Sure-thing Principle Research Articles

What is Quantum in probabilistic explanations of the sure-thing principle violation?

The Prisoner’s Dilemma game (PDG) is one of the simple test-beds for the probabilistic nature of the human decision-making process. Behavioral experiments have been conducted on this game for decades and show a violation of the so-called sure-thing principle, a key principle in the rational theory of decision. Quantum probabilistic models can explain this violation as a second-order interference effect, which cannot be accounted for by classical probability theory. Here, we adopt the framework of generalized probabilistic theories and approach this explanation from the viewpoint of quantum information theory to identify the source of the interference. In particular, we reformulate one of the existing quantum probabilistic models using density matrix formalism and consider different amounts of classical and quantum uncertainties for one player’s prediction about another player’s action in PDG. This enables us to demonstrate that what makes possible the explanation of the violation is the presence of quantum coherence in the player’s initial prediction and its conversion to probabilities during the dynamics. Moreover, we discuss the role of other quantum information-theoretical quantities, such as quantum entanglement, in the decision-making process. Finally, we propose a three-choice extension of the PDG to compare the predictive powers of quantum probability theory and a more general probabilistic theory that includes it as a particular case and exhibits third-order interference.

Mixture independence foundations for expected utility

An alternative preference foundation for expected utility is provided. Our segregated approach considers four logically independent implications of the classic von Neumann–Morgenstern independence axiom. The monotonicity principle is, for a transitive relation, equivalent to monotonicity with respect to first-order stochastic dominance. Rank-dependent separability is similar to the comonotonic sure-thing principle used in the ambiguity literature. The remaining two properties are weak formulations of the independence principle which invoke the latter only for probability mixtures with the extreme, that is the best, respectively, the worst outcome. These four implications of independence, together with completeness, transitivity and continuity of a preference relation, characterize expected utility. Furthermore, if rank-dependent separability is dropped, expected utility still holds on each subset of three-outcomes lotteries that give positive probability to both the best and the worst outcomes.

The Sure-Thing Principle

The Sure-Thing Principle famously appears in Savage’s axiomatization of Subjective Expected Utility. Yet Savage introduces it only as an informal, overarching dominance condition motivating his separability postulate P2 and his state-independence postulate P3. Once these axioms are introduced, by and large, he does not discuss the principle any more. In this note, we pick up the analysis of the Sure-Thing Principle where Savage left it. In particular, we show that each of P2 and P3 is equivalent to a dominance condition; that they strengthen in different directions a common, basic dominance axiom; and that they can be explicitly combined in a unified dominance condition that is a candidate formal statement for the Sure-Thing Principle. Based on elementary proofs, our results shed light on some of the most fundamental properties of rational choice under uncertainty. In particular they imply, as corollaries, potential simplifications for Savage’s and the Anscombe-Aumann axiomatizations of Subjective Expected Utility. Most surprisingly perhaps, they reveal that in Savage’s axiomatization, P3 can be weakened to a natural strengthening of so-called Obvious Dominance.

Preference for Knowledge

We examine the subjective value of gaining knowledge in a version of Savage's model for decisions under uncertainty in which the received outcome provides information about which event has obtained. Decision makers commonly value such knowledge either because they want to use it in future decisions or because they are personally interested in it. We find that in our model, the sure-thing principle and several other axioms of Savage are inconsistent with this value for knowledge about events. We provide a representation theorem for a subjective value of knowledge consisting of the sum of expected utility and a function of the information partition generated by the outcomes of an act. We characterize when the value of knowledge can be represented by a subjective value of knowledge about an information partition plus a Shannon entropy cost of processing information. Our results also provide a novel critique of the necessity of Savage's axioms for rational decisions under uncertainty.

Who accepts Savage’s axiom now?

We report the results of an experimental test of whether preaching the normative appeal of the sure-thing principle leads decision-makers to make choices that satisfy it. We use Allais-type decision problems to observe the incentive-compatible choices of 147 subjects, which either violate the sure-thing principle or adhere to it. Subjects are presented with normative arguments that support the counterfactual behaviour and then repeat their decisions. We observe violations of the sure-thing principle are robust to its normative justification. This result replicates a famous small-sample observation using hypothetical tasks that was published by Paul Slovic and Amos Tversky almost half a century ago. We argue that this finding is as relevant now as it was then and that their design can be usefully applied to address contemporary issues in behavioural economics.

An experimental investigation of the Allais paradox with subjective probabilities and correlated outcomes

Skew-symmetric additive decision theories such as salience theory assume that individuals adhere to Savage’s sure-thing principle. The present paper investigates that assumption in an incentivized lab experiment using Allais-type choice problems with subjective probabilities. Real-world events are employed to implement an easy-to-understand correlation structure between outcomes. We control for the role of display formats by presenting each choice problem in a coalesced format and an event-splitting format that clearly reveals the state space and the underlying correlation structure to subjects. We find the Allais paradox to be present in both display formats, which contradicts the rationale of skew-symmetric additive models such as salience theory. Because of significant event-splitting effects, Allais-type preferences are more pronounced in the coalesced format. The event-splitting effects obtained suggest that subjects assign a higher value to a lottery when the event of disbursement of its upside payoff is split into subevents. This holds both when event-splitting helps to unveil the state space and when it does not. Previous findings on correlated versions of the Allais paradox can be explained partially by event-splitting rather than correlation.

More Causes Less Effect: Destructive Interference in Decision Making.

We present a new experiment demonstrating destructive interference in customers’ estimates of conditional probabilities of product failure. We take the perspective of a manufacturer of consumer products and consider two situations of cause and effect. Whereas, individually, the effect of the causes is similar, it is observed that when combined, the two causes produce the opposite effect. Such negative interference of two or more product features may be exploited for better modeling of the cognitive processes taking place in customers’ minds. Doing so can enhance the likelihood that a manufacturer will be able to design a better product, or a feature within it. Quantum probability has been used to explain some commonly observed “non-classical” effects, such as the disjunction effect, question order effect, violation of the sure-thing principle, and the Machina and Ellsberg paradoxes. In this work, we present results from a survey on the impact of multiple observed symptoms on the drivability of a vehicle. The symptoms are assumed to be conditionally independent. We demonstrate that the response statistics cannot be directly explained using classical probability, but quantum formulation easily models it, as it allows for both positive and negative “interference” between events. Since quantum formalism also accounts for classical probability’s predictions, it serves as a richer paradigm for modeling decision making behavior in engineering design and behavioral economics.

The impossibility of agreeing to disagree: An extension of the sure-thing principle

The impossibility of agreeing to disagree in the non-probabilistic setup means that agents cannot commonly know their decisions unless they are all the same. We study the relation of this property to the sure thing principle when it is expressed in epistemic terms. We show that it can be presented in two equivalent ways: one is in terms of knowledge operators, which we call the principle of follow the knowledgeable, the other is in terms of kens—bodies of agents' knowledge—which we call independence of irrelevant knowledge. The latter can be easily extended to a property which is equivalent to the impossibility of agreeing to disagree.

A Parsimonious Theory of Subjective Probability

We axiomatize subjective probabilities on finite domains without requiring richness in the outcome space or restrictions on risk preference using Event Exchangeability (Chew and Sagi, 2006), which has been implicit in the prior literature (Savage, 1954; Machina and Schmeidler, 1992; Grant, 1995). In three successively stronger theorems, we characterize a probability representation of the exchangeability relation, followed by characterizing a unique subjective probability, and finally endowing this subjective probability with the property of reduction consistency—acts inducing the same lottery are indifferent. This subjective probability can serve as foundation to derive expected utility and rank linear utility by imposing the sure-thing principle and its comonotonic variant. Moreover, our finite-domain setting reveals a novel possibility of state dependence arising from our subjective probability not being reduction consistent. Our axiomatic treatment can be further adapted to smaller collections of events and deliver small worlds probabilistic sophistication.

Replication: Revisiting Tversky and Shafir’s (1992) Disjunction Effect with an extension comparing between and within subject designs

Abstract Does uncertainty about an outcome influence decisions? The sure-thing principle (Savage, 1954) posits that it should not, but Tversky and Shafir (1992) found that people regularly violate it in hypothetical gambling and vacation decisions, a phenomenon they termed “disjunction effect”. Very close replications and extensions of Tversky and Shafir (1992) were conducted in this paper (N = 890, MTurk). The target article demonstrated the effect using two paradigms in a between-subject design: here, an extension also testing a within-subject design, with design being randomly assigned was added. These results were consistent with the original findings for the “paying to know“ problem (original: Cramer’s V = 0.22, 95% (CI) [0.14, 0.32]; replication: Cramer’s V = 0.30, 95% CI [0.24, 0.37]), yet not for the “choice under risk” problem (original: Cramer’s V = 0.26, 95% CI [0.14, 0.39]; replication: Cramer’s V = 0.11, 95% CI [−0.07, 0.20]). The within-subject extension showed very similar results. Implications for the disjunction effect and judgment and decision-making theory are discussed, and a call for improvements on the statistical understanding of comparisons of between-subject and within-subject designs is introduced. All materials, data, and code are available on https://osf.io/gu58m/ .

Relative utilitarianism under uncertainty

In the context of uncertainty, belief-weighted relative utilitarianism consists in comparing acts according to a weighted sum of the (0, 1) -normalized subjective expected utilities they yield. The weights may change with the profile of beliefs but do not depend upon the profile of individual utilities for the outcomes. This class of social welfare functions is characterized by the Pareto principle, the sure-thing principle, a continuity condition, and an independence condition requiring that the social ranking of two acts is unaffected by the addition of an outcome that leaves everyone’s best and worst outcomes unchanged. The weights must be constant if the social ranking of constant acts is independent of individual beliefs. Anonymity then pins down plain relative utilitarianism: acts are compared according the sum of (0, 1)-normalized subjective expected utilities they generate.

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.

The sure-thing principle and P2

This paper offers a fine analysis of different versions of the well known sure-thing principle. We show that Savage’s formal formulation of the principle, i.e., his second postulate (P2), is strictly stronger than what is intended originally.

Counterfactuals in “agreeing to disagree” type results

Moses and Nachum (1990) identified conceptual flaws (later echoed by Samet, 2010) in Bacharach’s (1985) generalization of Aumann’s (1976) seminal “agreeing to disagree” result by demonstrating that the crucial assumptions of like-mindedness and the Sure-Thing Principle are not meaningfully expressible in standard partitional information structures. This paper presents a new agreement theorem couched in “counterfactual information structures” that resolves these conceptual flaws. The new version of the Sure-Thing Principle introduced here, which accounts for beliefs at counterfactual states, is also shown to sit well with the intuition of the original version proposed by Savage (1972).

Bayesian learning with multiple priors and nonvanishing ambiguity

The existing models of Bayesian learning with multiple priors by Marinacci (Stat Pap 43:145–151, 2002) and by Epstein and Schneider (Rev Econ Stud 74:1275–1303, 2007) formalize the intuitive notion that ambiguity should vanish through statistical learning in an one-urn environment. Moreover, the multiple priors decision maker of these models will eventually learn the “truth.” To accommodate nonvanishing violations of Savage’s (The foundations of statistics, Wiley, New York, 1954) sure-thing principle, as reported in Nicholls et al. (J Risk Uncertain 50:97–115, 2015), we construct and analyze a model of Bayesian learning with multiple priors for which ambiguity does not necessarily vanish in an one-urn environment. Our decision maker only forms posteriors from priors that survive a prior selection rule which discriminates, with probability one, against priors whose expected Kullback–Leibler divergence from the “truth” is too far off from the minimal expected Kullback–Leibler divergence over all priors. The “stubbornness” parameter of our prior selection rule thereby governs how much ambiguity will remain in the limit of our learning model.

Editorial: Quantum Structures in Cognitive and Social Science.

Editorial: Quantum Structures in Cognitive and Social Science.

Harsanyi’s theorem without the sure-thing principle: On the consistent aggregation of Monotonic Bernoullian and Archimedean preferences

This paper studies the extension of Harsanyi’s theorem (Harsanyi, 1955) in a framework involving uncertainty. It seeks to extend the aggregation result to a wide class of Monotonic Bernoullian and Archimedean preferences (Cerreia-Vioglio et al., 2011) that subsumes many models of choice under uncertainty proposed in the literature. An impossibility result is obtained, unless we are in the specific framework where all individuals and the social observer are subjective expected utility maximizers sharing the same beliefs. This implies that non-expected utility preferences cannot be aggregated consistently.

Choquet expected utility with affine capacities

This paper studies decisions under ambiguity when attention is paid to extreme outcomes. In a purely subjective framework, we propose an axiomatic characterization of affine capacities, which are Choquet capacities consisting in an affine transformation of a subjective probability. Our main axiom restricts the well-known Savage’s Sure-Thing Principle to a change in a common intermediate outcome. The representation result is then an affine combination of the expected utility of the valued act and its maximal and minimal utilities.

Costs of abandoning the Sure-Thing Principle

Risk-weighted expected utility theory (REU theory for short) permits preferences which violate the Sure-Thing Principle (STP for short). But preferences that violate the STP can lead to bad decisions in sequential choice problems. In particular, they can lead decision-makers to adopt a strategy that is dominated – i.e. a strategy such that some available alternative leads to a better outcome in every possible state of the world.