A broad range of concepts which have been developed to both describe and prescribe the process of decision making, where a choice is made from a finite set of possible alternatives. Normative decision theory describes how decisions should be made in order to accommodate a set of axioms believed to be desirable; descriptive decision theory deals with how people actually make decisions; and prescriptive decision theory formulates how decisions should be made in realistic settings. Thus, this field of study involves people from various disciplines: behavioral and social scientists and psychologists who generally attempt to discover elaborate descriptive models of the decision process of real humans in real settings; mathematicians and economists who are concerned with the axiomatic or normative theory of decisions; and engineers and managers who may be concerned with sophisticated prescriptive decision-making procedures.
Categories of problems
Efforts in decision theory may be divided into five categories:
Decision under certainty issues are those in which each alternative action results in one and only one outcome and where that outcome is sure to occur.
Decision under probabilistic uncertainty issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with known probabilities. There are outcome uncertainties, and the probabilities associated with these are known precisely.
Decision under probabilistic imprecision issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with unknown or imprecisely specified probabilities. There are outcome uncertainties, and the probabilities associated with the uncertainty parameters are not all known precisely.
Decision under information imperfection issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with imperfectly specified probabilities. There are outcome uncertainties, and the probabilities associated with these are not all known precisely. Imperfections in knowledge of the utility of the various event outcomes may exist as well.
Decision under conflict and cooperation issues are those in which there is more than a single decision maker, and where the objectives and activities of one decision maker are not necessarily known to all decision makers. Also, the objectives of the decision makers may differ.
Problems in any of these groupings may be approached from a normative, descriptive, or prescriptive perspective. Problems in category 5 are game theory-based problems. Most decision theory developments are concerned with issues in category 2. See also Game theory.
Bases of normative decision theory
The general concepts of axiomatic or normative decision theory formalize and rationalize the decision-making process. Normative decision theory depends on the following assumptions:
Past preferences are valid indicators of present and future preferences.
People correctly perceive the values of the uncertainties that are associated with the outcomes of decision alternatives.
People are able to assess decision situations correctly, and the resulting decision situation structural model is well formed and complete.
People make decisions that accurately reflect their true preferences over the alternative courses of action, each of which may have uncertain outcomes.
People are able to process decision information correctly.
Real decision situations provide people with decision alternatives that allow them to express their true preferences.
People accept the axioms that are assumed to develop the various normative theories.
People make decisions without being so overwhelmed by the complexity of actual decision situations that they would necessarily use suboptimal decision strategies.
Given these necessary assumptions, there will exist departures between normative and descriptive decision theories. A principal task of those aiding others in decision making is to retain those features from the descriptive approach which enable an acceptable transition from normative approaches to prescriptive approaches. The prescriptive features should eliminate potentially undesirable features of descriptive approaches, such as flawed judgment heuristics and information processing biases, while retaining acceptable features of the normative approaches. See also Decision analysis; Decision support system.
Determination of utility
When choosing among alternatives, the decision maker must be able to indicate preferences among decisions that may result in diverse outcomes. In simple situations when only money is involved, an expected-value approach might be suggested, in which a larger expected amount of money is preferred to a smaller amount. However, in many situations the utility associated with money is not a linear function of the amount of money involved.
According to expected utility theory, the decision maker should seek to choose the alternative ai which makes the resulting expected utility the largest possible. The utility uij, of choosing decision ai and obtaining outcome event ej, will also depend upon the particular value of the probabilistically uncertain random variable ej as conditioned on the decision path that is selected. So, the best that the decision maker can do here is to maximize some function, such as the expected value or utility (EU), as shown below, where the maximization is carried out over all 
Systematic approach to making decisions especially under uncertainty. Although statistics such as Expected Value and Standard Deviation are essential for choosing the best course of action, the decision problem can best be approached, using what is referred to as a payoff table (or decision matrix), which is characterized by: (1) the row representing a set of alternative Courses of Action available to the decision maker; (2) the column representing the State of Nature or conditions that are likely to occur and over which the decision maker has no control; and (3) the entries in the body of the table representing the outcome of the decision, known as payoffs, which may be in the form of costs, revenues, profits, or cash flows. By computing expected value of each action, we will be able to pick the best one.
Example 1: Assume the following probability distribution of daily demand for strawberries:
Also assume that unit cost = $3, selling price = $5 (i.e., profit on sold unit = $2), and salvage value on unsold units = $2 (i.e., loss on unsold unit = $1). We can stock either 0, 1, 2, or 3 units. The question is: How many units should be stocked each day? Assume that units from one day cannot be sold the next day. Then the payoff table can be constructed as follows:
*Profit for (stock 2, demand 1) equals (no. Of units sold) (profit per unit) - (no. Of units unsold)(loss per unit) = (1)($5 - 3) - (1)($3 - 2) = $1
**Expected value for (stock 2) is: -2(.2) + 1(.3) + 4(.3) + 4(.2) = $1.90. The optimal stock action is the one with the highest Expected Monetary Value, i.e., stock 2 units.
Suppose the decision maker can obtain a perfect prediction of which event (state of nature) will occur. The Expected Value With Perfect Information would be the total expected value of actions selected on the assumption of a perfect forecast. The Expected Value of Perfect Information can then be computed as:
Expected value with perfect information minus the expected value with existing information.
Example 2: From the payoff table in Example 1, the following analysis yields the expected value with perfect information:
State of Nature Demand 0 1 2 3 Expected Stock (.2) (.3) (.3) (.2) Value
0 $0 $0
Actions 1 2 .6
2 4 1.2
3 6 n1.2
$3.00
With existing information, the best that the decision maker could obtain was select (stock 2) and obtain $1.90. With perfect information (forecast), the decision maker could make as much as $3. Therefore, the expected value of perfect information is $3.00 - $1.90 = $1.10. This is the maximum price the decision maker is willing to pay for additional information.
The theory of how rational individuals (should) behave under risk and uncertainty. One branch deals with the individual against an uncertain environment (‘Nature’); the other, game theory, with the interactions of rational individuals who jointly produce an outcome that no one can control. Decision theory uses a set of axioms about how rational individuals behave which has been widely challenged on both empirical and theoretical grounds, but there is no agreed substitute for them.
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The theory of choices made when each option is associated with a risk, or expectation of gain or loss, where the expectation is a function of the probability of some outcome and the total gain or loss involved. Decision theory may be pursued with the aim of finding results about how decisions ought to be made (normative decision theory) or with the aim of finding out how they are actually made (empirical decision theory). See dominance, expected utility, game theory.
Decision theory is an area of study of discrete mathematics, related to and of interest to practitioners in all branches of science, engineering and in all human social activities. It is concerned with how real or ideal decision-makers make or should make decisions, and how optimal decisions can be reached.
Normative and descriptive decision theory
Most of decision theory is normative or prescriptive, i.e. it is concerned with identifying the best decision to take, assuming an ideal decision maker who is fully informed, able to compute with perfect accuracy, and fully rational. The practical application of this prescriptive approach (how people should make decisions) is called decision analysis, and aimed at finding tools, methodologies and software to help people make better decisions. The most systematic and comprehensive software tools developed in this way are called decision support systems.
Since it is obvious that people do not typically behave in optimal ways, there is also a related area of study, which is a positive or descriptive discipline, attempting to describe what people will actually do. Since the normative, optimal decision often creates hypotheses for testing against actual behaviour, the two fields are closely linked. Furthermore it is possible to relax the assumptions of perfect information, rationality and so forth in various ways, and produce a series of different prescriptions or predictions about behaviour, allowing for further tests of the kind of decision-making that occurs in practice.
What kinds of decisions need a theory?
Choice between incommensurable commodities
This area is concerned with the decision whether to have, say, one ton of guns and three tons of butter, or two tons of guns and one ton of butter. This is the classic subject of study of microeconomics and is rarely considered under the heading of decision theory, but such choices are often in fact part of the issues that are considered within decision theory.
Choice under uncertainty
This area represents the heartland of decision theory. The procedure now referred to as expected value was known from the 17th century. Blaise Pascal invoked it in his famous wager (see below), 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. 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 also 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, when it is known that there is a 5% chance that the ship and cargo will be lost. 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 concerns of orthodox statistical theory at that time, namely statistical hypothesis testing and statistical estimation theory, could both be regarded as particular special cases of the more general decision problem. This paper introduced much of the mental landscape of modern decision theory, including loss functions, risk functions, admissible decision rules, a priori distributions, Bayes decision rules, and minimax decision rules. The phrase "decision theory" itself was first used in 1950 by E. L. Lehmann.
The rise 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 only subjective probabilities are available. At this time it was generally assumed in economics that people behave as rational agents and thus expected utility theory also provided a theory of actual human decision-making behaviour under risk. The work of Maurice Allais and Daniel Ellsberg showed that this was clearly not so. The prospect theory of Daniel Kahneman and Amos Tversky placed behavioural economics on a more evidence-based footing. It emphasized that in actual human (as opposed to normatively correct) decision-making "losses loom larger than gains", people are more focused on changes in their utility states than the states themselves and estimation of subjective probabilities is severely biased by anchoring.
Castiglione and LiCalzi(1996), Bordley and LiCalzi (2000) recently showed that maximizing expected utility is mathematical equivalent to maximizing the probability that the uncertain consequences of the decision are preferable to uncertain benchmark (e.g., the probability that a mutual fund strategy outperforms the S&P 500 or that a firm outperforms the uncertain future performance of a major competitor.) This reinterpretation relates to psychological work suggesting that individuals seek to achieve fuzzy aspiration levels (Lopes & Oden) which may vary from choice context to choice context. Hence it shifts the focus from utility to the individual's uncertain reference point.
Pascal's Wager
Pascal's Wager is a classic example of a choice under uncertainty. The uncertainty, according to Pascal, is whether or not God exists. Belief or non-belief in God is the choice to be made. However, the reward for belief in God if God actually does exist is infinite. Therefore, however small the probability of God's existence, the expected value of belief exceeds that of non-belief, so it is better to believe in God. (There are several criticisms of the argument.)
Intertemporal choice
This area 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.
Competing decision makers
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 the business of game theory, and is not normally considered part of decision theory, though it is closely related. 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/emergergency/crisis situations. The signal detection theory is based on the Decision theory.
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. In such cases the issue is not the deviation between real and optimal behaviour, but the difficulty of determining the optimal behaviour in the first place. The Club of Rome, for example, developed a model of economic growth and resource usage that helps politicians make real-life decisions in complex situations.
Paradox of choice
Observed in many cases is the paradox that more choices may lead to a poorer decision or a failure to make a decision at all. It is sometimes theorized to be caused by analysis paralysis, real or perceived, or perhaps from rational ignorance. A number of researchers including Sheena S. Iyengar and Mark R. Lepper have published studies on this phenomenon. (Goode, 2001) A popularization of this analysis was done by Barry Schwartz in his 2004 book, The Paradox of Choice.
Statistical decision theory
Several statistial tools and methods are available to organize evidence, evaluate risks, and aid in decision making. The risks of Type I and type II errors can be quantified and rational decision making is improved.
One example shows a structure for deciding guilt in a criminal trial:
| Actual condition | |||
|---|---|---|---|
| Guilty | Not guilty | ||
| Decision | Verdict of 'guilty' | True Positive | False Positive (i.e. guilt reported unfairly) Type I error |
| Verdict of 'not guilty' | False Negative (i.e. guilt not detected) Type II error | True Negative | |
Alternatives to probability theory
A highly controversial issue is whether one can replace the use of probability in decision theory by other alternatives. The proponents of fuzzy logic, possibility theory, 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. Work by Yousef and others advocate exotic probability theories using complex-valued probability theories based on the probability amplitudes developed and validated by Birkhoff and Von Neumann in quantum physics.
Advocates 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 to
- the complete class theorems which show that all admissible decision rules are equivalent to a Bayesian decision rule with some prior distribution (possibly improper) and some utility function. Thus, for any decision rule generated by non-probabilistic methods, either there is an equivalent rule derivable by Bayesian means, or there is a rule derivable by Bayesian means which is never worse and (at least) sometimes better.
See also
References
- Paul Anand, "Foundations of Rational Choice Under Risk", Oxford, Oxford University Press (an overview of the philosophical foundations of key mathematical axioms in subjective expected utility theory - mainly normative) 1993 repr 1995 2002
- Sven Ove Hansson, "Decision Theory: A Brief Introduction", http://www.infra.kth.se/~soh/decisiontheory.pdf (an excellent non-technical and fairly comprehensive primer)
- Paul Goodwin and George Wright, Decision Analysis for Management Judgment, 3rd edition. Chichester: Wiley, 2004 ISBN 0-470-86108-8 (covers both normative and descriptive theory)
- Robert Clemen. Making Hard Decisions: An Introduction to Decision Analysis, 2nd edition. Belmont CA: Duxbury Press, 1996. (covers normative decision theory)
- D.W. North. "A tutorial introduction to decision theory". IEEE Trans. Systems Science and Cybernetics, 4(3), 1968. Reprinted in Shafer & Pearl. (also about normative decision theory)
- Glenn Shafer and Judea Pearl, editors. Readings in uncertain reasoning. Morgan Kaufmann, San Mateo, CA, 1990.
- Howard Raiffa Decision Analysis: Introductory Readings on Choices Under Uncertainty. McGraw Hill. 1997. ISBN 0-07-052579-X
- Morris De Groot Optimal Statistical Decisions. Wiley Classics Library. 2004. (Originally published 1970.) ISBN 0-471-68029-X.
- 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.
- J.Q. Smith Decision Analysis: A Bayesian Approach. Chapman and Hall. 1988. ISBN 0-412-27520-1
- Akerlof, George A. and Janet L. YELLEN, Rational Models of Irrational Behavior
- Arthur, W. Brian, Designing Economic Agents that Act like Human Agents: A Behavioral Approach to Bounded Rationality
- James O. Berger Statistical Decision Theory and Bayesian Analysis. Second Edition. 1980. Springer Series in Statistics. ISBN 0-387-96098-8.
- Goode, Erica. (2001) In Weird Math of Choices, 6 Choices Can Beat 600. The New York Times. Retrieved May 16, 2005.
- Miller, L. (1985). Cognitive risk taking after frontal or temporal lobectomy I. The synthesis of fragmented visual information. Neuropsychologia, 23, 359 369.
- Miller, L., & Milner, B. (1985). Cognitive risk taking after frontal or temporal lobectomy II. The synthesis of phonemic and semantic information. Neuropsychologia, 23, 371 379.
- Anderson, Barry F. The Three Secrets of Wise Decision Making. Single Reef Press. 2002. ISBN 0-9722177-0-3.
