Two students are sitting in a pub arguing over who should buy the next round. To settle the argument they decide to play the following game. They place 21 pennies on the table and take turns to remove them. Each player can remove either 1, 2 or 3 coins when it is their turn. The person who picks up the last coin wins and the loser buys the next round.

i. If there are 7 coins left explain why it is optimal to take off 3 leaving 4. Why does this also mean that you want to leave your opponent with 8 coins? Continue this reasoning backwards to find the subgame perfect strategies.

ii. If both players use subgame perfect strategies who wins? ili. Extend this to the general case where you can remove up to I coins and you start with n (> 2) coins on the table and also to the case where you lose if you pick up the last coin.

iv. Is it reasonable to assume that we use backwards induction to work out the subgame perfect equilibrium when we play this game?