The Price of Anarchy in Bilateral Network Formation in an Adversary Model

We study network formation with the bilateral link formation rule (Jackson and Wolinsky 1996) with $n$ players and link cost $\alpha >0$. After the network is built, an adversary randomly destroys one link according to a certain probability distribution. Cost for player $v$ incorporates the expected number of players to which $v$ will become disconnected. This model was previously studied for unilateral link formation (K. 2011). We prove existence of pairwise Nash equilibria under moderate assumptions on the adversary and $n\ge 9$. As the main result, we prove bounds on the price of anarchy for two special adversaries: one destroys a link chosen uniformly at random, while the other destroys a link that causes a maximum number of player pairs to be separated. We prove bounds tight up to constants, namely $O(1)$ for one adversary (if $\alpha >1\mathrm{/}2$), and $\mathrm{\Theta }(n)$ for the other (if $\alpha >2$ considered constant and $n\ge 9$). The latter is the worst that can happen for any adversary in this model (if $\alpha =\mathrm{\Omega }(1)$).