Nash equilibrium in a stochastic model of two competing athletes

TL;DR

This paper models a competitive scenario between two athletes with unequal strength as a stochastic zero-sum game, explicitly constructing a mixed strategy Nash equilibrium that guides optimal energy investment strategies.

Contribution

It introduces a novel analytically tractable stochastic model of athlete competition and explicitly constructs a mixed strategy Nash equilibrium for the first time in such a setting.

Findings

Existence of a mixed strategy Nash equilibrium for all strength differences d in (0,1)

Equilibrium strategies involve a combination of continuous distributions and delta peaks

The model provides optimal strategies for both weaker and stronger athletes to maximize winning chances

Abstract

We propose a toy model for a stochastic description of the competition between two athletes of unequal strength, whose average strength difference is represented by a parameter $d$. The athletes interact through the choice of their strategies $x,y\in[0,1]$. These variables denote the amount of energy each invests in the competition, and determine the performance of each athlete. Each athlete picks his strategy based on his knowledge of his own and his competitor's performance distribution, and on his evaluation of the danger of exhaustion, which increases with the amount of invested energy. We formulate this problem as a zero-sum game. Mathematically it is in the class of "discontinuous games" for which a Nash equilibrium is not guaranteed in advance. We demonstrate by explicit construction that the problem has a mixed strategy Nash equilibrium $\big( f(x),g(y) \big)$ for arbitrary $0<d<1$. The probability distributions $f$ and $g$ appear to both be the sum of a continuous component and a Dirac delta peak. It is remarkable that this problem is analytically tractable. The Nash equilibrium provides both the weaker and the stronger athlete with the best strategy to optimize their chances to win.