N-person Non-cooperative Game Research Articles

Non-cooperative differential game and feedback Nash equilibrium analysis for real-time electricity markets

Continuous liberalization of electricity markets makes a strong correlation between the economic behaviors of market participants and their profits. To guide the production of generators before the market reaches equilibrium, a new framework is proposed to model the real-time electricity market with help of optimal control theory. The market price is described as the dynamic state using a sticky price model. We establish an N-person non-cooperative differential game model, where all participants try to maximize their profits independently only by observing power prices. The existence of feedback Nash equilibrium and uniqueness of optimal price trajectory are proved in detail, which will help other scholars build an effective differential game model. To protect the privacy of all generators, a distributed algorithm is proposed based on neurodynamic and consensus theory, which only requires information exchange among neighboring participants. Furthermore, a special case of a duopoly power market is investigated in detail and we provide a feedback time-continuous solution for each generator. Compared with the commercial Cplex solver, the proposed distributed algorithm performs better in convergence speed.

Liberalism, Rationality, and Pareto Optimality

Rational players in game theory are neoliberal in the sense that they can choose any available action so as to maximize their payoffs. It is well known that this can result in Pareto inferior outcomes (e.g. the Prisoner’s Dilemma). Classical liberalism, in contrast, argues that people should be constrained by a no-harm principle (NHP) when they act. We show, for the first time to the best of our knowledge, that rational players constrained by the NHP will produce Pareto efficient outcomes in n-person non-cooperative games. We also show that both rationality and the NHP are required for this result.

Existence of solution to n-person noncooperative games and minimax regret equilibria with set payoffs

In this paper, by virtue of the Kakutani–Fan–Glicksberg fixed point theorem and a nonlinear scalarization function, we first obtain an existence theorem of general n-person noncooperative games problems with set payoffs. Then, we also obtain existence theorems of minimax regret equilibria and ϵ-minimax regret equilibria with scalar set payoffs. Some examples are given to illustrate our results.

A multicast routing restoration algorithm based on the n-person non-cooperative game in multi-domain optical networks

Aiming at the restoration problems for multicast service in multi-domain optical networks, a multicast routing restoration algorithm in multi-domain optical networks based on the n-person non-cooperative game is proposed by adopting the Breadth First Search (BFS) strategy and n-person non-cooperative game theory. This algorithm optimizes the searching sequence of BFS algorithm and reduces the searching scope by the competitive selection relation of the game theory. The algorithm analysis and experimental results show that the algorithm has low time complexity, reduces the computational time of the restoration path, and reduces the blocking rate of the multicast restoration service.

Local dense solutions for equilibrium problems with applications to noncooperative games

In this work, we establish several results concerning the existence of solutions for set-valued and single-valued equilibrium problems in real Hausdorff topological vector spaces. Firstly we introduce some generalizations of convexity and continuity conditions to set-valued mappings and then apply them to special dense subsets of the domain to obtain the existence of local dense solutions of equilibrium problems. Then the existence of the global solutions follows from a condition that is weaker than semistrict quasiconvexity. Specifically, we give an existence theorem for noncooperative n-person games, under assumptions imposed on a locally segment-dense subset of the strategy set of each player.

On efficiency of some algorithms for solving finite non-cooperative n-person games in general and multi-matrix settings.

On efficiency of some algorithms for solving finite non-cooperative n-person games in general and multi-matrix settings.

Existence of Generalized Nash Equilibrium in n-Person Noncooperative Games under Incomplete Preference

To prove the existence of Nash equilibrium by traditional ways, a common condition that the preference of players must be complete has to be considered. This paper presents a new method to improve it. Based on the incomplete preference corresponding to equivalence class set being a partial order set, we translate the incomplete preference problems into the partial order problems. Using the famous Zorn lemma, we get the existence theorems of fixed point for noncontinuous operators in incomplete preference sets. These new fixed point theorems provide a new way to break through the limitation. Finally, the existence of generalized Nash equilibrium is strictly proved in the n-person noncooperative games under incomplete preference.

A half-space projection method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method.

Formulating an n-person noncooperative game as a tensor complementarity problem

In this paper, we consider a class of n-person noncooperative games, where the utility function of every player is given by a homogeneous polynomial defined by the payoff tensor of that player, which is a natural extension of the bimatrix game where the utility function of every player is given by a quadratic form defined by the payoff matrix of that player. We will call such a problem the multilinear game. We reformulate the multilinear game as a tensor complementarity problem, a generalization of the linear complementarity problem; and show that finding a Nash equilibrium point of the multilinear game is equivalent to finding a solution of the resulted tensor complementarity problem. Especially, we present an explicit relationship between the solutions of the multilinear game and the tensor complementarity problem, which builds a bridge between these two classes of problems. We also apply a smoothing-type algorithm to solve the resulted tensor complementarity problem and give some preliminary numerical results for solving the multilinear games.

Essential stability of $$\alpha $$ α -core

Using the existence results in Kajii (J Econ Theory 56:194–205, 1992), we identify a class of n-person noncooperative games containing a dense residual subset of games whose cooperative equilibria are all essential. Moreover, we show that every game in this collection possesses an essential component of the \(\alpha \)-core by proving the connectivity of minimal essential subsets of the \(\alpha \)-core.

A novel method to compute Nash equilibrium in non-cooperative n-person games based on differential evolutionary algorithm

Nash equilibrium constitutes a central solution concept in game theory. In this paper, a Nash equilibrium of a finite n-person non-cooperative game can be formulated to an non-linear optimal solution of the optimization model with zero optimal value. This paper investigates the effectiveness of differential evolutionary optimization to compute Nash equilibrium of non-cooperative n-person games, as global minimum of a real-valued function. The experiments demonstrate the feasibility of this approach for finding an equilibrium of an N-person game.

A Bidding Methodology by Nash Equilibrium for Finite Generators Participating in Imperfect Electricity Markets

This paper presents an application of finite n-person non-cooperative game theory for analyzing bidding strategies of generators in a deregulated energy marketplace with Pool Bilateral contracts so as to maximize their net profits. A new methodology to build bidding methodology for generators participating in oligopoly electricity market has been proposed in this paper. It is assumed that each generator bids a supply function. This methodology finds out the coefficients in the supply function of generators in order to maximize benefits in an environment of competing rival bidders. A natural choice for developing strategies is Nash Equilibrium (NE) model incorporating mixed strategies, for solving the bidding problem of electrical market. Associated optimal profits are evaluated for a combination of set of pure strategies of bidding of generators, and payoff matrix has been constructed. The optimal payoff is calculated by using NE. An attempt has also been made to minimize the gap between the optimal payoff and the payoff obtained by a possible mixed strategies combination. The algorithm is coded in MATLAB. A numerical example is used to illustrate the essential features of the approach and the results are proved to be the optimal values.

AN APPLICATION OF A DISCRETE FIXED POINT THEOREM TO A GAME IN EXPANSIVE FORM

In this paper, we first present a discrete fixed point theorem for contraction mappings from the product set of integer intervals into itself, which is an extension of Robert's discrete fixed point theorem. Next, we derive an existence theorem of a pure-strategy Nash equilibrium for a noncooperative n-person game from our fixed point theorem. Finally, we show that Kuhn's theorem for a game in expansive form can be explained by our existence theorem.

Generalized Knaster–Kuratowski–Mazurkiewicz Theorem Without Convex Hull

In this paper, a new notion of Knaster–Kuratowski–Mazurkiewicz mapping is introduced and a generalized Knaster–Kuratowski–Mazurkiewicz theorem is proved. As applications, some existence theorems of solutions for (vector) Ky Fan minimax inequality, Ky Fan section theorem, variational relation problems, n-person noncooperative game, and n-person noncooperative multiobjective game are obtained.

NECESSITY AND SUFFICIENCY FOR THE EXISTENCE OF A PURE-STRATEGY NASH EQUILIBRIUM

In this paper, we consider a non-cooperative n-person game in the strategic form. As is well known, the game has a mixed-strategy Nash equilibrium. However, it does not always have a pure-strategy Nash equilibrium. Wherein, Topkis (1979), Iimura (2003), and Sato and Kawasaki (2009) provided a sufficient condition for the game to have a pure-strategy Nash equilibrium. However, they did not consider necessary conditions. This paper has two aims. The first is to extend the authors' sufficient condition, which is based on monotonicity of the best responses. The second is to show that the existence of a pure-strategy Nash equilibrium implies the monotonicity of the best responses or an isolation of the equilibrium.

Stability of solutions for variational relation problems with applications

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

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player's cost function and the opponents' strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player's decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player's cost function is quadratic, and the uncertainty sets for the opponents' strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.

Essential Stability of Solutions for Maximal Element Theorem with Applications

In this paper, we prove that most of problems in maximal element theorem (in the sense of Baire category) are essential and that, for any problem in maximal element theorem, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan's points based on Ky Fan Minimax Inequality, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games and the existence of essential components of the set of solutions of vector Ky Fan Minimax Inequality.

Gross Interst-Environment Games and Environment Crisis Theorems

To study the interaction between the player's actions and environment in economic management game such as launching some projects with pollution in a certain area, we consider the relation between the player’s interests and the environment. We introduce a so-called gross interest- environment game and the concept of common hazard degree based on binary number and n-person non-cooperative game theory. It is studied that properties of players’ utility functions and common hazard degree. Basic on the concept of N-M stable set in set of Nash equilibria, we prove environment crisis theorems. Our main results are as follows: if it is an Nash equilibrium that every enterprise launching the project with pollution and it is not an Nash equilibrium that every not doing, then it is the most probable to realize Nash equilibrium with the greatest common hazard degree. Key words: gross interest-environment game; common hazard degree; N-M stable set; environment crisis theorem

Fuzzy Nash equilibrium of fuzzy n-person non-cooperative game

Fuzzy Nash equilibrium of fuzzy n-person non-cooperative game