# Game Theory Is there a dominant strategy for either of the two agents? Which strategies can be eliminated be

answer:

given the matrix,

there is no dominant stratergy as lets see agent 2 in the matrix, now will agent 2 ever prefer S^{b}_{3 to} S^{b}_{2} the answer is that one cannot decide because the payoff for a_{13} =3 and that of a_{12}=1 but a_{22}=3 and a_{23}=2 though

a_{13}>a_{12} but a_{22}>a_{23.} The same case follows everywhere for agent 2 and agent 1. hence there is no dominant startergy and no rational player will ever play a dominant stratergy and in this case there is no dominant stratergy.

now a nash equilibrium is a pair of stratergies we use the best response method is that for

for a fixed value of s^{a}_{1} agent 2 would choose a13 as it gives a payoff of 3 which is greater when agent 1’s response is fixed at s^{a}_{1.} similarly for fixed s^{a}_{2} agent 2 would choose a_{21} and for fixed s^{a}_{3} agent 2 would chose either a_{31} or a_{32}.

now keeping agent 2’s response fixed, for fixed s^{b}_{1} agent 1 would chose a_{21} and for fixed s^{b}_{2} agent 1 would choose a_{12} and for a fixed s^{b}_{3} agent 1 would chose a_{13.} putting the picture together the nash equilibrium is at a_{21} and a_{13} as both the agents maximize their payoff given the others response yielding the nash equilibrium