We prove distributional convergence for a family of random processes on $mathbb{Z}$, which we call cooperative motions. The model generalizes the totally asymmetric hipster random walk introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, 2020]. We present a novel approach based on connecting a temporal recurrence relation satisfied by the cumulative distribution functions of the process to the theory of finite difference schemes for Hamilton-Jacobi equations [Crandall and Lyons, 1984]. We also point out some surprising lattice effects that can persist in the distributional limit, and propose several generalizations and directions for future research.
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