Decision Theory

**Decision theory** (or the **theory of choice**) is the study of the reasoning underlying an agent's choices.^{[1]} Decision theory can be broken into two branches: normative decision theory, which gives advice on how to make the best decisions, given a set of uncertain beliefs and a set of values; and descriptive decision theory, which analyzes how existing, possibly irrational agents actually make decisions.

Closely related to the field of game theory,^{[2]} decision theory is concerned with the choices of individual agents whereas game theory is concerned with interactions of agents whose decisions affect each other. Decision theory is an interdisciplinary topic, studied by economists, statisticians, psychologists, biologists,^{[3]} political and social scientists, philosophers,^{[4]} and computer scientists.

Empirical applications of this rich theory are usually done with the help of statistical and econometric methods, especially via the so-called choice models, such as probit and logit models. Estimation of such models is usually done via parametric, semi-parametric and non-parametric maximum likelihood methods.^{[5]}

Normative or prescriptive decision theory is concerned with identifying the best decision to make, modelling an ideal decision maker who is able to compute with perfect accuracy and is fully rational. The practical application of this prescriptive approach (how people *ought to* make decisions) is called decision analysis, and is aimed at finding tools, methodologies and software (decision support systems) to help people make better decisions.

In contrast, positive or descriptive decision theory is concerned with describing observed behaviors under the assumption that the decision-making agents are behaving under some consistent rules. These rules may, for instance, have a procedural framework (e.g. Amos Tversky's elimination by aspects model) or an axiomatic framework, reconciling the Von Neumann-Morgenstern axioms with behavioural violations of the expected utility hypothesis, or they may explicitly give a functional form for time-inconsistent utility functions (e.g. Laibson's quasi-hyperbolic discounting).

The prescriptions or predictions about behaviour that positive decision theory produces allow for further tests of the kind of decision-making that occurs in practice. There is a thriving dialogue with experimental economics, which uses laboratory and field experiments to evaluate and inform theory. In recent decades, there has also been increasing interest in what is sometimes called "behavioral decision theory" and this has contributed to a re-evaluation of what rational decision-making requires.^{[6]}

The area of choice under uncertainty represents the heart of decision theory. Known from the 17th century (Blaise Pascal invoked it in his famous wager, which is contained in his *Pensées*, published in 1670), the idea of expected value is that, when faced with a number of actions, each of which could give rise to more than one possible outcome with different probabilities, the rational procedure is to identify all possible outcomes, determine their values (positive or negative) and the probabilities that will result from each course of action, and multiply the two to give an "expected value", or the average expectation for an outcome; the action to be chosen should be the one that gives rise to the highest total expected value. In 1738, Daniel Bernoulli published an influential paper entitled *Exposition of a New Theory on the Measurement of Risk*, in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. He gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter. In his solution, he defines a utility function and computes expected utility rather than expected financial value (see^{[7]} for a review).

In the 20th century, interest was reignited by Abraham Wald's 1939 paper^{[8]} pointing out that the two central procedures of sampling-distribution-based statistical-theory, namely hypothesis testing and parameter estimation, are special cases of the general decision problem. Wald's paper renewed and synthesized many concepts of statistical theory, including loss functions, risk functions, admissible decision rules, antecedent distributions, Bayesian procedures, and minimax procedures. The phrase "decision theory" itself was used in 1950 by E. L. Lehmann.^{[9]}

The revival of subjective probability theory, from the work of Frank Ramsey, Bruno de Finetti, Leonard Savage and others, extended the scope of expected utility theory to situations where subjective probabilities can be used. At this time, von Neumann and Morgenstern theory of expected utility^{[10]} proved that expected utility maximization followed from basic postulates about rational behavior.

The work of Maurice Allais and Daniel Ellsberg showed that human behavior has systematic and sometimes important departures from expected-utility maximization. The prospect theory of Daniel Kahneman and Amos Tversky renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. Kahneman and Tversky found three regularities - in actual human decision-making, "losses loom larger than gains"; persons focus more on *changes* in their utility-states than they focus on absolute utilities; and the estimation of subjective probabilities is severely biased by anchoring.

Pascal's Wager is a classic example of a choice under uncertainty.

Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realised at different points in time. If someone received a windfall of several thousand dollars, they could spend it on an expensive holiday, giving them immediate pleasure, or they could invest it in a pension scheme, giving them an income at some time in the future. What is the optimal thing to do? The answer depends partly on factors such as the expected rates of interest and inflation, the person's life expectancy, and their confidence in the pensions industry. However even with all those factors taken into account, human behavior again deviates greatly from the predictions of prescriptive decision theory, leading to alternative models in which, for example, objective interest rates are replaced by subjective discount rates.

Some decisions are difficult because of the need to take into account how other people in the situation will respond to the decision that is taken. The analysis of such social decisions is more often treated under the label of game theory, rather than decision theory, though it involves the same mathematical methods. From the standpoint of game theory most of the problems treated in decision theory are one-player games (or the one player is viewed as playing against an impersonal background situation). In the emerging socio-cognitive engineering, the research is especially focused on the different types of distributed decision-making in human organizations, in normal and abnormal/emergency/crisis situations.^{[11]}

Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, or the complexity of the organization that has to make them. Individuals making decisions may be limited in resources or are boundedly rational (have finite time or intelligence); in such cases the issue, more than the deviation between real and optimal behaviour, is the difficulty of determining the optimal behaviour in the first place. One example is the model of economic growth and resource usage developed by the Club of Rome to help politicians make real-life decisions in complex situations^{[]}. Decisions are also affected by whether options are framed together or separately; this is known as the distinction bias.

The heuristic approach to decision-making makes decisions based on routine thinking, which, while quicker than step-by-step processing, opens the risk of introducing inaccuracies, mistakes and fallacies, which may be easily disproved in a step-by-step process of thinking.^{[12]} One example of common and incorrect thought process is the gambler's fallacy, or believing that a random event is affected by previous random events (truth is, there is a fifty percent chance of a coin landing on heads even after a long sequence of tails). Another example is that decision-makers may be biased towards preferring moderate alternatives to extreme ones; the "Compromise Effect" operates under a mindset driven by the belief that the most moderate option, amid extremes, carries the most benefits from each extreme.^{[13]}

A highly controversial issue is whether one can replace the use of probability in decision theory by other alternatives.

Advocates for the use of probability theory point to:

- the work of Richard Threlkeld Cox for justification of the probability axioms,
- the Dutch book paradoxes of Bruno de Finetti as illustrative of the theoretical difficulties that can arise from departures from the probability axioms, and
- the complete class theorems, which show that all admissible decision rules are equivalent to the Bayesian decision rule for some utility function and some prior distribution (or for the limit of a sequence of prior distributions). Thus, for every decision rule, either the rule may be reformulated as a Bayesian procedure (or a limit of a sequence of such), or there is a rule that is sometimes better and never worse.

The proponents of fuzzy logic, possibility theory, quantum cognition, Dempster-Shafer theory, and info-gap decision theory maintain that probability is only one of many alternatives and point to many examples where non-standard alternatives have been implemented with apparent success; notably, probabilistic decision theory is sensitive to assumptions about the probabilities of various events, while non-probabilistic rules such as minimax are robust, in that they do not make such assumptions.

A general criticism of decision theory based on a fixed universe of possibilities is that it considers the "known unknowns", not the "unknown unknowns"^{[]}: it focuses on expected variations, not on unforeseen events, which some argue (as in black swan theory) have outsized impact and must be considered - significant events may be "outside model". This line of argument, called the ludic fallacy, is that there are inevitable imperfections in modeling the real world by particular models, and that unquestioning reliance on models blinds one to their limits.

- Bayesian statistics
- Causal decision theory
- Choice modelling
- Constraint satisfaction
- Decision making
- Evidential decision theory
- Game theory
- Multi-criteria decision making
- Operations research
- Optimal decision
- Decision quality
- Preference (economics)
- Quantum cognition
- Rationality
- Secretary problem
- Signal detection theory
- Small-numbers game
- Stochastic dominance
- Two envelopes problem

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