Recontracting and stochastic stability in cooperative games

An evolutionary style model of recontracting is given which guarantees convergence to core allocations of an underlying cooperative game. Unlike its predecessors in the evolution/learning literature, this is achieved without assumptions of convexity of the characteristic function or a reliance on random errors. The stochastic stability properties of the model are then examined and it is shown that stochastically stable states solve a simple and intuitive minimization problem which reduces to maximizing a Rawlsian SWF for a common class of utility functions. In contrast to previous analyses, the stochastically stable state is unique for a broad class of utility functions.     

Serial dictatorship and Pareto optimality

In a deterministic allocation problem in which each agent is entitled to receive exactly one object, an allocation is Pareto optimal if and only if it is the outcome of a serial dictatorship. We extend the definition of serial dictatorship to settings in which some agents may be entitled to receive more than one object, and study the efficiency and uniqueness properties of the equilibrium allocations. We prove that subgame perfect equilibrium allocations of serial dictatorship games are not necessarily Pareto optimal; and generally not all Pareto optima can be implemented as subgame perfect equilibrium allocations of serial dictatorship games, except in the 2-agent separable preference case. Moreover, serial dictatorship games do not necessarily have unique subgame perfect equilibrium allocations, except in the 2-agent case, hence their outcomes are indeterminate and manipulable.     

An informationally efficient equity criterion

A new equity criterion for exchange economies is proposed, an allocation being declared equitable if no agent prefers the average of what everyone else consumes to what he consumes. The logical relationship of this criterion with other criteria in the published literature is studied and its informational merits emphasized. Equitable allocations are shown to exist under standard assumptions. A complete geometric characterization of all equitable allocations is offered, with special attention being given to the case of large economies. A detailed analysis of the two person–two commodity case (then the criterion coincides with that of an envy-free allocation) reveals several properties of interest of the criterion.     

A Simple Supermodular Mechanism that Implements Lindahl Allocations

This paper introduces a new incentive‐compatible mechanism which, for general preference environments, implements Lindahl allocations as Nash equilibria. We provide an example that having a mechanism induce a supermodular game is not typically sufficient to guarantee dynamic stability of equilibrium. However, for the new mechanism, inducing a supermodular game guarantees that the best reply mapping is a contraction. Thus, this new mechanism provides a connection between the desirable welfare properties of Lindahl allocations and the theoretical/convergence properties of games whose best reply mappings are a contraction.     

Optimal growth with many consumers

A method is described for constructing all Pareto-optimal allocations for a dynamic economy with many heterogeneous consumers, under certainty, in which both the technology and consumer preferences are recursive but preferences need not be additively separable over time. Optimal (perfect foresight competitive equilibrium) allocations are obtained through the study of a dynamic program. For an economy with one consumption good, sufficient conditions are given for the existence of a unique interior stationary distribution of consumption and wealth. For a two-person exchange economy, sufficient conditions are given for the global asymptotic stability of the unique interior stationary point.     

An iterative allocation mechanism in the edgeworth box

This paper considers an iterative allocation mechanism which resolves the problem of multiplicity of allocations given by a mechanism in commodity space. At each stage, the average of the extreme allocations is taken and used as the starting point of the next stage. As long as the mechanism is individually rational and Pareto optimal, this iterative procedure yields a unique final allocation which is also individually rational and Pareto optimal. (JEL C63, C71)This paper was written while both authors were at the State University of New York at Buffalo.     

Dynamic adjustment of labor allocation between collective,state, or cooperative farming and private farming in a planned economy

How collective and state farm workers dynamically adjust their labor allocations between farm and private plot is analyzed by adopting a behavioralist (as opposed to a marginalist) approach. Farm workers shift their labor allocations for a given season toward that operation which rewarded them with a better rate of return during the preceding season. This approach is then applied to a hypothetical case where the farm is a cooperative venture between state and farm workers, and farm workers can decide to work exclusively on private plots, or on the farm, or divide their labor in some way between these two operations.     

Competitive equilibria with limited enforcement

We show how to decentralize constrained efficient allocations that arise from enforcement constraints between sovereign nations. In a pure exchange economy these allocations can be decentralized with private agents acting competitively and taking as given government default decisions on foreign debt. In an economy with capital these allocations can be decentralized if the government can tax capital income as well as default on foreign debt. The tax on capital income is needed to make private agents internalize a subtle externality. The decisions of the government can arise as an equilibrium of a dynamic game between governments.     

Characterization of the extreme core allocations of the assignment game

A characterization of the extreme core allocations of the assignment game is given in terms of the reduced marginal worth vectors. For each ordering in the player set, a payoff vector is defined where each player receives his or her marginal contribution to a certain reduced game played by his or her predecessors. This set of reduced marginal worth vectors, which for convex games coincide with the usual marginal worth vectors, is proved to be the set of extreme points of the core of the assignment game. Therefore, although assignment games are hardly ever convex, the same characterization of extreme core allocations is valid for convex games.     

A dual characterization of incentive efficiency

We show that incentive efficient allocations in economies with adverse selection and moral hazard problems can be determined as optimal solutions to a linear programming problem and we use duality theory to obtain a complete characterization of the optima. Our dual analysis identifies welfare effects associated with the incentives of the agents to truthfully reveal their private information. Because these welfare effects may generate non-convexities, incentive efficient allocations may involve randomization. Other properties of incentive efficient allocations are also derived.     

The Manipulability of Fair Solutions in Assignment of an Indivisible Object with Monetary Transfers

Public decision making often involves the problem of fairly assigning one indivisible object to agents with monetary transfers. An example is the choice of the location of a garbage incineration facility where the accepting district should receive fair compensations from other districts. In this problem, we show that for broad classes of solutions satisfying a welfare lower bound and an efficiency‐oriented condition, the set of equilibrium allocations in the manipulation game associated with a given solution coincides with the set of all envy‐free allocations. This generalizes Tadenuma and Thomson's equivalence result for a class of envy‐free solutions. Our result covers the Shapley value, which is not covered by Tadenuma and Thomson's result.     

Proportional income taxes and cores in a public goods economy

An equivalence relationship between cores and Pareto optimal allocations of a public goods economy where the public goods are to be provided through a proportional income tax is presented. For this purpose, the definition of the core is modified by allowing coalitions to tax their complements at any given rate. Also, a certain rule which specifies the rate is introduced.     

Beliefs and Pareto Efficient Sets: A Remark

We show that, in a two-period economy with uncertainty in the second period, if an allocation is Pareto optimal for a given set of beliefs and remains optimal when these beliefs are changed, then the set of optimal allocations of the two economies must actually coincide. We identify equivalence classes of beliefs, giving rise to the same set of Pareto optimal allocations. Journal of Economic Literature Classification Numbers: D51, D61.     

A Characterization of Robust Sunspot Equilibria

In nonconvex environments, a sunspot equilibrium can sometimes be destroyed by the introduction of new extrinsic information. We provide a simple test for determining whether or not a particular equilibrium survives, or is robust to, all possible refinements of the state space. We use this test to provide a characterization of the set of robust sunspot-equilibrium allocations of a given economy; it is equivalent to the set of equilibrium allocations of the associated lottery economy. Journal of Economic Literature Classification Numbers: D51, D84, E32.     

The efficiency of competitive mechanisms under private information

We consider the efficiency properties of exchange economies where privately informed traders behave strategically. Specifically, a competitive mechanism is any mapping of traders’ reports about their types to an equilibrium price vector and allocation of the reported economy. In our model, some traders may have non-vanishing impact on prices and allocations regardless of the size of the economy. Although truthful reporting by all traders cannot be achieved, we show that, given any desired level of approximation, there is such that any Bayesian-Nash equilibrium of any competitive mechanism of any private information economy with or more traders leads, with high probability, to prices and allocations that are close to a competitive equilibrium of the true economy. In particular, allocations are approximately efficient. A key assumption is that there is small probability that traders behave non-strategically.     

Fair allocations in large economies

The purpose of this paper is the study of problems of equity in an economy with many agents. Conditions are given under which every equitable and Pareto optimal allocation is a competitive equilibrium with equal incomes for all agents. The results are also of potential relevance for the literature on incentive compatibility since the allocations that are obained through incentive mechanisms in large economies turn out to be equitable.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Pareto-optimal Nash equilibria are competitive in a repeated economy

Consider a finite exchange economy first as a static, 1 period, economy and then as a repeated economy over T periods when the utility of each agent is the mean utility over T. A family of strategic games is defined via a set of six general properties the most distinct of which is the ability of agents to move commodities forward in time. Now consider Pareto optimal allocations in the T period economy which are also Nash equilibria in this family of strategic games. We prove that as T becomes large this set converges to the set of competitive utility allocations in the one period economy. The key idea is that a repetition of the economy when agents can move commodities forward in the time acts as a convexification of the set of individually feasible outcomes for player i holding all other strategies fixed.     

Pareto optima and equilibria when preferences are incompletely known

An exchange economy in which agents have convex incomplete preferences defined by families of concave utility functions is considered. Sufficient conditions for the set of efficient allocations and equilibria to coincide with the set of efficient allocations and equilibria that result when each agent has a utility in her family are provided. Welfare theorems in an incomplete preferences framework therefore hold under these conditions and efficient allocations and equilibria are characterized by first order conditions.     

Fair welfare maximization

We consider the general problem of finding fair constrained resource allocations. As a criterion for fairness we propose an inequality index, termed “fairness ratio,” the maximization of which produces Lorenz-undominated, Pareto-optimal allocations. The fairness ratio does not depend on the choice of any particular social welfare function, and hence it can be used for an a priori evaluation of any given feasible resource allocation. The fairness ratio for an allocation provides a bound on the discrepancy between this allocation and any other feasible allocation with respect to a large class of social welfare functions. We provide a simple representation of the fairness ratio as well as a general method that can be used to directly determine optimal fair allocations. For general convex environments, we provide a fundamental lower bound for the optimal fairness ratio and show that as the population size increases, the optimal fairness ratio decreases at most logarithmically in what we call the “inhomogeneity” of the problem. Our method yields a unique and “balanced” fair optimum for an important class of problems with linear budget constraints.