Rational Reserve Pricing in Sequential Auctions
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1

Rational Reserve Pricing in Sequential Auctions
A durable asset is sold at a first-price auction. The reservation price is not disclosed. If no bid exceeds the reservation price the asset is sequentially auctioned until it is sold. The undisclosed reservation prices may vary in sequential auctions.
(a) How does a risk-neutral seller adjust his undisclosed reservation prices in sequential auctions to optimize his expected return?
(b) How do optimal adjustments of sequential reservation prices affect the ultimate transaction price?
A rational risk-neutral seller will design an inter-temporal series of reservation prices in such a way as to maximize the discounted value of the expected transaction price. The seller estimates the probability density functions governing the maximum bids at each future auction date.
Key words: reservation prices, sequential auctions, probabilities
JEL Classification : D44
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Rational Reserve Pricing in Sequential Auctions
1 INTRODUCTION
This paper analyzes the management of risks faced by a risk-neutral seller of a durable asset in a series of sequential auctions. The property of durability in this context means that the attributes determining the asset’s real monetary value are assumed to be temporally invariant. The risks the seller contemplates are manifest in two dimensions: time and money. Here is the general scenario.
The seller makes it known to a population of potential bidders that he is auctioning the asset at a specified future date. He solicits bids to be tendered on or before the specified date. The auction is a sealed-bid protocol; the bidders do not know each other ’s bids and there are too many bidders to make collusion among them feasible. The seller informs the potential bidders that he reserves the right to refuse to sell the asset if he does not receive a bid at least as large as an undisclosed reservation price. If the asset is not sold at the specified auction date, it is auctioned at a subsequent date, subject to the same right-of-refusal with respect to an undisclosed reservation price.
Many kinds of assets are sold pursuant to the procedure adumbrated above; sales of privately-owned businesses 1 , auctions of art and wine 2 , auctions on e-Bay 3 , sales of real property 4 , transponder leases 5 , sales of mackerel 6 and sales of cattle 7 are a few examples of the sale scenario.
1 Povel and Singh [23].
2 See, for example, Ashenfelter [1] and Beggs and Grady [4].
3 The e-Bay platform is probably the best known of the on-line auctions. In on-line auctions, the opening price is the public reserve price. In e-Bay auctions the seller can set a secret reserve price in addition to the opening price by paying an extra fee. The seller announces that there is a secret reserve, but keeps the price a secret. During the auction the seller advises the bidders whether the secret reserve price has been met. If the highest bid received in the auction is below the secret reserve, then the object is not sold. See Hossain [17], Chuan-Hoo Tan [8] and Houser
[18] and Katkar [19].
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Rational Reserve Pricing in Sequential Auctions
Apart from the physical assets described above, the theory developed in this paper is general enough to apply to initial public offerings (hereafter IPOs) of securities. In the United States and increasingly throughout the world a common method of managing an IPO is a two-stage auction-like method called (by some)
“ bookbuilding .” 8 In this process the investment bank solicits investor indications of interest. These expressions of interest are used by the members of the underwriting syndicate as the basis for designing share allocations as well as the offering price. The issuer reserves the right to withdraw the equity from sale if the resulting offer prices are not at least as large as the issuer’s undisclosed reservation price.
9
The seller must adjust his reservation price(s) for the possibility that the asset will be offered at a series of discrete future dates before it is finally sold.
Hence the two manifestations of risk identified above. Each reservation price is hypothetical in the sense that it is necessarily conditioned by the hypothesis that the asset was not sold at a prior date. The problem faced by the seller is to determine a time series of optimal reservation prices to be set by him in hypothetical sequential sale dates.
This paper addresses the question of how a rational seller will adjust his reservation prices, if at all, in successive auctions.
4 Haurin [16 ]
5 Milgrom and Weber [21 ]
6 See Hauge [15]
7 See Zulehner [28 ]
8 See Ljungqvist [20] and Sherman [25]
9 Twenty percent of the IPOs filed with the SEC between 1985 and 2000 were later withdrawn.
Dunbar and Foster [9]
4

Rational Reserve Pricing in Sequential Auctions
2 THE CONTRIBUTION OF THIS PAPER
The literature on bidders’ strategies in auctions is large and getting larger.
However, I know of no published results addressing the question of how a seller will adjust his undisclosed reservation prices in sequential auctions of an asset.
The absence of published research addressing that issue is significant because in the architecture of sequential ascending first-price auctions, one of the few parameters the seller can control is the series of his sequential reservation prices.
10
In a single auction format the optimization problem faced by a risk-neutral seller is to ascertain the reservation price which will maximize the expected revenue. The seller’s information concerning the bidders’ valuations in those auctions is limited to what he can learn (or estimate) antedating the auction. A representative statement in a recently published manuscript is the following
Hauge [5, p. 1]
“A central question is how the seller should design his auction format in order to garner the most revenues… In order to implement the optimal reserve price, we need information concerning the distribution of private values on which bidders base their bids.”
In sequential auctions, by contrast, the seller’s information set as to the bidders ’ private valuations is augmented at each successive auction. The information can be used by the seller to adjust his reservation price(s) to
10
The paucity of analytical results was recognized by Xiao, Yang and Li [27, p. 116] Those authors surveyed the literature on reserve prices extant in 2009. Their particular interest was sponsored research advertising, but their comments apply to all sequential auctions: “To our knowledge, factors around reserve price for sponsored research advertisement have not been fu lly addressed in the literature.”
5

Rational Reserve Pricing in Sequential Auctions maximize the expected revenue at the next auction and all successive auctions, should there be any.
The bidders, on the other hand, cannot acquire much information about the distribution of bids in sequential auctions. Bidders know only three facts ex post each auction:
(a) Each bidder knows only his own history of unsuccessful bids.
(b) Each bidder knows that no rival’s bid exceeded the undisclosed reservation price in any auction.
(c) Each bidder knows that the seller’s undisclosed reservation price in every auction may vary.
The formal model developed in this paper is not addressed to the strategies of bidders. The bidders’ information set, described above, is meager relative to the information set of the seller. The seller has complete information as to the distribution of bids in each successive auction.
The theory focuses exclusively on the adaptions of the seller. The analytical results derived here rely on the seller’s observations of the distribution of bids over time. Moreover, the model developed in this paper does not rely on an assumption that the distributions of the bids are stationary.
3 BEHAVIORAL ASSUMPTIONS APPLYING TO THE SELLER
I assume that the seller is risk-neutral, at least with respect to this particular asset sale. That assumption motivates the definition of rationality applied in this paper. I assume th at the seller’s attitude of risk-neutrality motivates him to set a series of hypothetical reservation prices such that they
6

Rational Reserve Pricing in Sequential Auctions maximize the discounted value of the expected auction transaction price, defined over the time horizon of hypothetical auction dates.
The seller selects a time-invariant discount rate reflecting his intertemporal preference and his perception of the risks incident to the sale.
The seller ’s intertemporal adjustments of his reservation prices are assumed to be based, in part, on the sequentially revealed information concerning the distribution(s) of bid prices. Future flexibility is valuable in a stochastic world and should be optimized. I assume the seller can estimate the properties of the probability distributions governing the bids after each auction.
He sets his subsequent reservation prices accordingly. This assumption conforms to what has been called the rational expectations hypothesis.
4 THE FORMAL MODEL OF THE SEQUENTIAL AUCTIONS
Let p(t) represent the seller’s undisclosed reservation price at auction date t . At that date the seller solicits bids. If the seller receives a bid as least as large as his reservation price, he sells the asset at the largest offer. If he does not receive a bid at date t
at least as large as his reservation price at that date, he does not sell and the asset is carried over to the next sale date, designated as t
+ Δ t . I assume the direct costs to the seller of deferring the sale are insignificant. At the next auction date the seller adjusts his reservation price, represented by p ( t
+ Δ t ) . To simplify the model I assume the time elapsed between successive auctions is a constant, where
Δ t = 1.
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Rational Reserve Pricing in Sequential Auctions prices:
{
The seller contemplates a time series of hypothetical optimal reservation p ( t ) t
=
,
, 
}
. Each element in the series is conditioned on the event that the asset was not sold at an earlier auction date. His objective is to set each element in the series to maximize the discounted expected revenue from the sale.
I assume the largest bid at each sequential auction is a random variable, symbolized by p . Let f t
( p ) represent the probability density function governing p at auction date t
.
The probability density functions governing the series of maximum bids are assumed to constitute a stochastic process symbolized by the series: { f t
( p ), f t
+
1
( p ), f t
+
2
( p ), 
}
I do not assume the density functions in the series
{ f t t
=
1 , 2 , 
}
are stationary, although they may be. Inasmuch as the auctioned asset is durable, the population of bidders in each successive auction is dissimilar to the population in previous auctions. Bidders from earlier auctions may participate in later auctions if the asset is not sold; some bidders may drop out of an auction and resume bidding at a later auction; some drop-outs may never resume bidding; and first-time bidders may enter successive auctions. Thus, the changing population of bidders is consistent with an assumption that the elements in the series of density functions
{ } t
are statistically independent.
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Rational Reserve Pricing in Sequential Auctions
The probability that the asset will be sold at auction date t , if it has not been sold earlier, is a conditional probability. It is calculated as the area under upper tale of the density function f t
( p ) , bounded from below by the seller’s reservation price at that date. The conditional probability is represented as:
Pr[ asset is sold at t no sale antedating t ] =
Pr[ highest offer price at t
≥ reservatio n price at t no sale antedating t ] =
Pr
[ p
≥ p ( t ) no sale antedating t
]
=
∞
∫ p ( t ) f t
( x ) dx
≡
F t
( p )
The unconditional probability that the asset will remain unsold until the auction date t exploits the assumption of statistical independence of the elements of the stochastic process
{ } t
. It is the product of the series of complementary conditional probabilities antedating the auction at date t . That probability is calculated as:
Pr( no sale antedating t
)  t j


1

1

1 F
( j p
)

The conditional expected value of the sale price at auction date t , given that it was not sold before date t , is symbolized by the function CE
( t
)
. It is calculated as:
CE
( t p no sale antedating t
) 

 t
) x f
( t x
) dx (1)
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Rational Reserve Pricing in Sequential Auctions
The unconditional expected value of the price at auction date t is symbolized by E t
[ p
]
. It is the product of the conditional expectation and its conditioning joint probability:
E t

CE t
 p no sale antedating t

 Pr  no sale antedating t

It is calculated as:
E t






 t x
) f
( t x
) dx








 t j

1


1

1

F
( j p
)





(2)
The seller ’s time-invariant discount rate is symbolized by r . The present discounted value of the expected sale price over the horizon of the hypothetical future auction dates is the discounted sum of the mutually exclusive unconditional expected values at each hypothetical date 11 :
E
[ p
]  k

 
1

1
 r
  k
E k
[ p
] (3)
Equation (3) is the seller ’s objective function. Appendix I displays the derivation of the seller’s optimal reservation price at auction date t . It is: p
* ( t
)  k


 t

1

1
 r
 t
 k
CE
( p
) j k


 t
1

1

1

F j
( p
)

(4)
5 THE MEANING AND THE IMPLICATIONS OF THE ADJUSTMENTS TO
RESERVATION PRICES IN SEQUENTIAL AUCTIONS
The main analytical result of this paper is represented by equation (4).
That result can be stated as a proposition:
11 The definition of equation (3) as the seller’s objective function was suggested to me by the paper by Chow [7].
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Rational Reserve Pricing in Sequential Auctions
PROPOSITION:
If a risk-neutral seller can estimate the density functions governing the largest bids at each future auction date, his optimal reservation price at each sequential auction is equal to the discounted expected value of the asset if it is not sold on that date and the reservation prices are the optimal prices for all subsequent auctions .
The proposition exploits the fact that equation (4) is mathematically equivalent to equation (3), except that the summation in equation (3) runs from k

1 whereas the summation in equation (4) runs from k
 t

1 .
The proposition is an implication of the rational expectations hypothesis.
Generally speaking, in price theory rational expectations means that the current price of an asset has incorporated all the relevant information and that it is equal to the present discounted value.
12 In the context of the model in this paper, the relevant observable information at date t consists of the realizations of the random variables in the truncated series
 f i i

1
,
2
,
 t

. The seller uses that information to form expectations about the series
 f i i
 t

1
, t

2
,


The behavioral theory was expressed by Sargent [ 24, p. 23]:
“ Partly because it focuses on outcomes and does not pretend to have behavioral content, the hypothesis of rational expectations has proved to be a powerful tool for making precise statements about complicated dynamic economic systems.
”
Sargent’s observation is directly applicable to the main analytical result in this paper. The equations defining the seller’s series of reservation prices is an instance of an optimization problem embedded in a complicated dynamic system.
12 Gertchev [12, p. 318].
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Rational Reserve Pricing in Sequential Auctions
Moreover, the optimal reservation price at an arbitrary future auction date relies on adaptations based on the realizations of a stochastic process manifested as the bidding behavior antedating the auction date. The seller need not concern himself with estimation of the private valuations imputed to the population of buyers or with asymmetric information among the buyers.
The specific information required by the seller at an arbitrary date is knowledge of the probability density functions governing the offers for subsequent sequential auctions. The optimal reservation price at date t in equation (4) is expressed as an infinite sum. That property raises some obvious problematic issues respecting the information required by the seller as well as the mathematical issue of convergence.
The convergence of the sum in (4) is assured by any of three properties of the terms in the sum.
(a) The practical way of assuring convergence of an infinite sum is to truncate it to a reasonably approximate finite sum. The
“reasonableness” of the approximation means that truncated series does not result in a significant loss of information as to what the future may hold in terms of bids. Here is a practical example.
It may be that the seller is constrained by a finite horizon such that the asset must be sold by date t

T
. For example, that kind of constraint is exogenously imposed if the seller is acting in the capacity of Trustee in a bankruptcy liquidation proceeding. He may be required by a court order to sell an
12

Rational Reserve Pricing in Sequential Auctions asset on or before a fixed date T to satisfy the owner’s creditors 13 . In that case, the Trustee/seller can assure convergence of the sum by predetermining a value for his reservation price
( p T
) such that F
T
( p
) 
1 . Appendix II displays the derivation of an empirical cdf for a reservation price assuring a sale at date t

T .
(b) The discounting coefficient of each conditional expectation assures convergence of their infinite sum if the discounting factor is not offset by increasing conditional expectations. Thus, any (positive) discount rate will suffice to effect convergence of the sum if the density functions in the series
 f i i

1
,
2
,
 t

are stationary.
(c) The stochastic process
 f i i

1
,
2
,
 t

may be non-stationary. If the series of density functions shift downward over time to reflect a declining trend in the bids on the asset, equation (1) implies the series of inequalities

 CE t

CE t

1

CE t

2
 

. The inequalities in the series of conditional expectations imply convergence of their sum.
The property of a non-stationary series of density functions
  described above may be ascribed to the confluence of two effects; the effect of a series of non-disclosed reserve prices and the effect of sequential sales.
13
Chapter 7 of the Title 11 of the United States Code governs the process of liquidation under the bankruptcy law of the United States. When a troubled business is badly in debt and unable to service that debt or pay its creditors, it may file (or be forced by its creditors to file) for bankruptcy in a Federal court under Chapter 7 . A Chapter 7 filing means that the business ceases operations unless continued by the Chapter 7 Trustee. A Chapter 7 Trustee is appointed almost i mmediately, with broad powers to examine the business's financial affairs. The Trustee generally sells all the assets and distributes the proceeds to the creditors.
13

Rational Reserve Pricing in Sequential Auctions
A policy of disclosing the reservation price at the inception of each successive auction can be expected to affect the distribution of offers by eliminating potential bidders early on in the process. In that case, F
(
1 p
) 
1 and the series of density functions
 
collapses to one function at t

1 .
In contrast, an undisclosed reservation price is an effective shadow offer price throughout the process and each potential bidder recognizes that he might not get the asset even if he is the highest posted offering price. Hence, each potential buyer must form a prior estimate of the undisclosed reservation price at each sequential sale date. Katkar and Riley [19 ] found a significant negative effect of an undisclosed reserve price on the effective transaction price. Their finding suggests that an undisclosed reserve price is expected to cause the transaction price to decrease.
14 The logic of the auction’s architecture implies that if transaction prices tend to decrease in sequential auctions, the elements in the series

CE t
| t

1
,
2
,


are monotone decreasing.
There exists a growing body of empirical studies adducing evidence of what has been called the “ the declining price anomaly.
” That expression signifies an observed phenomenon in sequential sales of similar objects where prices decline as the sales progress.
15
14 Theoretical papers develop theories to explain why prices in sequential sales decline. See von der Fehr [26], Engelbrecht-Wiggans [10], Bernhardt and Scoones [5], Gale and Hausch [11],
Pezanis-Christou [22] and Ginsburgh [13]. A good synopsis of the theoretical research appears in
Ashenfelter and Graddy [3].
15 The declining price anomaly has been documented in sequential sales of livestock Buccola
[6]; condominia Ashenfelter and Genesove [2]; transponder leases Milgrom & Weber [21]; and
Chinese porcelain recovered from shipwrecks, Ginsburgh and van Ours [14]. See Ashenfelter and Graddy [3] for citations to other empirical studies of the declining price anomaly.
14

Rational Reserve Pricing in Sequential Auctions
Theory and evidence are consistent with a non-stationary tend in the elements of the stochastic process
 
such that they shift downward monotonically as t

1
,
2
,
 .
6 CONCLUDING REMARKS
This paper addressed two related questions:
(a) How does a risk-neutral seller adjust undisclosed reservation prices in sequential auctions of an asset?
(b) How do optimal adjustments of sequential reservation prices affect the transactions price?
The theory developed in the paper suggests answers to both questions. A rational risk-neutral seller will set a series of reservation prices at sequential auctions in such a way as to maximize the discounted value of the transaction price. In order to accomplish this, the seller estimates the probability density functions governing the bids at each future auction date.
The optimal series of reservation prices displays a downward trend, thereby explaining the socalled “declining price anomaly” observed in transaction prices in sequential auctions.
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Rational Reserve Pricing in Sequential Auctions
APPENDIX I
The maximization proceeds in the usual way, setting

E

[ t p
)
]

0
for all t

1
,
2
,

and solving the equations for the series of optimal reservation prices, symbolized by
 p
*
(
1
), p
*
(
2
),


.
The series of partial derivatives


E
[ p
] /  ( p t
) i

1
,
2
,


exploits the fact that the conditional expectations in the sum are mathematically independent.


E
[ t p
)
]
 k



1

1
 r
  k

E k
 (
( p t p
)
) for t

1
,
2
,
 [1]
The partial differentiation can be partitioned into three disjoint sets indexed on
 t
 k
, t
 k
, t
 k

.
(a) For all values of t
 k , the partial derivative is :


E k
( p
( t p
)
)


1
 r
  k
CE k
( p
) k j


 t
1 j

1

1

F
( j p
)







F t
(
( p t p
)
)


 where



F t
(
( p t p
)
)
 
 ( p
 t
)
( p
 t

) f t
( x
) dx
 f t
( p
)
(b) For the case where t
 k , the partial derivative is:
[2]


E
(
( t p t p
)
)
 
1
 r
  t

CE t
 ( t
 
) t j

1


1

1

F
( j p
)

[3]
16

Rational Reserve Pricing in Sequential Auctions where

CE t
 (
( t
) p
)





 ( p
 t
)
 t

) xf
( t
(c) For all values of t
 k ,


E
( p t p
)
) x
) dx





  ( p t
) f
( t p
)
0 [4]
Setting

E
 ( p
[ p
] t
)

0 for all t , the resulting equations are


E
[
( p p
] t
)
  
1
 r
  t t
) f t
( p
) t j

1


1

1

F
( j p
)

 k


 t

1

1
 r
  k
CE
( p
) f
( t p
) j k

1

 t

1

F
( j p
)


0
[5]
Rearranging the equations in [5] we have [6] below:

1
 r
  t ( p t
) f t
( p
) j t

1


1

1

F
( j p
)

 f t
( p
) k


 t

1

1
 r
  k
CE k
( p
) j k

1

 t

1

F
( j p
)

[6]
Factoring out the density functions f
( t p
)
, we can solve equation [6] for the optimal value of p
*
( t
) p
*
( t
) 

1
 r
 t k


 t

1

1
 r
  k
CE
( p
)







 k j

1

 t j

1


1 t

1


1

F
( j
F
( j p
) p
)










[7]
The ratio of the products in equation [7] factors to:
17

Rational Reserve Pricing in Sequential Auctions k j


 t
1 j t



1
1

1

F
( j

1

F
( j p
)
 p
)





 t j



1
1

1

F
( j p
) j t



1
1

1








 j k


1
 t

1

1


F
( j p
)

F
( j p
)





[8]
 j k

1

 t

1

1

F
( j p
)

Substituting equation [8] into equation [7] yields equation (4) in the text: p
*
( t
) 

1
 r
 t k


 t

1

1
 r
  k
CE k
( p
) j k

1

 t

1

1

F j
( p
)

[9]
18

Rational Reserve Pricing in Sequential Auctions
APPENDIX II
The derivation of the empirical conditional cdf at date t

T is based, in part, on the assumption that the seller knows (or can estimate) the density function for the offer prices at that date. Let the seller’s a priori probability density function governing p at date T be represented by g
T
( p
)
for p

0 .
The reservation price at date T is
( p T
)
. There is a constant c such that the conditional cdf assuring a sale at date T satisfies the equation: c


T
) g
T
( x
) dx

1 [1]
The cdf of g
T
( p
)
is G
T
( p
)
. The equation in [B.1] can be expressed as: c

G
T
(  ) 
G
T
(
T
) ) 

1 [2]
The property of the cdf G
T
assures G
T
(

)

1 . The conditioning constant is: c

1

G
T
1
 (
T
)  [3]
The time series of the highest prices observed by the seller at successive auctions antedating T is
 p
1
, p
2
,
 p
T

1

. Suppose the series
  converges in probability towards p max such that for any

0 , t lim
 
Pr
 p t
 p max



0
19

Rational Reserve Pricing in Sequential Auctions
The a posteriori density function at date t

T establishes an upper bound p max such that
Pr
 p
 p max no sale antedating T


0 .
The empirical conditional cdf at date t

T allows a determination of the reservation price at that auction that will maximize the probability of a sale at the maximum price is:
1

G
T
(
1
( p T
))
( p p max

T
) g
T
( x
) dx

G
T
(
1 p max

) 
G
T
(
G
T
(
( p T
( p
))
T
))
[4]
20

Rational Reserve Pricing in Sequential Auctions
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