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Computing approximate Nash equilibria in network congestion games with polynomially decreasing cost functions
Distributed Computing ( IF 1.3 ) Pub Date : 2020-06-22 , DOI: 10.1007/s00446-020-00381-4
Vittorio Bilò , Michele Flammini , Gianpiero Monaco , Luca Moscardelli

We consider the problem of computing approximate Nash equilibria in monotone congestion games (a class of games generalizing network congestion games) with polynomially decreasing cost functions. In particular, we consider the case in which each resource j has a cost $$c_j$$ c j and the cost that each player incurs when using j is given by $$c_j/x^{\beta }$$ c j / x β for some constant $${\beta }>0$$ β > 0 , where x is the number of players using j . Observe that, for $${\beta }=1$$ β = 1 , we obtain the fundamental Shapley cost sharing method. We design a parametric distributed algorithm for computing $${\alpha }$$ α -approximate Nash equilibria. The complexity of this algorithm depends on $${\alpha }$$ α , being polynomial for $${\alpha }=\varOmega (p^{\beta })$$ α = Ω ( p β ) , for every fixed $${\beta }>0$$ β > 0 , with p being the number of players. Our algorithm provides the first non-trivial approximability results for this class of games and achieves an almost tight performance for network games in directed graphs. For the case of ring networks, we show that an approximate equilibrium can be polynomially computed for every fixed $${\alpha } >1$$ α > 1 . On the negative side, we prove that the problem of computing a Nash equilibrium in Shapley network cost sharing games is PLS -complete even in undirected graphs, where previous hardness results where known only in the directed case.

中文翻译:

在具有多项式递减成本函数的网络拥塞博弈中计算近似纳什均衡

我们考虑在具有多项式递减成本函数的单调拥塞博弈(一类泛化网络拥塞博弈的博弈)中计算近似纳什均衡的问题。特别地,我们考虑每个资源 j 的成本为 $$c_j$$ cj 并且每个玩家在使用 j 时产生的成本由 $$c_j/x^{\beta }$$ cj / x β 给出的情况对于某些常量 $${\beta }>0$$ β > 0 ,其中 x 是使用 j 的玩家数量。观察到,对于 $${\beta }=1$$ β = 1 ,我们获得了基本的 Shapley 成本分摊方法。我们设计了一个参数分布式算法来计算 $${\alpha }$$ α -approximate Nash equilibria。该算法的复杂性取决于 $${\alpha }$$ α ,对于 $${\alpha }=\varOmega (p^{\beta })$$ α = Ω ( p β ) 是多项式,对于每个固定$${\beta }>0$$ β > 0 , p 是玩家的数量。我们的算法为此类游戏提供了第一个非平凡的近似性结果,并在有向图中的网络游戏中实现了几乎严格的性能。对于环形网络的情况,我们表明可以对每个固定的 $${\alpha } >1$$ α > 1 用多项式计算近似平衡。在消极方面,我们证明了在 Shapley 网络成本分摊游戏中计算纳什均衡的问题即使在无向图中也是 PLS 完全的,其中先前的硬度结果仅在有向情况下已知。我们表明,对于每个固定的 $${\alpha } >1$$ α > 1 可以多项式计算近似均衡。在消极方面,我们证明了在 Shapley 网络成本分摊游戏中计算纳什均衡的问题即使在无向图中也是 PLS 完全的,其中先前的硬度结果仅在有向情况下已知。我们表明,对于每个固定的 $${\alpha } >1$$ α > 1 可以多项式计算近似均衡。在消极方面,我们证明了在 Shapley 网络成本分摊游戏中计算纳什均衡的问题即使在无向图中也是 PLS 完全的,其中先前的硬度结果仅在有向情况下已知。
更新日期:2020-06-22
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