# On approximate pure Nash equilibria in weighted congestion games with polynomial latencies

## TL;DR

This paper investigates the existence and efficiency of approximate pure Nash equilibria in weighted congestion games with polynomial latency functions, providing bounds and potential functions for such equilibria.

## Contribution

It introduces a simple technique to establish the existence of $d$-approximate potential functions and demonstrates the existence of near-optimal approximate equilibria with bounded costs.

## Key findings

- Existence of a $d$-approximate potential function for degree $d$ polynomial latencies.
- Every sequence of $d$-approximate improvement moves converges to a $d$-approximate pure Nash equilibrium.
- Existence of a $(d+ ext{delta})$-approximate equilibrium with bounded cost relative to the optimal.

## Abstract

We study natural improvement dynamics in weighted congestion games with polynomial latencies of maximum degree $d\geq 1$. We focus on two problems regarding the existence and efficiency of approximate pure Nash equilibria, with a reasonable small approximation factor, in these games. By exploiting a simple technique, we firstly show that such a game always admits a $d$-approximate potential function. This implies that every sequence of $d$-approximate improvement moves by the players leads to a $d$-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, the game always admits a constant approximate potential function. Secondly, using a simple potential function argument, we are able to show that a $(d+\delta)$-approximate pure Nash equilibrium of cost at most $(d+1)/(d+\delta)$ times the cost of an optimal state always exists, for $\delta\in [0,1]$.