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

TL;DR

This work analyzes player-funded stochastic-differential tug-of-war on the line, introducing Brownian Boost (BB$\left(\rho\right)$) as a continuum limit of discrete Trail of Lost Pennies (TLP) games. It develops an explicit, one-parameter family of time-invariant Markov-perfect Nash equilibria described by a coupled ODE system for $(f,g)$, with $X_t$ driven by drift $(a^\rho-b^\rho)/(a^\rho+b^\rho)$ and diffusion, and a careful ABMN$(\kappa,\rho)$ framework that yields explicit solutions and asymptotics. The paper proves equivalence between ABMN solutions in the discrete model and BB equilibria in the high-noise limit, derives detailed asymptotics for stake profiles and battlefield structure, and investigates robustness via the discouragement effect, showing that equilibria are fragile under small incentive asymmetries. It also establishes a rigorous diffusion limit for BB as $\kappa\to0$ and provides comprehensive asymptotic analyses of the Mina margin and the maximal incentive ratio, $\lambda_{\max}(\kappa,\rho)$. These results bridge economics-based tug-of-war models with infinity-Laplacian/PDE analyses, providing explicit analytic tools to study strategic resource allocation in noisy competitive environments. The findings illuminate battlefield localization, asymmetric decay of stakes, and the nontrivial interaction between incentive differences and equilibrium existence, offering a quantitative framework for understanding high-noise strategic dynamics in continuous-time stochastic 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.

Figures (4)

• Figure 1.1: The curves $a= a_1(1,\cdot)$ and $f = f_1(1,\cdot)$, as specified by Definitions \ref{['d.fg']} and \ref{['d.arhobrho']}; $b$ and $n$ are given by reflecting in the vertical axis. Maxine's stake profile $a$ takes maximum value $0.57$ at $x = 0.25$ to two decimal places.
• Figure 2.1: The main contours on the $(\kappa,\rho)$-map. The curve $C_1$ is the locus of $\kappa^2 \rho =1$, which is the upper boundary of the region $W$ in (\ref{['e.weakregion']}); $C_2$ is the locus of $\kappa \rho = 1 + \sqrt{1-\kappa^2}$. The curves emanate from $(1,1)$, the point indexing the game studied in LostPennies and the upper-right corner of the unit box in which Nash-ABMN equivalence is established by Theorem \ref{['t.nashabmn']}. The $\mathrm{BB}(\rho)$-line $\kappa =0$ lies below $C_1$, which is indicative of how the ODE-pair Theorem \ref{['t.fg']} is valid for all $\rho \in (0,\infty)$.
• Figure 2.2: The finite-trail Mina margin map $x \to \mathcal{M}_{0.9,1}^{-9,9}(x)$ is plotted on $(1,145)$. The map rises rapidly to the right of the plotted range, and its values on $(0,1)$ are determined by the symmetry $\mathcal{M}(x) = \mathcal{M}(x^{-1})^{-1}$. There are twenty-one roots of $\mathcal{M}(x) = 1$, given by $x=1$ and ten pairs $(z,z^{-1})$.
• Figure 7.1: A numerical approximation of the curve $\rho \to \lambda_{\rm max}(1,\rho)$ for $\rho$-values in $(0.8,1)$. The curve shown has been interpolated from a sequence of points $(\rho,\lambda_{\rm max}(\rho) )$, where each $\lambda_{\rm max}(\rho)$ is approximated by maximizing over a fine mesh the values in the central domain $D = D_{\kappa,\rho}$ of the finite-trail Mina margin map $\mathcal{M}_{j,k}$ for suitably high $j,k \in \mathbb{N}$.

Theorems & Definitions (69)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Proposition 1.5
• Definition 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 1.10