# 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?

• Could you be a bit more explicit? What happens if there are several best responses? – Michael Greinecker Apr 12 at 23:08
• Suppose you just pick one at random. – user3285148 Apr 13 at 7:26

Given that the system does converge, i.e. $$a_i^t=a_i^{t+1}=a_i^*$$ for all $$i$$ after some $$T<\infty$$, 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.