Question

Game Theory Question

Chris and Pat play the game shown below, without communicating with each other. Christ is playing across the rows and Pat is playing across the columns. The payoffs are given as: (x,y) = (payoff to Chris, payoff to Pat). Can you predict the outcome of the game? Explain

New York San Francisco New York (3,2) (1,1) San Francisto (0,0) (2,3)

b. In the following 3-person simultaneous game player 1 chooses the row (U,M, D), player 2 chooses the column (L, R), and player 3 chooses the box (Box 1, Box 2). Can you predict the outcome of the game? Explain.

BOX 1

	L	R
U	1,0,2	1,1,1
M	2,0,2	1,1,1
D	1,1,1	0,2,2

BOX 2

	L	R
U	1,0,0	1,2,1
M	0,2,2	1,1,1
D	1,1,1	2,0,2

Answer 1

a)

If Pat choses New York, Chris is better off choosing New York. But if Pat choses San Francisco, Chris is better off choosing San Francisco. Hence, Chris do not have a dominant strategy.

If Chris choses New York, Pat is better off choosing New York. But if Chris choses San Francisco, Pat is better off choosing San Francisco. Hence, Pat also do not have a dominant strategy.

No dominant strategy and this game will not have a single Nash Equlibrium. Therefore, it is not possible to predict the outcome.

b) Let x₁, x₂, x₃ be choices made by player-1,2 and 3 respectively.

Consider player 3.

If (x₁,x₂) = (U,L), player-3 is better off choosing Box-1

If (x₁,x₂) = (U,R), player-3 is indifferent between Box-1 and 2

If (x₁,x₂) = (M,L), player-3 is indifferent between Box-1 and 2

If (x₁,x₂) = (M,R), player-3 is indifferent between Box-1 and 2

If (x₁,x₂) = (D,L), player-3 is indifferent between Box-1 and 2

If (x₁,x₂) = (D,R), player-3 is indifferent between Box-1 and 2

Thus Box-1 is a weakly dominated strategy for player-3. Hence player 3 will always choose Box-1

Now, consider payoffs for player-2 in Box-1

If x₁= U, player-2 is better off with strategy R

If x₁= M, player-2 is better off with strategy R

If x₁= D, player-2 is better off with strategy R

That is, in Box-2, R is a strictly dominant strategy for player-2

Hence, given the choice of player-3, R is a better strategy for player -2. Thus player-2 will choose strategy 'R'

Consider player- 1

Given that player-3 choose 'Box-1',

If Player-2 choose 'L', 'M' is a batter strategy for player-1

If Player-2 choose 'R', Player-1 is indifferent between 'U' and 'M'.

Hence, in Box-1, Player-1 is better off choosing 'M'

Therefore, the likely outcome of the game is (M, R, Box-1)