The War of Attrition in the Limit of Infinitely Many Players

TL;DR

This paper analyzes the asymptotic behavior of the N-player War of Attrition game as the number of players approaches infinity, revealing that two different models converge to the same limit and providing new insights into the fixed-strategy model.

Contribution

It establishes the limit behavior of two N-player War of Attrition models and introduces new results for the fixed-strategy case in the large-player limit.

Findings

Both models' time evolution coincide as the number of players tends to infinity.

The paper proves new results for the fixed-strategy model in the N-player setup.

The asymptotic behavior of the models is characterized in the infinite-player limit.

Abstract

The "War of Attrition" is a classical game theoretic model that was first introduced to mathematically describe certain non-violent animal behavior. The original setup considers two participating players in a one-shot game competing for a given prize by waiting. This model has later been extended to several different models allowing more than two players. One of the first of these N-player generalizations was due to J. Haigh and C. Cannings (Acta Appl. Math.14) where two possible models are mainly discussed; one in which the game starts afresh with new strategies each time a player leaves the game, and one where the players have to stick with the strategy they chose initially. The first case is well understood whereas, for the second case, much is still left open. In this paper we study the asymptotic behavior of these two models as the number of players tend to infinity and prove that their time evolution coincide in the limit. We also prove new results concerning the second model in the N-player setup.