Optimal Auction Design when Bidders are Loss Averse
Anna Dodonova
School of Management, University of Ottawa,
136 Jean-Jacques Lussier, Ottawa, ON, K1N 6N5, Canada
Tel.: 1-613-562-5800 ext.4912
E-mail: dodonova@management.uottawa.ca
and
Yuri Khoroshilov*
School of Management, University of Ottawa,
136 Jean-Jacques Lussier, Ottawa, ON, K1N 6N5, Canada
Tel.: 1-613-562-5800 ext.4768
E-mail: khoroshilov@management.uottawa.ca
*
Corresponding author
1
Optimal Auction Design when Bidders are Loss Averse
ABSTRACT
A lot of experimental evidence document that people are loss averse and
that the compensation that they require to give up a possession of a good
is higher than the amounts they agree to pay to get this possession. This
paper analyzes how optimal English auction design changes when we take
people’s loss aversion into account. It shows that the order in which
bidders place their bids does matter and that the first bidder always has an
advantage. We analyze how the optimal auction design (open vs. sealedbid) and the optimal reserve price depend on the degree of bidders’ loss
aversion and on seller’s valuation of the object. We show that it might be
optimal for a seller to set a reserve price below his own valuation of the
object. We also show that a seller who wants to maximize his expected
revenue should implement the open-bid English auction.
2
ACKNOWLEDGMENT
We are particularly grateful to Sugato Bhattacharyya, Yan Chen, Joshua Coval, Roger
Gordon, James Hines, David Laibson, Christopher Mayer, Vernon Smith, and Richard
Thaler for their invaluable support and comments.
3
1. Introduction
According to Myerson (1981) the optimal reserve price in an open-bid English auction
must be higher than the seller’s valuation of the object. Nevertheless, in reality we
observe auctions that have zero reserve prices even though sellers’ valuations of the
object being sold are clearly non-zero. For example, an on-line auction store Bidz.com
runs 3-minutes no-reserve English auctions where it sells its own jewelry with retail
value from several hundred to several thousand dollars1. Several other auctions (e.g.,
Firstauction.com) used to run 1-minute no-reserve auctions2.
It is well known that in the private-value auctions the bidders’ strategies in the open-bid
English auction and in the second-price sealed-bid auction are identical (Vickrey, 1961,
and Myerson, 1981). This result is true regardless of the bidders’ risk aversion as long as
bidders’ utility functions satisfy the axioms of expected utility theory proposed by Von
Neumann and Morgenstern (1947). Psychological evidence, however, shows that people
usually violate these axioms. The endowment effect, documented by a number of
studies3, is one of these violations. The endowment effect refers to the situation where
people do not like any changes and prefer to keep what they have. Kahneman, Knetsch
and Thaler (1990) run an experiment in which some subjects were given a decorative
mug. Later on in the experiment subjects without mugs had options to buy the mug, and
subjects with mugs had options to sell the mug. An estimated value of the mug by
“buyers” (people who did not have the mug) was $3.50 and an estimated value of the
mug by “sellers” (people with mugs) was two times higher ($7).
4
The endowment effect was one of the behavioral biases that lead to the development of
prospect theory (Kahneman and Tversky (1979, 1991, and 1992)), an alternative theory
of people’s decision making, which is currently one of the leading theories in Behavioral
Economics. The main feature of the prospect theory is loss aversion, an extension of the
endowment effect to choices under uncertainty. According to prospect theory people’s
utility from a gain w is lower than their disutility from the same loss w. Namely, if the
marginal utility from getting an object is ξ, then the marginal disutility from losing the
same object is λξ , where λ > 1 . Thus, if a person considers buying the mug, the price
that he is willing to pay for it (his “buyer’s valuation”) is lower than the price for which
he would be willing to sale the same mug if he had already owned it (his “seller’s
valuation)4.
In this paper we examine a seller’s and bidders’ behavior in the private-value open-bid
English auction when bidders are loss averse. The difference between the private-value
open-bid English auction and the sealed-bid auction is in how bidder’s attitude toward
the object changes during the auction. In the sealed-bid auctions the endowment effect
does not work because bidders submit just one sealed bid and, thus, they look at the
auction only as buyers. The bidders’ behavior in the open-bid English auction, however,
is different. In this auction, once a bid is placed, the possibility of getting the object
makes the bidder become emotionally attached to it. The bidder feels like he almost got
the object. According to the endowment effect, the bidder will be willing to bid more
because he does not want to lose this “almost his” object. Consider, for example, an
5
English auction in which bidders consecutively submit their bids with $1 bid increments.
If bidder A bids $1,000 and gets overbid by the next bid of $1,001, he thinks of
increasing his bid to $1,002 as paying just two more dollars, while dropping out of the
auction is actually considered by him to be losing the object and getting his $1,000 back.
If his “buyer’s” valuation of the object is s, while his “seller’s” valuation is αs (where
α > 1 ), he will submit higher bids as long as αs > p (where p is the bid that he must
submit; p=$1,002 in our example). So, bidders will bid more in open-bid auctions than in
sealed-bid auctions5. When a rational bidder is thinking about placing the first bid, he
knows that he is loss averse and, thus, after entering the auction, he will suffer from the
endowment bias and will tend to overbid (from his current point of view). Thus, he may
decide not to enter the auction even if his valuation of the object is higher than the
current highest bid.
In this paper we analyze English auctions with two loss-averse bidders in which bidders
submit bids consecutively with small bid increments. We show that the first bidder’s
expected profit is always higher than the expected profit of the second bidder. This is so
because after the first bidder places his bid, he becomes emotionally attached to the
object, his valuation of the object increases, and, as a result, he precommits himself to bid
aggressively. Such precommitment, in turn, deters the second bidder with mediocre
valuation from entering the auction. To encourage the competition (and, thus, to increase
the final sale price), the seller may want to place a lower reserve price. As the paper
shows, it might be optimal for the seller to set a reserve price even below his own
valuation of the object and incur a non-zero probability of selling the object at a loss.
6
This result can explain why some English auctions are run with a zero reserve price even
when objects that are sold on these auctions clearly have significant values for their
sellers. This paper also provides comparative analysis of English and sealed-bid auctions
and analyzes factors that affect the choice of the optimal auction design.
2. The model
Consider the independent, private value English auction with 2 bidders in which bidders
submit their bids consecutively with infinitesimal bid increments ∆ = o (1) . Bidder’s i
valuation of the object is s i ≥ 0 , i.e., bidder i is willing to pay up to si in order to get the
object. Bidders are loss averse and, as a result, bidder i requires αsi to compensate him
for the loss of the object, where α > 1 .6 If bidder i enters the auction and places a bid,
his “status quo” changes, he becomes emotionally attached to the object and he feels like
he almost owns it. Thus, if another bidder places a higher bid then bidder i treats losing
the auction as losing the object and getting back his bid, while submitting a higher bid is
treated only as losing an additional (small) amount of money. Therefore, if bidder i
enters the auction, he will be willing to pay up to $αsi .
Assume that s1 and s 2 are i.i.d. with probability distribution function f (s ) and the
seller’s valuation of the object is u ≥ 0 . The auction design is the following. First, the
seller sets a reserve price r. Then, a randomly chosen bidder (we will call him “bidder 1”)
decides whether he wants to enter the auction7. He observes that there are no bids have
been made yet and makes his decision based on this information. If he decides to enter,
7
then he places bid r. After that, the second bidder observes the existence or nonexistence
of a bid and, based on this information, makes his entry decision. If he decides to enter,
then he places bid r + ∆ (if bid r has been made already) or bid r (if no bids were made
yet). After both bidders make their decisions about entering the auction, they
consecutively increase their bids (by bid increment ∆ ) until one of them drops out of the
auction. Once one of the bidders drops out or decides not to participate in the auction, the
other bidder receives the object at the price of his last bid.8
The seller wants to maximize his expected utility, which is equal to the object’s price (if
the sale takes place) or to his valuation of the object u (if bidders do not place any bids).
Each of the bidders also wants to maximize his expected utility. Note that the bidder’s
utility function changes after he enters the auction. Before entering the auction any bidder
treats winning the auction as buying the object and, thus, his preferences at the moment
he makes the entry decision are determined by ex-ante utility function U ex − ante = s − p ,
where s is his valuation of the object and p is the price he pays. After entering the auction
he treats winning the auction as keeping the object and his new preferences are
determined by ex-post utility function U ex − post = αs − p , where α > 1 .
At any point of time rational bidders maximize their current utility functions. Since, after
entering the auction, a bidder’s utility function is U ex − post = αs − p , he will keep on
bidding as long as p < αs . Before entering the auction, however, the bidder uses his exante utility function U ex − ante = s − p to make his entry decision. When he makes his entry
8
decision he also takes into account his future behavior and understands that he will tend
to overbid (from his current point of view)9. To prevent the possible overbidding, a
rational bidder may decide not to enter the auction even if his valuation s is higher than
the reserve price r. To simplify further notations, let us denote π b ,i ( s i ) to be the expected
ex-ante (i.e., before entering the auction) utility of bidder i and π s to be the seller’s
expected utility. The following theorem describes bidders’ equilibrium strategies in the
described auction.
Theorem 1. (Equilibrium strategies in the English auction with loss averse bidders)
1) The following strategies constitute equilibrium in the English auction with loss averse
bidders:
•
Given reserve price r, a bidder, who makes his entry decision when there are no
bids are made yet, enters the auction and bids r if and only if his valuation is
s1 > r .
•
A bidder, who makes his entry decision when bid r has already been made, enters
the auction and bids r + ∆ if and only if his valuation s 2 satisfies the following
inequality:
π b,2 (s 2 ) ≡
s2
∫ (s
2
− αs1 ) f ( s1 )ds1 > 0
(1)
r
•
After bidders make their entry decisions, the auction continues as a standard
English auction in which bidders bid as long as αs i ≥ p .
2) For an auction with nonzero reserve price the equilibrium described above is the only
one that is robust to the introduction of infinitesimal entry costs, i.e., it is the only
9
equilibrium in which a bidder whose expected utility from entering the auction is zero
decides not to enter.
Proof: see appendix.
3. Analysis
The first apparent feature of the equilibrium described in Theorem 1 is the difference in
the first and the second bidders’ strategies. Indeed, given that the first bidder enters the
auction, his valuation of the object must be s1 > r . If, in addition, the reserve price is not
zero (r > 0 ) then the second bidder whose valuation of the object is just slightly above r
(e.g., αr > s 2 > r ) will either lose the auction or win it at a price that is above his ex-ante
valuation of the object10 s 2 . As a result, he optimally decides not to participate in the
auction. Given such behavior of the second bidder, the first bidder with valuation that is
just slightly above r (e.g, between r and αr ) will either win the auction at price r (when
the second bidder does not enter) or lose the auction to the second bidder with higher
valuation. But in no circumstances he will pay a price above his ex-ante valuation s1 . So,
he is safe to enter the auction. In effect, the first bidder behaves as a Stackelberg leader:
he enters the auction, he precommits to bid aggressively (because his preferences
change), and, as a result, he deters competitors with mediocre valuations. Thus, one may
expect the first bidder to have higher expected utility and the following proposition states
that this is, in fact, true.
Theorem 2. (First bidder’s advantage)
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In an auction with nonzero reserve price (r ≠ 0 ) the expected utility of the first bidder is
strictly higher than the expected utility of the second bidder, namely, for ∀s > r ,
π b ,1 ( s ) > π b , 2 ( s) . In an auction with zero reserve price (r = 0) the expected utility of the
first bidder is weakly higher than the expected utility of the second bidder, namely, for
∀s > r , π b ,1 ( s ) ≥ π b , 2 ( s ) .
Proof: see appendix.
The higher is the bidders’ loss aversion, the more they are willing to overbid after they
enter the auction. Rational bidders recognize that their incentives will change after they
enter the auction so that they will bid more than they should given their ex-ante
preferences. As a result, higher loss aversion will make bidders less willing to enter. To
increase participation and, thus, competition among the bidders, the seller needs to set a
lower reserve price. In some cases the non-participation problem is so severe that the
seller may be willing to set the reserve price lower than his own valuations of the object.
In particular, we may state the following theorem:
Theorem 3. (Zero reserve price)
There are conditions under which it might be optimal for a seller to set a zero reserve
price in an open-bid English auction even when his own valuation of the object is strictly
higher than zero and there is a non-zero probability that the final price may be lower than
the seller’s valuation (i.e., the seller may end up selling the item at a loss).
Proof: see appendix.
11
Another important feature that makes auctions with loss averse bidders differ from the
“classical” (Myerson, 1981) English auction is the effect of the seller’s “absolute”
valuation of the object on the final outcome of the auction. In the “classical” auction (i.e.,
when bidders are not loss averse, but may be risk averse), the outcome of the auction
depends only on the difference between the seller’s and the bidders’ valuation of the
object, and not on the seller’s valuation of the object itself. More precisely, applying a
linear operator Γ( x) = λx + µ to the seller’s and bidders’ valuation of the object results in
the
equilibrium
reserve
price
rΓ = r ⋅ λ + µ ,
the
seller’s
expected
utility
π s (Γ ) = π s ⋅ λ + µ , and the bidders’ expected utility π b ,i (Γ ) = π b ,i ⋅ λ . These properties,
however, do not hold if we take the bidders’ loss aversion into account.
To analyze how the outcome of the auction with loss averse bidders is affected by the
linear transformation of the players’ value functions, let us consider additive and
multiplicative components of this transformation separately. Let us denote the auction
defined in Part 2 by 1, and let r1 and π s (1) be the optimal reserve price and the expected
seller’s utility in that auction. Consider two sets of auctions. The first set of auctions,
numbered by µ , is the set of auctions that can be obtained from auction 1 by applying
additive operator Α( x ) = x + µ . Namely, in auction µ the seller’s valuation of the object
is
u µadd = u + µ
(2)
and the probability distribution function of the bidders’ valuation is
f µadd ( s ) = f (s − µ )
(3)
12
The second set of auctions, numbered by λ , is the set obtained from auction 1 using
multiplicative operator Μ ( x ) = x ⋅ λ in which the seller’s valuation of the object is
u λmult = u ⋅ λ
(4)
and the probability distribution function of the bidders’ valuation is
f λmult ( s ) =
f (s / λ )
(5)
λ
If the bidders are not loss averse (α = 1) , then all of these auctions are virtually identical.
In particular, the optimal reserve prices satisfy
add
rµ
−µ =
rλmult
λ
= r1 ,
(6)
and the seller’s expected utility satisfy
π
add
s
π sadd (λ )
(µ ) − µ =
= π s (1)
λ
(7)
Now, let us consider how equalities (6) and (7) change if we take the bidders’ loss
aversion into account (α > 1) . The multiplicative operator Μ (⋅) affects only the units in
which utilities and prices are measured. Indeed, if in the auction 1 all utilities and prices
were measured in dollars, then applying a multiplicative operator Μ ( x ) = x ⋅ 100 to
auction 1 will result in the same auction 1 in which all utilities and prices are measured in
cents. Since such artificial transformation hardly affects people’s behavior, we should
expect
rλmult
λ
= r1 and
π sadd (λ )
= π s (1) . The effect of the additive operator Α(⋅) is more
λ
complex. On the one hand, when a loss averse bidder makes his entry decision, he cares
13
not only about the difference between his valuation of the object and the reserve price
(s − r ) , but also about the absolute value of the reserve price
r . This is so because he
takes into account his future behavior, in particular, the fact that he will tend to bid more
than his ex ante valuation of the object. So, if the seller simply sets the reserve price
rµadd = r1 + µ (for µ > 0 ), then this reserve price will make the costs of overbidding
higher, thus, leading to the lower participation and to the lower expected profit of the
seller. On the other hand, if both bidders decide to enter the auction, an increase in
bidders’ valuation by µ will lead to an even higher increase (by α ⋅ µ ) in the final price,
which results in higher expected profit of the seller. The following theorem summarizes
the discussion above:
Theorem 4. (Linear transformation)
Denote the auction defined in Part 2 by 1, and let r1 and π s (1) be the optimal reserve
price and the expected seller’s profit in that auction. Consider two sets of auctions: the
first set is the set of auctions in which the seller’s and the bidders’ valuations satisfy (2)
and (3) and the second set is the set in which the seller’s and the bidders’ valuations
satisfy (4) and (5). Then
(a)
π sadd (λ )
= π s (1) for any λ > 0
λ
(b)
There are α > 1 , f (⋅) , u , µ > 0 , and µ > 0 such that π sadd (µ ) − µ < π s (1) and
()
π sadd µ − µ > π s (1) .
Proof: see appendix.
14
4. Open vs. sealed bid auctions: an example
If bidders are not loss averse, then their strategies in the open-bid English auction and
second-price sealed-bid auction are identical (Vickrey, 1961, and Myerson, 1981). Loss
averse bidders, however, may treat open-bid and sealed-bid auctions differently. While in
the open-bid English auctions bidders submit bids consequently and may get emotionally
attached to the object, the bidders in the sealed-bid second-price auction submit only one
bid. As a result, a bidder in the sealed-bid auction treats winning the auction as buying
the object and he places his bid based on his ex-ante utility function is U ex − ante = s − p .
So, the bidders’ loss aversion does not affect the result of the second-price sealed-bid
auction and the equilibrium strategies and expected utilities of the players in the secondprice sealed-bid auction with loss averse bidders (α > 1) are the same as in the English
auction with loss neutral bidders (α = 1) . Since bidders’ loss aversion affects the seller’s
expected utility, the seller is no longer indifferent between English and second-price
sealed-bid auctions designs.
The choice between the auction designs (open vs. sealed bid auction) depends on how
bidders’ loss aversion affects the seller’s expected utility. On the one hand, loss averse
bidders tend to bid more than they should have had if they were loss neutral. Such
overbidding is beneficial to the seller and may make him willing to choose the open-bid
English auction design. On the other hand, when a rational loss averse bidder in the
English auction makes his entry decision, he takes his future behavior into account.
15
Since, after entering the auction, loss averse bidders bid more than their ex ante valuation
of the object, a bidder may decide not to enter the English auction even if his valuation of
the object is higher than the current price. This participation problem has a negative
effect on the seller’s expected profit and may make him willing to choose the secondprice sealed-bid auction design.
In general, the choice between the open bid English auction and the second-price sealedbid auction is ambiguous and depends on the probability distribution functions of the
bidders’ valuations, on the degree of bidders’ loss aversion, and on the seller’s own
valuation of the object. To provide some practical recommendations for a seller who
needs to decide between the English and the second-price sealed-bid, we consider a
special case where the bidders’ valuation of the object is uniformly distributed on [0,1]
interval, i.e., where f (s ) = 1 for s ∈ [0,1] .
When bidders’ valuations are drawn from the uniform distribution, condition (1) of
Theorem 1 can be rewritten as (see Appendix for the proof):
T (α ), if
s2 > 
if
1,
α <2
α ≥2
(8)
where
 α

T (α ) = min 
r ,1
2 −α 
(9)
So, if α ≥ 2 , then the second bidder will never enter the auction and, as a result, the
16
second-price sealed-bid auction provides higher expected utility to the seller. Kahnaman
and Tversky (1992) and Kahneman, Knetch and Thaler (1990) estimate the parameter of
loss aversion α to be 2.25 and 2 respectively. Since the endowment effect for a person
who bids for an object in an English auction is not as strong as for a person who actually
owns the object, we are safe to assume that in the English auction the loss aversion
parameter α is less than 2.
Given the bidders’ behavior described in Theorem 1, the seller’s optimization problem
can be written as (see Appendix for the proof):

α
1

max π s (r ) ≡ ur 2 + r 1 − r 2 − (1 − r )(1 − T ) + (1 − T ) T 2 − r 2 + α  1 − T 3 − T 2 + T 3  (10)
r
2
3


(
)
(
)
(
)
where T is defined by (10)
(Figure 1 is about here)
Figure 1 presents the choice of the optimal auction design for different values of u and
α . As can be seeing from this figure, the seller should implement the open-bid English
auction when
(a) His valuation of the object is very low and/or he must sell the object and wants to
maximize the expected revenue ( u is low). The best examples of such auctions
are bankruptcy auctions and privatization auctions.
(b) The bidders are only moderately loss averse ( α is low), e.g., they are the dealers
who plan to resale the object later on
17
(c) The bidders extremely loss averse ( α is high) and the seller has a low or
moderate valuation of the object ( u is not too high). For example, when the
object for sale is a rare painting for which there is only an imperfect market and
the bidders are the final customers who want to buy the painting for their own
use.
In all other situations the seller is better of with the second-price sealed-bid auction than
with the open-bid English auction.
To understand the intuition behind this result, let us look at the tradeoff faced by the
seller. Once a loss averse bidder enters the English auction, he is willing to bid more than
he would have bid in the second-price sealed-bid auction. However, loss averse bidders
in the English auction are reluctant to enter the auction even if their valuation of the
object is higher than the reserve price. So, to increase participation, the seller may decide
to set a lower reserve price. In fact, the seller has a tradeoff: setting lower reserve price
increases the participation, but may also result in selling the object below the seller’s own
valuation. A seller who has a very low valuation of the object and/or who has to sell it do
not suffer too much from reducing the reserve price, so, we should expect him to
implement open-bid English auction. Similarly, when the bidders are only moderately
loss averse, the seller does not have to reduce the reserve price too much. When, on the
other hand, bidders are extremely loss averse, the reduction of reserve price provides the
most benefits since such bidders will overbid the most. So, if the seller’s valuation of the
object is not too high while the bidders are very much loss averse, the seller is better of
18
with the English auction.
At this point it might be interesting to compare the auctions with loss averse bidders with
auctions where bidders are loss neutral but there is an information externality between
them, i.e., the signal of one bidder affects the valuation of the other bidder (Milgrom and
Weber (1982)). Both loss aversion and information externality result in a situation when
an open-bid English auction and a second-price sealed-bid auction are no longer
equivalent. However, the implication about the best auction design is different for these
two models. Milgrom and Weber (1982) argue that in the presence of the information
externality the seller would always prefer to conduct an open-bid English auction. In our
paper we argue that the choice between the open-bid English auction and the secondprice sealed-bid auction is ambiguous and depends on the degree of loss aversion and on
the seller’s and bidders’ valuations of the object.
When the seller values the object a lot, he does not want to set a low reserve price. So,
even bidders with high valuation may decide not to enter the auction. Under such a
scenario, only one bidder will be able to enter the English auction and simply buy the
object at the reserve price. Figure 2 describes the values of u and α under which there
will be at most one bidder in the English auction. As one can expect, the region with at
most one bidder is very similar to the region where the seller prefers to implement the
second-price sealed-bid auction (Figure 1).
(Figure 2 is about here)
19
(Figure 3 is about here)
In theorem 3 we show that the seller may be willing to set a zero reserve price in the
English auction even when his own valuation of the object is strictly higher than zero.
Figure 3 depicts the parameters’ region where the seller chooses to set a zero reserve
price in the case when the bidders’ valuations are drawn from uniform [0,1] distribution.
As one can see from the figure, this region is quite large and consists of pairs (u , α )
where u is not too large and α is not too small. Indeed, high loss aversion (α ) makes
non-participation problem severe and requires the seller to substantially reduce the
reserve price. The seller, who faces the tradeoff between low participation and the
possibility to sell the object too cheap, is willing to set a low (or even zero) reserve price
only when his own valuation of the object (u ) is not too high. Nevertheless, if bidders are
sufficiently loss averse ( α is close to 2), the sellers with the valuation of up to 0.58 are
willing to set a zero reserve price!
3. Conclusion
This paper presents a model of independent private-value open-bid English auctions with
loss-averse bidders. The model incorporates the evidence of the endowment effect which
says that people’ valuations of an object are affected by whether they already possess it,
i.e., people demand higher compensation to give up the possession of the good than the
amount of money they agree to pay to obtain it. The paper shows that an English auction
20
with rational loss-averse bidders has an asymmetric equilibrium in which the first bidder
always has an advantage. Since it is hard to get loss-averse bidders to participate in the
auction, the seller needs to set a low reserve price, which, sometimes, may be even lower
that the seller’s own valuation of the object. This is so because after loss-averse bidders
start to bid, they tend to overbid and this overbidding may compensate the seller for the
possible losses from the low reserve price. Thus, the model allows one to explain the
existence of zero reserve price English auctions in which sellers clearly have significant
positive valuations of the objects they sell and the number of bidders is finite. By
investigating an English auction in which bidders’ valuations are drawn from the uniform
distribution, we analyze which factors affect the choice between English and sealed-bid
second price auctions. Among other things, we find that the open-bid auction is optimal
for a seller who has no value of the good he sells and wants to maximize expected
revenue from the auction (e.g., in bankruptcy or privatization auctions).
21
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3) Kahneman, D. and A. Tversky, (1984), “Choices, Values, and Frames,” The
American Psychologist, 39, 341-350.
4) Kahneman, D. and A. Tversky, (1991), “Loss Aversion in Riskless Choice: A
Reference-Dependent Model,” The Quarterly Journal of Economics, 106, 10391061.
5) Kahneman, D. and A. Tversky, (1992), “Advances in Prospect Theory:
Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty, 5,
297-323.
6) Knetsch, J.
(1989), “The Endowment effect and Evidence of Nonreversible
Indifference Curves,” American Economic Review, 79, 1277-1284
7) Knetsch, J. and J. Sinden (1984), “Willingness to Pay and Compensation
Demanded: Experimental Evidence of an Unexpected Disparity in Measures of
Value,” Quarterly Journal of Economics, 49, 507-521.
8) Milgrom, P. and R. Weber (1982), “A Theory of Auctions and Competitive
Bidding,” Econometrica, 50, 1089-1122
9) Myerson, R. (1981), “Optimal Auction Design”, Mathematics of Operations
22
Research, 6 (1), 58-73.
10) Samuelson, W. and R. Zeckhauser (1988), “Status Quo Bias in Decision
Making,” Journal of Risk and Uncertainty, 1, 7-59.
11) Thaler. R. H. (1985), “Mental Accounting and Consumer Choice,” Marketing
Science 4 199-214.
12) Vickrey, W. (1961), “Counterspeculation, Auctions, and Competitive Sealed
Tenders,” Journal of Finance, 16, 8-37
13) Von Neumann, J. and O. Morgenstern (1947), “Theory of Games and Economic
Behavior,” Ed. 2., Princeton University Press, Princeton, NJ.
23
Appendix
Proof of Theorem 1:
The fact that, after entering the auction, bidders bid as long as αs ≥ p follows directly
from the fact that bidders’ disutility from loosing the object is αs . We need to show that
the strategy of the second bidder is the best response to the strategy of the first bidder and
vice versa. To simplify notations we will assume that the bidders’ identities are known
ex-ante, but the same proof is also valid in the case when bidders do not know their
identity.
Given the first bidder’s strategy, if the first bidder does not enter the auction, then the
second bidder with value s 2 > r enters the auction and gets a positive utility of
π b , 2 ( s 2 ) = s 2 − r > 0 . If the first bidder enters the auction, then the expected profit of the
second bidder from entering the auction is given by
π b,2 (s 2 ) =
s2
∫ (s
2
− αs1 ) f ( s1 )ds1
(A1)
r
Thus, he will enter the auction if and only if (1) is satisfied.
Let us denote Ω 2 = {s : E (π b , 2 ( s ) | first bidder
enters ) > 0} to be the set of the second
bidder’s valuations when he decides to enter the auction after observing that one bid has
already been made. If Ω 2 is empty, then the first bidder with signal s1 > r will have
positive expected utility, and, thus, will enter the auction if and only if his signal s1 > r .
24
If Ω 2 is not empty, let us denote Q = inf {s : s ∈ Ω 2 } . Given the equilibrium strategy of
the second bidder, the expected utility of the first bidder with signal s1 > r satisfies the
following inequality:
s1
π b ,1 ( s1 ) = ∫ (s1 − y (s 2 )) f ( s 2 )ds 2 +
0
s1
∫ (s1 − r ) f (s 2 )ds 2 ≥ ∫ (s1 − y ) f (s 2 )ds 2
[s1 ,∞ )I Ω2
(A2)
0
where Ω 2 = {s : s ∉ Ω 2 } is a compliment set to Ω 2 and
αs , if s 2 ∈ Ω 2
y (s 2 ) =  2
if s 2 ∉ Ω 2
r ,
(A3)
is the price that the first bidder will have to pay if he wins the auction.
∫ (s
We have
1
[s1 ,∞ )I Ω2
− r ) f ( s 2 )ds 2 term in (A2) because it might be the case that the second
bidder with signal s 2 > s1 will not enter the auction, in which case the first bidder will
get the object at the reserve price r and get utility of s1 − r .
If s1 ∈ Ω 2 , then
s1
s1
s1
0
r
r
π b,1 ( s1 ) ≥ ∫ (s1 − y ) f ( s 2 )ds 2 ≥ ∫ (s1 − αs 2 ) f ( s 2 )ds 2 = ∫ (s 2 − αs ) f ( s)ds = π b, 2 ( s1 ) > 0
(A4)
If s1 ∉ Ω 2 , then denote x = sup{s : s ∈ Ω 2 and s < s1 } if s1 > Q or x = 0 if s1 ≤ Q . Since
π b, 2 ( s 2 ) is continuous function, we have π b, 2 ( x) = 0 . Thus,
25
s1
x
π b ,1 ( s1 ) ≥ ∫ (s1 − y (s 2 )) f ( s 2 )ds 2 + ∫ (s1 − r ) f ( s 2 )ds 2 ≥
0
x
x
s1
s1
0
x
x
(A5)
≥ ∫ ( x − y (s 2 )) f ( s 2 )ds 2 + ∫ (s1 − r ) f ( s 2 )ds 2 ≥ π b , 2 ( x) + ∫ (s1 − r ) f ( s 2 )ds 2 ≥ 0
Therefore, the first bidder will enter the auction if and only if his signal s1 > r .
s1
If
r ≠ 0,
then
we
have
either
∫ (s
1
− r ) f ( s 2 )ds 2 > 0
(for
x ≠ s1 )
or
for
x = s1 .
So,
x
s1
s1
0
r
π b ,1 ( s1 ) ≥ ∫ (s1 − y (s 2 )) f ( s 2 )ds 2 > ∫ (s1 − αs1 ) f ( s 2 )ds 2 = π b , 2 ( s1 ) ≥ 0
inequality (A5) is strict and π b ,1 ( s1 ) > 0 for any s1 > r . This fact, combined with (1),
implies that for an auction with nonzero reserve price the equilibrium described in Part 1
of Theorem 1 is robust to the introduction of infinitesimal entry costs.
To check the uniqueness, consider an equilibrium in which bidders who have zero
expected utility from entering the auction decide not to enter. Again, let
Ω 2 = {s : E (π b , 2 ( s ) | first bidder
enters ) > 0} be the set of the second bidder’s
valuations that gives him positive expected utility in the case when the first bidder
decides to enter the auction. Let Ω1 = {s : E (π b ,1 ( s ) | Ω 2 ) > 0} be the set of the first
bidder’s valuations that gives him positive expected utility given the strategy of the
second bidder Ω 2 . Ω1 and Ω 2 determine the proposed equilibrium entry strategy for
both bidders.
26
Note that if Ω1 = (r , ∞ ) , then the proposed equilibrium coincides with the equilibrium
described in the first part of the theorem. Thus, to complete the proof we need to show
that Ω1 = (r , ∞ ) .
If Ω 2 is empty (i.e., second bidder never enters the auction after observing that the first
bidder has placed a bid), then it is obvious that the first bidder will place a bid if and only
if s1 > r , i.e., Ω1 = (r , ∞ ) .
If Ω 2 is not empty then, as before, denote Q = inf {s : s ∈ Ω 2 } . Since r > 0 , the second
bidder with signal s 2 ∈ [0, αr ] has no chances to win the auction at the price below r
when the first bidder enters the auction. Therefore, Q ≥ αr . Since Ω 2 ⊂ [0, ∞ ) and is an
open set, we can portion Ω 2 into intervals I n = (a 2 n −1 , a 2 n ), n ≥ 1 where a 2 n < a 2 n +1 and
[a2 n , a 2 n+1 ] I Ω 2
is empty. Since π b , 2 ( s 2 ) is a continuous function, we have π b , 2 (ai ) = 0
and a1 ≥ αr .
Let us prove the rest of the theorem by induction. As the base of induction, note that by
construction for any s1 ∈ (r , a1 ] , the expected profit of the first bidder is
s1
π b ,1 ( s1 ) ≥ ∫ (s1 − r ) f ( s 2 )ds 2 > 0
(A6)
0
and, as a result, (r , a1 ] ∈ Ω1 .
To complete the proof, we need to show that if s1 ∈ Ω1 for any s1 ∈ (r , ai ] then s1 ∈ Ω1
27
for any s1 ∈ (r , ai +1 ]. By doing so, we will show that Ω1 = (r , ∞ ) .
If i is an odd number (i.e., if
(ai , ai +1 ) ⊂ Ω 2 )
then, by contradiction, assume that
∃s1 ∈ (ai , ai +1 ] such that s1 ∉ Ω1 and let x be the infinum of all such s1 , i.e.,
x = inf {s1 ∈ (ai , ai +1 ] : π b ,1 (s1 ) ≤ 0}. Using (A2) and (A3) one can show that the expected
profit of the first bidder with signal x satisfies
ai
x
x
π b ,1 ( x) ≥ ∫ ( x − y (s 2 )) f ( s 2 )ds 2 = ∫ ( x − y (s 2 )) f ( s 2 )ds 2 + ∫ (x − αs 2 ) f ( s 2 )ds 2
0
0
(A7)
ai
By construction, (ai , x ) ⊂ Ω1 and (ai , x ) ⊂ Ω 2 . Thus,
π b ,1 ( x) − π b , 2 ( x) ≥ π b ,1 (ai ) − π b , 2 (ai ) = π b ,1 (ai ) > 0
(A8)
Therefore,
π b ,1 ( x) > π b , 2 ( x) ≥ 0
(A9)
Since x = inf {s1 ∈ (ai , ai +1 ] : π b ,1 (s1 ) ≤ 0}, we have that π b ,1 ( x) ≤ 0 which contradicts to
(A9). Thus, s1 ∈ Ω1 for any s1 ∈ (r , ai +1 ].
Now, consider a case when i is an even number (i.e., when [ai , ai +1 ] I Ω 2 is empty). In
this case the expected profit of the first bidder with signal s1 ∈ (ai , ai +1 ] satisfies
ai
s1
s1
π b ,1 ( s1 ) ≥ ∫ (s1 − y (s 2 )) f ( s 2 ) ds 2 = ∫ (s1 − y (s 2 )) f ( s 2 ) ds 2 + ∫ ( x − r ) f ( s 2 ) ds 2 =
0
= π b ,1 (a i ) +
0
s1
∫ (s
1
ai
ai
(A10)
s1
− r ) f ( s 2 )ds 2 > ∫ (s1 − r ) f ( s 2 ) ds 2 > 0
ai
Thus, s1 ∈ Ω1 for any s1 ∈ (r , ai +1 ].
28
QED
Proof Theorem 2.
Using the notation from the proof of Theorem 1, the utility of the first bidder with signal
s can be written as:
s

s

r

π b ,1 ( s ) ≥ ∫ (s − y ) f ( s 2 )ds 2 ≥ max 0, ∫ (s − αs 2 ) f ( s 2 )ds 2  = π b , 2 ( s)
0
(A11)

If, in addition, r ≠ 0 , then
s
π b ,1 ( s ) ≥ ∫ (s − y ) f ( s 2 )ds 2 =
0
αr
∫ (αs
2
− r ) f ( s 2 )ds 2 +
r
≥
∫ (αs
2
r
s
∫ (s − r ) f (s 2 )ds 2 + ∫ (s − y ) f (s 2 )ds 2 + ∫ (s − y ) f (s 2 )ds 2 =
r
αr
0
αr
∫ (s − α s ) f ( s
2
r
αr
αr
r
2
)ds 2 + ∫ (s − y ) f ( s 2 )ds 2 +
0
s
∫ (s − y ) f ( s
α
2
)ds 2 ≥
(A12)
r
− r ) f ( s 2 )ds 2 + π b , 2 ( s ) > π b , 2 ( s)
r
QED.
Proof of Theorem 3:
Consider, for example, an auction where bidders’ values of the object are uniformly
distributed on [0,1] interval, the seller’s value of the object is u ∈ [0,1) , and the bidders
degree of loss aversion is α < 2 .11 In this case, the second bidder’s profit can be written
as
29
s1
π b , 2 ( s 2 ) ≡ ∫ (s 2 − αs1 )ds1 = s 2 (s 2 − r ) −
r
α
(s
2
2
2
  2 −α  α 
− r 2 = (s 2 − r ) s 2 
 − r 
  2  2 
)
(A13)
Thus, condition (1) can be rewritten as
s2 >
α
2 −α
r
(A14)
To simplify notation, let us denote
 α

T (α ) = min 
r ,1
2 −α 
(A15)
Given the reserve price r, the seller’s utility is equal to
 u
 αs

1
R ( s1 , s 2 ) = 
α min{s1 , s 2 }
 r
if s1 ≤ r and s 2 ≤ r
if r < s1 ≤ T and s 2 > T
if s1 > T and s 2 > T
otherwise
(A16)
Taking expectations of (A19) with respect to s1 and s 2 , one can get the seller’s expected
utility:
π s (r ) = ur 2 + r (1 − r 2 − (1 − r )(1 − T )) +
α
2
(1 − T )(T 2 − r 2 ) + α  1 (1 − T 3 ) − T 2 + T 3  (A17)
3

The problem of the seller is to maximize (A17) with respect to r. Consider a special case
when α = 1.6 and u = 0.3 .
First, assume that the seller decides to choose reserve price r ≥
2 −α
α
= 0.25 . In this case
T (α ) = 1 and the seller will chose the reserve price so that
∂π s (r )
= 0.6r + 1 − 3r 2 = 0
∂r
(A18)
30
The
solution
to
1

r = 0.1 +  0.01 + 
3

(A18)
that
maximizes
the
seller’s
utility
function
is
0.5
≅ 1.68595 . Using (A17), one can find the seller’s expected utility
to be equal to π s ≅ 0.50435 .
Now, consider a scenario in which the seller chooses reserve price r <
this case T =
α
2 −α


1
3
Taking the first derivative of (A19) one can find
2 −α
α
α
= 0.25 . In
r = 4r , and we can rewrite the seller’s problem (A17) as
π s = 0.3r 2 + 5r 2 (1 − r ) + 15r 2 (1 − 4r ) + 1.6 (1 − 64r 3 ) − 16r 2 + 64r 3  =
r<
2 −α
= 0.25 . Thus, if the seller chooses r <
2 −α
α
1.6
9.8 3
+ −5.3r 2 +
r
3
3
(A19)
∂π s (r )
= r (9.8r − 10.6 ) < 0 for all
∂r
= 0.25 then he should chose zero
reserve price (r = 0 ) . Using (A19), the sellers expected utility under this scenario is
equal to π s =
1.6
≅ 0.53333 .
3
Since 0.53333>0.50435, the optimal reserve price is r = 0 even though the seller’s
valuation of the object is u = 0.3 > 0 .
Q.E.D.
Proof of Theorem 4:
31
(a) Given the reserve price r, the seller’s expected utility in auction λ can be written as
π smult (λ , r ) = ∫
∞
∞
∫ p(s , s )df λ (s )df λ (s )
add
−∞ −∞
1
add
2
2
(A20)
1
where
u
r


p (s1 , s 2 ) = r


α ⋅ min{s1 , s 2 }

if
if
s1 ≤ r and
s1 ≤ r and
s2 ≤ r
s2 > r
if
s1 > r
∫ (s
and
s2
2
− αs1 ) f λadd (s1 )ds1 ≤ 0
2
− αs1 ) f λadd (s1 )ds1 > 0
r
s2
if
s1 > r
and
∫ (s
(A21)
r
Substituting (4), (5) and (A21) into (A20), one can find
r
π smult (λ , r ) = λ ⋅ π s  
λ
(A22)
Since r1 maximizes π s (r ) , equation (A22) implies that r1 ⋅ λ maximizes π smult (λ , r ) , i.e.,
rλmult = r1 ⋅ λ . Thus, π smult (λ ) = λ ⋅ π s (1) .
(b) Let 1 be the auction where bidders’ values of the object are uniformly distributed on
[0,1] interval and the seller’s value of the object is u = 0 . Let also α = 1.1 , µ = 0.1 , and
µ = 1 . Using the notation from the proof of Theorem 3, the seller’s maximization
problem can be written as
(
)
π s = (u + µ )(r − µ )2 + r 1 − (r − µ )2 − (1 + µ − r )(1 + µ − T ) +
+
α
2
(1 + µ − T )(T 2 − r 2 ) + α  1 (1 + µ )3 + 2 T 3 − (1 + µ )T 2 
3
3
(A23)

32
subject to
 α

⋅ max{r , µ },1 + µ 
T (α ) = min 
2 − α

(A24)
where µ is equal to 0, µ and µ for auctions 1, µ and µ respectively.
Solving (A23) and (A24), one can find the optimal reserve prices r1 ≅ 0.4666 ,
rµadd ≅ 0.5604 and rµadd = 1 , and the seller’s expected utilities π s (1) ≅ 0.4284046 ,
()
()
π sadd µ ≅ 0.5287212 and π sadd (µ ) ≅ 1.4192958 . Thus, π sadd (µ ) − µ < π s (1) < π sadd µ − µ .
Q.E.D.
Proof of formulas (8)-(10)
Formulas (8) and (9) directly follows from (1), (A13), (A14) and (A15). Formula (10)
directly follows from (A16).
Q.E.D.
33
FIGURE 1
English vs. sealed-bid second-price auction
This figure depicts the region of parameters (u , α ) (gray area) in which the seller prefers
to implement the open-bid English auction rather than the sealed-bid second-price
auction, and the region of parameters (u , α ) (white area) in which the seller prefers to
implement the sealed-bid second-price auction rather than the open-bid English auction.
34
FIGURE 2
Bidders’ participation in the optimal English auction
This figure shows the regions of parameters (u , α ) . Grey area is the area in which the
optimal reserve price in the English auction is such that both bidders may be willing to
enter the auction. White area is the area in which the second bidder never enters the
auction if he observes that the first bidder enters.
35
FIGURE 3
Optimal reserve price in the English auction
This figure depicts the region of parameters (u , α ) in which the seller will optimally set
zero reserve price in the English auction (grey area)
36
ENDNOTES
1
Several other auctions (e.g., Firstauction.com) used to run 1-minute no-reserve auctions.
2
The authors had experience of buying several gold jewelry items and high-quality household items from
Firstauction.com for $1. This experience rules out the possibility that the seller can also participate in the
auction as a buyer and buy back the items when the winning bids are too low (i.e., de facto impose a nonzero reserve price). So, one may be confident that the declared reserve price is the true reserve price.
3
See, e.g., Knetsch and Sinden (1984), Knetsch (1989), Samuelson and Zeckhauser (1988).
4
Kahneman and Tversky (1984) and Thaler (1985) show that people usually do not treat the price that they
pay as a loss while they do treat selling the object as the loss.
5
In the second-price sealed-bid auction bidders will bid p=s.
6
Kahneman and Tversky (1992) estimate the parameter of loss-aversion α = 2.25 . In the Kahneman,
Knetch and Thaler (1990) experiment, “seller’s” valuation of the object was 2 times higher than “buyer’s”
valuation.
7
It is not relevant to our analysis whether the bidders know their identity before entering the auction. I.e.,
one can assume that if a bidder who is making an entry decision observes that there are no bids made yet,
he may thing that he is the first bidder or he may think that he is the second bidder and the first bidder has
decided not to participate. Alternatively, one can assume that the bidders identity is known ex ante.
8
Alternatively, one may frame this auction design as a continuous ascending English auction design
(Myerson 1981), when, first, the auctioneer announce the reserve price, than bidders made their entry
decisions in the pre-specified order, and than the auctioneer continuously increase the price until one of the
bidder drops out of the auction.
9
After entering the auction the bidder bids up to p = αs and this behavior may make his ex ante utility to
be negative, i.e., it might be the case that
10
s − p < 0.
This is so because the first bidder will bid up to
αs1 > αr
and, thus, the second bidder cannot buy the
object for a price lower than p > αr > s2 .
11
In this example we restrict the loss aversion coefficient to α < 2 because when α ≥ 2 the loss aversion
effect is extremely large and the second bidder will decide not to participate in the auction regardless of his
own valuation if he observes that the firs bidder has placed a bid.
37