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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?

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

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