Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

The effect of decision weights in bargaining problems

Bargaining problems are considered where the preferences of the bargainers deviate from expected utility but can be modelled according to rank-dependent utility theory. Under rank-dependent utility both the utility function and the probability weighting function influence the risk attitude of a decision maker. The same definition of risk aversion leads to two forms of risk aversion: utility risk aversion and probabilistic risk aversion. The main finding is that these two forms can have surprisingly opposite consequences for bargaining solutions that exhibit a weak monotonicity property. In particular, in a large class of bargaining problems both increased utility risk aversion and decreased probabilistic risk aversion of the opponent are advantagous for a player. This is demonstrated for the Kalai-Smorodinsky bargaining solution. The Nash bargaining solution does not behave regularly in this respect.     

Nonconvex n-person bargaining: efficient maxmin solutions

Summary. This paper provides an axiomatic characterization of a family of so-called efficient maxmin solutions which can be seen as generalizations of the Kalai-Smorodinsky solution to nonconvex n-person bargaining problems. Moreover, it is shown that even though there are several efficient maxmin solutions for some bargaining problems, there is typically a unique efficient maxmin solution. Received: February 15, 2001; revised version: November 14, 2001     

Low information nash equilibria for oligopolistic markets

This paper deals with an infinite horizon n firm oligopoly in which firms are assumed to have incomplete information about one another's actions and profit functions. An equilibrium concept is defined that is similar to the Nash non-cooperative equilibrium, but is suitable for the information assumptions of the model. The equilibrium uses a type of bounded rationality which makes firms' computations relatively easy. This is due to an implicit assumption that computation is costly and a Bayesian approach is prohibitively costly. This low information Nash equilibrium is proved to exist, and, in addition, an adaptive expectations decision process is described which, if followed by all firms, leads to the low information Nash equilibrium.     

Implementing the Kalai-Smorodinsky bargaining solution

The bargaining game is proposed where the agents bid fractions of dictatorship that are used by all non-winners of the auction to threaten acceptance of the winner's proposal. That procedure non-cooperatively implements the Kalai-Smorodinsky bargaining solution.     

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.     

A Note on the Probability of k Pure Nash Equilibria in Matrix Games

A formula is derived for the probability that a "randomly selected" n-person matrix game has exactly k pure strategy equilibria. It is shown that for all n ≥ 2, this probability converges to e⁻¹/k! as the sizes of the strategy sets of at least two players increase without bound. Thus the number of pure strategy equilibria in large random n-person matrix games is approximately Poisson distributed with mean one. The latter is a known result obtained by a new proof in this note. Journal of Economic Literature Classification Number: C72.     

Homotopy methods to compute equilibria in game theory

This paper presents a survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present the relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke–Howson algorithm and the van den Elzen–Talman algorithm to compute Nash equilibria in 2-person games, and the Herings–van den Elzen, Herings–Peeters, and McKelvey–Palfrey algorithms to compute Nash equilibria in general n-person games. We explain how the main ideas can be extended to compute equilibria in extensive form and dynamic games, and how homotopies can be used to compute all Nash equilibria.     

Equilibrium Selection inn-Person Coordination Games

This paper investigates several approaches to equilibrium selection and the relationships between them. The class of games we study aren-person generalized coordination games with multiple Pareto rankable strict Nash equilibria. The main result is that all selection criteria select the same outcome (namely the risk dominant equilibrium) in two-person games, and that most equivalences break for games with more than two players. All criteria select the Pareto efficient equilibrium in voting games, of which pure coordination games are special cases.Journal of Economic LiteratureClassification Numbers: C70, C72, D82.     

An admissible set occurring in various bargaining situations

Given a set of alternatives S and a binary relation M on S the admissible set of the pair (S, M) is defined to be the set of maximal elements with respect to the transitive closure of M. It is shown that existing solutions in game theory and mathematical economics are special cases of this concept (they are admissible sets of a natural S and M). These include the core of an n-person cooperative game, Nash equilibria of a noncooperative game, and the max-min solution of a two-person zero sum game. The competitive equilibrium prices of a finite exchange economy are always contained in its admissible set. Special general properties of the admissible set are discussed. These include existence, stability, and a stochastic dynamic process which leads to outcomes in the admissible set with high probability.     

A differentiable homotopy to compute Nash equilibria of n-person games

Summary. The literature on the computation of Nash equilibria in n-person games is dominated by simplicial methods. This paper is the first to introduce a globally convergent algorithm that fully exploits the differentiability present in the problem. It presents an everywhere differentiable homotopy to do the computations. The homotopy path can therefore be followed by several numerical techniques. Moreover, instead of computing some Nash equilibrium, the algorithm is constructed in such a way that it computes the Nash equilibrium selected by the tracing procedure of Harsanyi and Selten. As a by-product of our proofs it follows that for a generic game the tracing procedure defines a unique feasible path. The numerical performance of the algorithm is illustrated by means of several examples. Received: December 21, 1999; revised version: December 27, 2000     

The cournot duopoly model reconsidered in the presence of costly communication and enforcement agreements

Recently I proposed a general notion of equilibrium for n-person games which is based on mutual threats and promises (responses), and which leads, in general, to a determinate solution with respect to the order of choice moves. The nature of the solution set in our model is a function of the action sets, the payoff functions, and the cost of ‘communication and enforcement of agreements’ (C&E). The main concern of this paper is to show that the Cournot duopoly solution, for the case of a linear demand function facing two identical firms, can be regenerated by our solution concept under a certain assumption about a high cost C&E. On the other hand, it is shown that under a relatively low cost of C&E the Cournot/Nash solution does not belong to our solution set, because it is Pareto inferior for both duopolists relative to some solutions in our set, among them, the standard monopoly solution.     

Risk attitudes and the stag-hunt game

A stag-hunt game (with the risky and safe actions) has two pure Nash equilibria that are Pareto-rankable. The risky action leads either to the Pareto-superior equilibrium (high payoff) or to out of equilibrium (low payoff) depending on the opponent’s action. Both players may want to obtain high payoff but uncertainty about the opponent’s action may prevent them to take such strategic risk. This paper investigates how information about the risk attitude of an opponent affects a player’s action choice in the stag-hunt game. I find that although a subject’s propensity to choose the risky action depends on her opponent’s risk attitude, it does not depend on her own risk attitude.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Two characterizations of the Raiffa solution

It is shown when and how the two-person bargaining solution proposed by Raiffa can be generalized to the n-person case. Two characterizations are proposed involving the familiar axioms of Pareto-optimality, Symmetry, and Invariance, as well as two new Monotonicity axioms.     

The number of commodities required to represent a market game

An n-person game with side payments that can arise from an economic market is called a market game. It is proved here that any n-person market game can be obtained from a market with at most n ? 1 commodities. Moreover, no smaller number will suffice in some cases (e.g., the unanimity game). This result is then applied to settle a question regarding market games with a continuum of traders.     

Substituting one risk increase for another: A method for measuring risk aversion

This paper defines the rate of substitution of one stochastic change to a random variable for another. It then focuses on the case where one of these changes is an nth degree risk increase, and the other is an m  th degree risk increase, where n>m?1$n>m?1$. The paper shows that the rate of substitution for these two risk increases can be used to provide a broader definition and two additional characterizations of the nth degree Ross more risk averse partial order. The implications for local intensity measures of nth degree risk aversion are also examined. The analysis organizes the existing results as well as generates new ones.     

Nash equilibrium search for the best alternative

In a recent paper, Weitzman (Econometrica47 (1979) 641–654) described a policy of “optimal search for the best alternative.” The present paper is concerned with the development and characterization of a policy of “Nash equilibrium search for the best alternative.” Specifically, it is shown that, under certain monotonicity assumptions, and under the assumption that firms have incomplete information regarding the results of rivals' search behavior, a Nash equilibrium search policy exists and has the same form as Weitzman's optimal search policy.