A mechanism for eliciting the mean and quantiles of a random variable

We present two Becker–DeGroot–Marschak type incentive compatible elicitation mechanisms. The first can be used to elicit an agent’s belief about the mean of a random variable while the second elicits the quantiles.     

An index of loss aversion

To a considerable extent, risk aversion as it is commonly observed is caused by loss aversion. Several indexes of loss aversion have been proposed in the literature. The one proposed in this paper leads to a clear decomposition of risk attitude into three distinct components: basic utility, probability weighting, and loss aversion. The index is independent of the unit of payment. The main theorem shows how the indexes of different decision makers can be compared through observed choices.     

When is there state independence?

Whether a preference relation can be represented using state-independent utilities as opposed to state-dependent utilities may depend on which acts count as constant acts. This observation underlies an extension of Savage's expected utility theory to the state-dependent case that was proposed in this journal by Edi Karni. His result contains a condition requiring the existence of a set of acts which can play the role of constant acts and support a representation involving a state-independent utility function. This paper contains necessary and sufficient conditions on the preference relation for such a set of acts to exist. Results are obtained both for the Savage and the Anscombe and Aumann frameworks. Among the corollaries are representation theorems for state-dependent utilities. Relationships to Karni's work and extensions of the results are discussed.     

Agreeable bets with multiple priors

This paper considers a two-agent model of trade with multiple priors. Firstly, we characterize the existence of an agreeable bet on some event in terms of the set of priors. It is then shown that the existence of an agreeable bet on some event is a strictly stronger condition than the existence of an agreeable trade, whereas the two conditions are equivalent in the standard Bayesian framework. Secondly, we show that the two conditions are equivalent when the set of priors is the core of a convex capacity. These results are also related to the no trade theorems under asymmetric information.     

Negative recency, randomization device choice, and reduction of compound lotteries

We report an experiment in which subjects are not indifferent between real-money lotteries implemented with randomization devices that are equivalent under the Reduction Axiom. Instead, choice behavior is consistent with subjective distortion of conditional probability, and this persists in treatment conditions that control for (i) computational limitations and (ii) possible confounding by ratio bias.     

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.     

On p-rationalizability and approximate common certainty of rationality

This paper shows that common p-belief of rationality implies p-rationalizability for games with compact strategy sets. We also establish the Bayesian foundation for the perfect p-rationalizability for finite games. The p-rationalizability is then used to analyze the robustness of rationalizable sets. For any game with compact strategy sets, we show that the rationalizable set is robust, i.e., the strategies characterized by common p-belief of rationality are close to the rationalizable set when p→1.     

Dynamic variational preferences

We introduce and axiomatize 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.     

Solidarity among the poor

We conduct a field experiment with low-income subjects in Dallas, Texas. We examine voluntary, informal risk sharing using a visual representation of the solidarity game developed for low-literacy populations. We find substantially more ‘fixed gift to loser’ behavior and less ‘egotistical’ behavior than in previous studies. Individuals who display ‘egotistical’ behavior are more risk tolerant. The amount of the conditional gifts is positively related to age, income, and connection to the community. However, trust and empathy, which are commonly discussed as drivers for solidarity, are not significantly related to the amount given.     

A note on Kuhn’s Theorem with ambiguity averse players

Kuhn’s Theorem shows that extensive games with perfect recall can equivalently be analyzed using mixed or behavioral strategies, as long as players are expected utility maximizers. This note constructs an example that illustrates the limits of Kuhn’s Theorem in an environment with ambiguity averse players who use a maxmin decision rule and full Bayesian updating.     

Unintended hedging in ambiguity experiments

We describe an ambiguity hedging problem in Ellsberg experiments, where combinations of individually ambiguous bets eliminate aggregate ambiguity, and which may yield incorrect classifications of ambiguity averse subjects. We propose a new classification consistent with this hedging possibility.     

Preemption in a real option game with a first mover advantage and player-specific uncertainty

In this paper a two-player real option game with a first-mover advantage is analyzed, where payoffs are driven by a player-specific stochastic state variable. It is shown that there exists an equilibrium which has qualitatively different properties from those in standard real option games driven by common stochastic shocks. The properties of the equilibrium are four-fold: (i) preemption does not necessarily occur, (ii) if preemption takes place, the rent-equalization property holds, (iii) for almost all sample paths it is clear ex-ante which player invests first, and (iv) it is possible that both players invest simultaneously, even if that is not optimal. It is argued from simulations that real option games with a common one-dimensional shock do not provide a good approximation for games with player-specific uncertainty, even if these are highly correlated.     

Efficient assignment respecting priorities

A widespread practice in assignment of heterogeneous indivisible objects is to prioritize some recipients over others depending on the type of the object. Leading examples include assignment of public school seats, and allocation of houses, courses, or offices. Each object comes with a coarse priority ranking over recipients. Respecting such priorities constrains the set of feasible assignments, and therefore might lead to inefficiency, highlighting a tension between respecting priorities and Pareto efficiency. Via an easily verifiable criterion, we fully characterize priority structures under which the constrained efficient assignments do not suffer from such welfare loss, and the constrained efficient rule (CER) is indeed efficient. We also identify the priority structures for which the CER is singleton-valued and group strategy-proof.     

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.     

Risk taking by agents: The role of ex-ante and ex-post accountability

We study the role of accountability in situations where an agent makes risky decisions for a principal. We observe that in the absence of accountability, agents choose less risk averse investments for the principal than investors who invest for their own account. Accountability mitigates the observed decrease in risk aversion. Differences are observed between situations where agents are accountable for their decision (“ex-ante”) and where they are accountable for the outcome (“ex-post”).     

A simple preference foundation of cumulative prospect theory with power utility

Most empirical studies of rank-dependent utility and cumulative prospect theory have assumed power utility functions, both for gains and for losses. As it turns out, a remarkably simple preference foundation is possible for such models: Tail independence (a weakening of comonotonic independence which underlies all rank-dependent models) together with constant proportional risk aversion suffice, in the presence of common assumptions (weak ordering, continuity, and first stochastic dominance), to imply these models. Thus, sign dependence, the different treatment of gains and losses, and the separation of decision weights and utility are obtained free of charge.     

Stable schedule matching under revealed preference

Baiou and Balinski (Math. Oper. Res., 27 (2002) 485) studied schedule matching where one determines the partnerships that form and how much time they spend together, under the assumption that each agent has a ranking on all potential partners. Here we study schedule matching under more general preferences that extend the substitutable preferences in Roth (Econometrica 52 (1984) 47) by an extension of the revealed preference approach in Alkan (Econom. Theory 19 (2002) 737). We give a generalization of the Gale-Shapley algorithm and show that some familiar properties of ordinary stable matchings continue to hold. Our main result is that, when preferences satisfy an additional property called size monotonicity, stable matchings are a lattice under the joint preferences of all agents on each side and have other interesting structural properties.     

Risk taking under heterogenous revenue sharing

We examine the impact of informal risk sharing on risk taking incentives when transfers are organized through a social network. A bilateral partial sharing rule satisfies that neighbors share equally a part of their revenue. In such a society, correlated technologies generate interdependent risk levels. We obtain three findings. First, there is a unique and interior Nash-equilibrium risk profile, and it is in general differentiated and related to the Bonacich measure of the risk sharing network. Second, more revenue sharing enhances risk taking on average, although some agents may lower their risk level. Last, we find that under investment might often be observed.