Savage's Sure-thing Principle Research Articles

On the tradeoff between efficiency and strategyproofness

We study social decision schemes (SDSs), i.e., functions that map a collection of individual preferences over alternatives to a lottery over the alternatives. Depending on how preferences over alternatives are extended to preferences over lotteries, there are varying degrees of efficiency and strategyproofness. In this paper, we consider four such preference extensions: stochastic dominance (SD), a strengthening of SD based on pairwise comparisons (PC), a weakening of SD called bilinear dominance (BD), and an even weaker extension based on Savage's sure-thing principle (ST). While random serial dictatorships are PC-strategyproof, they only satisfy ex post efficiency. On the other hand, we show that strict maximal lotteries are PC-efficient and ST-strategyproof. We also prove the incompatibility of (i)PC-efficiency and PC-strategyproofness for anonymous and neutral SDSs, (ii) ex post efficiency and BD-strategyproofness for pairwise SDSs, and (iii) ex post efficiency and BD-group-strategyproofness for anonymous and neutral SDSs.

A Solution to Ellsberg's Paradox

Ellsberg's famous thought experiments demonstrate that most people prefer less ambiguous alternatives to more ambiguous ones. This apparently violates Savage's Sure-thing Principle. I provide a solution to Ellsberg's paradox. More precisely, I demonstrate that ambiguity aversion can be readily explained by subjectivistic decision theory. The given solution is simple and fits perfectly into Savage's subjectivistic framework. Since ambiguity aversion translates into the subjective probabilities of the decision-maker, they could even be used in order to quantify his ambiguity aversion.

A Contextual Risk Model for the Ellsberg Paradox

The Allais and Ellsberg paradoxes show that the expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions. The popular explanation in terms of ambiguity aversion is not completely accepted. On the other hand, we have recently introduced a notion of contextual risk to mathematically capture what is known as ambiguity in the economics literature. Situations in which contextual risk occurs cannot be modeled by Kolmogorovian classical probabilistic structures, but a non-Kolmogorovian framework with a quantum-like structure is needed. We prove in this paper that the contextual risk approach can be applied to the Ellsberg paradox, and elaborate a sphere model within our hidden measurement formalism which reveals that it is the overall conceptual landscape that is responsible of the disagreement between actual human decisions and the predictions of expected utility theory, which generates the paradox. This result points to the presence of a quantum conceptual layer in human thought which is superposed to the usually assumed classical logical layer.

MATHEMATICAL STRUCTURE OF QUANTUM DECISION THEORY

One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a "Quantum Decision Theory" of decision-making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to explain a variety of interesting fallacies and anomalies that have been reported to particularize the decision-making of real human beings. The theory describes entangled decision-making, non-commutativity of subsequent decisions, and intention interference of composite prospects. We demonstrate how the violation of the Savage's sure-thing principle (disjunction effect) can be explained as a result of the interference of intentions, when making decisions under uncertainty. The conjunction fallacy is also explained by the presence of the interference terms. We demonstrate that all known anomalies and paradoxes, documented in the context of classical decision theory, are reducible to just a few mathematical archetypes, all of which allow the finding of straightforward explanations in the frame of the developed quantum approach.

THE AMBIGUITY AVERSION LITERATURE: A CRITICAL ASSESSMENT

We provide a critical assessment of the ambiguity aversion literature, which we characterize in terms of the view that Ellsberg choices are rational responses to ambiguity, to be explained by relaxing Savage's Sure-Thing principle and adding an ambiguity-aversion postulate. First, admitting Ellsberg choices as rational leads to behaviour, such as sensitivity to irrelevant sunk cost, or aversion to information, which most economists would consider absurd or irrational. Second, we argue that the mathematical objects referred to as “beliefs” in the ambiguity aversion literature have little to do with how an economist or game theorist understands and uses the concept. This is because of the lack of a useful notion of updating. Third, the anomaly of the Ellsberg choices can be explained simply and without tampering with the foundations of choice theory. These choices can arise when decision makers form heuristics that serve them well in real-life situations where odds are manipulable, and misapply them to experimental settings.

INTRODUCTION TO THE SPECIAL ISSUE OF ECONOMICS AND PHILOSOPHY ON AMBIGUITY AVERSION

The paradigm for modelling decision-making under uncertainty has undoubtedly been the theory of Expected Utility, which was first developed by von Neumann and Morgenstern (1944) and later extended by Savage (1954) to the case of subjective uncertainty. The inadequacy of the theory of Subjective Expected Utility (SEU) as a descriptive theory was soon pointed out in experiments, most famously by Allais (1953) and Ellsberg (1961). The observed departures from SEU noticed by Allais and Ellsberg became known as “paradoxes”. The Ellsberg paradox gave rise, several years later, to a new literature on decision-making under ambiguity. The theoretical side of this literature was pioneered by Schmeidler (1989). This literature views the departures from SEU in situations similar to those discussed by Ellsberg as rational responses to ambiguity. The rationality is “recovered” by relaxing Savage's Sure-Thing principle and adding an ambiguity-aversion postulate. Thus the ambiguity-aversion literature takes a normative point of view and does consider Ellsberg-type choices as behavioural “anomalies”.

Pragmatic approach to decision making under uncertainty: The case of the disjunction effect

The disjunction effect (Tversky & Shafir, 1992) occurs when decision makers prefer option x (versus y) when knowing that event A occurs and also when knowing that event A does not occur, but they refuse x (or prefer y) when not knowing whether or not A occurs. This form of incoherence violates Savage's (1954) sure-thing principle, one of the basic axioms of the rational theory of decision making. The phenomenon was attributed to a lack of clear reasons for accepting an option (x) when subjects are under uncertainty. Through a pragmatic analysis of the task and a consequent reformulation of it, we show that the effect does not depend on the presence of uncertainty, but on the introduction of non-relevant goals into the text problem, in both the well-known Gamble problem and the Hawaii problem.

The Disjunction Effect and Reason-Based Choice in Games

This paper reports an experiment that extends previous find ings of the disjunction effect, sometimes described as a violation of Savage's sure-thing principle. Evidence of the disjunction ef fect is observed using elicited beliefs about others' actions (rather than controlled beliefs) in a prisoners' dilemma studied by Shafir and Tversky (1992) as well as in an asymmetric version of it and in a nondilemma game with a unique equilibrium. Debiasing tech niques as well as implications for these extensions are discussed.

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.

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.

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.

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.

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.

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.