N-person Games Research Articles

On Nash-solvability in pure stationary strategies of the deterministic [formula omitted]-person games with perfect information and mean or total effective cost

We study existence of Nash equilibria (NE) in pure stationary strategies in n-person positional games with no moves of chance, with perfect information, and with the mean or total effective cost function.We construct a NE-free three-person game with positive local costs, thus disproving the conjecture suggested in Boros and Gurvich (2003). Still, the following four problems remain open: Whether NE exist in all two-person games with total effective costs such that(I) all local costs are strictly positive or (II) there are no dicycles of the total cost zero?Whether NE exist in all n-person games with the terminal (transition-free) cost functions, provided all dicycles form a unique outcome c and(III) assuming that c is worse than any terminal outcome or (IV) without this assumption?For n=3 the case (I) (and hence (II)) is answered in the negative. This is the main result of the present paper. For n=2 the case (IV) (and hence (III)) was answered in the positive earlier.We briefly survey the above and some other negative and positive results on Nash-solvability in pure stationary strategies for the games under consideration.

Finding the Pure-Strategy Nash Equilibrium Using a Fixed-Point Algorithm

It is well known that determining whether a finite game has a pure-strategy Nash equilibrium is an NP-hard problem and it is an active research topic to find a Nash equilibrium recently. In this paper, an implementation of Dang's Fixed-Point iterative method is introduced to find a pure-strategy Nash equilibrium of a finite n-person game in normal form. There are two steps to find one pure-strategy Nash equilibrium in this paper. The first step is converting the problem to a mixed 0-1 linear programming formulation based on the properties of pure strategy and multilinear terms in the payoff function. In the next step, the Dangs method is used to solve the formulation generated in the former step. Numerical results show that this method is effective to find a pure-strategy Nash equilibrium of a finite n-person game in normal form.

Joint Effects of Asymmetric Payoff and Reciprocity Mechanisms on Collective Cooperation in Water Sharing Interactions: A Game Theoretic Perspective

Common-pool resource (CPR) dilemmas distinguish themselves from general public good problems by encompassing both social and physical features. This paper examines how a physical mechanism, namely asymmetric payoff; and a social mechanism, reciprocity; simultaneously affect collective cooperation in theoretical water sharing interactions. We present an iterative N-person game theoretic model to investigate the joint effects of these two mechanisms in a linear fully connected river system under three information assumptions. From a simple evolutionary perspective, this paper quantitatively addresses the conditions for Nash Equilibrium in which collective cooperation might be established. The results suggest that direct reciprocity increases every actor’s motivation to contribute to the collective good of the river system. Meanwhile, various upstream and downstream actors manifest individual disparities as a result of the direct reciprocity and asymmetric payoff mechanisms. More specifically, the downstream actors are less willing to cooperate unless there is a high probability that long-term interactions are ensured; however, a greater level of asymmetries is likely to increase upstream actors’ incentives to cooperate even though the interactions could quickly end. The upstream actors also display weak sensitivity to an increase in the total number of actors, which generally results in a reduction in the other actors’ motivation for cooperation. It is also shown that the indirect reciprocity mechanism relaxes the overall conditions for cooperative Nash Equilibrium.

Approximating Nash Equilibrium Using Genetic Algorithm

The problem of detecting the Nash equilibrium of a non cooperative n-person game is solved by introducing a non-linear optimization model that enables evolutionary search operators to converge towards Nash Equilibrium of a game.

Determinanty stosowanych strategii i osiąganych zysków w grach w postaci normalnej – podejście eksperymentalne

This study presents the results of an experiment on determinants of the strategy used by the participants and their profi tability. The study was based on N-person games in normal form 3x3. All games have clearly defined theoretical solution (the unique Nash equilibrium, which can be reached using the iterated elimination of strictly dominated strategy). The most important conclusion that can be drawn from this analysis is the failure of the theoretical model predictions based on Nash equilibrium. The use of non-equilibrium strategy for people with relatively little theoretical knowledge was due to the inability to find such a strategy. However, participants with a relatively high theoretical knowledge use non-equilibrium strategies deliberately expecting higher profi t. That is why theoretical knowledge is useful, but its use should be adapted to the situation in the real world. This enables players to achieve extra profits.

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.

Isoelastic oligopolies under uncertainty

Single-product oligopolies are examined with uncertain isoelastic price functions and linear cost functions. Each firm wants to maximize its expected profit and also wants to minimize its uncertainty by minimizing the variance. This multiobjective optimization problem is solved by the weighting method, where the utility function of each firm is a linear combination of the expectation and variance of its profit. The existence and uniqueness of the equilibrium of the resulting n-person game is proved and an efficient algorithm is suggested to compute the equilibrium. The asymptotic behavior of the equilibrium is also investigated. Complete stability and bifurcation analysis is presented. The theoretical results are verified by computer simulation.

A New Equilibrium for an n-Person Game with Fuzzy Parameters

SYNOPTIC ABSTRACT In this article, we consider an n-person strategic game involving fuzzy parameters. We introduce a concept of solution for this game that takes into account both the aspect of conflict and the aspect of decision making under uncertainty related to the fuzzy parameters. Then we give sufficient conditions for its existence and we propose a procedure for its determination. We conclude with an illustrative example.

The ex ante [formula omitted]-core for normal form games with uncertainty

In this paper we study the existence of the α-core for an n-person game with incomplete information. We follow a Milgrom–Weber–Balder formulation of a game with incomplete information. The players adopt behavioral strategies represented by Young measures. The game unrolls in one step at the ex ante stage. In this context, the mixed-extensions of the utility functions are not quasi-concave, and as a result the classical Scarf’s theorem cannot be applied. An approximation argument is used to overcome this lack of concavity.

A Closed-form Formula for the Fair Allocation of Gains in Cooperative N-person Games

Abstract This paper provides a closed-form optimal solution to the multi-objective model of the fair allocation of gains obtained by cooperation among all players. The optimality of the proposed solution is first proved. Then, the properties of the proposed solution are investigated. At the end, a numerical example in inventory control environment is given to demonstrate the application and to compare the results with the ones of an existing method.

Taylor approximation pricing for K-user multiple-input multiple-output (MIMO) interference channels

This study considers the immense problem of sum rate maximisation in K-user multiple-input multiple-output interference channels (IFC). In order to address this problem, the authors propose an iterative Taylor approximation algorithm to find the optimal transmit covariance matrices, where each user maximises the sum rate function through Taylor expansion. Owing to the distributed nature of the ITA algorithm, Taylor terms can be treated as prices, where each user collects current prices from the others to perform update. This process further leads us modelling it as a more general concave n-person game to analyse the existence and uniqueness of the Nash equilibrium (NE). Then an iterative covariance-price updating (ICP) algorithm is proposed for flexible updating and establishing a convergence process to obtain the optimal solution under NE uniqueness conditions. Simulation results show that both algorithms come with similar performance and both significantly outperform the conventional iterative waterfilling algorithm.

Evolution of quantum strategies on a small-world network

In this paper, quantum strategies are introduced within evolutionary games in order to inves- tigate the evolution of quantum strategies on a small-world network. Initially, certain quantum strategies are taken from the full quantum space at random and assigned to the agents who occupy the nodes of the network. Then, they play n-person quantum games with their neighbors according to the physical model of a quantum game. After the games are repeated a large number of times, a quantum strategy becomes the dominant strategy in the population, which is played by the majority of agents. However, if the number of strategies is increased, while the total number of agents remains constant, the dominant strategy almost disappears in the population because of an adverse environment, such as low fractions of agents with dif- ferent strategies. On the contrary, if the total number of agents rises with the increase of the number of strategies, the dominant strategy re-emerges in the population. In addition, from results of the evolution, it can be found that the fractions of agents with the dominant strategy in the population decrease with the increase of the number of agents n in a n-person game independent of which game is employed. If both classical and quantum strategies evolve on the network, a quantum strategy can outperform the classical ones and prevail in the population.

Investigation of N-Person Games by Agent-Based Modeling

The recent developments in agent-based modeling of several N-person games are reviewed and some features of the author’s software tools as well as some applications are described. After explaining the transition from two-person to N-person games, the classification and properties of N-person games, our software tools, and agent personalities, the following N-person games are considered: prisoners’ dilemma, chicken dilemma, battle of the sexes game, games with crossing payoff functions, a new game with parabolic payoff functions, transitions between different N-person games, and analytical solutions of N-person games. Several practical applications are then considered.

A STRATEGIC COOPERATIVE GAME-THEORETIC MODEL FOR MARKET SEGMENTATION WITH APPLICATION TO BANKING IN EMERGING ECONOMIES

Market segmentation is essential to target efficaciously core-segment customers and to obtain a competitive advantage. Firms when confronted by the range of market segments, have difficulty in deciding the core-segment customers who are the most probable purchasers of their product and services. We propose a novel fuzzy group multi-criteria method for market entry and segment evaluation and selection. This proposed method provides a comprehensive and systematic framework that combines bi-level multi-objective optimization with real option analysis (ROA) and fuzzy cooperative n-person game theory. The contribution of the proposed segment evaluation and selection method is fivefold: (1) it addresses the gaps in the marketing literature on the efficacious and effective assessment of market segments; (2) it provides a comprehensive and systematic framework that combines bi-level multi-objective optimization with ROA and fuzzy cooperative n-person game theory; (3) it considers fuzzy logic and fuzzy sets to represent ambiguous, uncertain or imprecise information; (4) it does not insist on consensus but synthesizes a representative outcome based on qualitative judgments and quantitative data; and (5) it is applicable to national and international market segmentation. The practical application of this proposed framework illustrates the efficacy of the procedures and algorithms.

The El Farol Bar Problem as an Iterated N-Person Game

The El Farol Bar Problem as an Iterated N-Person Game

Integrality gap analysis for bin packing games

A cooperative bin packing game is an N-person game, where the player set N consists of k bins of capacity 1 each and n items of sizes a1,…,an. The value v(S) of a coalition S of players is defined to be the maximum total size of items in S that can be packed into the bins of S. We analyze the integrality gap of the corresponding 0–1 integer program of the value v(N), thereby presenting an alternative proof for the non-emptiness of the 1/3-core for all bin packing games. Further, we show how to improve this bound ϵ≤1/3 (slightly) and point out that the conclusion in Matsui (2000) [9] is wrong (claiming that the bound 1/3 was tight). We conjecture that the true best possible value is ϵ=1/7. The results are obtained using a new “rounding technique” that we develop to derive good (integral) packings from given fractional ones.

Preference Coalition Formation Scheme for Buyer Coalition Services with Bundles of Items

Coalition stability is a major requirement in coalition formation. One important problem to achieve stability in n-person game theories is the assumption that the preference of each buyer is publicly known. The coalition is said to be stable if there are no objection by any subset of buyers according to their publicly known preferences. However, such assumption is often unrealistic in typical real-life situations. Individual buyers often have private preferences and make their decisions according to their own preferences instead. This study proposes a novel preference coalition formation scheme for buyer coalition services that attempts to consider private preference of individual buyers within the buyer coalition process. The theoretical foundations of the study are rooted in the fields of multi-criteria decision making, human practical reasoning, and n-person game theories, from which we design an appropriate scheme for our proposed buyer coalition framework with emphasis on private preferences of individual buyers. The authors validate their proposed scheme with simulation software developed to demonstrate results of a variety of practical situations.

Learning efficient Nash equilibria in distributed systems

An individualʼs learning rule is completely uncoupled if it does not depend directly on the actions or payoffs of anyone else. We propose a variant of log linear learning that is completely uncoupled and that selects an efficient (welfare-maximizing) pure Nash equilibrium in all generic n-person games that possess at least one pure Nash equilibrium. In games that do not have such an equilibrium, there is a simple formula that expresses the long-run probability of the various disequilibrium states in terms of two factors: (i) the sum of payoffs over all agents, and (ii) the maximum payoff gain that results from a unilateral deviation by some agent. This welfare/stability trade-off criterion provides a novel framework for analyzing the selection of disequilibrium as well as equilibrium states in n-person games.

Representation of transferable utility games by coalition production economies

We prove that, by the method of construction of a coalition production economy due to Sun et al. [Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n-person game. Journal of Mathematical Economics 44, 853–860], every transferable utility (TU) game can be generated by a coalition production economy. Namely, for every TU game, we can construct a coalition production economy that generates the given game. We briefly discuss the relationship between the core of a given TU game and the set of Walrasian payoff vectors for the induced coalition production economy.

Sequentially two-leveled egalitarianism for TU games: Characterization and application

A new linear value for cooperative transferable utility games is introduced. The recursive definition of the new value for an n-person game involves a sequential process performed at n−1 stages, applying the value to subgames with a certain size k,1⩽k<n, combining with the rule of two-leveled egalitarianism (additive normalization) in order to guarantee the efficiency property for the new value, sequentially two-leveled egalitarianism, shortly S2EG value, applied to subgames of size k+1. The new value will be characterized in various ways. The S2EG value differs from the Shapley value since, besides efficiency, linearity, and symmetry, it verifies an additional property with respect to so-called scale-dummy player (replacing dummy player property). Consequently, the S2EG value of a game may be determined as the solidarity value of the per-capita game (incorporating the proportional rule due to different levels of efficiency). Various potential representations of the new value are established. In the application to a land corn production economy, it yields allocations, in which the landlord’s interest coincides with striving for a maximum production level. For economies with the linear production function, not only the unique landlord but also all the workers have incentives to increase the scale of the economy.