The Nash Equilibrium With Inertia in Population Games

In the traditional game-theoretic set up, where agents select actions and experience corresponding utilities, a Nash equilibrium is a configuration where no agent can improve their utility by unilaterally switching to a different action. In this article, we introduce the novel notion of inertial Nash equilibrium to account for the fact that in many practical situations switching action does not come for free. Specifically, we consider a population game and introduce the coefficients
$c_{ij}$
describing the cost an agent incurs by switching from action
$i$
to action
$j$
. We define an inertial Nash equilibrium as a distribution over the action space where no agent benefits in switching to a different action, while taking into account the cost of such switch. First, we show that the set of inertial Nash equilibria contains all the Nash equilibria, is in general nonconvex, and can be characterized as a solution to a variational inequality. We then argue that classical algorithms for computing Nash equilibria cannot be used in the presence of switching costs. Finally, we propose a better-response dynamics algorithm and prove its convergence to an inertial Nash equilibrium. We apply our results to study the taxi drivers’ distribution in Hong Kong.