Decision theory

Decision theory is the systematic study of choice among alternative actions, particularly when their consequences are uncertain. It connects mathematics, statistics, economics, and philosophy by representing decisions through actions, possible outcomes, beliefs, and preferences. Normative theories specify standards for coherent choice; descriptive theories model how people actually choose. Applied decision analysis uses these frameworks to structure particular problems, rather than supplying a universal rule independent of the decision-maker’s objectives. (suppescorpus.stanford.edu)

Elements of a decision problem

A basic model distinguishes available actions, states of the world, and consequences. An action specifies what the decision-maker can do; a state describes relevant circumstances outside their control; an outcome results from their combination. A decision table displays these relationships. Preferences rank outcomes, while utility assigns numbers that represent relevant aspects of those preferences. Utility need not mean pleasure or monetary profit: it can represent whatever consequences the model treats as desirable. (suppescorpus.stanford.edu)

Under certainty, each action’s consequence is known. Decisions under risk involve outcomes with specified probabilities, whereas decisions under uncertainty may lack an agreed probability assignment. Terminology varies: “uncertainty” is also used broadly to include risk. Subjective models represent uncertainty through an agent’s degrees of belief, while other approaches avoid requiring a single precise probability model. (suppescorpus.stanford.edu)

Expected utility and rational preference

Expected utility theory evaluates an action by the expected value of its outcome utilities. For finitely many states,

where $p(s)$ is the probability of state $s$, $c(a,s)$ its consequence under action $a$, and $u$ the utility function. In this formulation, states are treated as independent of the chosen action. The preferred action maximizes $EU(a)$. Maximizing expected utility is not generally equivalent to maximizing expected money, because utility can be nonlinear in monetary outcomes. (plato.stanford.edu)

The von Neumann–Morgenstern representation theorem concerns preferences over lotteries with specified probabilities. Standard versions require completeness, transitivity, continuity, and an independence axiom. Completeness permits comparison of every pair; transitivity prevents preference cycles; continuity rules out certain discontinuous rankings; independence preserves rankings when both lotteries are mixed with the same third lottery. These conditions yield an expected-utility representation, unique up to a positive affine transformation. The independence axiom is a restriction on preferences, not statistical independence. (plato.stanford.edu)

Subjective expected utility extends the framework to situations without externally specified probabilities. Leonard Savage’s approach derives probability and utility representations from conditions on preferences over acts. Such representation results establish a mathematical relationship between assumptions and choice rules; whether the assumptions adequately characterize rationality remains a separate philosophical question. (suppescorpus.stanford.edu)

Statistical decisions and loss

Statistical decision theory treats inference as choosing an action after observing data. A decision rule $\delta(X)$ maps observations $X$ to actions, and a loss function $L(\theta,a)$ measures the cost of action $a$ when the unknown parameter is $\theta$. Its risk is

Risk averages loss over repeated samples at a fixed parameter value. This framework encompasses estimation, classification, and hypothesis testing, allowing procedures to be compared according to their consequences rather than accuracy alone. (stat.cmu.edu)

A Bayes rule minimizes risk averaged over a prior distribution. Equivalently, after observation it minimizes expected loss under the posterior distribution, connecting decisions to Bayesian inference. A minimax rule instead minimizes the largest risk across possible parameter values. A rule is admissible if no alternative has risk no greater everywhere and strictly smaller somewhere. Admissibility excludes dominated procedures but does not identify a uniquely best rule. (stat.cmu.edu)

These ideas also organize machine learning: predictive models can be evaluated by expected loss, with empirical risk minimization substituting observed average loss for an unknown population risk. The chosen loss determines which predictive errors matter most. (stat.cmu.edu)

Information and sequential choice

The value of information is the expected improvement in a decision obtainable from additional evidence. It compares optimal expected utility after receiving information with optimal expected utility before receiving it. In a standard expected-utility setting, costless information that can be ignored has nonnegative value; obtaining information may nevertheless be unattractive when its costs exceed its expected benefit. Information matters through its potential to change action, not simply through its quantity. (plato.stanford.edu)

Sequential decisions involve actions that affect later opportunities and observations. A Markov decision process models state transitions and rewards, while a policy specifies actions conditional on available state information. Dynamic programming uses a value function and the Bellman equation to relate present choices to future returns. Reinforcement learning addresses related problems when transition or reward information must be learned through interaction. (gradml.mit.edu)

Behavioral findings and boundaries

Prospect theory, introduced by Daniel Kahneman and Amos Tversky in 1979, provides a descriptive alternative for risky choice. It evaluates gains and losses relative to a reference point and uses decision weights rather than probabilities directly. Together with behavioral economics, it examines systematic departures from expected-utility predictions. Such findings challenge descriptive adequacy without automatically settling normative questions. (nobelprize.org)

Bounded rationality emphasizes limits on deliberation and information processing, rather than assuming unrestricted optimization. Decision theory typically analyzes an individual agent’s choice; game theory adds strategic interaction among agents whose outcomes depend on one another’s actions. Formal decision models also leave the substantive choice of objectives unresolved: consistency of preferences alone does not establish that those preferences are morally justified. (suppescorpus.stanford.edu)