Underestimation of probabilities modifications: characterization and economic implications

The aim of this paper is to propose a behavioral characterization of individuals who underestimate probabilities modifications and to characterize this behavior in the standard preferences representation models under risk (expected utility, dual theory, rank dependent utility theory and MaxMin expected utility). Our main results are the following. Underreaction to probabilities modifications is in general independent from standard risk aversion and prudence. In models involving probabilities transformation functions, it is characterized by the slope of the probability transformation function. In the MaxMin expected utility model under risk, it is related to the weights of the maximal and minimal consequences in the preferences representation function. Considering a simple prevention decision, consisting in the reduction in the probability of a monetary loss, we show that individuals who underreact to probabilities modifications, invest less in prevention than individuals who objectively evaluate these modifications. Underreaction to probabilities modification is thus a possible explanation for low investment in prevention.     

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.     

Decision-foundations for properties of nonadditive measures: general state spaces or general outcome spaces

This paper characterizes properties of chance attitudes (nonadditive measures). It does so for decision under uncertainty (unknown probabilities), where it assumes Choquet expected utility, and for decision under risk (known probabilities), where it assumes rank-dependent utility. It analyzes chance attitude independently from utility. All preference conditions concern simple violations of the sure-thing principle. Earlier results along these lines assumed richness of both outcomes and events. This paper generalizes such results to general state spaces as in Schmeidler's model of Choquet expected utility, and to general outcome spaces as in Gilboa's model of Choquet expected utility.     

A Definition of Uncertainty Aversion

A definition of uncertainty or ambiguity aversion is proposed. It is argued that the definition is well-suited to modelling within the Savage (as opposed to Anscombe and Aumann) domain of acts. The defined property of uncertainty aversion has intuitive empirical content, behaves well in specific models of preference (multiple-priors and Choquet expected utility) and is tractable. Tractability is established through use of a novel notion of differentiability for utility functions, called eventwise differentiability.     

On the correspondence between multivariate risk aversion and risk aversion with state-dependent preferences

This paper establishes the correspondence between multivariate risk aversion and risk aversion with state-dependent preferences. It shows that the prerequisite for comparability of risk aversion in the multivariate case, namely, identical ordinal preferences on the commodity space, corresponds to identical, properly defined, reference sets in the case of state-dependent preferences. For comparable decision makers the condition that the utility function of one is a concave transformation of that of the other on the commodity space corresponds to the condition that the expected utility of one is a concave transformation of that of the other on the reference set.     

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.     

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.     

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.     

Cobb-Douglas preferences under uncertainty

This paper axiomatizes Cobb-Douglas preferences under uncertainty. First, we extend the original Trockel (Econ Lett 30:7–10, 1989)’s axiomatic foundation to a general state space framework based on the Strong Homotheticity Axiom, obtaining also the incomplete case a la Bewley (Decis Econ Financ 25:79–110, 2002). We show that this key axiom for the Cobb-Douglas expected utility specification is refuted by Ellsberg’s uncertainty aversion behavioral pattern. Our main result provides a set of meaningful axioms characterizing Cobb-Douglas min-expected utility preferences, an important class of uncertainty averse preferences for studying the consequences of ambiguity in finance and other fields. Finally, we present briefly how to obtain more general representations like the variational case.     

Second-order ambiguous beliefs

This paper axiomatizes models of second-order ambiguous beliefs in the original domain of preferences of Anscombe and Aumann (Ann Math Stat 34:199–205, 1963) by weakening the first-stage independence postulate. The models we propose include the second-order subjective expected utility (SOSEU) of Seo (Econometrica 77:1575–1605, 2009) as a particular case. We characterize the intersection of our models of second-order ambiguity with the canonical models of (first-order) ambiguity aversion and provide a further generalization of SOSEU by relaxing the completeness axiom.     

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.     

Ambiguity,uncertainty aversion and equilibrium welfare

Summary. In order to analyse the effect of ambiguity and uncertainty aversion on equilibrium welfare, a two period, pure exchange one good economy is considered. Agents are Choquet-expected-utility maximizers with same convex capacity and strictly concave utility index. It is proven that equilibrium is indeterminate whenever several probabilities in the core of the capacity minimize the expected value of aggregate endowment and not all agents have same expected endowments under those probabilities. It is further shown that small changes in aggregate endowment may have drastic welfare implications. A more general model is considered in the case of no aggregate uncertainty: agents have a set of priors and are uncertainty averse as modelled by Gilboa-Schmeidler [1989]. In the case of complete markets, it is shown that assets have a spread of equilibrium prices similar to the spread of no-arbitrage prices compatible with absence of arbitrage in markets with imperfections.Received: 2 June 2000, Revised: 27 March 2003, JEL Classification Numbers: D46, D59,D60, G12.I have benefited from conversations with L. Epstein, F. Magnien and J. M. Tallon.     

Ambiguity aversion among student subjects: the role of probability interval and emotional parameters

In the literature on decision-making under uncertainty, it has been shown that decision-makers tend to prefer taking gambles with known-risk probabilities (pure risk) over equivalent gambles with ambiguous probabilities. This article contributes to the ongoing discussion in the literature on cognitive and non-cognitive covariates of ambiguity aversion. Through a series of experiments, it finds that subjects are more ambiguity-averse to prospects with wide probability intervals than to an equivalent prospect with narrow intervals, and that subjects’ inherent trust, happiness and level of optimism affect the level of ambiguity aversion.     

Range convexity and ambiguity averse preferences

We show that range convexity of beliefs, a `technical' condition that appears naturally in axiomatizations of preferences in a Savage-like framework, imposes some unexpected restrictions when modelling ambiguity averse preferences. That is, when it is added to a mild condition, range convexity makes the preferences collapse to subjective expected utility as soon as they satisfy structural conditions that are typically used to characterize ambiguity aversion. Received: February 25, 2000; revised version: April 17, 2000     

A new approach to modeling decision-making under uncertainty

This paper presents two axiomatic models of decision making under uncertainty that avoid the use of a state space. The first is a subjective expected utility model with action-dependent subjective probabilities and effect-dependent preferences (the case of effect-independent preferences is obtained as a special instance). The second is a nonexpected utility model involving well-defined families of action-dependent subjective probabilities on effects and a utility representation that is not necessarily linear in these probabilities (a probabilistic sophistication version of this model, with action-dependent subjective probabilities is obtained as a special case). The hospitality of EUREQua, University of Paris 1, and financial support by the National Science Foundation grant SES-0314249 are gratefully acknowledged.     

Ambiguity and the Value of Information: An Almost-objective Events Analysis

Using the concept of almost-objective acts, due to Machina, this paper shows that ambiguity aversion may be defined in terms of the value of information. In particular, for expected utility preferences, the value of information with respect to almost-objective acts is asymptotically equal to zero. I thank Simon Grant, Mark Machina, and an anonymous referee for helpful comments.     

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.     

Efficient allocations under ambiguity

Important implications of the expected utility hypothesis and risk aversion are that if agents have the same probability belief, then consumption plans in every efficient allocation of resources under uncertainty are comonotone with the aggregate endowment, and if their beliefs are concordant, then the consumption plans are measurable with respect to the aggregate endowment. We study these two properties of efficient allocations for models of preferences that exhibit ambiguity aversion using the concept of conditional beliefs, which we introduce in this paper. We provide characterizations of such conditional beliefs for the standard models of preferences used in applications.     

Subjective probabilities on “small” domains

The Savagian choice-theoretic construction of subjective probability does not apply to preferences, like those in the Ellsberg Paradox, that reflect a distinction between risk and ambiguity. We formulate two representation results—one for expected utility, the other for probabilistic sophistication—that derive subjective probabilities but only on a “small” domain of risky events. Risky events can be either specified exogenously or in terms of choice behavior; in the latter case, both the values and the domain of probability are subjective. The analysis identifies a mathematical structure—called a mosaic—that is intuitive for both exogenous and behavioral specifications of risky events. This structure is weaker than an algebra or even a λ-system.     

Trembles in extensive games with ambiguity averse players

We introduce and analyze three definitions of equilibrium for finite extensive games with imperfect information and ambiguity averse players. In a setting where players’ preferences are represented by maxmin expected utility, as characterized in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), our definitions capture the intuition that players may consider the possibility of slight arbitrary mistakes. This generalizes the idea leading to trembling-hand perfect equilibrium as introduced in Selten (Int J Game Theory 4(1):25–55, 1975), by allowing for ambiguous trembles characterized by sets of distributions. We prove existence for two of our equilibrium notions and relate our definitions to standard equilibrium concepts with expected utility maximizing players. Our analysis shows that ambiguity aversion can lead to behavioral implications that are distinct from those attained under expected utility maximization, even if ambiguous beliefs only arise from the possibility of slight mistakes in the implementation of unambiguous strategies.