View source for Decision theory

{{Short description|Branch of applied probability theory}}{{For|the [[positive economics|descriptive]] application of decision theory to modeling human behavior|Rational choice models}}{{Math topics TOC|exapnded=Relations}}
[[File:Jacques-Clément Wagrez - The Judgement of Paris, 1886.jpg|alt=Wagrez's "The Judgement of Paris": Paris, dressed in medieval livery and holding the apple of discord, chats with Athena, Aphrodite, and Hera.|thumb|The mythological [[judgement of Paris]] required selecting from three incomparable alternatives (the [[Greek Goddess|goddesses]] shown).  ]]
'''Decision theory''' or the '''theory of rational choice''' is a branch of [[probability theory|probability]], [[economics]], and [[analytic philosophy]] that uses the tools of [[expected utility]] and [[probabilities|probability]] to model how individuals would behave [[Rationality|rationally]] under [[uncertainty]].<ref>{{cite web |website=Dictionary.com |url=https://www.dictionary.com/browse/decision-theory |title=Decision theory Definition and meaning |access-date=2022-04-02}}</ref><ref>Hansson, Sven Ove. "Decision theory: A brief introduction". (2005) Section 1.2: A truly interdisciplinary subject.</ref> It differs from the [[Cognitive science|cognitive]] and [[Behavioural sciences|behavioral sciences]] in that it is mainly [[Prescriptive economics|prescriptive]] and concerned with identifying [[optimal decision|optimal decisions]] for a [[rational agent]], rather than [[Descriptive economics|describing]] how people actually make decisions. Despite this, the field is important to the study of real human behavior by [[Social science|social scientists]], as it lays the foundations for the [[Rational agent|rational agent models]] used to [[Mathematical model|mathematically model]] and analyze individuals in fields such as [[sociology]], [[economics]], [[criminology]], [[cognitive science]], and [[political 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 [[Rationality|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 system]]s) to help people make better decisions.<ref name=":5">{{cite book |last=MacCrimmon |first=Kenneth R. |chapter=Descriptive and normative implications of the decision-theory postulates |title=Risk and Uncertainty |publisher=Palgrave Macmillan |location=London |year=1968 |pages=3–32 |oclc=231114 }}</ref><ref name=":6">{{cite journal |last1=Slovic |first1=Paul |first2=Baruch |last2=Fischhoff |first3=Sarah |last3=Lichtenstein |title=Behavioral Decision Theory |journal=Annual Review of Psychology |volume=28 |issue=1 |year=1977 |pages=1–39 |doi=10.1146/annurev.ps.28.020177.000245 |hdl=1794/22385 |hdl-access=free }}</ref>

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 [[axiom]]atic framework (e.g. [[stochastic transitivity]] axioms), reconciling the [[Expected utility hypothesis#Von Neumann–Morgenstern utility theorem|Von Neumann-Morgenstern axioms]] with behavioral violations of the [[expected utility]] hypothesis, or they may explicitly give a functional form for [[Time inconsistency|time-inconsistent]] [[utility function]]s (e.g. Laibson's [[Hyperbolic discounting|quasi-hyperbolic discounting]]).<ref name=":5" /><ref name=":6" />

Prescriptive decision theory is concerned with predictions about behavior that positive decision theory produces to allow for further tests of the kind of decision-making that occurs in practice. In recent decades, there has also been increasing interest in "behavioral decision theory", contributing to a re-evaluation of what useful decision-making requires.<ref>For instance, see: {{cite book |last=Anand |first=Paul |year=1993 |title=Foundations of Rational Choice Under Risk |location=Oxford |publisher=Oxford University Press |isbn=0-19-823303-5 }}</ref><ref>{{cite journal | vauthors = Keren GB, Wagenaar WA |title=On the psychology of playing blackjack: Normative and descriptive considerations with implications for decision theory. |journal=Journal of Experimental Psychology: General |date=1985 |volume=114 |issue=2 |pages=133–158 |doi=10.1037/0096-3445.114.2.133 }}</ref>

==Types of decisions==

===Choice under uncertainty=== <!-- This section is linked from [[Risk]] -->
{{Details|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 [[Pascal's Wager|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 [[Norm (philosophy)|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.<ref>For a review see {{cite journal |last=Schoemaker |first=P. J. |title=The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations|journal=Journal of Economic Literature|volume=20|year=1982|issue=2 |pages=529–563 |jstor=2724488}}</ref>

In the 20th century, interest was reignited by [[Abraham Wald|Abraham Wald's]] 1939 paper pointing out that the two central procedures of [[frequentist statistics|sampling-distribution-based]] statistical-theory, namely [[statistical hypothesis testing|hypothesis testing]] and [[estimation theory|parameter estimation]], are special cases of the general decision problem.<ref>{{cite journal | title = Contributions to the Theory of Statistical Estimation and Testing Hypotheses | last = Wald | first = Abraham | author-link = Abraham Wald | journal = [[Annals of Mathematical Statistics]] | volume = 10 | issue = 4 | pages = 299–326 | year = 1939 | doi = 10.1214/aoms/1177732144 | mr = 932 | doi-access = free }}</ref> Wald's paper renewed and synthesized many concepts of statistical theory, including [[loss function]]s, [[risk function]]s, [[admissible decision rule]]s, [[prior probability|antecedent distributions]], [[admissible decision rule#Bayes rules|Bayesian procedures]], and [[minimax]] procedures. The phrase "decision theory" itself was used in 1950 by [[E. L. Lehmann]].<ref>{{cite journal |vauthors=Lehmann EL |author-link=E. L. Lehmann |title=Some Principles of the Theory of Testing Hypotheses |journal=[[Annals of Mathematical Statistics]] |year=1950 |volume=21 |issue=1 |pages=1–26 |jstor=2236552 |doi=10.1214/aoms/1177729884 |doi-access=free}}</ref>

The revival of [[subjective probability]] theory, from the work of [[Frank P. Ramsey|Frank Ramsey]], [[Bruno de Finetti]], [[L. J. Savage|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]]<ref>{{Cite book|title=Theory of Games and Economic Behavior | edition = third |last1=Neumann|first1=John von|last2=Morgenstern|first2=Oskar | name-list-style = vanc |publisher=Princeton University Press|year=1953 |orig-year=1944|location=Princeton, NJ}}</ref> 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]]).<ref>{{Cite book|title=Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions of the Decisions Under Uncertainty with Allais' Rejoinder|last1=Allais|first1=M.|last2=Hagen|first2=G. M.|date=2013|publisher=Springer Science & Business Media|isbn=9789048183548|location=Dordrecht|pages=333}}</ref> The [[prospect theory]] of [[Daniel Kahneman]] and [[Amos Tversky]] renewed the empirical study of [[behavioral economics|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.<ref>{{Cite book|title=Judgment Under Uncertainty: Heuristics and Biases|last1=Morvan|first1=Camille|last2=Jenkins|first2=William J.|date=2017|publisher=Macat International Ltd.|isbn=9781912303687|location=London|pages=13}}</ref> Kahneman and Tversky found three regularities – in actual human decision-making, "losses loom larger than gains"; people 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 (cognitive bias)|anchoring]].

===Intertemporal choice===
{{Main|Intertemporal choice}}
Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realized at different stages over time.<ref>{{Cite book|title=Essays In Decision Making: A Volume in Honour of Stanley Zionts|last1=Karwan|first1=Mark|last2=Spronk|first2=Jaap|last3=Wallenius|first3=Jyrki|date=2012|publisher=Springer Science & Business Media|isbn=9783642644993|location=Berlin|pages=135}}</ref> 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.<ref>{{Cite book|title=Aging and Decision Making: Empirical and Applied Perspectives|last1=Hess|first1=Thomas M.|last2=Strough|first2=JoNell|last3=Löckenhoff|first3=Corinna|author3-link= Corinna Löckenhoff |date=2015|publisher=Elsevier|isbn=9780124171558|location=London|pages=21}}</ref> 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 [[interest rate|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 [[hyperbolic discounting|subjective discount rates]].

===Interaction of decision makers===
[[File:NWC wargame 1958.jpg|alt=An electronic simulation room at the Naval War College during a 1958 wargame: against the far wall, a large map shows the outline of landmasses and some firing solutions.  Suited men sit at desks on the floor, papers in front of them, most staring up at the map.  Against the right wall, uniformed ensigns plot ship locations on (washed-out) screens.  |thumb|[[Military planners]] often conduct extensive [[Simulation|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.<ref>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.</ref>

===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 [[Bounded rationality|boundedly rational]]; the issue is thus, more than the deviation between real and optimal behavior, the difficulty of determining the optimal behavior 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|Heuristics in judgment and decision-making}}
[[File:Roulette - detail.jpg|alt=A ball inside a spinning roulette wheel|thumb|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.  ]]
[[Heuristic]]s 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.<ref name=":1">{{cite journal | vauthors = Bobadilla-Suarez S, Love BC | title = Fast or frugal, but not both: Decision heuristics under time pressure | journal = Journal of Experimental Psychology: Learning, Memory, and Cognition | volume = 44 | issue = 1 | pages = 24–33 | date = January 2018 | pmid = 28557503 | pmc = 5708146 | doi = 10.1037/xlm0000419 | url = http://discovery.ucl.ac.uk/10038268/1/2017-23585-001.pdf }}</ref> While quicker than step-by-step processing, heuristic thinking is also more likely to involve fallacies or inaccuracies.<ref name=":0">{{cite journal |last1=Johnson |first1=Eric J. |last2=Payne |first2=John W. |name-list-style=vanc |date=April 1985 |title=Effort and Accuracy in Choice |journal=Management Science |volume=31 |issue=4 |pages=395–414 |doi=10.1287/mnsc.31.4.395}}</ref>

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.<ref name=":2">{{cite journal |last1=Roe |first1=Robert M. |last2=Busemeyer |first2=Jermone R. |last3=Townsend |first3=James T.  | name-list-style = vanc |title=Multialternative decision field theory: A dynamic connectionst model of decision making. |journal=Psychological Review |date=2001 |volume=108 |issue=2 |pages=370–392 |doi=10.1037/0033-295X.108.2.370 |pmid=11381834 }}</ref> 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.<ref name=":3">{{cite journal | vauthors = Xu J, Harvey N | title = Carry on winning: the gamblers' fallacy creates hot hand effects in online gambling | journal = Cognition | volume = 131 | issue = 2 | pages = 173–80 | date = May 2014 | pmid = 24549140 | doi = 10.1016/j.cognition.2014.01.002 | doi-access = free }}</ref> 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.<ref name=":4">{{cite journal |first1=Shih-Chieh |last1=Chuang |first2=Danny Tengti |last2=Kao |first3=Yin-Hui |last3=Cheng |first4=Chu-An |last4=Chou  | name-list-style = vanc |title=The effect of incomplete information on the compromise effect |journal=Judgment and Decision Making |volume=7 |issue=2 |date=March 2012 |pages=196–206 |doi=10.1017/S193029750000303X |url=http://journal.sjdm.org/11/11214/jdm11214.html |citeseerx=10.1.1.419.4767 |s2cid=9432630 }}</ref>

==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 rule|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 probability|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]], [[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 can sometimes be [[sensitivity analysis|sensitive]] to assumptions about the probabilities of various events, whereas non-probabilistic rules, such as [[minimax]], are [[robust statistics|robust]] in that they do not make such assumptions.

===Ludic fallacy===
{{Main|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]]":<ref> {{cite journal |last=Feduzi |first=A. |title= Uncovering unknown unknowns: Towards a Baconian approach to management decision-making |journal=Decision Processes|volume=124|year=2014|issue=2 |pages=268–283}} </ref> 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 ==
{{Wikiquote}}
{{cols|colwidth=14em}}
* [[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]]
{{colend}}

==References==
{{reflist|30em}}

== Further reading ==
{{refbegin|2}}
* {{cite journal |author1=Akerlof, George A. |author2=Yellen, Janet L. |title=Rational Models of Irrational Behavior |journal=The American Economic Review |volume=77 |issue=2 |pages=137–142 |date=May 1987 |jstor=1805441}}
* {{cite book |author=Anand, Paul |title=Foundations of Rational Choice Under Risk |publisher=Oxford University Press |location=Oxford |year=1993 |isbn=978-0-19-823303-9 |url-access=registration |url=https://archive.org/details/foundationsofrat00anan }} (''an overview of the philosophical foundations of key mathematical axioms in subjective expected utility theory – mainly normative'')
* {{cite journal |author=Arthur, W. Brian |title=Designing Economic Agents that Act like Human Agents: A Behavioral Approach to Bounded Rationality |journal=The American Economic Review |volume=81 |issue=2 |pages=353–9 |date=May 1991|url=http://multiagent.martinsewell.com/Arth91.pdf}}
* {{cite book | title = Statistical decision theory and Bayesian Analysis | first = James O. | last = Berger | author-link = James Berger (statistician) | year = 1985 | edition = 2nd | publisher = Springer-Verlag | location = New York | isbn = 978-0-387-96098-2 | mr = 0804611 }}
* {{cite book | title = Bayesian Theory | author-link = José-Miguel Bernardo | last1 = Bernardo | first1 = José M. | author-link2 = Adrian Smith (academic) | last2 = Smith | first2 = Adrian F. M.  | name-list-style = vanc | publisher = Wiley | year = 1994 | mr = 1274699 | isbn = 978-0-471-92416-6 }}
* {{cite book |author1=Clemen, Robert  |author2=Reilly, Terence |title=Making Hard Decisions with DecisionTools: An Introduction to Decision Analysis |publisher=Cengage|location=Stamford CT |year=2014 |edition=3rd |isbn=978-0-538-79757-3 }} ''(covers normative decision theory)''
* {{cite book| title=Decision-Making: An Experimental Approach| author=[[Donald Davidson (philosopher)|Donald Davidson]], [[Patrick Suppes]] and [[Sidney Siegel]]| publisher=[[Stanford University Press]]|year=1957}}
* {{cite journal |author=de Finetti, Bruno |author-link=Bruno de Finetti |title=Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science |journal=Erkenntnis |volume=31 |date=September 1989}} (translation of 1931 article)
* {{cite journal |author=de Finetti, Bruno |title=La Prévision: ses lois logiques, ses sources subjectives |journal=Annales de l'Institut Henri Poincaré |year=1937}}
* de Finetti, Bruno.  "Foresight: its Logical Laws, Its Subjective Sources", (translation of the [http://www.numdam.org/item?id=AIHP_1937__7_1_1_0 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.
* {{cite book |author=De Groot, Morris|author-link=Morris H. DeGroot|title= Optimal Statistical Decisions|publisher=Wiley Classics Library|year=2004  | isbn=0-471-68029-X}}(Originally published 1970)
* {{cite book |author1=Goodwin, Paul  |author2=Wright, George |title=Decision Analysis for Management Judgment |publisher=Wiley |location=Chichester |year=2004 |isbn=978-0-470-86108-0 |edition=3rd }} ''(covers both normative and descriptive theory)''
* {{cite web|author=Hansson, Sven Ove |title=Decision Theory: A Brief Introduction |url=http://www.infra.kth.se/~soh/decisiontheory.pdf |url-status=dead |archive-url=https://web.archive.org/web/20060705052730/http://www.infra.kth.se/~soh/decisiontheory.pdf |archive-date=July 5, 2006 }}
* Khemani, Karan, [https://papers.ssrn.com/soL3/papers.cfm?abstract_id=647666 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.
* {{cite book |author=Leach, Patrick |title=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 |publisher=Probabilistic |year=2006 |isbn=978-0-9647938-5-9 }} A rational presentation of probabilistic analysis.
* {{cite journal | vauthors = Miller L | title = Cognitive risk-taking after frontal or temporal lobectomy--I. The synthesis of fragmented visual information | journal = Neuropsychologia | volume = 23 | issue = 3 | pages = 359–69 | year = 1985 | pmid = 4022303 | doi = 10.1016/0028-3932(85)90022-3 | s2cid = 45154180 }}
* {{cite journal | vauthors = Miller L, Milner B | title = Cognitive risk-taking after frontal or temporal lobectomy--II. The synthesis of phonemic and semantic information | journal = Neuropsychologia | volume = 23 | issue = 3 | pages = 371–9 | year = 1985 | pmid = 4022304 | doi = 10.1016/0028-3932(85)90023-5 | s2cid = 31082509 }}
* {{cite book|author=Morgenstern, Oskar|year=1976|publisher=New York University Press|chapter=Some Reflections on Utility|pages=65–70|title=Selected Economic Writings of Oskar Morgenstern|editor=Andrew Schotter|isbn=978-0-8147-7771-8}}
* {{cite journal | last = North | first = D.W. | title = A tutorial introduction to decision theory | journal = IEEE Transactions on Systems Science and Cybernetics | volume = 4 | issue = 3 | year = 1968 | pages = 200–210 | doi = 10.1109/TSSC.1968.300114 | citeseerx = 10.1.1.352.8089 }} Reprinted in Shafer & Pearl. ''(also about normative decision theory)''
* {{cite journal| author=[[Peirce, Charles Sanders]] and [[Joseph Jastrow]]|year=1885|url=http://psychclassics.yorku.ca/Peirce/small-diffs.htm|title=On Small Differences in Sensation| journal=Memoirs of the National Academy of Sciences|volume=3|pages=73–83}} http://psychclassics.yorku.ca/Peirce/small-diffs.htm
* {{cite book |author=Peterson, Martin |title=An Introduction to Decision Theory |publisher=Cambridge University Press |year=2009 |isbn=978-0-521-71654-3 }}
* {{cite book|author=Pfanzagl, J|year=1967|publisher=Princeton University Press|chapter=Subjective Probability Derived from the [[Oskar Morgenstern|Morgenstern]]-[[John von Neumann|von Neumann]] [[Expected utility|Utility Theory]]|pages=[https://archive.org/details/essaysinmathemat0000shub/page/237 237–251]|title=Essays in Mathematical Economics In Honor of Oskar Morgenstern|url=https://archive.org/details/essaysinmathemat0000shub|url-access=registration|editor=[[Martin Shubik]] }}
* {{cite book|author=Pfanzagl, J. in cooperation with V. Baumann and H. Huber|year=1968|publisher=Wiley|chapter=Events, Utility and Subjective Probability|pages=195–220|title=Theory of Measurement}}
* {{cite book |author=Raiffa, Howard |title=Decision Analysis: Introductory Lectures on Choices Under Uncertainty |publisher=McGraw Hill |year=1997 |isbn=978-0-07-052579-5 }}
* [[Frank P. Ramsey|Ramsey, Frank Plumpton]]; "Truth and Probability" ([https://web.archive.org/web/20030605070939/http://cepa.newschool.edu/het/texts/ramsey/ramsess.pdf PDF]), Chapter VII in ''The Foundations of Mathematics and other Logical Essays'' (1931).
* {{cite book | last = Robert | first = Christian | title = The Bayesian Choice | publisher = Springer | location = New York | year = 2007 | edition = 2nd | doi = 10.1007/0-387-71599-1 | isbn = 978-0-387-95231-4 | mr = 1835885 | series = Springer Texts in Statistics }}
* {{cite book |editor1=Shafer, Glenn |editor2=Pearl, Judea |title=Readings in uncertain reasoning |url=https://archive.org/details/readingsinuncert00shaf |url-access=registration |publisher=Morgan Kaufmann |location=San Mateo, CA |year=1990 |isbn=9781558601253 }}
* {{cite book |author=Smith, J.Q. |title=Decision Analysis: A Bayesian Approach |publisher=Chapman and Hall |year=1988 |isbn=978-0-412-27520-3 }}

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