<p>So, I have the following 9 by 9 probability matrix. I want to solve it for a nash equilibrium.
<a href="https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing" rel="nofollow">https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing</a></p>

<p>For anyone who might recognize it, the probabilities are from Hearthstone tournament data.</p>

<p>We can eliminate 4 strategies by dominance, leaving us with a 5 by 5 matrix, but when we try to solve in mathematica:</p>

<pre><code>data1 = {
{50, 56, 33, 44, 39},
{44, 50, 75, 69, 44},
{67, 25, 50, 25, 69},
{56, 31, 75, 50, 39},
{61, 56, 31, 61, 50}}
prob = List[m, pr, r, s, wr]

NSolve[{
data1[[All, 1]].prob == data1[[All, 2]].prob, 
data1[[All, 2]].prob == data1[[All, 3]].prob, 
data1[[All, 3]].prob == data1[[All, 4]].prob, 
data1[[All, 4]].prob == data1[[All, 5]].prob, 
Total[prob] == 1}, prob]
</code></pre>

<p>So I'm using the standard MSNE strategy, taking the dot product of the probability vector and player 2'd payoffs, then setting them equal. The output is:</p>

<pre><code>{m -&gt; 0.447102, pr -&gt; 0.5299, r -&gt; 0.286108, s -&gt; -0.241951, wr -&gt; -0.0211592}
</code></pre>

<p>But we can't have negative probabilities. This would mean an MSNE cannot occur, which would imply a PSNE, but that can't happen either with the way the matrix is set up.</p>

<p>So what's the deal? What's the NE in this matrix?</p>

CC BY-SA 3.0
2016-02-24T17:32:39.650Z
Note that even with a 5-by-5 game, the players need not mix over all of the 5 pure strategies. So you'd have to inspect the numbers more carefully. Also, please familiarize yourself with the community's policy on homework questions: http://meta.economics.stackexchange.com/q/1465/42
Herr K.
CC BY-SA 3.0
2016-02-24T18:25:18.153Z
@HerrK. This does not some like a homework question to me. But your remark about not mixing over all 5 strategies is spot on.
Giskard
CC BY-SA 3.0
2016-02-24T18:26:10.960Z
To calculate a Nash-equilibrium you can use this tool. http://banach.lse.ac.uk/ Data input is a little tedious but it has worked well for me.
Giskard
CC BY-SA 3.0
2016-02-24T19:00:30.327Z
@denesp: Thanks. I agree that this is most likely not a homework. I included the homework policy comment because I saw someone else had initiated a vote to close and I figured the policy was probably the reason. At any rate, there should be no harm to inform a new user of the site's various policies.
Herr K.
CC BY-SA 3.0
2016-02-25T02:08:02.913Z
@denesp: How would prefer to input payoffs (e.g., a pairs of payoffs directly)? I wrote banach.lse.ac.uk 15 years ago, but we still work on interfaces to game solvers and on algorithms themselves.
Rahul Savani
CC BY-SA 3.0
2016-02-25T02:09:51.703Z
@user7448: If your game is symmetric (i.e. $A=B^\top$ which I think yours is, but I didn't check completely), then you can enter only one matrix on banach.lse.ac.uk and use the symmetric game setting.
Rahul Savani
CC BY-SA 3.0
2016-02-25T05:04:06.447Z
@RahulSavani Thank you for the tool. I think being able to set the row and column separators would be nice.
Giskard
CC BY-SA 3.0
2016-02-25T09:00:20.613Z
@denesp: That's a good idea. It will probably never make it to banach.lse.ac.uk, but we are working on a new version of Game Theory Explorer (http://gte.csc.liv.ac.uk/gte/builder/ is the current version) using just HTML5 and javascript and there we welcome all feature requests; I will keep this one in mind. FYI, GTE also solves extensive form games. Cheers, Rahul.
Rahul Savani
<game-theory><nash-equilibrium><probability>
Can't solve this matrix for Nash Equilibrium?
user7448
# Can't solve this matrix for Nash Equilibrium?

(Asked by: username_0 on 2016-02-24)

So, I have the following 9 by 9 probability matrix. I want to solve it for a nash equilibrium.
https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing
For anyone who might recognize it, the probabilities are from Hearthstone tournament data.
We can eliminate 4 strategies by dominance, leaving us with a 5 by 5 matrix, but when we try to solve in mathematica:
data1 = {
{50, 56, 33, 44, 39},
{44, 50, 75, 69, 44},
{67, 25, 50, 25, 69},
{56, 31, 75, 50, 39},
{61, 56, 31, 61, 50}}
prob = List[m, pr, r, s, wr]

NSolve[{
data1[[All, 1]].prob == data1[[All, 2]].prob, 
data1[[All, 2]].prob == data1[[All, 3]].prob, 
data1[[All, 3]].prob == data1[[All, 4]].prob, 
data1[[All, 4]].prob == data1[[All, 5]].prob, 
Total[prob] == 1}, prob]

So I'm using the standard MSNE strategy, taking the dot product of the probability vector and player 2'd payoffs, then setting them equal. The output is:
{m -> 0.447102, pr -> 0.5299, r -> 0.286108, s -> -0.241951, wr -> -0.0211592}

But we can't have negative probabilities. This would mean an MSNE cannot occur, which would imply a PSNE, but that can't happen either with the way the matrix is set up.
So what's the deal? What's the NE in this matrix?

**Tags:** game-theory, nash-equilibrium, probability
> **Comments:**
> **username_1**: Note that even with a 5-by-5 game, the players need not mix over all of the 5 pure strategies. So you'd have to inspect the numbers more carefully. Also, please familiarize yourself with the community's policy on homework questions: http://meta.economics.stackexchange.com/q/1465/42  
> **username_2**: @HerrK. This does not some like a homework question to me. But your remark about not mixing over all 5 strategies is spot on.  
> **username_2**: To calculate a Nash-equilibrium you can use this tool. http://banach.lse.ac.uk/ Data input is a little tedious but it has worked well for me.  
> **username_1**: @denesp: Thanks. I agree that this is most likely not a homework. I included the homework policy comment because I saw someone else had initiated a vote to close and I figured the policy was probably the reason. At any rate, there should be no harm to inform a new user of the site's various policies.  
> **username_3**: @denesp: How would prefer to input payoffs (e.g., a pairs of payoffs directly)? I wrote banach.lse.ac.uk 15 years ago, but we still work on interfaces to game solvers and on algorithms themselves.  
> **username_3**: @user7448: If your game is symmetric (i.e. $A=B^\top$ which I think yours is, but I didn't check completely), then you can enter only one matrix on banach.lse.ac.uk and use the symmetric game setting.  
> **username_2**: @RahulSavani Thank you for the tool. I think being able to set the row and column separators would be nice.  
> **username_3**: @denesp: That's a good idea. It will probably never make it to banach.lse.ac.uk, but we are working on a new version of Game Theory Explorer (http://gte.csc.liv.ac.uk/gte/builder/ is the current version) using just HTML5 and javascript and there we welcome all feature requests; I will keep this one in mind. FYI, GTE also solves extensive form games. Cheers, Rahul.  

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## 0 answers

