Decision theory

The mythological judgement of Paris required selecting from three incomparable alternatives (the goddesses shown).

Decision theory (or the theory of choice) is a branch of applied probability theory and analytic philosophy concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical consequences to the outcome.

There are three branches of decision theory:

1. 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.
2. 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.
3. Descriptive decision theory: Analyzes how individuals actually make the decisions that they do.

Decision theory is a broad field from management sciences and is an interdisciplinary topic, studied by economists, management scientists, medical researchers, mathematicians, data scientists, psychologists, biologists, social scientists, philosophers and computer scientists.

Empirical applications of this theory are usually done with the help of statistical and discrete mathematical approaches from computer science.

Normative and descriptive

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.

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).

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.

Types of decisions

Choice under uncertainty

Further information: Expected utility hypothesis

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.

In the 20th century, interest was reignited by Abraham Wald's 1939 paper 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.

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 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 (Allais paradox and Ellsberg paradox). 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. 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

Main article: Intertemporal choice

Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realised at different stages over time. 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. 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.

Interaction of decision makers

Military planners often conduct extensive simulations to help predict the decision-making of relevant actors.

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 often treated under decision theory, though it involves mathematical methods. 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.

Complex decisions

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. Decisions are also affected by whether options are framed together or separately; this is known as the distinction bias.

Heuristics

Main article: Heuristics in judgment and decision-making

The gambler's fallacy: even when the roulette ball repeatedly lands on red, it is no more likely to land on black the next time.

Heuristics are procedures for making a decision without working out the consequences of every option. Heuristics decrease the amount of evaluative thinking required for decisions, focusing on some aspects of the decision while ignoring others. While quicker than step-by-step processing, heuristic thinking is also more likely to involve fallacies or inaccuracies.

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 flips of a fair coin give repeated tails, the coin still has the same probability (i.e., 0.5) of tails in future turns, though intuitively it might seems that heads becomes more likely. In the long run, heads and tails should occur equally often; people commit the gambler's fallacy when they use this heuristic to predict that a result of heads is "due" after a run 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 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.

Alternatives

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

Probability theory

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.

Alternatives to probability theory

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.

Ludic fallacy

Main article: Ludic fallacy

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 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.

See also

• Bayesian epistemology
• Bayesian statistics
• Causal decision theory
• Choice modelling
• Constraint satisfaction
• Daniel Kahneman
• Decision making
• Decision quality
• Emotional choice theory
• Evidential decision theory
• Functional decision theory
• Game theory
• Multi-criteria decision making
• Newcomb's paradox
• Operations research
• Optimal decision
• Preference (economics)
• Prospect theory
• Quantum cognition
• Rational choice theory
• Rationality
• Secretary problem
• Signal detection theory
• Small-numbers game
• Stochastic dominance
• TOTREP
• Two envelopes problem

Further reading

• Akerlof, George A.; Yellen, Janet L. (May 1987). "Rational Models of Irrational Behavior". The American Economic Review. 77 (2): 137–142. JSTOR 1805441.
• Anand, Paul (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press. ISBN 978-0-19-823303-9. (an overview of the philosophical foundations of key mathematical axioms in subjective expected utility theory – mainly normative)
• Arthur, W. Brian (May 1991). "Designing Economic Agents that Act like Human Agents: A Behavioral Approach to Bounded Rationality" (PDF). The American Economic Review. 81 (2): 353–9.
• Berger, James O. (1985). Statistical decision theory and Bayesian Analysis (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-96098-2. MR 0804611.
• Bernardo JM, Smith AF (1994). Bayesian Theory. Wiley. ISBN 978-0-471-92416-6. MR 1274699.
• Clemen, Robert; Reilly, Terence (2014). Making Hard Decisions with DecisionTools: An Introduction to Decision Analysis (3rd ed.). Stamford CT: Cengage. ISBN 978-0-538-79757-3. (covers normative decision theory)
• Donald Davidson, Patrick Suppes and Sidney Siegel (1957). Decision-Making: An Experimental Approach. Stanford University Press.
• de Finetti, Bruno (September 1989). "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science". Erkenntnis. 31. (translation of 1931 article)
• de Finetti, Bruno (1937). "La Prévision: ses lois logiques, ses sources subjectives". Annales de l'Institut Henri Poincaré.
• de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources", (translation of the 1937 article in French) in H. E. Kyburg and H. E. Smokler (eds), Studies in Subjective Probability, New York: Wiley, 1964.
• de Finetti, Bruno. Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974-5.
• De Groot, Morris (2004). Optimal Statistical Decisions. Wiley Classics Library. ISBN 0-471-68029-X.(Originally published 1970)
• Goodwin, Paul; Wright, George (2004). Decision Analysis for Management Judgment (3rd ed.). Chichester: Wiley. ISBN 978-0-470-86108-0. (covers both normative and descriptive theory)
• Hansson, Sven Ove. "Decision Theory: A Brief Introduction" (PDF). Archived from the original (PDF) on July 5, 2006.
• Khemani, Karan, Ignorance is Bliss: A study on how and why humans depend on recognition heuristics in social relationships, the equity markets and the brand market-place, thereby making successful decisions, 2005.
• Klebanov, Lev. B., Svetlozat T. Rachev and Frank J. Fabozzi, eds. (2009). Non-Robust Models in Statistics, New York: Nova Scientific Publishers, Inc.
• Leach, Patrick (2006). Why Can't You Just Give Me the Number? An Executive's Guide to Using Probabilistic Thinking to Manage Risk and to Make Better Decisions. Probabilistic. ISBN 978-0-9647938-5-9. A rational presentation of probabilistic analysis.
• Miller L (1985). "Cognitive risk-taking after frontal or temporal lobectomy--I. The synthesis of fragmented visual information". Neuropsychologia. 23 (3): 359–69. doi:10.1016/0028-3932(85)90022-3. PMID 4022303. S2CID 45154180.
• Miller L, Milner B (1985). "Cognitive risk-taking after frontal or temporal lobectomy--II. The synthesis of phonemic and semantic information". Neuropsychologia. 23 (3): 371–9. doi:10.1016/0028-3932(85)90023-5. PMID 4022304. S2CID 31082509.
• Morgenstern, Oskar (1976). "Some Reflections on Utility". In Andrew Schotter (ed.). Selected Economic Writings of Oskar Morgenstern. New York University Press. pp. 65–70. ISBN 978-0-8147-7771-8.
• North, D.W. (1968). "A tutorial introduction to decision theory". IEEE Transactions on Systems Science and Cybernetics. 4 (3): 200–210. CiteSeerX 10.1.1.352.8089. doi:10.1109/TSSC.1968.300114. Reprinted in Shafer & Pearl. (also about normative decision theory)
• Peirce, Charles Sanders and Joseph Jastrow (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences. 3: 73–83. http://psychclassics.yorku.ca/Peirce/small-diffs.htm
• Peterson, Martin (2009). An Introduction to Decision Theory. Cambridge University Press. ISBN 978-0-521-71654-3.
• Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern-von Neumann Utility Theory". In Martin Shubik (ed.). Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press. pp. 237–251.
• Pfanzagl, J. in cooperation with V. Baumann and H. Huber (1968). "Events, Utility and Subjective Probability". Theory of Measurement. Wiley. pp. 195–220.
• Raiffa, Howard (1997). Decision Analysis: Introductory Lectures on Choices Under Uncertainty. McGraw Hill. ISBN 978-0-07-052579-5.
• Ramsey, Frank Plumpton; "Truth and Probability" (PDF), Chapter VII in The Foundations of Mathematics and other Logical Essays (1931).
• Robert, Christian (2007). The Bayesian Choice. Springer Texts in Statistics (2nd ed.). New York: Springer. doi:10.1007/0-387-71599-1. ISBN 978-0-387-95231-4. MR 1835885.
• Shafer, Glenn; Pearl, Judea, eds. (1990). Readings in uncertain reasoning. San Mateo, CA: Morgan Kaufmann. ISBN 9781558601253.
• Smith, J.Q. (1988). Decision Analysis: A Bayesian Approach. Chapman and Hall. ISBN 978-0-412-27520-3.