Riemannian game dynamics

Abstract : We study a class of evolutionary game dynamics defined by bal- ancing a gain determined by the game’s payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation and global convergence in potential games. Moreover, when the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics pre- serve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforce- ment learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.
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Journal of Economic Theory, Elsevier, 2018, 177, pp.315-364. 〈10.1016/j.jet.2018.06.002〉
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Panayotis Mertikopoulos, William H. Sandholm. Riemannian game dynamics. Journal of Economic Theory, Elsevier, 2018, 177, pp.315-364. 〈10.1016/j.jet.2018.06.002〉. 〈hal-01382281〉

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