Studying the impact of cooperation in strategic settings is one of the cornerstones of algorithmic game theory. Intuitively, allowing more cooperation yields equilibria that are more beneficial for the society of agents. However, for many games it is still an open question how much cooperation is actually needed to ensure socially good equilibria. We contribute to this research endeavor by analyzing the benefits of cooperation in a network formation game that models the creation of communication networks via the interaction of selfish agents. In our game, agents that correspond to nodes of a network can buy incident edges of a given weighted host graph to increase their centrality in the formed network. The cost of an edge is proportional to its length, and both endpoints must agree and pay for an edge to be created. This setting is known for having a high price of anarchy. To uncover the impact of cooperation, we investigate the price of anarchy of our network formation game with respect to multiple solution concepts that allow for varying amounts of cooperation. On the negative side, we show that on host graphs with arbitrary edge weights even the strongest form of cooperation cannot improve the price of anarchy. In contrast to this, as our main result, we show that cooperation has a significant positive impact if the given host graph has metric edge weights. For this, we prove asymptotically tight bounds on the price of anarchy via a novel proof technique that might be of independent interest and can be applied in other models with metric weights.

翻译：研究合作在策略性环境中的影响是算法博弈论的基石之一。直观而言，允许更多合作能够产生对社会主体更有利的均衡。然而，对于许多博弈而言，究竟需要多少合作才能确保社会性良好的均衡仍然是一个悬而未决的问题。我们通过分析网络形成博弈中合作的效益，为这一研究领域做出贡献。该博弈通过自私主体的互动来模拟通信网络的创建过程。在我们的博弈中，对应于网络节点的主体可以通过购买给定加权宿主图的关联边来提高其在形成网络中的中心性。边的成本与其长度成正比，且必须由两端点共同同意并支付费用才能创建边。已知该设定具有较高的无政府代价。为揭示合作的影响，我们基于允许多种合作程度的解概念，研究了该网络形成博弈的无政府代价。在负面结果方面，我们证明在具有任意边权重的宿主图上，即使最强形式的合作也无法改善无政府代价。与此形成对比的是，作为主要结果，我们证明若给定宿主图具有度量边权重，合作将产生显著的积极影响。为此，我们通过一种新颖的证明技术给出了无政府代价的渐近紧界，该技术可能具有独立价值，并可应用于其他具有度量权重的模型。