<blockquote>
<p>why would a player want to balance out the payoffs of another player</p>
</blockquote>
<p>I don't think anyone is saying that a player <strong>wants</strong> to do this. But in mixed equilibrium their strategy is such that this property holds. Without this property, any mixed strategy of the other player would be suboptimal.</p>

CC BY-SA 4.0
2020-07-16T14:09:15.820Z
Why is this strategy considered optimal? I just can't wrap my mind around this fact.
Harshit Joshi
CC BY-SA 4.0
2020-07-16T13:38:16.373Z
2020-07-16T13:38:16.373Z
Giskard
<p>The question being investigated by the video is the existence of Nash equilibria, not the optimal choices by the players.</p>
<p>There are two obvious pure Nash equilibrium joint strategies, namely both play B or both play F, since in either case a deviation from the strategy by one of the players brings  a negative expected effect for that play is the other goes on with the strategy.</p>
<p>The question being addressed is whether there is also a mixed Nash equilibrium joint strategy.  It will be a Nash equilibrium if neither player can improve their own outcome by changing strategy while the other's strategy stays the same.  So if a player decides to find a mixed Nash equilibrium (rather than the more normal maximising expectation), then that player's approach is to find a way of making the result indifferent to the other player's strategy.</p>
<p>That is the answer to your question.  But it is not a particular good approach for that player from the start. The outcome is an expected gain of  <span class="math-container">$\frac23$</span> for each player, while the coordinated strategies Nash equilibria give <span class="math-container">$2$</span> or <span class="math-container">$1$</span>.  The one merit of this mixed strategy Nash equilibrium is that it reduces the risk of opposing strategies, which could be worse.</p>

CC BY-SA 4.0
2020-07-16T16:55:00.547Z
@Tomcat How is $(2,2)$ even a possible outcome here?
Henry
CC BY-SA 4.0
2020-07-16T16:26:13.740Z
2020-07-16T16:26:13.740Z
Henry
<p>Notice that if a player is indifferent between two strategies, she gets the same payoff from either strategy. This means that <em>anything</em> is optimal (a best response): Playing either pure strategy or any mixed strategy. This includes the mixed strategy which makes her opponent indifferent.</p>
<p>Of course, if the player is <em>not</em> indifferent, she will choose either one pure strategy or the other. But this cannot be the case in a Nash equilibrium where both players use mixed strategies.</p>

CC BY-SA 4.0
2020-07-17T04:29:12.303Z
Why do you suggest that if "if anything is possible, it is the best response"?
Harshit Joshi
CC BY-SA 4.0
2020-07-16T23:32:16.343Z
2020-07-16T23:32:16.343Z
pegasus
<p>Here's the <a href="https://youtu.be/7jBf5fzGBlk?t=164" rel="nofollow noreferrer">video</a> I will be referring too.</p>
<p>Now I am a complete beginner in game theory, so sorry if this sounds like a stupid question, but why would a player want to balance out the payoffs of another player through his strategy.</p>
<p>Like it doesn't make much intuitive sense to me as to why he would want to do that. How exactly is this the best response to the situation?</p>
<p>Edit:</p>
<p>Sorry for not making the question self contained. So the game concerned is the Battle of the Sexes game, and this is the payoff matrix.
<a href="https://i.sstatic.net/oMUNW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oMUNW.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2020-07-16T13:35:48.890Z
Please edit any information you deem important into your question, I do not want to watch a video and write a transcript.
Giskard
CC BY-SA 4.0
2020-07-16T13:37:15.987Z
"Two player zero-sum" matters for this conclusion
Henry
CC BY-SA 4.0
2020-07-16T14:08:22.007Z
@Giskard Made an edit.
Harshit Joshi
CC BY-SA 4.0
2020-07-16T14:08:45.227Z
@Henry The game is not a zero sum game.
Harshit Joshi
<game-theory><nash-equilibrium>
Doubt about Mixed Strategy Nash Equilibrium
Harshit Joshi
# Doubt about Mixed Strategy Nash Equilibrium

(Asked by: username_0 on 2020-07-16)

Here's the [video](https://youtu.be/7jBf5fzGBlk?t=164) I will be referring too.

Now I am a complete beginner in game theory, so sorry if this sounds like a stupid question, but why would a player want to balance out the payoffs of another player through his strategy.

Like it doesn't make much intuitive sense to me as to why he would want to do that. How exactly is this the best response to the situation?

Edit:

Sorry for not making the question self contained. So the game concerned is the Battle of the Sexes game, and this is the payoff matrix.
[![enter image description here](https://i.sstatic.net/oMUNW.png)](https://i.sstatic.net/oMUNW.png)

**Tags:** game-theory, nash-equilibrium
> **Comments:**
> **username_1**: Please edit any information you deem important into your question, I do not want to watch a video and write a transcript.  
> **username_2**: "Two player zero-sum" matters for this conclusion  
> **username_0**: @Giskard Made an edit.  
> **username_0**: @Henry The game is not a zero sum game.  

---

## 3 answers

### Answer by username_1 on 2020-07-16. Score: 1

> why would a player want to balance out the payoffs of another player

I don't think anyone is saying that a player **wants** to do this. But in mixed equilibrium their strategy is such that this property holds. Without this property, any mixed strategy of the other player would be suboptimal.

> **Comments:**
> **username_0**: Why is this strategy considered optimal? I just can't wrap my mind around this fact.  

---

### Answer by username_2 on 2020-07-16. Score: 0

The question being investigated by the video is the existence of Nash equilibria, not the optimal choices by the players.

There are two obvious pure Nash equilibrium joint strategies, namely both play B or both play F, since in either case a deviation from the strategy by one of the players brings a negative expected effect for that play is the other goes on with the strategy.

The question being addressed is whether there is also a mixed Nash equilibrium joint strategy. It will be a Nash equilibrium if neither player can improve their own outcome by changing strategy while the other's strategy stays the same. So if a player decides to find a mixed Nash equilibrium (rather than the more normal maximising expectation), then that player's approach is to find a way of making the result indifferent to the other player's strategy.

That is the answer to your question. But it is not a particular good approach for that player from the start. The outcome is an expected gain of $\frac23$ for each player, while the coordinated strategies Nash equilibria give $2$ or $1$. The one merit of this mixed strategy Nash equilibrium is that it reduces the risk of opposing strategies, which could be worse.

> **Comments:**
> **username_2**: @Tomcat How is $(2,2)$ even a possible outcome here?  

---

### Answer by username_3 on 2020-07-16. Score: 0

Notice that if a player is indifferent between two strategies, she gets the same payoff from either strategy. This means that *anything* is optimal (a best response): Playing either pure strategy or any mixed strategy. This includes the mixed strategy which makes her opponent indifferent.

Of course, if the player is *not* indifferent, she will choose either one pure strategy or the other. But this cannot be the case in a Nash equilibrium where both players use mixed strategies.

> **Comments:**
> **username_0**: Why do you suggest that if "if anything is possible, it is the best response"?  

---

