可行策略对应的图像拓扑下广义博弈Nash平衡的稳定性

以往关于广义博弈Nash平衡的稳定性的研究,均利用可行策略映射之间的一致度量.现考虑在更弱的度量下,利用可行策略映射图像之间的Hausdorff距离定义度量.在此弱图像拓扑下,证明了广义博弈空间的完备性,以及Nash平衡映射的上半连续性和紧性,进而得到广义博弈Nash平衡的通有稳定性.即在Baire分类的意义下,大多数的广义博弈都是本质的.     

Embedding processes in combinatorial game theory

Berlekamp asked the question “What is the habitat of ∗2?” (See Guy, 1996 [6].) It is possible to generalize the question and ask “For a game G, what is the largest n such that ∗n is a position of G?” This leads to the concept of the nim dimension. In Santos and Silva (2008) [8] a fractal process was proposed for analyzing the previous questions. For the same purpose, in Santos and Silva (2008) [9], an algebraic process was proposed. In this paper we implement a third idea related to embedding processes. With Alan Parr’s traffic lights, we exemplify the idea of estimating the “difficulty” of the game and proving that its nim dimension is infinite.     

Occupation games on graphs in which the second player takes almost all vertices

Given a connected graph G=(V,E), two players take turns occupying vertices v∈V by placing black and white tokens so that the current vertex sets B,W⊆V are disjoint, B∩W=0?, and the corresponding induced subgraphs G[B] and G[W] are connected any time. A player must pass whenever (s)he has no legal move. (Obviously, after this, the opponent will take all remaining vertices, since G is assumed connected.) The game is over when all vertices are taken, V=B^∗∪W^∗. Then, Black and White get b=|B^∗|/|V| and w=|W^∗|/|V|, respectively. Thus, the occupation game is one-sum, b+w=1, and we could easily reduce it to a zero-sum game by simply shifting the payoffs, b^′=b−1/2,w^′=w−1/2. Let us also notice that b≥0 and w≥0; moreover, b>0 and w>0 whenever |V|>1.[Let us remark that the so-called Chinese rules define similar payoffs for the classic game of GO, yet, the legal moves are defined in GO differently.]Like in GO, we assume that Black begins. It is easy to construct graphs in which Black can take almost all vertices, more precisely, for each ε>0 there is a graph G for which b>1−ε. In this paper we show that, somewhat surprisingly, there are also graphs in which White can take almost all vertices.     

Convex and exact games with non-transferable utility

We generalize exactness to games with non-transferable utility (NTU). A game is exact if for each coalition there is a core allocation on the boundary of its payoff set.     

Distributionally robust Optimal Control and MDP modeling

In this paper we discuss Optimal Control and Markov Decision Process (MDP) formulations of multistage optimization problems when the involved probability distributions are not known exactly, but rather are assumed to belong to specified ambiguity families. The aim of this paper is to clarify a connection between such distributionally robust approaches to multistage stochastic optimization.     

Adaptive allocation rules for hypergraph games

Cooperative games with hypergraph structure, or hypergraph games, assume that all players in a hyperlink or conference have to be present before communication. Contrary to this situation, assuming that whenever players leave a conference the remaining players can still communicate with each other, adaptive allocation rules for hypergraph games, being alternative extensions of the Myerson value and the position value respectively, are introduced in this paper. Axiomatic characterizations are also provided by considering players' absence.     

考虑边流量有限的网络路径博弈问题研究

考虑每条边有流量约束的网络路径博弈问题, 根据收益函数单调递增的特点分析其内在零和性质, 并建模为存在公共边的路径博弈模型。在寻找均衡解的过程中, 首先考虑非合作的情形, 在局中人风险中性的假设下, 给出了求Nash均衡流量分配的标号法并证明该均衡分配的唯一性。接着进一步考虑局中人合作的可能性, 给出模型求得所有局中人的整体最大收益, 并基于纳什谈判模型给出目标函数为凸函数的数学模型确定唯一收益分配方案。事实上, 该方案是对剩余价值的平均分配。最后给出一个算例, 验证本文理论和方法的可行性。关键词:流量约束; 均衡流量; 网络路径博弈; 收益分配     

连续区间策略下二人零和对策问题研究

考虑连续区间策略下的二人零和对策问题,研究其均衡策略的存在性。首先分析了完全信息下的二人零和对策问题,证明了该问题均衡策略的存在性并给出求解方法。然后进一步研究了收益函数不确定的不完全信息二人零和对策问题,在各局中人都认为对方是风险厌恶型的假设下,分析该类对策纯策略均衡的存在性,并通过研究纯策略均衡存在的充要条件给出判断并寻找纯策略均衡解的方法。最后给出一个数值算例,验证本文所提出方法的可行性。     

Equilibrium programming using proximal-like algorithms

We compute constrained equilibria satisfying an optimality condition. Important examples include convex programming, saddle problems, noncooperative games, and variational inequalities. Under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization. In the main algorithm each stage of computation requires two proximal steps, possibly using Bregman functions. One step serves to predict the next point; the other helps to correct the new prediction. To enhance practical applicability we tolerate numerical errors. Research supported partly by the Norwegian Research Council, project: Quantec 111039/401.