Utility theory

Utility theory is a field within decision theory that studies the quantitative representation of a decision-maker's (DM) preferences with the goal of identifying the best alternatives under conditions of certainty, uncertainty, or risk. It is based on constructing a numerical function, called a utility function, which reflects the subjective value of the possible outcomes of a choice.

General Characteristics

Utility theory is based on the concept of rational choice, where the DM aims to maximize the subjective utility of the outcome. This function can be:

• a value function — under conditions of complete certainty;
• a utility function — under conditions of probabilistic uncertainty.

Each alternative is assigned a numerical value that reflects its relative desirability for the DM. Comparing the function's values allows for ordering the options and selecting the most preferred one.

Axiomatic Approach

A utility function is introduced based on a system of axioms that reflect the requirements for rational behavior. The classical axiomatic system includes:

• Axiom of Completeness — any two options can be compared;
• Axiom of Transitivity — preferences among options are logically consistent;
• Axiom of Continuity — between any two options, a lottery can be found that is equivalent to a third;
• Axiom of Independence — the preference between options is preserved when they are included in compound lotteries.

Satisfying these conditions makes it possible to construct a numerical utility function that is invariant under linear transformations, which means it can be used on an interval scale.

Types of Utility Functions

1. Linear Utility Function

The linear form is used when:

• outcomes are expressed on a single numerical scale (e.g., monetary values),
• the DM's preferences exhibit constant marginal utility,
• there is no interaction between criterial attributes.

2. Convex and Concave Utility Functions

• A concave function reflects risk aversion: utility increases with value, but with diminishing returns.

• A convex function indicates risk-seeking (risk affinity): the DM prefers potentially more profitable but less reliable options.

The shape of the utility function makes it possible to account for the DM's attitude toward uncertainty when choosing between reliable and risky alternatives.

3. Step (Discrete) Utility Function

Used when preferences are expressed on qualitative scales or in situations with a limited number of discrete outcomes (e.g., approval/disapproval, quality levels). In such a function, each alternative is assigned a fixed utility value without considering intermediate gradations. Step functions are applied in verbal methods, expert systems, and logico-linguistic choice models.

4. Additive Utility Function (MAUT)

Used in multi-criteria choice problems where preferences for each criterion are independent. The total utility is represented as the sum of partial utilities for each criterion, multiplied by weight coefficients.

5. Multiplicative Utility Function

Used if there are interactions between criteria (e.g., one criterion enhances or diminishes the importance of another).

6. Utility Functions on Probability Spaces

Applied in decision-making under risk problems. Each outcome is assigned not only a value but also a probability of its occurrence. The utility function is used to calculate expected utility—the mathematical expectation of the subjective value of the outcome.

One-Dimensional and Multi-Dimensional Utility Theory

In a one-dimensional model, utility is determined for each option as a whole.

In a multi-dimensional model (MAUT — Multi-Attribute Utility Theory), partial evaluations across several criteria are considered. The overall utility is formed based on:

• an additive model, if preferences across criteria are independent;
• a multiplicative model, if there is interaction between criteria.

Example: the utility of a decision may depend simultaneously on time, cost, and risk; in such a case, partial utility functions and weight coefficients are assessed for each criterion.

Subjective and Objective Utility

Utility theory distinguishes between:

• objective probabilities (e.g., from statistics),
• subjective probabilities, defined by the DM.

Subjective utility takes into account personal confidence in the occurrence of events and preferences, which allows the theory to be used in real-world problems where complete information is lacking.

Prospect Theory and Critique of the Axiomatic System

In practice, it has been established that a DM's behavior does not always conform to the classical axiomatic system. Paradoxes have been identified (e.g., the Allais paradox) that demonstrate deviations from assumed rationality.

In response to these limitations, prospect theory was developed (by D. Kahneman and A. Tversky), which takes into account:

• the asymmetry in the perception of gains and losses;
• the overestimation of the significance of certain outcomes;
• the non-linear evaluation of probabilities and consequences.

Method for Assessing a Utility Function

One method is the standard gamble method. The DM compares a deterministic option with a lottery between the best and worst outcomes to determine the point of indifference. This allows for calculating the utility function value for the option, based on the subjective probability of this lottery.