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
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b. In the following 3-person 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 x1, x2, x3 be choicesmade by player-1,2 and 3 respectively.
Consider player3.
If (x1,x2) = (U,L), player-3 is better offchoosing Box-1
If (x1,x2) = (U,R), player-3 isindifferent between Box-1 and 2
If (x1,x2) = (M,L), player-3 isindifferent between Box-1 and 2
If (x1,x2) = (M,R), player-3 isindifferent between Box-1 and 2
If (x1,x2) = (D,L), player-3 isindifferent between Box-1 and 2
If (x1,x2) = (D,R), player-3 isindifferent 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 payoffsfor 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 forplayer-2
Hence, given the choice of player-3, R is a betterstrategy 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 forplayer-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)
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