Definitions of ambiguous events and the smooth ambiguity model

We examine a variety of preference-based definitions of ambiguous events in the context of the smooth ambiguity model. We first consider the definition proposed in Klibanoff et?al. (Econometrica 73(6):1849?C1892, 2005) based on the classic Ellsberg two-urn paradox (Ellsberg Q J Econ 75:643?C669, 1961) and show that it satisfies several desirable properties. We then compare this definition with those of Nehring (Math Soc Sci 38(2):197?C213, 1999), Epstein and Zhang (Econometrica 69:265?C306, 2001), Zhang (Econ Theory 20:159?C181, 2002), and Ghirardato and Marinacci (J Econ Theory 102:251?C289, 2002). Within the smooth ambiguity model, we show that Ghirardato and Marinacci (J Econ Theory 102:251?C289, 2002) would identify the same set of ambiguous and unambiguous events as our definition while Epstein and Zhang (Econometrica 69:265?C306, 2001) and Zhang (Econ Theory 20:159?C181, 2002) would yield a different classification. Moreover, we discuss and formally identify two key sources of the differences compared to Epstein and Zhang (Econometrica 69:265?C306, 2001) and Zhang (Econ Theory 20:159?C181, 2002). The more interesting source is that these two definitions can confound non-constant ambiguity attitude and the ambiguity of an event.     

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.     

Unambiguous events and dynamic Choquet preferences

This paper explores the relationship between dynamic consistency and existing notions of unambiguous events for Choquet expected utility preferences. A decision maker is faced with an information structure represented by a filtration. We show that the decision maker’s preferences respect dynamic consistency on a fixed filtration if and only if the last stage of the filtration is composed of unambiguous events in the sense of Nehring (Math Social Sci 38:197–213, 1999). Adopting two axioms, conditional certainty equivalence consistency and constrained dynamic consistency to filtration measurable acts, it is shown that the decision maker respects these two axioms on a fixed filtration if and only if the last stage of the filtration is made up of unambiguous events in the sense of Zhang (Econ Theory 20:159–181, 2002).     

A note on recursive multiple-priors

This note fills a gap in the proof of recursive multiple-priors utility [L. Epstein, M. Schneider, Recursive multiple priors, J. Econ. Theory 113 (2003) 1-31].     

Differentiating ambiguity: an expository note

Ghirardato et al. J Econ Theory 118:133–173, 2004 propose a method for distinguishing between perceived ambiguity and the decision-maker’s reaction to it. They study a general class of preferences which they refer to as invariant biseparable. This class includes CEU and MEU. This note presents some examples which illustrate their results. Research supported by ESRC grant no. RES-000-22-0650. For comments and discussion we would like to thank Klaus Nehring and a seminar audience at ESEM, Vienna 2006.     

Hierarchies of ambiguous beliefs

We present a theory of interactive beliefs analogous to Mertens and Zamir [Formulation of Bayesian analysis for games with incomplete information, Int. J. Game Theory 14 (1985) 1-29] and Brandenburger and Dekel [Hierarchies of beliefs and common knowledge, J. Econ. Theory 59 (1993) 189-198] that allows for hierarchies of ambiguity. Each agent is allowed a compact set of beliefs at each level, rather than just a single belief as in the standard model. We propose appropriate definitions of coherency and common knowledge for our types. Common knowledge of coherency closes the model, in the sense that each type homeomorphically encodes a compact set of beliefs over the others’ types. This space universally embeds every implicit type space of ambiguous beliefs in a beliefs-preserving manner. An extension to ambiguous conditional probability systems [P. Battigalli, M. Siniscalchi, Hierarchies of conditional beliefs and interactive epistemology in dynamic games, J. Econ. Theory 88 (1999) 188-230] is presented. The standard universal type space and the universal space of compact continuous possibility structures are epistemically identified as subsets.     

Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents

We provide several different generalizations of Debreu’s social equilibrium theorem by allowing for asymmetric information and a continuum of agents. The results not only extend the ones in Kim and Yannelis (J Econ Theory 77:330–353, 1977), Yannelis and Rustichini (Stud Econ Theory 2:23–48, 1991), but also new theorems are obtained which allow for a convexifying effect on aggregation (non-concavity assumption on the utility functions) and non-convex strategy sets (pure strategies). This is achieved by imposing the assumption of “many more agents than strategies” (Rustichini and Yannelis in Stud Econ Theory 1:249–265, 1991; Tourky and Yannelis in J Econ Theory 101:189–221, 2001; Podczeck in Econ Theory 22:699–725, 2003). To the memory of Gerard Debreu. A preliminary draft was presented in Paris, in April of 2005. I have benefited from the discussion, comments and questions of Erik Balder, Jean-Marc Bonnisseu, Bernard Cornet and Filipe Martins Da-Rocha and Conny Podczeck. A careful and knowledgeable referee made several useful comments and rescued me from a mishap.     

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.     

Rational preferences under ambiguity

This paper analyzes preferences in the presence of ambiguity that are rational in the sense of satisfying the classical ordering condition as well as monotonicity. Under technical conditions that are natural in an Anscombe?CAumann environment, we show that even for such a general preference model, it is possible to identify a set of priors, as first envisioned by Ellsberg (Q J Econ 75:643?C669, 1961). We then discuss ambiguity attitudes, as well as unambiguous acts and events, for the class of rational preferences we consider.     

Extreme inequality aversion without separability

Hammond (J Econ Theory 11, 465–467, 1975), Meyer (J Econ Theory 11, 119–132, 1975), and Lambert (The distribution and redistribution of income Manchester University Press, Manchester, 2001) provide the formal result connecting leximin and the idea of extreme inequality aversion for social preferences of the expected utility type. Using an analogous approach, we show that for social preferences not necessarily satisfying the separability axiom that underlies expected utility theory, the case of extreme inequality aversion is covered by the class of weakly maximin social preferences—i.e., the class of social preferences that give priority to the worst off in all cases in which the worst off is not indifferent. I wish to thank Bart Capéau, Frank Cowell, Peter Lambert, Luc Lauwers, Erik Schokkaert, Frans Spinnewyn, and Bertil Tungodden for valuable comments. Remaining shortcomings are mine. Financial support from the Fund for Scientific Research - Flanders (grant G.0005.04) and the Interuniversity Attraction Poles network funded by the Federal Public Planning Service, Belgian Science Policy (grant P5/21-A) is gratefully acknowledged.     

Preference aggregation under uncertainty: Savage vs. Pareto

Following Mongin [J. Econ. Theory 66 (1995) 313; J. Math. Econ. 29 (1998) 331], we study social aggregation of subjective expected utility preferences in a Savage framework. We argue that each of Savage's P3 and P4 are incompatible with the strong Pareto property. A representation theorem for social preferences satisfying Pareto indifference and conforming to the state-dependent expected utility model is provided.     

Ambiguity attitude, R&D investments and economic growth

The process aimed at discovering new ideas is an economic activity the returns from which are intrinsically uncertain. We extend the neo-Schumpeterian growth framework to investigate the role of strong uncertainty in the innovative process. In particular, we postulate that, when deciding upon R&D efforts, investors hold ‘ambiguous beliefs’ about the exact probability of arrival of the next vertical innovation, and that they face ambiguity via the α −MEU decision rule (Ghirardato et al., J Econ Theory 118:133–173, 2004). Along the balanced growth path, the higher the agent’s ambiguity aversion (α), the lower the R&D efforts and the economic performance. Consistent with cross-country empirical evidence, this causal mechanism suggests that, together with the profitability conditions of the economy, different ‘cultural’ attitudes towards ambiguity may help explain the different R&D efforts observed across countries.     

Are there cycles in realistic overlapping cohorts models?

Following Gale, 1973 (J. Econ. Theory 4, 110–137), Benhabib and Day, 1982 (J. Econ. Dyn. Control 4, 37–55) and Grandmont, 1985 (Econmetrica 45, 995–1045) proved the existence of P-cycles in Overlapping Generations (OLG) models. Aiyagari, 1989 (Q. J. Econ 104, 163–185) considered Overlapping Cohorts (OLC) models, with very long-lived agents of positively discounted utility functions and proved nonexistence of 2-cycles. We study OLC models with agents of normal life spans and utilities and show that: (1) for constant relative risk aversion (CRRA) utility (except for Leontief case), no 2-cycle exists; and (2) for Cobb-Douglas utility, no P-cycle exists.     

A belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring

This paper extends the belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring. We first find that the previous belief-based techniques [T. Sekiguchi, Efficiency in repeated prisoners' dilemma with private monitoring, J. Econ. Theory 76 (1997) 345-361; V. Bhaskar, I. Obara, Belief-based equilibria in the repeated prisoners' dilemma with private monitoring, J. Econ. Theory 102 (2002) 40-69] cannot succeed when players' private monitoring technologies are sufficiently different. We then modify the previous belief-based approach by letting the player with smaller observation errors always randomize between cooperate and defect along the cooperative path of the play. We show that with vanishing observation errors, efficiency and a folk theorem can be approximated using our modified belief-based strategies.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Correlated Nash equilibrium

We modify the epistemic conditions for Nash equilibrium only to accommodate Gilboa and Schmeidler's [I. Gilboa, D. Schmeidler, Maxmin expected utility with nonunique prior, J. Math. Econ. 18 (1989) 141-153] maxmin expected utility preferences, and identify the equilibrium concept in n-player strategic games that characterizes the modified epistemic conditions. The epistemic characterization supports the equilibrium concept as a minimal generalization of Nash equilibrium, in the sense that it deviates from Nash equilibrium only in terms of players' attitude towards ambiguity. Consequently, comparing it with Nash equilibrium constitutes a ceteris paribus study of the effects of ambiguity on how a game is played. For example, with ambiguity, (beliefs about) action choices are in general correlated.     

Efficiency and consistency for locating multiple public facilities

In the problem of locating multiple public facilities studied by Barberà and Beviá [Self-selection consistent functions, J. Econ. Theory 105 (2002) 263-277; Locating public facilities by majority: stability, consistency and group formation, Games Econ. Behav. 56 (2006) 185-200], we offer simple necessary and sufficient conditions for efficiency, decentralizability of efficient decisions in a game of community division and local public goods provision, and a constructive algorithm for efficient and consistent decisions.     

Corrigendum to “When is efficiency separable from distribution in the provision of club goods?” [Journal of Economic Theory 90 (2000) 204-221]

This note corrects a slip in Fraser's (J. Econ. Theory 90 (2000) 204-221) treatment of the necessary condition for partial separation of efficiency from distribution in providing a club good. His corrected Theorem 1 shows that many families of utility function can generate partial (and total) separation, thereby considerably extending the scope of his analysis.     

The Zilcha criteria for dynamic inefficiency reconsidered

We reconsider necessary and sufficient conditions for dynamic inefficiency given in Zilcha (J Econ Theory 52:364–379, 1990, J Econ Theory 55:1–16, 1991) and a critique by Rangazas and Russell (2005). First, we show that the characterization given in Zilcha (1990) for nonstationary economies is correct and correct Zilcha’s proof. Second, using this insight, we complement Rangazas and Russell’s (Econ Theory 26:701–716, 2005) discussion of the counterexamples to Zilcha (J Econ Theory 55:1–16, 1991). Third, we discuss consequences of our results for applied tests of (in-)efficiency based on the Zilcha criteria. We would like to thank Itzhak Zilcha, and in particular Peter Rangazas and Steve Russell for detailed and very helpful comments.     

Large economies with differential information and without free disposal

We consider exchange economies with a continuum of agents and differential information about finitely many states of nature. It was proved in Einy et al. (Econ Theory 18, 321–332, 2001) that if we allow for free disposal in the market clearing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equilibrium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incentive compatible and contracts may not be enforceable (see e.g. Glycopantis et al. in Econ Theory 21, 495–526, 2002). This is the main motivation for considering equilibrium solutions with exact feasibility. We first prove that the results in Einy et al. (Econ Theory 18, 321–332, 2001) are still valid without free-disposal. Then, motivated by the issue of contracts’ execution, we adapt the incentive compatibility property introduced in Krasa and Yannelis (Econometrica 62, 881–900, 1994) and we prove that every Pareto optimal exact feasible allocation is incentive compatible, implying that contracts of competitive or core allocations are enforceable. We would like to thank two anonymous reviewers and the Associate Editor for their valuable suggestions and remarks. This work was partially done while V.F. Martins-da-Rocha was visiting the Dipartimento di Matematica e Informatica of the Università degli Studi di Perugia. We thank the audience of the First General Equilibrium Workshop at Rio. Section 6 dealing with contract enforcement and coalitional incentive compatibility has benefited from discussions with J. Correia-da-Silva, W. Daher, F. Forges, C. Hervès-Beloso, E. Moreno-García, K. Podczeck, Y. Vailakis and N.C. Yannelis.