A Bidding Game Players 1 and 2 bid for an object that has value 2 for each of them. They both have wealth 3 and are not allowed to bid higher than this amount. Each bid must be a nonnegative integer...

A Bidding Game

Players 1 and 2 bid for an object that has value 2 for each of them. They both have wealth 3 and are not allowed to bid higher than this amount. Each bid must be a nonnegative integer amount. Besides bidding, each player, when it is his turn, has the options to pass (P) or to match (M) the last bid, where the last bid is set at zero at the beginning of the game. If a player passes (P), then the game is over and the other player gets the object and pays the last bid. If a player matches (M), then the game is over and each player gets the object and pays the last bid with probability 1 2 . Player 1 starts, and the players alternate until the game is over. Each new bid must be higher than the last bid.

(a) Draw the game tree of this extensive form game.

(b) How many strategies does player 1 have? Player 2?

(c) How many subgame perfect equilibria does this game have? What is (are) the possible subgame perfect equilibrium outcome(s)?

(d) Describe all (pure strategy) Nash equilibria of the game (do not make the strategic form). Is there any Nash equilibrium that does not result in a subgame perfect equilibrium outcome?