Nash Neural Networks

We introduce Nash Neural Networks (N³) to learn the cost functions that determine how rational individuals behave within a differential game with a Nash equilibrium. The N³ are constructed to respect the Hamiltonian equations that would be derived from the (unknown) utility, together with any dynamical constraints on the state variables of the system. For this, we build on recently developed Lagrangian/Hamiltonian Neural Networks, which have successfully been used to learn Lagrangians/Hamiltonians from dynamical data, while respecting the symmetries and conservation laws of the system. Thus, models trained on our N³ respect the game dynamics and are able to learn the utility in an unsupervised manner. We apply these N³ to an epidemic, in order to infer how the individuals react and contribute to the evolution of the disease, using social distancing as a proxy for the control variable in the underlying utility optimization problem. In this study, we focus on artificial data generated from an explicit solution of the Hamiltonian equations for the optimal control of an SIR model. We use the N³ formalism to recover the hidden utility underlying these solutions. We believe this approach will have wider applications for inferring utilities from behavioral data.