Sure-thing Principle Research Articles

State Dependent Expected Utility for Savage's State Space

This paper generalizes the Debreu/Gorman characterization of additively decomposable functionals and separable preferences to infinite dimensions. The first novelty concerns the very definition of additively decomposable functionals for infinite dimensions. For decision under uncertainty, our result provides a state-dependent extension of Savage's expected utility. A characterization in terms of preference conditions identifies the empirical content of the model; it amounts to Savage's axiom system with P4 (likelihood ordering) dropped. Our approach does not require that a (probability) measure on the state space be given a priori, or can be derived from extraneous conditions outside the realm of decision theory. Bayesian updating of new information is still possible, even though no prior probabilities are given. The finding suggests that the sure-thing principle, rather than prior probability, is at the heart of Bayesian updating.

10 Development of a clinical prediction model for an ordinal diagnostic outcome

For choice with deterministic consequences, the standard rationality hypothesis is ordinality, i.e., maximization of a weak preference ordering. For choice under risk (resp. uncertainty), preferences are assumed to be represented by the objectively (resp. subjectively) expected value of a von Neumann-Morgenstern utility function. For choice under risk, this implies a key independence axiom; under uncertainty, it implies some version of Savage’s sure-thing principle. This chapter investigates the extent to which ordinality, independence, and the sure thing principle can be derived from more fundamental axioms concerning behavior in decision trees. Following Cubitt (1996), these principles include dynamic consistency, separability, and reduction of sequential choice, which can be derived in turn from one consequentialist hypothesis applied to continuation subtrees as well as entire decision trees. Examples of behavior violating these principles are also reviewed, as are possible explanations of why such violations are often observed in experiments.

The sure thing principle and the value of information

This paper examines the relationship between Savage's sure thing principle and the value of information. We present two classes of results. First, we show that, under a consequentialist axiom, the sure-thing principle is neither sufficient nor necessary for perfect information to be always desirable: specifically, under consequentialism, the sure thing principle is not implied by the condition that perfect information is always valuable; moreover, the joint imposition of the sure thing principle, consequentialism and either one of two state independence axioms does not imply that perfect information is always desirable. Second, we demonstrate that, under consequentialism, the sure thing principle is necessary for a nonnegative value of possibly imperfect information (though of course the principle is still not sufficient). One implication of these results is that the sure thing principle, under consequentialism, plays a somewhat different role in ensuring dynamic consistency in decision making under uncertainty than does the independence axiom in decision making under risk.

The tortoise and the prisoners' dilemma

Simon Blackburn has claimed that on a theory of revealed preference. it is tautological that any eligible player-one whose preferences are consistent-plays hawk, chooses a strategy of non-cooperation with the other prisoner. This claim is examined and rejected. A prisoner in the appropriate sense is defined not by preferences over actions (playing hawk or playing dove) but by preferences over the four possible outcomes which result from the players' actions jointly (both playing hawk, both playing dove, the first playing hawk and the second dove, and vice versa). It is not inconsistent to suppose that a prisoner has the appropriate preferences over outcomes but nevertheless chooses to play dove. This can happen either if the prisoner believes for some reason that the full range of outcomes is not available, or if the prisoner is simply a bad strategic thinker. It is perhaps tautological that a player with the appropriate preferences over outcomes and with suitable beliefs will, if rational, choose to play hawk. However, rationality would then involve an appeal to a further prinicple, the sure-thing principle: if a prisoner would choose hawk knowing that the other will choose hawk, and would also choose hawk knowing that the other will choose dove, then hawk should be chosen when it is uncertain what the other will choose.

Rationality and the sure-thing principle

Rationality and the sure-thing principle

The Testing Principle: Inductive Reasoning and the Ellsberg Paradox

We postulate the Testing Principle : that individuals ''act like statisticians'' when they face uncertainty in a decision problem, ranking alternatives to the extent that available evidence allows. The Testing Principle implies that completeness of preferences, rather than the sure-thing principle, is violated in the Ellsberg Paradox. In the experiment, subjects chose between risky and uncertain acts in modified Ellsberg-type urn problems, with sample information about the uncertain urn. Our results show, consistent with the Testing Principle, that the uncertain urn is chosen more often when the sample size is larger, holding constant a measure of ambiguity (proportion of balls of unknown colour in the urn). The Testing Principle rationalises the Ellsberg Paradox. Behaviour consistent with the principle leads to a reduction in Ellsberg-type violations as the statistical quality of sample information is improved, holding ambiguity constant. The Testing Principle also provides a normative rationale for the Ellsberg paradox that is consistent with procedural rationality.

The comonotonic sure-thing principle

This article identifies the common characterizing property, the comonotonic sure-thing principle, that underlies the rank-dependent direction in non-expected utility. This property restricts Savage's sure-thing principle to comonotonic acts, and is characterized in full generality by means of a new functional form—cumulative utility—that generalizes the Choquet integral. Thus, a common generalization of all existing rank-dependent forms is obtained, including rank-dependent expected utility, Choquet expected utility, and cumulative prospect theory.

The sure-thing principle and the comonotonic sure-thing principle: An axiomatic analysis

This paper compares classical expected utility with the more general rank-dependent utility models. It shows that the difference between the sure-thing principle for preferences of expected utility and its comonotonic generalization in rank-dependent utility provides the exact demarcation between expected utility and rank-dependent models.

The Invention of the Independence Condition for Preferences

This paper discusses the history and interrelations of three central ideas in preference theory: the independence condition in decision under risk, the sure-thing principle in decision under uncert...

The Invention of the Independence Condition for Preferences

This paper discusses the history and interrelations of three central ideas in preference theory: the independence condition in decision under risk, the sure-thing principle in decision under uncertainty, and conjoint independence for multiattribute decisions and consumer theory. Independence was recognized as an important component of decision under risk in the late 1940s by Jacob Marschak, John Nash, Herman Rubin, and Norman Dalkey, and first appeared in publication in Marschak (Marschak, J. 1950. Rational behavior, uncertain prospects, and measurable utility. Econometrica 18 111–141.) and Nash (Nash, J. F. 1950. The bargaining problem. Econometrica 18 155–162.). The sure-thing principle can be credited to Savage (Savage, L. J. 1953. Une Axiomatisation du Comportement Raisonnable Face à l'Incertitude. Colloq. Internal. Centre National Rech. Sci. 40 Econométrie 29–40; Savage, L. J. 1954. The Foundations of Statistics. Wiley, New York (2nd edn., Dover, New York, 1972.). Conjoint independence for consumer theory was introduced by Sono (Sono, M. 1943. The effect of price changes on the demand and supply of separable goods (in Japanese). Kokumin Keisai Zasshi 74 1–51.) and Leontief (Leontief, W. W. 1947a. A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives. Bull. Amer. Math. Soc. 53 343–350; Leontief, W. W. 1947b. Introduction to a theory of the internal structure of functional relationships. Econometrica 51 361–373.); a form of it can also be recognized in Samuelson (Samuelson, P. A. 1947. Foundations of Economic Analysis. Harvard University Press, Cambridge, MA.), presented earlier in Samuelson (Samuelson, P. A. 1940. Foundations of analytical economics, the observational significance of economic theory. Ph.D. dissertation, Harvard University, Boston, MA.). Independence and the sure-thing principle are equivalent for decision under risk, but in a less elementary way than has sometimes been thought. The sure-thing principle for decision under uncertainty and conjoint independence are identical in a mathematical sense. The mathematics underlying our three preference conditions has an older history. The independence condition for decision under risk can be recognized in the characterization of “associative means,” and conjoint independence for multiattribute decisions in solutions to the “generalized associativity functional equation.”

Uncertainty and the difficulty of thinking through disjunctions

This paper considers the relationship between decision under uncertainty and thinking through disjunctions. Decision situations that lead to violations of Savage's sure-thing principle are examined, and a variety of simple reasoning problems that often generate confusion and error are reviewed. The common difficulty is attributed to people's reluctance to think through disjunctions. Instead of hypothetically traveling through the branches of a decision tree, it is suggested, people suspend judgement and remain at the node. This interpretation is applied to instances of decision making, information search, deductive and inductive reasoning, probabilistic judgement, games, puzzles and paradoxes. Some implications of the reluctance to think through disjunctions, as well as potential corrective procedures, are discussed.

Decision making with belief functions: Compatibility and incompatibility with the sure-thing principle

This article studies situations in which information is ambiguous and only part of it can be probabilized. It is shown that the information can be modeled through belief functions if and only if the nonprobabilizable information is subject to the principles of complete ignorance. Next the representability of decisions by belief functions on outcomes is justified by means of a neutrality axiom. The natural weakening of Savage's sure-thing principle to unambiguous events is examined and its implications for decision making are identified.

Dynamically Consistent Beliefs Must Be Bayesian

Experimental evidence such as the Ellsberg Paradox contradicts the Savage model of decision making under uncertainty, since the representation of beliefs underlying preferces by a single probability measure leaves no room for the degree of imprecision in information to affect decisions. Proceeding axiomatically, this paper shows that the existence of a Bayesain prior is implied, even if Savage′s Sure-Thing Principle is deleted, if preferences (i) are "based on beliefs" and (ii) admit dynamically consistent updating in response to new information. The result raises questions about the appeal of models of preference that feature a separation of tastes and beliefs. Journal of Economic Literature Classification Number: D81.

Choice under Partial Uncertainty

This paper analyzes problems of choice under uncertainty where a decisionmaker does not use subjective probabilities. The decisionmaker has a set of beliefs about which states are more likely than others, but his beliefs cannot be represented as subjective probabilities. Three main kinds of decision rules are possible in this framework. These are maximin-type, maximax-type, and choosing that action that gives the highest payoff in the state, which the decisionmaker believes to be most likely. The author replaces the commonly used 'merger of states' axiom with a version of the sure-thing principle. Copyright 1993 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Thinking through uncertainty: Nonconsequential reasoning and choice

When thinking under uncertainty, people often do not consider appropriately each of the relevant branches of a decision tree, as required by consequentialism. As a result they sometimes violate Savage's sure-thing principle. In the Prisoner's Dilemma game, for example, many subjects compete when they know that the opponent has competed and when they know that the opponent has cooperated, but cooperate when they do not know the opponent's response. Newcomb's Problem and Wason's selection task are also interpreted as manifestations of nonconsequential decision making and reasoning. The causes and implications of such behavior, and the notion of quasi-magical thinking, are discussed.

The Disjunction Effect in Choice under Uncertainty

One of the basic axioms of the rational theory of decision under uncertainty is Savage's (1954) sure-thing principle (STP) It states that if prospect x is preferred to y knowing that Event A occurred, and if x is preferred to y knowing that A did not occur, then x should be preferred to y even when it is not known whether A occurred We present examples in which the decision maker has good reasons for accepting x if A occurs, and different reasons for accepting x if A does not occur Not knowing whether or not A occurs, however, the decision maker may lack a clear reason for accepting x and may opt for another option We suggest that, in the presence of uncertainty, people are often reluctant to think through the implications of each outcome and, as a result, may violate STP This interpretation is supported by the observation that STP is satisfied when people are made aware of their preferences given each outcome

A More Robust Definition of Subjective Probability

The goal of choice-theoretic derivations of subjective probability is to separate a decisionmaker's underlying beliefs (subjective probabilities of events) from their preferences (attitudes toward risk). Classical derivations have all relied upon some form of the Marschak-Samuelson Independence Axiom or the Savage Sure-Thing Principle, which imply that preferences over lotteries conform to the expected utility hypothesis. This paper presents a choice-theoretic derivation of subjective probability in a Savage-type setting of purely subjective uncertainty, which neither assumes nor implies that the decisionmaker's preferences over lotteries necessarily conform to the expected utility hypothesis. Copyright 1992 by The Econometric Society.

A note on Savage's theorem with a finite number of states

This article gives a preference-based characterization of subjective expected utility for the general equilibrium model with a finite number of states. The characterization follows Savage (1954) as closely as possible but has to abandon his axiom (P6), atomlessness of events, since this requires an infinite state space. To introduce continuity we replace (P6) with a continuity assumption on the set of consequences and assume the preferences are smooth. Then we apply Savage's sure-thing principle and his state-independence axiom to get an additively separable utility representation. Finally, to separate subjective probabilities from basic tastes, we apply a new axiom, which states that for each pair of states the marginal rate of substitution is constant along the certainty line.

The delusion of certainty in Savage's sure-thing principle

In risky choice situations Savage's sure-thing principle is regularly advocated as a rational way to clarify one's preferences. It is also used to justify controversial features of the expected utility procedure. This paper sets out explicitly the four steps of Savage's sure-thing principle and argues that the principle is irrational and internally inconsistent. Its repeated use can lead to a truncated probability distribution and a delusion of certainty.

Comment on Noll and Krier, "Some Implications of Cognitive Psychology for Risk Regulation"

We have known about systematic violations of the expected utility (EU) theory of choice for almost forty years, since Maurice Allais got Jimmie Savage to violate his own sure-thing principle (or independence axiom) while making hypothetical choices over lunch in Paris. Savage was victimized by some combination of wine and intuition. The wine's effect is gone, but the intuition is not: devotion to EU sometimes produces unappealing choices.