论文标题
关于平均场和稀疏政权大图的随机游戏的案例研究
A case study on stochastic games on large graphs in mean field and sparse regimes
论文作者
论文摘要
我们研究一类线性二次随机差异游戏，其中每个玩家只能在给定图中与最近的邻居直接互动。我们发现，根据该图的归一化拉普拉斯矩阵的经验特征值分布，我们发现了任何横向图的马尔可夫平衡。这促进了各种图序列的大型人群渐近学，详细讨论了几个稀疏和密集的例子。特别是，平均字段游戏仅在密集的图形案例中才是正确的限制，即，当该学位在适当的意义上差异时。尽管平衡策略是非本地的，但取决于所有参与者的行为，我们使用相关衰减估算来证明混乱的传播导致密集和稀疏的制度，由于典型顶点之间的较大距离，因此稀疏的情况。如果不假设图是及传递的，我们还可以显示平均场游戏解决方案可用于在任何足够致密的图序列上构建分散的近似平衡。
We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semi-explicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph's normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.
