From tug-of-war to Brownian Boost: explicit ODE solutions for player-funded stochastic-differential games

Abstract

Brownian Boost is a one-parameter family of stochastic differential games played on the real line in which players spend at rates of their choosing in an ongoing effort to influence the drift of a randomly diffusing point particle~$X$. One or other player is rewarded, at time infinity, according to whether~$X$ tends to plus or minus infinity. Each player's net receipt is the final reward (only for the victor) minus the player's total spend. We characterise and explicitly compute the time-homogeneous Markov-perfect Nash equilibria of Brownian Boost, finding the derivatives of the players' expected payoffs to solve a pair of coupled first-order non-linear ODE. Brownian Boost is a high-noise limit of a two-dimensional family of player-funded tug-of-war games, one of which was studied in~\cite{LostPennies}. We analyse the discrete games, finding them, and Brownian Boost, to exemplify key features studied in the economics literature of tug-of-war initiated by~\cite{HarrisVickers87}: a battlefield region where players spend heavily; stakes that decay rapidly but asymmetrically in distance to the battlefield; and an effect of discouragement that makes equilibria fragile under asymmetric perturbation of incentive. Tug-of-war has a parallel mathematical literature derived from~\cite{PSSW09}, which solved the scaled fair-coin game in a Euclidean domain via the infinity Laplacian PDE. By offering an analytic solution to Brownian Boost, a game that models strategic interaction and resource allocation, we seek to build a bridge between the two tug-of-war literatures.