Strategy Nash Equilibrium Of Game Research Articles

Existence and uniqueness of Nash equilibrium in discontinuous Bertrand games: a complete characterization

Since (Reny in Econometrica 67:1029–1056, 1999) a substantial body of research has considered what conditions are sufficient for the existence of a pure strategy Nash equilibrium in games with discontinuous payoffs. This work analyzes a general Bertrand game, with convex costs and an arbitrary sharing rule at price ties, in which tied payoffs may be greater than non-tied payoffs when both are positive. On this domain, necessary and sufficient conditions for (i) the existence of equilibrium (ii) the uniqueness of equilibrium are presented. The conditions are intuitively easy to understand and centre around the relationships between intervals of real numbers determined by the primitives of the model.

Game theory without theory: Interactive choice in pigeons, humans and machines

A unique approach to the experimental analysis of interactive choice behavior is described. Pairs of subjects were coupled in such a manner that each subject’s choice determined either the probability of reinforcement or the points awarded to the other subject’s choice. In this manner, several of game theory’s familiar games, like Prisoner’s Dilemma, were created. With pigeons as subjects, with humans playing against pigeons or humans playing each other (either in the laboratory or on the Internet), and even with a simple computer simulation, the results were clear. In every instance, players’ choices gravitated to the game’s pure strategy Nash equilibrium in games that have them and to a near equality of responding in the game Rock-Paper-Scissors, which does not. The principle of positive reinforcement entirely accounts for the results. There is no need for appeal to complex mathematical models or higher-order concepts of strategy, optimality, rational expectations and the like.

Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

On strategic complementarities in discontinuous games with totally ordered strategies

This paper studies the existence of a pure strategy Nash equilibrium in games with strategic complementarities where the strategy sets are totally ordered. By relaxing the conventional conditions related to upper semicontinuity and single crossing, we enlarge the class of games to which monotone techniques are applicable. The results are illustrated with a number of economics-related examples.

On Strategic Complementarities in Discontinuous Games

This paper studies the existence of a pure strategy Nash equilibrium in games with strategic complementarities where the strategy sets are totally ordered. By relaxing the conventional conditions related to upper semicontinuity and single crossing, we enlarge the class of games to which monotone techniques are applicable. The results are illustrated with a number of economics-related examples.

On the learning and stability of mixed strategy Nash equilibria in games of strategic substitutes

This paper analyzes the learning and stability of mixed strategy Nash equilibria in games of strategic substitutes (GSS), complementing recent work done in the case of strategic complements (GSC). Mixed strategies in GSS are of particular interest because it is well known that such games need not exhibit pure strategy Nash equilibria. First, we establish bounds on the strategy space which indicate where randomizing behavior may occur in equilibrium. Second, we show that mixed strategy Nash equilibria are generally unstable under a wide variety of learning rules. Multiple examples are given.

On the existence of equilibria in games with arbitrary strategy spaces and preferences

This paper provides necessary and sufficient conditions for the existence of pure strategy Nash equilibria by replacing the assumptions concerning continuity and quasiconcavity with a unique condition, passing strategy space from topological vector spaces to arbitrary topological spaces. Preferences may also be nontotal/nontransitive, discontinuous, nonconvex, or nonmonotonic. We define a single condition, recursive diagonal transfer continuity (RDTC) for aggregator payoff function and recursive weak transfer quasi-continuity (RWTQC) for individuals’ preferences, respectively, which establishes the existence of pure strategy Nash equilibria in games with arbitrary (topological) strategy spaces and preferences without imposing any kind of quasiconcavity-related conditions.

On Uniform Conditions for the Existence of Mixed Strategy Equilibria

Embarking from the concept of uniform payoff security (Journal of Econonomic Theory, Vol. 134, 2007), we introduce two other uniform conditions and then study the existence of mixed strategy Nash equilibria in games where the sum of the payoff functions is not necessarily upper semicontinuous.

Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasiconcavity of functions and convexity or compactness of choice sets. A new condition, called “local dominatedness property”, is shown to be necessary and further, under some mild continuity condition, sufficient for the existence of equilibrium. We then apply the basic result obtained in the paper to generalize the existing theorems on the existence of saddle points, fixed points, and coincidence points without convexity or compactness assumptions. As an application, we also characterize the existence of pure strategy Nash equilibrium in games with discontinuous and non-quasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.

A Note on the Existence of Nash Equilibrium in Games with Discontinuous Payoffs

This paper generalises the approach taken by Dasgupta & Maskin (1986) and Simon (1989) and provides necessary and sufficient conditions for the existence of pure and mixed strategy Nash equilibrium in games with continuous strategy spaces and discontinuous payoff functions. The conditions can be applied widely, and examples for existence of pure strategy and monotonic equilibria in First-Price auctions are provided. The conditions are also appropriate for ensuring that computer generated equilibrium solutions can be extended to continuous strategy spaces.

Undominated Equilibria in Games with Strategic Complementarities

We analyze the problem of existence of undominated pure strategy Nash equilibria in games with strategic complementarities. A Nash equilibrium is undominated if no player's equilibrium strategy is weakly dominated by another pure strategy. We show the existence of the least and the greatest undominated equilibrium. The equilibrium strategies of these extremal equilibria are undominated against mixed strategies as well and have nice comparative statics properties.Journal of Economic LiteratureClassification Number: C72.