Name Key

Chris 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.

Mary 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 as:

1. At node C what strategy should Mary play? $1.50

2. At node A what strategy should Tom play? Pass

3. At node B what strategy should Tom play? $0.50

4. At the root what should Mary do? $0.50

5. Who wins the auction as it is configured? Mary

6. What is the payoff to the winner? $0.75

7. What is the payoff to the loser? $0.00

8. How much revenue accrues to the auctioneer? $0.50

9. Suppose Tom bids first, who wins the auction? Whoever bids first wins: Tom is the winner.

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? No

11. Does Tom have a dominated strategy? Yes, $1.50

12. Will either Tom or Mary ever bid $1.50 in the sealed bid all-pay auction? No. This is a dominated strategy for each of them.

13. Find the mixed strategy that Tom/Mary will play in this auction.  Fill in the blanks:

There is symmetry in the game so we only need to do the deed for one of them. Suppose Tom is playing the columns. He must pick p₁ to play column one, p₂ to play column two and (1-p₁-p₂) to play column three in a fashion that makes Mary indifferent between her three undominated strategies. So, Mary's expected payoff from any pair of her pure strategies must be equal. If Mary is indifferent between Pass and a bid of $0.5 then .625p₁=.75p₁+.125 p₂-.5(1-p₁-p₂)Similarly, being indifferent between Pass and a bid of $1.50 requires .625p₁=.25p₁+.25 p₂-.375(1-p₁-p₂). Doing the algebra yields p₁=.2. Plug your answer for p₁ back into the first relation to find p₂=.6.

14. Using her mixed strategy, what does Mary expect to pay for the token? (3/5)*(.50)+(1/5)*(1) = 0.50

15. What is the expected revenue accruing to the auctioneer? $1.00