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]

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?

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    $\begingroup$ 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: meta.economics.stackexchange.com/q/1465/42 $\endgroup$ – Herr K. Feb 24 '16 at 17:32
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    $\begingroup$ To calculate a Nash-equilibrium you can use this tool. banach.lse.ac.uk Data input is a little tedious but it has worked well for me. $\endgroup$ – Giskard Feb 24 '16 at 18:26
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    $\begingroup$ @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. $\endgroup$ – Herr K. Feb 24 '16 at 19:00
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    $\begingroup$ @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. $\endgroup$ – Rahul Savani Feb 25 '16 at 2:08
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    $\begingroup$ @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 (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. $\endgroup$ – Rahul Savani Feb 25 '16 at 9:00

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