Ranking Games that have Competitiveness-based Strategies

TL;DR

This paper investigates the computational complexity of Nash equilibria in discretized contest games with competitiveness-based strategies, providing polynomial-time algorithms for certain cases and approximation schemes for others.

Contribution

It introduces polynomial-time algorithms for exact equilibria in 2-player tie-breaking contests and characterizes equilibria in multi-player cases, along with approximation schemes for games with ties.

Findings

Polynomial-time algorithm for 2-player tie-breaking contests.

Characterization of Nash equilibria in multi-player contests.

Approximation schemes for contests with ties under specific conditions.

Abstract

An extensive literature in economics and social science addresses contests, in which players compete to outperform each other on some measurable criterion, often referred to as a player's score, or output. Players incur costs that are an increasing function of score, but receive prizes for obtaining higher score than their competitors. In this paper we study finite games that are discretized contests, and the problems of computing exact and approximate Nash equilibria. Our motivation is the worst-case hardness of Nash equilibrium computation, and the resulting interest in important classes of games that admit polynomial-time algorithms. For games that have a tie-breaking rule for players' scores, we present a polynomial-time algorithm for computing an exact equilibrium in the 2-player case, and for multiple players, a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2-player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give approximation schemes for two special cases: constant-sized set of strategies, and constant number of players.