Network games provide a natural machinery to compactly represent strategic interactions among agents whose payoffs exhibit sparsity in their dependence on the actions of others. Besides encoding interaction sparsity, however, real networks often exhibit a multi-scale structure, in which agents can be grouped into communities, those communities further grouped, and so on, and where interactions among such groups may also exhibit sparsity. We present a general model of multi-scale network games that encodes such multi-level structure. We then develop several algorithmic approaches that leverage this multi-scale structure, and derive sufficient conditions for convergence of these to a Nash equilibrium. Our numerical experiments demonstrate that the proposed approaches enable orders of magnitude improvements in scalability when computing Nash equilibria in such games. For example, we can solve previously intractable instances involving up to 1 million agents in under 15 minutes.

中文翻译：

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多尺度游戏：在具有组结构的网络上表示和解决游戏

网络游戏提供了一种自然的机制，可以紧凑地表示代理商之间的战略互动，这些代理商的收益在依赖他人行为方面表现出稀疏。但是，除了编码交互性稀疏性之外，真实网络还经常表现出多尺度的结构，在这种结构中，可以将代理分组为社区，将这些社区进一步分组，依此类推，并且这些组之间的交互也可能表现出稀疏性。我们提出了编码这种多层次结构的多尺度网络游戏的通用模型。然后，我们开发了几种利用这种多尺度结构的算法方法，并得出了将它们收敛到纳什均衡的充分条件。我们的数值实验表明，在此类游戏中计算Nash平衡时，所提出的方法可实现可扩展性的数量级提高。例如，我们可以在15分钟内解决涉及多达100万名特工的棘手实例。