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CHAPTER 16
Bidding Strategy and Auction Design
Teaching Suggestions
Most students nowadays will have used some Internet auction sites, certainly as
buyers, and several as sellers. You can start a discussion related to their own experiences and
bring in the appropriate terminology as appropriate. In this way, you should be able to address
the differences between common-value and private-value auctions and between sealed bid and
open outcry; students may also be familiar with first and second price, as well as ascending and
descending auction mechanisms. And you will be able to elicit many of the strategies, fair and
fraudulent, that are used to manipulate the outcomes of auctions.
For the winner’s curse, you can play the auction as a penny-jar game, described below,
or think of a different example in which the low bid wins. House painters and other service
providers often participate in such low-bid auctions. A painter will come up with her estimate
of the time necessary to do the job, add in the cost of the predicted necessary materials, and
submit a bid. In these situations, the painter (and the purchaser of the service) can verify the
quantity of materials used, so she can ask that these costs be covered in full even if they
exceed (by some limited amount) the amount stated in the original bid. But the painter cannot
ask her client to cover the costs of additional time under most such bids, because of the moral
hazard problems involved. Suppose the true time needed to sand and paint a house is 14
person-days of time (assuming every painter has the same abilities). Then each painter makes
an estimate of the time it will take her, and each estimate will come with some error (within 2
or 3 days on either side of 14, for example). If lots of painters make bids and these are all
pooled, their arithmetic average would be an unbiased and probably pretty accurate indicator of
the true time needed to do the job. But each painter has only her own estimate, and the lowest
of these will be biased on the short side (2 or 3 days short of 14). Thus, if a homeowner
chooses the lowest bid, the winning painter is likely to have bid too little and will find that she
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
has to use more time than anticipated in completing the job. Here the painter wins a job only
when her rivals make estimates higher than hers. You could ask your students about ways to
combat this problem. For instance, what if all house painters looked ahead to the winner’s
curse and adjusted their bids upward to compensate for the bias?
The revenue equivalence outcome is another example that can be used successfully in
class. The text notes that expected outcomes under all four primary types of auctions are
identical when bidders are risk neutral and their valuations are independent, but goes on to note
that more-advanced mathematical techniques are needed to prove this result.
In an English auction, the person who places the higher value on the item, call this
value VA, will win it. Given that VA is the higher valuation, the other bidder’s valuation VB is
equally likely to be anywhere between 0 and VA. On average, therefore, the lower valuation VB
= VA/2. Since the lower bidder drops out of the auction at VB = VA/2, the bidder with the higher
valuation wins the item, pays (on average) VA/2, and receives an expected profit equal to VA –
VA/2 = VA/2.
Now, suppose the same people are bidding on the item using a first-price sealed-bid
auction. Obviously, the bidders will shade their bids below their true values. The question is By
how much? Since the bidders are risk neutral, each will act in a way that maximizes expected
profit = (person’s probability of winning)(person’s value – person’s bid). Without deriving the
optimal bids from scratch, it can be shown that a Nash equilibrium bidding outcome entails
each bidder’s submitting a bid equal to half his valuation.
To show that this strategy is a Nash equilibrium, assume that Bidder 2 uses it, and
therefore submits a bid equal to B2 = V2/2. We must then demonstrate that Bidder 1’s optimal
response is to adopt the same strategy. To show this, note that Bidder 1’s probability of
winning the auction equals Prob(B1 > B2). Given Bidder 2’s assumed strategy, this probability
equals Prob(B1 > V2/2), which can be rewritten as Prob(V2 < 2B1). Since we’ve assumed that
valuations are equally likely to be anywhere between 0 and 1, the probability that V2 < 2B1
(assuming that B1 is less than or equal to 1/2) equals 2B1. Thus (given Bidder 2’s assumed
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
strategy), the probability that Bidder 1’s bid of B1 will be the higher bid (and will thus win the
auction) equals 2B1. If Bidder 1 does submit the higher bid, her profit from doing so will be (V1
– B1). We can therefore say that Bidder 1’s expected profit = (probability of winning)(value –
bid) = 2B1(V1 – B1) = 2(B1V1 – B12). The remaining step is for Bidder 1 to choose B1 in a way that
maximizes her expected winnings. (One could employ a calculus-based approach, using a
simple derivative, to show that Bidder 1’s expected winnings are maximized when V1 – 2B1 = 0,
or B1 = V1/2. Such an approach has a critical flaw, however, because the bidder’s payoff
function is not accurate at any outcome that would entail a bid over 0.5. The formula is based
on the probability of winning the auction equaling 2V1; obviously, this formula is in error for B1
> 0.5, although it turns out that there is never any reason for Bidder 1 to bid more than 0.5. The
optimality of the B = V/2 strategy is not altered by this complication, but a correct derivation of
the optimal bidding strategy is more complicated than the simple derivative would suggest.)
To avoid the difficulties inherent in applying calculus to the problem, a series of tables
of possible bids can be used to show that the optimal bid is B1 = V1/2. The procedure is to
assume that Bidder 2 uses the bidding strategy B2 = V2/2, assume a particular value for V1, and
determine Bidder 1’s optimal bid. Suppose, for example, that B2 = V2/2, V1 = 0.4, and Bidder 1
bids 0.1. Bidder 1 will win the auction with probability 0.2 (since B1 = 0.1 > B2 only if V2 < 0.2)
and will collect earnings of 0.4 – 0.1 = 0.3 if she does. Her expected earnings using this bid are
thus (0.2)(0.3) = 0.6. The following table shows how Bidder 1 would do if she used some other
possible bids:
Bid
Probability
of win
Profit
Expected profit =
(probability of win)(profit)
0.10 0.2 0.20 0.060
0.15 0.3 0.25 0.075
0.20 0.4 0.20 0.080
0.25 0.5 0.15 0.075
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
0.30 0.6 0.10 0.060
Under these assumptions, it is clear that B1 = 0.2 is the best bid; note that this bid follows the
rule B1 = V1/2.
Now, consider other possible values for V1; for example, V1 = 0.6. In this case (again
assuming that B2 = V2/2), a table like the above reads:
Bid
Probability
of win
Profit
Expected profit =
(probability of win)(profit)
0.20 0.4 0.40 0.160
0.25 0.5 0.35 0.175
0.30 0.6 0.30 0.180
0.35 0.7 0.25 0.175
0.40 0.8 0.20 0.160
Again, the optimal bid follows the rule B1 = V1/2.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
One last example, assuming that V1 = 0.9:
Bid
Probability
of win
Profit
Expected profit =
(probability of win)(profit)
0.35 0.7 0.55 0.385
0.40 0.8 0.50 0.400
0.45 0.9 0.45 0.405
0.50 1.0 0.40 0.400
Again, the optimal bid is B1 = V1/2. (Note that the above table only has four entries because
[given Bidder 2’s assumed strategy] there is never any reason for Bidder 1 to bid more than 0.5;
that bid guarantees that her bid will be the higher.)
In all of these examples, we see that Bidder 1’s optimal strategy is to bid B1 = V1/2. The
same conclusion holds for other possible values of V1. We can thus conclude that an outcome in
which each person bids an amount equal to half of her evaluation is a Nash equilibrium; when
B2 = V2/2, Bidder 1 wants to set B1 = V1/2 (and Bidder 2’s incentive is the same).
Finally, consider a bidder’s expected winnings in this Nash equilibrium. Since both
bidders are following the same strategy, the bidder who places the highest valuation on the
good will win it. Furthermore, since the winning bid is equal to half the person’s valuation, the
person with the higher valuation will always collect winnings equal to exactly half his
valuation.
Remember our conclusion from the English auction case analyzed above. The bidder
(A) with the higher valuation (VA) wins the item, pays (on average) VA/2, and receives an
expected amount of profit equal to VA – VA/2 = VA/2. In the simple case we have analyzed,
therefore, we can see that (in equilibrium) the English auction and the first-price sealed-bid
auction produce identical outcomes in terms of expected value. The person who values the
object the most always wins it and pays, either on average (in the English auction) or with
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
certainty (in the first-price sealed-bid auction), a price equal to half of her valuation of the
object.
We already know that with risk-neutral bidders and independent valuations (in
equilibrium), the first-price, sealed-bid auction produces an outcome identical to that of the
Dutch auction, while the English auction produces an outcome identical to that of a sealed-bid
second-price auction. The conclusion that the first-price sealed-bid auction and the English
auction produce identical outcomes (in this case) thus establishes the equivalence of all four
auction designs.
If you want to go more deeply into the Internet auction case study, you may want to
distribute funds to your students to use to purchase an actual item on eBay or some other
online auction site. Many, many items can be purchased for surprisingly small amounts
(although shipping fees can add considerably to the final price). Another possibility is to
discuss the bidding program used at eBay. The instructions at the site describe the bidding
program (“proxy bidding”) as a way to simulate an English auction (although eBay doesn’t use
that term) between interested bidders. The eBay procedure can (not surprisingly) also be
viewed as a way to conduct a second-price sealed-bid auction with an increment added to the
second-highest bid. Students could be asked to write (for hypothetical posting on the eBay site)
a new description that explains the eBay bidding procedure in terms of a second-price
sealed-bid auction. For listings and information about other auction sites on the Internet, see
http://online-auction-sites.toptenreviews.com or www.internetauctionlist.com.
For those who want to consider other auction-related issues, another possibility is to
pursue in greater detail the Clark-Groves public goods valuation revelation scheme to
supplement or buttress the discussion of Vickrey and second-price auctions. This connection is
mentioned briefly in the text, but the details of the Clark-Groves scheme are found in many
intermediate microeconomics texts; for example, see H. Varian, Intermediate Microeconomics,
9th ed. (New York: W. W. Norton, 2014), pp. 730–735. We also provide a number of different
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
in-class games that can be used to address a variety of topics from Vickrey’s truth serum to the
derivation of the optimal bid in an all-pay auction.
Game Playing in Class
GAME 1—Auctioning a Penny Jar (Winner’s Curse)
This game was discussed as a possible game for the first or second day of class (see
Chapters 1 and 2). Show a jar of pennies; pass it around so each student can have a closer look
and form an estimate of the contents. Show the students a stack of 100 pennies to give them a
better idea of what the jar might contain. While the jar is going around, explain the rules.
Everyone plays and each submits a sealed bid; hand out blank cards and ask students to write
down their names and bids and return the cards. The winner will pay his bid and get money
(paper and silver, not pennies) equal to that in the jar. Ties for a positive top bid split both prize
and payment equally. When you explain the rules, emphasize that the winner must pay his bid
on the spot in cash.
After you have collected and sorted the cards, write the whole distribution of bids on
the board. Our experience is that if the jar contains approximately $5.00, the bids average to
$3.50 (including a few zeros). Thus the estimates are on the average conservative. But the
winner usually bids about $6.00. Hold a brief discussion with the goal of getting across the idea
of the winner’s curse.
GAME 2—All-Pay Auction of $10
This game was also discussed as a possible game for the first or second day of class.
Everyone plays. Show students a $10 bill, and announce that it is the prize; the known value of
the prize guarantees that there is no winner’s curse. Hand out cards. Ask each student to write
down his name and a bid (in whole quarters). Collect the cards. The highest posi tive bid wins
$10; if two or more tie with the highest positive bids, they share the $10 equally. All players
pay the instructor what they bid, win or lose.
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Be sure to emphasize before bids are submitted that “This is for real money; you must
pay your bid in cash on the spot. You can make sure of not losing money by writing $0.00. But
of course if almost everyone does that, then someone can win with $0.25 and walk away with a
tidy profit of $9.75.”
Once you have collected the cards, write the distribution of bids on the board. Hold a
brief discussion about the distribution and the value of the optimal bid. This game usually leads
to gross overbidding; a profit of $50 in a class or section of 20 is not uncommon. If that
happens, you will have to find ways of returning the profit to the class; we have done this by
having a party if the sum is large enough or by bringing cookies for the next meeting if the sum
is small. Of course, do not announce this plan in advance.
You can follow this game with a relatively in-depth discussion of the optimal bid in an
all-pay auction. If your students are comfortable with the necessary mathematics, you could go
on to derive the formula for the optimal bid.
GAME 3—Common-Value Asset Auction
Tell your students to imagine that you own an asset. The value to you of this asset is
some number between 0 and 100 points. All values from 0 to 100 are equally likely to be the
asset’s value.
Whatever this asset is worth to the instructor, it is worth 1.5 times that amount to each
student. Thus, the value to the student of this asset is some number between 0 and 150 points
(each possible value is equally likely). Students are asked to make a bid for this asset. The
instructor will sell the asset to a student if and only if the student’s bid is larger than the asset’s
value to the instructor.
Students can receive points (or cash) from this game in two ways: the first outcome
will involve luck, the second will be the average outcome. For the first outcome, the instructor
will use a random-number table to choose a number between 0 and 100, which will be the
value to her of the asset. Anybody who bids more than that value buys the asset and pays the
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
number of points (or pennies) he bid; anybody who bids less than the amount found in the table
does not buy the asset. If a student buys the asset, the gain or loss of points (or pennies) is
computed as the asset’s value to the student minus the amount paid to buy it. (Instructors can
use the same asset value to compute each student’s gain or loss of points [or pennies]. A
student’s gain or loss depends on the amount bid and on the randomly selected value of the
asset.) For the second outcome, students will receive the expected value of their bids, which is
the average number of points per game that they would be expected to have if this game were
repeated a number of times.
The results from this game can be used to lead a discussion about issues related to
common-value auctions. (There are also ways in which this game can be construed as a type of
coordination game if students are concerned not only about the outcome from their own bid but
about how all other students fare in acquiring points from the game.)
GAME 4—Puppy Auction
The following is based on an auction experiment used by Bob Weber in a course on
strategic decision making at Northwestern University. Tell students:
Suppose that you are one of two collectors involved in the sealed-bid
auction of a classic 1997 “Spot” Beanie Baby puppy. This Spot is worth $V to
you (you will determine V precisely below). Thus, you would be indifferent
between losing Spot to your rival bidder and paying $V for Spot.
Having sized up your opponent, you think Spot could be worth
anything between $0 and $100 to her. (She has probably sized you up
similarly.) The two valuations of the bidders in this auction are subjective—
primarily matters of taste—and therefore we will assume that they are
independent. This means that the value you assign to Spot will not affect your
assessment of your rival’s valuation. (For instance, if you have a valuation of
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$95, this does not change your opinion about your rival’s being equally likely
to value Spot at any value from $0 to $100.)
The seller (the instructor) will unseal both bids, and will sell Spot to
the high bidder. We will consider two different pricing rules: (1) under the first
set of rules, the seller will collect from the winner a price (in points or pennies)
equal to that bidder’s bid; and (2) under the second set of rules, the seller will
collect from the winner a price (in points or pennies) equal to the lower, losing
bid.
To make it worthwhile to win the auction, all winning bidders will
receive 10 points for each dollar of profit made on their purchase. Profit is
defined as the difference between your V and the price you pay if you win the
auction. In addition, you will all calculate a pair of values and will make two
bids in each auction; this is to avoid discriminating against those with high
valuations.
You will be asked to use a piece of personal information to determine
the actual value of Spot to you. You will then be asked to consider several
questions pertaining to each type of auction.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
HANDOUT FOR THIS GAME
Name: _____________________________________
Student ID number: __________________________
Your value 1 is found by taking the last two digits of your ID number.
Value 1: _______
Your value 2 is found by subtracting value 1 from 99.
Value 2: _______
First Auction
This is a standard sealed-bid auction, where the higher of the two submitted bids wins, and the
winner pays the amount she bid. Assume the value of Spot to you is value 1.
What will you bid for Spot? _________
How likely do you think it is that your bid will win? _______
(There are no points associated with the second question here.)
Now assume the value of Spot to you is value 2.
What will you bid for Spot? _________
How likely do you think it is that your bid will win? _______
Second Auction
This is an auction in which the higher of the two submitted bids wins, but the winner only pays
the amount of the (lower) losing bid.
Assume the value of Spot to you is value 1.
What will you bid for Spot? _________
How likely do you think it is that your bid will win? _______
Now assume that the value of Spot to you is value 2.
What will you bid for Spot? _________
How likely do you think it is that your bid will win? _______
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Discussion of the results from these auction questions can be used to compare bidding
results in different auction types. Students can also consider the amount of shading that occurs
in different auctions and how bids may vary depending on the valuations of the bidders.
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