If a best-response dynamic converges, does it converge to a Nash equilibrium? Consider a game with a finite number of players and finite action space. Suppose we consider a sequential iterative game-playing process in which, in
each period, players myopically select actions that are a best-response to the actions last
chosen by all other players.
Question: suppose that after a certain number of iterations no player wants to change action.
Hence, the best-response dynamic has converged to some action profile. Is this action profile necessarily a pure strategy Nash equilibrium?
Given that the system does converge, i.e. $a_i^t=a_i^{t+1}=a_i^*$ for all $i$ after some $T&lt;\infty$, and that $a_i^{t+1}\in BR_i(a_{-i}^{t})$, it follows that $a_i^*\in BR_i(a_{-i}^*)$ for all $i$. Hence $a^*$ is a Nash equilibrium of the underlying game.