Title: Risk, Ambiguity and the Savage Axioms

Reading List

Author: ELLSBERG, D.

Source: The Quarterly Journal of Economics, Vol. 75, 643–669, 1961

Authors: ANSCOMBE, F. J., AND R. J. AUMANN

Title: A Definition of Subjective Probability

Source: The Annals of Mathematical Statistics, 34, 199–205, 1963

Authors: Gilboa I. and Schmeidler D.

Title: Max min Expected Utility with non unique prior

Source: Journal of Mathematical Economics Vol. 18, pp.141-153, 1989

Abstract: Acts are functions from states of nature into finite-support distributions over a set of ‘deterministic outcomes’. We characterize preference relations over acts which have a numerical representation by the functional J(f)=min>{∫u f dP¦PεC} where f is an act, u is a von Neumann- Morgenstern utility over outcomes, and C is a closed and convex set of finitely additive probability measures on the states of nature. In addition to the usual assumptions on the preference relation as transitivity, completeness, continuity and monotonicity, we assume uncertainty aversion and certainty-independence. The last condition is a new one and is a weakening of the classical independence axiom: It requires that an act f is preferred to an act g if and only if the mixture of f and any constant act h is preferred to the same mixture of g and h. If non-degeneracy of the preference relation is also assumed, the convex set of priors C is uniquely determined. Finally, a concept of independence in case of a non-unique prior is introduced.

Authors: CAMERER, C; WEBER, M

Title: RECENT DEVELOPMENTS IN MODELING PREFERENCES - UNCERTAINTY AND AMBIGUITY

Source: JOURNAL OF RISK AND UNCERTAINTY, 5 (4): 325-370 OCT 1992

Author Keywords: AMBIGUITY; UNCERTAINTY; ELLSBERG PARADOX; NONEXPECTED UTILITY

Abstract: 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)

1 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.

Author: MACHINA, M. J., AND D. SCHMEIDLER

Title: A More Robust Definition of Subjective Probability

Source: Econometrica, 60, 745–780, 1992

Authors: CASADESUS-MASANELL, R., P. KLIBANOFF, AND E. OZDENOREN

Title: Maxmin Expected Utility over Savage Acts with a Set of Priors

Source: Journal of Economic Theory, 92, 35–65, 2000

Author Keywords: uncertainty aversion; ambiguity; expected utility; set of priors; Knightian uncertainty

Abstract: This paper provides an axiomatic foundation for a maxmin expected utility over a set of priors (MMEU) decision rule in an environment where the elements of choice are Savage acts. This characterization complements the original axiomatization of MMEU developed in a lottery-acts (or Anscombe–Aumann) framework by I. Gilboa and D. Schmeidler (1989, J. Math. Econ.18, 141–153). MMEU preferences are of interest primarily because they provide a natural and tractable way of modeling decision makers who display an aversion to uncertainty or ambiguity. The novel axioms are formulated using standard sequence techniques, which allow cardinal properties of utility to be expressed directly through preferences.

Authors: Eichberger, J; Kelsey, D

Title: Strategic complements, substitutes, and ambiguity: The implications for public goods

Source: JOURNAL OF ECONOMIC THEORY, 106 (2): 436-466 OCT 2002

Author Keywords: ambiguity; strategic complements; public goods; Choquet integral; free rider

Abstract: We examine the effect of ambiguity in symmetric games with aggregate externalities. We find that ambiguity will increase/decrease the equilibrium strategy in games with strategic complements/substitutes and positive externalities. These effects are reversed in games with negative externalities. We consider some economic applications of these results to Cournot oligopoly, bargaining, macroeconomic coordination, and voluntary donations to a public good. In particular we show that ambiguity may reduce free-riding. Comparative statics analysis shows that increases in uncertainty will increase donations, to a public good.

2 Authors: Mukerji S.; Tallon J.-M.

Title: Ellsberg's two-color experiment, portfolio inertia and ambiguity

Source: JOURNAL OF MATHEMATICAL ECONOMICS, 39 (3-4): 299-315 JUN 2003

Author Keywords: Ellsberg paradox; portfolio inertia; testing for ambiguity aversion; uncertainty aversion; unforeseen contingencies; subjective state space

Abstract: Results in this paper relate the observation of an interval of prices at which a decision maker (DM) strictly prefers to hold a zero position on an asset (termed "portfolio inertia") to the DM's perception of the underlying payoff relevant events as ambiguous, as the term is defined in [Econometrica 69 (2001) 265]. The connection between portfolio inertia and ambiguity is established without invoking a parametric preference form, such as the Choquet expected utility or the max-min multiple priors model. This allows us to draw an observable distinction between portfolio inertia that may arise purely due to first-order risk aversion type effects, such as those which could arise even if preferences were probabilistically sophisticated, and portfolio inertia that involves ambiguity perceptions.

Authors: Ghirardato, P; Maccheroni, F; Marinacci, M

Title: Differentiating ambiguity and ambiguity attitude

Source: JOURNAL OF ECONOMIC THEORY, 118 (2): 133-173 OCT 2004

Author Keywords: ambiguity; unambiguous preference; Clarke differentials; alpha-maxmin expected utility

Abstract: The objective of this paper is to show how ambiguity, and a decision maker (DM)'s response to it, can be modelled formally in the context of a general decision model. We introduce a relation derived from the DM's preferences, called "unambiguous preference", and show that it can be represented by a set of probabilities. We provide such set with a simple differential characterization, and argue that it is a behavioral representation of the "ambiguity" that the DM may perceive. Given such revealed ambiguity, we provide a representation of ambiguity attitudes. We also characterize axiomatically a special case of our decision model, the "alpha-maxmin" expected utility model.

Authors: Eichberger, J; Grant, S; Kelsey, D

Title: CEU preferences and dynamic consistency

Source: MATHEMATICAL SOCIAL SCIENCES, 49 (2): 143-151 MAR 2005

Author Keywords: ambiguity; Choquet integral; capacity; dynamic consistency

3 Abstract: This paper investigates the dynamic consistency of Choquet Expected Utility (CEU) preferences. A decision-maker is faced with an information structure represented by a fixed filtration. If beliefs are represented by a convex capacity, we show that a necessary and sufficient condition for dynamic consistency is that beliefs be additive over the final stage in the filtration.

Authors: Ghirardato, P; Maccheroni, F; Marinacci, M

Title: Certainty independence and the separation of utility and beliefs

Source: JOURNAL OF ECONOMIC THEORY, 120 (1): 129-134 JAN 2005

Author Keywords: certainty independence; separable representations; invariant biseparable preferences

Abstract: Economists often operate under an implicit assumption that the tastes of a decision maker are quite stable, while his beliefs change with the availability of new information. We show that for a general class of preferences, a separation of a key component of tastes, the utility function, from the other components of the representation is possible only if the decision maker's preferences satisfy a mild but not completely innocuous condition, called 'certainty independence'. We also outline the axiomatic characterization of the preferences that obtain such separation, which are a subset of the biseparable preferences.

Authors: Klibanoff, P; Marinacci, M; Mukerji, S

Title: A smooth model of decision making under ambiguity

Source: ECONOMETRICA, 73 (6): 1849-1892 NOV 2005

Author Keywords: ambiguity; uncertainty; Knightian uncertainty; ambiguity aversion; uncertainty aversion; Ellsberg paradox; ambiguity attitude

Abstract: We propose and characterize a model of preferences over acts such that the decision maker prefers act f to act g if and only if E(mu)phi (E(pi)u circle f) >= E(mu)phi (E(pi)u circle g), where E is the expectation operator, u is a von Neumann-Morgenstern utility function, phi is an increasing transformation, and mu is a subjective probability over the set Pi of probability measures pi that the decision maker thinks are relevant given his subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective beliefs, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes toward pure risk are characterized by the shape of u, as usual, while attitudes toward ambiguity are characterized by the shape of phi. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures Pi. One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's paradox). The maxmin expected utility model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as a limiting case of our model with infinite ambiguity aversion. Two illustrative portfolio choice examples are offered.

4 Authors: Maccheroni, F; Marinacci, M; Rustichini, A

Title: Dynamic variational preferences

Source: JOURNAL OF ECONOMIC THEORY, 128 (1): 4-44 MAY 2006

Author Keywords: ambiguity aversion; model uncertainty; recursive utility; robust control; time consistency

Abstract: We introduce and aximatize dynamic variational preferences, the dynamic version of the variational preferences we axiomatized in [F. Maccheroni, M. Marinacci, A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Mimeo, 2004], which generalize the multiple priors preferences of Gilboa and Schmeidler [Maxmin expected utility with a non-unique prior, J. Math. Econ. 18 (1989) 141-153], and include the Multiplier Preferences inspired by robust control and first used in macroeconomics by Hansen and Sargent (see [L.P. Hansen, T.J. Sargent, Robust control and model uncertainty, Amer. Econ. Rev. 91 (2001) 60-66]), as well as the classic Mean Variance Preferences of Markovitz and Tobin. We provide a condition that makes dynamic variational preferences time consistent, and their representation recursive. This gives them the analytical tractability needed in macroeconomic and financial applications. A corollary of our results is that Multiplier Preferences are time consistent, but Mean Variance Preferences are not.

Authors: Halevy, Y

Title: Ellsberg revisited: An experimental study

Source: ECONOMETRICA, 75 (2): 503-536 MAR 2007

Author Keywords: uncertainty aversion; probabilistic sophistication; reduction of compound lotteries; nonexpected utility; maxmin expected utility; anticipated utility; rank dependent utility; recursive utility; compound independence; bundling; rule rationality

Abstract: An extension to Ellsberg's experiment demonstrates that attitudes to ambiguity and compound objective lotteries are tightly associated. The sample is decomposed into three main groups: subjective expected utility subjects, who reduce compound objective lotteries and are ambiguity neutral, and two groups that exhibit different forms of association between preferences over compound lotteries and ambiguity, corresponding to alternative theoretical models that account for ambiguity averse or seeking behavior.

Author: Ahn, DS

Title: Ambiguity without a state space

5 Source: REVIEW OF ECONOMIC STUDIES, 75 (1): 3-28 JAN 2008

Abstract: 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

Author: Faro, JH

Title: Variational Bewley Preferences

Source: http://www.kellogg.northwestern.edu/research/math/seminars/200809/Faro050809.pdf

Authors: Eichberger, J; Kelsey, D; Schipper, BC

Title: Granny versus game theorist: Ambiguity in experimental games

Source: THEORY AND DECISION, 64 (2-3): 333-362 MAR 2008

Author Keywords: Choquet expected utility; equilibrium under ambiguity; experiments; Knightian uncertainty; strategic uncertainty

Abstract: We report on an experiment in which subjects choose actions in strategic games with either strategic complements or substitutes against a granny, a game theorist or other subjects. The games are selected in order to test predictions on the comparative statics of equilibrium with respect to changes in strategic ambiguity. We find that subjects face higher ambiguity while playing against the granny than playing against the game theorist if we assume that subjects are ambiguity averse. Moreover, under the same assumption, subjects choose more secure actions in games more prone to ambiguity which is in line with the predictions.

Authors: Gajdos, T; Hayashi, T; Tallon, JM; Vergnaud, JC

Title: Attitude toward imprecise information

Source: JOURNAL OF ECONOMIC THEORY, 140 (1): 27-65 MAY 2008

Author Keywords: imprecise information; imprecision aversion; multiple priors; steiner point

Abstract: This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. Information is assumed to take the form of a probability-possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P and is assumed to rank pairs of the form (P, f) where f is an act mapping states into outcomes. The key representation result delivers maxmin expected utility (MEU) where the min operator ranges over a set of probability priors-just as in the MEU representation result of Gilboa and Schmeidler [Maxmin expected utility with a non-unique 6 prior, J. Math. Econ. 18 (1989) 141-153]. However, unlike the MEU representation, the representation here also delivers a mapping, phi, which links the probability-possibility set, describing the available information, to the set of revealed priors. The mapping phi is shown to represent the decision maker's attitude to imprecise information: under our axioms, the set of representation priors is constituted as a selection from the probability-possibility set. This allows both expected utility when the selected set is a singleton and extreme pessimism when the selected set is the same as the probability-possibility set, i.e., phi is the identity mapping. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more specific functional form, in which the set of selected probability distributions is obtained by (i) solving for the "mean value" of the probability-possibility set, and (ii) shrinking the probability-possibility set toward the mean value to a degree determined by preferences.

Authors: Gilboa, I, Postlewaite, AW, Schmeidler, D

Title: Probability and uncertainty in economic modeling

Source: JOURNAL OF ECONOMIC PERSPECTIVES, 22 (3): 173-188 SUM 2008

Author: Siniscalchi M.

Title: Vector Expected Utility and Attitudes Toward Variation

Source: Econometrica, Vol. 77, Issue 3, pp. 801-855, 2009

Author Keywords: Ambiguity, attitudes toward variability, reference prior

Abstract: This paper proposes a model of decision under ambiguity deemed vector expected utility, or VEU. In this model, an uncertain prospect, or Savage act, is assessed according to (a) a baseline expected-utility evaluation, and (b) an adjustment that reflects the individual's perception of ambiguity and her attitudes toward it. The adjustment is itself a function of the act's exposure to distinct sources of ambiguity, as well as its variability. The key elements of the VEU model are a baseline probability and a collection of random variables, or adjustment factors, which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events. The adjustment to the baseline expected-utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor, which reflects exposure to the corresponding ambiguity source. A behavioral characterization of the VEU model is provided. Furthermore, an updating rule for VEU preferences is proposed and characterized. The suggested updating rule facilitates the analysis of sophisticated dynamic choice with VEU preferences.

LEARNING UNDER AMBIGUITY 7 Authors: Epstein, LG; Schneider, M

Title: Recursive multiple-priors

Source: JOURNAL OF ECONOMIC THEORY, 113 (1): 1-31 NOV 2003

Author Keywords: ambiguity; multiple-priors; dynamic consistency; Ellsberg paradox; robust control; updating beliefs; recursive utility

Abstract: This paper axiomatizes an intertemporal version of multiple-priors utility. A central axiom is dynamic consistency, which leads to a recursive structure for utility, to 'rectangular' sets of priors and to prior-by-prior Bayesian updating as the updating rule for such sets of priors. It is argued that dynamic consistency is intuitive in a wide range of situations and that the model is consistent with a rich set of possibilities for dynamic behavior under ambiguity.

Author: Wang, T

Title: Conditional preferences and updating

Source: JOURNAL OF ECONOMIC THEORY, 108 (2): 286-321 FEB 2003

Author Keywords: updating rules; learning; uncertainty; ambiguity

Abstract: This paper axiomatizes updating rules for preferences that are not necessarily in the expected utility class. Two sets of results are presented. The first is the axiomatization of conditional preferences. The second consists of the axiomatization of three updating rules: the traditional Bayes rule, the Dempster-Shafer rule, and the generalized Bayes rule. The last rule can be regarded as the updating rule for the multi-prior expected utility (Gilboa and Schmeidler, J. Math. Econom. 18 (1989) 141). Operationally, it is equivalent to updating each prior by the traditional Bayes rule.

Authors: Eichberger, J; Grant, S; Kelsey, D

Title: Updating choquet beliefs

Source: JOURNAL OF MATHEMATICAL ECONOMICS, 43 (7-8): 888-899 SEP 2007

Author Keywords: updating ambiguous beliefs; full Bayesian updating; choquet expected utility

Abstract: We apply Pires's coherence property between unconditional and conditional preferences that admit a CEU representation. In conjunction with consequential ism (only those outcomes on states which are still possible can matter for conditional preference) this implies that the conditional preference may be obtained from the unconditional preference by taking the Full Bayesian Update of the capacity.

8 Authors: Epstein, LG; Schneider

Title: Learning under ambiguity

Source: REVIEW OF ECONOMIC STUDIES, 74 (4): 1275-1303 OCT 2007

Abstract: This paper considers learning when the distinction between risk and ambiguity matters. It first describes thought experiments, dynamic variants of those provided by Ellsberg, that highlight a sense in which the Bayesian learning model is extreme-it models agents who are implausibly ambitious about what they can learn in complicated environments. The paper then provides a generalization of the Bayesian model that accommodates the intuitive choices in the thought experiments. In particular, the model allows decision-makers' confidence about the environment to change-along with beliefs-as they learn. A portfolio choice application compares the effect of changes in confidence under ambiguity vs. changes in estimation risk under Bayesian learning. The former is shown to induce a trend towards more stock market participation and investment even when the latter does not.