Department of Economics
Principles of Microeconomics, Honors
T. Southeby was doing some housekeeping last weekend and found a subway token
for which he has no use. He has decided to hold an auction in order to
sell the token to one of his two friends, Tom and Mary. Chris has decided that
he will use an ALL PAY auction in which both the winner and loser pays his
bid. Chris' predicament is that he must choose between a sequential, open,
oral bid format and a sealed bid format.
and Tom each have $1.50 with which to purchase an envelope containing the subway
token worth $1.25. They are bidding against one another in increments of $0.50
in an ALL PAY auction. The auction is oral and open. Tom and Mary bid
sequentially. Mary bids first; she
may bid either $0.50 or pass. If
Mary passes on the first round then bidding goes to Tom who can get the token
with a minimum bid if he chooses. After Mary's
first bid, any subsequent bid must be exactly $0.50 higher than the preceding
bid. Subsequent to the first round, bidding ends as soon as either
of them passes. The game tree is depicted
1. At node C what strategy should Mary play?
2. At node A what strategy should Tom play?
3. At node B what strategy should Tom play?
4. At the root what should Mary do?
5. Who wins the auction as it is configured?
6. What is the payoff to the winner?
7. What is the payoff to the loser?
8. How much revenue accrues to the auctioneer?
9. Suppose Tom bids first, who wins the auction?
Let's change the auction design somewhat. It is still an
ALL PAY auction. Mary and Tom each have $1.50 with which to purchase an envelope
containing a subway token worth $1.25. Tom and Mary submit sealed bids to
the auctioneer. Their bids must be in multiples of $0.50. The highest bidder
wins the token, but the loser also pays. If there is a tie bid then the token is
awarded with the flip of a coin and both Tom and Mary pay their bid.
On a separate piece of paper, for your own use, write out a
normal form game that characterizes this auction. You should have four
strategies for each player. In your game have Tom play the columns and
Mary play the rows.
10. Does Tom have a dominant strategy?
11. Does Tom have a dominated strategy?
12. Will either Tom or Mary ever bid $1.50 in the sealed
bid all-pay auction?
13. Find the mixed strategy that Tom/Mary will play in this
auction. Fill in the blanks:
14. Using her mixed strategy, what does Mary expect to pay
for the token?
15. What is the expected revenue accruing to the