Sure-thing Principle Research Articles

Subjective expected utility with nonincreasing risk aversion

It is shown that assumptions about risk aversion, usually studied under the pre-supposition of expected utility maximization, have a surprising extra merit at an earlier stage of the measurement work: together with the sure-thing principle, these assumptions imply subjective expected utility maximization for monotonic continuous weak orders.

Expectation and Variation in Multi-Period Decisions

Multi-period decisions are decisions which determine an individual's payoffs in several periods in the future. This paper examines the theoretical foundations of the prevalent weighted average assumption. More specifically, we use a multi-period interpretation of the famous Ellsberg paradox in decision under uncertainty to show that in many cases of interest additively-separable functionals (in general) and weighted average ones (in particular) do not seem appropriate for the representation of the decision maker's preferences. We then suggest replacing the sure-thing principle, which may be used to axiomatize a weighted average functional, by a weaker version of it. Using the weakened axiom in Schmeidler's nonadditive measure model (reinterpreted for the multi-period context) yields an axiomatization of a larger class of decision rules which are representable by a weighted average of the utility in each period und the utility variation between each two consecutive periods. The weighted average assumption is a special case of the generalized model, a case in which the decision maker is variation neutral. Similarly, we define and characterize variation aversion and variation liking, and show an example of the economic implications of these properties.

Consequentialist foundations for expected utility

Behaviour norms are considered for decision trees which allow both objective probabilities and uncertain states of the world with unknown probabilities. Terminal nodes have consequences in a given domain. Behaviour is required to be consistent in subtrees. Consequentialist behaviour, by definition, reveals a consequence choice function independent of the structure of the decision tree. It implies that behaviour reveals a revealed preference ordering satisfying both the independence axiom and a novel form of sure-thing principle. Continuous consequentialist behaviour must be expected utility maximizing. Other plausible assumptions then imply additive utilities, subjective probabilities, and Bayes' rule.

Disappointment and Dynamic Consistency in Choice under Uncertainty

The central proposition of disappointment theory is that an individual forms expectations about uncertain prospects, and that if the actual consequence turns out to be worse than (or better than) that expectation, the individual experiences a sensation of disappointment (or elation) generating a decrement (or increment) of utility which modifies the basic utility derived from the consequence. By incorporating a simple disappointment-elation function into a model of individual choice, many observed violations of conventional expected utility axioms-including violations of Savage's sure-thing principle and the isolation effect-can be predicted and defended as rational and dynamically consistent behaviour.

Timing contradictions in von Neumann and Morgenstern's axioms and in savage's ?sure-thing? proof

At the point of choice, let N be the delay in learning the outcome. Then von Neumann and Morgenstern's postulates contradictorily imply that N = 0 and N > 0. As a consequence, Savage's ‘sure-thing’ proof, which has bestowed on expected utility theory most of its normative appeal, depends on inconsistent assumptions. Further, the validity of Savage's proof cannot be retrieved by minimizing N > 0, by making the delay a mere moment or so. The historical origins of these contradictions are traced to (i) von Neumann and Morgenstern inadvertently limiting their risk model to the certain period, that is the period after gamblers learn the outcome(s), and (ii) Savage's use of the sure-thing principle for analysing “atemporally but also quite formally” compound gambles [Savage, 1954, p. 23].

A method of weighing qualitative preference axioms

A method is developed for determining the absolute and relative strengths of qualitative preference axioms in normative Bayesian decision theory. These strengths are calculated for the three most common qualitative axioms; transitivity, the sure-thing principle, and dominance. The relative strength of the latter two axioms with respect to transitivity is calculated for special cases, and a bound is derived which is applicable to a larger class of decision problems. Possible implications of this theoretical work for decision heuristics are discussed.

The inadequacy of Bayesian decision theory

From the arguments presented in this paper I trust that we are in a position to see that Bayesian decision theory is inadequate both from the descriptive and normative standpoints. As a descriptive theory it is just false. As a normative theory we can say that there is a presumption in favour of the view that it is not irrational to value certainty as such (because this is in accord with ordinary intuition) and that no argument has been presented — and there seems little prospect of such an argument being presented — that would force us to abandon that presumption. That is, from the normative standpoint Bayesian decision theory is prima facie implausible. We have located the problem for Bayesianism in one of its rationality postulates, viz., the Sure-Thing Principle. Of course, this paper does not go very far in suggesting what is to supplant Bayesian decision theory as a theory of rational decision making. Nonetheless, I take the arguments in this paper to present a compelling reason for embarking on such a program.

Limited resoluteness and strategic voting: The case of linear sincere preference orderings

This note is concerned with the strategic manipulation of the aggregation procedures, which for every profile of individual preference orderings and every set of alternatives (presented for choice) specify a non-empty subset of alternatives. In the context of multi–valued aggregation procedures, the strategic behaviour of an individual crucially depends on the criterion of comparing different subsets of alternatives. It is shown that when the sincere (or true) preference orderings are restricted to be strict, the requirement of resoluteness in pairwise choice is the frontier between the manipulability and non-manipulability results in social choice as long as individuals adopt some versions of the sure-thing principle in comparing different subsets of alternatives.

Nonlinear social welfare functions

It is argued that Bayesian decision theory is a solution of an important philosophical problem, viz. the problem of how to define rational behavior under risk and uncertainty. The author has shown in earlier papers that if we take the Bayesian rationality postulates seriously, and take an individualistic point of view about social welfare, then our social welfare function must be a linear function of individual utilities: indeed, it must be their arithmetic mean. The present paper criticizes Diamond's and Sen's contention that one of the Bayesian postulates (viz. the sure-thing principle) does not apply to social decisions, even though it may apply to individual decisions. It also criticizes Sen's proposal of making social welfare a nonlinear function of individual utilities. The social welfare function proposed by the author depends on interpersonal utility comparisons. The use of such comparisons is defended. It is also argued that anybody who feels that the utilitarian (i.e., linear) form of the social welfare function is not egalitarian enough, should reject the author's individualism axiom, instead of trying to reject the Bayesian rationality axioms. However, this would be equivalent to giving egalitarian considerations a priority in many cases over humanitarian considerations. Finally, the paper discusses the reasons why even full agreement on the mathematical form of the social welfare function would not give rise to a Utopian state of moral consensus: moral controversies arising from disagreements about what predictions to make about future empirical facts would still remain.

Minimal preferences on lotteries

Several important implications of the von Neumann-Morgenstern preference axioms follow from significantly weaker and psychologically more realistic assumptions. Four axioms that use convex linear combinations of probability distributions are discussed and criticized. Two of these, which permit indifference to be intransitive and do not imply a dilution form of the sure-thing principle, imply that the set of most preferred distributions includes at least one distribution that has probability 1 for a basic outcome and that every distribution not in the most preferred set is less preferred than some distribution in the most preferred set. Comparisons are made with previous weakenings of the von Neumann-Morgenstern axioms. The basic outcome set is assumed to be finite.

On Simpson's Paradox and the Sure-Thing Principle

Abstract This paradox is the possibility of P(A|B) <P(A|B') even though P(A|B)≥P(A| B') both under the additional condition C and under the complement C' of that condition. Details are given on why this can happen and how extreme the inequalities can be. An example shows that Savage's sure-thing principle (“If you would definitely prefer g to f, either knowing that the event C obtained, or knowing that C did not obtain, then you definitely prefer g to f.”) is not applicable to alternatives f and g that involve sequential operations.

On Simpson's Paradox and the Sure-Thing Principle

On Simpson's Paradox and the Sure-Thing Principle