We are asked to find a Nash equilibrium in which there is no weakly dominant strategy. Given the bimatrix below:

S DS

K -1 , 1 1, - 1

D 0, 2 0, 2

I found that DS is weakly dominated by S for Column player. if Row plays K, any weighted probability combination that places a positive probability on S will yield a higher payoff to Column than Column choosing DS. If Row plays D, any weighted probability combination that places a positive probability on Swill yield a payoff to Column equal to choosing DS.

The Nash equilibrium here is (D,S). However S, a weakly dominant strategy, is included in the Nash.

How do I construct a new Nash without a weakly dominant strategy. I tried constructing a mixed strategy for row (0.5,0.5) and for column (0.5,0.5) below:

S DS M

K -1 , 1 1, - 1 0,0

D 0, 2 0, 2 0,2

M -0.5, 1.5 0.5,0.5 0,1

With this new table, my Nash is (D,S) and (D,M). I still have the same problem. The equilibira contain a weakly dominant strategy (S for Column and D for Row).

My next question is, how do I extend this game and solve for a Bayesian Nash Equilibrium? I'm confused on the probability greeks in textbooks.