Question

5. Solve for Nash Equilibrium in the following 3 Player Game

Player 3: Chooses Enter

		Player 2
		High	Low
Player 1	Low	2,0,4	1,1,1
	Medium	3,2,3	0,1,0
	High	1,0,2	0,0,3

Player 3: Chooses Opt Out

		Player 2
Player 1		High	Low
	Low	2,0,3	4,1,2
	Medium	1,3,2	2,2,2
	High	0,0,0	3,0,3

Answer 1

Here, in this given game the first entries are of Player 3.

Hence, when Player 3 chooses to enter the game, then the game as is follows:

If player 3 chooses high (as 4>1) the best response for Player 1 is Low (2>1) and Player 2 is high (as 1>0). So, no Nash Equilibrium in this row

In the second row, when player 3 chooses high (as 3>0), player 1 chooses high as 3>2 and the best response for Player 2 is also high (as 2>1) and thus, there's Nash equilibrium in this row i.e 3,2,3

In the third row, when player 3 chooses low (as 3>2), player 1 chooses High (as 1>0) and player 2 is indifferent as both payoffs are same hence no Nash Equilibrium

Player 3: Chooses Enter

		Player 2
		High	Low
Player 1	Low	2,0,4	1,1,1
	Medium	3,2,3	0,1,0
	High	1,0,2	0,0,3

Thus, the Nash equilibrium in this case when the Player 3 chooses to enter the game is 3,2,3

Now Similarly when Player 3 chooses to opt out from the game, then the game is as follows:

If Player 3 chooses from first row, he opts for high (3) as it is greater than low (2), but player 1 will choose low(4) as his payoff is greater than high and player 2 will also choose low(1) as his payoff will be greater than high(0). Hence, no Nash equlibrium in the first row

Now if Player 3 chooses from second row, he is indifferent between playing high and low as the payoff for both is same i.e. 2, Player 1 chooses low (as 2>1) and player 2 chooses high (as 3>2). Thus, no Nash equilibrium in this row

Similarly, for the third row player 3 chooses low as 3>0 , Player 1 chooses Low as again 3>0 and player 2 is indifferent between High and Low as both payoffs are 0.

The Nash Equlibrium in this case is undetermined even though player 1 and player 3 choose the same strategy this is beacuse player 2 is indifferent about choosing High and Low. Hence the equlibrium is not certain.

Player 3: Chooses Opt Out

		Player 2
Player 1		High	Low
	Low	2,0,3	4,1,2
	Medium	1,3,2	2,2,2
	High	0,0,0	3,0,3

Hence, the only Nash Equilibrium in this game is 3,2,3 when player 3 chooses to play.