1. (40) Suppose that Betty and Veronica are participating in an ascending price, all-pay auction of a
v = 5 dollar bill. Each player starts at 0 and alternate turns bidding, with Betty going first. In each
turn, the active bidder may either outbid their rival by the minimum increment of $\epsilon$ = 1 and then pass
to the other player, or drop out and end the auction. If bidding ends with the high bidder at b and the
low bidder at b - 1, then both players pay their bids but only the winner receives the prize, receiving
payoffs of v - b and -(b - 1), respectively. If Betty declines to bid initially, then Veronica wins by
default.
(a) (10) Suppose that bidding goes on for up to n = 5 turns. Draw the game tree.
(b) (10) Using backward induction, find the subgame perfect Nash equilibrium.
(c) (10) Would it make sense to extend the bidding to a 6th turn? If we did, what would be the
subgame perfect Nash equilibrium?
(d) (10) For any finite number of turns n, find the subgame perfect Nash equilibrium, and explain
why the last turn is relevant even if it is never reached.
2. (30) Now suppose that the auction does not have a set number of turns, and that both players can
continue bidding up as high as they want until one of them drops out. While each player has an
expansive strategy set to choose from, the only meaningful choice is to set your maximum bid, as the
game ends and the winner determined once one of the players reaches their limit and drops out. Let
$b_1$ be Betty's maximum bid and $b_2$ Veronica's. Note that since the players alternate rounds, $b_1$ must
be odd and $b_2$ even, and the auction cannot end in a tie unless Betty refuses to bid initially.
(a) (10) Find each player's payoff as a function of the maximum bids $u_i(b_i, \bar{b}_{-i})$.
(b) (10) Find each player's best response to the other's maximum bid $b_i = BR_i(\bar{b}_{-i})$.
(c) (10) Find all of the pure strategy Nash equilibria in the infinite game.
3. (30) Now consider the possibility of bidders playing mixed strategies. Again, as the full strategy set
is too expansive to consider, we will focus on a subset of mixed strategies where on each of player i's
turns, she drops out with the same probability $\sigma_i$ and bid up with probability 1 - $\sigma_i$.
(a) (10) Suppose that it is Betty's turn with the high bid currently at b, and that she knows Veronica
will drop out next turn with probability $\sigma_2$. Find her expected payoffs to dropping out $E[u_1(b-
1,b)]$ and bidding up $E[u_1(b+1,b)]$.
(b) (10) Find a symmetric mixed strategy Nash equilibrium in the infinite game.
(c) (10) How does the equilibrium bidding strategy change if we were to change the value of the prize
v or the minimum bid increment $\epsilon$?
