Utility Functions: A Comprehensive Guide

Utility Functions, introduced by John von Neumann and Oskar Morgenstern in their 1944 book "Theory of Games and Economic Behavior," are central to Game Theory and decision theory. They mathematically represent preferences over outcomes, quantifying satisfaction, benefit, or value. By assigning numerical values, utility functions enable the analysis of strategic choices in scenarios involving certainty or uncertainty.

This MathMultiverse guide provides an in-depth exploration of utility functions, covering their definitions, types (e.g., linear, logarithmic), examples, visualizations, and applications in economics, psychology, and strategic analysis. From consumer behavior to game-theoretic strategies, utility functions bridge subjective preferences with objective decision-making.

Definition and Types of Utility Functions

Utility functions map outcomes to real numbers, reflecting preference rankings or intensities. They vary by structure and risk attitude, making them versatile for modeling decision-making.

Basic Definition

A utility function $ u: O \to \mathbb{R} $ assigns a real number to each outcome $ o \in O $, where $ u(o_1) > u(o_2) $ implies $ o_1 $ is preferred over $ o_2 $:

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

Ordinal Utility

Ranks preferences without quantifying intensity. Monotonic transformations (e.g., $ v(o) = a u(o) + b $, $ a > 0 $) preserve order:

\[ u(o_1) > u(o_2) \iff o_1 \succ o_2 \]

Cardinal Utility

Measures preference intensity, invariant under affine transformations:

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

Expected Utility (Von Neumann-Morgenstern)

For lotteries $ L = (p_1, o_1; p_2, o_2; \ldots) $, where $ \sum p_i = 1 $:

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

This framework models decisions under uncertainty.

Risk Attitudes

The second derivative determines risk preference:

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

Common Forms

Key utility functions include:

Linear: $ u(x) = a x + b $, risk-neutral.
Logarithmic: $ u(x) = \ln(x) $, risk-averse, modeling diminishing returns.
Exponential: $ u(x) = 1 - e^{-kx} $, often risk-averse.

Utility Functions Comparison

Graphs of $ u(x) = x $ (linear, risk-neutral) and $ u(x) = \ln(x) $ (logarithmic, risk-averse) for $ x \in [0.1, 100] $.

Examples

These examples illustrate utility functions in practical scenarios.

Monetary Choice

Choosing between $10 or $20, linear utility $ u(x) = x $:

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

Risk-averse variant, $ u(x) = \sqrt{x} $:

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

Expected Utility

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

\[ u(L) = 0.5 \cdot \sqrt{100} + 0.5 \cdot \sqrt{0} = 0.5 \cdot 10 + 0 = 5 \]

Cobb-Douglas Utility

Two goods $ x $ and $ y $, $ u(x, y) = x^\alpha y^{1-\alpha} $, $ \alpha = 0.5 $, $ x = 4 $, $ y = 9 $:

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

Risk-Seeking

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

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

Multi-Attribute

Utility over time $ t $ and money $ m $, $ u(t, m) = w_1 t + w_2 m $, $ 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

Utility functions are pivotal in various fields:

Economics

Optimize consumer demand subject to budget constraints:

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

Game Theory

Model payoffs in strategic interactions, e.g., Prisoner’s Dilemma:

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

Psychology

Prospect Theory for gains and losses:

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

Negotiations

Maximize joint utility in bargaining:

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

Finance

Portfolio optimization using expected utility:

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