Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games

We consider team zero-sum network congestion games with $n$ senders playing against $k$ interceptors over a graph $G$. The senders aim to minimize their collective cost of sending messages over paths in $G$, which is an aggregation of edge costs, while the interceptors aim to maximize the collective cost by increasing some of these edge costs. To evade the interceptors, the senders will usually use randomized strategies. We consider two cases, the correlated case when senders have access to a shared source of randomness, and the uncorrelated case, when each sender has access to only its own source of randomness. We study the additional cost that uncorrelated senders have to bear, specifically by comparing the costs incurred by senders in cost-minimal Nash Equilibria when senders can and cannot share randomness. We prove that for an intuitive strict subset of cost functions, the ratio between correlated and uncorrelated costs at equilibrium is $O(\min(m_c(G),n))$, where $m_c(G)$ is the mincut size of $G$. This bound is much milder compared to the most general case, where an upper bound of $\Omega((m_c(G))^{n-1})$ on the ratio is known. We show that the senders can approximate their optimal play by playing simple strategies which select paths uniformly at random from subsets of disjoint paths. We then focus on two natural cost functions. For the first, we prove that one of the simple strategies above is an optimal strategy for senders over graphs with disjoint paths. In complete contrast, for the second cost function we prove that none of these simple strategies is optimal for the senders over these graphs, unless the game instance admits a trivial optimal senders strategy.