Under a specific informational framework, we compare the seller's expected revenue from a first-price auction and a second-price auction when bidders are risk averse and have private affiliated values.auctions, risk aversion, private affiliated values

François Maréchal

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 2 
  
First-price vs second-price auctions under risk aversion and private affiliated 
values 
  
 
 
François Maréchal  
CRESE, University of Franche-Comté (France) 
Abstract 
Under a specific informational framework, we compare the seller's expected revenue from a first-price auction and a 
second-price auction when bidders are risk averse and have private affiliated values.
 
Citation: François Maréchal, (2011) ''First-price vs second-price auctions under risk aversion and private affiliated values'', Economics 
Bulletin, Vol. 31 no.2 pp. 1765-1771. 
Submitted: May 18 2011.   Published: June 16, 2011. 
 
     Economics Bulletin, 2011, Vol. 31 no.2 pp. 1765-1771
1 Introduction
The pioneer article of Vickrey (1961) has established that the four main auction mech-
anisms1 yield the same expected revenue for the seller. However, the Vickrey theorem
only holds under the specic assumptions of the independent private value (hereafter
IPV) paradigm. Namely, bidders are assumed to adopt symmetric strategies, are risk
neutral and the value of the auctioned object for each bidder is known to himself but not
to other bidders whose valuations are independent. The IPV model can be restrictive in
practice as bidders may have private values related to each other. Besides, when bidders
are risk averse e.g., the Vickrey theorem no longer holds since the rst-price sealed-bid
auction (hereafter FPA) or the dutch auction yields a higher expected revenue for the
seller than the second-price sealed-bid auction (hereafter SPA) or the english auction.2
Another polar case mainly analyzed in the literature is the common value (hereafter
CV) model. This model assumes that the value of the auctioned object is the same
but unknown to all bidders who have dierent (private) estimates of this common value.
Although the IPV model recognizes dierences in individuals' preferences, the CV model
does not. Thus, as noticed by Campo et al. (2003) and Li et al. (2002) many auction
situations may not correspond to either one of these two polar cases. Your value for a
painting e.g. may depend on both a private information (how much you like it) and on
others private valuations because this aects the resale value.3
Milgrom and Weber (1982) provide a more realistic and general model of auctions that
allows for aliated values. Positive aliation e.g. means that higher values of the item
for one bidder make higher values for other bidders more likely. Namely, they show that
when bidders are risk neutral and have aliated values, the SPA (or the english auction)
yields a higher expected revenue than the FPA (or the dutch auction). However, in their
general model, no clear qualitative comparison can be made between the FPA and the
SPA in the presence of risk aversion and aliated values. The comparison between the
FPA and the SPA in terms of expected revenue is the main goal of this paper.
We analyze this comparison in a setting where bidders have aliated valuations.
However, we consider that these valuations are private. This means that each bidder has
perfect information regarding his own value for the object at auction, but higher values
of the item for one bidder make higher values for other bidders more likely. As noted by
Kagel et al. (1987) \a simple example of an auction with aliated private values would be
a charity fundraiser of consumer perishables, where an item unusually appealing to you
is tipically more appealing to other bidders as well". The aliated private values model4
seems to be quite realistic since it allows for dependence among bidders private values
while retaining a bidders private valuation for the auctioned object. We also consider that
bidders are risk averse. This assumption seems to be reasonable since bidders face many
uncertainties. It has been tested extensively on experimental data. Bajari and Hortacsu
(2005) e.g. show that the risk aversion model provides the best t for experimental data.
Perrigne and Vuong (2007) provide an estimation of bidders' risk aversion in rst-price
auctions. Namely, they nd signicant risk aversion in the case of timber auction data.
Following Von Ungern-Sternberg (1991) (hereafter VUS), we adopt a simple infor-
1The rst-price sealed-bid auction, the dutch auction, the second-price sealed-bid auction and the
english auction.
2see e.g. Milgrom and Weber (1982), Maskin and Riley (1984) or Matthews (1983).
3See e.g. Klemperer (1999) for a survey on auction theory.
4See. e.g. Li et al. (2002) for a structural estimation of the symmetric aliated private value model.
1
1766Economics Bulletin, 2011, Vol. 31 no.2 pp. 1765-1771
mational paradigm. VUS's model actually combines the simplifying properties of the
standard independent private values model and the common value model. Indeed, each
bidder learns his own valuation with certainty and, as in the simplest version of the
common value model, each bidder has no grounds for believing his own valuation esti-
mate to be higher or lower on average than his competitor's valuations. Formally, VUS
models this by assuming that the dierent bidders' valuations are independent drawings
from a known distribution with an unknown mean.5 Then, since the bidders know the
functional form of the distribution but are uncertain about the mean, they will infer this
mean from their own private valuation.
VUS's model considers symmetric risk neutral bidders. In this paper, we extend the
analysis of VUS to the case of risk averse buyers. Then we compare the seller's expected
revenue from an FPA and an SPA. The paper is organized as follows. In the next section,
we detail the model assumptions. In section 3, we analyze the FPA. In section 4, we
compute the expected revenue from an SPA. Section 5 yields a comparison between both
auction mechanisms. Section 6 oers concluding remarks.
2 The assumptions of the model
We consider a model with n  2 bidders. Each bidder has a private valuation v. All
bidders are ex ante symmetric relative to the informational knowledge and believe that
their private valuations are drawn from a distribution F over [   a; + a]; where a is
common knowledge whereas the mean  is unknown. Thus, when a bidder, say i, learns
his own valuation vi; he can infer that  2 [vi   a;vi + a] according to the cumulative
F with corresponding density f: Since all the bidders are ex ante symmetric relative to
the informational knowledge, bidder i then infers that the evaluation of an opponent, say
j, vj 2 [vi   2a;vi + 2a] according to the cumulative distribution Fj with corresponding
density fj: In order to highlight the impact of risk aversion on bidding strategies, let us
assume that each bidder i is characterized by a constant relative risk aversion (CRRA)
utility function U(xi) = x

i (with 0 <   1), where 1  is the CRRA parameter of each
bidder and xi is bidder i's income.
3 The rst-price sealed-bid auction
We rst consider an FPA. As noted by VUS, the assumption that a bidder's valuation is
drawn from a known distribution of unknown mean introduces a substantial simplication
relative to the standard IPV model (where the mean of the distribution is assumed to
be known). In the IPV model each bidder's own valuation tells him something about his
position relative to his competitors. As a result, the bidder chooses a mark-up (or prot)
which depends on his own valuation. In VUS's model, where the mean of the distribution
is not known, the fact that a bidder privately knows his own valuation does not reveal
him anything about his relative position and so does not aect his winning probability.
Therefore, there is no reason why he should let his strategic mark-up depend on his own
valuation. His optimal bidding strategy is therefore very simple: bid his own valuation
minus a constant mark-up (which is independent of bidders' valuations). Hence, if we
consider the case of bidder i, we can assume that his equilibrium bid has the following
form
5This prior belief is described by Biais and Bossaerts (1998) as the \average opinion rule."
2
1767Economics Bulletin, 2011, Vol. 31 no.2 pp. 1765-1771
Bi (vi) = vi   bi;
where bi represents his markup (or prot). Then, under an FPA, bidder i's expected
utility is given by
EUi = (bi)
P(bi;bj); (1)
where P(bi;bj) reects his probability of winning when he chooses a strategic mark-up bi;
while the other bidders choose bj: Dierentiating (1) with respect to bi yields the optimal
markup for bidder i
@EUi
@bi
= (bi)
P
0(bi;bj) + bi
( 1)P(bi;bj) = 0
, biP
0(bi;bj) + P(bi;bj) = 0
, bi =  
P(bi;bj)
P 0(bi;bj)
: (2)
Let us now derive the probability of winning. Bidder i wins against bidder j if vi   bi >
vj bj, i.e. if vj < vi bi+bj, which occurs with probability Fj(vi bi+bj): Then, ex ante,
bidder i wins against (n   1) other bidders with probability P (dened in expectation
over the unknown mean ) such that
P =
Z

(Fj(vi   bi + bj))
n 1f()d:
In order to provide an explicit form of the probability of winning, we now consider
the special case of a uniform distribution for F (and therefore for Fj andF). Then, we
have
P =
Z


a   bi + bj + vi   )
2a
n 1 1
2a
d:
To compute this probability, assume6 e.g. that bi  bj. Then P  0 if   vi+a bi+bj
and P  1 if   vi   a   bi + bj. Since bi  bj and given that  lies over the interval
[vi   a;vi + a], one thus has to integrate only over the interval [vi   a;vi + a   bi + bj].
Then P becomes
P =
Z vi+a bi+bj
vi a
1
2a

a   bi + bj + vi   )
2a
n 1
d
=
1
n

2a   (bi   bj)
2a
n
: (3)
Given (3) and the rst order condition (2), the unique symmetric equilibrium for the
strategic markups satises7
bi = bj =
2a
n
:
Note that the optimal strategic markup of a bidder is increasing with respect to the
uncertainty parameter a, decreasing with his risk aversion level and decreasing with the
6The same reasoning can be applied with bi < bj.
7We check that the second order conditions are satised.
3
1768Economics Bulletin, 2011, Vol. 31 no.2 pp. 1765-1771
number of bidders. These results are consistent with intuition and conventional results
in auction theory.8
Given the bidders' optimal strategies, we can now compute the seller's expected rev-
enue from an FPA
ERFPA =
Z +a
 a
vnf (v)(F (v))
n 1 dv  
2a
n
=  + a  
2a
n + 1
 
2a
n
; (4)
where nf (v)(F (v))
n 1 is the frequency distribution of the highest valuation among n
bidders.
4 The second-price sealed-bid auction
Under an SPA, the winner pays a price equal to the second highest bid. Given this
payment rule, it is well known that it is a dominant strategy for a bidder to place a
bid equal to his true valuation. Therefore, the seller gets a price equal to the expected
second-highest valuation. The frequency distribution of the second-highest valuation is
n(n   1)f (v)(1   F (v))(F (v))
n 2 :
Then, the expected revenue from an SPA can be written as
ERSPA =
Z +a
 a
vn(n   1)f (v)(1   F (v))(F (v))
n 2 dv
=  + a  
4a
n + 1
: (5)
5 The revenue comparison
We now compare the seller's expected revenue from both auctions. We rst consider the
case of risk neutral bidders (i.e.  = 1). We can compute
ERFPA   ERSPA =
 2a
n(n + 1)
< 0: (6)
Notice that even if the seller's expected revenue from an FPA or an SPA depends
on the true value of the mean, the dierence ERFPA   ERSPA does not depend on this
parameter. So, when the bidders observe the procedure chosen by the seller, they cannot
infer anything concerning : Therefore, the strategies depicted in the previous sections
remain the optimal ones. From (6), we have the following lemma:
Lemma 1 In a setting with risk neutral bidders whose valuations are drawn from a uni-
form distribution, the SPA yields a higher expected revenue than the FPA.
This lemma can be easily interpreted. In an FPA, when  = 1, each bidder chooses
a markup equal to 2a=n. Then, from (4), the expected revenue from an FPA is equal
to the expected highest valuation minus 2a=n. In an SPA, the seller's expected revenue
8See among others Klemperer (2001) or Krishna (2002).
4
1769Economics Bulletin, 2011, Vol. 31 no.2 pp. 1765-1771
is equal the expected second highest valuation among n bidders whose valuations are
uniformly distributed in the interval [   a; + a] of length 2a. This expected second
highest valuation is equal to the expected highest valuation minus 2a=(n + 1). Since
2a=n > 2a=(n + 1), the seller's expected revenue is higher in an SPA than in an FPA.
The result of Lemma 1 is consistent with the more general result of Milgrom and
Weber (1982) which shows that when bidders are risk neutral and have aliated values,
the SPA yields a higher expected revenue than the FPA. However, as noted by Milgrom
and Weber (1982), there is no clear qualitative comparison between FPA and SPA in the
presence of risk aversion. In order to compare these two auction mechanisms in the case
of risk averse bidders (i.e. 0 <  < 1) having private aliated values, we can compute
ERFPA   ERSPA = 2a

1
n + 1
 

n

:
Then, we have the following proposition:
Proposition 1 Suppose that bidders have a CRRA utility function and that their valu-
ations are drawn from a uniform distribution. When  < (resp: >) n
n+1, the FPA yields a
higher (resp. lower) expected revenue than the SPA.
This result is consistent with the experiments of Kagel et al. (1987) which show
that the revenue-raising possibilities inherent in second price auction, relative to a rst
price auction, with positively aliated values, are severely compromised by the potential
for risk averse bidding. Besides, we can give an interpretation of proposition 1. In an
SPA, the seller's expected revenue does not depend on the bidders' level of risk aversion.
However, in an FPA, when bidders are risk averse, the markup is now equal to 2a=n;
raising one's bid slightly in an FPA is analogous to buying partial insurance. Following
the same interpretation as for lemma 1, the FPA yields a lower expected revenue for the
seller if
2a
n < 2a
n+1. Notice that the cut-o value of  (i.e. n
n+1) is increasing with n.
Actually, when n is increasing, the bidders' markup are decreasing. This is the result of
a competition eect. So, the dierence between the seller's expected revenues from both
auctions is also decreasing. Then, with a large number of bidders, the seller is better o
with an FPA even if the bidders exhibit a low level of risk aversion.
6 Conclusion
This paper has considered a revenue comparison between an FPA and an SPA when
bidders are risk averse and have private aliated values. Under a specic informational
framework which combines the independent private value model and the common value
model, we have provided conditions under which one auction mechanism out-performs
the other.
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6
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First-price vs second-price auctions under risk aversion and private affiliated values

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