Consider the following game tree representing a simple dollar auction game.

 

The following describes the extensive-form game in detail.

THE PLAYERS: Bidders 1 and 2, each with a wealth level of $3.

AVAILABLE ACTIONS: At each of their decision nodes, a bidder must decide whether to pass or to bid, where the bids they can choose from are all those multiples of $1 that exceed the current high bid by $1 or more. For example, if bidder j's bid $1 is the current high bid, then player i's set of actions is {Pass,$2,$3} (where i,j=1,2 and j≠i).

SEQUENCE OF MOVES: Bidders take turns bidding. If a bidder chooses Pass when it is their turn, the game ends and the other player receives the 'prize' of $2 provided that they have previously submitted a bid. In any outcome of the auction, both bidders pay their last bids (if any). If bidder 1 chooses Pass at the start of the game, we assume that the auction ends and neither bidder receives the prize and no payments are made; if bidder 1 bids $1, bidder 2 bids $2 and bidder 1 subsequently chooses Pass, then bidder 2 receives the prize and pays $2 for it, whereas bidder 1 pays $1 and receives nothing in return.

AVAILABLE INFORMATION: First mover (bidder 1) obviously does not know bidder 2's bidding decision when they choose their Stage 1 action; however, the second mover (bidder 2) observes bidder 1's action and takes it into account when choosing their own action, and so on.

PAYOFFS: Each bidder i's payoff is given by their net gain (i.e., what they have won in the auction minus their most recent bid).

Question

Find all backwards induction solutions to this game.
 

8 0 Pass Pass 0 0 Pass Pl. 2 Pl. 1 $2 $1 $3 $3 -2 -1 Pl. 1 $2 Pass 8 0 Pl. 2 $3 $3 -2 -1 O 0