Dynamic Game Model of Endogenous Growth with Consumption Externalities

This paper introduces consumption externalities into an endogenous growth model of common capital accumulation and characterizes balanced growth equilibria. Contrary to the standard argument in previous studies, we show that the growth rate in a feedback Nash equilibrium can be higher than that in an open-loop Nash equilibrium if agents strongly admire the consumption of others. This result is irrelevant to whether preferences exhibit “keeping up with the Joneses” or “running away from the Joneses”.     

A Strong Anti-Folk Theorem

We study the properties of finitely complex, symmetric, globally stable, and semi-perfect equilibria. We show that: (1) If a strategy satisfies these properties then players play a Nash equilibrium of the stage game in every period; (2) The set of finitely complex, symmetric, globally stable, semi-perfect equilibrium payoffs in the repeated game equals the set of Nash equilibria payoffs in the stage game; and (3) A strategy vector satisfies these properties in a Pareto optimal way if and only if players play some Pareto optimal Nash equilibrium of the stage game in every stage. Our second main result is a strong anti-Folk Theorem, since, in contrast to what is described by the Folk Theorem, the set of equilibrium payoffs does not expand when the game is repeated.This paper is a revised version of Chapter 3 of my Ph.D. thesis, which has circulated under the title “An Interpretation of Nash Equilibrium Based on the Notion of Social Institutions”.     

Generalized correlated equilibrium for two-person games in extensive form with perfect information

A correlation scheme (leading to a special equilibrium called “soft” correlated equilibrium) is applied for two-person finite games in extensive form with perfect information. Randomization by an umpire takes place over the leaves of the game tree. At every decision point players have the choice either to follow the recommendation of the umpire blindly or freely choose any other action except the one suggested. This scheme can lead to Pareto-improved outcomes of other correlated equilibria. Computational issues of maximizing a linear function over the set of soft correlated equilibria are considered and a linear-time algorithm in terms of the number of edges in the game tree is given for a special procedure called “subgame perfect optimization”.     

Negotiation-proof Nash equilibrium

This paper defines “negotiation-proof Nash equilibrium', a notion that applies to environments where players can negotiate openly and directly prior to the play of a noncooperative game. It recognizes the possibility that a group of self-interested players may choose, voluntarily and without binding agreement, to coordinate their choice of strategies and make joint objections; moreover, it takes the perfect foresight of rational players fully into account. The merit of the notion of negotiation-proof Nash equilibrium is twofold: (1) It offers a way to rectify the nestedness assumption and myopia embedded in the notion of coalition-proof Nash equilibrium. (2) The negotiation process is formalized by a “graph”, which serves as a natural extension to the approach that models preplay communication by an extensive game. Received: October 1998/Final version: May 2000     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

On consistent solutions for strategic games

Nash equilibria for strategic games were characterized by Peleg and Tijs (1996) as those solutions satisfying the properties of consistency, converse consistency and one-person rationality.  There are other solutions, like the ɛ-Nash equilibria, which enjoy nice properties and appear to be interesting substitutes for Nash equilibria when their existence cannot be guaranteed. They can be characterized using an appropriate substitute of one-person rationality. More generally, we introduce the class of “personalized” Nash equilibria and we prove that it contains all of the solutions characterized by consistency and converse consistency. Received January 1996/Final version December 1996     

Advertising Strategies in a Differential Game with Negative Competitor’s Interference

We consider a duopolistic industry where the current sales of each firm is proportional to its goodwill stock. The evolution of the latter depends positively on own advertising effort and negatively on competitor’s advertising. A standard assumption in the literature in differential games of advertising is that the players remain active throughout the whole (infinite) duration of the game. We relax this assumption and characterize the circumstances under which a firm finds it optimal to remain or exit the industry. Among other things, it is shown that, if both players are “strong”, then the unique Nash equilibrium is the same that one would obtain in the absence of interference from competitor’s advertising. Research supported by Ministry of University and Research of Italy, University of Padua and NSERC, Canada.     

Monotonicity in asymmetric first-price auctions with affiliation

I study monotonicity of equilibrium strategies in first-price auctions with asymmetric bidders, risk aversion, affiliated types, and interdependent values. Every mixed-strategy equilibrium is shown to be outcome-equivalent to a monotone pure-strategy equilibrium under the “priority rule” for breaking ties. This provides a missing link to establish uniqueness in the “general symmetric model” of Milgrom and Weber (Econometrica 50:1089–1122, 1982). Non-monotone equilibria can exist under the “coin-flip rule” but they are distinguishable: all non-monotone equilibria have positive probability of ties whereas all monotone equilibria have zero probability of ties. This provides a justification for the standard empirical practice of restricting attention to monotone strategies. Hendricks et al. (2003) provide an overview of recent empirical work. For a survey of experimental work, see Kagel and Levin (2002).     

The Existence and Uniqueness of Nash Equilibrium Point in an m-player Game “Shoot Later, Shoot First!”

We consider the following “silent duel” of m players with a possible economic interpretation. Each player has one “bullet”, which she can shoot at any time during the time interval [0,1]. The probability that the i-th player hits the “target” at moment t is given by an increasing accuracy function f _i (t). The winner is the player who hits the target first. Under natural assumptions on the functions f _i (t) we prove the existence and uniqueness of a Nash equilibrium point in this game, and we provide an explicit construction of this equilibrium. This construction allows us to obtain exact solutions for many specific examples. Some of them are presented.This work was partly supported by RBRF grants 03-01-00479.     

Trembling hand perfection for mixed quantal/best response equilibria

The quantal response equilibrium (QRE) is a powerful alternative to full rationality equilibrium concepts. At a QRE, all joint moves have non-zero probability. However in “mixed scenarios”, where some players use quantal response and some use best response, equilibrium strategy profiles can have joint moves with zero probability. This raises the question of applying the trembling hand refinement to such mixed scenarios. To address this I first show how to reformulate the QRE as a “best response” equilibrium where expected utilities are replaced by more general objective functions. I then show that under this reformulation the two popular types of trembling hand perfection can differ when some players use quantal response and some use best response. I end by showing that one of those types of trembling hand perfection cannot be used to remove certain troubling kinds of equilibrium in such mixed scenarios, while the other type can. The conclusion is that only the one type of trembling hand perfection should be applied when we allow some players to be quantal response and some to be best response.     

Equilibria of a two-stage market with chance outcome in the spot market

We address the problem of how to improve the efficiency of markets of similar goods (electric power, gas, and other resources). One way to undermine the market dominance of some companies is the possibility of forward contracts. Here a model of the spot and forward markets functioning as Curnout auctions is studied using the example of symmetrical oligopoly. Suppliers try to maximize their profit by this two-stage game’s strategies of traded subgame equilibrium (TSE). The conditions for equilibrium achieved by correlated mixed strategies are elucidated: either a “bull” or “bear” market is established according to a chance factor. The optimum strategies of rational bidders are found to depend on the reserve price and a risk-avoiding parameter. TSE is compared to the Nash equilibria for one-stage models.     

On the existence of almost-pure-strategy Nash equilibrium in <Emphasis Type="Italic">n</Emphasis>-person finite games

This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.     

On a Noncooperative Stochastic Game Played by Internally Cooperating Generations

We study a model of intergenerational stochastic game with general state space in which each generation consists of n players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite-horizon intergenerational game in which cooperation is assumed inside every generation. A suitable change in the terminology used in this paper provides a new equilibrium theorem for stochastic games with so-called “hyperbolic players”. A discussion of perfect equilibria in games of noncooperative generations is also given. Some applications to economic theory are included.     

A non-zero-zum War of Attrition

The division of a cake by two players is modelled by means of a game of timing in which the players have a probability of learning when their opponent acts. It is shown that the game has a unique Nash equilibrium when both players are non-noisy but that there are many Nash equilibria including pure ones when at least one of the players is noisy. Explicit expressions for the strategies used in these Nash equilibria are obtained.This work was carried out while Dr. Garnaev was visiting the University of Southampton on a Postdoctoral Fellowship of The Royal Society of London.     

Independent mistakes in large games

Economic models usually assume that agents play precise best responses to others' actions. It is sometimes argued that this is a good approximation when there are many agents in the game, because if their mistakes are independent, aggregate uncertainty is small. We study a class of games in which players' payoffs depend solely on their individual actions and on the aggregate of all players' actions. We investigate whether their equilibria are affected by mistakes when the number of players becomes large. Indeed, in generic games with continuous payoff functions, independent mistakes wash out in the limit. This may not be the case if payoffs are discontinuous. As a counter-example we present the n players Nash bargaining game, as well as a large class of “free-rider games.” Received: November 1997/Final version: December 1999     

Nash implementation of competitive equilibria in the job-matching market

This paper studies Nash implementation in the job-matching market where each worker works for only one firm and a firm hires as many workers as it wishes. We show that the competitive equilibrium correspondence (CEC) is the smallest Nash implementable correspondence satisfying individual rationality and Pareto indifference. Furthermore, the CEC is the minimal monotonic extension of the worker-optimal and firm-optimal subcorrespondences. We offer two “good” mechanisms that implement this correspondence in Nash equilibrium.     

“Costless” regulation of monopolies with large entry cost: A game theoretic approach

A major issue within the realm of Antitrust policy is the regulation of existing monopolies. We describe a new potential indirect scheme for regulating a natural monopoly that arises from high entry cost. The approach involves minimal government intervention, and it is based on encouraging entry by offering to subsidize entry cost for potential competitors. We pose this issue as a four-stage non-cooperative game. Analysis of sub-game perfect Nash equilibria of the game reveals that the first best outcome is achieved as the unique equilibrium in which the monopolist prices at marginal cost and there is no entry. The regulation is “costless,” since no entry will occur and hence no subsidy will be paid. Received June 1995     

Refinements of rationalizability for normal-form games

There exist three equivalent definitions of perfect Nash equilibria which differ in the way “best responses against small perturbations” are defined. It is shown that applying the spirit of these definitions to rationalizability leads to three different refinements of rationalizable strategies which are termed perfect (Bernheim, 1984), weakly perfect and trembling-hand perfect rationalizability, respectively. We prove that weakly perfect rationalizability is weaker than both perfect and proper (Schuhmacher, 1995) rationalizability and in two-player games it is weaker than trembling-hand perfect rationalizability. By means of examples, it is shown that no other relationships can be found. Received: January 1997/final version: August 1998     

Equilibrium selection and altruistic behavior in noncooperative social networks

Given their importance in determining the outcome of many economic interactions, different models have been proposed to determine how social networks form and which structures are stable. In Bala and Goyal (Econometrica 68, 1181–1229, 2000), the one-sided link formation model has been considered, which is based on a noncooperative game of network formation. They found that the empty networks, the wheel in the one-way flow of benefits case and the center-sponsored star in the two-way flow case play a fundamental role since they are strict Nash equilibria of the corresponding games for a certain class of payoff functions. In this paper, we first prove that all these network structures are in weakly dominated strategies whenever there are no strict Nash equilibria. Then, we exhibit a more accurate selection device between these network architectures by considering “altruistic behavior” refinements. Such refinements that we investigate here in the framework of finite strategy sets games have been introduced by the authors in previous papers.