# Computational Aspects of Equilibria in Discrete Preference Games

## TL;DR

This paper investigates the computational complexity of finding equilibria in discrete preference games, revealing PLS-completeness in general cases and polynomial-time solutions for specific metric spaces, while also discussing existence issues in directed networks.

## Contribution

It establishes the PLS-completeness of equilibrium computation in discrete preference games and identifies special cases where polynomial-time algorithms are possible.

## Key findings

- Equilibrium computation is PLS-complete in general cases.
- Polynomial-time algorithms exist when the metric space is a tree or a product of path metrics.
- Directed social networks may lack equilibria, indicating non-existence issues.

## Abstract

We study the complexity of equilibrium computation in discrete preference games. These games were introduced by Chierichetti, Kleinberg, and Oren (EC '13, JCSS '18) to model decision-making by agents in a social network that choose a strategy from a finite, discrete set, balancing between their intrinsic preferences for the strategies and their desire to choose a strategy that is `similar' to their neighbours. There are thus two components: a social network with the agents as vertices, and a metric space of strategies. These games are potential games, and hence pure Nash equilibria exist. Since their introduction, a number of papers have studied various aspects of this model, including the social cost at equilibria, and arrival at a consensus. We show that in general, equilibrium computation in discrete preference games is PLS-complete, even in the simple case where each agent has a constant number of neighbours. If the edges in the social network are weighted, then the problem is PLS-complete even if each agent has a constant number of neighbours, the metric space has constant size, and every pair of strategies is at distance 1 or 2. Further, if the social network is directed, modelling asymmetric influence, an equilibrium may not even exist. On the positive side, we show that if the metric space is a tree metric, or is the product of path metrics, then the equilibrium can be computed in polynomial time.