N-person Non-cooperative Game Research Articles

Stationary perfect equilibria of an n-person noncooperative bargaining game and cooperative solution concepts

This paper characterizes the stationary (subgame) perfect equilibria of an n-person noncooperative bargaining model with characteristic functions, and provides strategic foundations of some cooperative solution concepts such as the core, the bargaining set and the kernel. The contribution of this paper is twofold. First, we show that a linear programming formulation successfully characterizes the stationary (subgame) perfect equilibria of our bargaining game. We suggest a linear programming formulation as an algorithm for the stationary (subgame) perfect equilibria of a class of n-person noncooperative games. Second, utilizing the linear programming formulation, we show that stationary (subgame) perfect equilibria of n-person noncooperative games provide strategic foundations for the bargaining set and the kernel.

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.

Agent-Based Evolutionary Game-Theoretic Framework for Optimizing Investment Plans of Competitive Port Facilities

Abstract The current paper deals with a special case of the Network Design Problem (NDP) that arises when multiple authorities are controlling certain parts of a network and in particular when these are engaged into a deliberative environment, forming a competitive market for transportation services. This alteration departs significantly from the classical paradigm of the NDP extending the classical single leader-multiple followers Stackelberg game-theoretic formulation to its complete form of multiple leaders-multiple followers Competitive NDP (CNDP). In order to address the particular problem setup, a multilevel vector-optimization programming problem is formulated, which is tackled by a proposed novel Artificial Intelligence (AI)-based decentralized evolutionary optimization algorithm. The application of this framework is done over a freight network that is composed of multiple transportation means, and in particular over a complete transportation chain formed by shippers, maritime and land carriers and port authorities, where the latter are competing for profits. The market of maritime facilities is modeled as an n-person non-cooperative game among port authorities who control the attractiveness of their terminal facilities. By taking the above interdependencies into consideration, the estimation of the equilibrium point of the above formulation is made by creating a game-theoretic platform where Intelligent Agents (corresponding to port authorities) are evolving their strategies towards equilibrium points. Results from the application of the proposed framework into a realistic part of the East Mediterranean freight network are provided and discussed.

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.

A game-theory approach for job scheduling in networked manufacturing

This paper presents a new kind of scheduling solution for jobs in networked manufacturing environments. The main contributions of this study can be focused on three points: The first is to distinguish the concepts and requirements of job scheduling in the networked manufacturing environment form those in the traditional manufacturing environment. The second is to construct a game-theory mathematical model to deal with this new job scheduling problem. In this presented mathematical model, this new job scheduling problem is formulated as an N-person non-cooperative game with complete information. The players correspond to the jobs submitted, respectively, by related customers and the payoff of each job is defined as its makespan. Each player has a set of strategies which correspond to the feasible geographical distributive machines. Therefore, obtaining the optimal scheduling results is determined by the Nash equilibrium (NE) point of this game. In order to find the NE point, the last point is to design and develop a genetic algorithm (GA)-based solution algorithm to effectively solve this mathematical model. Finally, a numerical example is presented to demonstrate the feasibility of the approach.

Note on the equal split solution in an n-person noncooperative bargaining game

This note examines a noncooperative bargaining game model to implement the “equal split” solution in a transferable utility coalitional form game provided by Hart and Mas-Colell [Hart, S., Mas-Colell, A., 1996. Bargaining and value. Econometrica 64, 357–380]. We first clarify the relationship between the equal split solution and the Nash bargaining solution in a coalitional form game and extend the model to a nontransferable utility coalitional form game. We then provide a sufficient condition for generating the Nash bargaining solution payoff configuration and the equal split solution as the limit of the stationary subgame-perfect equilibrium payoffs of Hart and Mas-Colell's bargaining game when the probabilities of the breakdown of negotiations converge to zero.

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.

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.

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.

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.

Manipulation of Preferences and Relative Utilitarianism

Given n agents with von Neumann-Morgenstern utility functions who wish to divide m commodities, consider the n-person noncooperative game with strategies consisting of concave, increasing von Neumann-Morgenstern utility functions, and whose outcomes are the relative utilitarian solution. It is shown that any constrained equal-income competitive equilibrium allocation for the true utilities is a Nash equilibrium outcome for the noncooperative game. Conditions are presented under which these are the only pure strategy equilibrium outcomes.

Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games

The heart of the equilibrium selection theory of Harsanyi and Selten (1988, A General Theory of Equilibrium Selection in Games, Cambridge, MA: MIT Press) is given by the tracing procedure, a mathematical construction that adjusts arbitrary prior beliefs into equilibrium beliefs. Although the term “procedure” suggests a numerical approach, the tracing procedure itself is a nonconstructive method. In this paper we propose a homotopy algorithm that generates a path of strategies. By using lexicographic pivoting techniques, it can be shown that for the entire class of noncooperative N-person games, the path converges to an approximate Nash equilibrium, even when the starting point or the game is degenerate. The outcome of the algorithm is shown to be arbitrarily close to the equilibrium beliefs proposed by the tracing procedure. Therefore, the algorithm does not compute just any Nash equilibrium, but one with a sound game-theoretic underpinning. Like other homotopy algorithms, it is easily implemented on a computer. Journal of Economic Literature Classification Numbers: C63, C72.

Mathematical Economics and Descriptive Set Theory

The purpose of this paper is first to show that for any integer n≥2, there exist no Borel measurable necessary and sufficient conditions on n lower semi-continuous payoff functions defining a non-cooperative n-person game in strategic form which can assert the existence of non-cooperative Nash equilibria (in either pure or mixed strategies). Second we show that there exist no Borel measurable necessary and sufficient conditions on an exchange economy with preference relations that are represented by lower semi-continuous utility functions which can assert the existence of Walrasian equilibria. And third we show that there exist no Borel measurable necessary and sufficient conditions on a triple of a lower semi-continuous one-period return function, a compact-valued continuous constraint correspondence, and a discount factor defining a deterministic discrete infinite horizon macroeconomic model which can assert the existence of optimal plans starting at some point.

Manipulation of Preferences and Relative Utilitarianism

Given n agents with who wish to divide m commodities, consider the n-person noncooperative game with strategies consisting of concave increasing utility functions, and whose outcomes are the relative utilitarian solution. Any constrained equal-income competitive equilibrium allocation for the true utilities is shown to ba a Nash equilibrium outcome for the noncooperative game. Conditions are presented under which these are the only pure strategy equilibrium outcomes. Journal of Economic Literature Classification Numbers: C72, C78, D71.

Essential equilibria of n-person noncooperative games

In this paper, we prove that most of the games (in the sense of Baire Category) are essential or weakly essential for the following three models: (A) the games parametrized by payoffs; (B) games parametrized by payoffs and strategy spaces; and (C) games parametrized by payoffs and feasible strategy correspondences.

A convergence method for stochastic differential games with a small parameter

A stochastic differential game problem for wideband noise driven system is considered. Often in applications, one have a single realization, then expectation is not appropriate in the cost function. First we will consider the payoff structure in the pathwise but not necessarily in the expected value sense. For N-person noncooperative games, under very general conditions, it will be shown that the optimal equilibrium policies of the limit diffusion when applied to the physical processes, will be δ-equilibrium as the parameters ϵ > 0 and T→ ∞. A combination of direct averaging and perturbed test function techniques will be used in convergence analysis. Results are shown to hold when mathematical expectations are used in the payoff structure. Two person zero sum games can also be considered in this framework