We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either game and, thus, leads – via our first algorithm – to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd (Math Op Res 41(3):1125–1134 2016, https://doi.org/10.1287/moor.2015.0771).

我们研究了两个经典的资源分配博弈的均衡计算问题：原子可分裂拥塞博弈和多市场古诺寡头。对于具有单例策略和特定于玩家的仿射成本函数的原子可分裂拥塞游戏，我们设计了第一个多项式时间算法来计算纯Nash平衡。我们的算法是组合的，并在有理输入的情况下计算精确的平衡。这个想法是为一个相关的整体可分裂计算一个平衡单例拥塞游戏，在这种游戏中，玩家只能将他们的需求分解为一个通用数据包大小的整数倍。尽管以前在文献中曾考虑过积分博弈，但尚无计算平衡的多项式时间算法。同样对于此类，我们设计了第一个多项式时间算法，并将其用作主要算法的构建块。然后，我们开发了多项式时间可计算的变换，将具有公司特定的仿射价格函数和二次成本的多市场古诺竞争游戏映射到相关的原子可分裂拥塞游戏，如上所述。转换可以在任何一个游戏中都保持平衡，因此，通过我们的第一个算法，导致了计算古诺平衡的多项式时间算法。最后，我们对可拆分可分博弈的分析暗示了实古诺均衡与整数古诺均衡之间差异的新界限。界限可以看作是托德获得的单一市场寡头的最新界限的概括（Math Op Res 41（3）：1125-1134 2016，https://doi.org/10.1287/moor.2015.0771）。