  Art heritage cities are popular tourist destinations but for many of them
overcrowding is becoming an issue. In this paper, we address the problem of
modeling and analytically studying the flow of tourists along the narrow alleys
of the historic center of a heritage city. We initially present a mean field
game model, where both continuous and switching decisional variables are
introduced to respectively describe the position of a tourist and the point of
interest that he/she may visit. We prove the existence of a mean field
equilibrium. A mean field equilibrium is Nash-type equilibrium in the case of
infinitely many players. Then, we study an optimization problem for an external
controller who aims to induce a suitable mean field equilibrium.
