Game Theory Question Chris and
Game Theory Question
Chris and Pat play the game shown below, withoutcommunicating with each other. Christ is playing across the rowsand Pat is playing across the columns. The payoffs are given as:(x,y) = (payoff to Chris, payoff to Pat). Can you predict theoutcome of the game? Explain

b. In the following 3person simultaneous game player 1chooses the row (U,M, D), player 2 chooses the column (L, R), andplayer 3 chooses the box (Box 1, Box 2). Can you predict theoutcome 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:
a)
If Pat choses New York, Chris is better off choosing New York.But if Pat choses San Francisco, Chris is better off choosing SanFrancisco. 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 SanFrancisco. Hence, Pat also do not have a dominant strategy.
No dominant strategy and this game will not have a single NashEqulibrium. Therefore, it is not possible to predict theoutcome.
b) Let x_{1}, x_{2}, x_{3} be choicesmade by player1,2 and 3 respectively.
Consider player3.
If (x_{1},x_{2}) = (U,L), player3 is better offchoosing Box1
If (x_{1},x_{2}) = (U,R), player3 isindifferent between Box1 and 2
If (x_{1},x_{2}) = (M,L), player3 isindifferent between Box1 and 2
If (x_{1},x_{2}) = (M,R), player3 isindifferent between Box1 and 2
If (x_{1},x_{2}) = (D,L), player3 isindifferent between Box1 and 2
If (x_{1},x_{2}) = (D,R), player3 isindifferent between Box1 and 2
Thus Box1 is a weakly dominated strategy for player3.Hence player 3 will always choose Box1
Now, consider payoffsfor player2 in Box1
If x_{1}= U, player2 is better off with strategy R
If x_{1}= M, player2 is better off with strategy R
If x_{1}= D, player2 is better off with strategy R
That is, in Box2, R is a strictly dominant strategy forplayer2
Hence, given the choice of player3, R is a betterstrategy for player 2. Thus player2 will choose strategy’R’
Consider player1
Given that player3 choose ‘Box1′,
If Player2 choose ‘L’, ‘M’ is a batter strategy forplayer1
If Player2 choose ‘R’, Player1 is indifferent between ‘U’ and’M’.
Hence, in Box1, Player1 is better off choosing ‘M’
Therefore, the likely outcome of the game is (M, R,Box1)