? E XERCISE 173.2 (Dollar auction) An object that two people each value at v (a pos- itive…

? EXERCISE 173.2 (Dollar auction) An object that two people each value at v (a pos- itive integer) is sold in an auction. In the auction, the people alternately have

the opportunity to bid; a bid must be a positive integer greater than the previous bid. (In the situation that gives the game its name, v is 100 cents.) On her turn, a player may pass rather than bid, in which case the game ends and the other player receives the object; both players pay their last bids (if any). (If player 1 passes ini- tially, for example, player 2 receives the object and makes no payment; if player 1 bids 1, player 2 bids 3, and then player 1 passes, player 2 obtains the object and pays 3, and player 1 pays 1.) Each person’s wealth is w, which exceeds v; neither player may bid more than her wealth. For v = 2 and w = 3 model the auction as an extensive game and find its subgame perfect equilibria. (A much more ambitious project is to find all subgame perfect equilibria for arbitrary values of v and w.)