Question

Find all of the pure and mixed strategy Nash equilibrium of the following game:

Left 5,10 3.7 2,5 Top 1 Middle Bottom 2 Center Right 4,4 8.8 6.5 2,7 2,2 2,1 Figure 1: A Random Game

Answer 1

The Pure strategy nash equilibrium is a combination of strategies in which none of the players would gain by deviating from their chosen strategy, given that all other players do not deviate. It is an equilibrium or optimal solution, in which all other players are playing pure strategies.

So, in this case, when player 1 will choose 'Top', player 2 will choose 'Left' as 10 is its highest payoff. When player 1 will choose 'Middle', player 2 will choose 'Center' as 8 is highest. When player 1 will choose 'Bottom', player 2 will choose 'Center' as 7 is highest.

Similarly, when player 2 will choose 'Left', player 1 will choose 'Top' as 5 is its highest payoff, When player 2 will choose 'Center', player 1 will choose 'Middle' as 8 is highest and when player 2 will choose 'Right', player 1 will choose 'Middle' as 6 is highest.

So, the two choices which are common or mutual are (Top, Left) and (Middle,Center) or (5,10) and (8,8) So, these two are pure strategy nash equilibrium and when chosen, both players 1 and 2 won't deviate from them.

Now, coming to mixed strategy nash equilibrium, it is a case,where at least one of the players will play a randomized strategy and there is no player who can increase his or her expected payoff by playing any alternate strategy. The row's payoffs need to be equal for all the strategies that the row player chooses with positive probability.

So, let us denote the probabilities. Player 1 plays Top, Middle and Bottom with the probabilities p1, p2 and (1-p1-p2) respectively. Similarly, player 2 plays left, center and right with the probabilities q1, q2 and (1-q1-q2) respectively.

			Player : 2
		Left (q1)	Center (q2)	Right (1-q1-q2)
Player: 1	Top (p1)	(5,10)	(4,4)	(2,1)
	Middle (p2)	(3,7)	(8,8)	(6,5)
	Bottom (1-p1-p2)	(2,5)	(2,7)	(2,2)

Now, we will find the expected payoffs from playing top,middle and bottom for player 1.

E(Top) = 5*q1 + 4*q2 + 2(1-q1-q2)

E(Middle) = 3*q1 + 8*q2 + 6*(1-q1-q2)

E(Bottom) = 2*q1 + 2*q2 + 2*(1-q1-q2)

Now, player 1 will choose a randomized strategy if only the expected payoffs from each of the strategies are equal.

So, equating E(Top) and E(Middle):

5*q1 + 4*q2 + 2(1-q1-q2) = 3*q1 + 8*q2 + 6*(1-q1-q2)

3q1 + 2q2 + 2 = 2q2 - 3q1 +6

6q1 = 4, q1 = 4/6 = 2/3

Now, equating E(Middle) and E(Bottom) :

3*q1 + 8*q2 + 6*(1-q1-q2) = 2*q1 + 2*q2 + 2*(1-q1-q2)

2q2 -3q1 + 6 = 2

3q1 = 2q2 + 4,

Solving for q2 and (1-2/3-q2) will give the MSNE for player 1.

Similarly, for player 2:

E(Left) = 10*p1 + 7*p2 + 5(1-p1-p2)

E(Center) = 4*p1 + 8*p2 + 7(1-p1-p2)

E(Right) = 1*p1 + 5*p2 +2(1-p1-p2)

So, player 2 will choose a randomized strategy if only the expected payoffs from each of the strategies are equal.

Equating E(Right) with E(Center):

1*p1 + 5*p2 +2(1-p1-p2)   = 4*p1 + 8*p2 + 7(1-p1-p2)

3p2 - p1 + 2 = p2 -3p1 + 7

2p2 +2p1 = 5

Equating E(Left) with E(Right):

10*p1 + 7*p2 + 5(1-p1-p2) = 1*p1 + 5*p2 +2(1-p1-p2)

5p1 + 2p2 + 5 = 3p2 - p1 + 2

6p1 + 3 = p2

Now, solving for p1, p2 and (1-p1-p2) will give the MSNE for player 2.