**Decision theory** (or the **theory of choice**; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical consequences to the outcome.^{ [1] }

- Normative and descriptive
- Types of decisions
- Choice under uncertainty
- Intertemporal choice
- Interaction of decision makers
- Complex decisions
- Heuristics
- Alternatives
- Probability theory
- Alternatives to probability theory
- Ludic fallacy
- See also
- References
- Further reading

There are three branches of decision theory:

**Normative decision theory**: Concerned with the identification of optimal decisions, where optimality is often determined by considering an ideal decision-maker who is able to calculate with perfect accuracy and is in some sense fully rational.**Prescriptive decision theory**: Concerned with describing observed behaviors through the use of conceptual models, under the assumption that those making the decisions are behaving under some consistent rules.**Descriptive decision theory**: Analyzes how individuals actually make the decisions that they do.

Decision theory is closely related to the field of game theory ^{ [2] } and is an interdisciplinary topic, studied by economists, mathematicians, data scientists, psychologists, biologists,^{ [3] } political and other social scientists, philosophers^{ [4] } and computer scientists.

Empirical applications of this theory are usually done with the help of statistical and econometric methods.

Normative decision theory is concerned with identification of optimal decisions where optimality is often determined by considering an ideal decision maker who is able to calculate with perfect accuracy and is in some sense 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.^{ [5] }^{ [6] }

In contrast, descriptive decision theory is concerned with describing observed behaviors often under the assumption that those making decisions 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 (e.g. stochastic transitivity axioms), reconciling the Von Neumann-Morgenstern axioms with behavioral 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).^{ [5] }^{ [6] }

Prescriptive decision theory is concerned with predictions about behavior that positive decision theory produces to allow for further tests of the kind of decision-making that occurs in practice. In recent decades, there has also been increasing interest in "behavioral decision theory", contributing to a re-evaluation of what useful decision-making requires.^{ [7] }^{ [8] }

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.^{ [9] }

In the 20th century, interest was reignited by Abraham Wald's 1939 paper^{ [10] } 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.^{ [11] }

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 the time, von Neumann and Morgenstern's theory of expected utility ^{ [12] } 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.^{ [13] } The prospect theory of Daniel Kahneman and Amos Tversky renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. It describes a way by which people make decisions when all of the outcomes carry a risk.^{ [14] } 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.

Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realised at different stages over time.^{ [15] } It is also described as cost-benefit decision making since it involves the choices between rewards that vary according to magnitude and time of arrival.^{ [16] } 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 field of 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.^{ [17] }

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 are limited in resources (i.e. time and intelligence) and are therefore boundedly rational; the issue is thus, more than the deviation between real and optimal behaviour, 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^{[ citation needed ]}. Decisions are also affected by whether options are framed together or separately; this is known as the distinction bias.

Heuristics in decision-making is the ability of making decisions based on unjustified or routine thinking. While quicker than step-by-step processing, heuristic thinking is also more likely to involve fallacies or inaccuracies.^{ [18] } The main use for heuristics in our daily routines is to decrease the amount of evaluative thinking we perform when making simple decisions, making them instead based on unconscious rules and focusing on some aspects of the decision, while ignoring others.^{ [19] } One example of a common and erroneous thought process that arises through heuristic thinking is the Gambler's Fallacy — believing that an isolated random event is affected by previous isolated random events. For example, if a fair coin is flipped to tails for a couple of turns, it still has the same probability (i.e., 0.5) of doing so in future turns, though intuitively it seems more likely for it to roll heads soon.^{ [20] } This happens because, due to routine thinking, one disregards the probability and concentrates on the ratio of the outcomes, meaning that one expects that in the long run the ratio of flips should be half for each outcome.^{ [21] } Another example is that decision-makers may be biased towards preferring moderate alternatives to extreme ones. The *Compromise Effect* operates under a mindset that the most moderate option carries the most benefit. In an incomplete information scenario, as in most daily decisions, the moderate option will look more appealing than either extreme, independent of the context, based only on the fact that it has characteristics that can be found at either extreme.^{ [22] }

A highly controversial issue is whether one can replace the use of probability in decision theory with something else.

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, whereas 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":^{ [23] } it focuses on expected variations, not on unforeseen events, which some argue 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.

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**Bayesian probability** is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.

As a topic of economics, **utility** is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a **utility function** that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful for ethical decisions.

**Bounded rationality** is the idea that rationality is limited when individuals make decisions. In other words, humans' "preferences are determined by changes in outcomes relative to a certain reference level". Limitations include the difficulty of the problem requiring a decision, the cognitive capability of the mind, and the time available to make the decision. Decision-makers, in this view, act as satisficers, seeking a satisfactory solution, rather than an optimal solution. Therefore, humans do not undertake a full cost-benefit analysis to determine the optimal decision, but rather, choose an option that fulfils their adequacy criteria.

**Behavioral economics** studies the effects of psychological, cognitive, emotional, cultural and social factors on the decisions of individuals and institutions and how those decisions vary from those implied by classical economic theory.

**Prospect theory** is a theory of behavioral economics and behavioral finance that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics.

In mathematical optimization and decision theory, a **loss function** or **cost function** is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An **objective function** is either a loss function or its opposite, in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy.

The **expected utility hypothesis** is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and preferences.

In decision theory, **subjective expected utility** is the attractiveness of an economic opportunity as perceived by a decision-maker in the presence of risk. Characterizing the behavior of decision-makers as using subjective expected utility was promoted and axiomatized by L. J. Savage in 1954 following previous work by Ramsey and von Neumann. The theory of subjective expected utility combines two subjective concepts: first, a personal utility function, and second a personal probability distribution.

**Decision analysis** (**DA**) is the discipline comprising the philosophy, methodology, and professional practice necessary to address important decisions in a formal manner. Decision analysis includes many procedures, methods, and tools for identifying, clearly representing, and formally assessing important aspects of a decision; for prescribing a recommended course of action by applying the maximum expected-utility axiom to a well-formed representation of the decision; and for translating the formal representation of a decision and its corresponding recommendation into insight for the decision maker, and other corporate and non-corporate stakeholders.

The **Ellsberg paradox** is a paradox of choice in which people's decisions produce inconsistencies with subjective expected utility theory. The paradox was popularized by Daniel Ellsberg in his 1961 paper “Risk, Ambiguity, and the Savage Axioms”, although a version of it was noted considerably earlier by John Maynard Keynes. It is generally taken to be evidence for ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown, incalculable risks.

The **Allais paradox** is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory.

**Formal epistemology** uses formal methods from decision theory, logic, probability theory and computability theory to model and reason about issues of epistemological interest. Work in this area spans several academic fields, including philosophy, computer science, economics, and statistics. The focus of formal epistemology has tended to differ somewhat from that of traditional epistemology, with topics like uncertainty, induction, and belief revision garnering more attention than the analysis of knowledge, skepticism, and issues with justification.

An **optimal decision** is a decision that leads to at least as good a known or expected outcome as all other available decision options. It is an important concept in decision theory. In order to compare the different decision outcomes, one commonly assigns a utility value to each of them.

**David Schmeidler** was an Israeli mathematician and economic theorist. He was a Professor Emeritus at Tel Aviv University and the Ohio State University.

In decision theory, the **von Neumann–Morgenstern** (**VNM**) **utility theorem** shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann–Morgenstern utility function. The theorem is the basis for expected utility theory.

In expected utility theory, a **lottery** is a discrete distribution of probability on a set of *states of nature*. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, e.g.. Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

In marketing, Bayesian inference allows for decision making and market research evaluation under uncertainty and with limited data.

**Ecological rationality** is a particular account of practical rationality, which in turn specifies the norms of rational action – what one ought to do in order to act rationally. The presently dominant account of practical rationality in the social and behavioral sciences such as economics and psychology, rational choice theory, maintains that practical rationality consists in making decisions in accordance with some fixed rules, irrespective of context. Ecological rationality, in contrast, claims that the rationality of a decision depends on the circumstances in which it takes place, so as to achieve one's goals in this particular context. What is considered rational under the rational choice account thus might not always be considered rational under the ecological rationality account. Overall, rational choice theory puts a premium on internal logical consistency whereas ecological rationality targets external performance in the world. The term ecologically rational is only etymologically similar to the biological science of ecology.

**Behavioral game theory** analyzes interactive strategic decisions and behavior using the methods of game theory, experimental economics, and experimental psychology. Experiments include testing deviations from typical simplifications of economic theory such as the independence axiom and neglect of altruism, fairness, and framing effects. As a research program, the subject is a development of the last three decades.

**Intuitive statistics**, or **folk statistics**, refers to the cognitive phenomenon where organisms use data to make generalizations and predictions about the world. This can be a small amount of sample data or training instances, which in turn contribute to inductive inferences about either population-level properties, future data, or both. Inferences can involve revising hypotheses, or beliefs, in light of probabilistic data that inform and motivate future predictions. The informal tendency for cognitive animals to intuitively generate statistical inferences, when formalized with certain axioms of probability theory, constitutes statistics as an academic discipline.

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