Question

1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players take turns. When it is a player's turn, she can remove any number of matches from either pile. Each player is required to remove some number of matches if either pile has matches remaining, and can only remove matches from one pile at a time. Whichever player removes the last match wins the game. Winning gives a player a payoff equal to 1, and losing gives a player a payoff equal to 0. The initial configuration of the piles has one match in one of the piles, and two in the other one. a) Write down the game tree for this sequential game. [3 points] b) What is the Nash equilibrium? [3 points]

Answer 1

Brief expalantion of the game tree : If player 1 chooses pile1 and removes the single matchstick then player 2 has two options i.e either to remove 1 or 2 matchstick from pile 2. If he removes 1 match, then player 1 is left with the last match and thus he wins and the payoff is (1,0). If player 2 removes 2 matchsticks then he wins and payoff is (0,1). Similar game is followed if player 1 chooses the 2nd pile. Through backward induction we can see that for player 1 to ensure his win, he must remove 1 match from pile 2nd and then he wins in any case.

b.) Assuming that both the players are rational , from the table we can see that the optimal strategy for player 1 is to remove 1 matchstick from pile 2 and by playing this strategy, he'll always win. In general terms, whoever starts the game should remove 1 match from the 2nd pile. Therefore the nash equilibrium is Player 1 removing one matchstick from pile 2 .