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.     

Stochastic revealed preference and rationalizability

This article explores rationalizability issues for finite sets of observations of stochastic choice in the framework introduced by Bandyopadhyay et al. (Journal of Economic Theory, 84(1), 95–110, 1999). It is argued that a useful approach is to consider indirect preferences on budgets instead of direct preferences on commodity bundles. A new rationalizability condition for stochastic choices, “rationalizable in terms of stochastic orderings on the normalized price space” (rsop), is defined. rsop is satisfied if and only if there exists a solution to a linear feasibility problem. The existence of a solution also implies rationalizability in terms of stochastic orderings on the commodity space. Furthermore it is shown that the problem of finding sufficiency conditions for binary choice probabilities to be rationalizable bears similarities to the problem considered here.     

Risky investment decisions: How are individuals influenced by their groups?

We investigate the effect of group versus individual decision-making in the context of risky investment decisions in which all subjects are fully informed of the probabilities and payoffs. Although there is full information, the lottery choices pose cognitive challenges so that people may not be sure of their expected utility-maximizing choice. Making such decisions in a group context provides real-time information in which group members can observe others’ choices and revise their own decisions. Our experimental results show that simply observing what others in the group do has a significant impact on behavior. Coupling real-time information with group decisions based on the median value, i.e., majority rule, makes the median investment choice focal, leading people with low values to increase investments and those with high values to decrease investments. Group decisions based on the minimum investment amount produce more asymmetric effects.     

Empirical tests of stochastic binary choice models

This paper provides an experimental test of stochastic choice models of decisions. Models that admit Fechnerian structure are tested through the repeated pairwise choice problems. Results refute the Fechner hypothesis that characterizing the probability of selecting a given prospect increases in how strongly it is preferred to alternative choices. However, the experimental data lend support to characterizing an individual’s binary choice probability as some scalable functions of the von Neumann–Morgenstern utilities in the risky context.

     

A Deeper Look at Hyperbolic Discounting

We conduct an experiment to investigate the degree to which deviations from exponential discounting can be accounted for by the hypothesis of hyperbolic discounting. Subjects are asked to choose between an earlier or later payoff in a series of 40 choice questions. Each question consists of a pair of monetary amounts determined by compounding a given base amount at a constant rate per period. Two bases (8 and 20 dollars), three compounding rates (low, medium and high) and three delays (2, 4, and 6 weeks) are each used. There are also 2 initial periods (Today and 2 weeks) and there are two separate questionnaires, one with lower “realistic” compounding rates and the other with higher compounding rates, typical of those used in previous studies. We analyze the detailed patterns of choice in 6 groups of 6 related questions each (in which the base and rate is fixed but the initial period and delay varies), documenting the frequency of patterns consistent with exponential discounting and with hyperbolic discounting. We find that exponential discounting is the clear modal choice pattern in virtually all cases. Hyperbolic discounting is never the modal pattern (except in the sense that constant discounting is a special case of hyperbolic discounting). We also estimate a linear probability model that takes account of individual heterogeneity. The estimates show substantial increases in the probability of choosing the later option when the compounding rate increases, as one would expect. There are small, sometimes significant, increases in this probability when the delay is increased or the initial period is in the future. Such behavior is consistent with hyperbolic discounting, but can account for only a small proportion of choices. Overall, deviations from exponential discounting appear to be due to error, or to other effects not accounted for by hyperbolic discounting. Principal among these is an increase in later choices when the base is larger.     

Justice in an uncertain world: Evidence on donations to cancer research

The paper uses information on actual and hypothetical charitable contributions to cancer research in the United Kingdom to elicit information on justice principles endorsed by donors. They face a choice between fund-raising contributions for several hereditary and lifestyle-related cancers. Donors’ choices of how much to donate to different cancers reveal how they view luck vis-a-vis risky individual choices. The estimation results reveal that donations are smaller for cancers with higher prevention rates, which is the probability that the potential cancer victim can avoid the cancer in question by some choice. We also find that provision of information on lifestyle-related causes of cancer adversely affects contributions. In contrast, information on hereditary causes has a positive effect on donations. Furthermore, a large share of donors indicated in their feedback that they chose donations to a hereditary over a lifestyle-related cancer to “punish” poor individual choices. These findings suggest that many donors lean toward choice egalitarianism, which conditions donations on the potential beneficiaries’ choices.

     

Estimating individual and group preference functionals using experimental data

In this paper, the empirical performance of several preference functionals is assessed using individual and group experimental data. We investigate if there is a risky choice theory that fits group decisions better than alternative theories, and if there are significant differences between individual and group choices. Experimental findings reported in this paper provide answers to both of those questions showing that expected utility gains a “winning” position over higher-level functionals (we considered disappoint aversion and two variants of rank-dependent utility) when risky choices are undertaken by individuals as well as by small groups. However, in the group experiment, alternatives (and, most notably, disappoint aversion) improve their relative performance, a fact that hints at the existence of differences between individual and group choices. We interpreted this result as evidence that feelings-like disappointment aversion become stronger in group decision.     

Is Context-Based Choice due to Context-Dependent Preferences?

The rationalization of context-based choice is usually based on the assumption that preferences are context-dependent. In this paper, we show that context-based choice can be due to the characteristics of the choice procedure applied by the individual and not to the dependence of preferences (stochastic or deterministic) on the context. Our arguments are illustrated focusing on the much-studied dominated-alternative effects.     

Choice with imprecise information: an experimental approach

This article provides an experimental analysis of attitude toward imprecise and variable information. Imprecise information is provided in the form of a set of possible probability values, such that it is virtually impossible for the subjects to guess or estimate, which one in the set is true or more likely to be true. We investigate how geometric features of such information pieces affect choices. We find that the subjects care about more features than the pairs of best-case and worst-case, which is a counter-evidence to the well-known models, maximin and α-maximin. We find that presence of nonextreme points in the set affects choice, which suggests that attitude toward imprecision is ‘nonlinear.’ We also obtain an observation, though not significant, that information pieces have a complementarity that may not be explained by the Bayesian view.     

Individual vs. group decision-making: an experiment on dynamic choice under risk and ambiguity

This paper focuses on the comparison of individual and group decision-making, in a stochastic inter-temporal problem in two decision environments, namely risk and ambiguity. Using a consumption/saving laboratory experiment, we investigate behaviour in four treatments: (1) individual choice under risk; (2) group choice under risk; (3) individual choice under ambiguity and (4) group choice under ambiguity. Comparing decisions within and between decision environments, we find an anti-symmetric pattern. While individuals are choosing on average closer to the theoretical optimal predictions, compared to groups in the risk treatments, groups tend to deviate less under ambiguity. Within decision environments, individuals deviate more when they choose under ambiguity, while groups are better planners under ambiguity rather than under risk. Our results extend the often observed pattern of individuals (groups) behaving more optimally under risk (ambiguity), to its dynamic dimension.

     

Behavioral patterns and reduction of sub-optimality: an experimental choice analysis

This paper attempts to identify behavioral patterns and compare their average success considering several criteria of bounded rationality. Experimentally observed choice behavior in various decision tasks is used to assess heterogeneity in how individual participants respond to 15 randomly ordered portfolio choices, each of which is experienced twice. Treatments differ in (not) granting probability information and in (not) eliciting aspirations. Since in our setting neither other regarding concerns nor risk attitude matter and probability of the binary chance move is (optimal) choice irrelevant, categorizing decision types relies on parameter dependence and choice adaptations. We find that most participants reduce systematically sub-optimality when following the identified criteria.     

An experimental study of the generosity game

We study ultimatum and dictator variants of the generosity game. In this game, the first mover chooses the amount of money to be distributed between the players within a given interval, knowing that her own share is fixed. Thus, the first mover is not confronted with the typical trade-off between her own and the other’s payoff. For each variant of the game, we study three treatments that vary the range of potential pie sizes so as to assess the influence of these changes on the first movers’ generosity. We find that removing the trade-off inspires significant generosity, which is not always affected by the second mover’s veto power. Moreover, the manipulation of the choice set indicates that choices are influenced by the available alternatives.     

Competence effects for choices involving gains and losses

We investigate how choices for uncertain gain and loss prospects are affected by the decision maker’s perceived level of knowledge about the underlying domain of uncertainty. Specifically, we test whether Heath and Tversky’s (J Risk Uncertain 4:5–28, 1991) competence hypothesis extends from gains to losses. We predict that the commonly-observed preference for high knowledge over low knowledge prospects for gains reverses for losses. We employ an empirical setup in which participants make hypothetical choices between gain or loss prospects in which the outcome depends on whether a high or low knowledge event occurs. We infer decision weighting functions for high and low knowledge events from choices using a representative agent preference model. For gains, we replicate the results of Kilka and Weber (Manage Sci 47:1712–1726, 2001), finding that decision makers are more attracted to choices that they feel more knowledgeable about. However, for losses, we find limited support for our extension of the competence effect.     

A Note on the Portfolio Selection Problem

In this note we provide new results of interest in the portfolio choice problem when the risky opportunities are correlated: for a general vector (X₁, X₂,..., X_n) of risky opportunities we give new conditions for stochastic comparison among different portfolios choices and new necessary and sufficient conditions to characterize the portfolio which gives the maximal expected utility.     

The Experimetrics of Public Goods: Inferring Motivations from Contributions

In public goods experiments, stochastic choice, censoring and motivational heterogeneity give scope for disagreement over the extent of unselfishness, and whether it is reciprocal or altruistic. We show that these problems can be addressed econometrically, by estimating a finite mixture model to isolate types, incorporating double censoring and a tremble term. Most subjects act selfishly, but a substantial proportion are reciprocal with altruism playing only a marginal role. Isolating reciprocators enables a test of Sugden’s model of voluntary contributions. We estimate that reciprocators display a self-serving bias relative to the model.      

The econometric modelling of social preferences

Experimental data on social preferences present a number of features that need to be incorporated in econometric modelling. We explore a variety of econometric modelling approaches to the analysis of such data. The approaches under consideration are: the Random Utility approach (in which it is assumed that each possible action yields a utility with a deterministic and a stochastic component, and that the individual selects the action yielding the highest utility); the Random Behavioural approach (which assumes that the individual computes the maximum of a deterministic utility function, and that computational error causes their observed behaviour to depart stochastically from this optimum); and the Random Preference approach (in which all variation in behaviour is attributed to stochastic variation in the parameters of the deterministic component of utility). These approaches are applied in various ways to an experiment on fairness conducted by Cappelen et al. (Am Econ Rev 97(3):818–827, 2007). Various models that we estimate succeed in capturing the key features of the dataset. Conclusions concerning fairness-related behaviour depend crucially on the choice of econometric model.     

Games of Competition in a Stochastic Environment

The paper presents a set of games of competition between two or three players in which reward is jointly determined by a stochastic biased mechanism and players’ choices. More specifically, a resource can be found with unequal probabilities in one of two locations. The first agent is rewarded only if it finds the resource and avoids being found by the next agent in line; the latter is rewarded only if it finds the former. Five benchmarks, based on different psychological and game-theoretic assumptions are derived and their predictions compared to actual behavior of 120, 40, and 48 participants playing repeatedly. Of the five benchmarks—the unique (Nash) equilibrium, reinforcement learning, trust-based efficiency, maximum unpredictability, and regret-based (Impulse Balance) equilibrium—regret for missed opportunities best accounts for the qualitative aspect of participants’ behavior and regret attenuated by randomization best accounts for the quantitative aspect of behavior.     

Implementation without rationality assumptions

Hurwicz (Social Choice and public decision making. Essays in honor of Kenneth J. Arrow, Cambridge University Press, Cambridge, 1986) was the first to study an approach to implementation theory based on choice functions instead of preference relations. We argue that the solution concept used by him, the generalized Nash equilibrium, is not really compatible with the idea that individual behavior is describable by a choice function. A new solution concept that better fits the choice function framework is then introduced. Using this, we investigate what behavioral assumptions are needed for the full characterizations of Nash implementable social choice correspondences to still hold. We will show that a condition known as Property α is central.     

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.     

Political influence in multi-choice institutions: cyclicity,anonymity, and transitivity

We study political influence in institutions where each member chooses a level of support for a collective goal. These individual choices determine the degree to which the goal is reached. Influence is assessed by newly defined binary relations, each of which ranks members on the basis of their relative performance at a corresponding level of participation. For institutions with three options (e.g., voting games in which each voter may vote “yes”, “abstain”, or vote “no”), we obtain three influence relations, and show that their strict components may be cyclic. This latter property describes a “paradox of power” which contrasts with the transitivity of the unique influence relation of binary voting games. Weak conditions of anonymity suffice for each of these relations to be transitive. We also obtain a necessary and sufficient condition for each of these relations to be complete. Further, we characterize institutions in which the rankings induced by these relations, and the Banzhaf–Coleman and Shapley–Shubik power indices coincide. We argue that extending the influence relations to firms would be useful in efficiently assigning workers to different units of production. Finally, we provide applications to various forms of political and economic organizations.