Utility Functions: A Comprehensive Guide

Utility Functions are a foundational concept in Game Theory and decision theory, providing a mathematical representation of preferences over a set of outcomes. Introduced by John von Neumann and Oskar Morgenstern in their 1944 book "Theory of Games and Economic Behavior," they quantify satisfaction, benefit, or value that a player derives from different scenarios. These functions enable the analysis of strategic choices by assigning numerical values to reflect desirability.

Utility Functions bridge subjective preferences with objective analysis, allowing players to rank outcomes and make decisions under certainty or uncertainty. They vary widely—linear for simplicity, logarithmic for diminishing returns, or complex forms for risk attitudes—making them versatile tools across disciplines. This guide offers an exhaustive exploration of Utility Functions, detailing their definitions, types, examples, equations, and applications in economics, psychology, and beyond.

Whether modeling consumer behavior or predicting game outcomes, Utility Functions are key to understanding rational decision-making in interdependent systems.

Definition and Types of Utility Functions

Utility Functions come in various forms, each suited to different preference structures and risk profiles. Below, we define and categorize them.

Basic Definition

A utility function $ u: O \to \mathbb{R} $ maps a set of outcomes $ O $ to real numbers, where $ u(o_1) > u(o_2) $ implies $ o_1 $ is preferred over $ o_2 $:

\[ u(o) \in \mathbb{R}, \ \forall o \in O \]

Ordinal Utility

Ranks preferences without magnitude:

\[ u(o_1) > u(o_2) \text{ if and only if } o_1 \succ o_2 \]

Monotonic transformations preserve order, e.g., $ v(o) = a u(o) + b $, $ a > 0 $.

Cardinal Utility

Measures intensity of preference, invariant only under affine transformations:

\[ u'(o) = a u(o) + b, \ a > 0 \]

Expected Utility (Von Neumann-Morgenstern)

For lotteries $ L = (p_1, o_1; p_2, o_2; \ldots) $:

\[ u(L) = \sum_{i} p_i u(o_i) \]

Where $ \sum p_i = 1 $.

Risk Attitudes

Concavity determines risk preference:

\[ u''(x) < 0 \text{ (risk-averse)}, \ u''(x) = 0 \text{ (risk-neutral)}, \ u''(x) > 0 \text{ (risk-seeking)} \]

Common Forms

Linear: $ u(x) = a x + b $

Logarithmic: $ u(x) = \ln(x) $

Exponential: $ u(x) = 1 - e^{-kx} $

Detailed Examples of Utility Functions

Let’s explore various utility functions with practical scenarios.

Example 1: Monetary Choice

Player chooses $10 or $20:

\[ u(\$10) = 10, \ u(\$20) = 20 \]

Risk-averse variant:

\[ u(x) = \sqrt{x}, \ u(\$10) = \sqrt{10} \approx 3.16, \ u(\$20) = \sqrt{20} \approx 4.47 \]

Example 2: Expected Utility

Lottery: 50% chance of $100, 50% of $0, $ u(x) = x^{0.5} $:

\[ u(L) = 0.5 u(100) + 0.5 u(0) \]
\[ = 0.5 \sqrt{100} + 0.5 \sqrt{0} = 5 \]

Example 3: Cobb-Douglas Utility

Two goods $ x $ and $ y $:

\[ u(x, y) = x^\alpha y^{1-\alpha}, \ 0 < \alpha < 1 \]

For $ x = 4, y = 9, \alpha = 0.5 $:

\[ u(4, 9) = 4^{0.5} \cdot 9^{0.5} = 2 \cdot 3 = 6 \]

Example 4: Risk-Seeking

Exponential utility, $ u(x) = x^2 $, for $5 or $10:

\[ u(5) = 5^2 = 25, \ u(10) = 10^2 = 100 \]

Example 5: Multi-Attribute

Utility over time $ t $ and money $ m $:

\[ u(t, m) = w_1 t + w_2 m \]

For $ t = 2, m = 50, w_1 = 0.3, w_2 = 0.7 $:

\[ u(2, 50) = 0.3 \cdot 2 + 0.7 \cdot 50 = 0.6 + 35 = 35.6 \]

Applications of Utility Functions

Utility Functions drive analysis across multiple domains.

Economics

Consumer demand, maximize:

\[ \max u(x, y) \text{ s.t. } p_x x + p_y y = I \]

Game Theory

Payoff in Prisoner’s Dilemma:

\[ u_i(s_i, s_{-i}) \]

Psychology

Prospect Theory utility:

\[ v(x) = \begin{cases} x^\alpha & \text{if } x \geq 0 \\ -\lambda (-x)^\beta & \text{if } x < 0 \end{cases} \]

Negotiations

Bargaining utility:

\[ \max (u_1(x) - d_1)(u_2(1 - x) - d_2) \]

Finance

Portfolio optimization:

\[ E[u(W)] = \sum p_i u(w_i) \]