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?