Kindness,confusion, or … ambiguity?

Other-regarding preferences or decision errors are the main explanations put forward to justify contributions exceeding the non-cooperative optimum in VCM games. An alternative rationale relies on ambiguity aversion. Ambiguity aversion increases the perceived marginal benefit of own contributions, which in equilibrium will exceed the Nash level. We present a series of experiments testing this hypothesis. To control for other-regarding preferences, we run a two-player game in which a human player plays with a virtual agent. Players are assigned either to a risky setting (known probabilities of opponent’s choices) or to an ambiguity setting (probabilities of opponent’s contribution are vague). Results show that ambiguity affects contributions. However, attitude to ambiguity appears to be affected by the location of the aggregate Nash optimum inside the decision space.     

Ambiguity Without a State Space

Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyses a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of objective ambiguity without explicit reference to any state space. We characterize a representation that integrates a non-linear transformation of first-order expected utility with respect to a second-order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion.     

EXPLAINING ATTITUDES TOWARDS AMBIGUITY: AN EXPERIMENTAL TEST OF THE COMPARATIVE IGNORANCE HYPOTHESIS

Many theories have been put forward to explain attitudes towards ambiguity. This paper reports on an experiment designed to test for the existence of Comparative Ignorance when it is tested over events with a range of different likelihoods. A total of 93 subjects valued a series of gambles, one of which was played out for real. The results do not lend support to the theory, although the relationship between risk and ambiguity does appear to correspond with other theories and previous empirical work.     

The dual theory of the smooth ambiguity model

This paper studies the “dual” theory of the smooth ambiguity model introduced by Klibanoff et al. (Econometrica 73:1849–1892, 2005). Unlike the original model, we characterize attitudes toward ambiguity captured by second-order probabilities. First, we give a set of axioms to derive a dual representation of the smooth ambiguity model. Second, we present a characterization of ambiguity aversion. Last, as an application of our dual model to a portfolio problem, we conduct comparative static predictions which give sufficient conditions to guarantee that an increase in smooth ambiguity aversion decreases the optimal portfolio.     

Ambiguity Made Precise: A Comparative Foundation

The theory of subjective expected utility has been recently extended to allow ambiguity to matter for choice. We propose a notion of absolute ambiguity aversion by building on a notion of comparative ambiguity aversion. We characterize it for a preference model which encompasses some of the most popular models in the literature. We next build on these ideas to provide a definition of unambiguous act and event and show the characterization of the latter. As an illustration, we consider the classical Ellsberg 3-color urn problem and find that the notions developed in the paper provide intuitive answers. Journal of Economic Literature Classification Number: D81.     

A theory of subjective compound lotteries

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.     

Marginal risk aversion and preferences in a betting market

An individual's behavioural attitudes toward variance and non-symmetry in the payoff distributions of pari-mutuel gambles are empirically examined using the von Neumann - Morgenstern expected utility of wealth paradigm. Preferences over payoff distributions for a representative bettor are estimated from observed payoffs at a greyhound racetrack. The results indicate that the representative bettor exhibits increasing absolute risk aversion and, given that the representative bettor is locally non-satiated with regard to wealth, exhibits preference for variance and aversion to positive skewness in the payoff distributions of the gambles examined.     

Ambiguous act equilibria

A game-theoretic framework that allows for explicitly randomized strategies is used to study the effect of ambiguity aversion on equilibrium outcomes. The notions of “independent strategies” as well as of “common priors” are amended to render them applicable to games in which players lack probabilistic sophistication. Within this framework the equilibrium predictions of two-player games with ambiguity-averse and with ambiguity-neutral players are observationally equivalent. This equivalence result does not extend to the case of games with more than two players. A translation of the concept of equilibrium in beliefs to the context of ambiguity aversion yields substantially different predictions – even for the case with just two players.     

An Experimental Study of Risk Aversion in Decision-making Under Uncertainty

This paper presents the findings of an experimental study of risk aversion in decision making under uncertainty. When presented with a series of gambles, subjects determined the certainty-equivalent wealth of each gamble. Risk aversion was measured by the Markowitz risk premium of the decision. The fixed effects regression model indicates the significant influence of the first three moments of a probability distribution in determining the risk premium. These results lend support to the rules of mean-variance and third-degree stochastic dominance. The extent of influence is also affected by the individual’s age, but not by gender, wealth or schooling.     

Risk, ambiguity, and state-preference theory

The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling ??smooth?? ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state dependence of utility and unobservable background risk. The decision maker??s local revealed beliefs are encoded in her risk-neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk-neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow?CPratt measure for approximating local risk premia, and its structure reveals whether the decision maker??s risk preferences are ambiguity averse as well as risk-averse. Two versions of the smooth ambiguity model are analyzed??the source-dependent risk aversion model and the second-order uncertainty (KMM) model??and it is shown that in both cases, the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium.     

“Probability of risk aversion” and other applications of derivatives of the certainty equivalent

“Probability of risk” aversion is principally concerned with reactions to scaling up of probabilities of non-zero values of a non-positive random variable by a common factor. Decreasing probability-of-risk aversion is defined and shown to be equivalent to ordinary risk aversion. Implications of this for insurance are pointed out. The sort of scaling involved is the same as that involved in “self-protection,” and it is shown that, for any expenditure on self-protection, say x, a concave utility function will prefer a coinsurance policy, costing x, which leaves probabilities unchanged, but scales down loss amounts by the same proportion as probabilities are scaled under self-protection. Properties of several comparative concepts of decreasing risk aversion are established. Derivatives of the certainty equivalent (CE) are used to elucidate well-known comparative static results in models of expected utility maximization. Finally, the study proves that concavity of the CE implies convexity of the coefficient of absolute risk aversion and examines the role of curvature of the CE in exploring relationships between properties of risk vulnerability, properness, and standardness.

Scientific Ambiguity and Climate Policy

Economic evaluation of climate policy traditionally treats uncertainty by appealing to expected utility theory. Yet our knowledge of the impacts of climate policy may not be of sufficient quality to be described by unique probabilistic beliefs. In such circumstances, it has been argued that the axioms of expected utility theory may not be the correct standard of rationality. By contrast, several axiomatic frameworks have recently been proposed that account for ambiguous knowledge. In this paper, we apply static and dynamic versions of a smooth ambiguity model to climate mitigation policy. We obtain a general result on the comparative statics of optimal abatement and ambiguity aversion, and then extend our analysis to a more realistic, dynamic setting, where we introduce scientific ambiguity into the well-known DICE model of the climate-economy system. For policy-relevant exogenous mitigation policies, we show that the value of emissions abatement increases as ambiguity aversion increases, and that this ‘ambiguity premium’ can in some plausible cases be very large. In these cases the effect of ambiguity aversion on welfare is comparable to that of other much studied welfare parameters. Thus ambiguity aversion may be an important neglected aspect of climate change economics, and seems likely to provide another argument for strong abatement policy.     

A brain imaging study of the choice procedure

We study the behavior of subjects facing choices between certain, risky, and ambiguous lotteries. Subjects' choices are consistent with the economic theories modeling ambiguity aversion. Our results support the conjecture that subjects face choice tasks as an estimation of the value of the lotteries, and that the difficulty of the choice is an important explanatory variable (in addition to risk and ambiguity aversion).The brain imaging data suggest that such estimation is of an approximate nature when the choices involve ambiguous and risky lotteries, as the regions in the brain that are activated are typically located in parietal lobes. Thus such choices require mental faculties that are shared by all mammals, and in particular are independent of language. In contrast, choices involving partial ambiguous lotteries additionally produce an activation of the frontal region, which indicates a different, more sophisticated cognitive process.     

Dragon Slaying with Ambiguity: Theory and Experiments

This paper studies the impact of ambiguity in the best‐shot and weakest‐link models of public good provision. The models are first analyzed theoretically. Then, we conduct experiments to study how ambiguity affects behavior in these games. We test whether subjects' perception of ambiguity differs between a local opponent and a foreign one. We find that an ambiguity‐safe strategy is often chosen by subjects. This is compatible with the hypothesis that ambiguity aversion influences behavior in games. Subjects tend to choose contributions above (respectively, below) the Nash equilibrium in the best‐shot (respectively, weakest‐link) model.     

The effect of ambiguity aversion on reward scheme choice

We test the implications of ambiguity aversion in a principal–agent problem with multiple agents. Models of ambiguity aversion suggest that, under ambiguity, comparative compensation schemes may become more attractive than independent wage contracts. We test this by presenting agents with a choice between comparative reward schemes and independent contracts, which are designed such that under uncertainty about output distributions (that is, under ambiguity), ambiguity averse agents should typically prefer comparative reward schemes, independent of their degree of risk aversion. We indeed find that the share of agents who choose the comparative scheme is higher under ambiguity.     

Ambiguity aversion and trade

What is the effect of ambiguity aversion on trade? Although in a Bewley??s model, ambiguity aversion always leads to less trade; in other models, this is not always true. However, we show that if the endowments are unambiguous, then more ambiguity aversion implies less trade for a very general class of preferences. The reduction in trade caused by ambiguity aversion can be as severe as to lead to no trade. In an economy with MEU decision makers, we show that if the aggregate endowment is unanimously unambiguous, then every Pareto optima allocation is also unambiguous. We also characterize the situation in which every unanimously unambiguous allocation is Pareto optimal. Finally, we show how our results can be used to explain the home-bias effect. As a useful result for our methods, we also obtain an additivity theorem for CEU and MEU decision makers that does not require comonotonicity.     

A SMOOTH AMBIGUITY MODEL OF THE COMPETITIVE FIRM

This paper examines the optimal production decision of the competitive firm under price uncertainty when the firm's preferences exhibit smooth ambiguity aversion. Ambiguity is modeled by a second‐order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the price risk. Ambiguity preferences are modeled by the (second‐order) expectation of a concave transformation of the (first‐order) expected utility of profit conditional on each plausible subjective distribution of the price risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity‐averse firm optimally produces less in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. In the case that the price risk is binary, we show that ambiguity and greater ambiguity aversion always adversely affect the firm's production decision.     

Optimal greenhouse gas emissions under various assessments of climate change ambiguity

The probabilities associated with global warming damage are likely to be continuously revised in the light of new information. Such revisions of probability are the defining characteristic of ambiguity, as opposed to risk. This paper examines how climate change ambiguity may affect optimal greenhouse gas emission strategies, via the decision maker's attitude towards anticipated changes of damage probabilities. Two conceptualizations of ambiguity are distinguished, according to the emphasis placed on the ambiguity of priors or on the ambiguity of news, respectively. It is shown that the way in which ambiguity is viewed and the attitude taken towards it have a substantial influence on the optimal emission trajectory.     

Security Returns and Tax Aversion Bias: Behavioral Responses to Tax Labels

This article studies behavioral responses to taxes in financial markets. It is motivated by recent puzzling empirical evidence of taxable municipal bond yields significantly exceeding the level expected relative to tax exempt bonds. A behavioral explanation is a tax aversion bias, the phenomenon that people perceive an additional burden associated with tax payments. We conduct market experiments on the trading of differently taxed and labeled securities. The data show an initial overvaluation of tax payments that diminishes when subjects gain experience. The tax deduction of expenses is valued more than an equivalent tax exemption of earnings. We find that the persistence of the tax aversion bias critically depends on the quality of feedback. This suggests that tax aversion predominantly occurs in one-time, unfamiliar financial decisions and to a lesser extent in repetitive choices.     

Utility of gambling II: risk,paradoxes, and data

We specialize our results on entropy-modified representations of event-based gambles to representations of probability-based gambles by assuming an implicit event structure underlying the probabilities, and adding assumptions linking the qualitative properties of the former and the latter. Under segregation and under duplex decomposition, we obtain numerical representations consisting of a linear weighted utility term plus a term corresponding to information-theoretical entropies. These representations accommodate the Allais paradox and most of the data due to Birnbaum and associates. A representation of mixed event-and probability-based gambles accommodates the Ellsberg paradox. We suggest possible extensions to handle the data not accommodated.