Editing Decision theory (section)

===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]].