We consider the question of whether, and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider a sequence of games with an increasing number of players where each player's weight tends to zero. We prove that all (mixed) Nash equilibria of the finite games converge to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider again an increasing number of players but now each player has a unit weight and participates in the game with a probability tending to zero. In this case, the Nash equilibria converge to the set of Wardrop equilibria of a different nonatomic game with suitably defined costs. The latter can also be seen as a Poisson game in the sense of Myerson (1998), establishing a precise connection between the Wardrop model and the empirical flows observed in real traffic networks that exhibit stochastic fluctuations well described by Poisson distributions. In both settings we give explicit upper bounds on the rates of convergence, from which we also derive the convergence of the price of anarchy. Beyond the case of congestion games, we establish a general result on the convergence of large games with random players towards Poisson games.

中文翻译：

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大型原子拥塞博弈的收敛

我们考虑的问题是，Wardrop 均衡是否以及在何种意义上为具有大量小玩家的原子不可分裂拥塞博弈中的纳什均衡提供了良好的近似。我们研究了小玩家的两种不同定义。在第一个设置中，我们考虑一个游戏序列，其中每个玩家的权重都趋于零。我们证明有限博弈的所有（混合）纳什均衡收敛到相应非原子极限博弈的 Wardrop 均衡集。在第二个设置中，我们再次考虑越来越多的玩家，但现在每个玩家都有一个单位权重，并且以趋于零的概率参与游戏。在这种情况下，纳什均衡收敛到具有适当定义成本的不同非原子博弈的 Wardrop 均衡集。后者也可以看作是 Myerson (1998) 意义上的泊松博弈，在 Wardrop 模型和实际交通网络中观察到的经验流量之间建立了精确的联系，这些流量网络表现出泊松分布很好地描述的随机波动。在这两种情况下，我们都给出了收敛速度的明确上限，从中我们还得出了无政府状态价格的收敛性。除了拥塞博弈的情况外，我们还建立了具有随机玩家的大型博弈向泊松博弈收敛的一般结果。在这两种情况下，我们都给出了收敛速度的明确上限，从中我们还得出了无政府状态价格的收敛性。除了拥塞博弈的情况外，我们还建立了具有随机玩家的大型博弈向泊松博弈收敛的一般结果。在这两种情况下，我们都给出了收敛速度的明确上限，从中我们还得出了无政府状态价格的收敛性。除了拥塞博弈的情况外，我们还建立了具有随机玩家的大型博弈向泊松博弈收敛的一般结果。