Population Games and Discrete Optimal Transport

Abstract

We propose a new evolutionary dynamics for population games with a discrete strategy set, inspired by the theory of optimal transport and Mean field games. The dynamics can be described as a Fokker-Planck equation on a discrete strategy set. The derived dynamics is the gradient flow of a free energy and the transition density equation of a Markov process. Such process provides models for the behavior of the individual players in population, which is myopic, greedy and irrational. The stability of the dynamics is governed by optimal transport metric, entropy and Fisher information.Recently, a new viewpoint has been brought into the realm of population games based on optimal transport, see Villani's book [3,38] and mean field games in the series work of Larsy, Lions [6,13,19]. The mean field games have continuous strategy sets and infinite players [4,5]. Each player is assumed to make decisions according to a stochastic process instead of making a one-shot decision. More specifically, individual players change

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“…This structure can be made mathematically rigorous and general to work for probability measures with finite second moments by using tools in metric geometry [5]. Under such a setting, the gradient flow with respect to the L 2 -Wasserstein metric tensor, known as the Wasserstein gradient flow, is welldefined and has been seen deep connections to fluid dynamics [11] [40], differential geometry [25] and mean-field games [16] [17].…”

Age‐optimal design for UAV‐assisted grant‐free non‐orthogonal massive access: Mean‐field game approach

“…\end{equation}According to the results in ref. [36], the inner product of two potential gradient $$\nabla \Phi , \nabla \hat{\Phi }$$ over $$\bm {\rho }(t)$$ can be written as $$\begin{equation} (\nabla \Phi ,\nabla \hat{\Phi })_{{\bm {\rho }(t)}} = \frac{1}{2}\sum \limits _{(i,j) \in \mathcal {E}}(\Phi ^i - \Phi ^j)(\hat{\Phi }^i - \hat{\Phi }^j)\theta ^{ij}({\bm {\rho }(t)}), \end{equation}$$where $$\frac{1}{2}$$ is because each edge is summed twice. Here, $$\theta ^{ij}({\bm {\rho }(t)})$$ is the weight for each edge, i.e.…”

“…Therefore, a metric, i.e. Wasserstein metric [36], is adopted to quantify the performance over the strategy graph $$\mathcal {G}$$.…”

Modeling COVID-19 with mean field evolutionary dynamics: Social distancing and seasonality

“…With the state space in (2) and the inner product in Definition 2, we can construct the Riemannian manifold (P o (S), g W ) [30], [27]. In this regard, we can give the proof of Theorem 1 as follows:…”