Non-cooperative N-person Research Articles

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.

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.

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.

$N$-Person Card Game Approach for Solving SET $K$-COVER Problem in Wireless Sensor Networks

Solutions to SET K-COVER problem can prolong the lifetime of wireless sensor networks (WSNs) by partitioning the sensors into sets. In this paper, a novel purely distributed method for solving SET K-COVER problem is introduced based on the game theory, in which we consider the SET K-COVER problem as a noncooperative N-person card game. The sensors in WSN are considered as the players, the cover sets chosen by N sensors are considered as the strategies, and the sensing area covered alone is considered as the payoff function for each sensor. After the gaming process, the best strategies that all of the players have chosen constitute the Nash equilibrium. In addition, we analyze the effects of the initial strategies on the gaming result and propose a solution to avoid this effect in order to obtain a better coverage performance. We also extend the optimality of Nash equilibrium in the coverage game to a more general case. Besides that, the analysis of the convergence performance and the message complexity of the proposed algorithm are also presented. Extensive simulations have been conducted to show the superiority of the proposed algorithm in convergence, robustness, and coverage, comparing with the random, K-COVER, and synchronous Nash equilibria convergence algorithms. Finally, based on the results from real experiments on both small- and large-scale WSNs, we conclude that the proposed algorithm can be applied in real application environment and is also of good performance in both convergence and coverage. Furthermore, we also provide the comparisons between results from simulations and real experiments when some practical issues are considered.

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.

Deterministic time-inconsistent optimal control problems — an essentially cooperative approach

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N-person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.

Adaptive Noncooperative $N$-Person Games With Unknown General Quadratic Objectives

Motivated by the problems of unknown objectives in strategic bidding in the restructured electricity power markets where in practice, not all participants have full access to competitors' information, this paper investigates a class of noncooperative N-person games with unknown objectives with linear and cross terms in controls. We first formulate and explicitly derive the Nash equilibrium solution for this type of games, then incorporate the concept of fictitious play with the technique of adaptive control to estimate multiple controller strategies, which are not necessarily a Nash solution. We show both theoretically and numerically that the adaptation procedure converges under certain conditions. The proposed scheme is applicable to all systems with similar mathematical structures and objectives.

EQUILIBRIUM IN n-PERSON GAME OF SHOWCASE SHOWDOWN

In this article we consider a noncooperative n-person optimal stopping game of Showcase Showdown, in which each player observes the sum of independent and identically distributed random variables uniformly distributed in [0, 1]. Players can decide to stop the draw in each moment. The objective of a player is to get the maximal number of scores that does is not exceeded level 1. If the scores of all players exceed 1, then the winner is the player whose score is closest to 1. We derive the equilibrium in this game on the basis of the dynamic programming approach.

DISCRETE FIXED POINT THEOREMS AND THEIR APPLICATION TO NASH EQUILIBRIUM

In this paper, we present discrete fixed point theorems. They are based on monotonicity of the mapping. We apply them to a non-cooperative n-person game and give an existence theorem of a Nash equilibrium of pure strategies. As a special case, we consider bimatrix games.

Dynamic Volunteer's Dilemmas over a Finite Horizon

Volunteer's dilemmas that evolve over time are presented and modeled as noncooperative n-person games in extensive form with symmetric players, discrete time, finite horizon, and complete information. Volunteering is costly, thereby giving rise to free riding. Reflecting on the observation that in many naturally occurring social dilemmas it is beneficial to volunteer earlier than later, the model assumes that the payoff to the volunteer and the (higher) payoff to each of the nonvolunteers decrease monotonically over time. The authors construct symmetric and asymmetric subgame perfect equilibria to the game. An experimental study shows that financially motivated subjects who are rewarded contingent on their performance volunteer more readily when the cost of volunteering is relatively low; that they largely fail to coordinate on any of the asymmetric equilibria in which only a single subject volunteers immediately; that they volunteer, on average, earlier than predicted; and that they vary considerably from one another in their inclination to free ride.

Multi-Index Cooperative Mixed Strategy for Service Selection Problem in Service-Oriented Architecture

Under the environment of Service-Oriented Architecture (SOA), service users have to decide how to choose the proper provider from the candidates to obtain optimal service level. The existing approaches based on the optimization theories usually ignore the decision conflicts among users, which result in periodic QoS fluctuation and poor system performance. This paper treated the service selection problem as an inconstant-sum non-cooperative n-person dynamic game with incomplete information. The mechanism that generated the decision conflicts is discussed at first, and then Multi-index Cooperative Mixed Strategy (MCMS) with corresponding algorithm for finding its optimal solution is developed. Simulation shows that MCMS offers service users an optimal strategy that reduces the fluctuation of QoS and improves performance of SOA. Moreover, the stability and equilibrium of MCMS make it acceptable to most of the users, thus MCMS is able to improve predictability and controllability of SOA significantly.

Successful strategies in repeated minority games

The minority game is an important example of a non-cooperative n-person game, which can be applied on different situations with social and economic contexts. We analysed the minority game as an elementary traffic scenario in which human participants had to choose 100 times between a road A and a road B. In each period, the road, which was chosen by the minority of players won. At first view, there seems to be no outstandingly advisable strategy for the participants to enhance their payoffs because both roads have the same properties. However, an important observation is that the number of road changes of a participant is negatively correlated to his/her cumulative payoff. On average, subjects with high numbers of road changes received less money than participants who stoically chose the same road. Furthermore one could increase the coordination of the players by providing the players distribution on both roads in the last period.

Existence of nash equilibria for constrained stochastic games

In this paper, we consider constrained noncooperative N-person stochastic games with discounted cost criteria. The state space is assumed to be countable and the action sets are compact metric spaces. We present three main results. The first concerns the sensitivity or approximation of constrained games. The second shows the existence of Nash equilibria for constrained games with a finite state space (and compact actions space), and, finally, in the third one we extend that existence result to a class of constrained games which can be “approximated” by constrained games with finitely many states and compact action spaces. Our results are illustrated with two examples on queueing systems, which clearly show some important differences between constrained and unconstrained games.

A dynamical method for the calculation of Nash-equilibria in n-person games

In this paper, we are concerned with the calculation of Nash-equilibria in non-cooperative n-person games. For this purpose, we construct a continuous mapping of the Cartesian product of the strategy sets of the players into itself such that the fixed points of this mapping are Nash-equilibria. This gives rise to an iteration method for the calculation of fixed points of this mapping which leads to Nash-equilibria, if it converges. As important special cases Bi-matrix games and evolution matrix games are considered.

Joining a Queue or Staying Out: Effects of Information Structure and Service Time on Arrival and Staying Out Decisions

We study a class of single-server queueing systems with a finite population size, FIFO queue discipline, and no balking or reneging. In contrast to the predominant assumptions of queueing theory of exogenously determined arrivals and steady state behavior, we investigate queueing systems with endogenously determined arrival times and focus on transient rather than steady state behavior. When arrival times are endogenous, the resulting interactive decision process is modeled as a non-cooperative n-person game with complete information. Assuming discrete strategy spaces, the mixed-strategy equilibrium solution for groups of n=20 agents is computed using a Markov chain method. Using a 2x2 between-subject design (private vs. public information by short vs. long service time), arrival and staying out decisions are presented and compared to the equilibrium predictions. The results indicate that players generate replicable patterns of behavior that are accounted for remarkably well on the aggregate, but not individual, level by the mixed-strategy equilibrium solution unless congestion is unavoidable and information about group behavior is not provided.

Equilibrium play in single-server queues with endogenously determined arrival times

We study a class of queueing problems with endogenous arrival times formulated as non-cooperative n-person games in normal form. With multiple equilibria in pure strategies, these queueing games give rise to problems of tacit coordination. We first describe a Markov chain algorithm used to compute the symmetric mixed-strategy equilibrium solution, and then report the results of an experimental study of a large-scale ( n=20) queueing game with fixed service time, FIFO queue discipline, and no early arrivals. Our results show consistent and replicable patterns of arrival that provide strong support for mixed-strategy equilibrium play on the aggregate but not individual level.

Undecidability of the existence of pure Nash equilibria

The purpose of this paper is to show that for any positive integer n, there exists no algorithm which decides for each non-cooperative n-person game in strategic form with partially computable payoff functions whether it has a pure Nash equilibrium or not.

Two noncooperative games between a coalition and its surrounding in a class of n-person games with constant sum

A cooperative game engendered by a noncooperative n-person game (the master game) in which any subset of n players may form a coalition playing an antagonistic game against the residual players (the surrounding) that has a (Nash equilibrium) solution, is considered, along with another noncooperative game in which both a coalition and its surrounding try to maximize their gains that also possesses a Nash equilibrium solution. It is shown that if the master game is the one with constant sum, the sets of Nash equilibrium strategies in both above-mentioned noncooperative games (in which a coalition plays with (against) its surrounding) coincide.

Neural nets in a group decision process

In recent years there has been some interest in applying Artificial Adaptive Agents (AAA) to the study of complex adaptive systems, especially economic systems. Neural networks are frequently employed as AAA. Artificial neural nets mimic certain aspects of the physical structure and information processing of the human brain and their most attractive characteristic is their ability to learn a pattern from a given set of examples. In this study, we investigated the ability of neural nets to model human behavior in a group decision process. The context was a market entry game with a linear payoff function and binary decisions. The players had to decide, for each trial, whether or not to enter a market whose capacity is public knowledge. Human behavior in this situation has been modeled and empirically validated by the Nash equilibrium for noncooperative n-person games. A simulation of the game was performed with neural nets instead of human subjects. The nets were trained using the results of the games in which they participated. The simulation with groups of neural nets exhibits phenomena very similar to those observed in groups of human players.