(Player A is on the left, Player B is on the top.)
I am trying to find a mixed strategy Nash equilibrium for a $5\times 3$ matrix (table below). I've only gone as far as proving that the one strategy that is never a best response (strategy 1/4/6) does not get strictly dominated by any mixed strategy $\sigma_{l1} = (p, 1-p)$. The way I did that was that I set three inequalities: $u_a(\sigma_{l1} [\text{mixed strategy}], S_{acf(/cdf/cef separately)}) > u_a(S_{146}, S_{acf/cdf/cef})$, and I always got contradicting equations (or rather inequalities where there simply was no $p$ that fulfilled all three equations). Now I am stuck, though, as to what to do next. I have tried to set inequalities where S_146 was the best response to a mixed strategy from Player B and plotted those results on a graph, but that didn't seem to make much sense to me. I seriously don't know how to progress. Can anybody provide me with a push into the right direction?
| acf | cdf | cef | |
|---|---|---|---|
| 1 2 4 | 0.45,0.55 | 0.75,0.25 | 0.2,0.8 |
| 1 3 6 | 0.5,0.5 | 0.55,0.45 | 0.65,0.35 |
| 1 4 6 | 0.5,0.5 | 0.65,0.35 | 0.55,0.45 |
| 3 4 6 | 0.8,0.2 | 0.6,0.4 | 0.6,0.4 |
| 3 5 6 | 0.75,0.25 | 0.750.25 | 0.45,0.55 |