GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 271

Reserve Price When Bidders are Asymmetric

Hikmet Gunay, Xin Meng and Mark Nagelberg

June 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Reserve Price When Bidders are Asymmetric1
Hikmet Gunay, Xin Meng, and Mark Nagelberg2
University of Manitoba
Department of Economics
Southwestern University of Finance and Economics
Research Institute of Economics and Management
PRA
June 29, 2012

1 Part of this research has been done while H. Gunay was visiting Osaka University-ISER; their

hospitality is much appreciated. H. Gunay and X. Meng acknowledge the ﬁnancial support of Social
Sciences and Humanities Research Council (SSHRC) and the University of Manitoba. X.Meng also
acknowledges the support of Project 211(Phase III) of the Southwestern University of Finance andEconomics. We received useful suggestions and feedback from our colleagues. Their names will be
added later.
2 Contact author is Hikmet Gunay, University of Manitoba, Economics, Winnipeg, MB R3T
5V5, Canada. E-mail: gunay@cc.umanitoba.ca. Fax: (204) 4747681, Tel: (204) 474-8915

Abstract
We analyze the optimal reserve price in a second price auction when there are 𝑁 types ofbidders whose valuations are drawn from diﬀerent distribution functions. The seller cannot
determine the speciﬁc type of each bidder. First, we show that the number of bidders aﬀects
the reserve price. Second, we give the suﬃcient conditions for the uniqueness of the optimalreserve price. Third, we ﬁnd that if a bidder is replaced by a stronger bidder, the optimalreserve price may decrease. Finally, we give suﬃcient conditions that ensure the seller will
not use a reserve price; hence, the auction will be eﬃcient.
JEL Codes: D44.
Keywords: Auction, Reserve (Reservation) Price, Asymmetric Bidders.

1Introduction

Auctions are widely used by business, government and citizens to buy and sell goods. For
example, ﬁrms use auctions for initial public oﬀerings, governments use auctions to sell
spectrum licenses and the general public has been increasingly using auction sites suchas E-bay. In many of these auctions, single goods are sold and sellers use reserve prices.
Accordingly, there has been much research on the eﬀect of reserve prices in single object
auctions (e.g., Maskin and Laﬀont (1980), Riley and Samuelson (1981), Englebrecht-Wiggans(1987), McAfee and McMillan (1987), Levin and Smith (1996), McAfee and Vincent (1997),
Gershkov (2009), Lu (2010), Hu et. al. (2010)). However, all of these papers assume
symmetric bidders. In this paper, we relax this commonly-used assumption by allowing
the independent private valuations to be drawn from diﬀerent distribution functions, thenanalyze how this will aﬀect optimal reserve prices.
In our model, there is one good that is auctioned oﬀ using a second price auction. The
seller knows that there are 𝑁 types of bidders whose valuations are drawn from diﬀerent
distribution functions but he cannot determine the speciﬁc type of each bidder,1 or equivalently we assume that the seller has to use a unique reserve price.2 We give conditions
that determine whether or not the seller will use a reserve price. If a reserve price is used,
we analyze its properties for cases where the bidders’ valuations are drawn from common
supports. Our assumptions diﬀer from Myerson (1981) who studies the revenue maximizing
mechanism under the assumption that seller can determine the types.3 As a result, he ﬁnds
a “discriminatory reserve price”; i.e., an optimal reserve price for each type. Our motivation
comes from the fact that in practice, almost all auctions have a single reserve price (or noreserve price at all).4 We show that the optimal reserve price is determined when bidders’
1This is an assumption commonly used in industrial organization. Firms are aware of the distributionof types across consumers but they cannot price discriminate because they do not know which speciﬁc
consumers have each given type.
2
For example, when government agencies cannot use diﬀerent reserve price for diﬀerent bidders; they may
be accused of corruption, if they do so.
3
Myerson (1981) also shows that there exists a unique optimal reserve price when bidders are symmetric.
4
Englmaier and Schmöller (2012) document that many players do not use reserve price while playing theonline game HATTRICK.

1

weighted average of virtual valuations is equal to the seller’s valuation. In Myerson (1981),
the optimal reserve price is found by equating the virtual valuation of each bidder to theseller’s valuation; hence, there is a diﬀerent reserve price for each type of bidder.5
Our ﬁrst contribution is showing that, under these assumptions, the number of bidders
aﬀects the optimal reserve price. This is in contrast with Maskin and Laﬀont (1980) andRiley and Samuelson (1981). This result is in line with Levin and Smith (1996) but theirmodel assumes aﬃliated valuations. In a private valuation model, a reserve price aﬀects
revenue when there is exactly one agent bidding above the reserve price. In the case ofsymmetric bidders, the marginal cost and marginal revenue of changing the reserve pricewill be the same regardless of the identity of this bidder since all bidders are identical. In
the case of asymmetric bidders, however, the identity of the bidders will have an eﬀect.
Changing the number of agents of each type will inﬂuence the probability of who may bethat critical agent and, as a result, the reserve price will be aﬀected.
Our second contribution is to show that if a weak bidder is replaced with a stronger bidder(“strongness” is determined by the ﬁrst order stochastic dominance), the optimal reserve
price may decrease. Hence, there is no monotonic relation between ﬁrst order stochastic
dominance and the reserve price. Hu et. al. (2010) show that the optimal reserve price will
decrease if the seller becomes more risk averse since the seller does not like the risk of not
selling the object. In our model, the seller is risk-neutral but lowers the reserve price due toa change in the type of bidders.
Our third contribution is giving suﬃcient conditions for the uniqueness of the optimalreserve price. We prove this by showing that the revenue function is quasi-concave when
there are bidders that can be ranked via likelihood ratio dominance.
Our fourth contribution is showing when the seller will use (eﬀective) reserve prices,and when she will not.6 We show that when the virtual valuation of each bidder at the5

It must be clear by writing “bidders who have diﬀerent types” we mean that their valuations are drawn
from diﬀerent distribution functions. Of course, bidders who have the same type will have diﬀerent valuations
with probability measure 1 since support is continuous.
6
By eﬀective reserve price, we mean a reserve price strictly greater than the lower bound of buyer’svaluations.

2lower bound of the support is greater than the seller’s valuation and Myerson’s regularity
condition holds, then the seller will not use an eﬀective reserve price. While there are otherpapers in the literature that ﬁnd the same result (e.g. Englebrecht-Wiggans (1987), McAfee
and McMillan (1987), and Lu (2010)), they all assume that symmetric bidders have to pay
some participation (or information) cost/fee before entering the auction or non quasi linear
utility functions (Dastidar (2010)). In our model, asymmetric bidders do not have cost ofparticipation nor do they buy information. We ﬁnd suﬃcient conditions for cases when theseller uses and does not use an (eﬀective) reserve price. When the seller does not use reserve
price, the auction is eﬃcient. The implicit assumption in the literature (e.g Myerson (1981))
is that the seller is committed not to re-auction the object if he fails to sell it in his ﬁrst
attempt (McAfee and Vincent (1997) is one of the exceptions in the literature). By ﬁnding
the conditions that make using the reserve price redundant, we ensure that the commitment
problem is not an issue.
There are a few papers in the literature that assume asymmetric bidders other thanMyerson (1981). Plum (1992) and Lebrun (1999) show the existence of equilibrium bids.
Maskin and Riley (2000) and Kirkegaard (2010) compare revenues of the English (second
price) auction and the high-bid (ﬁrst price) auction. Cantillon (2008) investigates the eﬀectof bidder asymmetries on revenue of diﬀerent auction formats compared to a symmetric case(geometric average of the asymmetric distributions). Kirkegaard (2005) shows that using
reserve price is better than using entry fees. None of these papers analyze the optimal reserve
price.

2

Asymmetric Distributions with Common Support

Assume that there are 𝑁 ≥ 2 risk neutral bidders, and one risk-neutral seller with one object.
Bidders’ valuations are independently distributed with cumulative distribution functions 𝐹𝑖 ,𝑖 = 1, 2, ..𝑁 with support [𝜈, 𝜈] (with 𝜈 ≥ 0). All distribution and density functions arecontinuously diﬀerentiable and density function 𝑓𝑖 are positive everywhere.7 The distribu7

This implies continuous diﬀerentiability of the expected revenue function.

3

tions are common knowledge but the valuations are private information. The seller cannot
determine the speciﬁc type for each bidder.8 We assume that we have at least one 𝑖 suchthat 𝐹𝑖 ∕= 𝐹𝑗 ; hence, we have asymmetric bidders. The value of the object to the seller is
𝑥0 < 𝜈. If the inequality does not hold, the seller will never sell the object.
We will calculate the optimal reserve price that will be set by the seller in a second-priceauction. It is well-known that the bidders bid truthfully in a second price auction even when
they know the existence of asymmetric bidders (e.g. see Krishna (2009)). First, we haveto calculate the seller’s revenue function, 𝐸Π, when facing asymmetric bidders. For this,we will ﬁx a bidder 𝑖, and calculate the ﬁrst (i.e., highest) order statistic of the remaining∏
bidders; 𝐻𝑖 (𝑥) = 𝑗∕=𝑖 𝐹𝑗 (𝑥). We let ℎ𝑖 denote the corresponding density function. The
expected revenue function is given by:𝐸Π = 𝑥0𝑁∏∫ 𝜈 ∫ 𝑥𝑖𝑁∑
𝐹𝑖 (𝑟) +[(1 − 𝐹𝑖 (𝑟))𝐻𝑖 (𝑟)𝑟 +(𝑦ℎ𝑖 (𝑦)𝑑𝑦)𝑓𝑖 (𝑥𝑖 )𝑑𝑥𝑖 ]

𝑖

𝑟

𝑖

𝑟

The ﬁrst term with 𝑥0 shows that the case where all bidders bid below the reserve price𝑟. The ﬁrst term in the brackets of summation shows the case where a given bidder 𝑖 wins
the auction and all other bidders bid below the reserve price 𝑟, or equivalently, the otherbidders do not enter the auction. The second term is the expected payment of bidder 𝑖 giventhat he wins the auction, and at least one other bidder bids above 𝑟. Finally, to get the
expected revenue function, we have to sum over all 𝑁 bidders.
By changing the order of integration, we can write this as:𝐸Π = 𝑥0

𝑁
∏
𝑖

∫ 𝜈
𝑁
∑
𝑦ℎ𝑖 (𝑦)(1 − 𝐹𝑖 (𝑦))𝑑𝑦]
𝐹𝑖 (𝑟) +[(1 − 𝐹𝑖 (𝑟))𝐻𝑖 (𝑟)𝑟 +𝑟𝑖

By taking the derivative of this function with respect to 𝑟, we have the following equation:
8

This is analogous to a ﬁrm facing two or more diﬀerent types that it cannot distinguish. One can also
assume that the seller can distinguish the types but is required to use a unique reserve price by law. For
example, a government agency cannot treat bidders diﬀerently and use diﬀerent reserve price for diﬀerent
bidders.

4′𝐸Π = 𝑥0𝑁∑𝑓𝑖 (𝑟)𝐻𝑖 (𝑟) +𝑖𝑁∑

𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)𝑟)

𝑖

=

𝑁
∑

𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ))

(1)

𝑖
𝑖 (𝑥)
in the equation, then we
We use Myerson’s virtual valuation which is 𝐽𝑖 (𝑥) = 𝑥 − 1−𝐹𝑓𝑖 (𝑥)have:𝑁∑𝐸Π′ = −[
𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟) − 𝑥0 )](2)𝑖

So we ﬁnd an equation that the optimal reserve price must satisfy. Our ﬁrst lemma
summarizes this result.
Lemma 1 Any interior optimal reserve price must satisfy the implicit equation:
𝑁
∑

𝑁
(∑
)
𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) = −𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟) − 𝑥0 ) = 0

𝑖

(3)

𝑖

The ﬁrst order condition shows that bidders’ weighted average of virtual valuations is 𝑥0 at
the optimal reserve price, since one can re-write the ﬁrst order condition as
0⇒

∑𝑁
𝑖

𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟)−𝑥0 )∑𝑁=
𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)∑𝑁
𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟))∑𝑁
= 𝑥0 . In Myerson (1981), when bidders are asymmetric but their dis𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)tributions are known, a diﬀerent reserve price is determined for each bidder by setting theirvirtual valuation equal to the seller’s valuation. This may result in an ineﬃcient allocation since the bidder who values the object most may lose the auction.9 As we discussed
before, setting a diﬀerent reserve price for diﬀerent bidders is not practical. For example,
government agencies cannot set diﬀerent reserve price for diﬀerent bidders. If one uses aunique reserve price, the ineﬃcient allocation result of Myerson (1981) also disappears (atthe expense of seller’s revenue). That is, if the object is sold, the winner would be the bidder
who values it most in our second price “unique” reserve price auction.
9

The ineﬃciency in Myerson’s paper can be, other than the reserve price, due to the allocation rule; one
with the highest virtual valuation gets the object. His mechanism favors the weak.

5

If we had symmetric bidders (i.e, 𝐹𝑖 = 𝐹 , 𝐻𝑖 = 𝐻, and 𝐽𝑖 = 𝐽 for all 𝑖), equation 3
would be equal to (1 − 𝑓 (𝑟)(𝑟 − 𝑥0 )) = 𝐹 (𝑟) or 𝐽(𝑟) = 𝑥0 , and hence, the optimal reserve
price would not depend on the number of bidders (Maskin and Laﬀont (1980), Riley andSamuelson (1981), and Krishna (2009)). However, in our case, the optimal reserve price does
depend on the number of bidders.
Why does the result change? One has to understand that the reserve price matters when
there is exactly one bidder bidding above the reserve price. Suppose that with symmetricbidders, we increase the reserve price slightly (by one very small unit). The marginal cost ofincreasing 𝑟 is the net expected revenue forgone by not selling the object. In equation 3, thisis the part (𝑟 − 𝑥0 )𝑓 (𝑟)𝐻(𝑟) where the seller loses 𝑟 with probability 𝑓 (𝑟)𝐻(𝑟) but still owns
the object so enjoys 𝑥0 . The net marginal beneﬁt of increasing 𝑟 is (1 − 𝐹 (𝑟))𝐻(𝑟)(𝑟 − 𝑥0 ).
The winning bidder pays the new higher reserve price when she is the only person bidding
higher than 𝑟 and no other players bids above r. The probability of this is (1 − 𝐹 (𝑟))(𝐻(𝑟)).
Since the loser sells the object, he loses 𝑥0 so this amount is subtracted. With symmetricbidders, it does not matter which bidder was bidding above the reserve price. The marginal
cost and beneﬁt is the same for any given bidder.
Now consider our asymmetric case and for the sake of example assume there are two(types of) bidders. Increasing the reserve price will give diﬀerent marginal costs and beneﬁts
since it now matters whether the bidder bidding above the reserve price is the ﬁrst or thesecond type. As the number of bidders changes, it aﬀects the probability that the bidder
bidding above the reserve price is the ﬁrst or second type, and the reserve price is aﬀected.
However, note that if we double the number of all types of bidders, we ﬁnd the same optimalreserve price. On the other hand, if we add more of one type of bidder, the optimal reserve
price still changes.10
Example 2 Suppose there are 𝑁 bidders whose valuations are drawn from a uniform distribution 𝐹 , and 𝑀 bidders whose valuations are drawn from the distribution function
10

Assume that an E-bay seller faces bidders from two countries (i.e types). If she expects that bidders
from one country would increase, then she should set a reserve price accordingly.

6𝐹𝑗 (𝑥) = 𝐹 1/2 with support on [0, 1], the seller values the object at 𝑥0 = 0. The optimal uniquereserve price is determined by the following equation: (2𝑁 + 𝑀 + 𝑀/2)𝑟 − 𝑀 (𝑟)1/2 − 𝑁 = 0.
When 𝑁 = 𝑀 = 1, the optimal reserve price is 0.4846; when 𝑁 = 2, 𝑀 = 1, the optimalreserve price is 0.4910.
Levin and Smith (1996), under the aﬃliated valuations assumption show that the numberof bidders aﬀects the reserve price. In their paper, the probability of everyone bidding below𝑟, and hence, not selling the good may not go to zero in the limit (as 𝑁 increases) due tothe aﬃliated values. In our private value settings, the probability of not selling the good iszero in the limit. Yet, we ﬁnd the same result.
Now, we will ﬁnd conditions under which the seller will set an eﬀective reserve price𝑟 > 𝜈, and when he will not. In the theorem below, note that we do not assume regularity
condition in part a; i.e, we do not assume that 𝐽𝑖 functions are increasing.
𝑖 (𝑥)𝑓𝑖 (𝑥)Theorem 3 Let 𝑡𝑖 (𝜈) = 𝑙𝑖𝑚𝑥↓𝜈 Σ𝑁𝐻𝐻
exist.
𝑖 (𝑥)𝑓𝑖 (𝑥)𝑖a) If Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) < 0 then 𝑟 > 𝜈.
b) Assume 𝐽𝑖 is increasing for each 𝑖. If Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) ≥ 0 then there will be noeﬀective reserve price.11
Proof Part a) Suppose 𝑚 = Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) < 0. Note that by the properties ofthe limit function and 𝐽 is a continuous function (𝑓 and 𝐹 are continuous and 𝑓 is positive
everywhere) we have𝑁𝑚 = Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝑙𝑖𝑚𝑥↓𝜈 𝐽𝑖 (𝑥) − 𝑥0 ) = Σ𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) < 0

Then, by deﬁnition of the limit, for all 𝜖 > 0, there exists 𝛿 > 0 such that for all
Σ𝑁 𝐻 (𝑥)𝑓 (𝑥)𝑥 ∈ (𝜈, 𝜈 + 𝛿), 𝑚 − 𝜖 < Σ𝑖𝑁 𝐻𝑖𝑖 (𝑥)𝑓𝑖𝑖 (𝑥) (𝐽𝑖 (𝑥) − 𝑥0 ) < 𝑚 + 𝜖. Since 𝑚 < 0, we can pick 𝜖 small𝑖𝑁enough so that Σ𝑁
𝑖 𝐻𝑖 (𝑥)𝑓𝑖 (𝑥)(𝐽𝑖 (𝑥) − 𝑥0 ) < 0, and thus −Σ𝑖 𝐻𝑖 (𝑥)𝑓𝑖 (𝑥)(𝐽𝑖 (𝑥) − 𝑥0 ) > 0 for

all 𝑥 ∈ (𝜈, 𝜈 + 𝛿). But this last expression is the derivative of the revenue function, so a seller
will strictly increase revenues by setting 𝑟 > 𝜈.
11

We are grateful to an anonymous referee whose suggestions helped us to write this generalized theorem.

7

Suppose 𝑚 = Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) < 0. Then, by deﬁnition of the limit, for all 𝜖 > 0,
there exists 𝛿 > 0 such that for all 𝑥 ∈ (𝜈, 𝜈 + 𝛿), 𝑚 − 𝜖 < Σ𝑁
𝑖 𝐻𝑖 (𝑥)𝑓𝑖 (𝑥)(𝐽𝑖 (𝑥) − 𝑥0 ) < 𝑚 + 𝜖.
Since 𝑚 < 0, we can pick 𝜖 small enough so that Σ𝑁
𝑖 𝐻𝑖 (𝑥)𝑓𝑖 (𝑥)(𝐽𝑖 (𝑥) − 𝑥0 ) < 0, and thus
−Σ𝑁
𝑖 𝐻𝑖 (𝑥)𝑓𝑖 (𝑥)(𝐽𝑖 (𝑥)−𝑥0 ) > 0 for all 𝑥 ∈ (𝜈, 𝜈 +𝛿). But this last expression is the derivativeof the revenue function, so a seller will strictly increase revenues by setting 𝑟 > 𝜈.
Part b) Assume that 𝐽𝑖 is increasing for each 𝑖 and Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) > 0 and aneﬀective reserve price 𝑟 > 𝜈 exists. By Lemma 1, this 𝑟 must satisfy Σ𝑁
𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟) −𝑥0 ) = 0. But then for all 𝑥 ∈ (𝜈, 𝑟) we have Σ𝑁
𝑖 𝐻𝑖 (𝑥)𝑓𝑖 (𝑥)(𝐽𝑖 (𝑥) − 𝑥0 ) < 0 by the fact that 𝐽𝑖
are increasing, and 𝐻(.) and 𝑓 (.) are positive. But then the assumption Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈)−𝑥0 ) >0 cannot hold. This is a contradiction; hence, no eﬀective reserve price exists.
Now suppose Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈)−𝑥0 ) = 0 and an eﬀective reserve price 𝑟 > 𝜈 exists. Again, by𝑁
Lemma 1, this 𝑟 must satisfy Σ𝑁
𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟)−𝑥0 ) = 0. But we have Σ𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟)−𝑥0 ) > 0 by the fact that Σ𝑁𝑖 𝑡𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 ) = 0, 𝐽𝑖 are increasing, and 𝐻(.) and 𝑓 (.) are
positive. Again, this is a contradiction; hence, no eﬀective reserve price exists.

Before discussing the implications of the theorem, we give a corollary providing easiersuﬃcient conditions to check.
Corollary 4 a) If 𝐽𝑖 (𝜈) < 𝑥0 for each 𝑖 then 𝑟 > 𝜈.
b) Assume that 𝐽𝑖 are increasing and 𝐽𝑖 (𝜈) > 𝑥0 for each 𝑖 then 𝑟 = 𝜈. That is, eﬀectively,
there is no reserve price.
Proof Part a) If the seller values the object at 𝑥0 , the derivative of the expected revenue
function becomes:
𝑁𝑥0 Σ𝑁𝑖 𝑓𝑖 (𝑟)𝐻𝑖 (𝑟) + Σ𝑖 𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)𝑟)𝑁= Σ𝑁
𝑖 𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) = −[Σ𝑖 𝐻𝑖 (𝑟)𝑓𝑖 (𝑟)(𝐽𝑖 (𝑟) − 𝑥0 )](4)

By the assumption 𝐽𝑖 (𝜈) < 𝑥0 and continuity of revenue function and the fact thatderivative of the revenue function is zero exactly at 𝜈, we can ﬁnd a small 𝜖 > 0, −Σ𝑁𝑖 [(𝐽𝑖 (𝜈 +8𝜖) − 𝑥0 )𝑓𝑖 (𝜈 + 𝜖)𝐻𝑖 (𝜈 + 𝜖) > 0 without making any assumption on the curvature of 𝐽𝑖 . But
then, by setting the reserve price slightly more than 𝜈 + 𝜖 increases revenue. This completes
the proof.
Part b) The ﬁrst order derivative of expected revenue function,−[Σ𝑁
𝑖 𝐻𝑖 (𝜈)𝑓𝑖 (𝜈)(𝐽𝑖 (𝜈) − 𝑥0 )], is zero. For any 𝑟 > 𝜈 in the support, the ﬁrst orderderivative is negative since 𝐽(.) is increasing and 𝐽𝑖 (𝜈) > 𝑥0 , and 𝐻 and 𝑓 are positive.
Since increasing the reserve price decreases the revenue, the optimal reserve price should be
set at 𝑟 = 𝜈.12
Corollary 4 implies that when 𝑥0 > 𝜈 (and 𝑓𝑖 are positive as the standard assumption),setting 𝑟 > 0 (i.e., using an eﬀective reserve price) is always optimal when the support is on[0, 𝜈]. Therefore, we generalize this result in the literature (e.g. Krishna (2009)) with our
proposition above.
When will the seller not use the reserve price? 𝐽𝑖 (𝜈) > 𝑥0 implies that 1 < 𝑓𝑖 (𝜈)(𝜈 − 𝑥0 )which in turn implies that 𝜈 > 𝑥0 . Hence, a necessary condition not to use reserve priceis 𝜈 > 𝑥0 . Also, 𝑓𝑖 (𝜈) should be big enough. This means, there has to be an enough massof consumers with valuations close to 𝜈. Note that Myerson’s optimal auction will not use
eﬀective reserve price under these conditions. Our contribution is to explicitly state thesuﬃcient conditions for not using the eﬀective reserve price.
We believe that there are certain cases in which 𝑥0 can be negative. For example, there
may be a storage cost of keeping the object to the seller or a ﬁrm that is making a loss under
the current management may have negative values to the seller. In these cases, the seller
may not use the reserve price (or equivalently set the reserve price equal to 𝜈).
Example (No reserve price): Assume two bidders. Their valuation is drawn fromthe support [4, 5]. The ﬁrst one has a uniform density function 𝑓1 = 1. The second one hasa density function 𝑓2 = −3.5 + 𝑥. The seller values the object at 𝑥0 = 1. Then, the seller
would not set an eﬀective reserve price.
Englebrecht-Wiggans (1987), by assuming that bidders have a cost to participate in the
12

The seller may set any reserve price less than or equal to 𝜈 but this is eﬀectively using no reserve price.

9

auction, and Dastidar (2010), by assuming a model with non-quasi linear utility functions,show that the optimal reserve price will be equal to zero (eﬀectively no reserve price). Also,
McAfee and McMillan (1987) and Lu (2010) show that sellers may not use reserve price but
the “endogenous number” of bidders have to buy information/pay entry fee. Unlike all these
papers, we use asymmetric bidders and ﬁnd conditions when the seller will use the reserve
price and when it will not use it. If the reserve price is not used (like in the example above),this implies that the auction is eﬃcient since the object will always be sold. When using areserve price is not optimal in a second price auction, we do not have to be concerned withthe assumption of whether the seller can commit not to sell the object in a second auction.13

2.1

Stochastic Dominance with N types and a Reserve Price

We will introduce 𝑁 types that we will order from “strongest” to “weakest” in the senseof reverse hazard rate and hazard rate dominance.14 Speciﬁcally, we assume that 𝐹1 /𝑓1 <𝐹2 /𝑓2 < ... < 𝐹𝑁 /𝑓𝑁 in the interior of the support.15 In addition, we assume 𝑓𝑖 nondecreasing
which in turn implies that 𝐽𝑖 are increasing. The proof of the proposition shows that thederivative of the revenue function is single-crossing zero from above –for the interior values–. This implies that the revenue function is quasi-concave, and hence, the reserve price isunique.
Proposition 5 Suppose that all 𝑓𝑖 are non-decreasing and 𝐽𝑖 (𝜈) < 𝑥0 for each 𝑖 (i.e., 𝑟 >𝜈). There are 𝑁 types of bidders ranked from strongest to weakest in the sense of both
1 (𝑥)
2 (𝑥)
𝑁 (𝑥)
> 1−𝐹
> ... > 1−𝐹
and reverse hazard rate dominance,
hazard rate dominance, 1−𝐹
𝑓1 (𝑥)
𝑓2 (𝑥)𝑓𝑁 (𝑥)𝐹1 (𝑥)/𝑓1 (𝑥) < 𝐹2 (𝑥)/𝑓2 (𝑥) < ... < 𝐹𝑁 (𝑥)/𝑓𝑁 (𝑥) for 𝑥 ∈ (𝜈, 𝜈) and 𝑀𝑖 bidders for each type.
Then, there is a unique reserve price 𝑟 that maximizes revenue which can be calculated from
13

With the exception of McAfee and Vincent (1997) who assume seller will re-auction the object if it is
not sold, the literature generally assumes that seller will make this “not re-auction” commitment credibly(e.g. Myerson (1981).
14
If one assumes likelihood ratio dominance, it would imply both reverse hazard rate and hazard rate
dominance (Krishna (2009)).
15
Note that the way we write the inequalities are such that 𝐹𝑖 dominates 𝐹𝑗 for 𝑖 < 𝑗 in terms of thereverse hazard rate. Also note that 𝐹𝑖 (𝜈) = 0 and hence the whole term will be equal to zero for all 𝑖 so wedeﬁne inequalities only in the interior of the support.

10equation 3.
Proof of Proposition 5: From equation 3 in Lemma 1, the optimal reserve pricesatisﬁes:
Σ𝑁
𝑖=1 𝑀𝑖 𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) = 0.
First note that likelihood ratio dominance implies hazard rate dominance (i.e. 𝜆𝑖 (𝑟) =(𝑓𝑖 (𝑟)/(1 − 𝐹𝑖 (𝑟)) < 𝜆𝑗 (𝑟)) and ﬁrst order stochastic dominance (i.e., 𝐹𝑖 (𝑟) < 𝐹𝑗 (𝑟)) in theinterior of the support. This implies that (1 − 𝜆𝑖 (𝑟)(𝑟 − 𝑥0 ))(1 − 𝐹𝑖 (𝑟)) > (1 − 𝜆𝑗 (𝑟)(𝑟 −𝑥0 ))(1 − 𝐹𝑗 (𝑟)) ⇔ (1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) > (1 − 𝐹𝑗 (𝑟) − 𝑓𝑗 (𝑟)(𝑟 − 𝑥0 )) for 𝑖 < 𝑗. Therefore,
there cannot be a unique 𝑟 that makes 1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ) = 0 for all 𝑖.
It is clearly not the case that all terms in the ﬁrst order condition sum, Σ𝑁
𝑖=1 𝑀𝑖 𝐻𝑖 (𝑟)(1 −𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )), are positive or negative. So for some 𝑗 ∈ {2, ..., 𝑁 − 1}, (1 − 𝐹𝑘 (𝑟) −𝑓𝑘 (𝑟)(𝑟 − 𝑥0 )) > 0 for all 1 ≤ 𝑘 ≤ 𝑗 − 1 and 1 − 𝐹𝑘 (𝑟) − 𝑓𝑘 (𝑟)(𝑟 − 𝑥0 ) < 0 for 𝑗 ≤ 𝑘 ≤ 𝑁 .
From now on, 𝑗 will denote that cutoﬀ agent.
Take the derivative of this ﬁrst order equation and get:′Σ𝑁
𝑖=1 [𝑀𝑖 ℎ𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) + 𝑀𝑖 𝐻𝑖 (𝑟)(−2𝑓𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ))].

Since the density functions are nondecreasing, we have′Σ𝑁
𝑖=1 𝑀𝑖 𝐻𝑖 (𝑟)(−2𝑓𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) < 0, so we will ignore this part of the sum andfocus on showing that (*) Σ𝑁
𝑖=1 [𝑀𝑖 ℎ𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) < 0.
We solve for 1 − 𝐹𝑗 (𝑟) − 𝑓𝑗 (𝑟)(𝑟 − 𝑥0 ) from the ﬁrst order condition and substitute this
into (*) to get:𝑁Σ𝑁𝑖∕=𝑗 𝑀𝑖 ℎ𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 )) + 𝑀𝑗 ℎ𝑗 (𝑟)(−Σ𝑖=2 𝑀𝑖 𝐻𝑖 (𝑟)(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 −𝑥0 ))/𝑀𝑗 𝐻𝑗 (𝑟))We do not have to be concerned with division by zero since our assumptions ensure that𝑟 > 𝜈 by proposition 4. After cancelling 𝑀𝑗 in the right hand term and combining bothterms, we are left with:
Σ𝑁𝑖∕=𝑗 𝑀𝑖 𝐻𝑖 (𝑟)(ℎ𝑖 (𝑟)/𝐻𝑖 (𝑟) − ℎ𝑗 (𝑟)/𝐻𝑗 (𝑟))(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ))By using ℎ𝑘 (𝑟)/𝐻𝑘 (𝑟) = Σ𝑙∕=𝑖 𝑓𝑙 (𝑟)/𝐹𝑙 (𝑟), we can re-write the term above as:
Σ𝑁𝑖∕=𝑗 𝑀𝑖 𝐻𝑖 (𝑟)(𝑓𝑗 (𝑟)/𝐹𝑗 (𝑟) − 𝑓𝑖 (𝑟)/𝐹𝑖 (𝑟))(1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ))11This is less than 0: The ﬁrst 𝑗 − 1 terms are less than 0 because for each 𝑖 such that1 ≤ 𝑖 ≤ 𝑗 − 1, we have 𝑓𝑗 /𝐹𝑗 − 𝑓𝑖 /𝐹𝑖 < 0 (by reverse hazard rate assumption) and 1 −𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ) > 0 (since 𝑗 is the cutoﬀ agent). The 𝑗 + 1 to nth terms are lessthan 0 because for each 𝑖 such that 𝑗 + 1 ≤ 𝑖 ≤ 𝑁 , 𝑓𝑗 (𝑟)/𝐹𝑗 (𝑟) − 𝑓𝑖 (𝑟)/𝐹𝑖 (𝑟) > 0 and
1 − 𝐹𝑖 (𝑟) − 𝑓𝑖 (𝑟)(𝑟 − 𝑥0 ) < 0. Therefore, any 𝑟 that satisﬁes the ﬁrst order condition mustalso satisfy the second order condition (concavity). This implies uniqueness of an optimalreserve price. To see why, suppose there are two distinct values 𝑟1 and 𝑟2 that maximize
expected revenue and suppose without loss of generality that 𝑟1 < 𝑟2 . Both of these points
must satisfy the ﬁrst order condition and hence the second order condition, as we have justshown. But by the diﬀerentiability (and hence, continuity) of the expected revenue function,
this would imply that there is a point 𝑟3 ∈ (𝑟1 , 𝑟2 ) such that there is a local minimum at 𝑟3
and the ﬁrst order condition is therefore satisﬁed at 𝑟3 .16 But this is a contradiction because
we have just shown that any point that satisﬁes the ﬁrst order condition must be a local
maximum. Therefore, the optimal reserve price is unique.
For example, let 𝐹𝑁 = 𝑥 be uniform distribution function on [0, 1], 𝐹𝑁 −1 = (𝐹𝑁 )2 =𝑥2 ,...,𝐹1 = (𝐹𝑁 )𝑁 = 𝑥𝑁 . This is the power distribution widely used in asymmetric bidders
literature. (see Cantillon (2008), for example). This example satisﬁes all conditions of our
proposition 5; hence, the reserve price that satisﬁes equation 3 is the optimal unique reserve
price. More examples can be created by replacing the uniform distribution in the example
with any other distribution that has a non-decreasing density function.
Next, we will show that under some conditions, replacing a weaker bidder with a stronger
one may lower the optimal reserve price.
Normally, one would expect that if a stronger bidder joins the auction, the seller should
increase the reserve price. Next, we will give an example that this is not correct; thatis, there is not a monotonic relation between ﬁrst order stochastic dominance and optimalreserve price. Let us emphasize that in this example, some 𝑓𝑖 are decreasing.
16

Since 𝑟1 and 𝑟2 satisfy the second order condition, the expected revenue function cannot be a line between
𝑟1 and 𝑟2 .

12

Example 6 Suppose there are 𝑁 bidders whose valuations are drawn from a uniform distribution 𝐹 and 𝑀 bidders whose valuations are drawn from the distribution function 𝐹𝑗 (𝑥) =𝐹 𝛼 with support on [0, 1]. For any 𝛼 > 0, the optimal unique reserve price is determined by
the following equation: (2𝑁 + 𝑀 + 𝑀 𝛼)𝑟 − 𝑀 (𝑟)1−𝛼 − 𝑁 = 0. When 0 < 𝛼 < 1, the second
order condition holds and 𝑀 bidders are weak bidders in the sense of ﬁrst order stochastic
dominance, and as 𝛼 increases, bidders become stronger. Figure 1 shows the optimal reserve
price as 𝛼 changes. That is, we keep 𝑁 bidders in the auction but replace 𝛼 = 0.1 bidders
with 𝛼 = 0.2 bidders, then replace 𝛼 = 0.2 bidders with 𝛼 = 0.3 bidders and so on. Figure 1
shows that the optimal reserve price ﬁrst decreases, then increases.
Plot of Optimal Reserve Prices
0.498

0.496

Reserve Price

0.494

0.492

0.49

0.488

0.486

0.484
0.1

0.2

0.3

0.4

0.5
alpha

0.6

0.7

0.8

0.9

Figure 1: 𝑁 = 𝑀 = 5. When 0 < 𝛼 < 1, optimal reserve price may decrease.

To understand why there is no monotonic relation between 𝛼 and 𝑟, suppose that 𝛼 isextremely low; that is, there is a very high probability that the weak bidder’s valuation isclose to zero. Hence, the seller is better oﬀ by selecting a reserve price similar to what theywould choose if they faced only the strong bidder. As a result, they will either sell the object
to the strong bidder at the reserve price or will not sell the object at all. Now, if we increase13

𝛼 slightly, the weak bidder may bid above the reserve price at a time when the strong bidder
turns out to have a low valuation that is below 𝑟. Lowering 𝑟 makes sense since the seller ismore likely to sell the good to one of the types. The marginal beneﬁt of lowering 𝑟 is sellingto the weak type at the new 𝑟. The marginal cost is selling the good to the strong type atthe new low reserve price. The ﬁrst eﬀect must be dominating at low 𝛼 so that the seller
decreases the reserve price. As the weak type becomes stronger (we increase 𝛼 further), thenthe risk of not selling the object diminishes considerably; hence, there is no need to decrease
the reserve price, rather it is better to increase it.
Hu et. al (2010) shows that the more risk averse the seller is the lower the reserve price(since a more risk averse seller does not want to face the risk of not selling the object). In
our paper, we show that the optimal reserve price will be lower when the seller is risk-neutral
but the type of bidders change.

3

Conclusion

We characterized the optimal reserve price for asymmetric bidders. First, we showed that, incontrast to the symmetric case, the number of bidders changes the reserve price. When thebidders’ types are known, the seller should set a diﬀerent reserve price for each bidder wherethe virtual valuations of each bidder is equal to seller’s valuation at the optimal reserve price.
This may create an ineﬃcient allocation (Myerson (1981)) in the sense a bidder that values
the object less may win the auction.17 In our model, we show that the unique reserve priceis found by setting a weighted average of bidders’ virtual valuations equal to the seller’svaluation of the object. Hence, the allocation is more eﬃcient; the bidder who values theobject most will win the auction as long as his valuation is above the reserve price. Second,
we gave suﬃcient conditions for the uniqueness of the optimal reserve price. Third, with an
example, we showed that if a weak bidder is replaced with a strong bidder in an auction,
the optimal reserve price may decrease. Hence, there is no monotonic relation between ﬁrst
17

We once again acknowledge that the ineﬃciency in Myerson’s paper can be due to the allocation rule;
one with the highest virtual valuation gets the object. The mechanism favors the weak.

14

order stochastic dominance and the reserve price. Finally, we showed when the seller will use
a reserve price in an asymmetric auction and when she will not. The conditions that makethe seller abstain from using a reserve price are suﬃcient to make the auction an eﬃcient
one, since the object will always be sold to the bidder who values it most.
One potential avenue for future research is ﬁnding the optimal reserve price for the ﬁrstprice auction under these asymmetric bidders assumption. Unfortunately, this is complicatedsince it is well-known that ﬁnding equilibrium explicitly is valid only for limited cases even
without reserve price. In addition, Maskin and Riley (2000) and Cantillon (2008) ﬁnd theirresults for ﬁrst price auctions with only two bidders.

4

References

Cantillon, E.(2008), “The eﬀect of bidders’ asymmetries on expected revenue in auctions,”
Games and Economic Behavior 62, 1-25.
Dastidar, K. G.(2010), “Auctions where incomes are private information and preferences (nonquasi-linear) are common knowledge,” Discussion Paper No. 790, ISER, Osaka University.
Englebrecht-Wiggans, R.(1987), “On optimal reservation price in auctions,” Management
Science 33, 763-770.
Englemaier, F., and A.Schmöller(2012), “Determinants and eﬀects of reserve prices in online
auctions,” Mimeo.
Gershkov, Alex.(2009), “Optimal auctions and information disclosure,” Review of Economic
Design 13, 335-344.
Hu, A., Matthews S.A., and L. Zou(2010), “Risk aversion and optimal reserve price in ﬁrst
and second-price auctions,” Journal of Economic Theory 145, 1188-1202.
Kirkegaard, R.(2005), “Participation fees vs. reserve prices in auctions with asymmetric or
colluding buyers, Economics Letters 89, 328-332.
Kirkegaard, R.(2009), “Asymmetric ﬁrst price Auctions,” Journal of Economic Theory 144,
15

1617-1635.
Kirkegaard, R.(2012), “Ranking asymmetric auctions using the dispersive order,” Working
Paper. University of Guelph.
Krishna, V.(2009), Auction Theory 2nd.ed. Academic Press.
Lebrun, B.(1999), “First price auctions in the asymmetric N bidder case,” International
Economic Review 40, 125-142.
Levin, D., and J.L.Smith(1996), “Optimal reservation prices in auctions,” Economic Journal
106, 1271-1283.
Lu, J.(2010), “Entry coordination and auction design with private costs of information acquisition,” Economic Inquiry 48, 274-289.
Maskin, E., and J.J.Laﬀont(1980), “Optimal reservation price in the Vickery auction,” Economics Letters 6, 309-313.
Maskin, E., and J. Riley(2000), “Asymmetric auctions,” Review of Economic Studies 67,
413-438.
McAfee, P., and J. McMillan(1987), “Auctions with entry,” Economics Letters 23, 343-347.
McAfee, P., and D.Vincent(1997), “Sequentially optimal auctions,” Games and Economic
Behavior 18, 246-276.
Menezes F. M., and P.K. Monteiro(2004), An Introduction to Auction Theory, Oxford University Press.
Myerson, R.B.(1981), “Optimal auction design,” Mathematics of Operation Research 6, 5873.
Plum, M.(1992), “Characterization and computation of Nash-equilibria for auctions withincomplete information. International Journal of Game Theory 20, 393-418.
Riley, J.G., and W.F.Samuelson(1981), “Optimal Auctions,” American Economic Review
71, 381-392.
16