Mixed Strategy Nash Equilibrium Research Articles

NASH EQUILIBRIUM WITH LOWER PROBABILITIES

We generalize the concept of Nash equilibrium in mixed strategies for strategic form games to allow for ambiguity in the players' expectations. In contrast to other contributions, we model ambiguity by means of so-called lower probability measures or belief functions, which makes it possible to distinguish between a player's assessment of ambiguity and his attitude towards ambiguity. We also generalize the concept of trembling hand perfect equilibrium. Finally, we demonstrate that for certain attitudes towards ambiguity it is possible to explain cooperation in the one-shot Prisoner's Dilemma in a way that is in accordance with some recent experimental findings.

Finitely additive and epsilon Nash equilibria

We prove the existence of a mixed strategy Nash equilibrium in normal form games when the space of mixed strategies consists of finitely additive probability measures. It is then proved that from this result an existence result for epsilon equilibria with countably additive mixed strategies can be obtained. These results are applied to the classic Cournot game.

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.

Mixed-Strategy Nash Equilibrium Based upon Expected Utility and Quadratic Utility

The notion of a mixed-strategy Nash equilibrium suffers from three inherent difficulties. First, given the equilibrium strategies of other players, there are many best replies. Second, the equilibrium is unstable. Third, comparative statics results are counterintuitive. We demonstrate that these difficulties all have their origin in von Neumann and Morgenstern′s expected utility. In contrast, players with "quadratic utility" have unique best replies and the Nash equilibrium appears to yield intuitive comparative statics results. Journal of Economic Literature Classification Number: C72.

The solution to the Tullock rent-seeking game when R>2: Mixed-strategy equilibria and mean dissipation rates

In Tullock's rent-seeking model, the probability a player wins the game depends on expenditures raised to the power R. We show that a symmetric mixed-strategy Nash equilibrium exists when R>2, and that overdissipation of rents does not arise in any Nash equilibrium. We derive a tight lower bound on the level of rent dissipation that arises in a symmetric equilibrium when the strategy space is discrete, and show that full rent dissipation occurs when the strategy space is continuous. Our results are shown to be consistent with recent experimental evidence on the dissipation of rents.

A simple extension of the Dasgupta-Maskin existence theorem for discontinuous games with an application to the theory of rent-seeking

I state that upper-semi-continuity of the sum of payoffs is not necessary for the classical existence result in symmetric games by Dasgupta and Maskin. This is then applied to the theory of rent-seeking to show that there always exists some Nash equilibrium in mixed strategies in the simultaneous-move game by Tullock in which no `overdissipation can occur.

Bertrand-Edgeworth Competition in Experimental Markets

The Bertrand-Edgeworth model describes competition among price setting sellers with production capacity constraints. The authors report on laboratory experiments that permit evaluation of different theories of Bertrand-Edgeworth competition: competitive pricing, Edgeworth cycles in prices, mixed strategy Nash equilibrium pricing, and tacit collusion. Each of the theories helps to explain some aspects of the data. However, none of these theories are completely consistent with the data. In relative terms, the Edgeworth cycle theory provides better predictions of key aspects of the data than the other theories. Coauthors are Stephen Rassenti, Stanley S. Reynolds, and Vernon L. Smith. Copyright 1994 by The Econometric Society.

Competitive Spatial Price Discrimination with Capacity Constraints

A model of duopolistic spatial price discrimination on a network is extended by adding capacity constraints on firms' outputs. With zero-one consumer demands, some configurations of capacity levels and locations do not have pure strategy Nash equilibria in price schedules, in particular when firms have some excess capacity but not enough to serve the entire market individually. An alternative to the mixed strategy Nash equilibrium in price schedules—the core of the market game among producers and consumers—is described and shown to have allocations corresponding to pure strategies for all location and capacity configurations. If both firms have some excess capacity, price schedules are identical to those in the uncapacitated problem. Equilibrium prices are no higher than the lowest prices in the mixed strategy Nash equilibria.

A game theoretical approach to the international debt overhang

We consider an international financial problem called debt overhang, by which we mean a situation where a sovereign country has borrowed money from foreign banks and has been unable to fulfill the scheduled repayments for some period. The problem is formulated as a noncooperative game withn lender banks as players where each decides either to sell its loan exposure to the debtor country at the present price of debt on the secondary market, or to wait and keep its exposure. This game has many pure and mixed strategy Nash equilibria. We show, however, that in any Nash equilibrium, the resulting secondary market price remains almost the same as the present price for a large number of banks. We also obtain the comparative statics result that in a mixed strategy equilibrium, a bank with a smaller loan exposure has a greater tendency to sell than one with a larger loan exposure. We discuss the implications of these results for the functioning of the secondary market and the resolution of debt overhang.

Are Equilibrium Strategies Unaffected by Incentives?

In a mixed-strategy Nash equilibrium, changing one player's payoffs affects only the other player's equilibrium strategy mix. This `Payoff Irrelevance Proposition' (PIP) appears to undercut the main foundations of economic policy analysis since, allegedly, equilibrium behavior will not respond to changes in incentives. We show, in contrast, that: (1) When the policy-maker has the first move in a sequential-move game, the PIP does not hold. (2) Even in a simultaneous-move game, the PIP holds only when the policy space is discrete, and for sufficiently small payoff revisions. Thus, incentives do generally affect behavior in equilibrium.

Necessary conditions of parametric extremum for nash equilibrium in mixed strategies in differential games

Necessary conditions of parametric extremum for nash equilibrium in mixed strategies in differential games

Equilibrium without independence

Because players whose preferences violate the von neumann-Morgenstern independence axiom may be unwilling to randomize as mixed-strategy Nash equilibrium would require, a Nash equilibrium may not exist without independence. This paper generalizes Nash's definition of equilibrium, retaining its rational-expectations spirit but relaxing its requirement that a player must bear as much uncertainty about his own strategy choice as other players do. The resulting notion, “equilibrium in beliefs”, is equivalent to Nash equilibrium when independence is satisfied, but exists without independence. This makes it possible to study the robustness of equilibrium comparative statics results to violations of independence.

On the instability of mixed-strategy Nash equilibria

What behavior processes will lead to a Nash equilibrium of an N-person game? Two very general processes are examined. For finite games with finite-horizon generalized fictitious play, strict mixed-strategy Nash equilibria are shown to be unstable. For almost all finite games with discrete or continuous time generalized gradient processes, mixed-strategy Nash equilibria are shown to be unstable.

Bertrand-Edgeworth Oligopoly in Large Markets

The relation between perfectly competitive and monopolistically competitive equilibria is analysed for a Bertrand-Edgeworth model of a single market in which capacity constrained firms choose prices as strategies. The market always has a Nash equilibrium in pure or mixed strategies. As the number of firms increases, the corresponding equilibria converge in distribution to a perfectly competitive price. This result provides a justification for perfect competition that is based on an explicit account of price formation. However, monopoly prices persist with a positive but vanishing probability. Regularity or well defined inverse demand functions are not required.

Distributional Strategies for Games with Incomplete Information

We study games with incomplete information from a point of view which emphasizes the empirical predictions arising from game-theoretic models. Using the notion of “distributional” strategies, we prove four main theorems: (i) a mixed-strategy Nash equilibrium existence theorem, (ii) a pure-strategy equilibrium existence theorem, (iii) a pure-strategy ϵ-equilibrium existence theorem, and (iv) a theorem describing how the set of equilibria of a game varies with the parameters of the game.

Learning behavior and mixed-strategy Nash equilibria

The paper considers whether an adaptive justification, like those commonly available for non-interactive optimization models, can be found for the mixed-strategy Nash equilibrium. Although it is known that such a justification is frequently available for pure-strategy equilibria, it is shown that all members of a wide class of behaviorally plausible learning mechanisms must fail to converge in ‘almost all’ games in which the equilibrium involves mixed strategies. An exact formal analogy is developed, which allows these learning mechanisms to be viewed as discrete taˆtonnement processes in properly chosen competitive exchange economies. In the analogy, the instability derives from the independence of excess demand functions (except at the boundaries) from ‘own’ prices. This independence arises because of the linearity in probabilities of von Neumann-Morgenstern risk preferences, and therefore does not extend to pure-strategy equilibria in general. Thus, assuming that agents will play equilibrium strategies implicitly assumes more sophistication when the equilibrium involves mixed strategies than when it involves only pure strategies.

Approximate Purification of Mixed Strategies

Under relatively weak conditions on the information structure of a game it is shown that any mixed strategy of a player can be replaced by a certain pure strategy without affecting any player’s expected payoff appreciably, no matter what strategies the other players use. Under even weaker conditions the existence of a Nash equilibrium in mixed strategies is shown to guarantee the existence of a pure-strategy combination which is an approximate equilibrium.

Existence and Computation of Mixed Strategy Nash Equilibrium for 3-Firms Location Problem

Location problems of firms on a closed interval were introduced by Hotelling [3] and later investigated by Eaton & Lipsey [2]. Firms sell a homogeneous product at a fixed price, customers distributed along the interval buy one unit each from the firm nearest to them and firms aim to maximize the number of their customers. It is well known that where the customers are evenly distributed along the interval and the number of firms is other than 3 the location problem has a pure strategy Nash equilibrium (see [2]). The case of 3 firms is peculiar in that no pure strategy Nash equilibrium exists, for in any situation one of the extreme firms (not located between the other two) can increase its revenue by either moving towards or away from the other two firms. The existence of mixed strategies is not obvious because the payoff functions are discontinuous whenever two or more firms are located at the same point. In a recent paper Dasgupta and Maskin [1] investigate a class of games, all derived from economic problems and all with some discontinuities in their payoff functions. They demonstrate that the discontinuities in the payoff functions do not prevent the existence of mixed strategies Nash equilibria in the games investigated. In particular, location problems of the type described earlier have such weak discontinuities and hence possess a mixed strategy Nash equilibrium. Although their proof is constructive, offering a method of approaching these equilibria as a limit of equilibria of finite games, the equations involved very quickly become unmanageable. In this note we compute a mixed strategy Nash equilibrium for the 3 firms location problem. Let the customers be evenly distributed along [0, 1] then a mixed strategy is a probability measure over [0, 1]. The symmetry of the problem suggests that the solution will be doubly symmetric in that all firms choose the same mixed strategy and that this strategy is symmetric around - (symmetric locations are given the same probabilities). Dasgupta and Maskin confine their analysis to doubly symmetric solutions and prove that the strategies in any doubly symmetric equilibrium are atomless. Given the symmetric nature of the game and following the analysis of Dasgupta and Maskin we shall confine ourselves here to such doubly sym