关联双矩阵博弈
Incidence Bimatrix Games
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中文总结 AI 辅助
本研究针对关联双矩阵博弈，证明无环图下玩家I的均衡策略唯一，玩家II的均衡策略为路径矩阵列向量的凸包，推广了Bapat与Tijs1997年的零和博弈结果。
中文摘要 AI 辅助
我们求解一类与图相关的自然双矩阵博弈。考虑有限有向图 $G=(V,E)$，其中玩家I的策略集为顶点集 $V$，玩家II的策略集为边集 $E$。存在两组正权重 $\{\{\beta_e\rbrace\rbrace_{e\in E}$ 和 $\{\beta_e\rbrace\rbrace_{e\in E}$。若玩家I选择顶点 $v$，玩家II选择边 $e$，当 $v$ 与 $e$ 不关联时，双方收益均为0；当边 $e$ 从 $v$ 出发时，玩家I获得 $\beta_e$，玩家II获得 $-\beta_e$；当边 $e$ 终止于 $v$ 时，玩家I获得 $-\beta_e$，玩家II获得 $\beta_e$。该博弈的收益矩阵是图 $G$ 的加权关联矩阵。我们证明，当图为无环图时，玩家I在任何均衡中都有唯一策略，在该策略下，每个顶点被选中的概率与从该顶点出发的所有有向路径的最大长度成正比。定义图的路径矩阵后，可证明玩家II的所有均衡策略集合是路径矩阵列向量的凸包。本研究推广了Bapat和Tijs（1997）针对零和博弈的早期结果。
英文摘要
We solve a natural bimatrix game related to graphs. We consider a finite directed graph $G=(V,E),$ where the strategy set of Player I is the set of vertices $V$ and that of Player II is the set of edges $E.$ There are two sets of positive weights ${\{α_e\}}_{e\in E}$ and ${\{β_e\}}_{e\in E}.$ If Player I chooses a vertex $v$ and Player II chooses an edge $e,$ then the payoff to both players is zero if $v$ and $e$ are not incident. If $e$ originates from $v,$ then Player I obtains $α_e$ and Player II obtains $-β_e.$ If $e$ terminates at $v,$ then Player I obtains $-α_e$ and Player II obtains $β_e.$ For this game the payoff matrices are weighted incidence matrices of the graph $G.$ We show that when the graph is acyclic, Player I has a unique strategy in any equilibrium. At this strategy, every vertex is chosen with a probability that is proportional to the maximum length over all directed paths originating from that vertex. Defining the path matrix of the graph, it is shown that the set of all equilibrium strategies of Player II is the convex hull of the column vectors of the path matrix. This work extends earlier results of Bapat and Tijs (1997) for zero-sum games.
