Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games

TL;DR

This work studies the price of uncorrelation (PoU) in team congestion games where a group of $n$ agents competes against $k$ interceptors on a directed acyclic network from $s$ to $t$, under SUM and MAX cost variants and two randomness regimes (correlated vs uncorrelated). The authors establish a linear PoU bound $\Theta(\min(m_c(G),n))$ tied to the graph's min-cut $m_c(G)$, improving the general exponential bounds known for normal-form games. They introduce and analyze uniform strategies $\mathcal{U}_n$ and the corresponding uniform PoU $r_U(\alpha)$, proving that for SUM on graphs with disjoint $s$-$t$ paths, uniform strategies are effectively optimal, while for MAX they are not generally optimal. The paper shows $r_U(\alpha)$ is continuously increasing in $\alpha$ and computable in polynomial time via min-cost flow, and it relates cost and payoff formulations to align with existing PoU frameworks, enhancing understanding of correlation’s value in congestion settings.

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 $Θ((m_c(G))^{n-1})$ 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.