The insurance market is considered to be a competitive market, so in order to study competition to determine a competitive premium, game theory seems to be a useful tool for studying those situations.
I've seen so many articles in which the objective is to calculate the Nash equilibrium (which represents the equilibrium premium of the insurance market), but I'm not sure why this is of interest.
For my understanding of the insurance concept, it doesn't seem to be a good idea since (I'm not sure!) insurance companies do not choose their premiums at the same time (Stackelberg equilibrium may be a good thing).
So my question is, why would they want to calculate the Nash equilibrium?
Nash equilibrium is both a useful and convenient concept as it essentially says no one has an incentive to change behavior without coordination. The Nash equilibrium is unique for some problems, but in general is a set of outcomes.
Because insurance companies are free to change their premiums at any time, only at a Nash equilibrium will prices settle, and if we are away from the Nash equilibrium at least one agent will want to change their behavior.