Elements of Bayesian Games

Introduction

Welcome back! In our last lesson, we established a crucial distinction between imperfect information (uncertainty about others' actions) and incomplete information (uncertainty about others' payoffs or 'types'). We also saw John Harsanyi's ingenious insight: any game of incomplete information can be converted into a game of complete but imperfect information by introducing a fictional player, "Nature," who makes a random move at the start to determine players' types.

Today, we will formalize this idea. Our goal is to define the essential components of these games of incomplete information, which are known as Bayesian games. By the end of this lesson, you will be able to define and identify a game's types, beliefs, and state-dependent payoffs. This structured framework is the foundation for finding solutions in strategic situations where players have private information, bringing us closer to your goal of analyzing complex, "chaotic" scenarios.

From Intuition to a Formal Model

Before diving into the math, let's build the intuition. In previous modules, we assumed players' payoff matrices were common knowledge. Now, we're relaxing that. Imagine you are bidding in an auction. You know your own valuation for the item, but you don't know how much your rivals value it. Your uncertainty isn't about what action they will take (their bid), but about a fundamental characteristic that drives that action—their valuation.

This is the core problem that Bayesian games are designed to model. The following video provides an excellent intuitive explanation of how we translate this uncertainty into a workable game-theoretic structure.

1. Introduction to Bayesian Games (Game Theory Playlist 9)

In this video, Selcuk Ozyurt uses an auction example to explain why we need to move beyond complete information and how we can start to model uncertainty about others' payoffs.

Please watch from 20:59 to 33:22. As you watch, focus on how the lecturer formalizes the informal idea of 'not knowing the opponent's valuation': Private Information (27:47): What term is introduced to represent a player's private information, like their valuation? Modeling Uncertainty (28:54): How do we formally represent a player's uncertainty about their opponent's private information?

The video introduces the two central concepts we need:

Type ( $θ_{i}$ ): A player's type is their private information. It encapsulates everything a player knows about the game that other players do not. In the auction example, a player's valuation $v_{i}$ is their type. A player always knows their own type.
Beliefs: Since a player doesn't know their opponents' types, they must have beliefs about them. A belief is a probability distribution over the possible types of the other players. For our analysis to be tractable, we rely on Harsanyi's idea of a common prior. This means we assume that before the game begins and before any player learns their own type, there is a single, common probability distribution over all possible combinations of types that all players agree on. A player then uses this common prior and their own type to form updated beliefs about their opponents' types via conditional probability.

With these concepts, we can now construct the formal definition of a Bayesian game.

The Five Components of a Bayesian Game

Your interest in the mathematical structure of theories will be well-served here. A Bayesian game is formally defined by a collection of five components. We can represent it as a tuple: $(N, A, Θ, p, u)$ .

Let's break this down piece by piece, using the formal definition found in the resource below.

[PDF] Stochastic Games and Bayesian Games

This lecture slide from the University of British Columbia provides a concise, formal definition of a Bayesian game using the 'epistemic type' formulation, which is the standard in modern game theory.

Please read the section titled 'Definition 3: Epistemic Types'. It presents the Bayesian game as a tuple. We will dissect each part of this definition.

Here are the components from the definition:

Players $N$ : This is the familiar set of players, $N = {1, 2, ..., n}$ .
Actions $A$ : This is the set of all possible action profiles, $A = A_{1} \times A_{2} \times \dots \times A_{n}$ , where $A_{i}$ is the set of actions available to player $i$ .
Types $Θ$ : This is the set of all possible type profiles, $Θ = Θ_{1} \times Θ_{2} \times \dots \times Θ_{n}$ , where $Θ_{i}$ is the type space for player $i$ . Nature chooses a type profile $θ = (θ_{1}, \dots, θ_{n})$ according to the probability distribution $p$ . Player $i$ is told their own type $θ_{i}$ , but not the types of the other players, $θ_{- i}$ .
Beliefs $p$ : This is the common prior probability distribution over the set of types $Θ$ . For any given combination of types $θ \in Θ$ , $p (θ)$ is the probability that this is the actual combination of types chosen by Nature.
Payoffs $u$ : This is a set of payoff functions $u = (u_{1}, u_{2}, \dots, u_{n})$ . Each player's payoff function $u_{i} : A \times Θ \to R$ depends on the action profile $a \in A$ and the type profile $θ \in Θ$ . This is what we call state-dependent payoffs: the outcome depends on the "state of the world," which is determined by the players' types.

An Example to Put It All Together

These abstract definitions become much clearer with a concrete example. The following image presents a simple 2-player Bayesian game and breaks down its formal components.

Bayesian Game Example: Definition and Initial Steps Towards BNE — This image illustrates a 2x2 Bayesian game. Player 1 has types $\{t_1, t'_1\}$ and Player 2 has types $\{t_2, t'_2\}$. The table at the top left shows the common prior probability for each type-pair. The matrices show the payoffs for each action profile, which depend on the players' types.

Let's use this image to identify the five components of our formal definition $(N, A, Θ, p, u)$ :

Players $N$ : There are two players, so $N = {1, 2}$ .
Actions $A$ : Player 1 can choose Up (U) or Down (D), and Player 2 can choose Left (L) or Right (R). So, $A_{1} = {U, D}$ and $A_{2} = {L, R}$ .
Types $Θ$ : Player 1 has two possible types, $Θ_{1} = {t_{1}, t_{1}^{'}}$ . Player 2 also has two possible types, $Θ_{2} = {t_{2}, t_{2}^{'}}$ .
Beliefs $p$ : The common prior is given in the top-left table. For instance, the probability that Player 1 is type $t_{1}$ and Player 2 is type $t_{2}$ is $p (t_{1}, t_{2}) = 1/2$ . The probability of the type profile $(t_{1}^{'}, t_{2})$ is $p (t_{1}^{'}, t_{2}) = 1/6$ .
State-Dependent Payoffs $u$ : The payoff matrices show how outcomes depend on both actions and types.
- If the type profile is $(t_{1}, t_{2})$ and the action profile is $(U, L)$ , the payoffs are $u_{1} = 2$ and $u_{2} = 1$ .
- However, if the type profile is $(t_{1}^{'}, t_{2})$ (Player 1's type has changed) and the actions are the same $(U, L)$ , the payoffs are now $u_{1} = 2$ and $u_{2} = 2$ .
- This clearly demonstrates that the payoffs are a function of the players' types—they are state-dependent.

A key part of the reasoning in these games involves updating beliefs. For example, if Player 1 learns they are type $t_{1}$ , what is their belief about Player 2's type? They use conditional probability (Bayes' Rule):

P (Player 2 is t_{2} ∣ Player 1 is t_{1}) = \frac{p ( t _{1} , t _{2} )}{p ( t _{1} )}

The total probability of Player 1 being type $t_{1}$ is $p (t_{1}) = p (t_{1}, t_{2}) + p (t_{1}, t_{2}^{'}) = 1/2 + 1/6 = 2/3$ .
Therefore, Player 1's belief is:

P (t_{2} ∣ t_{1}) = \frac{1/2}{2/3} = \frac{3}{4}

So, a type- $t_{1}$ Player 1 believes there is a 75% chance their opponent is type $t_{2}$ and a 25% chance they are type $t_{2}^{'}$ . This calculation is central to deciding which action is best.

Conclusion

In this lesson, we have constructed the formal apparatus for analyzing games of incomplete information. We moved from the intuitive problem of not knowing an opponent's preferences to a rigorous mathematical structure.

Key Takeaways:

A Bayesian game is defined by five components: Players ( $N$ ), Actions ( $A$ ), Types ( $Θ$ ), a Common Prior belief ( $p$ ), and State-Dependent Payoffs ( $u$ ).
A player's type ( $θ_{i}$ ) is their private information, which they know but others don't.
Beliefs are probability distributions over opponents' types, derived from a common prior and conditioned on a player's own type.
State-dependent payoffs $u_{i} (a, θ)$ mean that the utility from an outcome depends on the actions of all players and the types of all players.

We have successfully defined the game. The next logical question is: how do we solve it? A player's strategy can no longer be a simple choice of action. Instead, it must be a contingent plan: a function that maps each possible type a player could be to a chosen action. In our next lesson, we will define this new concept of strategy and introduce its corresponding equilibrium concept: the Bayesian Nash Equilibrium.

Recommended resources

Introduction

From Intuition to a Formal Model

The Five Components of a Bayesian Game

An Example to Put It All Together

Conclusion