Exercise 2 Solution

Page 1

GAME THEORY 2017-2018 5 cfu 2 Games in extensive form and combinatorial games 2.1 Games in extensive form Exercise 7. Consider the following two games: 1. The popular game of Rock-Paper-Scissors. 2. Job offer in Paris : Anne and Mark are working together in Milan. Anne is offered a job in Paris. If she accepts the offer, Mark has to decide whether to stay in Milan or to go to Paris with his girlfriend. Anne prefers to be in Paris rather than in Milan. Mark prefers to be in Paris with her rather than in Milan. How would you represent the two games? Solution The Rock-Paper-Scissors game can be represented by a matrix, where the first column (row) represents the strategy “play rock”, the second column (row) represents the strategy “play paper” and the last column (row) represents the strategy “play scissors”:   0 1 −1 −1 0 1 1 −1 0   Since this is a zero sum game, it is sufficient to write a matrix whose entries describe the payoffs of player I: we write “1” when the first player wins, “-1” if the second player wins and “0” if there is a tie. The second game, instead, can be easily represented by a tree where Anne is player I and Mark player II (we suppose that the utilities of players are 1 if they are in Milan and 2 if they are in Paris.): I II (1, 1) leave stay leave (2, 2) stay (2, 1) 1 Note that the payoffs might change if we make more assumptions, for example if we assume that Anne prefers to be in Paris with mark rather than in Paris alone. Exercise 8. Do the trees in Figure 1a and 1b represent games? Why (or why not)? c d a e f g b I II II 1 2 3 4 5 (a) Exercise 8 c d a e f g b I II II 1 2 3 4 5 (b) Exercise 8 Figure 1: Exercise 8 Solution The tree in Figure 1b does not represent a game. Since player II does not know the choice of player I, he cannot distinguish…