player, the expected payoff from $A$ is $0.5(2-0.5)+0.5(0-0.5)=0.5$. The payoff from $B$ is 0, so $H$ prefers $A$. For type $L$, the expected payoff from $A$ is $0.5(2-1.5)+0.5(0-1.5)=-0.5$. Since $-0.5<0$, $L$ prefers $B$. These calculations apply symmetrically to both players, verifying a pure-strategy Bayesian Nash equilibrium.

The same method works when a player has more than two possible types or actions: calculate conditional expected payoffs for each type, compare all available actions, and check that the proposed strategy assigns an optimal action to every type. An action that is best for one type need not be best for another.

Correlated Types Change Beliefs

Worked Example: Updating Beliefs from a Correlated Type

Two players each have type $H$ or $L$, with joint type probabilities shown below. Each chooses $A$ or $B$. Choosing $A$ gives a payoff of 2 if the opponent chooses $A$, and 0 otherwise, minus a cost of 1 for type $H$ or 0.5 for type $L$. Choosing $B$ gives 0.

Player 1 type	Player 2 type $H$	Player 2 type $L$
$H$	0.40	0.10
$L$	0.10	0.40

The probabilities sum to 1. Each player’s marginal probability of type $H$ is $0.50$. But conditional beliefs differ: for either player, the probability the opponent is type $H$, given own type $H$, is $0.40/0.50=0.80$. Given own type $L$, it is $0.10/0.50=0.20$.

Consider the strategy “choose $A$ when type $H$, and $B$ when type $L$” for both players. A type-$H$ player expects the opponent to choose $A$ with probability $0.80$. Its expected payoff from $A$ is $0.80(2-1)+0.20(0-1)=0.60$, greater than the payoff 0 from $B$. A type-$L$ player expects $A$ with probability $0.20$. Its expected payoff from $A$ is $0.20(2-0.5)+0.80(0-0.5)=0.30-0.40=-0.10$, less than the payoff 0 from $B$. The symmetric strategy is therefore a pure-strategy Bayesian Nash equilibrium.

Using the unconditional probability $0.50$ for both types would give incorrect beliefs in this example. The relevant probability depends on the type each player has observed.

Why the Equilibrium Conditions Matter

A Bayesian Nash equilibrium is a consistency requirement between beliefs, strategies, and incentives. Beliefs come from the prior and a player’s information; strategies say how every type acts; and, given those strategies and beliefs, each type must find its prescribed action optimal. It is not enough to show that the average payoff across types is high, or that the strategy is optimal for a “typical” type. The condition is checked type by type.

This structure matters in structural models because observed behavior can depend on information that the analyst does not observe. A model’s predicted actions depend on both the payoff parameters and the beliefs implied by its information assumptions. Treating a private type as if every player observed it can change the predicted strategic incentives, even when the payoff table itself is unchanged.

Common Mistakes

Giving a strategy as one action. A strategy in a Bayesian game specifies an action for each possible own type. Write, for example, “$H$ invests and $L$ stays out,” not merely “the firm invests.”
Using the prior instead of a conditional belief. The prior describes uncertainty before observing one’s type. A player with private information uses beliefs conditional on that information; correlated types make the distinction consequential.
Checking only one type’s incentives. An equilibrium requires every positive-probability type to prefer its prescribed action to every alternative, given opponents’ strategies.
Confusing an expected payoff with a realized payoff. A type’s expected payoff averages over uncertainty about opponents’ types. It is not the payoff that must occur in every type profile.
Assuming a type-contingent profile is automatically an equilibrium. A proposed mapping from types to actions is only a candidate until all best-response inequalities have been checked.

Key takeaway: To verify a pure-strategy Bayesian Nash equilibrium, derive each type’s conditional beliefs, calculate its expected payoff from every available action, and confirm that the prescribed action maximizes that payoff for every player and type.

Check Your Understanding

Use the definitions and calculations in this tutorial to answer the following questions.

What information does a pure strategy specify in a Bayesian game, and how does that differ from choosing one action?
If player types are independent, how does observing one’s own type affect beliefs about an opponent’s type?
In the correlated-types example, why is the probability of an opponent being type $H$ different after observing own type $H$ versus own type $L$?
What inequalities must hold for a proposed pure-strategy profile to be a Bayesian Nash equilibrium?
Suppose a type’s expected payoff from $A$ is $-0.2$ and its payoff from $B$ is 0. Which action is a best response for that type?