Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games

Abstract: We consider team zero-sum network congestion games with $n$ agents playing against $k$ interceptors over a graph $G$. The agents aim to minimize their collective cost of sending traffic 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 agents will usually use randomized strategies. We consider two cases, the correlated case when agents have access to a shared source of randomness, and the uncorrelated case, when each agent has access to only its own source of randomness. We study the additional cost that uncorrelated agents have to bear, specifically by comparing the costs incurred by agents in cost-minimal Nash equilibria when agents can and cannot share randomness. We consider two natural cost functions on the agents, which measure the invested energy and time, respectively. We prove that for both of these cost functions, the ratio of uncorrelated cost to correlated cost at equilibrium is $O(\min(m_c(G),n))$, where $m_c(G)$ is the mincut size of $G$. This bound is much smaller than the most general case, where a tight, exponential bound of $\Theta((m_c(G))^{n-1})$ on the ratio is known. We also introduce a set of simple agent strategies which are approximately optimal agent strategies. We then establish conditions for when these strategies are optimal agent strategies for each cost function, showing an inherent difference between the two cost functions we study.