<p>As you have suggested in a comment, $M$ will not be played with positive probability in any Nash equilibrium. This is true for any strictly dominated action. </p>

<p>Given that $M$ has zero probability, the rest of your question is answered in another question of yours in another thread at 
<a href="https://economics.stackexchange.com/q/8423/5739">How infinite Nash equilibria are possible in a game?</a></p>

CC BY-SA 3.0
2015-09-28T17:43:13.843Z
How do we conform that M will not be played with positive probability. I am interested in the calculation of that.
Sub-Optimal
CC BY-SA 3.0
2015-09-28T19:20:09.473Z
A strategy profile is a Nash equilibrium if for each player, the strategy specified for that player yields at least as good utility as any other alternative strategy given the rest of the strategies of others. One corollary of this is that every strategy that is played with positive probability must be a best response to others' strategies. M is strictly dominated as you have calculated, meaning that whatever the other players choose, it yields a lower payoff for player 1 (compared to the one you calculated), implying that it is not a best response for any possible strategy profile of others.
ramazan
CC BY-SA 3.0
2015-09-28T20:07:44.950Z
In the above structure let the prob. of L,M,R be p1,p2,p3 respectively for row player and prob. of L, R for column player be q and q-1. Now if we try to calculate the the probabilities here, what will be the solution. I am trying to find out the mixed equilibrium in this way. Ideally it should come out to be that p2=0, p1 and p2 can be anything from 0 to 1 and q=4/7. As I calculated in the other question. So can we find out the same solution by solving for p1,p2,p3 and q.
Sub-Optimal
CC BY-SA 3.0
2015-09-29T00:34:21.680Z
You cannot proceed further by trying to prove it that way. The guaranteed method here is case analysis. You would say:
Case1 : Is there an equilibrium with p1>0, p2>0, p3=0?
Case2 : Is there an equilibrium with p1>0, p2=0, p3>0?
Case3 : Is there an equilibrium with p1=0, p2>0, p3>0?
Case4 : Is there an equilibrium with p1>0, p2>0, p3>0?
Case5 : Is there a pure strategy equilibrium?

However, recognizing that p2=0 due to strict dominance lets you eliminate a big chunk of your cases, which is good. But if you happen to not realize that shortcut, you are still fine. More work but still will work.
ramazan
CC BY-SA 3.0
2015-09-28T04:29:37.347Z
2015-09-28T04:29:37.347Z
2017-04-13T12:51:38.253Z
ramazan
<p>How to find the mixed strategy equilibrium in the following game:</p>

<pre><code>                     P2

               L            R

     L      (3,1)         (0,1)
P1   M      (1,1)         (1,1)
     R      (0,1)         (4,1)
</code></pre>

CC BY-SA 3.0
2015-09-27T19:13:12.007Z
We can eliminate M for P1 but even without eliminating it we should be able to calculate the same mixed strategy equilibrium. How to do it ?
Sub-Optimal
CC BY-SA 3.0
2015-09-27T20:32:21.007Z
I'm voting to close this question as off-topic because it is a basic question and there is no effort shown at a solution. Because the site is not aimed at solving homework problems we try to avoid these questions. If you write down what you have tried so far the question may be reopened and you will probably get helpful feedback.
Giskard
CC BY-SA 3.0
2015-09-27T20:39:29.517Z
I have tried solving this question by eliminating the dominated strategy. For player P1, M is a strictly dominated strategy because if he randomizes between L and M with 0.5 probability associated with each then the expected payoffs for that mixed strategy will be (1.5,1) and (2,1) which are in any case better for P1 than what he gets by playing M. So, M can be eliminated. And then we get an infinite number of Nash's equilibria in the game. However, if I do not want to eliminate M beforehand but solve this game with this given structure only then I am unable to find the solution. Please help.
Sub-Optimal
<game-theory><nash-equilibrium>
Finding Nash's Equilibrium in mixed strategies
Sub-Optimal
# Finding Nash's Equilibrium in mixed strategies

(Asked by: username_0 on 2015-09-27)

How to find the mixed strategy equilibrium in the following game:
                     P2

               L            R

     L      (3,1)         (0,1)
P1   M      (1,1)         (1,1)
     R      (0,1)         (4,1)

**Tags:** game-theory, nash-equilibrium
> **Comments:**
> **username_0**: We can eliminate M for P1 but even without eliminating it we should be able to calculate the same mixed strategy equilibrium. How to do it ?  
> **username_1**: I'm voting to close this question as off-topic because it is a basic question and there is no effort shown at a solution. Because the site is not aimed at solving homework problems we try to avoid these questions. If you write down what you have tried so far the question may be reopened and you will probably get helpful feedback.  
> **username_0**: I have tried solving this question by eliminating the dominated strategy. For player P1, M is a strictly dominated strategy because if he randomizes between L and M with 0.5 probability associated with each then the expected payoffs for that mixed strategy will be (1.5,1) and (2,1) which are in any case better for P1 than what he gets by playing M. So, M can be eliminated. And then we get an infinite number of Nash's equilibria in the game. However, if I do not want to eliminate M beforehand but solve this game with this given structure only then I am unable to find the solution. Please help.  

---

## 1 answers

### Answer by username_2 on 2015-09-28. Score: 1 | Accepted answer

As you have suggested in a comment, $M$ will not be played with positive probability in any Nash equilibrium. This is true for any strictly dominated action.

Given that $M$ has zero probability, the rest of your question is answered in another question of yours in another thread at
[How infinite Nash equilibria are possible in a game?](https://economics.stackexchange.com/q/8423/5739)

> **Comments:**
> **username_0**: How do we conform that M will not be played with positive probability. I am interested in the calculation of that.  
> **username_2**: A strategy profile is a Nash equilibrium if for each player, the strategy specified for that player yields at least as good utility as any other alternative strategy given the rest of the strategies of others. One corollary of this is that every strategy that is played with positive probability must be a best response to others' strategies. M is strictly dominated as you have calculated, meaning that whatever the other players choose, it yields a lower payoff for player 1 (compared to the one you calculated), implying that it is not a best response for any possible strategy profile of others.  
> **username_0**: In the above structure let the prob. of L,M,R be p1,p2,p3 respectively for row player and prob. of L, R for column player be q and q-1. Now if we try to calculate the the probabilities here, what will be the solution. I am trying to find out the mixed equilibrium in this way. Ideally it should come out to be that p2=0, p1 and p2 can be anything from 0 to 1 and q=4/7. As I calculated in the other question. So can we find out the same solution by solving for p1,p2,p3 and q.  
> **username_2**: You cannot proceed further by trying to prove it that way. The guaranteed method here is case analysis. You would say: Case1 : Is there an equilibrium with p1>0, p2>0, p3=0? Case2 : Is there an equilibrium with p1>0, p2=0, p3>0? Case3 : Is there an equilibrium with p1=0, p2>0, p3>0? Case4 : Is there an equilibrium with p1>0, p2>0, p3>0? Case5 : Is there a pure strategy equilibrium?  However, recognizing that p2=0 due to strict dominance lets you eliminate a big chunk of your cases, which is good. But if you happen to not realize that shortcut, you are still fine. More work but still will work.  

---

