We consider auction games where, prior to the auction, bidders spend resources to increase their valuations. The market game is solved by solving an equivalent auxiliary social choice problem. We show that standard auctions are fully efficient, whereas reserve price requirements entail a double inefficiency. Moreover, we explain how optimal auctions differ from the well-known static optimum, and sketch the impact of information spillovers.Auctions, procurement

Elmar G. Wolfstetter

Thomas D. Jeitschko

English

Research Papers in Economics

CESifo Working Paper Series
March 2000
CESifo
Poschingerstr. 5
81679 Munich
Germany
Phone: +49 (89) 9224-1410/1425
Fax: +49 (89) 9224-1409
http://www.CESifo.de
________________________
* Financial support was received by the Deutsche Forschungsgemeinschaft, SFB 373
(“Quantifikation und Simulation Ökonomischer Prozesse”), Humboldt-Universität zu Berlin.
AUCTIONS WHEN BIDDERS
PREPARE BY INVESTING IN IDEAS
Thomas D. Jeitschko
Elmar Wolfstetter*
Working Paper No. 258CESifo Working Paper No. 258
March 2000
AUCTIONS WHEN BIDDERS PREPARE BY
INVESTING IN IDEAS
Abstract
We consider auction games where, prior to the auction, bidders
spend resources to increase their valuations. The market game is
solved by solving an equivalent auxiliary social choice problem. We
show that standard auctions are fully efficient, whereas reserve price
requirements entail a double inefficiency. Moreover, we explain how
optimal auctions differ from the well-known static optimum, and
sketch the impact of information spillovers.
Keywords: Auctions, procurement
JEL Classification: D44
Thomas D. Jeitschko
Texas A&M University
Department of Economics
College Station
Texas 77843
USA
e-mail: thosd@econ.tamu.edu
Elmar Wolfstetter
Humboldt University
Department of Economics
Spandauer Str. 1
10178 Berlin
Germany
e-mail: wolf@wiwi.hu-berlin.de1 Introduction
In many auction settings bidders not only compete for the allocation of a
given good at given valuations, but they also spend substantial resources
to increase their valuations. For example:
  Contractors engage in research and development in order to lower
their cost or to be able to supply goods with more desirable quality
characteristics.
  Bidders for real estate property employ architects to create innova-
tive plans to remodel buildings in order to more accurately assess
their potential rental value.
  Bidders for spectrum licenses invest in transmission technology,
retail networks, and communication services in order to increase
the value of radio frequencies.
The present paper contributes to analyzing the interrelationship be-
tween auctions and investment in a simple private values framework,
assuming investments are made prior to bidding and are not observed
by rival bidders.
There is a small literature on investment and bidding, mainly in the
context of regulation (see the survey in Laﬀont and Tirole (1993), Ch. 7).
While most contributions focus on investments that take place after the
winner has been selected, Piccione and Tan (1996) consider investment
in R&D before procurement takes place, and independently model the
impact of investments on costs essentially in the same way as we do
here.
The plan of the paper is as follows. We state the model (Section 2),
state and solve an auxiliary social choice problem (Sections 3), and use
it to solve and the market game (Section 4). Then, we spell out how
optimal auctions deviate from the well–known static optimum (Section
5), sketch how results change if investment is subject to information
spillovers (Section 6), and close with a discussion.
2 The Model
There are n ≥ 2 risk neutral bidders who compete for an indivisible
good in an auction. Bidders’ have independent private valuations, V,
drawn from a ﬁxed support that is normalized to [0,1]. The auction-
eer’s own valuation is equal to zero. If the auctioneer requires a reserve
1price above his own valuation, bidders do not participate below a certain
critical valuation v0 ∈ [0,1] (see Riley and Samuelson (1981)).
The market game is structured as follows: The auctioneer commits
to an auction rule that awards the item to the highest bidder, subject to
some reserve price requirement. After observing these rules, each bid-
der chooses his investment expenditure a which shifts the probability
distribution of that bidder’s valuation, F(v,a), and makes high valua-
tions more likely. After investments are made, each bidder draws his
valuation from F, and then makes his bid. Bidders do not observe each
other’s investment expenditures. Finally, the auction awards the item to
the highest bidder subject to a reserve price requirement.
The investment technology has the following properties: 1) Invest-
ment implies a ﬁrst–order stochastic dominance shift of the own prob-
ability distribution:1 ∀v,∀a : Fa(v,a) < 0. 2) Investment is subject to
diminishing returns to scale, in the sense of the “convexity of the distri-
bution function condition:” ∀a : Faa(v,a) > 0.
3) The probability distribution of the highest order statistic from a
sample of n independently drawn valuations, V(n), F(n)(v,a1,...,a n),i s
strictly concave in investments. This assures that uniform investments
are desirable. 4) For all v: lima→∞ Fa(v,a,...,a)= 0. This assures that
it pays to engage in some investment. 5) The hazard rate f(v)/(1−F(v)
is monotone increasing (Myerson’s (1981) “regular case”).
3 Socially Optimal Investments
To prepare the solution of the market game we construct an equivalent
auxiliary social choice problem. The welfare optimum is obtained as a
special case.
Suppose a social planner chooses investments a := (a1,...,a n) to
maximize the “as–if social surplus”
S(v0,a):= EV(n)≥v0[V(n) | a] −

i
ai + v0F(n)(v0,a) (1)
Evidently, S is the social surplus from awarding the object to the bidder
with the highest valuation if the auctioneer’s own valuation is equal to
v0. However, the seller’s valuation is equal to zero. Therefore, welfare
maximization is obtained only for v0 = 0. We denote the as–if welfare
maximizer by a∗(v0) and the true welfare maximizer by a∗(0).
1A function with a subscript denotes a partial derivative with respect to the sub-
scripted variable.
2Proposition 1 (As–IfWelf are Maximum) The solution of the as–if wel-
fare maximization problem is symmetric, unique, and positive unless v0
is “large.” Moreover, a∗(v0) is strict monotone decreasing if a∗(v0)>0,
and a∗(0)>0.
Proof As one can easily verify, S(v0,a 1,...,a n) is strictly concave in
investment expenditures, by the strict convexity of F(n)(v,a1,...,a n).
Hence, if a solution exists, it must be unique. Suppose the solution were
asymmetric. Then, any reassignment of optimal investment expendi-
tures across bidders is also a solution, since all bidders have the same
investment technology. Therefore, if a solution exists, it is symmetric.
Hence, F(n)(v,a1,...,a n) = F(v,a)n, and with slight abuse of notation
S(v0,a)=
 1
v0
vdF(v,a)
n − na + v0F(v0,a)
n
=1 −
 1
v0
F(v,a)
ndv − na (2)
which gives rise to the Kuhn–Tucker conditions
Sa(v0,a
∗) ≤ 0 and a
∗Sa(v0,a
∗) = 0( 3)
Sa(v0,a
∗) = n

−
 1
v0
(F(v,a
∗))
n−1 Fa(v,a
∗)dv − 1

(4)
If no solution exists, S(v0,a) must be strictly monotone increasing
in a. However, as one can easily show, this contradicts assumption 4).
Therefore Sa(v0,a)is either positive for small a and negative for large
a or nonpositive for all a.
Using assumption 4) it follows that a∗(0)>0. As one increases v0,
the S∗
a function shifts down. Therefore, a∗ is a decreasing function of
v0.I f v0 is suﬃciently large, Sa(v0,a)becomes negative, which entails
the corner solution a∗(v0) = 0. 
We mention that it is not socially optimal to restrict the number of
participating investors/bidders. This follows immediately from the fact
that the solution is symmetric for all n.
4 Equilibrium Investments
We now turn to the market game where bidders independently choose
their investment expenditures, draw their private values, and make their
bids. Remarkably, we ﬁnd:
3Proposition 2 (Market Game) The solution of the market game, ¯ a(v0),
coincideswiththesolutionofthe“as–ifwelfaremaximum”, ¯ a(v0) = a∗(v0).
Moreover, the market game implements the eﬃcient allocation of the good
and of investments (full eﬃciency) iﬀ v0 = 0.
Proof Suppose every bidder assumes that all rival bidders have chosen
the same investment level ¯ a. Then, by a standard result from auction
theory, the equilibrium payoﬀ of a bidder with valuation v ≥ v0 from
participating in the auction is
u(v, ¯ a) =
 v
v0
F(y,¯ a)
n−1dy, (5)
and zero otherwise. Therefore bidders’ ex ante expected payoﬀ when
choosing the investment expenditure a can be written as follows (using
integration by parts)
U(a, ¯ a) :=EV≥v0[u(V, ¯ a)] − a
=
 1
v0
(1 − F(v,a))F(v,¯ a)
n−1dv − a (6)
A symmetric equilibrium exists if and only if ¯ a is a mutual best reply:
¯ a = arg maxa U(a, ¯ a). Since Fa < 0,F aa > 0 the function U(a, ¯ a) is
strictly concave, and therefore the symmetric equilibrium investment ex-
penditure ¯ a has to solve the conditions
Ua(¯ a, ¯ a) ≤ 0 and ¯ aUa(¯ a, ¯ a) = 0 (7)
Ua(a, ¯ a) =−
 1
v0
Fa(v,a)F(v, ¯ a)
n−1dv − 1. (8)
Evidently, these conditions are equivalent to the Kuhn–Tucker condi-
tions that characterize the “as–if welfare maximizer” a∗(v0) (see (3)–(4)).
Therefore ¯ a(v0) = a∗(v0). 
Corollary 1 (Double Inefﬁciency) Suppose the auctioneer requires a
reserve price above his own valuation, v0 > 0.Then, bidders underinvest:
v0 > 0 ⇒ ¯ a(v0)<a ∗(0), and the allocation of the good is ineﬃcient.
Proof It is well–known that v0 > 0 implies an ineﬃcient allocation of
the good. By Proposition 2 ¯ a(v0) = a∗(v0) and by Proposition 1 ¯ a(v0)<
a∗(0). 
45 Investment Incentives and Optimal Auctions
The static optimal auction involves a reserve price requirement v
s
0 that
solves the condition ∂π/∂v0 = 0 (Myerson (1981) and Riley and Samuel-
son (1981)). However, as one takes into account the eﬀect of the reserve
price on investment incentives, the optimal auction diﬀers.
Proposition 3 (Optimal Auction) Compared to the static optimum the
optimal auction involves a lower reserve price requirement.
Proof Let π(v0, ¯ a(v0)) denote the auctioneer’s equilibrium expected
revenue. For given investment expenditures the static optimal reserve
price, v
s
0, solves the condition ∂π/∂v0 = 0. However, accounting for the
eﬀect of v0 on investments, one obtains
v0 = v
s
0 ⇒
dπ
dv0
=
∂π
∂v0
+
∂π
∂¯ a
¯ a
  =
∂π
∂¯ a
¯ a
 . (9)
We now show that dπ/dv0 is negative at v0 = v
s
0.
By Proposition 1 ¯ a (v0)<0. Therefore, it only remains to be shown
that ∂π/∂¯ a is positive at (v
s
0, ¯ a(v
s
0). By the Revenue Equivalence Theo-
rem the auctioneer’s expected revenue is the same as in a second–price
auction, which is equal to (where all probabilities and expected values
are conditional on a = ¯ a(v0))
π(v0, ¯ a(v0)) =v0

1 − Pr{V(n−1) >v 0}−Pr{V(n) <v 0}

+ E[V(n−1) | V(n−1) >v 0]Pr{V(n−1) >v 0}
=v0

1 − Pr{V(n) <v 0}

(10)
+

E[V(n−1) | V(n−1) >v 0] − v0

Pr{V(n−1) >v 0}.
Evidently, both terms on the RHS of this equation are strictly increasing
in ¯ a. Hence, dπ/dv0 < 0, and the revenue maximizing auctioneer should
set v0 below v
s
0. 
6Information Spillovers
Now suppose investment is subject to information spillovers. In particu-
lar, we stipulate that spillovers put this investor in the same position as
if he had invested the amount a+λ˜ a, where ˜ a is the amount invested by
his rivals. The spillover parameter is restricted to λ ∈ [0,n− 1], where
λ = n − 1 represents “complete spillovers” and λ = 0 “no spillovers”.
Spillovers reduce investment incentives. Not surprisingly, this im-
plies:
5Proposition 4 In the presence of information spillovers bidders invest
less.Underinvestment occurs even if the auctioneer does not use a reserve
price requirement, and the optimal auction involves an even lower reserve
price requirement than in the absence of spillovers.
The proof of these results is a straightforward extensions of the above
analysis.
7 Discussion
A critical ingredient of the present analysis is the concavity of the in-
vestment technology. This assumption makes it desirable that all bid-
ders engage in the same level of investment expenditures. If, instead,
investment is subject to increasing returns to scale, it becomes desirable
to concentrate investments, and the market game has asymmetric solu-
tions, where bidders draw their valuations from diﬀerent distributions.
Asymmetric auctions are generally not tractable with analytic meth-
ods (see Maskin and Riley (1996)). However, the optimal asymmetric auc-
tion problem has been solved for given distributions by Myerson (1981).
In the assumed “regular” case, when hazard rates are monotone increas-
ing, the static optimal auction involves handicapping the bidder who
draw his valuation from the more favorable distribution. This handi-
capping has an adverse eﬀect on investment since it eﬀectively penalizes
the bidder who bears the main burden of investment. This suggests that,
taking investment incentives into account, the optimal auction involves
less handicapping, possibly to the extent of a preference for the bidder
with the more favorable distribution.
6References
Laﬀont, J. J., and Tirole, J. (1993). A Theory of Incentives in Procurement
and Regulation. MIT Press.
Maskin, E., and Riley, J. G. (1996). Asymmetric auctions (Working Paper).
Harvard University.
Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations
Research, 6, 58-73.
Piccione, M., and Tan, G. (1996). Cost–reducing investment, optimal pro-
curement and implementation by auctions. International Economic
Review, 37, 662–685.
Riley, J. G., and Samuelson, W. F. (1981). Optimal auctions. American
Economic Review, 71, 381-392.
7

Auctions when Bidders Prepare by Investing in Ideas

http://www.cesifo-group.de/portal/page/portal/DocBase_Content/WP/WP-CESifo_Working_Papers/wp-cesifo-2000/wp-cesifo-2000-03/cesifo_wp258.pdf

Auctions when Bidders Prepare by Investing in Ideas

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics