Decision theory

Decision theory is a field of study concerned with the principles and methods used to make decisions under conditions of certainty, risk, and uncertainty. It combines concepts from mathematics, statistics, economics, philosophy, and psychology to analyse how individuals and organisations choose among alternative courses of action.

Decision theory examines the relationship between available information, possible outcomes, preferences, probabilities, and the consequences of decisions. It is applied in economics, finance, business management, public policy, medicine, engineering, military strategy, and artificial intelligence.

Decision theory is commonly divided into normative decision theory, which studies how decisions should be made according to specified principles, and descriptive decision theory, which examines how people actually make decisions. Prescriptive decision theory focuses on methods that help decision-makers improve their choices in practical settings.

History

The development of decision theory is closely associated with the history of probability, statistics, economics, and mathematical philosophy.

In the 17th century, mathematicians Blaise Pascal and Pierre de Fermat contributed to the development of probability theory through their correspondence on problems involving games of chance. These developments provided mathematical foundations for analysing uncertain outcomes.

In the 18th century, Daniel Bernoulli introduced the concept of expected utility in his analysis of the St. Petersburg paradox. His work demonstrated that the value people assign to an uncertain financial outcome need not be proportional to its monetary value.

During the 20th century, decision theory developed into a formal discipline. Frank Ramsey explored the relationship between subjective probability and preference in his 1926 essay Truth and Probability. John von Neumann and Oskar Morgenstern established an influential mathematical framework for expected utility in their 1944 work Theory of Games and Economic Behavior.

In 1954, Leonard Jimmie Savage published The Foundations of Statistics, which developed a framework for rational decision-making under uncertainty using subjective probabilities and preferences.

Later developments incorporated behavioural research, bounded rationality, game theory, information economics, and computational methods. The work of Herbert A. Simon highlighted the limitations of human information processing and the role of satisficing in practical decision-making.

Basic concepts

Alternatives and actions

A decision-maker selects from a set of possible actions or alternatives. These may include investment choices, business strategies, medical treatments, or public policy options.

States of nature

States of nature are possible conditions or events that influence the outcome of a decision but are not necessarily controlled by the decision-maker. Examples include future market conditions, weather, customer demand, and the behaviour of competitors.

Outcomes and consequences

An outcome is the result associated with a particular action under a particular state of nature. Outcomes may be evaluated in monetary terms, utility, time, health, welfare, or other measures relevant to the decision-maker.

Preferences and utility

Preferences represent how a decision-maker ranks possible outcomes. A utility function assigns numerical values to outcomes to represent their desirability within a decision model.

Utility is not necessarily equivalent to monetary value. A decision-maker may prefer a lower but more certain financial return over a higher return accompanied by substantial risk.

Probabilities and beliefs

Probabilities represent uncertainty about possible states or outcomes. They may be estimated from historical data, statistical models, expert judgement, or subjective beliefs.

The treatment of probabilities differs between decision frameworks, particularly when the available evidence is incomplete or the decision-maker cannot assign reliable probabilities to all possible outcomes.

Types of decision theory

Normative decision theory

Normative decision theory establishes principles for rational choice. It evaluates decisions against criteria such as consistency of preferences, probability assessments, expected utility, and the information available to the decision-maker.

In expected utility theory, a rational decision-maker selects an action that maximises expected utility, subject to the assumptions of the model.

Descriptive decision theory

Descriptive decision theory studies how people actually make decisions. It draws on psychology, behavioural economics, and empirical research to examine the influence of cognitive limitations, emotions, social pressures, habits, and biases.

Descriptive research has shown that actual human decisions may systematically differ from predictions generated by some normative models.

Prescriptive decision theory

Prescriptive decision theory develops practical tools and procedures to improve decision-making. These may include decision trees, sensitivity analysis, scenario planning, probability estimation, and structured comparison of alternatives.

Decision-making under different conditions

Decision-making under certainty

Decision-making under certainty occurs when the relevant consequences of each available action are known. The decision-maker can compare alternatives directly using an established objective or preference criterion.

Decision-making under risk

Decision-making under risk occurs when possible outcomes are uncertain but their probabilities can be estimated or specified. Expected utility and related methods can be used to compare alternative actions.

Decision-making under uncertainty

Decision-making under uncertainty occurs when the probabilities of possible outcomes are unknown, unreliable, or difficult to establish. Decision-makers may use scenario analysis, robust optimisation, minimax criteria, or other approaches that do not require precise probability estimates.

Decision-making under strategic interdependence

In strategic settings, the outcomes of a decision depend partly on the choices of other decision-makers. Game theory provides methods for analysing such situations, including competition between firms, bargaining, military strategy, and negotiations.

Expected utility theory

Expected utility theory is one of the principal frameworks in decision theory. It evaluates alternatives by calculating the probability-weighted sum of the utilities associated with their possible outcomes.

The expected utility of an action $a$ can be expressed as:

$E U (a) = \sum_{i = 1}^{n} P (s_{i} ∣ a) U (x_{i})$

where:

$E U (a)$ is the expected utility of action $a$ .
$P (s_{i} ∣ a)$ is the probability of state or outcome $i$ given action $a$ .
$U (x_{i})$ is the utility associated with outcome $x_{i}$ .
$n$ is the number of possible outcomes.

Under this framework, a decision-maker selects the action with the highest expected utility, provided the model's assumptions and objectives are appropriate.

For example, an investment with a 60% probability of producing a utility of 100 and a 40% probability of producing a utility of 20 has an expected utility of 68. Another investment can be evaluated using the same method, although the decision-maker's final preference may also depend on risk tolerance, liquidity needs, and other constraints.

Decision criteria

When probabilities or preferences are incomplete, decision-makers may use alternative criteria to compare available actions.

Expected value

Expected value calculates the probability-weighted average of the possible numerical outcomes. It is particularly useful when outcomes can be measured in monetary or other quantitative terms. However, maximising expected monetary value does not necessarily maximise expected utility when the decision-maker's preferences are sensitive to risk.

Maximin criterion

The maximin criterion selects the action whose worst possible outcome is better than the worst possible outcomes of the alternatives. It is a conservative approach that can be useful when avoiding severe losses is a priority.

Maximax criterion

The maximax criterion selects the action with the most favourable possible outcome. It represents an optimistic decision rule but does not account for the likelihood of achieving that outcome.

Minimax regret

The minimax regret criterion selects the action that minimises the maximum possible regret, where regret is the difference between the outcome obtained and the best outcome that could have been achieved under the same state of nature.

Bayesian decision analysis

Bayesian decision analysis combines prior beliefs, observed evidence, probability updating, and utility assessment. It is useful when new information can change the estimated likelihood of outcomes or the preferred course of action.

Decision analysis tools

Decision trees

A decision tree represents alternatives, uncertain events, and their possible consequences in a branching structure. It can be used to calculate expected values or expected utilities and compare sequential decisions.

Sensitivity analysis

Sensitivity analysis examines how a decision changes when assumptions, probabilities, costs, or other model inputs are varied. It helps identify which assumptions have the greatest influence on the result.

Scenario analysis

Scenario analysis evaluates decisions under a set of plausible future conditions. It is commonly used in strategic planning, investment analysis, and risk management when a single forecast is insufficient.

Value of information

The value of information measures the potential benefit of obtaining additional information before making a decision. Information is valuable when it can improve the choice of action enough to justify the cost of acquiring and interpreting it.

Multi-criteria decision analysis

Multi-criteria decision analysis evaluates alternatives against multiple objectives, such as cost, efficiency, safety, environmental impact, and strategic fit. It is particularly useful when no single measure adequately represents the decision-maker's goals.

Behavioural decision theory

Behavioural decision theory examines how psychological processes influence choices. Research in behavioural economics and cognitive psychology has identified systematic differences between actual human decisions and predictions made by some normative models.

Important concepts include:

Bounded rationality: The concept, associated with Herbert A. Simon, that decision-makers face limits in information, time, and cognitive capacity.
Prospect theory: A theory developed by Daniel Kahneman and Amos Tversky describing how people evaluate risky prospects relative to reference points, with different responses to gains and losses.
Loss aversion: The tendency for losses to have a greater psychological impact than comparable gains.
Framing effect: The influence of how a choice or its consequences are presented on the decision made.
Overconfidence bias: The tendency to overestimate the accuracy of one's knowledge, judgements, or predictions.
Status quo bias: A preference for maintaining an existing situation rather than changing it.

These concepts help explain why actual decisions may depart from predictions based on consistent preferences and probability-weighted utility.

Applications

Economics and finance

Decision theory is used to analyse consumer behaviour, investment selection, portfolio allocation, capital budgeting, insurance, and financial risk management. It provides a framework for evaluating uncertain returns and comparing alternatives with different risk profiles.

Business strategy

Organisations use decision analysis to assess market entry, product development, mergers and acquisitions, resource allocation, and competitive responses. Scenario planning and sensitivity analysis help evaluate strategic choices under uncertainty.

Public policy

Governments and public institutions use decision theory to compare policy alternatives, allocate limited resources, assess public risks, and evaluate the expected consequences of interventions.

Medicine and healthcare

Clinical decision analysis can compare diagnostic tests, treatments, and preventive interventions by considering their potential benefits, risks, costs, and patient preferences.

Artificial intelligence

Decision theory provides foundations for rational-agent models in artificial intelligence. An agent may select actions by maximising expected utility, subject to uncertainty, computational constraints, and the information available to it.

Military strategy

Decision theory is applied to operational planning, resource allocation, intelligence assessment, and strategic choices involving uncertain outcomes and the actions of adversaries. It is closely related to game theory, which examines decisions involving strategic interdependence.

Limitations

Decision theory provides structured methods for analysing choices, but its conclusions depend on the assumptions and information used in the model.

Uncertain probabilities: Reliable probabilities may be unavailable for novel or unprecedented events.
Difficulty measuring utility: Preferences involving welfare, reputation, safety, and ethical considerations may not be easily expressed numerically.
Computational complexity: Large decision problems with many alternatives, states, and time periods can be difficult to solve.
Behavioural limitations: People may not consistently follow the assumptions of normative models.
Model uncertainty: A model may omit relevant variables or incorrectly represent the relationships between actions and outcomes.
Conflicting objectives: Decisions involving multiple stakeholders may require trade-offs between incompatible goals.
Ethical considerations: Maximising a numerical objective does not automatically ensure fairness, legitimacy, or acceptable treatment of affected parties.

Decision theory is therefore best understood as a collection of analytical frameworks rather than a single method that guarantees an optimal decision in every circumstance.

Related fields

Decision theory is closely related to several disciplines:

Game theory studies strategic interactions among decision-makers.
Probability theory provides mathematical tools for representing uncertainty.
Operations research develops quantitative methods for optimising decisions and allocating resources.
Behavioural economics examines psychological influences on economic choices.
Information theory studies the representation and transmission of information.
Control theory analyses the behaviour of dynamic systems and the selection of actions to achieve specified objectives.