Uniqueness and efficiency of Nash equilibrium in a family of randomly generated repeated games

This paper describes the results of an analysis of the Nash equilibrium in randomly generated repeated games. We study two families of games: symmetric bimatrix games G(A, B) with B = A^⊤ and nonsymmetric bimatrix games (the first includes the classical games of prisoner dilemma, battle of the sexes, and chickens). We use pure strategies, implemented by automata of size two, and different strategy domination criteria. We observe that, in this environment, the uniqueness and efficiency of equilibria outcomes is the typical result.     

Flipping the winner of a poset game

Partially-ordered set games, also called poset games, are a class of two-player combinatorial games. The playing field consists of a set of elements, some of which are greater than other elements. Two players take turns removing an element and all elements greater than it, and whoever takes the last element wins. Examples of poset games include Nim and Chomp. We investigate the complexity of computing which player of a poset game has a winning strategy. We give an inductive procedure that modifies poset games to change the nim-value which informally captures the winning strategies in the game. For a generic poset game G, we describe an efficient method for constructing a game ¬G such that the first player has a winning strategy if and only if the second player has a winning strategy on G. This solves the long-standing problem of whether this construction can be done efficiently. This construction also allows us to reduce the class of Boolean formulas to poset games, establishing a lower bound on the complexity of poset games.     

Bargaining property of nucleolus and τ-value in a class of TU-games

First, we introduce pairwise-bargained consistency with a reference point, and use as reference points the maxmin and the minmax value within pure strategies of a certain constant-sum bimatrix game, and also the game value within mixed strategies of it. Second, we show that the pairwise-bargained consistency with reference point being the maxmin or the minmax value determines the nucleolus in some class of transferable utility games. (This result is known in the bankruptcy games and the pseudoconcave games with respect to supersets of the managers.) This class of games whose element we call a pseudoconcave game with respect to essential coalitions, of course, includes the bankruptcy games and the pseudoconcave games with respect to supersets of the managers. It is proved that this class of games is exactly the same as the class of games which have a nonempty core that is determined only by one-person and (n − 1)-person coalition constraints. And we give a sufficient condition which guarantees that the bargaining set coincides with the core in this class of games. Third, we interpret the τ-value of a quasibalanced transferable utility game by the pairwise-bargained consistency with reference point being the game value. Finally, by combining the second and the third results, if a transferable utility game in this class is also semiconvex, then the nucleolus and the τ-value are characterized by the pairwise-bargained consistency with different reference points which are given by the associated bimatrix game.     

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.     

基于粒子群优化的混合模糊双矩阵对策求解

针对具有模糊策略集与模糊支付值的不确定性冲突环境,建立了混合模糊双矩阵对策模型。在假定模糊支付值为三角模糊数的情形下,采用了基于单个截集的模糊数线性排序函数,将模型清晰化后转化为双矩阵对策,并应用粒子群优化算法求解。最后,给出一个军事例子说明了模型的实用有效性和粒子群优化算法的高效性。     

Bayesian equilibria for uncertain bimatrix game with asymmetric information

In an uncertain bimatrix game, there are two solution concepts of \((\alpha ,\beta )\)-optimistic equilibrium strategy and \((u,v)\)-maximum chance equilibrium strategy. This paper goes further by assuming that the confidence levels \(\alpha , \beta \) and payoff levels \(u, v\) are private information. Then, the so-called uncertain bimatrix game with asymmetric information is investigated. Two solution concepts of Bayesian optimistic equilibrium strategy and Bayesian maximum chance equilibrium strategy as well as their existence theorems are presented. Moreover, sufficient and necessary conditions are given for finding the Bayesian equilibrium strategies. Finally, a two-firm advertising problem is analyzed for illustrating our modelling idea.     

A Game-theoretic approach to analyze interacting actors in GRL goal models

Goal-oriented requirements engineering aims to capture desired goals and strategies of relevant stakeholders during early requirements engineering stages, using goal models. Goal-oriented modeling techniques support the analysis of system requirements (especially non-functional ones) from an operationalization perspective, through the evaluation of alternative design options. However, conflicts and undesirable interactions between requirements produced from goals are inevitable, especially as stakeholders often aim for different objectives. In this paper, we propose an approach based on game theory and the Goal-oriented Requirement Language (GRL) to reconcile interacting stakeholders (captured as GRL actors), leading to reasonable trade-offs. This approach consists in building a payoff bimatrix that considers all actor’s valid GRL strategies, and computing its Nash equilibrium. Furthermore, we use two optimization techniques to reduce the size of the payoff bimatrix, hence reducing the computational cost of the Nash equilibrium. The approach goes beyond existing work by supporting nonzero-sum games, multiple alternatives, and inter-actor dependencies. We demonstrate the applicability of our game-theoretic modeling and analysis approach using a running example and two GRL models from the literature, with positive results on feasibility and applicability, including performance results.

     

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

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.     

The complexity of stochastic games

We consider the complexity of stochastic games—simple games of chance played by two players. We show that the problem of deciding which player has the greatest chance of winning the game is in the class NP co-NP.     

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).     

Seeking the Equilibrium Situations in Bimatrix Games

Determination of the Nash-equilibria in the bimatrix game was considered by reducing it to its equivalent nonconvex problem of optimization solved by the algorithm of global search based on the theory of global extremum for this problem. Efficiency of this method was illustrated by numerical solution of bimatrix games of sufficiently high dimensions.     

A survey of partial-observation stochastic parity games

We consider two-player zero-sum stochastic games on graphs with ω-regular winning conditions specified as parity objectives. These games have applications in the design and control of reactive systems. We survey the complexity results for the problem of deciding the winner in such games, and in classes of interest obtained as special cases, based on the information and the power of randomization available to the players, on the class of objectives and on the winning mode. On the basis of information, these games can be classified as follows: (a) partial-observation (both players have partial view of the game); (b) one-sided partial-observation (one player has partial-observation and the other player has complete-observation); and (c) complete-observation (both players have complete view of the game). The one-sided partial-observation games have two important subclasses: the one-player games, known as partial-observation Markov decision processes (POMDPs), and the blind one-player games, known as probabilistic automata. On the basis of randomization, (a) the players may not be allowed to use randomization (pure strategies), or (b) they may choose a probability distribution over actions but the actual random choice is external and not visible to the player (actions invisible), or (c) they may use full randomization. Finally, various classes of games are obtained by restricting the parity objective to a reachability, safety, Büchi, or coBüchi condition. We also consider several winning modes, such as sure-winning (i.e., all outcomes of a strategy have to satisfy the winning condition), almost-sure winning (i.e., winning with probability 1), limit-sure winning (i.e., winning with probability arbitrarily close to 1), and value-threshold winning (i.e., winning with probability at least ν, where ν is a given rational).     

Fuzzy differential games for nonlinear stochastic systems:suboptimal approach

A fuzzy differential game theory is proposed to solve the n-person (or n-player) nonlinear differential noncooperative game and cooperative game (team) problems, which are not easily tackled by the conventional methods. In the paper, both noncooperative and cooperative quadratic differential games are considered. First, the nonlinear stochastic system is approximated by a fuzzy model. Based on the fuzzy model, a fuzzy controller is proposed to deal with the noncooperative differential game in the sense of Nash equilibrium strategies or with the cooperative game in the sense of Pareto-optimal strategies. Using a suboptimal approach, the outcomes of the fuzzy differential games for both the noncooperative and the cooperative cases are parameterized in terms of an eigenvalue problem. Since the state variables are usually unavailable, a suboptimal fuzzy observer is also proposed in this study to estimate the states for these differential game problems. Finally, simulation examples are given to illustrate the design procedures and to indicate the performance of the proposed methods     

The Complete Solution of the Competitive Rank Selection Problem

Two decision-makers A and B observe sequentially a given permutation of n uniquely rankable options. A and B have one choice each (without recall) and both must make a choice. At each step only the relative ranks are known, and A has the priority of choice. At the end the (absolute) ranks are compared and the winner is the one who has chosen the better rank. Extending results by Enns and Ferenstein [6] and Berry et al. [1] this article gives, for both A and B , the optimal strategy and the corresponding winning probabilities. We show in particular that the limiting winning probabilities for A and B do exist, which closes a most important gap in the work of previous authors. This also provides an algorithm for numerically computing the limiting value of these probabilities. Although our proof is analytic in a strong sense, it is interesting to see that it would have been very hard to assemble it without the help of computer algebra. The reason is that the functions we have to investigate display subranges of indices which contrast considerably with respect to error terms when certain terms are replaced by approximations, and that computations were very helpful to locate those ranges where a particularly fine tuning of error estimates turned out to be indispensable. Received October 1997; revised March 15, 1998.     

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.     

Symbolic computational techniques for solving games

Games are useful in modular specification and analysis of systems where the distinction among the choices controlled by different components (for instance, the system and its environment) is made explicit. In this paper, we formulate and compare various symbolic computational techniques for deciding the existence of winning strategies. The game structure is given implicitly, and the winning condition is either a reachability game of the form p until q (for state predicates p and q) or a safety game of the form Always p.For reachability games, the first technique employs symbolic fixed-point computation using ordered binary decision diagrams (BDDs) [9]. The second technique checks for the existence of strategies that ensure winning within k steps, for a user-specified bound k, by reduction to the satisfiability of quantified boolean formulas. Finally, the bounded case can also be solved by reduction to satisfiability of ordinary boolean formulas, and we discuss two techniques, one based on encoding the strategy tree and one based on encoding a witness subgraph, for reduction to Sat. We also show how some of these techniques can be adopted to solve safety games. We compare the various approaches by evaluating them on two examples for reachability games, and on an interface synthesis example for a fragment of TinyOS [15] for safety games. We use existing tools such as Mocha [4], Mucke [7], Semprop [19], Qube [12], and Berkmin [13] and contrast the results.     

An improved algorithm on the content of realizable fuzzy matrices

An n×nn{\times}n fuzzy matrix A is called realizable if there exists an n×tn{\times}t fuzzy matrix B such that A=B\odot B^T,A=B\odot B^{T}, where \odot\odot is the max–min composition. Let r(A)=min{p:A=B\odot B^T, B ? L^n×p}.r(A)={min}\{p:A=B\odot B^{T}, B\in L^{n\times p}\}. Then r(A)r(A) is called the content of A. Since 1982, how to calculate r(A) for a given n×nn{\times}n realizable fuzzy matrix A was a focus problem, many researchers have made a lot of research work. X. P. Wang in 1999 gave an algorithm to find the fuzzy matrix B and calculate r(A) within [r(A)]ⁿ²[r(A)]^{n^{2}} steps. Therefore, to find a simpler algorithm is a problem what we have to consider. This paper makes use of the symmetry of the realizable fuzzy matrix A to simplify the algorithm of content r(A)r(A) based on the work of Wang (Chin Ann Math A 6: 701–706, 1999).