Fixed-Prize Tournaments versus First-Price Auctions in
Innovation Contests∗
Anja Schöttner†
School of Business and Economics
Humboldt-University at Berlin
December 14, 2006
Abstract
This paper analyzes a procurement setting with identical firms and stochastic
innovations. In contrast to the previous literature, I show that a procurer who
cannot charge entry fees may prefer a fixed-prize tournament to a first-price
auction. The reason is that holding an auction may leave higher rents to firms
when the innovation technology is subject to large random factors.
Keywords: innovation contest, auction, tournament, quality
JEL classification: D44, H57, L15
∗
I would like to thank Dominique Demougin and Carsten Helm for helpful comments and
discussions. Financial support by the Deutsche Forschungsgemeinschaft through the SFB 649
“Economic Risk” is gratefully acknowledged.
†
Address: Ziegelstr. 13a, D-10099 Berlin, Germany, phone: +49 30 2093-1345, fax: +49 30
2093-1343, e-mail: [email protected]
1
Introduction
A buyer who wishes to procure an innovative good or service usually cares about the
quality of the innovation, which is affected by suppliers’ investments in R&D. Since
investments and quality are often non-contractible, procurers frequently hold contests among potential suppliers to induce investments in R&D. Two popular contest
mechanisms are fixed-prize tournaments and first-price auctions. In the tournament,
the best innovator receives an ex-ante fixed prize. In the auction, the buyer procures
the innovation from the firm that offers the most favorable combination of quality
and price. Both mechanisms prevent ex-post opportunistic behavior: The procurer
cannot lower payments to firms by understating quality, and firms do not benefit
from overstating their costs.
Two prominent papers on innovation contests, Fullerton et al. (2002) and Che
and Gale (2003), suggest that procurers prefer first-price auctions to fixed-prize
tournaments. Intuitively, under an auction mechanism, a firm with a low-quality
innovation can sustain the opportunity to win the contest by asking for a low price.
Thus, compared to a fixed-prize scheme, an auction creates a more competitive
situation between firms, which is beneficial from the procurer’s point of view.
Nevertheless, we do not only observe auctions but also fixed-prize tournaments in
R&D settings. A historic example is the 1829 contest where Liverpool and Manchester Railway announced a prize of £500 for the best performing engine for the
first passenger line between two British cities (Fullerton and McAfee 1999). More
recently, the Defense Advanced Research Projects Agency (DARPA) of the U.S.
Department of Defense sponsored the “Grand Challenge 2005” to promote R&D in
autonomous ground vehicle technology. A prize of $2 million was awarded to the
team whose vehicle completed a certain route the fastest (DARPA 2005). In the
1
private sector, the InnoCentive company provides an online forum where “seeker
companies” post R&D challenges in chemistry or biology for which scientists then
submit solutions. The best solution is rewarded by a prespecified prize (InnoCentive
2005).1
The purpose of this paper is to provide a possible answer to the question of when
a buyer may prefer a fixed-prize tournament to a first-price auction. To do so, I
compare both mechanisms in a situation in which two homogeneous firms are able
to supply the innovation that the buyer wishes to procure. The quality of a firm’s
innovation stochastically depends on the firm’s investment in R&D. Specifically,
quality is additively separable in investments and a random variable. Furthermore,
firms are liquidity-constrained,2 so that the buyer cannot charge entry fees and thus
firms may earn rents.
It turns out that, in the auction, if the qualities of firms’ innovations differ
significantly, the high-quality firm can demand a much higher price than the lowquality firm. Consequently, firms may earn higher rents under an auction than under
a prespecified fixed prize. As a result, the buyer prefers a fixed-prize tournament
to an auction when the innovation technology is subject to large random factors, so
that firms are likely to realize quite different innovations.
Formally, I measure “randomness” in terms of conditional stochastic dominance
(CSD).3 In my model, the quality difference between firms’ innovations determines
the price in the auction, and, therefore, the cumulative distribution function (cdf) of
this quality difference is of particular interest. If such a cdf F dominates another cdf
1
For further examples of fixed-prize tournaments, see, e.g., Windham (1999), Che and Gale
(2003), or Maurer and Scotchmer (2004).
2
Otherwise, the buyer would always implement efficient investments and extract rents ex ante
by charging an entry fee. In Fullerton et al. (2002) and Che and Gale (2003), entry fees are also
infeasible.
3
This concept is commonly used in the literature. See, e.g., Lebrun (1998), Maskin and Riley
(2000), or Arozamena and Cantillon (2004).
2
G in terms of CSD, then the expected price in the auction is, ceteris paribus, higher
under F than under G. I show that the buyer prefers to preannounce a fixed prize
if the cdf of the quality difference dominates an exponential distribution. Another,
stronger, sufficient condition for the optimality of a fixed prize is the log-convexity
of the cdf. On the other hand, if the cdf is log-concave, I find that an auction is
optimal.
My result that there are circumstances under which a tournament outperforms
a first-price auction differs from the findings in Fullerton et al. (2002) and Che and
Gale (2003). This is due to distinct assumptions on the firms’ innovation technology.
As in the present paper, Fullerton et al. compare a first-price auction and a fixed
prize in a stochastic environment with identical firms. Adopting the design by Taylor
(1995) to represent the innovation process, they find that the auction generally leads
to lower costs for the buyer. In Taylor’s model, firms decide in each of an ex-ante
specified number of periods whether to pay a fixed cost to obtain a draw from a
distribution of innovations. At the end of the last period, the best draw is submitted
to the buyer. By contrast, I investigate a one-period innovation process, where
expected quality increases in the amount of firms’ investments. Furthermore, while
Fullerton et al. consider a special class of log-concave cdfs for quality, I allow for
more general distributions. In particular, in my framework, for random influences
to be sufficiently significant so that the buyer’s expected costs are lower under a
fixed prize, it is necessary that the cdf of the quality difference is not log-concave.
In contrast to my setup, Che and Gale (2003) consider a deterministic innovation technology. Within a very general framework with an endogenously determined
number of possibly heterogeneous firms, they show that a first-price auction is optimal within the broad range of contest mechanisms in which only the winner receives
a prize. Since they assume a deterministic relationship between investments and
3
quality, the equilibrium in the investment stage is one in mixed strategies, implying
complete dissipation of expected rents when firms are identical. In my model, it is
crucial that, due to the stochastic environment leading to a pure investment strategy,
firms may earn rents, which differ under the two mechanisms under consideration.
Another of my assumptions departs from both Fullerton et al. (2002) and Che and
Gale (2003). I assume that, before a firm bids a price, the quality of its innovation
is not only observed by the buyer but also by the other firm.4 In the two other
papers, the quality that a firm can offer is observed only by the buyer. Imposing
a more general observability of quality is not essential for the results, but greatly
simplifies the analysis. In section 5, I briefly discuss the case in which the quality of
a firm’s innovation is known only to this firm and the buyer. In addition, I consider
the extension of the analysis to more than two and asymmetric firms.
The fixed-prize tournament model with an additive noise term that I use was
pioneered by Lazear and Rosen (1981) to analyze rank-order payment schemes as
an incentive device in labor contracts. Since then, it has been widely applied in
labor economics. For example, Green and Stokey (1983) analyze under which conditions a principal who employs risk-averse agents prefers a rank-order tournament
to independent contracts.5 Hvide (2002) uses the same approach to analyze risktaking, and Grund and Sliwka (2005) adopt it to investigate inequity aversion in
tournaments. I contribute to this strand of tournament literature by applying the
Lazear-Rosen framework to innovation contests and comparing it with a first-price
auction.
Among the first contributions to the analysis of innovative activity under uncer4
For example, this is the case if submitting innovations involves testing prototypes (e.g., of a
military plane), and employees of both firms are present when prototypes are tested.
5
Nalebuff and Stiglitz (1983) also compare relative and absolute payment schemes, but consider
an output function with a multiplicative common shock. Other prominent papers on tournaments
with alternative output technologies are Malcomson (1984), O’Keefe, Viscusi, and Zeckhauser
(1984), or Rosen (1986).
4
tainty are Loury (1979), Lee and Wilde (1980), and Dasgupta and Stiglitz (1980).
These authors analyze innovation races, i.e., contests in which the winner is the
party that first discovers an innovation of an ex-ante specified quality. By contrast, an innovation tournament, as considered in, e.g., Taylor (1995), Fullerton and
McAfee (1999), Fullerton et al. (1999), and the present paper, rewards the party
with the best innovation on a prespecified date. Innovation tournaments with a
deterministic invention process resemble first-price all-pay auctions with complete
information. Dasgupta (1986) was the first to analyze this class of games in the
context of technological competition. Baye, Kovenock, and de Vries (1996) give a
full characterization of equilibrium behavior for the all-pay auction with complete
information.
The remainder of the paper is organized as follows. In section 2, the model is
introduced. Section 3 analyzes the bidding and investment stage. In section 4, the
buyer’s optimization problem is investigated. Afterwards, I discuss some extensions
of the model. The last section concludes.
2
The model
A buyer holds a contest to procure an innovation from one of two ex-ante identical
firms, which are indexed by i, j ∈ {1, 2}, i 6= j. All parties are risk-neutral. The
buyer cannot charge entry fees. Her valuation of an innovation is equivalent to the
innovation’s quality. Firm i receives an innovation of quality
q i = x i + µi
5
(1)
if it invests c(xi ) + c̄, where xi , c̄ ≥ 0, and µi is the realization of a random variable.
I call xi firm i’s investment strategy. I assume that c(xi ) is strictly increasing,
strictly convex, and twice differentiable for all xi > 0. Furthermore, c(0) = 0,
limxi →+0 c0 (xi ) = 0, inf xi >0 c00 (xi ) > 0.6 If a firm decides to participate in the contest,
it must at least invest c̄ to be able to submit an innovation. Thus, c̄ captures all
fixed and opportunity costs from contest participation.
The random variables µ1 and µ2 are identically and independently distributed.
Additionally,
E[max{µ1 , µ2 }] − 2c̄ > ū,
(2)
where ū ∈ R+ denotes the buyer’s utility if she does not procure the innovation. This
assumption guarantees that the buyer ex ante benefits from holding the contest.7
Since the difference between the qualities of firms’ innovations will be crucial, I
define the random variable η := µ2 − µ1 with cdf G(η) and density g(η). Because
µ1 and µ2 are identically distributed, g(η) is symmetric around zero. Let S denote
the support of g. I assume that G(η) is continuous on R and both G(η) and g(η)
are differentiable on the interior of S.8
A firm’s investment strategy, investment costs, and realization of the random
variable are non-observable. Both qualities q1 , q2 are observed by the buyer and each
firm. However, qualities are non-verifiable. Third parties can only verify whether a
firm submitted an innovation, firms’ bids, from which firm the buyer procured the
innovation, and payments to firms.
Timing is as follows. In the first stage, the buyer specifies a fixed payment f ≥ 0
6
As will become clear in section 3, the last assumption guarantees the existence of a pure
strategy equilibrium in the investment stage.
7
The buyer can always implement xi = 0 if she commits to paying c̄ to each firm that submits
an innovation (assuming that the submission of an innovation is verifiable).
8
For many distributions of µi (e.g., the uniform distribution), g is not differentiable at zero,
which does, however, not affect the results.
6
to each firm submitting an innovation. Additionally, she commits to procuring one
of the submitted innovations, where the winning firm receives at least a minimum
price of p ≥ 0 and at most a maximum price of p ≥ p. In the second stage, firms
choose their investment strategies x1 , x2 . Afterwards, random variables µ1 , µ2 are
realized, firms submit their innovations, and qualities q1 , q2 are observed.9
In the last stage, the bidding takes place. Firms announce prices p1 and p2 ,
respectively, such that p ≤ pi ≤ p, i = 1, 2. Firm i wins if it offers the higher
surplus to the buyer, i.e., if qi − pi > qj − pj . If surpluses are identical, the firm with
the higher quality wins. If both surpluses and qualities are identical, the winner is
chosen by flipping a fair coin. If firm i wins the bidding, it receives pi + f , and firm
j receives f .
Observe that the mechanism nests both fixed-prize tournaments and first-price
auctions. Specifically, if the buyer chooses p = p, the mechanism amounts to a fixedprize tournament where the prize is awarded to the firm with the higher quality. In
contrast, with p = 0 and p = ∞, the mechanism is a first-price auction without a
minimum or maximum allowable price.
3
Firms’ decisions
The game is solved by backwards induction. In the last stage, when bidding occurs,
all parties involved know q1 and q2 . Suppose that qi ≥ qj . Then, in equilibrium,
firm j bids pj = p, and firm i bids
pi =


 p + (qi − qj )
if qi − qj ≤ ∆

 p+∆
if qi − qj > ∆
9
,
(3)
Alternatively, all parties involved could observe non-verifiable quality signals si = qi +²i , where
²i is some noise occurring when quality is measured.
7
where ∆ := p − p. Consequently, if qi − qj ≤ ∆, both firms offer the same surplus.
In this case, if qi > qj , firm i wins the bidding. If qi = qj , the winner is chosen
randomly. If qi − qj > ∆, firm i offers the higher surplus and therefore wins the
bidding.
Thus, the winning firm receives the minimum price p plus a quality premium
which is bounded from above by ∆ and equals |q1 − q2 | if |q1 − q2 | ≤ ∆. Given
investment strategies x1 and x2 , we have that q1 − q2 = x1 − x2 − η. Consequently,
the payment that firm 1 receives in the last stage of the game in addition to f is:



0





 1p
2


p + x1 − x2 − η





 p+∆
if
x1 − x2 < η
if
x1 − x2 = η
if
(4)
x1 − x2 − ∆ ≤ η < x 1 − x2
if
η < x 1 − x2 − ∆
Analogously, firm 2 obtains:



0





 1
2
if
p


p + x2 − x1 + η





 p+∆
η < x 1 − x2
if
x1 − x2 = η
if
x1 − x2 < η ≤ x1 − x2 + ∆
if
(5)
x1 − x2 + ∆ < η
In the investment stage, each firm chooses its investment strategy to maximize
its expected payment net of investment costs. Given that firm 2 adopts investment
strategy x2 , firm 1 chooses x1 to maximize
Z
Z
x1 −x2
x1 −x2 −∆
(p + x1 − x2 − η)g(η)dη +
x1 −x2 −∆
(p + ∆)g(η)dη − c(x1 ).
−∞
8
(6)
Given x1 , firm 2 maximizes
Z
Z
x1 −x2 +∆
∞
(p + x2 − x1 + η)g(η)dη +
x1 −x2
(p + ∆)g(η)dη − c(x2 ).
(7)
x1 −x2 +∆
N
The first-order conditions for a pure-strategy Nash-equilibrium (xN
1 , x2 ) are
0 N
N
N
N
N
N
pg(xN
1 − x2 ) + [G(x1 − x2 ) − G(x1 − x2 − ∆)] = c (x1 ),
(8)
0 N
N
N
N
N
N
pg(xN
1 − x2 ) − [G(x1 − x2 ) − G(x1 − x2 + ∆)] = c (x2 ).
(9)
For the purpose of this paper (i.e., to provide a possible intuition for when a
tournament may outperform an auction), and given that firms are identical, I focus
on the analytically more tractable case of symmetric pure-strategy Nash-equilibria
N
10
xN
Symmetry of g(η) implies that G(−∆) = 1 − G(∆) and G(0) = 21 ,
1 = x2 =: x.
so that the first-order conditions simplify to
¸
1
pg(0) + G(∆) −
= c0 (x).
2
·
(10)
Starting from identical investment strategies, the left-hand side of (10) gives a
firm’s marginal benefit from increasing investments. The term pg(0) indicates the
higher probability of winning the bidding and obtaining at least p. The term in
square brackets represents the increase in the expected quality premium that the
winning firm receives in addition to p.
Naturally, high values of p and ∆ provide strong investment incentives. Furthermore, investments increase in g(0) and, given ∆, in G(∆). A large g(0) indicates
that the probability of winning responds strongly to changes in investments. A large
G(∆) reflects that the quality premium is likely to equal the difference between qual10
Depending on G(.), c(.), p, and p, asymmetric equilibria may exist.
9
ities (instead of ∆). Both properties imply that the outcome of the mechanism is
relatively sensitive to changes in investments.
Firms’ objective functions (6) and (7) are not necessarily concave, so that we
need further assumptions to ensure that x as given by (10) is indeed an equilibrium.
A sufficient condition for the strict concavity of (6) in x1 (given an arbitrary x2 ) is
that11
sup {pg 0 (η) + g(η)} < inf c00 (x),
x>0
η∈intS
(11)
i.e., random influences are sufficiently significant (g is sufficiently “flat”).
Because g(η) is symmetric around zero, this condition also guarantees strict
concavity of (7) in x2 (given an arbitrary x1 ). Since condition (11) should be satisfied
for all p that the buyer might choose, we need to specify an upper bound on p. Such
an upper bound can be determined by deriving an upper bound on the investment
strategies that the buyer may wish to implement, denoted x̄. Since the buyer will
never induce investments for which the resulting expected quality is smaller than her
expected payments to firms, which must be at least as high as firms’ expenditures,
x̄ can be implicitly defined as
x̄ + E[max{µ1 , µ2 }] = 2(c(x̄) + c̄).
(12)
Due to assumption (2) and strict convexity of c(x), this definition is unique. Using
(10) we obtain p ≤ c0 (x̄)/g(0). Thus, I henceforth assume that
½
sup
η∈intS
¾
c0 (x̄) 0
g (η) + g(η) < inf c00 (x).
x>0
g(0)
(13)
That is, I restrict attention to the class of problems for which the exogenously given
11
intS denotes the interior of S.
10
functions g(η) and c(x) are such that a firm’s objective function is concave for all p
and ∆ that the buyer might choose.
The socially optimal investment level (given that two firms invest), denoted x∗ ,
is given by
x∗ := argmaxx x + E[max{µ1 , µ2 }] − 2(c(x) + c̄),
(14)
i.e., c0 (x∗ ) = 1/2. By (10), each firm chooses x∗ if the buyer holds a first-price
auction without minimum and maximum price (p = 0, p = ∞).12 However, as will
be shown in the next section, this choice of p and p will in general not be optimal
from the buyer’s point of view.
4
The buyer’s problem
I now analyze the first stage in which the buyer specifies f, p, and ∆. Instead
of maximizing the buyer’s expected surplus, I consider the problem of minimizing
her expected costs for implementing a given investment strategy x. This greatly
simplifies the analysis. Moreover, the conditions under which the buyer prefers a
fixed-prize tournament will be independent of x. Therefore, to characterize these
conditions, it is also not necessary to determine the surplus-maximizing investment
strategy.
In the last stage, assuming identical investments, the price that the buyer has to
pay is:


 p + |η|
if |η| ≤ ∆

 p+∆
otherwise
(15)
Let Ḡ denote the cdf of |η| and ḡ the corresponding density function. It is easily
12
This mechanism maximizes the buyer’s expected surplus in Che and Gale (2003) when firms
are homogeneous.
11
verified that Ḡ(y) = 2G(y) − 1 for all y ≥ 0.13 Then, the expected price before
observing qualities is
Z
∆
P (p, ∆) := p +
yḡ(y)dy + (1 − Ḡ(∆))∆.
(16)
0
This leads to the following optimization problem for the buyer:
s.t.
C(x) := min P (p, ∆) + 2f
f,p,∆
·
¸
1
− c0 (x) = 0
pg(0) + G(∆) −
2
1
f + P (p, ∆) − c(x) ≥ c̄
2
(18)
f, p, ∆ ≥ 0
(20)
(17)
(19)
Equation (18) is the incentive compatibility constraint, inequality (19) is a firm’s
participation constraint, and (20) are the non-negativity constraints. By eliminating
p using (18) and observing that the buyer will choose the smallest non-negative f
that satisfies (19), the problem becomes
h
n
oi
C(x) = min max 2(c(x) + c̄), P̂ (∆; x)
s.t. 0 ≤ ∆, Ḡ(∆) ≤ 2c0 (x),
∆
(21)
where
¤
1 £ 0
2c (x) − Ḡ(∆) +
P̂ (∆; x) :=
ḡ(0)
Z
∆
yḡ(y)dy + (1 − Ḡ(∆))∆.
(22)
0
P̂ (∆; x) is the expected price that the buyer has to pay depending on her choice of
∆ and given x. The buyer minimizes her expected costs by choosing ∆ to minimize
13
Also note that |η| = µ(2) − µ(1) , where µ(2) , µ(1) denote the highest and second highest order
statistic of the sample µ1 , µ2 .
12
P̂ (∆; x). Let ∆∗ denote an optimal choice of ∆, i.e.,
Ḡ(∆)
∆ ∈ arg min −
+
∆
ḡ(0)
Z
∆
∗
yḡ(y)dy + (1 − Ḡ(∆))∆.
(23)
0
Two cases can be distinguished. If 2(c(x) + c̄) > P̂ (∆∗ ; x), the expected price in
the auction is smaller than firms’ investment and opportunity costs. Thus, f must
be positive to make firms participate in the contest. The buyer chooses f such that
firms are just compensated for their costs, i.e., firms’ participation constraints are
binding.14 On the other hand, if 2(c(x) + c̄) ≤ P̂ (∆∗ ; x), fixed payments are zero
and firms earn (ex-ante) rents.
A sufficient condition for the optimality of a fixed-prize tournament (∆∗ = 0) is
that P̂ (∆; x) always increases in ∆, i.e.,
∂ P̂ (∆; x)
ḡ(∆)
=−
+ [1 − Ḡ(∆)] ≥ 0 for all ∆ ≥ 0.
∂∆
ḡ(0)
(24)
Raising ∆ while holding x constant has two effects. First, the minimum price p
decreases (given by − ḡ(∆)
). Second, the expected quality premium that the winning
ḡ(0)
firm receives in addition to p increases (given by [1 − Ḡ(∆)]). The second effect
always dominates the first one if
λḠ (∆) :=
ḡ(∆)
≤ ḡ(0) for all ∆ ≥ 0.
1 − Ḡ(∆)
(25)
Consequently, ∆∗ = 0 for all x if the hazard rate of Ḡ, λḠ (∆), is sufficiently small.
On the other hand, a sufficient condition for the optimality of an auction scheme
is that P̂ (∆; x) strictly decreases in ∆ over some interval (0, a], a > 0. The optimal
14
Actually, in this case, the buyer is indifferent between all ∆ for which 2(c(x) + c̄) > P̂ (∆; x),
i.e., minimizing P̂ (∆; x) is sufficient but not necessary for minimizing procurement costs.
13
minimum and maximum price then depends on Ḡ and x.15 In the special case in
which P̂ (∆; x) always decreases in ∆, i.e.,
λḠ (∆) ≥ ḡ(0) for all ∆ ≥ 0,
(26)
the buyer minimizes expected procurement costs by setting either (i) p = 0 and
∆ = Ḡ−1 (2c0 (x)), or (ii) p = (2c0 (x) − 1)/ḡ(0) and ∆ = ∞. Case (i) occurs when
the buyer wishes to implement an investment strategy x ≤ x∗ . Then, she uses
an auction without a minimum price and sets the maximum price such that firms
implement x. Case (ii) occurs when the buyer wants firms to choose an investment
strategy x > x∗ . Under these circumstances, she does not announce a maximum
price and determines p such that firms choose x.
I now apply the concept of conditional stochastic dominance (CSD) to classify
distributions of |η| with respect to the optimality of a fixed-prize tournament or an
auction.
Definition 1 Consider two cdfs Ḡ1 and Ḡ2 . Ḡ1 dominates Ḡ2 in terms of conditional stochastic dominance, denoted Ḡ1 º Ḡ2 , if
λḠ1 (y) ≤ λḠ2 (y)
for all y for which λḠ1 (y) and λḠ2 (y) are well defined.
This means that, conditional on any maximum quality premium ∆, this maximum quality premium is more likely to be paid under Ḡ1 than under Ḡ2 . It also
implies that Ḡ1 (y) ≤ Ḡ2 (y) for all y, i.e., Ḡ1 first-order stochastically dominates
15
In particular, since P̂ (∆; x) is in general not convex in ∆, the first-order condition for minimizing P̂ (∆; x) is not sufficient to characterize ∆∗ .
14
Ḡ2 .16
Inequality (25) is binding at ∆ = 0 for every distribution of |η|. Furthermore, if
|η| is exponentially distributed,17 we have λḠ (∆) = ḡ(0) for all ∆ ≥ 0 so that (25)
is always binding. Therefore, we obtain the following proposition.
Proposition 1 Let H(y) := 1 − exp[−ḡ(0)y].
(i) If Ḡ º H, then the buyer uses a fixed-prize tournament.
(ii) If H º Ḡ, then the buyer uses an auction.
Thus, if it is possible to rank Ḡ in terms of CSD relative to an exponential
distribution with the same marginal probability of having the higher quality under
identical investments (H 0 (0) = Ḡ0 (0)), we can decide whether the buyer should use
a tournament or an auction scheme. This optimal decision on the procurement
mechanism is independent of the investment strategy x that the buyer wants to
implement. In the special case of Ḡ = H, all combinations of p and ∆ that implement
x lead to the same expected price in the auction and thus to the same expected costs
for the buyer.
The intuition of proposition 1 is as follows. If Ḡ º H, the realization of |η|
is likely to be relatively large, i.e., the values of the innovations for the buyer will
probably differ greatly. As a result, in an auction, the high-quality firm can demand
a large quality premium, i.e., the buyer is likely to pay p + ∆. Setting incentives
through an auction is then too expensive from the buyer’s point of view since it
leaves higher rents to firms than a fixed prize.
The concept of CSD allows to capture the intuition for the superiority of a
fixed-prize tournament under certain probability distributions. However, we do not
16
17
See, e.g., Krishna (2002), p. 260.
This is the case for exponentially distributed µ1 and µ2 , see Sukhatme (1937).
15
know yet which of the common distributions (e.g., normal or uniform distribution)
favor an auction or a fixed-prize scheme. To answer this question, note that the
optimality condition for a tournament, inequality (25), holds if λḠ (y) is monotone
decreasing. By contrast, a monotone increasing λḠ (y) (i.e., inequality (26) holds)
favors an auction. Monotonicity of λḠ turns out to be equivalent to the log-concavity
or log-convexity of G on S ∩ R− .
Definition 2 A function F : R → (0, ∞) is log-concave on the interval (a, b) ⊆ R
if the function ln F is concave on (a, b) and log-convex if ln F is convex on (a, b).
Thus, G is log-convex (log-concave) on a certain interval if and only if g(y)/G(y)
is increasing (decreasing) on this interval. For y ≥ 0, we have that
λḠ (y) =
ḡ(y)
g(y)
g(−y)
=
=
,
1 − G(y)
G(−y)
1 − Ḡ(y)
(27)
which implies that λḠ is monotone decreasing (increasing) if and only if G is logconvex (log-concave) on S ∩ R− . This yields the following proposition.
Proposition 2
(i) If G is log-convex on S ∩ R− , then the buyer uses a fixed-prize
tournament.
(ii) If G is log-concave on S ∩ R− , then the buyer uses an auction.
As is well known from the literature, most “named” distributions are log-concave
(see, e.g., An (1998) or Bagnoli and Bergstrom (2005)). However, cdfs that exhibit
log-convexity are easy to construct. For example,



G(y) :=
1
2(1−y)

 1−
1
2(1+y)
16
y≤0
y>0
(28)
is log-convex on R− so that, under this distribution function, a fixed-prize tournament dominates an auction.18
Log-convexity of G, or, equivalently, a decreasing λḠ means that the instantaneous probability that a certain quality difference |η| is realized, given that the
quality difference is at least |η|, decreases in |η|. This is a strong requirement. However, by (25), it is sufficient for a tournament to be superior that λḠ (y) decreases
for small y, while it may increase for relatively large y as long as it does not exceed
λḠ (0). Intuitively, λḠ (y) decreases for small y if it becomes more likely that firms’
innovations differ relatively strongly, given that they have not found very similar
innovations. This may be the case if the innovation technology is subject to large
random factors.
5
Extensions
The extension of the analysis to the case of n identical firms is straightforward.
The details are in the appendix. Focussing on a symmetric equilibrium in the
investment stage, firm i ∈ {1, . . . , n} =: N wins the contest if maxj∈N \i µj < µi . Let
g(.|n) and G(.|n) denote the density and cdf, respectively, of the random variable
maxj∈N \i µj − µi .19
The price the buyer has to pay is p + min{µ(n) − µ(n−1) , ∆}, where µ(n) and
µ(n−1) denote the highest and second highest order statistic, respectively, of the
sample µ1 , . . . , µn . Defining Ḡ(.|n) as the cdf of µ(n) − µ(n−1) and ḡ(.|n) as the
18
It is easily verified that, for this distribution function, condition (13) holds if 1.5 < inf x c00 (x).
In general, log-concavity or log-convexity of the distribution function does not contradict (13).
19
Since the random variables µi , i = 1, . . . , n, are identically distributed, g(.|n) and G(.|n) are
independent of i.
17
corresponding density, it turns out that the buyer will choose ∆ = 0 if
g(0|n) ≥
g(−∆|n)
1 − Ḡ(∆|n)
for all ∆ ≥ 0.
(29)
It can be shown that inequality (29) binds for all ∆ if the µj ’s are exponentially distributed.20 Thus, for all n, the exponential distribution marks a boundary
between the distributions that favor a first-price auction and those that favor a
fixed-prize tournament.
However, for n > 2, condition (29) cannot be stated in terms of CSD. The reason
is that g(.|n) is symmetric around zero if and only if n = 2. As a result, if n > 2,
there is no straightforward relationship between g(.|n) and ḡ(.|n). Therefore, we
cannot express (29) solely in terms of the distribution of µ(n) − µ(n−1) , as it was done
in (25) for the case of two firms. For the same reason, proposition 2 does not extend
to the n-firm case.
Whether (29) is more or less likely to hold as n increases should depend on the
specific distribution of the noise terms. Intuitively, one would expect the marginal
probability of having the highest quality under identical investments, g(0|n), to be
decreasing in n, implying that investment incentives in the tournament decrease.
However, to the best of my knowledge, such a result has not been established (see
also McLaughlin (1988), fn. 9). Furthermore, it is also not clear whether the righthand side of (29) is decreasing or increasing in n since both the numerator and
denominator will in general decrease in n.
With asymmetric firms, the problem becomes considerably more complex. Basically, there are two issues that complicate the analysis: firms’ asymmetric investment
20
Due to Sukhatme (1937), if the µj ’s are exponentially distributed with mean λ, the random
variable µ(n) − µ(n−1) is also exponentially distributed with mean λ. Furthermore, it is straightforward to verify that, in this case, g(−∆|n) = nλ e−λ∆ .
18
strategies, and the fact that it is often not possible to implement the same investment
strategies in both mechanisms. Then, comparing the mechanisms by considering the
buyer’s cost minimization problem for given investments is no longer feasible.
Furthermore, an increasing asymmetry of firms will usually affect the performance of both mechanisms negatively. In general, under the tournament, investment incentives decrease for each given prize. Firms anticipate that the outcome
of the tournament becomes less responsive to changes in investments since the firm
with the better investment technology is likely to obtain a superior innovation.21
In the auction, firms’ investments tend to differ more strongly, which increases the
expected difference in innovations and therefore the expected price the buyer has
to pay. Therefore, neither mechanism has a clear intuitive advantage if firms are
heterogeneous. Additionally, the performance of each mechanism is likely to be
improved by discriminating between firms. This can be achieved by differentiating
fixed payments, prizes in the tournament, or the range of allowable bids in the auction. For example, in Che and Gale (2003), when firms have different technologies,
the buyer optimally handicaps the more efficient firm by a maximum allowable price
in the auction while the less efficient firm’s bid is not restricted.
Despite the fact that several new issues arise when firms are heterogeneous, so
that the specific results from propositions 1 and 2 do no longer apply, the “randomness” of the innovation technology should remain relevant and have a similar impact
as with homogeneous firms. However, a thorough analysis of this case is beyond the
scope of this paper and subject to future research.
So far I assumed that all parties involved observe qualities before the bidding
process. As noted by Fullerton et al. (2002), if instead only the buyer and firm i
observe qi under a first-price auction without any minimum or maximum price, firm
21
For example, Kräkel and Sliwka (2004) derive such a result.
19
i bids qi − E[qj |qj < qi ].22 It can be shown that this leads to the same investments
of firms and expected costs for the buyer as in the case analyzed above. Intuitively,
although optimal bidding strategies change, parties’ expected payoffs in the stages
before qualities are observed remain the same.
On the other hand, with a fixed prize, it does not matter whether a firm can
observe the quality that the other contestant supplies. Therefore, the buyer still
prefers a fixed prize if the expected price in the auction, E[µ(2) − µ(1) ], is so high
that firms earn large rents.
6
Conclusion
This paper shows that a fixed-prize tournament can dominate a first-price auction in
procurement settings. The key point leading to this result is that, under stochastic
innovations and in the absence of entry fees, holding an auction may leave higher
rents to firms than announcing a fixed prize. The technical results on CSD and logconvexity versus log-concavity, however, are presumably particular to the way the
stochastic innovation technology is modelled, namely, that an increase in a firm’s
investment shifts its cdf of quality to the right.
It is often argued that holding an auction (without a minimum or maximum
price) has a substantial advantage over announcing a fixed-prize tournament since
the latter requires more knowledge on the side of the buyer. To calculate an appropriate fixed prize, the buyer has to know firms’ costs and innovation technologies. In
this paper, if the buyer conducts an auction without restricting the set of allowable
prices, firms even choose efficient investments. However, this is often not optimal
from the buyer’s point of view. Then, in the auction, the buyer also needs detailed
22
This optimal bid is a transformation of the optimal bidding function (3) in Fullerton et al.
(2002). It is derived by partially integrating (3) for the case of two firms.
20
knowledge on investment costs and innovation technologies to calculate the appropriate maximum and/or minimum price. Therefore, as soon as the buyer wishes
to direct firms’ investment behavior, the informational advantage of the auction
disappears.
Appendix
The n-firm case. The timing and rules of the game are identical to the two-firm
case. First consider the last stage. Assume that qi ≥ qj for all j ∈ N , and define
q m := maxj∈N \i qj . Then, firm i bids
pi =


 p + (qi − q m )

 p+∆
if qi − q m ≤ ∆
.
(30)
m
if qi − q > ∆
All other firms bid p.23 If qi > q m , firm i wins the bidding. If qi = q m and
k ≥ 2 denotes the number of firms with quality q m , each of these firms wins with
probability 1/k.
Restricting attention to a symmetric equilibrium in the investment stage, xj = x for
all j ∈ N , the first-order condition for the equilibrium investment can be derived as
in section 3. We obtain:
pg(0|n) + [G(0|n) − G(−∆|n)] = c0 (x).
(31)
Note that g(.|n) is symmetric around zero if and only if n = 2. Furthermore,
since G(0|n) denotes the probability of having the highest quality in a symmetric
equilibrium with n firms, we obtain G(0|n) = n1 .
23
There are additional, payoff-equivalent equilibria in which each firm j with qj < q m bids an
arbitrary price from [p, p̄].
21
Analogously to (13), we can derive a sufficient condition for the existence of a purestrategy equilibrium:
½
sup
η
¾
c0 (x̄) 0
g (η|n) + g(η|n) < inf c00 (x).
x>0
g(0|n)
(32)
In the auction stage, the buyer pays the price:


 p + µ(n) − µ(n−1)
if µ(n) − µ(n−1) ≤ ∆

 p+∆
otherwise
(33)
Using the same analytical approach as in section 4 yields the following costs for the
buyer:
h
n
oi
C(x) = min max n(c(x) + c̄), P̂ (∆; x)
∆
s.t. 0 ≤ ∆, G(−∆|n) ≥
1
− c0 (x),
n
(34)
(35)
where
·
µ
¶¸
1
1
0
P̂ (∆; x) :=
c (x) −
− G(−∆|n)
g(0|n)
n
Z ∆
+
yḡ(y|n)dy + [1 − Ḡ(∆|n)]∆.
(36)
0
It is easily verified that
∂ P̂ (∆;x)
∂∆
≥ 0 for all ∆ if and only if (29) holds.
References
An, M. Y. (1998). Logconcavity versus logconvexity: A complete characterization.
Journal of Economic Theory 80, 350–369.
22
Arozamena, L. and E. Cantillon (2004). Investment incentives in procurement
auctions. Review of Economic Studies 71, 1–18.
Bagnoli, M. and T. Bergstrom (2005). Log-concave probability and its applications. Economic Theory 26, 445–469.
Baye, M. R., D. Kovenock, and C. G. de Vries (1996). The all-pay auction with
complete information. Economic Theory 8, 291–305.
Che, Y.-K. and I. Gale (2003). Optimal design of research contests. American
Economic Review 93, 646–671.
Dasgupta, P. (1986). The theory of technological competition. In J. E. Stiglitz and
G. F. Mathewson (eds.) New Developments in the analysis of market structure,
pp. 519–547. Cambridge: MIT Press.
Dasgupta, P. and J. Stiglitz (1980). Uncertainty, industrial structure, and the
speed of R&D. Bell Journal of Economics 11, 1–28.
Defense Advanced Research Projects Agency (2005). DARPA Grand Challenge.
http://www.darpa.mil/grandchallenge/.
Fullerton, R. L., B. G. Linster, M. McKee, and S. Slate (1999). An experimental
investigation of research tournaments. Economic Inquiry 37, 624–636.
Fullerton, R. L., B. G. Linster, M. McKee, and S. Slate (2002). Using auctions to
reward tournament winners: Theory and experimental investigations. RAND
Journal of Economics 33, 62–84.
Fullerton, R. L. and R. P. McAfee (1999). Auctioning entry into tournaments.
Journal of Political Economy 107, 573–605.
Green, J. R. and N. L. Stokey (1983). A comparison of tournaments and contracts.
Journal of Political Economy 91, 349–364.
23
Grund, C. and D. Sliwka (2005). Envy and compassion in tournaments. Journal
of Economics and Management Strategy 14, 187–207.
Hvide, H. K. (2002). Tournament rewards and risk taking. Journal of Labor Economics 20, 877–898.
InnoCentive (2005). About InnoCentive. http://www.innocentive.com/about/.
Krishna, V. (2002). Auction Theory. Academic Press.
Kräkel, M. and D. Sliwka (2004). Risk taking in asymmetric tournaments. German
Economic Review 5, 103–116.
Lazear, E. P. and S. Rosen (1981). Rank-order tournaments as optimum labor
contracts. Journal of Political Economy 89, 841–864.
Lebrun, B. (1998). Comparative statics in first price auctions. Games and Economic Behavior 25, 97–110.
Lee, T. and L. L. Wilde (1980). Market structure and innovation: A reformulation.
Quarterly Journal of Economics 94, 429–4 36.
Loury, G. C. (1979). Market structure and innovation. Quarterly Journal of Economics 93, 395–410.
Malcomson, J. M. (1984). Work incentives, hierarchy, and internal labor markets.
Journal of Political Economy 92, 486–507.
Maskin, E. and J. Riley (2000). Asymmetric auctions. Review of Economic Studies 67, 413–438.
Maurer, S. M. and S. Scotchmer (2004). Procuring knowledge. In Intellectual
Property and Entrepreneurship, Volume 15 of Advances in the Study of Entrepreneurship, Innovation, and Economic Growth, pp. 1–31. Elsevier Ltd.
24
McLaughlin, K. J. (1988). Aspects of tournament models: A survey. Research in
Labor Economics 9, 225–256.
Nalebuff, B. and J. Stiglitz (1983). Prizes and incentives: Towards a general theory
of compensation and competition. Bell Journal of Economics 14, 21–43.
O’Keefe, M., W. K. Viscusi, and R. J. Zeckhauser (1984). Economic contests:
Comparative reward schemes. Journal of Labor Economics 2, 27–56.
Rosen, S. (1986). Prizes and incentives in elimination tournaments. American
Economic Review 76, 701–715.
Sukhatme, P. V. (1937). Tests of significance for samples of the χ2 -population
with two degrees of freedom. Annals of Eugenics 8, 52–56.
Taylor, C. R. (1995). Digging for golden carrots: An analysis of research tournaments. American Economic Review 85, 872–890.
Windham, P. H. (1999). A taxonomy of technology prizes and contests. In Concerning Federally Sponsored Inducement Prizes in Engineering and Science,
pp. 21–34. National Academies Press.
25