<p>Yes. This follows from the following three facts:</p>
<ol>
<li>Any two Nash equilibria of a two-player zero-sum game have the same expected payoff- the value.</li>
<li>Every finite game has a Nash equilibrium in undominated strategies. In two-player games, these are exactly the (trembling hand) perfect equilibria.</li>
<li>If you take any Nash equilibrium in a finite game and remove all strategies that are not used in this equilibrium, the equilibrium is still one in the reduced game. Indeed, the defining inequalities that no player wants to deviate  become only easier.</li>
</ol>
<p>So, yes you can remove weakly dominated strategies and can even do so iteratively.</p>

CC BY-SA 4.0
2022-10-19T14:16:36.043Z
2022-10-19T14:16:36.043Z
Michael Greinecker
<p>I am trying to find the value of a game with the following matrix:
<span class="math-container">$$
\begin{matrix}
1 &amp; 3 &amp; 2 &amp; 5\\
0 &amp; -1 &amp; 6 &amp; 7\\
3 &amp; 4 &amp; 2 &amp; 3\\
-7 &amp; 2 &amp; 2 &amp; 1
\end{matrix}
$$</span>
I am wondering if I can eliminate the fourth row since it is weakly dominated by the third row for the row player, and then the fourth column since it is strictly dominated for the column player,..., and then find the value of the game after all dominated strategies are eliminated?<br />
I know we can do this for strictly dominated strategies, but how about those weakly dominated?<br />
Thanks.</p>

CC BY-SA 4.0
2022-10-19T16:20:08.847Z
You can do it, but the result winds up depending on the order in which you eliminate strategies. (I might be able to post a full answer about this later today or in the next few days.)
Dave
<game-theory>
Can we eliminate weakly dominated strategies when finding value of a game?
WinnieXi
# Can we eliminate weakly dominated strategies when finding value of a game?

(Asked by: username_0 on 2022-10-19)

I am trying to find the value of a game with the following matrix:
$$
\begin{matrix}
1 & 3 & 2 & 5\\
0 & -1 & 6 & 7\\
3 & 4 & 2 & 3\\
-7 & 2 & 2 & 1
\end{matrix}
$$
I am wondering if I can eliminate the fourth row since it is weakly dominated by the third row for the row player, and then the fourth column since it is strictly dominated for the column player,..., and then find the value of the game after all dominated strategies are eliminated?  
I know we can do this for strictly dominated strategies, but how about those weakly dominated?  
Thanks.

**Tags:** game-theory

> **Comments:**
> **username_1**: You can do it, but the result winds up depending on the order in which you eliminate strategies. (I might be able to post a full answer about this later today or in the next few days.)  

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## 1 answers

### Answer by username_2 on 2022-10-19. Score: 1 | Accepted answer

Yes. This follows from the following three facts:

1. Any two Nash equilibria of a two-player zero-sum game have the same expected payoff- the value.
2. Every finite game has a Nash equilibrium in undominated strategies. In two-player games, these are exactly the (trembling hand) perfect equilibria.
3. If you take any Nash equilibrium in a finite game and remove all strategies that are not used in this equilibrium, the equilibrium is still one in the reduced game. Indeed, the defining inequalities that no player wants to deviate become only easier.

So, yes you can remove weakly dominated strategies and can even do so iteratively.


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