Expected Utility in Bayesian Games

Introduction

Hello! In our last lesson, we defined Bayesian Nash Equilibrium (BNE) as the central solution concept for games of incomplete information. We established that a BNE is a strategy profile where each player's strategy, which is a complete plan for every possible type they could be, is a best response to the strategies of others.

The critical element in finding this best response is the maximization of expected utility. Since players are uncertain about their opponents' types, they can't simply maximize their direct payoff. Instead, they must make the choice that yields the highest payoff on average, weighted by their beliefs about the state of the world.

Today, we will move from the definition to the mechanics. Our goal is to formulate the expected utility calculation for a player's type in a Bayesian game. This is the fundamental building block for solving for BNE, and mastering this calculation is essential for analyzing any game with private information.

The Logic of Expected Utility

Before diving into the formal mathematics, let's establish the core intuition. Imagine you are a player in a Bayesian game. You don't know your opponent's type, but you have a belief about the probability of each type. Your opponent has a strategy, which specifies what action they will take for each of their possible types.

To calculate your expected utility for taking a specific action, you must:

Consider the first possible type your opponent could be.
Identify the action their strategy tells them to play as that type.
Find the payoff you would receive if you take your chosen action and they take theirs.
Multiply this payoff by the probability that your opponent is this first type.
Repeat this for all possible types your opponent could be.
Sum up all these probability-weighted payoffs.

The result is your expected utility for that single action. You would then repeat this entire process for every other action available to you to find which one is your best response.

The Formal Calculation

This intuitive process can be expressed with mathematical precision. Given your interest in rigor, let's look at the formal definition.

[PDF] Bayesian Games - Game Theory - Indian Institute of Science

The following document from the Indian Institute of Science provides the exact mathematical formula for calculating the expected payoff for a specific type of a player.

Please read section 3.2, 'Payoff Computation in Selten Game', and focus on the formula presented at the very end of the section, just before the heading for section 4. This is the generalized formula for what we're about to practice.

As the document shows, the conditional expected utility for player $i$ of a specific type $θ_{i}$ , when playing action $s_{θ_{i}}$ against the opponents' strategy profile $s_{- i}$ , is given by:

u_{θ_{i}} (s_{θ_{i}}, s_{- i}) = θ_{- i} \in Θ_{- i} \sum p_{i} (θ_{- i} ∣ θ_{i}) u_{i} (θ_{i}, θ_{- i}, s_{θ_{i}}, s_{θ_{- i}})

Let's break this down:

$u_{θ_{i}} (\cdot)$ : The expected utility for player $i$ given their type is $θ_{i}$ .
$Θ_{- i}$ : The set of all possible type combinations for all other players.
$\sum_{θ_{- i} \in Θ_{- i}}$ : The sum over all these possible type combinations.
$p_{i} (θ_{- i} ∣ θ_{i})$ : Player $i$ 's belief; the probability that the other players have types $θ_{- i}$ , given player $i$ knows their own type is $θ_{i}$ .
$u_{i} (θ_{i}, θ_{- i}, s_{θ_{i}}, s_{θ_{- i}})$ : The actual payoff player $i$ gets when the types are $(θ_{i}, θ_{- i})$ and the actions played are $(s_{θ_{i}}, s_{θ_{- i}})$ .

A Worked Example: Calculating Expected Utility

This formula is best understood through application. Let's watch a detailed walk-through of a calculation.

4. An Example for Bayesian Nash Equilibrium (Game Theory Playlist 9)

In this video, Selcuk Ozyurt analyzes a game where Player 2 is uncertain about Player 1's cost (type). He meticulously calculates Player 2's expected utility for her actions, demonstrating the formula we just saw.

Watch the segment from 05:18 to 07:32. Pay close attention to how the expected utility of Player 2 playing 'Enter' is computed. Notice how it's a sum of two terms: the payoff from the first game state multiplied by its probability (1/2), and the payoff from the second game state multiplied by its probability (1/2).

Let's deconstruct the calculation for Player 2's expected utility of playing 'Enter' as shown in the video.

Setup:

Player 1's Types: $Θ_{1} = {c = 1.5, c = 3}$
Player 2's Beliefs: $P (c = 1.5) = 1/2$ , $P (c = 3) = 1/2$
Player 1's Assumed Strategy $s_{1}$ :
- If type is $c = 1.5$ , play 'Fight'. So, $s_{1} (c = 1.5) = Fight$ .
- If type is $c = 3$ , play 'Don't Fight'. So, $s_{1} (c = 3) = DF$ .
Player 2's Action under consideration: 'Enter'

Calculation:
Player 2's expected utility is the sum of payoffs from each of Player 1's types, weighted by the probability of that type occurring.

$E [u_{2} (Enter, s_{1})] = P (c = 1.5) \times u_{2} (Enter, s_{1} (c = 1.5)) + P (c = 3) \times u_{2} (Enter, s_{1} (c = 3))$

Let's plug in the actions and payoffs from the video's game matrices:

$E [u_{2} (Enter, s_{1})] = P (c = 1.5) \times u_{2} (Enter, Fight) + P (c = 3) \times u_{2} (Enter, DF)$

E [u_{2} (Enter, s_{1})] = (\frac{1}{2}) \times (- 1) + (\frac{1}{2}) \times (1) = - \frac{1}{2} + \frac{1}{2} = 0

This is precisely how you formulate and compute the expected utility for a player facing uncertainty. You would do the same for Player 2's other action, 'Don't Enter', to see which is the better choice.

Visualizing the Uncertainty

The structure of this uncertainty can be visualized with a game tree where "Nature" makes the first move.

Extensive-Form Game with Incomplete Information — This game tree illustrates the structure of a Bayesian game. 'Nature' makes the first move, determining Player 1's type (e.g., with probability 1/2 each). Player 2 does not observe Nature's move and thus does not know which decision node they are at (indicated by the dashed line connecting the nodes, forming an 'information set'). When Player 2 calculates their expected utility for an action (like l2 or h2), they must average the payoffs from both possible nodes within their information set, weighted by their beliefs about Player 1's type.

A Second Perspective

Let's see one more quick example to solidify the concept.

Game Theory 101 (#64): Bayesian Nash Equilibrium

William Spaniel's 'Game Theory 101' series provides another concise demonstration of this calculation.

Watch from 07:21 to 07:59. Here, Player 2's belief is represented by a variable 'p'. Observe how the expected utility for choosing 'Left' is constructed as a weighted sum of the payoffs against Player 1's 'A type' and 'B type', with weights p and (1-p) respectively.

The calculation shown in the video for Player 2 choosing 'Left' is:

E [u_{2} (Left)] = p \times (3) + (1 - p) \times (2)

This is the same fundamental logic: $(probability of type A) \times (payoff vs type A) + (probability of type B) \times (payoff vs type B)$ .

Conclusion

Today we've moved from the abstract definition of BNE to the concrete mechanics of its core component. You have learned how to formally set up and compute the expected utility for a player's type in a Bayesian game.

Key Takeaways:

The expected utility of an action is a weighted average of the payoffs that action can yield in different states of the world.
The weights in this average are the player's beliefs (probabilities) about the other players' types.
The formal calculation involves summing the payoffs from each possible opponent type profile, weighted by the conditional probability of that profile occurring.
This calculation is the essential tool for a rational player to make decisions under incomplete information.

We now have all the necessary components. In our next lesson, we will put them together to achieve our main goal: computing the complete Bayesian Nash Equilibrium for a 2x2 game. We will use the expected utility calculations we practiced today to find the full set of strategies that are mutual best responses.

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