<p>Given that the system does converge, i.e. <span class="math-container">$a_i^t=a_i^{t+1}=a_i^*$</span> for all <span class="math-container">$i$</span> after some <span class="math-container">$T&lt;\infty$</span>, and that <span class="math-container">$a_i^{t+1}\in BR_i(a_{-i}^{t})$</span>, it follows that <span class="math-container">$a_i^*\in BR_i(a_{-i}^*)$</span> for all <span class="math-container">$i$</span>. Hence <span class="math-container">$a^*$</span> is a Nash equilibrium of the underlying game.</p>

CC BY-SA 4.0
2021-04-12T23:41:11.203Z
2021-04-12T23:41:11.203Z
Herr K.
<p>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.</p>
<p><strong>Question:</strong> 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?</p>

CC BY-SA 4.0
2021-04-12T23:08:19.847Z
Could you be a bit more explicit? What happens if there are several best responses?
Michael Greinecker
CC BY-SA 4.0
2021-04-13T07:26:20.937Z
Suppose you just pick one at random.
Star
<game-theory><nash-equilibrium>
If a best-response dynamic converges, does it converge to a Nash equilibrium?
Star
# If a best-response dynamic converges, does it converge to a Nash equilibrium?

(Asked by: username_0 on 2021-04-12)

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?

**Tags:** game-theory, nash-equilibrium
> **Comments:**
> **username_1**: Could you be a bit more explicit? What happens if there are several best responses?  
> **username_0**: Suppose you just pick one at random.  

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## 1 answers

### Answer by username_2 on 2021-04-12. Score: 4 | Accepted answer

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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