Many real-world networks, like the Internet, are not the result of central design but instead the outcome of the interaction of local agents who are selfishly optimizing for their individual utility. The famous Network Creation Game [Fabrikant et al., PODC 2003] enables us to understand such processes, their dynamics, and their outcomes in the form of equilibrium states. In this model, agents buy incident edges towards other agents for a price of $\alpha$ and simultaneously try to minimize their buying cost and their total hop distance. Since in many real-world networks, e.g., social networks, consent from both sides is required to maintain a connection, Corbo and Parkes [PODC 2005] proposed a bilateral version of the Network Creation Game, in which mutual consent and payment are required in order to create edges. It is known that the bilateral version has a significantly higher Price of Anarchy, compared to the unilateral version. This is counter-intuitive, since cooperation should help to avoid socially bad states. We investigate this phenomenon by analyzing the Price of Anarchy of the bilateral version with respect to different solution concepts that allow for various degrees of cooperation among the agents. With this, we provide insights into what kind of cooperation is needed to ensure that socially good networks are created. We present a collection of asymptotically tight bounds on the Price of Anarchy that precisely map the impact of cooperation on the quality of tree networks and we find that weak forms of cooperation already yield a significantly improved Price of Anarchy. Moreover, for general networks we show that enhanced cooperation yields close to optimal networks for a wide range of edge prices.

翻译：许多现实世界中的网络（如互联网）并非中心化设计的结果，而是源于自私优化自身效用的局部代理者之间相互作用的产物。著名的网络创建博弈 [Fabrikant et al., PODC 2003] 使我们能够理解此类过程及其动态机制，以及它们以均衡状态呈现的最终结果。在该模型中，代理者以价格为$\alpha$购买通向其他代理者的直达边，同时试图最小化其购买成本与总跳数距离。在诸多现实网络（例如社交网络）中，连接需要双方同意才能维持，因此 Corbo 和 Parkes [PODC 2005] 提出了网络创建博弈的双边版本，其中边的创建需要双方同意并共同支付费用。已知与单边版本相比，双边版本的无政府代价显著更高。这一现象有悖直觉，因为合作本应有助于避免社会性恶劣状态。我们通过分析双边版本在不同允许代理者之间不同程度合作的解概念下的无政府代价来研究这一现象。借此，我们揭示了确保创建社会性优良网络所需的具体合作类型。我们提出了一系列渐近紧确的无政府代价边界，精确刻画了合作对树形网络质量的影响，并发现弱形式的合作已能显著改善无政府代价。此外，针对一般网络，我们证明增强型合作能在广泛的边价格范围内生成接近最优的网络。