[论文解读] On Dynamics in Selfish Network Creation

Bernd Kawald, Pascal Lenzner|arXiv (Cornell University)|Dec 19, 2012

Game Theory and Applications参考文献 18被引用 4

一句话总结

本文研究了自私网络构建博弈的动态收敛性，重点关注个体在未协调、局部改进动作下是否能收敛到稳定均衡。研究证明，尽管在树状网络结构的交换博弈中收敛是保证的，但当网络中存在环路时，收敛将完全失败——即使仅存在一条非树边也会破坏收敛性，且在非树网络中，任何动作策略都无法确保收敛。

ABSTRACT

We consider the dynamic behavior of several variants of the Network Creation Game, introduced by Fabrikant et al. [PODC'03]. Equilibrium networks in these models have desirable properties like low social cost and small diameter, which makes them attractive for the decentralized creation of overlay-networks. Unfortunately, due to the non-constructiveness of the Nash equilibrium, no distributed algorithm for finding such networks is known. We treat these games as sequential-move games and analyze whether (uncoordinated) selfish play eventually converges to an equilibrium state. Thus, we shed light on one of the most natural algorithms for this problem: distributed local search, where in each step some agent performs a myopic selfish improving move. We show that fast convergence is guaranteed for all versions of Swap Games, introduced by Alon et al. [SPAA'10], if the initial network is a tree, and show that this process can be sped up to an almost optimal number of moves. For non-tree networks we show the surprising result that even one non-tree edge suffices to destroy the convergence guarantee and no move policy can enforce convergence. This answers an open problem from Ehsani et al. [SPAA'11] in the negative. We extend our negative results to the well-studied original version and prove that there is no convergence guarantee -- even if all agents play optimally. Furthermore, we show the quite surprising result that employing cost-sharing yields even worse dynamic behavior. Finally, we contrast our mostly negative theoretical results by a careful empirical study. Our simulations indicate two positive facts: (1) The non-convergent behavior seems to be confined to a small set of pathological instances and is unlikely to show up in practice. (2) In all our simulations we observed a remarkably fast convergence towards a stable network in O(n) steps, where n is the number of agents.

研究动机与目标

• 理解未协调的、自私的局部搜索在网络构建博弈中是否收敛到稳定均衡。
• 分析网络拓扑结构——特别是环的存在——对收敛保证的影响。
• 评估成本分摊机制是否能改善网络构建博弈中的动态行为。
• 确定最优策略或特定动作策略是否能在各种网络构建博弈变体中强制实现收敛。
• 对比理论上的负面结果与实际中观察到的收敛行为。

提出的方法

• 将网络构建建模为顺序移动博弈，其中个体执行短视的、自私的改进动作。
• 分析交换博弈变体，其中个体交换边，重点关注从树状初始状态出发的收敛性。
• 提出并评估一种自然动作策略，以加速基于树的网络中的收敛。
• 采用最优响应动态模拟个体行为，识别循环或收敛现象。
• 构建病态反例，证明在非树网络中即使采用最优策略，收敛也无法保证。
• 将分析扩展至原始网络构建博弈（买入/删除边）以及需要双方同意的成本分摊变体。

实验结果

研究问题

• RQ1 当初始网络为树时，交换博弈中未协调的自私行为是否收敛到纳什均衡？
• RQ2 在交换博弈中，是否存在任何动作策略可确保非树网络的收敛？
• RQ3 在原始网络构建博弈中，即使采用最优策略，是否仍能保证收敛（允许买入和删除边）？
• RQ4 引入需要双方同意的成本分摊机制是否能改善网络构建博弈的收敛行为？
• RQ5 实际中的模拟结果与理论收敛保证相比如何？

主要发现

• 当初始网络为树时，交换博弈可保证在 O(n) 步内快速收敛至均衡。
• 一种自然动作策略可显著加快基于树的交换博弈的收敛速度，实现接近最优的性能。
• 即使初始网络中仅存在一条非树边，交换博弈的收敛保证也会被破坏。
• 通过病态反例可证明，任何动作策略都无法在非树网络中强制实现交换博弈的收敛。
• 在原始网络构建博弈中，即使采用最优策略，收敛也并非保证，且某些实例中不存在导致稳定的改进序列。
• 出人意料的是，成本分摊变体的动态行为比原始博弈更差，且无任何收敛保证。