| Abstract: |
| In this talk, we consider a novel field of application of replicator dynamics by proposing the discrete-time replicator equations for studying optimal transport networks with congestion. We first introduce the concept of a Wardrop optimal network that admits Wardrop optimal flows that are both Nash equilibrium and system optimum, and are the only networks with the price of anarchy exactly equal to its least value of 1. Then we present a novel dynamical model of optimal flow distribution on Wardrop optimal networks, using the ideas of evolutionary game theory, which unlike the classical game theory, focuses on the dynamics of strategy change. Our dynamical model is based on discrete-time mean-field replicator equations defined over probability simplices, generated by nonlinear order-preserving mappings. In particular, we study replicator dynamics induced by convex differentiable functions and Schur-convex potential functions. As examples, we employ complete symmetric functions, gamma functions, and symmetric gauge functions in generating replicator dynamics. We analyze the dynamic behavior of these systems, focusing on convergence and stability properties. Using techniques from dynamical systems theory, including Lyapunov functions, we examine Nash equilibria, convergence to fixed points, and conditions for asymptotic stability. For the replicator equations under consideration, the Nash equilibrium, the Wardrop equilibrium, and the system optimum coincide, thus representing the same point in the state space. Certain affine and nonlinear deformations of networks that preserve the property of Wardrop optimality and stochastic method of the construction of Wardrop optimal networks will be presented. |
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