Loss Averse Behavior

A behavioral condition of loss aversion is proposed and tested. Forty-nine students participated in experiments on binary choices among lotteries involving small scale real gains and losses. At the aggregate level, a significant proportion of the choices are in the direction predicted by loss aversion. Individuals can be classified as loss averse (28 participants), gain seeking (12), and unclassified (9). A comparison with risk behavior for binary choices on lotteries involving only gains shows that risk attitudes vary across these domains of lotteries. A gender effect is also observed: proportionally more women are loss averse. In contrast to the predictions of comonotonic independence, the size of common outcomes has systematic influence on choice behavior. JEL Classification: D81, C91     

When Allais meets Ulysses: Dynamic axioms and the common ratio effect

We report experimental findings about subjects’ behavior in dynamic decision problems involving multistage lotteries with different timings of resolution of uncertainty. Our within-subject design allows us to study violations of the independence and dynamic axioms: Dynamic Consistency, Consequentialism and Reduction of Compound Lotteries. We investigate the effects of changes in probability and outcome levels on the pattern of choices observed in the Common Ratio Effect (CRE) and in the Reverse Common Ratio Effect (RCRE) and on their dynamic counterparts. We find that the probability level plays an important role in violations of Reduction of Compound Lottery and Dynamic Consistency and the outcomes levels in violations of Consequentialism. Moreover, more than one quarter of our subjects satisfy the Independence axiom but violate two dynamic axioms. We thus suggest that there is a greater dissociation that might have been expected between preferences captured by dynamic axioms and those observed over single-stage lotteries.     

An additive representation on the product of complete,continuous extensive structures

This article develops an axiom system to justify an additive representation for a preference relation \({\succsim}\) on the product \({\prod_{i=1}^{n}A_{i}}\) of extensive structures. The axiom system is basically similar to the n-component (n ≥ 3) additive conjoint structure, but the independence axiom is weakened in the system. That is, the axiom exclusively requires the independence of the order for each of single factors from fixed levels of the other factors. The introduction of a concatenation operation on each factor A _i makes it possible to yield a special type of restricted solvability, i.e., additive solvability and the usual cancellation on \({\prod_{i=1}^{n}A_{i}}\). In addition, the assumption of continuity and completeness for A _i implies a stronger type of solvability on A _i . The additive solvability, cancellation, and stronger solvability axioms allow the weakened independence to be effective enough in constructing the additive representation.     

Ordinal independence in nonlinear utility theory

Individual behavior under uncertainty is characterized using a new axiom, ordinal independence, which is a weakened form of the von Neumann-Morgenstern independence axiom It states that if two distributions share a tail in common, then this tail can be modified without altering the individual's preference between these distributions. Preference is determined by the tail on which the distributions differ. This axiom implies an appealing and simple functional form for a numerical representation of preferences. It generalizes the form of anticipated utility, and it explains some well-known forms of behavior, such as the Friedman-Savage paradox, that anticipated utility cannot.     

Risk aversion elicitation: reconciling tractability and bias minimization

Risk attitude is known to be a key determinant of various economic and financial choices. Behavioral studies that aim to evaluate the role of risk attitudes in contexts of this type, therefore, require tools for measuring individual risk tolerance. Recent developments in decision theory provide such tools. However, the methods available can be time consuming. As a result, some practitioners might have an incentive to prefer “fast and frugal” methods to clean but more costly methods. In this article, we focus on a tractable procedure initially proposed by Holt and Laury (2002) to elicit risk attitude. We generalize this method to measure utility and risk aversion as follows. First, we allow measurement of probabilistic risk attitude through violations of expected utility due to probability weighting. Second, we use the outcome scale rather than the probability scale in the menu of choices. Third, we compare sure payoffs with lotteries instead of comparing non-degenerate lotteries. A within-subject experimental study illustrates the gains in tractability and bias minimization that can result from such an extension.     

Expected discounted utility

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann (Ann Math Stat 34(1):199–205, 1963, p. 201).

     

Stochastic Choice and Consistency in Decision Making Under Risk: An Experimental Study

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time.     

Violations of betweenness and choice shifts in groups

In decision theory, the betweenness axiom postulates that a decision maker who chooses an alternative A over another alternative B must also choose any probability mixture of A and B over B itself and can never choose a probability mixture of A and B over A itself. The betweenness axiom is a weaker version of the independence axiom of expected utility theory. Numerous empirical studies documented systematic violations of the betweenness axiom in revealed individual choice under uncertainty. This paper shows that these systematic violations can be linked to another behavioral regularity—choice shifts in a group decision making. Choice shifts are observed if an individual faces the same decision problem but makes a different choice when deciding alone and in a group.     

Axiomatization of a Preference for Most Probable Winner

In binary choice between discrete outcome lotteries, an individual may prefer lottery L₁ to lottery L₂ when the probability that L₁ delivers a better outcome than L₂ is higher than the probability that L₂ delivers a better outcome than L₁. Such a preference can be rationalized by three standard axioms (solvability, convexity and symmetry) and one less standard axiom (a fanning-in). A preference for the most probable winner can be represented by a skew-symmetric bilinear utility function. Such a utility function has the structure of a regret theory when lottery outcomes are perceived as ordinal and the assumption of regret aversion is replaced with a preference for a win. The empirical evidence supporting the proposed system of axioms is discussed.     

Independence of irrelevant alternatives revisited

This paper aims to reexamine the axiom of the independence of irrelevant alternatives in the theory of social choice. A generalized notion of independence is introduced to clarify an informational requirement of binary independence which is usually imposed in the Arrovian framework. We characterize the implication of binary independence.     

Strategic games with security and potential level players

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies (Nash, Proceedings of the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players.      

Security level,potential level,expected utility: A three-criteria decision model under risk

Among the violations of expected utility (E.U.) theory which have been observed by experimenters, the violations of its independence axiom is, by far, the most common. It seems that, in many cases, these inconsistencies can be ascribed to the desire for security - called the security factor by L. Lopes (1986) - which makes people attach special importance to the worst outcomes of risky decisions as well as to the sole outcomes of riskless decisions (certainty effect). J.-Y. Jaffray (1988) has proposed a model which generalizes E.U. theory by taking into account this factor and is then able to account for certain violations. However, especially in experiments on choice involving prospective losses, violations of the von Neumann-Morgenstern independence axiom cannot be explained by the security factor alone and have to be partially ascribed to the potential factor (L. Lopes, 1986) which reflects heightened attention to the best outcomes of decisions, especially when the best outcome is the status quo. In this paper, we construct an axiomatic model for subjects taking into account simultaneously or alternatively the security factor and the potential factor. For this, as in Jaffray's model, it has been necessary to weaken not only the standard independence axiom but also the continuity axiom and, in the same time, to reinforce the dominance axiom. In the resulting model, choices are partially determined by the mere comparison of the (security level, potential level) (i.e. the (worst outcome, best outcome)) pairs offered, and completed by the maximization of an affine function of the expected utility, the coefficients of which depend on both the security level and potential level.In this model, a decision maker who (i) has constant marginal utility for money, (ii) is sensitive to the security factor alone in the domain of gains, (iii) is sensitive to the potential factor alone in the domain of losses, behaves as a risk averter for gains and a risk seeker for losses.     

Stochastic expected utility theory

This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When distorted by errors, the expected utility of a lottery never exceeds (falls below) the utility of the highest (lowest) outcome. This assumption implies that errors are likely to overvalue (undervalue) lotteries with expected utility close to the utility of the lowest (highest) outcome. Proposed theory explains many stylized empirical facts such as the fourfold pattern of risk attitudes, common consequence effect (Allais paradox), common ratio effect and violations of betweenness. Theory fits the data from ten well-known experimental studies at least as well as cumulative prospect theory.

The curse of hope

In K?szegi and Rabin’s (Q J Econ 1133–1165, 2006, Am Econ Rev 97:1047–1073, 2007) reference-dependent model of preferences, the chance of obtaining a better outcome can reduce an agent’s expected utility through an increase in the stochastic reference point. This means that individuals may prefer stochastically dominated lotteries. In this sense, hope, understood as a small probability of a better outcome, can be a curse. While K?szegi and Rabin focus on a linear specification of the utility function, we show that this effect occurs more broadly. Using fairly plausible assumptions and parameter values, we specify the conditions under which it occurs, as well as the type of lotteries in which this should be expected. We then show that while a simple subjective transformation of probability into weights of the reference point may in some cases mitigate the issue, in others, it can intensify it or even generate new ones. Finally, we extend the model by adding the individual’s current reference point (status quo) to the stochastic reference point. We show that this modification can reconcile K?szegi and Rabin’s model with the apparent empirical infrequency of stochastically dominated choices while maintaining its main qualitative results.     

Two simple characterizations of the Nash bargaining solution

We provide two alternative characterizations of the Nash bargaining solution. We introduce new simple axioms, strong undominatedness by the disagreement point, and egalitarian Pareto optimality. First, we prove that the Nash solution is characterized by symmetry, scale invariance, independence of irrelevant alternatives, and strong undominatedness by the disagreement point. Second, we replace the independence of irrelevant alternatives axiom with the sandwich axiom (Rachmilevitch in Theory Decis 80:427–442, 2016) and egalitarian Pareto optimality. We then demonstrate that the Nash solution is characterized by symmetry, scale invariance, strong undominatedness by the disagreement point, the sandwich axiom, and egalitarian Pareto optimality.     

Myopic risk-seeking: The impact of narrow decision bracketing on lottery play

In two experiments conducted with low-income participants, we find that individuals are more likely to buy state lottery tickets when they make several purchase decisions one-at-a-time, i.e. myopically, than when they make one decision about how many tickets to purchase. These results extend earlier findings showing that “broad bracketing” of decisions encourages behavior consistent with expected value maximization. Additionally, the results suggest that the combination of myopic decision making and the “peanuts effect”—greater risk seeking for low stakes than high stakes gambles—can help explain the popularity of state lotteries.

Signed orders in linear and nonlinear utility theory

Self-reflecting signed orders on a set A and its anti-set A ^* were introduced previously as a way to account for negative as well as positive feelings about the inclusion of items in A in potential subsets of choice. The present paper extends the notion of signed orders to lotteries on A A ^*, describes reflection axioms for the lottery context, and shows how these axioms simplify utility representations for preference between lotteries. The simplified representations are then used to guide procedures for extending preferences from A A ^* and its lotteries to preferences between subsets of items.     

Ambiguity framed

In his exposition of subjective expected utility theory, Savage (1954) proposed that the Allais paradox could be reduced if it were recast into a format which made the appeal of the independence axiom of expected utility theory more transparent. Recent studies consistently find support for this prediction. We consider a salience-based choice model which explains this frame-dependence of the Allais paradox. We then derive the novel prediction that the presentation format responsible for reductions in Allais-style violations of expected utility theory will also reduce Ellsberg-style violations of subjective expected utility theory. This format makes the appeal of Savage’s “sure thing principle” more transparent. We design an experiment to test this prediction and find strong support for such frame-dependence of ambiguity aversion in Ellsberg-style choices. In particular, we observe markedly less ambiguity-averse behavior in Savage’s matrix format than in a more standard “prospect” format. This finding poses a new challenge for the leading models of ambiguity aversion.