广义Nash平衡点和切换控制在对策论中的应用

通过把平衡点和决策者的动机耦合的方法,提出了广义纳什平衡点这一新概念.决策者的动机通常有两类：一是最大化自己的利益,另一则是最大化对手的利益.如果每一个决策者的动机都是第一类,一个理性的群体就会形成,整个系统最终会达到第一类平衡点（也就是经典的纳什平衡点）.如果每一个决策者的动机都是第二类,一个有智慧的群体就会形成,整个系统最终会达到第二类平衡点.同时,切换控制被用来帮助决策者确定他们的动机.     

Consensus in Noncooperative Dynamic Games: A Multiretailer Inventory Application

We focus on Nash equilibria and Pareto optimal Nash equilibria for a finite horizon noncooperative dynamic game with a special structure of the stage cost. We study the existence of these solutions by proving that the game is a potential game. For the single-stage version of the game, we characterize the aforementioned solutions and derive a consensus protocol that makes the players converge to the unique Pareto optimal Nash equilibrium. Such an equilibrium guarantees the interests of the players and is also social optimal in the set of Nash equilibria. For the multistage version of the game, we present an algorithm that converges to Nash equilibria, unfortunately, not necessarily Pareto optimal. The algorithm returns a sequence of joint decisions, each one obtained from the previous one by an unilateral improvement on the part of a single player. We also specialize the game to a multiretailer inventory system.     

Learning through reinforcement for N-person repeated constrained games

The design and analysis of an adaptive strategy for N-person averaged constrained stochastic repeated game are addressed. Each player is modeled by a stochastic variable-structure learning automaton. Some constraints are imposed on some functions of the probabilities governing the selection of the player's actions. After each stage, the payoff to each player as well as the constraints are random variables. No information concerning the parameters of the game is a priori available. The "diagonal concavity" conditions are assumed to be fulfilled to guarantee the existence and uniqueness of the Nash equilibrium. The suggested adaptive strategy which uses only the current realizations (outcomes and constraints) of the game is based on the Bush-Mosteller reinforcement scheme in connection with a normalization procedure. The Lagrange multipliers approach with a regularization is used. The asymptotic properties of this algorithm are analyzed. Simulation results illustrate the feasibility and the performance of this adaptive strategy.     

Graphical Congestion Games

We consider congestion games with linear latency functions in which each player is aware only of a subset of all the other players. This is modeled by means of a social knowledge graph G in which nodes represent players and there is an edge from i to j if i knows j. Under the assumption that the payoff of each player is affected only by the strategies of the adjacent ones, we first give a complete characterization of the games possessing pure Nash equilibria. Namely, if the social graph G is undirected, the game is an exact potential game and thus isomorphic to a classical congestion game. As a consequence, it always converges and possesses Nash equilibria. On the other hand, if G is directed an equilibrium is not guaranteed to exist, but the game is always convergent and an equilibrium can be found in polynomial time if G is acyclic, even if finding the best equilibrium remains an intractable problem.     

Constrained Games: The Impact of the Attitude to Adversary's Constraints

We consider properties of constrained games, where the strategy set available to a player depends on the choice of strategies made by other players. We show that the utilities of each player associated with that player's own performance and constraints are not sufficient to model a constrained game and to define equilibria; for the latter, one also needs to model how a player values the fact that other players meet their constraints. We study three different approaches to other players' constraints, and show that they exhibit completely different equilibrium behaviors. Further, we study a general class of stochastic games with partial information, and focus on the case where the players are indifferent to whether the constraints of other players hold.     

Equilibria problems on games: Complexity versus succinctness

We study the computational complexity of problems involving equilibria in strategic games and in perfect information extensive games when the number of players is large. We consider, among others, the problems of deciding the existence of a pure Nash equilibrium in strategic games or deciding the existence of a pure Nash or a subgame perfect Nash equilibrium with a given payoff in finite perfect information extensive games. We address the fundamental question of how can we represent a game with a large number of players? We propose three ways of representing a game with different degrees of succinctness for the components of the game. For perfect information extensive games we show that when the number of moves of each player is large and the input game is represented succinctly these problems are PSPACE-complete. In contraposition, when the game is described explicitly by means of its associated tree all these problems are decidable in polynomial time. For strategic games we show that the complexity of deciding the existence of a pure Nash equilibrium depends on the succinctness of the game representation and then on the size of the action sets. In particular we show that it is NP-complete, when the number of players is large and the number of actions for each player is constant, and that the problem is -complete when the number of players is a constant and the size of the action sets is exponential in the size of the game representation. Again when the game is described explicitly the problem is decidable in polynomial time.     

Matrix games with missing,interval, and ambiguous lottery payoffs of pure strategy profiles and compound strategy profiles

In a matrix game, the interactions among players are based on the assumption that each player has accurate information about the payoffs of their interactions and the other players are rationally self‐interested. As a result, the players should definitely take Nash equilibrium strategies. However, in real‐life, when choosing their optimal strategies, sometimes the players have to face missing, imprecise (i.e., interval), ambiguous lottery payoffs of pure strategy profiles and even compound strategy profile, which means that it is hard to determine a Nash equilibrium. To address this issue, in this paper we introduce a new solution concept, called ambiguous Nash equilibrium, which extends the concept of Nash equilibrium to the one that can handle these types of ambiguous payoff. Moreover, we will reveal some properties of matrix games of this kind. In particular, we show that a Nash equilibrium is a special case of ambiguous Nash equilibrium if the players have accurate information of each player's payoff sets. Finally, we give an example to illustrate how our approach deals with real‐life game theory problems.     

On the complexity of constrained Nash equilibria in graphical games

A widely accepted rational behavior for non-cooperative players is based on the notion of Nash equilibrium. Although the existence of a Nash equilibrium is guaranteed in the mixed framework (i.e., when players select their actions in a randomized manner) in many real-world applications the existence of “any” equilibrium is not enough. Rather, it is often desirable to single out equilibria satisfying some additional requirements (in order, for instance, to guarantee a minimum payoff to certain players), which we call constrained Nash equilibria.In this paper, a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games, where a player p may directly be interested in some of the other players only, called the neighbors of p. This setting is very useful for modeling large population games, where typically each player does not directly depend on all the players, and representing her utility function extensively is either inconvenient or infeasible.Based on this framework, the complexity of deciding the existence and of computing constrained equilibria is then investigated, in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players’ interactions. The analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies, i.e., when players are forced to deterministically choose the action to perform. In particular, for this latter case, restrictions on players’ interactions and on constraints are identified, that make the computation of Nash equilibria an easy problem, for which polynomial and highly-parallelizable algorithms are presented.     

Quantum games and quaternionic strategies

It is well-known that the phenomenon of entanglement plays a fundamental role in quantum game theory. Occasionally, games constructed via maximally entangled initial states (MEIS) will have new Nash equilibria yielding to the players higher payoffs than the ones they receive in the classical version of the game. When examining these new games for Nash equilibrium payoffs, a fundamental question arises; does a suitable choice of an MEIS improve the lot of the players? In this paper, we show that the answer to this question is yes for at least the case of a variant of the well-known two player, two strategy game of Chicken. To that end, we generalize Landsburg’s quaternionic representation of the payoff function of two player, two strategy maximally entangled states to games where the initial state is chosen arbitrarily from a circle of maximally entangled initial states and for the corresponding quantized games show the existence of superior Nash equilibrium payoffs when an MEIS is appropriately chosen.     

Computation of Multi-facility Location Nash Equilibria on a Network Under Quantity Competition

We deal with the location-quantity problem for competing firms when they locate multiple facilities and offer the same type of product. Competition is performed under delivered quantities that are sent from the facilities to the customers. This problem is reduced to a location game when the competing firms deliver the Cournot equilibrium quantities. While existence conditions for a Nash equilibrium of the location game have been discussed in many contributions in the literature, computing an equilibrium on a network when multiple facilities are to be located by each firm is a problem not previously addressed. We propose an integer linear programming formulation to fill this gap. The formulation solves the profit maximization problem for a firm, assuming that the other firms have fixed their facility locations. This allows us to compute location Nash equilibria by the best response procedure. A study with data of Spanish municipalities under different scenarios is presented and conclusions are drawn from a sensitivity analysis.

     

Coordination via genetic learning

This paper uses genetic algorithms to model the solution to a coordination problem through learning. The coordination problem arises out of the following two stage game. At core of the two stage game is a model of social interaction in which players use visible goods as signals about the identity of other players. If these signals are informative enough, players use them to condition their social interaction. Importantly, accurate signals are mutually beneficial. This game is then wrapped in another in which players choose their visible goods. There are many types of players and many visible goods that could be used to signal type. There are many equilibria of the two stage game, some of which allow individuals to perfectly signal their type in all social interactions, and others of which do not. The perfect signaling equilibria Pareto dominate the others, but since there are many of them, the players face a difficult coordination problem. We approach this coordination problem using genetic algorithms to simulate learning in this game. A player is a genetic code that evolves via selection.Questions of primary interest concern the set of parameter values such that the players manage to solve the coordination problem. The results of simulations indicate that the convergence of the genetic algorithm to a perfect signaling equilibrium depends on the values of the parameters that determine players' payoffs.We analyze two different scenarios. In one scenario, each player makes one (unconditional) decision to either use the visible goods displayed by other players as type signals, or to ignore the visible goods they display. In the other scenario, the decision to regard a player's visible good as a type signal is conditional on the good displayed. Our results indicate that it is easier for players to solve the coordination problem under the second scenario.     

Complexity and stochastic evolution of dyadic networks

A strategic model of network formation is developed which permits unreliable links and organizational costs. Finding a connected Nash network which guarantees a given payoff to each player proves to be an NP-hard problem. For the associated evolutionary game with asynchronous updating and logit updating rules, the stochastically stable networks are characterized.     

Bounded budget betweenness centrality game for strategic network formations

In computer networks and social networks, the betweenness centrality of a node measures the amount of information passing through the node when all pairs are conducting shortest path exchanges. In this paper, we introduce a strategic network formation game in which nodes build connections subject to a budget constraint in order to maximize their betweenness in the network. To reflect real world scenarios where short paths are more important in information exchange in the network, we generalize the betweenness definition to only count shortest paths with a length limit ? in betweenness calculation. We refer to this game as the bounded budget betweenness centrality game and denote it as ?- B³C game, where ? is the path length constraint parameter.We present both complexity and constructive existence results about Nash equilibria of the game. For the nonuniform version of the game where node budgets, link costs, and pairwise communication weights may vary, we show that Nash equilibria may not exist and it is NP-hard to decide whether Nash equilibria exist in a game instance. For the uniform version of the game where link costs and pairwise communication weights are one and each node can build k links, we construct two families of Nash equilibria based on shift graphs, and study the properties of Nash equilibria. Moreover, we study the complexity of computing best responses and show that the task is polynomial for uniform 2- B³C games and NP-hard for other games (i.e. uniform ?- B³C games with ?≥3 and nonuniform ?- B³C games with ?≥2).     

On Equilibria in Quantitative Games with Reachability/Safety Objectives

In this paper, we study turn-based multiplayer quantitative non zero-sum games played on finite graphs with reachability objectives. In this framework each player aims at reaching his own goal as soon as possible. We focus on existence results for two solution concepts: Nash equilibrium and secure equilibrium. We prove the existence of finite-memory Nash (resp. secure) equilibria in n-player (resp. 2-player) games. For the case of Nash equilibria, we extend our result in two directions. First, we show that finite-memory Nash equilibria still exist when the model is enriched by allowing n-tuples of positive costs on edges (one cost by player). Secondly, we prove the existence of Nash equilibria in quantitative games with both reachability and safety objectives.     

Reaching pareto-optimality in prisoner’s dilemma using conditional joint action learning

We consider the learning problem faced by two self-interested agents repeatedly playing a general-sum stage game. We assume that the players can observe each other’s actions but not the payoffs received by the other player. The concept of Nash Equilibrium in repeated games provides an individually rational solution for playing such games and can be achieved by playing the Nash Equilibrium strategy for the single-shot game in every iteration. Such a strategy, however can sometimes lead to a Pareto-Dominated outcome for games like Prisoner’s Dilemma. So we prefer learning strategies that converge to a Pareto-Optimal outcome that also produces a Nash Equilibrium payoff for repeated two-player, n-action general-sum games. The Folk Theorem enable us to identify such outcomes. In this paper, we introduce the Conditional Joint Action Learner (CJAL) which learns the conditional probability of an action taken by the opponent given its own actions and uses it to decide its next course of action. We empirically show that under self-play and if the payoff structure of the Prisoner’s Dilemma game satisfies certain conditions, a CJAL learner, using a random exploration strategy followed by a completely greedy exploitation technique, will learn to converge to a Pareto-Optimal solution. We also show that such learning will generate Pareto-Optimal payoffs in a large majority of other two-player general sum games. We compare the performance of CJAL with that of existing algorithms such as WOLF-PHC and JAL on all structurally distinct two-player conflict games with ordinal payoffs.     

WSN中利用贝叶斯博弈的数据传递策略

在无线传感器网络中,数据的传递策略对网络的能量损耗具有重要的影响,为此,提出了一个基于贝叶斯博弈的数据传递模型。在该模型中网络节点为了获取最大的收益,在考虑自身能量水平的基础上,适当的调整发送/转发的数据量。当节点发送/转发的数据满足一定条件时,网络存在均衡状态。仿真结果表明,该基于博弈论的数据传递策略在均衡状态下能够明显降低能量损耗,延长网络的使用寿命。     

Equilibrium Trajectories in Dynamical Bimatrix Games with Average Integral Payoff Functionals

This paper considers models of evolutionary non-zero-sum games on the infinite time interval. Methods of differential game theory are used for the analysis of game interactions between two groups of participants. We assume that participants in these groups are controlled by signals for the behavior change. The payoffs of coalitions are defined as average integral functionals on the infinite horizon. We pose the design problem of a dynamical Nash equilibrium for the evolutionary game under consideration. The ideas and approaches of non-zero-sum differential games are employed for the determination of the Nash equilibrium solutions. The results derived in this paper involve the dynamic constructions and methods of evolutionary games. Much attention is focused on the formation of the dynamical Nash equilibrium with players strategies that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. An application of the minimax approach for constructing optimal control strategies generates dynamical Nash equilibrium trajectories yielding better results in comparison to static solutions and evolutionary models with the replicator dynamics. Finally, we make a comparison of the dynamical Nash equilibrium trajectories for evolutionary games with the average integral payoff functionals and the trajectories for evolutionary games with the global terminal payoff functionals on the infinite horizon.     

Random Bimatrix Games Are Asymptotically Easy to Solve (A Simple Proof)

We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player’s payoffs are bounded and independent random variables, not necessarily identically distributed, but with almost common expectations. We show that the completely mixed uniform strategy profile, i.e., the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a $\sqrt{\frac{\ln n}{n}}$ -Nash equilibrium and a $\sqrt{\frac{3\ln n}{n}}$ -well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ?-well-supported Nash equilibrium where ? tends to zero as n tends to infinity.     

无线多跳网络基于位势博弈的传输调度方法*

为了进一步提高无线多跳网络的吞吐量,采用物理干扰模型,考虑链路速率可以随信干噪比(signal tointerference plus noise ratio,SINR)动态可调,提出了基于位势博弈的传输调度算法。通过设计合适的位势函数,使得纳什均衡点的存在性和收敛性都得到保证。同时,每个参与者在最小化自己支付的同时,使全局函数达到最优。仿真结果表明,该算法具有较好的吞吐量性能,而且有较快的收敛速度。     

A Numerical Toolbox to Solve <Emphasis Type="Italic">N</Emphasis>-Player Affine LQ Open-Loop Differential Games

We present an algorithm and a corresponding MATLAB numerical toolbox to solve any form of infinite-planning horizon affine linear quadratic open-loop differential games. By rewriting a specific application into the standard framework one can use the toolbox to calculate and verify the existence of both the open-loop non-cooperative Nash equilibrium (equilibria) and cooperative Pareto equilibrium (equilibria). In case there is more than one equilibrium for the non-cooperative case, the toolbox determines all solutions that can be implemented as a feedback strategy. Alternatively, the toolbox can apply a number of choice methods in order to discriminate between multiple equilibria. The user can predefine a set of coalition structures for which they would like to calculate the non-cooperative Nash solution(s). It is also possible to specify the relative importance of each player in any coalition structure. Furthermore, the toolbox offers plotting facilities as well as other options to analyse the outcome of the game. For instance, it is possible to disaggregate each player’s total loss into its contributing elements. The toolbox is available as a freeware from the authors of this paper.