Dynamics of Three Agent Games

TL;DR

This paper analyzes the long-term score distributions in three-agent competitive games, revealing six distinct asymptotic behaviors depending on winning probabilities, supported by analytical and numerical evidence.

Contribution

It provides a comprehensive classification of score distribution regimes in three-agent games and extends the model to include decline rates and n-agent scenarios.

Findings

Six qualitatively different asymptotic score distributions identified

Analytical results validated by numerical simulations

Score dynamics can be generalized to n-agent games with decline rates

Abstract

We study the dynamics and resulting score distribution of three-agent games where after each competition a single agent wins and scores a point. A single competition is described by a triplet of numbers $p$, $t$ and $q$ denoting the probabilities that the team with the highest, middle or lowest accumulated score wins. We study the full family of solutions in the regime, where the number of agents and competitions is large, which can be regarded as a hydrodynamic limit. Depending on the parameter values $(p,q,t)$, we find six qualitatively different asymptotic score distributions and we also provide a qualitative understanding of these results. We checked our analytical results against numerical simulations of the microscopic model and find these to be in excellent agreement. The three agent game can be regarded as a social model where a player can be favored or disfavored for advancement, based on his/her accumulated score. It is also possible to decide the outcome of a three agent game through a mini tournament of two-a gent competitions among the participating players and it turns out that the resulting possible score distributions are a subset of those obtained for the general three agent-games. We discuss how one can add a steady and democratic decline rate to the model and present a simple geometric construction that allows one to write down the corresponding score evolution equations for $n$-agent games.