Learning under compound risk vs. learning under ambiguity – an experiment

We design and conduct an economic experiment to investigate the learning process of agents under compound risk and under ambiguity. We gather data for subjects choosing between lotteries involving risky and ambiguous urns. Agents make decisions in conjunction with a sequence of random draws with replacement, allowing us to estimate the agents’ beliefs at different moments in time. For each type of urn, we estimate a behavioral model for which the standard Bayesian updating model is a particular case. Our findings suggest an important difference in updating behavior between risky and ambiguous environments. Specifically, even after controlling for the initial prior, we find that when learning under ambiguity, subjects significantly overweight the new signal, while when learning under compound risk, subjects are essentially Bayesian.     

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).

     

Elicitation of preferences under ambiguity

This paper is about behaviour under ambiguity—that is, a situation in which probabilities either do not exist or are not known. Our objective is to find the most empirically valid of the increasingly large number of theories attempting to explain such behaviour. We use experimentally-generated data to compare and contrast the theories. The incentivised experimental task we employed was that of allocation: in a series of problems we gave the subjects an amount of money and asked them to allocate the money over three accounts, the payoffs to them being contingent on a ‘state of the world’ with the occurrence of the states being ambiguous. We reproduced ambiguity in the laboratory using a Bingo Blower. We fitted the most popular and apparently empirically valid preference functionals [Subjective Expected Utility (SEU), MaxMin Expected Utility (MEU) and α-MEU], as well as Mean-Variance (MV) and a heuristic rule, Safety First (SF). We found that SEU fits better than MV and SF and only slightly worse than MEU and α-MEU.     

Recursive expected utility and the separation of attitudes towards risk and ambiguity: an experimental study

We use the multiple price list method and a recursive expected utility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over gains and are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent.      

Sampling experience reverses preferences for ambiguity

People often need to choose between alternatives with known probabilities (risk) and alternatives with unknown probabilities (ambiguity). Such decisions are characterized by attitudes towards ambiguity, which are distinct from risk attitudes. Most studies of ambiguity attitudes have focused on the static case of single choice, where decision makers typically prefer risky over ambiguous prospects. However, in many situations, decision makers may be able to sample outcomes of an ambiguous alternative, allowing for inferences about its probabilities. The current paper finds that such sampling experience reverses the pattern of ambiguity attitude observed in the static case. This effect can only partly be explained by the updating of probabilistic beliefs, suggesting a direct effect of sampling on attitudes toward ambiguity.     

Ellsberg paradox: Ambiguity and complexity aversions compared

We present a simple model where preferences with complexity aversion, rather than ambiguity aversion, resolve the Ellsberg paradox. We test our theory using laboratory experiments where subjects choose among lotteries that “range” from a simple risky lottery, through risky but more complex lotteries, to one similar to Ellsberg’s ambiguity urn. Our model ranks lotteries according to their complexity and makes different—at times contrasting—predictions than most models of ambiguity in response to manipulations of prizes. The results support that complexity aversion preferences play an important and separate role from beliefs with ambiguity aversion in explaining behavior under uncertainty.     

Recent developments in modeling preferences: Uncertainty and ambiguity

In subjective expected utility (SEU), the decision weights people attach to events are their beliefs about the likelihood of events. Much empirical evidence, inspired by Ellsberg (1961) and others, shows that people prefer to bet on events they know more about, even when their beliefs are held constant. (They are averse to ambiguity, or uncertainty about probability.) We review evidence, recent theoretical explanations, and applications of research on ambiguity and SEU.Thanks to Jonathan Baron, James Dow, Peter Fishburn, Itzhak Gilboa, Gordon Hazen, Howard Kunreuther, Tomas Phillipson, David Schmeidler, Amos Tversky, the editor, and several anonymous referees for corrections and helpful comments. Camerer's contribution to this work was supported by the National Science Foundation, grant no. SES 88-09299. Weber's contribution was supported by the Deutsche Forschungsge-meinschaft, grant no. WE 993/5-1.     

Effects of Outcome and Probabilistic Ambiguity on Managerial Choices

Information ambiguity is prevalent in organizations and likely influences management decisions. This study examines, given imprecise probabilities and outcomes, how managers make choices when they are provided with single-figure benchmarks. Seventy-nine MBA students completed two experiments. We found that, in a decision framed as a decision under certainty involving an ambiguous outcome, the majority of the subjects were ambiguity prone in the loss condition and switched to ambiguity aversion in the gain condition. However, in the presence of probabilistic ambiguity in a decision under risk, this expected switching pattern was shown only when the difference in riskiness between the two choice options (in the loss condition) was perceived to be relatively small. In a companion study, we used a written protocol approach to identify factors that affect decision makers' investment choices when faced with ambiguous outcomes. Protocols frequently mentioned that the ambiguous outcome option was risky, even in the case which was framed as a decision under certainty in the problem statement. In a decision under risk with ambiguous outcomes, the combination of probabilistic risk and outcome ambiguity was seen as even more risky.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Preference and belief: Ambiguity and competence in choice under uncertainty

We investigate the relation between judgments of probability and preferences between bets. A series of experiments provides support for the competence hypothesis that people prefer betting on their own judgment over an equiprobable chance event when they consider themselves knowledgeable, but not otherwise. They even pay a significant premium to bet on their judgments. These data connot be explained by aversion to ambiguity, because judgmental probabilities are more ambiguous than chance events. We interpret the results in terms of the attribution of credit and blame. The possibility of inferring beliefs from preferences is questioned.¹     

When is ambiguity–attitude constant?

This paper studies how updating affects ambiguity attitude. In particular we focus on generalized Bayesian updating of the Jaffray–Philippe sub-class of Choquet Expected Utility preferences. We find conditions for ambiguity attitude to be the same before and after updating. A necessary and sufficient condition for ambiguity attitude to be unchanged when updated on an arbitrary event is for the capacity to be neo-additive. We find a condition for updating on a given partition to preserve ambiguity attitude. We relate this to necessary and sufficient conditions for dynamic consistency. Finally, we study whether ambiguity increases or decreases after updating.     

Betting on Machina’s reflection example: an experiment on ambiguity

In a recent article, Machina (Am Econ Rev forthcoming, 2008) suggested choice problems in the spirit of Ellsberg (Q J Econ 75:643–669, 1961), which challenge tail-separability, an implication of Choquet expected utility (CEU), to a similar extent as the Ellsberg paradox challenged the sure-thing principle implied by subjective expected utility (SEU). We have tested choice behavior for bets on one of Machina’s choice problems, the reflection example. Our results indicate that tail-separability is violated by a large majority of subjects (over 70% of the sample). These empirical findings complement the theoretical analysis of Machina (Am Econ Rev forthcoming, 2008) and, together, they confirm the need for new approaches in the analysis of ambiguity for decision making.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

Preferences for information on probabilities versus prizes: The role of risk-taking attitudes

Many real-world decisions entail choices between information on either probabilities or payoffs (i.e., prizes). Simplified versions of such decisions are examined to gain insight into preferences for different types of information as a function of risk-attitudes. General and simple decision rules are derived for cases where the utility function is concave (or convex) over the relevant payoff interval.The article further describes several experiments to test business students' intuitions concerning these optimal decision rules. In general, risk-taking attitudes did not correlate significantly with subjects' preferences for information, in violation of theorems regarding mean-preserving spreads of risk. Other tests, e.g., narrowing certain probability ranges, also resulted in preferences contrary to expected utility (EU) theory.     

Ambiguity aversion and the propensities for self-insurance and self-protection

Two models of ambiguity preferences that permit comparative statics analysis of greater ambiguity aversion yield definite predictions concerning propensities for self-insurance and self-protection: The levels of both activities that are optimal for an ambiguity-averse decision maker are higher in the presence of ambiguity than in its absence, and demands for both activities increase with greater ambiguity aversion. The reason is that, at levels optimal for one decision maker, an increase in either activity results in a mean-preserving contraction in the distribution of expected utility in the presence of ambiguity, which is valuable to anyone with the same risk preferences who is more ambiguity averse.     

Savage vs. Anscombe-Aumann: an experimental investigation of ambiguity frameworks

The Savage and the Anscombe–Aumann frameworks are the two most popular approaches used when modeling ambiguity. The former is more flexible, but the latter is often preferred for its simplicity. We conduct an experiment where subjects place bets on the joint outcome of an ambiguous urn and a fair coin. We document that more than a third of our subjects make choices that are incompatible with Anscombe–Aumann for any preferences, while the Savage framework is flexible enough to account for subjects’ behaviors.