Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

Non-cooperative n-person game with a stopped set

A continuous time non-cooperative n-person Markov game with a stopped set is studied in this paper. We prove that, in the game process with or without discount factor, there exists an optimal stationary point of strategies, called the equilibrium point, and each player has his equilibrium stationary strategy, such that the total expected discounted or non-discounted gain are maximums.     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points

Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any gameΓ the players' uncertainty about the other players' exact payoffs can be modeled as a disturbed gameΓ ^*, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point inΓ, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed gameΓ ^* when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs willmake them use their pure strategies approximately with the prescribed probabilities.     

Equilibrium payoffs of dynamic games

We give a characterization of the equilibrium payoffs of a dynamic game, which is a stochastic game where the transition function is either one or zero and players can only use pure actions in each stage. The characterization is in terms of convex combinations of connected stationary strategies; since stationary strategies are not always connected, the equilibrium set may not be convex. We show that subgame perfection may reduce the equilibrium set.     

Applications of Ky Fan's inequality on σ-compact set to variational inclusion and n-person game theory

In this paper, Ky Fan's inequality on σ-compact set is applied to variational inclusions and n-person game theory. We give results of some variational inclusions and existence of non-cooperative equilibrium in n-person game on σ-compact set.     

Non-cooperative stochastic dominance games

A non-cooperative stochastic dominance game is a non-cooperative game in which the only knowledge about the players' preferences and risk attitudes is presumed to be their preference orders on the set ofn-tuples of pure strategies. Stochastic dominance equilibria are defined in terms of mixed strategies for the players that are efficient in the stochastic dominance sense against the strategies of the other players. It is shown that the set of SD equilibria equals all Nash equilibria that can be obtained from combinations of utility functions that are consistent with the players' known preference orders. The latter part of the paper looks at antagonistic stochastic dominance games in which some combination of consistent utility functions is zero-sum over then-tuples of pure strategies.     

A note on correlated equilibrium

The set of correlated equilibria for a bimatrix game is a closed, bounded, convex set containing the set of Nash equilibria. We show that every extreme point of a maximal Nash set is an extreme point of the above convex set. We also give an example to show that this result is not true in the payoff space, i.e. there are games where no Nash equilibrium payoff is an extreme point of the set of correlated equilibrium payoffs.     

A fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers

The aim of this paper is to develop an effective method for solving matrix games with payoffs of triangular fuzzy numbers (TFNs) which are arbitrary. In this method, it always assures that players’ gain-floor and loss-ceiling have a common TFN-type fuzzy value and hereby any matrix game with payoffs of TFNs has a TFN-type fuzzy value. Based on duality theorem of linear programming (LP) and the representation theorem for fuzzy sets, the mean and the lower and upper limits of the TFN-type fuzzy value are easily computed through solving the derived LP models with data taken from 1-cut set and 0-cut set of fuzzy payoffs. Hereby the TFN-type fuzzy value of any matrix game with payoffs of TFNs can be explicitly obtained. Moreover, we can easily compute the upper and lower bounds of any Alfa-cut set of the TFN-type fuzzy value for any matrix game with payoffs of TFNs and players’ optimal mixed strategies through solving the derived LP models at any specified confidence level Alfa. The proposed method in this paper is demonstrated with a numerical example and compared with other methods to show the validity, applicability and superiority.     

Nash equilibrium and the law of large numbers

Pascoa (1993a) showed that the failure of the law of large numbers for a continuum of independent randomizations implies that Schmeidler's (1973) concept of a measure-valued profile function in equilibrium might not coincide with the concept of mixed strategies equilibrium of a nonatomic game. The latter should be defined as a probability measure on pure strategies profiles which is induced by the product measure of players' mixed strategies. This paper addresses existence and approximate purification of the latter and presents an assumption on continuity of payoffs that guarantees the equivalence between the two equilibrium concepts.     

Stability of solutions for Ky Fan's section theorem with some applications

In this paper, we prove that “most of” problems in Ky Fan's section theorem (in the sense of Baire category) are essential and that for any problem in Ky Fan's section theorem, there exists at least one essential component of its solution set. As applications, we deduce both the existence of essential components of the set of Ky Fan's points based on Ky Fan's minimax inequality theorem and the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games with non-concave payoffs.     

On Nash Equilibrium Strategy of Two-person Zero-sum Games with Trapezoidal Fuzzy Payoffs

In this paper, we investigate Nash equilibrium strategy of two-person zero-sum games with fuzzy payoffs. Based on fuzzy max order, Maeda and Cunlin constructed several models in symmetric triangular and asymmetric triangular fuzzy environment, respectively. We extended their models in trapezoidal fuzzy environment and proposed the existence of equilibrium strategies for these models. We also established the relation between Pareto Nash equilibrium strategy and parametric bi-matrix game. In addition, numerical examples are presented to find Pareto Nash equilibrium strategy and weak Pareto Nash equilibrium strategy from bi-matrix game.     

Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers

The purpose of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of trapezoidal fuzzy numbers (TrFNs), which are a type of two-person non-cooperative games with payoffs expressed by TrFNs and players’ strategies being constrained. In this methodology, it is proven that any Alfa-constrained matrix game has an interval-type value and hereby any constrained matrix game with payoffs of TrFNs has a TrFN-type value. The auxiliary linear programming models are derived to compute the interval-type value of any Alfa-constrained matrix game and players’ optimal strategies. Thereby the TrFN-type value of any constrained matrix game with payoffs of TrFNs can be directly obtained through solving the derived four linear programming models with data taken from only 1-cut and 0-cut of TrFN-type payoffs. Validity and applicability of the models and method proposed in this paper are demonstrated with a numerical example of the market share game problem.     

FiniteN-person non-cooperative games with unique equilibrium points

A necessary and sufficient condition is found for a given completely mixed strategyN-tuple to be the unique equilibrium point of some finiteN-person non-cooperative game.     

The use of public randomization in discounted repeated games

This paper presents an example where the set of subgame-perfect equilibrium payoffs of the infinitely repeated game without public randomization is not convex, no matter how large the discount factor is. Also, the set of pure-strategy equilibrium payoffs is not monotonic with respect to the discount factor in this example. These results are in sharp contrast to the fact that the equilibrium payoff set is convex and monotonic if public randomization is available.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

Optimal strategic reasoning with McNaughton functions

The aim of the paper is to explore strategic reasoning in strategic games of two players with an uncountably infinite space of strategies the payoff of which is given by McNaughton functions—functions on the unit interval which are piecewise linear with integer coefficients. McNaughton functions are of a special interest for approximate reasoning as they correspond to formulas of infinitely valued Lukasiewicz logic. The paper is focused on existence and structure of Nash equilibria and algorithms for their computation. Although the existence of mixed strategy equilibria follows from a general theorem (Glicksberg, 1952) [5], nothing is known about their structure neither the theorem provides any method for computing them. The central problem of the article is to characterize the class of strategic games with McNaughton payoffs which have a finitely supported Nash equilibrium. We give a sufficient condition for finite equilibria and we propose an algorithm for recovering the corresponding equilibrium strategies. Our result easily generalizes to n-player strategic games which don't need to be strictly competitive with a payoff functions represented by piecewise linear functions with real coefficients. Our conjecture is that every game with McNaughton payoff allows for finitely supported equilibrium strategies, however we leave proving/disproving of this conjecture for future investigations.     

Games with frequency-dependent stage payoffs

Games with frequency-dependent stage payoffs (FD-games), are infinitely repeated non-cooperative games played at discrete moments in time called stages. The stage payoffs depend on the action pair actually chosen, and on the relative frequencies with which all actions were chosen before. We assume that players wish to maximize their expected (limiting) average rewards over the entire time-horizon. We prove an analogy to, as well as an extension of the (perfect) Folk Theorem. Each pair of rewards in the convex hull of all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Moreover, each pair of rewards in same set giving each player strictly more than the threat-point-reward, can be supported by a subgame-perfect equilibrium. Under a pair of jointly-convergent strategies, the relative frequency of each action pair converges in the long run. Received: March 2002/Revised: January 2003     

On constrained maxima of convex functions

This paper states and proves Kuhn-Tucker necessary conditions for a maximum point of a convex function subject to convex constraints. Also presented are conditions which imply that the optimal policy set of this type program is a continuous point-to-set mapping of the resource vector.     

On a Parameterized System of Nonlinear Equations with Economic Applications

We study a parameterized system of nonlinear equations. Given a nonempty, compact, and convex set, an affine function, and a point-to-set mapping from the set to the Euclidean space containing the set, we constructively prove that, under certain (boundary) conditions on the mapping, there exists a connected set of zero points of the mapping, i.e., the origin is an element of the image for every point in the connected set, such that the connected set has a nonempty intersection with both the face at which the affine function is minimized and the face at which that function is maximized. This result generalizes and unifies several well-known existence theorems including Browder??s fixed point theorem and Ky Fan??s coincidence theorem. An economic application with constrained equilibria is also discussed.