Adaptive learning in extensive form games and sequential equilibrium

Summary. This paper studies adaptive learning in extensive form games and provides conditions for convergence points of adaptive learning to be sequential equilibria. Precisely, we present a set of conditions on learning sequences such that an assessment is a sequential equilibrium if and only if there is a learning sequence fulfilling the conditions, which leads to the assessment. Received: November 5, 1996; revised version: May 28, 1997     

Reallocation of an infinitely divisible good

Summary. We consider the problem of reallocating the total initial endowments of an infinitely divisible commodity among agents with single-peaked preferences. With the uniform reallocation rule we propose a solution which satisfies many appealing properties, describing the effect of population and endowment variations on the outcome. The central properties which are studied in this context are population monotonicity, bilateral consistency, (endowment) monotonicity and (endowment) strategy-proofness. Furthermore, the uniform reallocation rule is Pareto optimal and satisfies several equity conditions, e.g., equal-treatment and envy-freeness. We study the trade-off between properties concerning variation and properties concerning equity. Furthermore, we provide several characterizations of the uniform reallocation rule based on these properties. Received: August 29, 1995; revised version June 26, 1996     

A pitchfork bifurcation in the tatonnement process★

Summary . This note extends the example of Gale (1963) by considering the continuous time tatonnement process for a class of two agent, two commodity exchange economies, parametrized by a number μ∈(0,1). We demonstrate that as the parameter passes a threshold value μ^* the unique, globally stable competitive equilibrium loses local stability while two new locally stable equilibria appear. Intuitively, as μ increases the income effect become increasingly more important relative to substitution effect, and eventually overwhelms the latter. As the parameter μ approaches 1, the economy tends to the example considered by Gale, as does the limiting behavior of the tatonnement. Received: February 28, 1996; revised version August 5, 1996     

On the uniqueness of competitive equilibrium

Summary . The paper is concerned with the following question: in addition to local uniqueness, what other conditions must be imposed to ensure global uniqueness of competitive equilibrium? The answer is provided within a standard framework involving excess demand functions. Conditions are identified which have the nice property that they are true close to a regular equilibrium. Uniqueness is established by considering an adjustment process and showing that under the mentioned conditions, every equilibrium is locally asymptotically stable and moreover the process itself is globally stable; uniqueness follows from an Arrow and Hahn (1971) result. Alternatively, the paper may be seen as identifying conditions under which there is a vectorfield satisfying the requirements of an uniqueness result due to Dierker (1974). Received: July 1, 1996; revised version October 7, 1996     

A resolution of N-person prisoners' dilemma

Summary. The purpose of this paper is to show that cooperation is possible in N-person prisoners' dilemma if players have incomplete information on the order of their moves. We consider a modified version of N-person prisoners' dilemma in which players sequentially move in the order determined by Nature, and during the play they get some information about others. We provide an information partition and a condition of payoffs for which cooperation is attained in an equilibrium. Further, for the case that full cooperation is not attained, we examine the largest partial cooperation which is achieved in an equilibrium. Received: March 5, 1995; revised version August 11, 1996     

Coalitional stability under perfect foresight

Summary. We analyze strategic social environments where coalitions can form through binding or nonbinding agreements and actions of a coalition may impose externalities upon the welfare of the rest of the players. We define a solution concept that (1) captures the perfect foresight of the players that has been overlooked in the literature (e.g., Harsanyi [10] and Chwe [6]) and (2) identifies the coalitions that are likely to form and the “stable” outcomes that will not be replaced by any coalition of rational (and hence farsighted) players. The proposed solution concept thereby offers a notion of agreements and coalition formation in complex social environments. Received: February 12, 1996; revised version: March 3, 1997     

Markov-perfect Nash equilibria in a class of resource games

Summary. A general model of non-cooperating agents exploiting a renewable resource is considered. Assuming that the resource is sufficiently productive we prove that there exists a continuum of Markov-perfect Nash equilibria (MPNE). Although these equilibria lead to over-exploitation one can approximate the efficient solution by MPNE both in the state space and the payoff space. Furthermore, we derive a necessary and sufficient condition for maximal exploitation of the resource to qualify as a MPNE. This condition is satisfied if there are sufficiently many players, or if the players are sufficiently impatient, or if the capacity of each player is sufficiently high.Received: November 1, 1996This revised version was published online in February 2005 with corrections to the cover date.     

An extremely simple proof of the K-K-M-S Theorem

Summary.  An extremely simple proof of the K-K-M-S Theorem is given involving only Brouwer’s fixed point theorem and some elementary calculus. A function is explicitly given such that a fixed point of it yields an intersection point of a balanced collection of sets together with balancing weights. Moreover, any intersection point of a balanced collection of sets together with balancing weights corresponds to a fixed point of the function. Furthermore, the proof can be used to show -balanced versions of the K-K-M-S Theorem, with -balancedness as introduced in Billera (1970). The proof makes clear that the conditions made with respect to by Billera can be even weakened. Received: January 22, 1996; revised version June 9, 1996     

A comparison between the core and the monopoly solutions in a mixed exchange economy

Summary. Using a mixed market model for analyzing imperfectly competitive economies, we maximize the oligopolists' Welfare Function, given individual rationality and feasibility constraints. We prove that solutions belong to the core for a large class of economies. This class includes, in particular, every monopoly having a single type of small traders. Note that all such solutions yield the large trader, utility-wise, strictly more than at any monopoly solution, where the monopolist plays strategically, and the ocean of small traders act as being as price-takers. Received: March 4, 1996; revised version July 7, 1996     

Asymmetric information, strategic behavior, and discrimination in the labor market

Summary  The neoclassical model of labor market discrimination assumes the presence of either prejudiced preferences, biased assessments of worker productivity, or monopsony power. We show that when market agents control asymmetric information, strategic behavior can induce discriminatory hiring practices even when these market features are absent. Moreover, strategic interaction many distort public policies to the point of harming the segments of the work force they were designed to support. Received: January 3, 1996 revised version April 29, 1996     

Serial cost sharing of excludable public goods: general cost functions

Summary. A group of individuals meet to share the cost and determine output allocations of a partial-excludable public good. We demonstrate that, for general cost functions and preferences that satisfy the Spence-Mirlees sorting condition, the serial cost-sharing formula (Moulin, 1994) has remarkable incentive properties. First, a direct economic mechanism that uses the serial formula is coalition strategy-proof, envy-free and satisfies the stand-alone property. Second, the serial mechanism involves partial exclusion, which is important for the reduction of the free-rider problem. Received: June 10, 1996; revised version; February 11, 1997     

Sunspot fluctuations in dynamics with predetermined variables

Summary. In this paper, I study the existence of Sunspot Equilibria in a general framework whose dynamics allow for the presence of predetermined variables in the system. The main motivation for this research comes from the fact that previous studies did not allow for such predetermined variables which, nevertheless, appear quite naturally in economic models. I show, for a non-negligible subset of dynamics with predetermined variables verifying usual assumptions, the existence of Stationary Sunspot Equilibria fluctuating between an arbitrary finite number of states arbitrarily close to a steady state. Received: March 1, 1995; revised version September 18, 1996     

Quasitransitive social preference: why some very large coalitions have very little power

Summary. If only the strict part of social preference is required to be transitive then Independence of Irrelevant Alternatives implies that there is a coalition containing all but one individual that cannot force x to be socially ranked above y for at least half of the pairs of alternatives (x,y). Received: August 29, 1996; revised version: March 24, 1997     

Efficient delays in a stochastic model of bargaining

Summary. We consider a k-player sequential bargaining model in which both the cake size and the identity of the proposer are determined by a stochastic process. For the case where the cake is a simplex (of random size) and the players share a common discount factor, we establish the existence of a unique stationary subgame perfect payoff which is efficient and characterize the conditions under which agreement is delayed. We also investigate how the equilibrium payoffs depend on the order in which the players move and on the correlation between the identity of the proposer and the cake size.Received: November 5, 1996; revised version: December 31, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Matching and bargaining models of markets: approximating small markets by large markets

Summary. We show that the equilibrium of a matching and bargaining model of a market in which there is a finite number of agents at each date need not be near the equilibrium of a market with a continuum of agents, although matching probabilities are the same in both markets. Holding the matching process fixed, as the finite market becomes large its equilibrium approaches the equilibrium of its continuum limit.Received: January 22, 1996; revised version: September 24, 1996This revised version was published online in February 2005 with corrections to the cover date.     

On capital accumulation paths in a neoclassical stochastic growth model

Summary. Boldrin and Montrucchio [2] showed that any twice continuously differentiable function could be obtained as the optimal policy function for some value of the discount parameter in a deterministic neoclassical growth model. I extend their result to the stochastic growth model with non-degenerate shocks to preferences or technology. This indicates that one can obtain complex dynamics endogenously in a wide variety of economic models, both under certainty and uncertainty. Further, this result motivates the analysis of convergence of adaptive learning mechanisms to rational expectations in economic models with (potentially) complicated dynamics. Received: June 21, 1996; revised version: October 31, 1996     

Uniqueness of the index for Nash equilibria of two-player games

Summary. Given a map whose roots are the Nash equilibria of a game, each component of the equilibrium set has an associated index, defined as the local degree of the map. This note shows that for a two-player game, every map with the same roots induces the same index. Moreover, this index agrees with the Shapley index constructed from the Lemke-Howson algorithm. Received: May 30, 1996; revised version June 25, 1996