Two players, A and B, play the following game: On the table is a pile of 4 stones. The stones are alphabetically marked, meaning "a" is on the top, below is "b", then "c", and on the bottom is "d". In the game, players take turns to take stones from the pile, starting at the top. Each player can take either one or two stones. Winner is the player who takes the last stone. Let us define the payoff of winning as 1 and of loosing as 0.

a)Give the extensive-form representation of the game.

b)Give the normal-form representation of the game and find all (pure-strategy) Nash equilibria.

c)Find all subgame perfect Nash equilibria.

d)Is there awinning strategyfor either player?

e)Assume the pile includes only 3 stones. Give the extensive-form representationand find all subgame perfect Nash equilibria. Is there a winning strategy for either player?f)Which player has a winning strategy if the game starts with 5 stones? Can you work out a general rule for winning strategies for n stones?