We study a coordination game motivated by the formation of Internet Exchange Points (IXPs), in which agents choose which facilities to join. Joining the same facility as other agents you communicate with has benefits, but different facilities have different costs for each agent. Thus, the players wish to join the same facilities as their "friends", but this is balanced by them not wanting to pay the cost of joining a facility. We first show that the Price of Stability ($PoS$) of this game is at most 2, and more generally there always exists an $\alpha$-approximate equilibrium with cost at most $\frac{2}{\alpha}$ of optimum. We then focus on how better stable solutions can be formed. If we allow agents to pay their neighbors to prevent them from deviating (i.e., a player $i$ voluntarily pays another player $j$ so that $j$ joins the same facility), then we provide a payment scheme which stabilizes the solution with minimum social cost $s^*$, i.e. PoS is 1. In our main technical result, we consider how much a central coordinator would have to pay the players in order to form good stable solutions. Let $\Delta$ denote the total amount of payments needed to be paid to the players in order to stabilize $s^*$, i.e., these are payments that a player would lose if they changed their strategy from the one in $s^*$. We prove that there is a tradeoff between $\Delta$ and the Price of Stability: $\frac{\Delta}{cost(s^*)} \le 1 - \frac{2}{5} PoS$. Thus when there are no good stable solutions, only a small amount of extra payment is needed to stabilize $s^*$; and when good stable solutions already exist (i.e., $PoS$ is small), then we should be happy with those solutions instead. Finally, we consider the computational complexity of finding the optimum solution $s^*$, and design a polynomial time $O(\log n)$ approximation algorithm for this problem.

中文翻译：

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在受 Internet 交换点 (IXP) 启发的组队游戏中形成更好的稳定解决方案

我们研究了由互联网交换点 (IXP) 的形成所激发的协调博弈，其中代理选择加入哪些设施。加入与您交流的其他座席相同的设施有好处，但不同的设施对每个座席的成本不同。因此，玩家希望加入与他们的“朋友”相同的设施，但这通过他们不想支付加入设施的成本来平衡。我们首先证明这个游戏的稳定价格（$PoS$）至多为 2，更一般地说，总是存在一个 $\alpha$-近似均衡，成本最大为 $\frac{2}{\alpha}$的最佳。然后我们专注于如何形成更好的稳定解决方案。如果我们允许代理人向他们的邻居支付费用以防止他们偏离（即，一个玩家 $i$ 自愿支付另一个玩家 $j$ 以便 $j$ 加入同一个设施），然后我们提供了一个支付方案，以最小的社会成本 $s^*$ 稳定解决方案，即 PoS 为 1。在我们的主要的技术结果，我们考虑中央协调员需要支付给玩家多少费用才能形成良好的稳定解决方案。让 $\Delta$ 表示为稳定 $s^*$ 需要支付给玩家的总金额，即，如果玩家改变策略从 $s^ *$。我们证明 $\Delta$ 和稳定价格之间存在权衡：$\frac{\Delta}{cost(s^*)} \le 1 - \frac{2}{5} PoS$。因此当没有好的稳定解决方案时，只需要少量的额外支付来稳定$s^*$；并且当良好的稳定解决方案已经存在时（即，$PoS$ 很小），那么我们应该对这些解决方案感到满意。最后，我们考虑了寻找最优解$s^*$的计算复杂度，并针对这个问题设计了一个多项式时间$O(\log n)$的近似算法。