Limited observability as a constraint in contract design

Limited observability is the assumption that economic agents can only observe a finite amount of information. Given this constraint, contracts among agents are necessarily finite and incomplete in comparison to the ideal complete contract that we model as infinite in detail. We consider the extent that finite contracts can approximate a complete contract. The objectives of the paper are: (i) to identify properties of agents’ preferences that determine whether or not finiteness of contracts causes significant inefficiency; (ii) to evaluate the performance of finite contracts against the ideal optimal contract in a bilateral bargaining model.     

Formation of cartels in large markets

The von Neumann-Morgenstern concept of solution is applied to nonatomic pure exchange markets consisting of a finite number of different types of trader, initially owning disjoint sets of goods. It is proved that if all traders of each type form a cartel and behave like a single (atomic) trader (their “representative”), then one gets solutions of the original market from those of the finite market of the representatives, and all solutions are obtained in this manner—i.e., the market must organize itself in cartels, and this will indeed be stable.     

Simple Strategy-Proof Approximately Walrasian Mechanisms

This note provides an indirect analysis of the incentive properties of the Walrasian mechanism. It presents mechanisms under which truth-telling is a dominant strategy in finite exchange economies (in contrast to the Walrasian mechanism) and whose outcomes (generically) approximate Walrasian ones for large economies. These mechanisms provide new insights on the well-know trade-off between efficiency and incentive compatibility in finite economies. Journal of Economic Literature Classification Numbers: D50, D71, D78.     

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.     

Ethically flexible gini indices for income distributions in the continuum

A single-parameter generalization of the Gini coefficient (S-Gini) is presented for income distributions defined in the continuum. Special cases are Dorfman's formula for the Gini, and the authors' S-Ginis for finite populations. We exploit the duality between indices of relative inequality and homothetic social-evaluation functions to construct these new indices.     

Existence and fairness of value allocation without convex preferences

Shafer (Econometrica, 48 (1980), 467–476) proved that in a finite exchange economy value allocations exist, provided that each agent has convex, complete, transitive, compact and monotone preferences. However, if preferences are not convex, then value allocations may not exist. To remedy this difficulty we enlarge the set of value allocations by introducing the concept of approximate value allocations, and show that in a finite exchange economy approximate value allocations exist, even if preferences are not convex, or compact, or monotone. This value existence result can be used to provide a very general value existence theorem for a sequence of finite economies. Further, we show that value allocations may discriminate in favor of or against a coalition of agents.     

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.     

Median-based extensions of an ordering over a set to the power set: An axiomatic characterization

In this paper we consider the problem of inducing an ordering over the set of all non-empty subsets of a finite set X of alternatives, given an ordering R over X. Assuming R to be antisymmetric and X to have at least six elements, we provide a set of independent, necessary, and sufficient conditions for the induced ordering to be “median-based” (so that every non-empty subset of X is “indifferent” to its own median set defined in terms of R).     

Existence of perfect equilibria: a direct proof

We provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. It is done by constructing a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by a generalization of Kakutani theorem, which is proved in this paper. This work offers a new approach to perfect equilibria, which would hopefully facilitate further study on this topic. We also hope our direct proof would be the first step toward building an algorithm to find the set of all perfect equilibria of a strategic game.     

Maximum entropy regressions

Maximum entropy (ME) regression is compared to ordinary regression in the case of two observations on two normally distributed variables (one dependent and one explanatory) with correlation coefficient ρ. ME regressions have the smaller risk under quadratic loss if ? lies in the interval ±0.95. In the case of two explanatory variables and two observations, ordinary regression is not possible but ME regressions do exist and have finite risk.     

Asynchronous Choice and Markov Equilibria

We provide a theoretical foundation for the use of Markov strategies in repeated games with asynchronous moves. If admissible strategies must display finite (arbitrarily long) memory and each player incurs a “complexity cost” which depends on the memory length required by her strategy, then every Nash equilibrium must be in Markovian strategies. If, in addition, admissible strategies have uniformly bounded memory, every rationalizable strategy must be Markovian. These results are robust to considerations of perfection and also yield interesting implications for equilibrium selection in simple contexts. Journal of Economic Literature Classification Numbers: C72, C73.     

Characterizing the Common Prior Assumption

Logical characterizations of the common prior assumption (CPA) are investigated. Two approaches are considered. The first is called frame distinguishability, and is similar in spirit to the approaches considered in the economics literature. Results similar to those obtained in the economics literature are proved here as well, namely, that we can distinguish finite spaces that satisfy the CPA from those that do not in terms of disagreements in expectation. However, it is shown that, for the language used here, no formulas can distinguish infinite spaces satisfying the CPA from those that do not. The second approach considered is that of finding a sound and complete axiomatization. Such an axiomatization is provided; again, the key axiom involves disagreements in expectation. The same axiom system is shown to be sound and complete both in the finite and the infinite case. Thus, the two approaches to characterizing the CPA behave quite differently in the case of infinite spaces. Journal of Economic Literature Classification Numbers: C70, D80.     

Characterization of optimal plans for stochastic dynamic programs

This paper provides general techniques for the characterization of optimal plans resulting from stochastic dynamic programming. We show that under standard assumptions the optimal plans in both finite and infinite horizon problems can be obtained by an application of the Implicit Function Theorem to first order conditions. Further, we show that under certain checkable conditions, optimal plans and value functions are p-times differentiable for any integer p ? 0. Finally, we apply our technique to obtain a C^p plan and value function in a one sector infinite horizon growth problem under uncertainty.     

An encompassing test for non-nested quantile regression models

We propose an encompassing test for non-nested linear quantile regression models and show that it has an asymptotic χ² distribution. It is also shown that the proposed test is a regression rank score test in a comprehensive model under conditional homogeneity. Our simulation results indicate that the proposed test performs very well in finite samples.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

Self-control,revealed preference and consumption choice

We provide a time-consistent model that addresses the preference reversals that motivate the time-inconsistency literature. The model subsumes the behavior generated by the time-inconsistency approach in finite settings but, unlike the time-inconsistent models, allows for self-control. This paper provides a brief summary of theoretical results shown elsewhere [Gul and Pesendorfer, Econometrica 69 (2001) 1403; Econometrica, (2002), in press; Rev. Econ. Stud. (2002), in press] and contrasts the predictions and welfare implications of our model and the time-inconsistent β−δ model.     

Strategy-Proof Probabilistic Decision Schemes for One-Dimensional Single-Peaked Preferences

Collective decision problems are considered with a finite number of agents who have single-peaked preferences on the real line. A probabilistic decision scheme assigns a probability distribution over the real line to every profile of reported preferences. The main result of the paper is a characterization of the class of unanimous and strategy-proof probabilistic schemes with the aid of fixed probability distributions that play a role similar to that of the phantom voters in H. Moulin (Public Choice35 (1980), 437-455). Thereby, the work of Moulin (1980) is extended to the probabilistic framework. Journal of Economic Literature Classification Numbers: D71, D81.     

Moral Hazard and Private Monitoring

We clarify the role of mixed strategies and public randomization (sunspots) in sustaining near-efficient outcomes in repeated games with private monitoring. We study a finitely repeated game, where the stage game has multiple equilibria and show that mixed strategies can support partial cooperation, but cannot approximate full cooperation even if monitoring is “almost perfect.” Efficiency requires extensive form correlation, where strategies can condition upon a sunspot at the end of each period. For any finite number of repetitions, we approximate the best equilibrium payoff under perfect monitoring, assuming that monitoring is sufficiently accurate and sunspots are available. Journal of Economic Literature Classification Numbers: C73, D82.     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

Belief consistency and trade consistency

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agentsʼ beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinsteinʼs email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.