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Encyclopedia > Decision theory

Decision theory is an area of study of discrete mathematics that models human decision-making in science, engineering and indeed 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. Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ... Part of a scientific laboratory at the University of Cologne. ... Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ... Decision making is the cognitive process of selecting a course of action from among multiple alternatives. ... Look up decision in Wiktionary, the free dictionary. ...

## Normative and descriptive decision theory GA_googleFillSlot("encyclopedia_square");

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. In philosophy, normative is usually contrasted with positive, descriptive or explanatory when describing types of theories, beliefs, or statements. ... In linguistics, prescription is the laying down or prescribing of normative rules of the language. ... Rational may be: the adjective for the state of rationality acting according to the philosophical principles of rationalism a mathematical term for certain numbers; the rational numbers the software company Rational Software; now owned by IBM, and formerly Rational Software Corporation This is a disambiguation page &#8212; a navigational aid... Decision analysis (DA) is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner. ... Decision support systems are a class of computer-based information systems including knowledge based systems that support decision making activities. ...

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. In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... In linguistics, prescription is the laying down or prescribing of normative rules for a language. ...

## What kinds of decisions need a theory?

### Choice between incommensurable commodities

Decision Making Theory becames a very important part in our daily lives due to the decision we make that lead us to believe the theory if its rational or irrational.

### 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 probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... Blaise Pascal (pronounced ), (June 19, 1623 â€“ August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... The PensÃ©es (literally, thoughts) represented an apology for the Christian religion by Blaise Pascal, the renowned 17th century philosopher and mathematician. ... Daniel Bernoulli Daniel Bernoulli (February 8, 1700 â€“ March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland where he died. ... In probability theory and decision theory the St. ... In philosophy, normative is usually contrasted with positive, descriptive or explanatory when describing types of theories, beliefs, or statements. ... For other uses, see Amsterdam (disambiguation). ... Saint Petersburg (Russian: &#1057;&#1072;&#1085;&#1082;&#1090;-&#1055;&#1077;&#1090;&#1077;&#1088;&#1073;&#1091;&#769;&#1088;&#1075;, English transliteration: Sankt-Peterburg), colloquially known as &#1055;&#1080;&#1090;&#1077;&#1088; (transliterated Piter), formerly known as Leningrad (&#1051;&#1077;&#1085;&#1080;&#1085;&#1075;&#1088;&#1072;&#769;&#1076;, 1924&#8211;1991) and... This article is about utility in economics and in game theory. ... The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ...

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. Abraham Wald (October 31, 1902 Kolozsvár, Hungary (now Cluj, Romania) - December 13, 1950 India) was a mathematician who contributed to decision theory, geometry, and econometrics, and founded the field of statistical sequential analysis. ... John Venn. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. ... In statistics, decision theory and economics, a loss function is a function that maps an event (technically an element of a sample space) onto a real number representing the economic cost or regret associated with the event. ... In decision theory, the risk of an estimator the expected value of the loss function as a function on the unknown underlying state of nature: . Categories: Decision theory ... In classical (frequentist) decision theory, an admissible decision rule is a rule for making a decision that is better in some sense than any other rule that may compete with it. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... Bayesian probability is an interpretation of probability suggested by Bayesian theory, which holds that the concept of probability can be defined as the degree to which a person believes a proposition. ... â€œMinmaxâ€ redirects here. ...

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. Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesian inference. ... Frank Plumpton Ramsey (February 22, 1903 â€“ January 19, 1930) was a British mathematician who, in addition to mathematics, made significant contributions in philosophy and economics. ... Bruno de Finetti (Innsbruck, June 13, 1906 - Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the operational subjective conception of probability. ... Leonard Jimmie Savage (20 November 1917 - 1 November 1971) was a US mathematician and statistician. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ... Maurice Allais (born May 31, 1911) was the 1988 winner of The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for his pioneering contributions to the theory of markets and efficient utilization of resources. ... Daniel and Patricia Marx Ellsberg - 2006 Jacob Appelbaum Daniel Ellsberg (born April 7, 1931) is a former American military analyst employed by the RAND Corporation who precipitated a national uproar in 1971 when he released the Pentagon Papers, the U.S. militarys account of activities during the Vietnam War... Prospect theory was developed by Daniel Kahneman and Amos Tversky in 1979 as a psychologically realistic alternative to expected utility theory. ... Daniel Kahneman Daniel Kahneman (born March 5, 1934 in Tel Aviv, in the then British Mandate of Palestine, now in Israel), is a key pioneer and theorist of behavioral finance, which integrates economics and cognitive science to explain seemingly irrational risk management behavior in human beings. ... Amos Tversky (March 16, 1937 - June 2, 1996) was a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. ... Nobel Prize in Economics winner Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ... Evidence-based policy is the idea that all public policy should be informed by rigorously established objective evidence. ... Anchoring or focalism is a term used in psychology to describe the common human tendency to rely too heavily, or anchor, on one trait or piece of information when making decisions. ...

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.) Pascals Wager (or Pascals Gambit) is the application by the French philosopher Blaise Pascal of decision theory to the belief in God. ... Blaise Pascal (pronounced ), (June 19, 1623 â€“ August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... This article discusses the term God in the context of monotheism and henotheism. ... Pascals Wager (or Pascals Gambit) is the application by the French philosopher Blaise Pascal of decision theory to the belief in God. ...

### 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. Game theory is a branch of applied mathematics that is often used in the context of economics. ... Signal detection theory, or SDT, is a means to quantify the ability to discern between signal and noise. ...

### 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. The Club of Rome is a global think tank that deals with a variety of international political issues. ...

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. Analysis paralysis is an informal phrase applied to when the opportunity cost of decision analysis exceeds the benefits. ... Rational ignorance is a term most often found in economics, particularly public choice theory, but also used in other disciplines which study rationality and choice, including philosophy (epistemology) and game theory. ...

## Statistical decision theory

Several statistical 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. Type I errors (or Î± error, or false positive) and type II errors (Î² error, or a false negative) are two terms used to describe statistical errors. ...

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. Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. ... The Dempster-Shafer theory is a mathematical theory of evidence [SH76] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. ... Info-gap decision theory is a non-probabilistic decision theory seeking to optimize robustness to failure, or opportunity of windfall. ...

Advocates of probability theory point to:

• 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.

Richard Threlkeld Cox (1898 - May 2, 1991) was a professor of physics at Johns Hopkins University, known for Coxs theorem relating to the foundations of probability. ... In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, no matter what the outcome of the gamble. ... Bruno de Finetti (Innsbruck, June 13, 1906 - Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the operational subjective conception of probability. ... In classical (frequentist) decision theory, an admissible decision rule is a rule for making a decision that is better in some sense than any other rule that may compete with it. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... Bayesian refers to probability and statistics -- either methods associated with the Reverend Thomas Bayes (ca. ...

## 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
• Lev Virine and Michael Trumper. Project Decisions: The Art and Science, Vienna, VA: Management Concepts, 2008. ISBN 978-1567262179
• 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.

Results from FactBites:

 Decision theory - Wikipedia, the free encyclopedia (1490 words) Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. 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. 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.
 : Decision Theory (695 words) One approach to decision theory made popular by Cyert and March (he of the "garbage model of organizations") is the consideration of the organization as a coalition of individuals, in which goals are arrived at by a bargaining process and change over time. Decisions are dependent on the amount and kind of information available and the expectations of those involved. Decisions made under this approach often favor s ecurity and the status quo and are risk averse.
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