Question

Consider a variant of the Nim game called "Stones". Suppose that initially there is a single pile of 5 stones and two players, I and II. Each player takes turns picking up either 1 or 2 stones from the pile. Player I moves first, then Player II, then Player I, etc. until all stones have been picked up 3. Assuming that the loser is the player who picks up the last stone, write the game of Stones out in extensive (game-tree) form labeling strategies in terms of remaining stones (as in the Nim game discussed in class). Be sure to assign payoffs to the terminal nodes Who has a winning strategy, Player I or Player II and what is it? Explain. Redo parts a-b under the assumption that the winner is the player who picks up the last stone. a. b. c.

Answer 1

L = loss w = win Player 1 has winning strategy as if player 1 plays with 1 pebble then player 2 can either choose 1 or 2. Therefore if player 2 chooses 1 pebble then remaining pebbles are 3 to win player 1 will always choose 2 pebbles [If pl 2 choose 2 pebble then pl 1 choose 1 pebble] player 1 can surely win pl 2 has to choose last pebble If player 1 starts with 2 then player 2 can choose 2 pebbles Therefore player 1 will loose for sure. Therefore player 1 has winning strategy of choosing pebble 1 no matter what. \r\n (who picks last pebble i winner) Player 1 has winning strategy by playing 2 pebbles in stage 1. If player 2 will choose 1 pebble then player 1 choose 2 pebbles 2 & If player 2 choose 2 pebbles then player 1 will play 1 pebble. Player 1