N-person Games Research Articles

Existence of equilibria for N-person games

Existence of equilibria for N-person games

On the Felsenthal Power Index

The paper that introduces the Felsenthal index is titled: ‘A well-behaved index of a priori P-Power for simple n-person games.’ In 2016, Felsenthal introduced his index for simple games. His definition does not base on the axiomatic approach. Then, Felsenthal regarded some properties and proved that his index satisfies a list of six reasonable and compelling postulates. Three of the properties that he regarded refer to the weighted games, but this fact does not reduce the definition of his index to weighted games. He proves that none of seven well-known efficient power indices proposed to date satisfies the list of postulates, indicating for each of them which of the six postulates violate. In this paper we extend some of his postulates, originally defined for weighted games, to simple games. The main objective of the article is to answer three open questions motivated in his article. In particular, we prove that his index may not be the unique one fulfilling the six proposed postulates, provide an axiomatic characterization for his index and, propose an impossibility result, which is obtained by adding a new postulate to a sublist of the postulates he considered. We also remark the existence of some alternative compelling postulates which are not satisfied for the index.

On the strong $$\beta$$-hybrid solution of an N-person game

On the strong $$\beta$$-hybrid solution of an N-person game

On the strong hybrid solution of an n-person game

We propose a new notion of coalitional equilibrium, the strong hybrid solution, which is a refinement of Zhao’s hybrid solution. It is well suited to study situations where people cooperate within coalitions but where coalitions compete with one another. In the strong hybrid solution, as opposed to the hybrid solution, the strategy profile assigned to each coalition is strongly Pareto optimal. We show that there exists a strong hybrid solution whenever preferences are partially quasi-transferable.

A Constructive Approach to Existence of Equilibria in Time-Inconsistent Stochastic Control Problems

We extend the construction of equilibria for linear-quadratic and mean-variance portfolio problems available in the literature to a large class of mean-field time-inconsistent stochastic control problems in continuous time. Our approach relies on a time discretization of the control problem via n-person games, which are characterized via the maximum principle using Backward Stochastic Differential Equations (BSDEs). The existence of equilibria is proved by applying weak convergence arguments to the solutions of n-person games. A numerical implementation is provided by approximating n-person games using finite Markov chains.

Efficiency of the Nash equilibrium search algorithm for n-person games in the general formulation

The efficiency of using the local search algorithm (а cutting plane algorithm ) to find the Nash equilibrium for n person games in a general formulation using Python software environments is studied. The use of multiplication of a multidimensional matrix by a vector in the local search procedure proved to be an effective tool.

Weak selection helps cheap but harms expensive cooperation in spatial threshold dilemmas

Public Goods Games (PGGs) are n-person games with dependence of individual fitness benefits on the collective investment by the players. We have studied a simple PGG scenario played out by cooperating (C) and defecting (D) agents, applying the highly nonlinear threshold benefit function in an individual-based lattice model. A semi-analytical approximation of the lattice model has been developed and shown to describe the dynamics fairly well in the vicinity of the steady state. Besides the expected outcomes (i.e., the negative effect on cooperator persistence of higher cooperation costs and/or more intensive mixing of the population) we have found a surprising, counter-intuitive effect of the strength of selection on the steady state of the model. The effect is different at low and high cooperation costs, and it shows up only in the lattice model, suggesting that stochastic effects and higher order spatial correlations due to the emergent spatial clustering of cooperators (not taken into account in the semi-analytical approximation) must be responsible for the unexpected results for which we propose an intuitive explanation, present a tentative demonstration, and shortly discuss their biological relevance.

Retraction

Abstract Original Article Title: Existence of k-People Stable Alliance in n-Player Cooperative Games Authors: Zheng Tao,1 Yuanwei Hu,2 Ziyi Pan,2 Junyi Zhao,2 and Dongao Xing2 This article has been retracted upon request from the authors due to errors they found in the conclusions. ASTM International February 2022

Discrete stop-or-go games

Dubins and Savage (How to gamble if you must: inequalities for stochastic processes, McGraw-Hill, New York, 1965) found an optimal strategy for limsup gambling problems in which a player has at most two choices at every state x at most one of which could differ from the point mass delta (x). Their result is extended here to a family of two-person, zero-sum stochastic games in which each player is similarly restricted. For these games we show that player 1 always has a pure optimal stationary strategy and that player 2 has a pure epsilon -optimal stationary strategy for every epsilon > 0. However, player 2 has no optimal strategy in general. A generalization to n-person games is formulated and epsilon -equilibria are constructed.

Game-Theoretic Demand Side Management of Thermostatically Controlled Loads for Smoothing Tie-Line Power of Microgrids

Thermostatically controlled loads (TCLs) are regarded as one of the promising resources for suppressing power fluctuations due to renewable energy (RENs). However, due to the great burdens of fully considering each users' characteristics, it is difficult to achieve unity of the individual optimal decisions and global optimum in demand-side management (DSM). This paper proposes a game-theoretic DSM that can optimize the global power consumption schedule by individual TCL user's optimization. By integrating the prediction of REN outputs into the pricing mechanism, the proposed DSM can guide the users to make their best power consumption schedules along with intermittent REN generations, thus to smooth the tie-line power of microgrids. In this paper, a novel pricing mechanism is firstly developed based on the concave N-person game theory, which is more adaptive and flexible compared with existing game-theoretic DSM. Then, an individual's power consumption optimization and its simplified model are developed considering the constraints of the TCL model and personal preferences. The simplified model can be easily solved and achieve a fast solution in the power consumption game. An implementation framework of the proposed DSM is further developed for practical application. Finally, numerical studies verify the effectiveness of the proposed models and methods.

Essential stability of the alpha cores of finite games with incomplete information

We introduce a variant of Milgrom and Weber’s (1985) model of n-person games with incomplete information (games for short) and define a correspondence that maps each game to its α-core (the α-core correspondences). Our main objective is to prove such a correspondence to be generically lower semicontinuous. For a multi-valued solution correspondence, the lower semicontinuity is relevant as a theoretical base for predicting outcomes using game-theoretic models. We introduce a family of games parametrized by both payoff functions and information structures (common priors), which allows simultaneous perturbations in those two parameters. We then appeal to Fort’s (1951) theorem to conclude that generic games are essential relative to the parameter space.

A Note: Nash Equilibrium Local Search Algorithm for n-Person Games in Multi-Matrix Settings

The effectivity of using the local search algorithm (mountain climbing) for searching Nash equilibrium in n-person games in multi-matrix settings using the Python software environment has been studied.

Terminate to Cooperate: Repeated Games in Self-Sustained Communities

In the current mobile world, repeated relationships (communities) must be self-sustained. We formulate a framework in which some or all players strategically choose whether to terminate or repeat an N-person game. A dynamic game ends when a certain number of players choose termination. To sustain the maximal set of outcomes while maintaining the community, the players may need to end the interaction as an equilibrium punishment. However, since termination is an absorbing state, players cannot reward one another for the appropriate use of termination. We show that voluntary termination is always incentive-compatible when players vote on game continuation/termination simultaneously under non-unanimous ending rules. For other cases, we construct a new mechanism to make termination incentive-compatible. The frontier of equilibrium outcomes can be larger than that of ordinary repeated games without the termination option, but, in that case, the set of equilibrium outcomes expands in the direction of unequal payoffs.

The Shapley value of coalitions to other coalitions

The Shapley value for an n-person game is decomposed into a 2n × 2n value matrix giving the value of every coalition to every other coalition. The cell ϕIJ(v, N) in the symmetric matrix is positive, zero, or negative, dependent on whether row coalition I is beneficial, neutral, or unbeneficial to column coalition J. This enables viewing the values of coalitions from multiple perspectives. The n × 1 Shapley vector, replicated in the bottom row and right column of the 2n × 2n matrix, follows from summing the elements in all columns or all rows in the n × n player value matrix replicated in the upper left part of the 2n × 2n matrix. A proposition is developed, illustrated with an example, revealing desirable matrix properties, and applicable for weighted Shapley values. For example, the Shapley value of a coalition to another coalition equals the sum of the Shapley values of each player in the first coalition to each player in the second coalition.

Fictitious play in networks

This paper studies fictitious play in networks of noncooperative two-person games. We show that continuous-time fictitious play converges to the set of Nash equilibria if the overall n-person game is zero-sum. Moreover, the rate of convergence is 1/τ, regardless of the size of the network. In contrast, arbitrary n-person zero-sum games with bilinear payoff functions do not possess the continuous-time fictitious-play property. As extensions, we consider networks in which each bilateral game is either strategically zero-sum, a weighted potential game, or a two-by-two game. In those cases, convergence requires a condition on bilateral payoffs or, alternatively, that the network is acyclic. Our results hold also for the discrete-time variant of fictitious play, which implies, in particular, a generalization of Robinson's theorem to arbitrary zero-sum networks. Applications include security games, conflict networks, and decentralized wireless channel selection.

Subgame Consistent Cooperative Behavior in an Extensive form Game with Chance Moves

We design a mechanism of the players’ sustainable cooperation in multistage n-person game in the extensive form with chance moves. When the players agreed to cooperate in a dynamic game they have to ensure time consistency of the long-term cooperative agreement. We provide the players’ rank based (PRB) algorithm for choosing a unique cooperative strategy profile and prove that corresponding optimal bundle of cooperative strategies satisfies time consistency, that is, at every subgame along the optimal game evolution a part of each original cooperative trajectory belongs to the subgame optimal bundle. We propose a refinement of the backwards induction procedure based on the players’ attitude vectors to find a unique subgame perfect equilibrium and use this algorithm to calculate a characteristic function. Finally, to ensure the sustainability of the cooperative agreement in a multistage game we employ the imputation distribution procedure (IDP) based approach, that is, we design an appropriate payment schedule to redistribute each player’s optimal payoff along the optimal bundle of cooperative trajectories. We extend the subgame consistency notion to extensive-form games with chance moves and prove that incremental IDP satisfies subgame consistency, subgame efficiency and balance condition. An example of a 3-person multistage game is provided to illustrate the proposed cooperation mechanism.

On nonlinear complementarity problems with applications

In this paper, it is shown that nonlinear complementarity problems has a unique solution if function is continuous and strongly monotone of higher order on nonnegative orthant of finite dimensional Euclidean space. The uniqueness of solution is discussed for both primal and dual for a symmetric dual nonlinear program. A unique nonnegative saddle point for a differentiable scalar function is obtained. Further, we obtained a unique equilibrium point of an n-person game.

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 the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games: 25 years later

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Existence of Equilibrium Strategies in Fuzzy Stochastic Games with Finite Sets of States and Decisions

Noncooperative discounted stochastic n-person games are considered; the payoffs at each step are represented by trapezoidal fuzzy numbers. The existence of stationary Nash equilibrium strategies is proved.