Defining Utility Functions in Game Theory

This article provides an overview of how utility functions are defined and applied within game theory. It explains how individual preferences are converted into mathematical values, distinguishes between ordinal and cardinal utility, and demonstrates how these functions are used to model strategic decision-making and analyze player payoffs in various game scenarios.

The Core Concept of Utility in Game Theory

In game theory, a utility function is a mathematical representation of a player’s preferences over a set of possible outcomes. If a player prefers outcome \(A\) to outcome \(B\), the utility function assigns a higher numerical value to \(A\) than to \(B\).

Formally, let \(X\) represent the set of all possible outcomes in a game. A utility function \(u\) maps each outcome \(x\) in \(X\) to a real number:

\[u: X \to \mathbb{R}\]

If a player strictly prefers outcome \(x\) to outcome \(y\) (denoted as \(x \succ y\)), then:

\[u(x) > u(y)\]

If the player is indifferent between the two outcomes (denoted as \(x \sim y\)), the utility function assigns them equal value:

\[u(x) = u(y)\]

These numerical values, often referred to as payoffs, allow game theorists to analyze complex strategic interactions using mathematical optimization.

Ordinal vs. Cardinal Utility

Utility functions are classified into two primary types depending on the nature of the decisions being modeled:

1. Ordinal Utility

Ordinal utility only reflects the sequential ranking of preferences. It indicates which outcomes are preferred over others, but the actual numerical differences between the payoffs carry no meaning. For example, if \(u(A) = 3\), \(u(B) = 2\), and \(u(C) = 1\), we only know that \(A\) is preferred to \(B\), and \(B\) is preferred to \(C\). We cannot claim that \(A\) is three times better than \(C\). Ordinal utility is sufficient for analyzing deterministic games where players choose strategies with certain outcomes.

2. Cardinal Utility

When games involve uncertainty, risk, or mixed strategies (where players randomize their choices), ordinal utility is insufficient. Instead, game theory relies on cardinal utility, specifically von Neumann-Morgenstern (vNM) utility functions. Cardinal utility measures the intensity of preferences, meaning the differences between numerical values are meaningful. This allows for the calculation of expected utility.

Expected Utility Theory

Under uncertainty, players do not choose direct outcomes; instead, they choose strategies that result in probability distributions over outcomes (known as lotteries).

If a strategy leads to outcome \(x_1\) with probability \(p_1\) and outcome \(x_2\) with probability \(p_2\), the expected utility (\(EU\)) is the weighted sum of the utilities of each outcome:

\[EU = p_1 \cdot u(x_1) + p_2 \cdot u(x_2)\]

According to the Expected Utility Theorem, a rational player will always choose the strategy that maximizes their expected utility. This framework requires the utility function to satisfy four key axioms: * Completeness: The player can compare and rank any two outcomes. * Transitivity: If \(A\) is preferred to \(B\), and \(B\) to \(C\), then \(A\) must be preferred to \(C\). * Continuity: There exists a probability at which a player is indifferent between a middle outcome and a lottery between the best and worst outcomes. * Independence: If a player prefers \(A\) to \(B\), they will also prefer a lottery between \(A\) and \(C\) over a lottery between \(B\) and \(C\), assuming equal probabilities.

Utility Functions in Game Representation

Once defined, utility functions are used to construct the payoffs in game representations:

By defining preferences through these mathematical functions, game theory can identify stable strategic profiles, such as the Nash Equilibrium, where no player can increase their utility by unilaterally changing their strategy.