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(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
Surge
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MGMatthew F
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  • You should provide all the information needed to answer the question in the text on this website. Please consider type-setting your table as well. Links to external sites may die, which will make the work of answering the question not usable by others in the future. In addition, you should provide some of the steps you tried in solving this problem, and where exactly you are stuck. – Surge Jan 03 '24 at 11:00
  • @Surge edited the problem – MGMatthew F Jan 03 '24 at 11:29
  • I've fixed your tex layout a bit, but you should work on it more if you want people to read your question and work on it. – Surge Jan 03 '24 at 11:39
  • There's software that does this sort of thing. Do you want to know how to do it by hand? – joriki Jan 03 '24 at 17:46

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