Strategy Set Of Player Research Articles

Computing Nash Equilibria in Potential Games with Private Uncoupled Constraints

We consider the problem of computing Nash equilibria in potential games where each player's strategy set is subject to private uncoupled constraints. This scenario is frequently encountered in real-world applications like road network congestion games where individual drivers adhere to personal budget and fuel limitations. Despite the plethora of algorithms that efficiently compute Nash equilibria (NE) in potential games, the domain of constrained potential games remains largely unexplored. We introduce an algorithm that leverages the Lagrangian formulation of NE. The algorithm is implemented independently by each player and runs in polynomial time with respect to the approximation error, the sum of the size of the action-spaces, and the game's inherit parameters.

Optimality Conditions and Constraint Qualifications for Generalized Nash Equilibrium Problems and Their Practical Implications

Generalized Nash equilibrium problems (GNEPs) are a generalization of the classic Nash equilibrium problems (NEPs), where each player's strategy set depends on the choices of the other players. In ...

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.

Complexity of Rational and Irrational Nash Equilibria

We introduce two new natural decision problems, denoted as ? RATIONAL NASH and ? IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ? RATIONAL NASH and ? IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. The problem NASH-EQUIVALENCE asks whether or not the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ? RATIONAL NASH. The problem NASH-REDUCTION asks whether or not there is a so called Nash reduction: a suitable map between corresponding strategy sets of players that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of NASH-REDUCTION that witnesses ? IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co- $\mathcal{NP}$ -hard and ? RATIONAL NASH is $\mathcal{NP}$ -hard, and (ii) NASH-REDUCTION and ? IRRATIONAL NASH are both $\mathcal{NP}$ -hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm (Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765---771, 2003; Games Econ. Behav. 63(2), 621---641, 2008).

Methods for Solving Generalized Nash Equilibrium

The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash equilibrium problem (NEP), in which each player's strategy set may depend on the rival player's strategies. In this paper, we present two descent type methods. The algorithms are based on a reformulation of the generalized Nash equilibrium using Nikaido-Isoda function as unconstrained optimization. We prove that our algorithms are globally convergent and the convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of the GNEP.

Rationalizability in Continuous Games

Define a continuous game to be one in which every player's strategy set is a Polish space, and the payoff function of each player is bounded and continuous. We prove that in this class of games the process of sequentially eliminating never-best-reply strategies does not terminate after the first uncountable ordinal, and that this bound is tight. Also, we examine the connection between this process and common belief of rationality in the universal type space of Mertens and Zamir .

Rationalizability in continuous games

Define a continuous game to be one in which every player's strategy set is a Polish space, and the payoff function of each player is bounded and continuous. We prove that in this class of games the process of sequentially eliminating “never-best-reply” strategies terminates before or at the first uncountable ordinal, and this bound is tight. Also, we examine the connection between this process and common belief of rationality in the universal type space of Mertens and Zamir (1985).

Collusive game solutions via optimization

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported.

An epistemic analysis of the Harsanyi transformation

Harsanyi (1967–68) proposed a method for transforming uncertainty over the strategy sets of players into uncertainty over their payoffs. The transformation appears to rely on an assumption that the players are rational, or, indeed, that they are rational and that there is common belief of rationality. Such an assumption would be awkward from the perspective of the epistemic program, which is often interested in the implications of irrationality or a lack of common belief of rationality. This paper shows that without common belief of rationality, such implications are not necessarily maintained under a Harsanyi transformation. The paper then shows how, with the belief-system model of Aumann and Brandenburger (1995), such implications can be maintained in the absence of common belief of rationality.

On the computability of Nash equilibria

Abstract We present some algorithmic unsolvability and incompleteness results in game theory and discuss their significance. The main theorem presents a class of n -person games, where each player's strategy set is the real line and payoffs are continuous functions, for which there could not possibly exist an algorithm to compute either a Nash equilibrium or an ϵ-equilibrium. Conditions sufficient to ensure solvability are also discussed.

The Cost of Derandomization: Computability or Competitiveness

Recently, much work has been done in game theory towards understanding the bounded rationality of players in infinite games. This requires the strategies of realistic players to be restricted to have bounded resources of reasoning. (See [H. Simon, Decision and Organization, North--Holland, Amsterdam, 1972, pp. 161--176] for an extensive discussion; also see [X. Deng and C. H. Papadimitriou, Math. Oper. Res., 19 (1994), pp. 257--266], [C. Futia, J. Math. Econom., 4 (1977), pp. 289--299], [V. Knoblauch, Games Econom. Behav., 7 (1994), pp. 381--389], [E. Kalai and W. Stanford, Econometria, 56 (1988), pp. 397--410], [A. Neyman, Econom. Lett., 19 (1985), pp. 227--229], and [C. H. Papadimitriou, Game Theory Econom. Behav., 4 (1992), pp. 122--131].) In this paper, we discuss infinite two-person games, focusing on the case where our player follows a computable strategy and the adversary may use any strategy, which formulates the notion of computer against extremely formidable nature. In this context, we say that an infinite game is semicomputably determinate if either the adversary has a winning strategy or our player has a computable winning strategy. We show that, whereas all open games are semicomputably determinate, there is a semicomputably indeterminate closed game. Since we want to prove an indeterminacy result for closed games and since the adversary's strategy set is uncountable and our player's strategy set is countable, our proof for the indeterminacy result requires a new diagonalization technique, which might be useful in other similar cases. Our study of semicomputable games was inspired by online computing problems. In this direction, we discuss several possible applications to derandomization in online computing, with the restriction that the strategies of our player should be computable. We also study the power of randomization for the classical case where our player is allowed to play according to unrestricted strategies. An indeterminate game is obtained for which both players have a simple randomized winning strategy against all of the deterministic strategies of the opponent.

Fixed points, Nash games and their organizations

The concepts of $(S, \sigma)$-invariance and $(S, \sigma, R, M)$-invariance are introduced and are used to prove two existence theorems of equilibrium in the sense of Berge [2] and Nash [1, 2] using fixed point arguments. Radjef's results [8] have been extended. Conditions under which these equilibria are Nash are also shown. Assuming that each player's strategy set is a subset of a reflexive Banach space and that the strategies can be partitioned in such a way that the argmax of each player's objective over an element of the considered partition is unique and satisfies one of the invariance properties, equilibria exist. Similar results are obtained for games with an infinite number of players.

Games of stopping with infinite horizon

We consider two-person zero-sum games of stopping: two players sequentially observe a stochastic process with infinite time horizon. Player I selects a stopping time and player II picks the distribution of the process. The pay-off is given by the expected value of the stopped process. Results of Irle (1990) on existence of value and equivalence of randomization for such games with finite time horizon, where the set of strategies for player II is dominated in the measure-theoretical sense, are extended to the infinite time case. Furthermore we treat such games when the set of strategies for player II is not dominated. A counterexample shows that even in the finite time case such games may not have a value. Then a sufficient condition for the existence of value is given which applies to prophet-type games.

Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities

We study a rich class of noncooperative games that includes models of oligopoly competition, macroeconomic coordination failures, arms races, bank runs, technology adoption and diffusion, R&D competition, pretrial bargaining, coordination in teams, and many others. For all these games, the sets of pure strategy Nash equilibria, correlated equilibria, and rationalizable strategies have identical bounds. Also, for a class of models of dynamic adaptive choice behavior that encompasses both best-response dynamics and Bayesian learning, the players' choices lie eventually within the same bounds. These bounds are shown to vary monotonically with certain exogenous parameters. WE STUDY THE CLASS of (noncooperative) supermodular games introduced by Topkis (1979) and further analyzed by Vives (1985, 1989), who also pointed out the importance of these games in industrial economics. Supermodular games are games in which each player's strategy set is partially ordered, the marginal returns to increasing one's strategy rise with increases in the competitors' strategies (so that the game exhibits strategic complementarity2) and, if a player's strategies are multidimensional, marginal returns to any one com- ponent of the player's strategy rise with increases in the other components. This class turns out to encompass many of the most important economic applications of noncooperative game theory. In macroeconomics, Diamond's (1982) search model and Bryant's (1983, 1984) rational expectations models can be represented as supermodular games. In each of these models, more activity by some members of the economy raises the returns to increased levels of activity by others. In oligopoly theory, some models of Bertrand oligopoly with differentiated products qualify as supermodu- lar games. In these games, when a firm's competitors raise their prices, the marginal profitability of the firm's own price increase rises. A similar structure is present in games of new technology adoption such as those of Dybvig and Spatt (1983), Farrell and Saloner (1986), and Katz and Shapiro (1986). When more users hook into a communication system or more manufacturers adopt an interface standard, the marginal return to others of doing the same often rises. Similarly, in some specifications of the bank runs model introduced by Diamond and Dybvig (1983), when more depositors withdraw their funds from a bank, it is more worthwhile for other depositors to do the same. In the warrant exercise

Strategy additions and equilibrium selection

In games with multiple equilibria, if one of the players is given the option of burning an appropriately chosen sum of money before the game and the iterated elimination of dominated strategies is used to select strategies for play, then a unique equilibrium can often be made to appear. The usefulness of an equilibrium selection device depends upon the limits on its ability to select equilibria. This paper shows that given any pure strategy Nash equilibrium in a normal form game, then by adding at most one strategy to each player's strategy set and appropriately defining payoffs, one can make that equilibrium the unique outcome surviving the iterated elimination of dominated strategies.

Convex semi-infinite games

This paper introduces a generalization of semi-infinite games. The pure strategies for player I involve choosing one function from an infinite family of convex functions, while the set of mixed strategies for player II is a closed convex setC inRn. The minimax theorem applies under a condition which limits the directions of recession ofC. Player II always has optimal strategies. These are shown to exist for player I also if a certain infinite system verifies the property of Farkas-Minkowski. The paper also studies certain conditions that guarantee the finiteness of the value of the game and the existence of optimal pure strategies for player I.