Computing Nash equilibrium in multi-agent games is a longstanding challenge at the interface of game theory and computer science. It is well known that a general normal form game in N players and k strategies requires exponential space simply to write down. This Curse of Multi-Agents prompts the study of succinct games which can be written down efficiently. A canonical example of a succinct game is the graphical game which models players as nodes in a graph interacting with only their neighbors in direct analogy with markov random fields. Graphical games have found applications in wireless, financial, and social networks. However, computing the nash equilbrium of graphical games has proven challenging. Even for polymatrix games, a model where payoffs to an agent can be written as the sum of payoffs of interactions with the agent's neighbors, it has been shown that computing an epsilon approximate nash equilibrium is PPAD hard for epsilon smaller than a constant. The focus of this work is to circumvent this computational hardness by considering average case graph models i.e random graphs. We provide a quasipolynomial time approximation scheme (QPTAS) for computing an epsilon approximate nash equilibrium of polymatrix games on random graphs with edge density greater than poly(k, 1/epsilon, ln(N))$ with high probability. Furthermore, with the same runtime we can compute an epsilon-approximate Nash equilibrium that epsilon-approximates the maximum social welfare of any nash equilibrium of the game. Our primary technical innovation is an "accelerated rounding" of a novel hierarchical convex program for the nash equilibrium problem. Our accelerated rounding also yields faster algorithms for Max-2CSP on the same family of random graphs, which may be of independent interest.
翻译:多试玩游戏中的计算纳什平衡是游戏理论和计算机科学界面的长期挑战。 众所周知, 在游戏理论和计算机科学的界面中, 普通正常形式游戏在 N 玩家和 k 战略中需要光写下来的指数空间。 多代理者的诅咒促使了对简洁游戏的研究, 并且可以高效地写下来。 一个简单游戏的典型例子是图形游戏, 将玩家模拟为在图中与邻居直接比对 markov 随机字段进行互动的节点。 图形游戏在无线、 金融和社交网络中找到了应用。 然而, 计算图形游戏的纳什 quilbrium 已经证明具有挑战性。 即使对于多功能游戏来说, 向代理人支付的报酬可以作为与代理人的邻居进行互动的补偿总和。 简洁游戏的一个简单明的例子是, 将精度大约的纳什平衡计算成一个硬度的图形, 比常数小。 这项工作的重点是, 通过考虑普通的直径直径直径直径直径直径的直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径的游戏, 。 我们的直径直的直直距直径直距直直直距直直距的直直直直直直直直直直直直距直距直距距直的直距直距直距直距距距直距直距直距直距的直距运动直距运动直距运动直距的直距的直距的直距的直距距距距距距距距距距距距距的直距距距距的直距的直距的直的直的直的直的直距直距直距直距的直距的直距的直距的直距的直距比直距的直距的距的直距比直距距距距的距的距的直距的直距直距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距的距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距距