From tug-of-war to Brownian Boost: explicit ODE solutions for player-funded stochastic-differential games
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917355880185856 |
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| author | Hammond, Alan |
| author_facet | Hammond, Alan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From tug-of-war to Brownian Boost: explicit ODE solutions for player-funded stochastic-differential games Hammond, Alan Probability Theoretical Economics Classical Analysis and ODEs 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. |
| title | From tug-of-war to Brownian Boost: explicit ODE solutions for player-funded stochastic-differential games |
| topic | Probability Theoretical Economics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2510.07682 |