arXiv:2611.01234 · v1

How Much Can Approximate Equilibria Change the Social Cost of Selfish Routing?

首次提交 2026-11-01 04:01

版本更新 2026-11-01 04:01

雷达观测 2026-11-04 23:43

摘要

We study approximate Nash equilibria in nonatomic congestion games, with the goal of understanding how much the social cost can change when the equilibrium condition is relaxed. We consider $ε$-approximate Nash equilibria ($ε$-NE), in which each player uses a strategy whose cost is at most $1 + ε$ times the minimum available cost. We compare the social costs of $ε$-NE and exact Nash equilibria. This lets us quantify, for example, how much a navigation app can reduce total travel time when drivers tolerate routes slightly slower than the fastest available route. For degree-$d$ polynomial latencies in both general nonatomic congestion games and single-commodity routing games on series-parallel networks, we give tight lower and upper bounds on the relative change in social cost from an exact Nash equilibrium to any $ε$-NE, with matching instances for every $ε$. For small $ε$, the largest possible relative reduction and increase in general congestion games are both $Θ(\sqrt{dε})$, independently of the number of resources and populations. In contrast, on series-parallel networks of depth $h$, the largest possible relative reduction is $Θ(hε)$, while the largest possible relative increase is known to be $ε$. For the classes of games we study, these results give a tight and quantitative characterization of the possible changes in social cost and establish a separation between general and series-parallel networks.