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**Decision theory** (or the **theory of choice** not to be confused with choice theory) is the study of an agent's choices.^{ [1] } Decision theory can be broken into two branches: normative decision theory, which analyzes the outcomes of decisions or determines the optimal decisions given constraints and assumptions, and descriptive decision theory, which analyzes *how* agents actually make the decisions they do.

- 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

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

Empirical applications of this rich 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, positive or descriptive decision theory is concerned with describing observed behaviors often 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 (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] }

The prescriptions or predictions about behavior that positive decision theory produces allow for further tests of the kind of decision-making that occurs in practice. In recent decades, there has also been increasing interest in what is sometimes called "behavioral decision theory" and 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 (see^{ [9] } for a review).

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 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^{[ citation needed ]}. Decisions are also affected by whether options are framed together or separately; this is known as the distinction bias. In 2011, Dwayne Rosenburgh explored and showed how decision theory can be applied to complex decisions that arise in areas such as wireless communications.^{ [18] }

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.^{ [19] } 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.^{ [20] } 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 coin is flipped to tails for a couple of turns, it still has the same probability of doing so; however it seems more likely, intuitively, for it to roll heads soon.^{ [21] } 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.^{ [22] } 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.^{ [23] }

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"^{[ citation needed ]}: 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.

Wikiquote has quotations related to: Decision theory |

- 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
- TOTREP
- Two envelopes problem
- Daniel Kahneman
- Prospect theory

**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.

Within economics, the concept of **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 satisfaction within 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 consumer's preference ordering over a choice set. It is devoid of its original interpretation as a measurement of the pleasure or satisfaction obtained by the consumer from that choice.

**Rationality** is the quality or state of being rational – that is, being based on or agreeable to reason. Rationality implies the conformity of one's beliefs with one's reasons to believe, and of one's actions with one's reasons for action. "Rationality" has different specialized meanings in philosophy, economics, sociology, psychology, evolutionary biology, game theory and political science.

**Bounded rationality** is the idea that rationality is limited, when individuals make decisions, by the tractability of the decision problem, the cognitive limitations 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 one.

The **prospect theory** is an economics theory developed by Daniel Kahneman and Amos Tversky in 1979. It challenges the expected utility theory, developed by John von Neumann and Oskar Morgenstern in 1944, and earned Daniel Kahneman the Nobel Memorial Prize in Economics in 2002. It is the founding theory of behavioral economics and of behavioral finance, and constitutes one of the first economic theories built using experimental methods.

* Theory of Games and Economic Behavior*, published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinary research field of game theory. In the introduction of its 60th anniversary commemorative edition from the Princeton University Press, the book is described as "the classic work upon which modern-day game theory is based."

In economics, game theory, and decision theory, the **expected utility hypothesis**—concerning people's preferences with regard to choices that have uncertain outcomes (gambles)—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. The introduction of St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis. This hypothesis has proven useful to explain some popular choices that seem to contradict the expected value criterion, such as occur in the contexts of gambling and insurance.

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 action 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 stakeholders.

The **Ellsberg paradox** is a paradox in decision theory in which people's choices violate the postulates of subjective expected utility. It is generally taken to be evidence for ambiguity aversion. The paradox was popularized by Daniel Ellsberg, although a version of it was noted considerably earlier by John Maynard Keynes.

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.

**Generalized expected utility** is a decision-making metric based on any of a variety of theories that attempt to resolve some discrepancies between expected utility theory and empirical observations, concerning choice under risky (probabilistic) circumstances.

In decision theory and economics, **ambiguity aversion** is a preference for known risks over unknown risks. An ambiguity-averse individual would rather choose an alternative where the probability distribution of the outcomes is known over one where the probabilities are unknown. This behavior was first introduced through the Ellsberg paradox.

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. If there is uncertainty as to what the outcome will be, then under the von Neumann–Morgenstern axioms the optimal decision maximizes the expected utility.

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

In decision theory, the **von Neumann-Morgenstern****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. Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

**Trade-off talking rational economic person** (**TOTREP**) is one term, among others, used to denote, in the field of choice analysis, the rational, human agent of economic decisions.

**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.

**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. Traditional game theory focuses on mathematical equilibriums, utility maximizing, and rational choice; in contrast, behavioral game theory focuses on choices made by participants in studies and is game theory applied to experiments. Choices studied in behavioral game theory are not always rational and do not always represent the utility maximizing choice.

- ↑ Steele, Katie and Stefánsson, H. Orri, "Decision Theory", The Stanford Encyclopedia of Philosophy (Winter 2015 Edition), Edward N. Zalta (ed.), URL =
- ↑ Myerson, Roger B. (1991). "1.2: Basic concepts of Decision Theory".
*Game theory analysis of conflict*. Cambridge, Massachusetts: Harvard University Press. ISBN 9780674728615. - ↑ Habibi I, Cheong R, Lipniacki T, Levchenko A, Emamian ES, Abdi A (April 2017). "Computation and measurement of cell decision making errors using single cell data".
*PLoS Computational Biology*.**13**(4): e1005436. doi:10.1371/journal.pcbi.1005436. PMC 5397092 . PMID 28379950. - ↑ Hansson, Sven Ove. "Decision theory: A brief introduction." (2005) Section 1.2: A truly interdisciplinary subject.
- 1 2 MacCrimmon, Kenneth R. "Descriptive and normative implications of the decision-theory postulates."
*Risk and uncertainty*. Palgrave Macmillan, London, 1968. 3-32. - 1 2 Slovic, Paul, Baruch Fischhoff, and Sarah Lichtenstein. "Behavioral decision theory."
*Annual review of psychology*28.1 (1977): 1-39. - ↑ For instance, see: Anand, Paul (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press.
- ↑ Keren GB, Wagenaar WA (1985). "On the psychology of playing blackjack: Normative and descriptive considerations with implications for decision theory".
*Journal of Experimental Psychology: General*.**114**(2): 133–158. doi:10.1037/0096-3445.114.2.133. - ↑ Schoemaker PJ (1982). "The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations".
*Journal of Economic Literature*.**20**: 529–563. - ↑ Wald, Abraham (1939). "Contributions to the Theory of Statistical Estimation and Testing Hypotheses".
*Annals of Mathematical Statistics*.**10**(4): 299–326. doi:10.1214/aoms/1177732144. MR 0000932. - ↑ Lehmann EL (1950). "Some Principles of the Theory of Testing Hypotheses".
*Annals of Mathematical Statistics*.**21**(1): 1–26. doi:10.1214/aoms/1177729884. JSTOR 2236552. - ↑ Neumann Jv, Morgenstern O (1953) [1944].
*Theory of Games and Economic Behavior*(third ed.). Princeton, NJ: Princeton University Press. - ↑ Allais, M.; Hagen, G. M. (2013-03-14).
*Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions of the Decisions Under Uncertainty with Allais' Rejoinder*. Dordrecht: Springer Science & Business Media. p. 333. ISBN 9789048183548. - ↑ Morvan, Camille; Jenkins, William J. (2017-07-05).
*Judgment Under Uncertainty: Heuristics and Biases*. London: Macat International Ltd. p. 13. ISBN 9781912303687. - ↑ Karwan, Mark; Spronk, Jaap; Wallenius, Jyrki (2012).
*Essays In Decision Making: A Volume in Honour of Stanley Zionts*. Berlin: Springer Science & Business Media. p. 135. ISBN 9783642644993. - ↑ Hess, Thomas M.; Strough, JoNell; Löckenhoff, Corinna (2015).
*Aging and Decision Making: Empirical and Applied Perspectives*. London: Elsevier. p. 21. ISBN 9780124171558. - ↑ Crozier, M. & Friedberg, E. 1995. "Organization and Collective Action. Our Contribution to Organizational Analysis" in Bacharach S.B, Gagliardi P. & Mundell P. (Eds). Research in the Sociology of Organizations. Vol. XIII, Special Issue on European Perspectives of Organizational Theory, Greenwich, CT: JAI Press.
- ↑ Rosenburgh, Dwayne, "Decision Theory with its Applications in Wireless Communication" in Zhang, Y. (Ed.), GUIZANI, M. (Ed.). (2011). Game Theory for Wireless Communications and Networking. Boca Raton: CRC Press. ISBN 9781439808894
- ↑ Johnson EJ, Payne JW (April 1985). "Effort and Accuracy in Choice".
*Management Science*.**31**(4): 395–414. doi:10.1287/mnsc.31.4.395. - ↑ Bobadilla-Suarez S, Love BC (January 2018). "Fast or frugal, but not both: Decision heuristics under time pressure" (PDF).
*Journal of Experimental Psychology: Learning, Memory, and Cognition*.**44**(1): 24–33. doi:10.1037/xlm0000419. PMC 5708146 . PMID 28557503. - ↑ Roe RM, Busemeyer JR, Townsend JT (2001). "Multialternative decision field theory: A dynamic connectionst model of decision making".
*Psychological Review*.**108**(2): 370–392. doi:10.1037/0033-295X.108.2.370. - ↑ Xu J, Harvey N (May 2014). "Carry on winning: the gamblers' fallacy creates hot hand effects in online gambling".
*Cognition*.**131**(2): 173–80. doi:10.1016/j.cognition.2014.01.002. PMID 24549140. - ↑ Chuang S, Kao DT, Cheng Y, Chou C (March 2012). "The effect of incomplete information on the compromise effect".
*Judgment and Decision Making*.**7**(2): 196–206. CiteSeerX 10.1.1.419.4767 .

- Akerlof, George A.; Yellen, Janet L. (May 1987). "Rational Models of Irrational Behavior" (PDF).
**77**(2): 137–142.Cite journal requires`|journal=`

(help) - 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)* - De Groot, Morris,
*Optimal Statistical Decisions*. Wiley Classics Library. 2004. (Originally published 1970.) ISBN 0-471-68029-X. - 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.
- 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. - 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. - 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)* - Peterson, Martin (2009).
*An Introduction to Decision Theory*. Cambridge University Press. ISBN 978-0-521-71654-3. - Raiffa, Howard (1997).
*Decision Analysis: Introductory Lectures on Choices Under Uncertainty*. McGraw Hill. ISBN 978-0-07-052579-5. - 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. - Smith, J.Q. (1988).
*Decision Analysis: A Bayesian Approach*. Chapman and Hall. ISBN 978-0-412-27520-3. - Charles Sanders Peirce 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 - Ramsey, Frank Plumpton; "Truth and Probability" (PDF), Chapter VII in
*The Foundations of Mathematics and other Logical Essays*(1931). - 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. - Donald Davidson, Patrick Suppes and Sidney Siegel (1957).
*Decision-Making: An Experimental Approach*. Stanford University Press. - 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. - 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. - Non-Robust Models in Statistics by Lev B. Klebanov, Svetlozat T. Rachev and Frank J. Fabozzi, Nova Scientific Publishers, Inc. New York, 2009.

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