Understanding Bayesian Nash Equilibrium

From Nash Equilibrium to Bayesian Nash Equilibrium

Let's quickly recall the definition of a Nash Equilibrium (NE) in a complete information game. A strategy profile $s^{\ast }=(s_1^{\ast },\dots ,s_n^{\ast })$ is a Nash Equilibrium if no single player $i$ can get a better payoff by unilaterally changing their strategy, given that all other players $-i$ stick to their strategies $s_{-i}^{\ast }$. Formally:

This logic hinges on players knowing the full payoff structure of the game. But in a Bayesian game, a player doesn't know their opponents' types, and therefore doesn't know which exact payoff matrix is being played. How can a player choose a best response?

The answer has two parts:

1. Redefining Strategy: A player's strategy can no longer be a single action. It must be a complete contingent plan. A strategy for player $i$ in a Bayesian game is a function, $s_i({\theta }_i)$, that specifies an action for each of the player's possible types ${\theta }_i\in {\Theta }_i$.
2. Maximizing Expected Utility: Since a player is uncertain about their opponents' types, they cannot maximize their utility directly. Instead, they maximize their expected utility, where the expectation is taken over their beliefs about the distribution of their opponents' types.

Combining these ideas gives us the Bayesian Nash Equilibrium.

Defining Bayesian Nash Equilibrium

The most direct way to understand BNE is to see it as a direct parallel to NE, but with "player" replaced by "type of player".

As the video highlights, the core definition is:

A Bayesian Nash Equilibrium is a set of strategies, one for each type of player, such that no type has an incentive to change their strategy, given their beliefs about the types of the other players and the strategies those other types are playing.

This is the central idea. Now, let's build on this with the mathematical formalism you appreciate. The following resources provide the precise definitions.

First, let's formalize what a strategy is in this context.

To consolidate this, the following document provides the definition in a clear, textual format.

Let's synthesize these resources into a single, formal definition.

A strategy profile $s^{\ast }=(s_1^{\ast }(\cdot ),s_2^{\ast }(\cdot ),\dots ,s_n^{\ast }(\cdot ))$, where each $s_i^{\ast }(\cdot )$ is a function mapping types to actions, is a Bayesian Nash Equilibrium if for every player $i$ and for every one of their possible types ${\theta }_i\in {\Theta }_i$, the action $a_i^{\ast }=s_i^{\ast }({\theta }_i)$ specified by their strategy maximizes player $i$'s expected utility.

This means that for any player $i$ of type ${\theta }_i$, switching to any other action $a_i\in A_i$ will not yield a higher expected payoff. The condition is:

Expanding the expectation, this becomes:

Let's break down this inequality:

• $s_i^{\ast }({\theta }_i)$: The action prescribed for player $i$ by their equilibrium strategy when their type is ${\theta }_i$.
• $s_{-i}^{\ast }({\theta }_{-i})$: The actions prescribed for all other players by their equilibrium strategies, given their types ${\theta }_{-i}$.
• $p({\theta }_{-i}|{\theta }_i)$: Player $i$'s belief—the conditional probability of opponents having types ${\theta }_{-i}$, given player $i$ knows their own type is ${\theta }_i$.
• The summation ${\sum }_{{\theta }_{-i}\in {\Theta }_{-i}}$ is the process of averaging the payoffs over all possible combinations of types for the other players, weighted by player $i$'s beliefs.

BNE as a Nash Equilibrium of an "Expanded Game"

John Harsanyi's critical insight was that a BNE isn't a new concept from scratch. It is simply the Nash Equilibrium of a cleverly constructed "expanded game" of complete (but imperfect) information.

How do we construct this expanded game?

1. The "Players": The players are not $1,2,\dots ,n$. Instead, the players are every possible type of every original player. If Player 1 has two types, ${\theta }_{1A}$ and ${\theta }_{1B}$, we treat them as two separate "players" in the expanded game: "Player 1-Type-A" and "Player 1-Type-B".
2. The Actions: The actions for these new "players" are the same as the original actions. For example, "Player 1-Type-A" chooses an action from $A_1$.
3. The Payoffs: The payoff for each of these new "players" is their expected utility from the original Bayesian game, as defined in the inequality above.

In this expanded game, every "player" (i.e., every type) simultaneously chooses an action to maximize their expected utility, given the actions chosen by all other "players" (types). A profile of actions, one for each "player-type," is a Nash Equilibrium of this expanded game if and only if no "player-type" has a profitable unilateral deviation. This is precisely the definition of a Bayesian Nash Equilibrium.

This perspective is extremely useful. It shows that by correctly framing the problem—treating each type as a strategic agent maximizing expected utility—we can apply the familiar logic of Nash Equilibrium to the complex world of incomplete information.

Conclusion

In this lesson, we have formally defined the primary solution concept for static games of incomplete information. You can now see how the principles of rationality and strategic thinking are extended to environments with private information.

Key Takeaways:

• A Bayesian Nash Equilibrium (BNE) is a profile of strategies, where a strategy for each player is a function mapping each of their types to an action.
• The core condition of BNE is that for every player, the action prescribed for each of their types must be a best response to the equilibrium strategies of the other players.
• This best response is defined in terms of maximizing expected utility, calculated by averaging over a player's beliefs about their opponents' types.
• Fundamentally, a BNE is a Nash Equilibrium of an "expanded game" where each player-type is considered a separate player maximizing their expected utility.

We have defined the what. In our next lesson, we will focus on the how. We will take the expected utility formula we've seen today and apply it step-by-step to compute the BNE in simple games.