Designing Innovation Tournaments to Maximize Their Value - MISRC
Transcript of Designing Innovation Tournaments to Maximize Their Value - MISRC
Designing Innovation Tournaments to MaximizeTheir Value: An Economic Perspective
Eric K. Clemons, M. Fazil Pac, Sergei SavinThe Wharton School, University of Pennsylvania, [email protected], [email protected],
We analyze open innovation tournaments to see when they are stable, can attract sellers, and can provide
attractive prices to seekers (buyers). Although these tournaments are usually analyzed under the restrictive
condition that only a single seller wins and receives a fixed and pre-announced level of compensation from
the seeker, we examine more general settings in which in settings where tournament participants can derive
extra value beyond the announced reward from sources other than the seeker if they win, and can also
retain a fraction of the value of their submissions even if they lose. We show that contrary to (implicit)
assumptions in prior research, in the absence of these additional incentives open innovation tournaments may
become unsustainable in the presence of high number of potential participants. On the other hand, when
either the salvage value from unsuccessful submissions or the extra compensation derived by a tournament
winner, or both, are high enough, tournaments become an attractive mode of procuring innovative solutions
even in the presence of high number of potential solvers. In particular, we describe settings in which a firm
holding a tournament can entice potential solvers to participate without itself offering any direct monetary
compensation to a potential winner.
History : December 5, 2010
1. Introduction
Like many of the internet business models that have captured the attention of the public, jour-
nalists, and enthusiastic investors, online innovation tournaments (Terwiesch and Ulrich (2009))
have much to recommend them, although they may actually be less universally applicable than the
enthusiasm would suggest. There are areas where they can be and indeed have been enormously
successful. Unfortunately, there are also areas, even some that have been publicly embraced, where
the potential benefits offered by the innovation tournaments may be limited. We focus here only on
open innovation (Chesbrough (2003), Von Hippel (2005)), or innovation tournaments that involve
1
2 Clemons, Pac and Savin Designing Innovation Tournaments
purchase from an outside supplier; there is of course a large literature on internal innovation tour-
naments (see, for example, Dahan and Mendelson (2001), MacCormack et al. (2001), Sommer and
Loch (2004)), which we will not address.
Under typical innovation tournament, a firm (we follow Terwiesch and Xu (2008) in using the
term “seeker”) faced with an innovation problem sets up a solution contest involving a number of
potential solution providers (“solvers”), in which a fixed, known, and pre-announced reward is paid
to the solver with the best solution. Innovation tournaments have three possible justifications:
(1) they can reduce the cost and uncertainty of evaluation, since it is much easier to evaluate a
piece of software by running it or a new fighter aircraft by flying it than it is to make the same
evaluation from the sellers’ proposal documents
(2) they can reduce the search cost associated with finding solutions, since sellers with available
expertise and nearly-off-the-shelf offerings will respond without the buyer having to conduct an
extensive search for them
(3) they can accomplish the first or the second at an acceptable purchase price, since prices will
be driven down by competition among a large number of competing sellers
Our first concern is that innovation tournaments may have unfavorable implications for the
seeker’s prices, since development costs are incurred by more firms and must be covered somehow
if the market is to survive; for this reason we address costs and prices first. Solvers’ behavior in
equilibrium has been addressed (Terwiesch and Xu (2008)), including solvers’ decisions on prices
and on whether to participate or not, but the previous model does not yet provide enough detail to
allow seekers to determine if the tournament will be their best procurement strategy. Our analysis
provides an extension to this work.
An innovation tournament differs from a traditional request for a proposal (or RFP) because it
requests a finished product (since completed products are much bigger than proposals let’s call a
tournament an RFMBP). With an RFP, lots of firms compete, the winner gets a contract, and
only the winner incurs development costs and gets paid for a final deliverable. With an innovation
tournament, all competing sellers submit a completed product, and therefore all incur development
Clemons, Pac and Savin Designing Innovation Tournaments 3
costs; once again, only the winner gets paid. When a traditional RFP is generated the cost of
responding to the RFP is buried in the profit margin of the bidding firm; there is no line item
for preparing the proposal, but the costs of these proposals must be covered. A firm that bids
unsuccessfully on too many RFPs, like a cheetah that hunts unsuccessfully too many times, must
starve. While preparing a response to an RFP is relatively cheap, responding to an RFMBP is
relatively expensive. This means that one of the following conditions must hold:
(1) the solver must expect to win a significant proportion of the time it enters (high probability
of success), or
(2) the solver must attempt to recover the development costs from several failed sub-missions in
the profit margins on its few successful entries (high ability to include development costs of several
projects in a few successful submissions), or
(3) the solver must not care if it recovers its development costs from seekers (low development
costs).
There really is no other alternative; either you win often, or you charge a lot when you do when
to cover the high costs of failed entries, or you don’t need to charge the seeker enough to cover
your costs. (Many internal innovation tournaments use staged competitions and filters, which can
reduce the amount of redundant effort, but does not significant alter the nature of the problem;
we ignore staged innovation tournaments in this paper.)
Given the social costs associated with redundant development efforts, only some of which ever
get directly paid by winning, we address two issues that determine the applicability of innovation
tournaments and indeed their long-term viability as procurement mechanisms:
(1) if these costs are recouped by solvers directly through the purchase price, how does this affect
the seeker’s price of items obtained through an innovation tournament?
(2) if these costs cannot be recouped by solvers directly through the purchase price, what does
this say about the conditions under which innovation tournaments still remain viable?
Below we introduce our model (Section 2), discuss our results (Section 3), and provide overview
of potential directions for future research (Section 3).
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2. Model
In our analysis we focus on a “fixed-price winner-take-all expertise-based” open innovation tourna-
ment (Terwiesch and Xu (2008)), in which a firm looking for a solution to an innovation problem
(seeker) announces a fixed reward A to be collected by the contest winner. We consider n identical
risk-neutral firms that can potentially participate in the contest (solvers), and assume that each
solver incurs the cost c if it decides to participate in the contest. In our model, we consider a
one-dimensional representation of the quality of the resulting submission, similar to Dahan and
Mendelson (2001) and Terwiesch and Loch (2004). Our analysis is based on the following assump-
tions:
A1. The number of potential participants in the tournament, n, is common knowledge for every
potential participant (solver) as well as for the seeker.
A2. Solver firm i at the time of deciding whether to enter the contest, and before the solution
process starts, can only estimate the final quality of competitors’ submissions vj, j 6= i as being
random i.i.d. variables. With respect to the knowledge that the solver firm i has regarding the
quality of its own submission vi, we consider two alternative informational settings. In the first
setting, we assume that firm i knows only the distribution of vi, while the second setting describes
the situation in which firm i knows the exact value of vi. The first setting applies to tournaments
with high degree of information uncertainty, while the second one applies to high-certainty tour-
naments; in some sense, a high degree of uncertainty may often correspond to a tournament with
a high degree of innovation and research, while a low degree of uncertainty may often correspond
to a tournament with less innovation and a high degree of development or commercialization of
known ideas.
A3. We assume that for a given n, the values of vi, i = 1, ..., n are i.i.d. uniform random variables
with support on [vmin, vmax]. Note that such a set up represents a combination, or hybrid design,
somewhere between an “ideation” and a “trial-and-error” project, as described in Terwiesch and
Xu (2008).
Clemons, Pac and Savin Designing Innovation Tournaments 5
A4. The expected reward for a solver that wins the contest is a sum of the fixed reward, A, paid
by the seeker, and the “external” future rewards, R≥ 0, paid by outside parties (other than the
seeker, as with the benefits received by participants in the American Idol and similar contests).
A5. An unsuccessful entry of quality vi from solver i results in the reward of αvi, where α∈ [0,1]
is the salvage value parameter. Of course, the salvage value is obtained from the market, and is
not paid by the seeker.
Under this set of assumptions, the expected profit derived from the tournament by the seeker is
given by
Π(A) = E
[max
i=1,...,nvi
]−A, (1)
where the expectation is taken over the distributions of quality of submissions. The seeker’s objec-
tive is to maximize his or her profit, Π(A), from the innovation tournament by selecting the optimal
value of the reward A. Clearly, A is dependent upon solvers’ behavior; an A that is too low will
result in no submissions, while an A that is too high will have the seeker paying more than is
necessary.
Below we consider the two settings described in the assumption A2 separately.
2.1. High-Uncertainty Setting
Suppose that when deciding to develop a potential solution, each solver ex ante knows the quality
of his or her final submission as well as the quality of competing submissions only in distribution.
For the analysis below it is convenient to define
A∗ (n,vmin, vmax,R,α, c) = max(
0, nc−α
((vmin + vmax
2
)(1− 1
n+1
)+
vmin
n+1
)(n− 1)−R
),
(2)
and
αl =2c
vmax + vmin
, (3)
αh =4c
vmax +3vmin
. (4)
6 Clemons, Pac and Savin Designing Innovation Tournaments
Note that αl represents the ratio of the cost of participating in a tournament, c, to the expected
value of the submission, vmax+vmin2
. As will be established below, this parameter plays an important
role in determining the impact of the salvage value α on the optimal profit of the seeker firm. As
follows from (3) and (4), αl < αh whenever vmin < vmax.
Using the definitions (2)-(4), we can proceed to describe the properties of innovation tournaments
in this environment.
Proposition 1. a) In the high-uncertainty setting the tournament can exist if and only if
A∗ (n,vmin, vmax,R,α, c)≤ vmax− vmax− vmin
n+1. (5)
Under (5), the optimal reward set by the seeker is A∗ (n,vmin, vmax, r,α, c), and all potential
solvers participate in the tournament.
b) For A∗ (n,vmin, vmax,R,α, c) > 0, let
Π∗ = Π(A∗ (n,vmin, vmax,R,α, c)) (6)
be the optimal seeker’s profit value. Then, for αl < 1, Π∗ is increasing (decreasing) in the number
of potential solvers, n, if α≥ αl
(α < max
(0, αh +1− αh
αl
)). If α∈
[max
(0, αh +1− αh
αl
), αl
),
Π∗ is increasing (decreasing) in the number of potential solvers, n, for n≤ n∗(α) (n > n∗(α)),
where
n∗(α) =
⌊2
√(1−α) (αh−αl)
αh(α−αl)
⌋− 1. (7)
For αl ≥ 1, Π∗ is always decreasing in the number of potential solvers n.
c) If α < min(1, αl), the number of potential solvers that a tournament can support is bounded
from above by a finite threshold n(α).
Part a) of Proposition 1 implies that when solvers have identical properties, with no ex ante
private information to help them assess their likelihood of winning, an A is either sufficient to
attract all solvers or no solvers; the optimal A from the perspective of the seeker is the lowest that
will attract bidders. Figure 1 illustrates how the optimal reward value (2) depends on the number
Clemons, Pac and Savin Designing Innovation Tournaments 7
Number of potential solvers, n
1.0
5.0 !
7.0 !
c
A*
Figure 1: The ratio of the optimal reward to the fixed participation cost, A∗c
as a function of the number of potential
solvers n for different values of the salvage value parameter α (vmin = 0, vmax = 1, R = 1, c = 0.4).
of potential tournament participants for different values of the salvage value α. Note that in this
setting the reward that is needed to entice solvers to participate in the tournament grows nearly
linearly with n, with higher levels of salvage value allowing seeker to offer lower rewards.
Monotonicity properties of the optimal seeker’s profit Π∗ with respect to the number of potential
solvers for different levels of salvage value α are illustrated in Figure 2. Note that for low values
of α, the higher the number of potential solvers, the less attractive a tournament is for the seeker,
since as we have seen an increase in the number of seekers requires an increase in A∗. On the other
hand, sufficiently high values of α imply that the benefits of a tournament for the seeker increase
as the number of potential solvers grows; this follows directly from the specification of the model,
8 Clemons, Pac and Savin Designing Innovation Tournaments
since as α increases the necessary payoff A∗ decreases, and as the number of solvers increases the
expected value of the best solution approaches the maximum value that could be achieved by any
solver. For intermediate values of α, there exists an optimal number of potential solvers beneficial
for the seeker - but as the number of solvers grows, a tournament eventually becomes unsustainable
since A∗ will have to increase beyond the seeker’s willingness to pay. As part c) of Proposition 1
states, αl plays a role of an upper bound of the range of salvage values α for which a tournament
can only support a limited number of potential solvers.
Part b) of Proposition 1 implies that the presence of salvage value α and the external reward
component R may allow the seeker to hold a zero-reward tournament that would still be attractive
to solvers as long as the values of α and R are high enough. Intuitively, this means that with
sufficiently high R the tournament will be attractive to solvers even with a large number of solvers,
and with sufficiently high R the high degree of compensation needed for a large number of solvers
does not need to be paid by the seeker. The solvers get their high reward, needed to compensate
them for a small (1/n) chance of winning, and they remain in the market; and since the high
reward is not paid by the seeker the high reward does not drive the seeker out of the market. The
market does not fail. We can state this result formally as follows.
Proposition 2. For any α∈ (αl, αh), define
n(α) =
⌊2
√α(αh−αl)αh(α−αl)
⌋− 1. (8)
It is optimal for the seeker not to offer any reward to solvers, as long as the external reward
component R is at or above the threshold
R (n,vmin, vmax,α, c) = max(0, nc−α
((vmin + vmax
2
)(1− 1
n+1
)+
vmin
n+1
)(n− 1)), (9)
which is an increasing function of c and a decreasing function of α. In addition, the zero-reward
threshold R (n,vmin, vmax, α, c) is a increasing function of n for α ≤ min(1, αl), a non-increasing
function of n for α > min(1, αh). For min(1, αl) < α≤min(1, αh) it is an increasing function of n
for n≤ n(α) and a non-increasing function of n for n > n(α).
Clemons, Pac and Savin Designing Innovation Tournaments 9
!
"#$#%&'
Number of potential solvers, n
!
Number of potential solvers, n
!"!#$%&
!
Number of potential solvers, n
!"!#$%&
Figure 2: The optimal seeker’s profit Π as a function of the number of potential solvers n for α = 0.1, 0.65, and 0.75.
(vmin = 0, vmax = 1, R = 0.3, c = 0.34).
10 Clemons, Pac and Savin Designing Innovation Tournaments
Number of potential solvers, n
!"!#$%c
R
!"!#$%&
!"!#$'&
Figure 3: Zero-reward threshold, Rc
as a function of the number of potential solvers n for different values of the
salvage value parameter α (vmin = 0, vmax = 1, R = 1, c = 0.2).
Figure 3 provides an example of the dependence of zero-reward threshold R (n,vmin, vmax, α, c)
on the number of potential solvers for different salvage values α. Note that as the salvage value
for the unsuccessful submissions increases, the solvers require progressively lower external reward
values R in order to participate in a zero-reward tournament. In particular, for high values of α,
zero-reward tournament is sustainable even in the absence of any external reward, once the number
of potential solvers becomes sufficiently high1.
1 This conclusion, of course, is based on the assumption that the salvage value does not depend on the number ofpotential solvers n. In real-life settings, as the number of solvers increases beyond some level alternative uses forsubmissions will not be sufficient to sustain a secondary salvage market.
Clemons, Pac and Savin Designing Innovation Tournaments 11
2.2. High-Certainty Setting
The high-certainty setting describes a tournament in which each solver i a) has a good idea about
the future quality, vi, of his or her final submission, but b) knows the distribution of the quality of
competing submissions. Knowing the quality of his or her submission, a solver decides whether to
participate in the tournament. The expected profit that solver i with submission quality vi derives
from participating in the tournament is given by
πi(vi) = (A+R) Prob (i wins)+αvi (1−Prob(i wins))− c
= (A+R) FU(vi)n−1 +αvi
(1−FU(vi)n−1
)− c, (10)
where
FU(x) =x− vmin
vmax− vmin
, x∈ [vmin, vmax] . (11)
To avoid trivial cases, we assume that the reward set by the seeker is high enough to ensure
the participation of a solver with a solution of the highest possible quality (vmax), i.e., πi(vmax) =
(A + R)− c≥ 0. This ensures that all solvers are better-off by winning the tournament, since the
total reward, A+R, earned by the winner covers the cost of participation c.
Proposition 3. For a given reward A, there exists a threshold vl(A), such that only solvers with
vi ≥ vl(A) participate in the tournament. The threshold vl(A) is decreasing in the salvage value,
α, and in the values of internal/external rewards, A and R, and is increasing in the number of
potential solvers, n.
Proposition 3 indicates that the innovation tournament becomes more attractive to solvers as
the salvage value, α, and/or the rewards, A and R, increase, which leads to a lower quality thresh-
old for solver participation. On the other hand, as the number of potential solvers, n, increases,
each solver’s probability of winning the tournament decreases, leading to a higher solution quality
participation threshold.
The seeker’s objective, as before, is to maximize his or her profit, Π(A), from the innovation
12 Clemons, Pac and Savin Designing Innovation Tournaments
tournament by selecting the optimal value of the reward A. When there are n potential solvers in
the market, the expected number of solvers participating in the tournament is given by
k(n,A) = n(1−FU(vl(A))
). (12)
For a given reward value A, the seeker’s expected profit from the innovation tournament is given
by
Π(A) = E
[max
i=1,...,k(n,A)vi
]−A
= vmax− vmax− vmin
k(n,A)+ 1−A. (13)
The following Proposition describes the properties of innovation tournaments in high-certainty
settings.
Proposition 4. a) The optimal reward, A∗(n,vmin, vmax,R,α, c) is decreasing, while the optimal
profit, Π∗ = Π(A∗(n,vmin, vmax,R,α, c)), is increasing in the salvage value, α, and the external
reward, R.
b) The expected number of solvers participating the tournament, k(n,A∗(n,vmin, vmax,R,α, c)) is
increasing in the salvage value, α, and the external reward, R.
c) For α > cvmax
, as the number of potential solvers becomes large (n→∞), the seeker can sustain
a tournament without offering positive reward, limn→∞A∗(n,vmin, vmax,R,α, c) = 0, earning the
maximum possible profit, limn→∞Π(A∗(n,vmin, vmax,R,α, c)) = vmax.
d) For α≤ cvmax
, there exists a threshold n(α) such that the tournament can exist only if the number
of potential solvers is lower than n(α). A threshold value n(α) is increasing in R and α.
Proposition 4 confirms that a number of properties of innovation tournaments derived for the high-
uncertainty setting, continues to hold for the high-certainty setting as well. In particular, as parts
c) and d) of Proposition 4 imply, an innovation tournament can support a large number of solvers
only if the salvage value α is sufficiently high.
Clemons, Pac and Savin Designing Innovation Tournaments 13
3. Discussion of Other Cases Where Innovation Tournaments Can Work
While tournaments are good sources of ideas from customers, and can work very well when innova-
tion involves recombination of prior technologies, they may not always provide the best mechanism
for cost-effective procurement. Our analysis shows that under the basic assumptions that a) the
tournament winner’s only compensation comes from the fixed reward paid by the seeker, and b)
losing submissions have no salvage value, the zero-profit price for a submitted bid to a tournament
should grow quickly as the number of sellers increases (Propositions 1 and 4). More significantly,
and as a direct consequence, the price that a seeker has to pay for obtaining a solution through an
innovation tournament can be higher than it would be with a traditional form of procurement based
upon RFP. Thus, with even a modest number of qualified solvers, prices obtained through tourna-
ments should become very high since the prices must cover the costs of unsuccessful development
efforts, not just the cost of preparing unsuccessful proposals. And yet, innovation tournaments
are gaining in popularity. How can we explain this? Basically, there appear to be at least three
ways that innovation tournaments can be successful: (1) the solver can obtain revenue outside the
tournament reward (either through future compensation for the winner not paid by the seeker,
or through the salvage value for unsuccessful submissions, or both; (2) the innovation contest can
attract extremely low-cost solvers, so that even when reflecting the lower probability of winning
and getting paid, the contest can still be attractive to the seeker; or (3) the higher reward that
the seeker (buyer) has to offer is more than off-set through other savings, such as reduction in
uncertainty about expected quality.
We examined (1) through our propositions above. In a number of real-life settings solver’s
rewards can be much higher than the actual award paid by the seeker. Reality television shows
like American Idol or Dancing with the Stars, or their counterparts in many other countries, are
extraordinarily cheap to produce because their producers do not have to pay for talent. The appeal
to solvers (participants) is real; in 2007, according to the Learning and Skills Council, one in seven
UK teenagers hoped to gain fame by appearing on reality television. Talented individuals volunteer
14 Clemons, Pac and Savin Designing Innovation Tournaments
because the future rewards for winning are enormous, and competitors have the opportunity to
compete because these future rewards are not paid by the buyer, keeping the buyer’s costs low.
Alternately, solvers may know that they will enjoy a high salvage value from unsuccessful submis-
sions: a software system developed for one bank or restaurant might later be offered to others. An
award paid by the seeker needs to cover development costs and the expected costs from unsuccessful
bids net of the benefits from salvaging unsuccessful bids.
We discuss the remaining two mechanisms below.
The settings with low-cost solvers can arise when solvers are either idle, with no other work to
do, or when solvers consider the tournament as “fun” rather than “work”, or when solvers are
re-using the material that they have developed for use elsewhere.
We argue that condition (2) is unlikely to be observed in real settings: under most economic
conditions, teams that are good enough to work will be working. If the demand for a category of
worker has dropped so low that their opportunity costs approach zero, but they are still of value to
the buyer running the tournament, then under those very rare conditions innovation tournaments
may succeed. Under normal conditions, however, the risk of adverse selection confronting the seeker
is just too great when facing solvers who have no opportunity costs. We would resist relying upon
a cardiovascular surgeon who was perpetually idle, and most firms would not want to accept a
significant piece of software from unemployed developers who submitted the work on speculation,
hoping that they might get paid. The second and the third condition, however, can be observed in
practice. There are numerous categories of innovation tournaments where solvers do not need to
get paid for their work. Sometimes this involves student competitions, especially classroom-based
competitions, like several of those reported on in the innovation tournament literature (Terwiesch
and Ulrich (2009)). Sometimes these involve amateurs competing for recognition, like Sam Adams
annual Home Brewers’ Long Shot Competition to develop a Fall beer offering. Such tournaments
can work, but only for those products that can be designed by amateurs working for their own
amusement. Alternatively, if the solver has already been paid for his submission, because he is using
it elsewhere, then bid prices can be quite low. Tournaments that attract reuse of new but not novel
Clemons, Pac and Savin Designing Innovation Tournaments 15
technology can succeed. In markets where there is strong fragmentation in the seeker’s industry,
and significant commonality in the needs among players in the seeker’s industry this can work
quite well. There are probably still thousands of community banks and thousands of high schools
in the United States, and how different can their needs be? It should indeed be possible to find
a complete software solution to almost any need of small banks and high schools. A tournament
provides the added advantage of reducing seekers’ search costs and of allowing the inspection of
completed work. However, it can be argued that the innovation content of the products delivered
in this instance may be low.
Condition (3) is problematic because it may make sense, but only when the cost of selecting
the wrong offering is extraordinarily high. In settings where costs associated with errors in quality
assessment are high, the seeker may be willing to pay a higher price for a completed product with
an observable quality than for a contract for a product to be delivered later. Buying software that
can be run and tested, with output that can be observed and performance that can be measured,
greatly reduces uncertainty compared to bidding based on the expected quality of a product that
will be delivered based on a proposal. However, the cost of software development “on spec”, with
each bidder expecting ex ante only a 1/n chance of winning, would render such tournaments unsus-
tainable. We suspect that the continued existence of software development tournaments is largely
due to the fact that (i) bidders frequently are tailoring an existing product rather than engaging
in truly innovative development and consequently (ii) generally have an accurate assessment ex
ante of the quality of their future submissions and (iii) expect that any development work for one
seeker will have a significant salvage value when offered to similar seekers in the marketplace.
4. Future Research
Our analysis opens up an avenue for exploration of other mechanisms for maximizing the value of
innovation tournaments. For example, it is likely that tournaments can be successful in settings
where solvers know that they have a high probability of winning some of the time in a repeated
game, and that they will be able to extract monopoly rents later, after they win. It can be argued
16 Clemons, Pac and Savin Designing Innovation Tournaments
that the US Department of Defense historically resorted to running innovation tournaments when
it sought bids for new fighter aircraft. Contractors submitted completed prototype aircraft, devel-
oped at enormous expense, and only the winning submission received payment. This innovation
tournament was the only mechanism for the DoD to learn the true quality of submitted entries:
by flying them and observing their behavior. Participants knew that development costs could be
hundreds of millions of dollars, which would not be recovered for unsuccessful bids. They also knew
that there were few competitors and that their probability of an eventual win is substantial if they
continued their participation in the future tournaments; they also knew that when they did win
they would have monopoly power over future pricing for their successful entry. This monopoly
power meant that the solver did not have to recover development costs fully in each initial bid,
since unrecovered costs could be covered through massive charges for change orders and, indeed,
for any modifications to the initial order. Thus, this combined the seeker’s higher willingness to
pay and the solver’s ability to obtain revenue outside the tournament price. DoD contracting pro-
cedures have since been significantly altered, because this form of tournament was certainly not
the lowest-cost way of achieving innovation. Unsuccessful bidders still learn a great deal about
how to deliver the desired product and are rewarded with revenue streams as subcontractors; this
avoids some of the social cost due to duplicated effort, reduces the amount of over-charging on
change orders needed to cover the expense of unsuccessful submissions, and reduces the market
power of the successful bidder. Note that with these adjustments, the resulting competition differs
in a substantial manner from the “classical” innovation tournament.
Future research on the subject of innovation tournaments may also need to address a number
of potential risks to the seekers that may be created by innovation tournaments. The solver must
receive far more information from the seeker before responding to an innovation tournament request
than is needed to respond to a traditional RFP. What happens when the solver knows a great deal
about the seeker’s needs and indeed may be able to reverse-engineer current business practices
or future strategy, and additionally needs to recover the costs associated with its unsuccessful
development effort? We suspect that this may lead to a form of reuse of the original submission, with
Clemons, Pac and Savin Designing Innovation Tournaments 17
unsuccessful solvers approaching direct competitors of the seeker (see, for example, Clemons and
Hitt (2005) and Clemons et al. (1993)). In addition, much remains to be done on the empirical side,
in particular, with respect to studying the actual reward amounts observed in actual innovation
tournaments, the advantages of uncertainty and search cost reduction, and the dynamics of seeker-
solvers interactions emerging in repeated innovation contests.
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Clemons, Pac and Savin Designing Innovation Tournaments 1
Appendix
Proof of Proposition 1
a) The expected profit for solver i is given by
πi = (A+R) Prob (i wins)+αE(vi|i loses) (1−Prob(i wins))− c. (A1)
Note that the CDF function of the quality of solver’s i prototype, conditional on the fact that i
loses the tournament, is given by
Fl(vi|i loses) =Prob(vi ≤ x|i loses)1−Prob(i wins)
=
1vmax−vmin
∫ x
vmindy
(1−
(y−vmin
vmax−vmin
)n−1)
1− 1n
, (A2)
and the corresponding pdf is
fl(vi|i loses) =
1vmax−vmin
(1−
(x−vmin
vmax−vmin
)k−1)
1− 1n
. (A3)
Thus, the conditional expectation E(vi|i loses) is given by
E(vi|i loses) =∫ vmax
vmin
xdxfl(vi|i loses) =vmax + vmin
2
(1− 1
n+1
)+
vmin
n+1, (A4)
so that (A1) becomes
πi = (A+R) Prob (i wins)+αE(vi|i loses) (1−Prob(i wins))− c
= (A+R)(
1n
)+α
((vmin + vmax
2
)(1− 1
n+1
)+
vmin
n+1
)(1− 1
n
)− c. (A5)
From (A5) it follows that the solver i will enter the tournament if and only if
(A+R)(
1n
)+α
((vmin + vmax
2
)(1− 1
n+1
)+
vmin
n+1
)(1− 1
n
)≥ c. (A6)
This expression can be re-written as
A≥A∗ (n,vmin, vmax,R,α, c) . (A7)
Thus, focusing on symmetric Nash equilibria, we conclude that all of the potential participants will
enter the tournament if (2) is satisfied - and none will enter if it isn’t. In particular, we note that
2 Clemons, Pac and Savin Designing Innovation Tournaments
in order for the tournament to be attractive for potential solvers for given values of A, n, vmin and
vmax, either the non-seeker reward fraction r or the savage value parameter α should be substantial
enough, or the participation cost c should be low enough.
Further, focusing on the seeker’s problem, we observe that, for A≥A∗ (n,vmin, vmax,R,α, c), the
expression for its expected profit is given by (1). Note that for any A≥A∗ (n,vmin, vmax,R,α, c) all
potential solvers enter the tournament, and E [maxi=1,...,n vi] does not depend on the amount of the
declared reward A. Thus, it is optimal for the seeker to set the reward at A∗ (n,vmin, vmax,R,α, c).
Further, note that the CDF of the maximum of n i.i.d. submissions is given by
Fn(x) = Prob(
maxi=1,...,n
vi ≤ x
)= (FU(x))n, (A8)
where
FU(x) =x− vmin
vmax− vmin
, x∈ [vmin, vmax] (A9)
is the CDF of the uniform distribution with the support on [vmin, vmax]. Then, for given k, the
expected value of maxi=1,...,n vi is given by∫ vmax
vmin
n
(x− vmin
vmax− vmin
)n−1x
vmax− vmin
dx = vmax− vmax− vmin
n+1. (A10)
Finally, (5) is obtained by requiring Π(A∗ (n,vmin, vmax,R,α, c))≥ 0.
b) A∗ (n,vmin, vmax,R,α, c) > 0 implies that R ≤ nc − α((
vmin+vmax
2
)(1− 1
n+1
)+ vmin
n+1
)(n− 1).
Then
Π∗ = vmax− vmax− vmin
n+1−nc+α
((vmin + vmax
2
)(1− 1
n+1
)+
vmin
n+1
)(n− 1)+R, (A11)
so that
∂Π∗
∂n= α
(vmax + vmin
2
)− c+(1−α)
vmax− vmin
(n+1)2
= c
(α
αl
− 1+4
(n+1)2(1−α)
(1αl
− 1αh
)). (A12)
Consider αl < 1. As it follows from (A12), Π∗ is an increasing function of n for α≥ αl. Note that
the expression on the right-hand side of (A12) has the highest value when n = 1. Thus,
α
αl
− 1+ (1−α)(
1αl
− 1αh
)< 0 (A13)
Clemons, Pac and Savin Designing Innovation Tournaments 3
implies that Π∗ is a decreasing function of n. (A13) is equivalent to
α < αh +1− αh
αl
. (A14)
Note that for αl < 1, αh +1− αhαl
< αl. Further, if α∈[max
(0, αh +1− αh
αl
),αl
)the sign of ∂Π(A∗)
∂n
switches from positive to negative at
n = 2
√(1−α) (αh−αl)
αh(α−αl)− 1. (A15)
Requiring that the number of potential solvers be integer, we obtain (7). Finally, for αl ≥ 1, we
have α≤ αl, and αh + 1− αhαl≥ αl. Then, (A14) and, thus, (A13) hold, so that ∂Π∗
∂nis negative for
any n.
c) As follows from the proof of part b), for α < min(1, αl) the sign of ∂Π∗∂n
is negative when
n is high enough (if αl ≥ 1, this sign is always negative, while if αl < 1, it becomes negative for
n > n∗(α)). Note that as n increases, the ∂Π∗∂n
approaches a negative constant ( ααl− 1). Thus, Π∗
will become negative if n is above a finite threshold value.
¤
Proof of Proposition 2
Note that, as follows from (2), A∗ (n,vmin, vmax,R,α, c) = 0 if and only if
R≥ nc−α
((vmin + vmax
2
)(1− 1
n+1
)+
vmin
n+1
)(n− 1) = R∗ (n,vmin, vmax, α, c) , (A16)
which is increasing in c and decreasing in α. Further, differentiating R∗ (n,vmin, vmax, α, c) with
respect to n, we obtain
∂R∗
∂n= c−α
(vmax + vmin
2
)+α
vmax− vmin
(n+1)2= c
(1− α
αl
+4α
(n+1)2
(1αl
− 1αh
)). (A17)
Note that the expression on the right-hand side of (A17) is non-negative (non-positive) for any
n if and only if α≤min(1, αl) (α > min(1, αh)). For α ∈ (min(1, αl),min(1, αh)], this expression is
positive (negative) if and only if n≤ n(α) (n > n(α)). Note that R∗ (n = 1, vmin, vmax, α, c) = cn > 0,
we obtain the monotonicity properties of R (n,vmin, vmax, α, c) = max(0,R∗ (n,vmin, vmax, α, c)). ¤
4 Clemons, Pac and Savin Designing Innovation Tournaments
Proof of Proposition 3
Below we show that the threshold vl(A) is the unique solution to
πi(v) = (A+R) FU(v)n−1 +αv(1−FU(v)n−1
)− c = 0. (A18)
To show that (A18) has a unique solution in (vmin, vmax), we rewrite (A18) as
(A+R−αv)FU(v)n−1 = c−αv. (A19)
The left-hand side and the right-hand side of (A19) “cross” only once for v ∈ (vmin, vmax). Indeed,
for v = vmin, the left-hand side of (A19) is equal to 0, while the right-hand side is equal to c > 0.
For v = vmax the left-hand side of (A19) is equal to A + R−αvmax, which is greater than or equal
to the right-hand side, c−αvmax, since A+R≥ c.
A solver’s payoff, πi(v), is increasing in the quality of his/her solution for v≤ A+Rα
as
dπi(v)dv
= (A+R−αv)(n− 1)FU(v)(n−2)fU(v)+α(1−FU(v)(n−1)
)> 0 (A20)
for v ≤ A+Rα
. The right-hand side of (A19) is equal to 0 for v = cα
and negative for v > cα, while
the left-hand side is non-negative at v = cα, since A + R ≥ c. As a result, the left-hand side of
(A19) “crosses” the right hand side exactly once for v ∈ (vmin,cα). The left-hand side of (A19) does
not “cross” the right hand side of (A19) for v ∈ (cα, vmax
]: (i) For v ∈ (
cα, A+R
α
)the left-hand side
is non-negative and the right-hand side is negative, hence the left-hand side is greater than the
right-hand side. (ii) For v ∈ [A+R
α, vmax
], the left-hand side is concave and the right-hand side is
linear. Since the left-hand side is greater than the right-hand side at v = A+Rα
, it can “cross” the
right-hand side at most once for v ≥ A+Rα
. Since the left-hand side is equal to A + R−αvmax and
the right-hand side is equal to c−αvmax for v = vmax, and A + R≥ c, we know that the left-hand
side cannot “cross” the right-hand side for v ∈ [A+R
α, vmax
]. Therefore, vl(A)≤ c
α.
vl(A) is decreasing in the salvage value, α, and internal and external rewards, A and R, since
(A18) is increasing in α, A and R for all v ∈ [vmin, vmax]. vl(A) is increasing in the number of
potential solvers n, since (A18) is decreasing in n for v≤ A+Rα
and since vl(A)≤ cα.
¤
Clemons, Pac and Savin Designing Innovation Tournaments 5
Proof of Proposition 4
a) To show that Π(A) has a unique maximizer A∗, we perform a change of variable and use
vl(A) instead of A. Note that vl(A) is a monotone function of A; hence, there exists a unique A∗
maximizing Π(A) if and only if there exists a unique vl maximizing Π(vl). The reward that will
induce a participation threshold vl, A(vl) is given by
A(vl) =c−αvl
FU(vl)(n−1)+αvl−R. (A21)
We can write the seeker’s profit as a function of vl:
Π(vl) = vmax− vmax− vmin
n(
vmax−vl
vmax−vmin
)+1
− c−αvl
FU(v)(n−1)+αvl +R. (A22)
Π(vl) is concave in vl as
d2Π(vl)d(vl)2
= − 2n2
(n(vmax−vl)vmax−vmin
+1)3
(vmax− vmin)
−(n− 1)(
vmax− vmin
vl− vmin
)(n−1) (2α(vl− vmin)+n(n− 1)(c−αvl)(vl− vmin)2
)≤ 0 (A23)
since vl ≥ vmin, n ≥ 1 and vl ≤ cα
from Proposition 3. As a result the seeker’s profit function is
concave around the global maximizer A∗.
To show that A∗(n,vmin, vmax,R,α, c) is decreasing in α we first show that the optimal threshold
vl(A∗) is decreasing in α. vl(A∗) is decreasing in α since
d2Π(vl)dvldα
=1
FU(vl)n−1
(vl
vl− vmin
− 1)− 1 < 0. (A24)
and since Π(vl) is concave in vl. The first order condition for Π(vl) is:
0 =−n
n(vmax−vl(A∗))vmax−vmin
+1+α
((vmax− vmin
vl(A∗)− vmin
)n−1
− 1
)(A25)
+(c−αvl(A∗))(n− 1)
(vl(A∗)− vmin)
(vmax− vmin
vl(A∗)− vmin
)n−1
. (A26)
Note that the first term, −nn(vmax−vl(A∗))
vmax−vmin+1
, in (A26) is negative, while the second term,
α
((vmax−vmin
vl(A∗)−vmin
)n−1
− 1)
, and the third term, (c−αvl(A∗)) (n−1)
(vl(A∗)−vmin)
(vmax−vmin
vl(A∗)−vmin
)n−1
, are pos-
itive. The first term is increasing in α since vl(A∗) is decreasing in α. Therefore the sum of the
6 Clemons, Pac and Savin Designing Innovation Tournaments
second and third terms must be decreasing in α for the first order condition to hold. The second
term, α
((vmax−vmin
vl(A∗)−vmin
)n−1
− 1)
, is increasing in α therefore the third term must be decreasing in
α.
If the third term (c − αvl(A∗)) (n−1)
(vl(A∗)−vmin)
(vmax−vmin
vl(A∗)−vmin
)n−1
is decreasing in α, then (c −
αvl(A∗))(
vmax−vminvl(A∗)−vmin
)n−1
is decreasing in α since (n−1)
(vl(A∗)−vmin)is positive and increasing in α. When
(c−αvl(A∗))(
vmax−vminvl(A∗)−vmin
)n−1
is decreasing in α, A(vl) = (c−αvl(A∗))(
vmax−vminvl(A∗)−vmin
)n−1
+ αvl −R
is decreasing in α. As a result, A∗(n,vmin, vmax,R,α, c) is decreasing in α.
The seeker’s profit, Π(A), is increasing in α, for any reward A≥ 0, since the expected number of
participants, k(n,A) = n(1−FU(vl(A))), is increasing in α as vl(A) is decreasing in α. Similarly,
Π(A) is increasing in R for any reward A≥ 0.
b) The expected number of solvers participating in the tournament is k(n,A∗) = n(1 −
FU(vl(A∗))). Now, vl(A∗) is decreasing in α since
d2Π(vl)dvldα
=1
FU(vl)n−1
(vl
vl− vmin
− 1)− 1 < 0. (A27)
and since Π(vl) is concave in vl.
c) As n→∞, the probability of a solver winning the tournament approaches 0. As a result, the
profits of all solvers approach limn→∞ πi(vi) = αvi− c. When α > cvmax
, there is positive probability
that a solver will have positive profit, and, hence, will join the tournament. Therefore, for α > cvmax
,
there will be infinitely many solvers participating in the innovation tournament, despite the seeker
offering no reward (A = 0), i.e., limn→∞ k(n,0) =∞. As a result, the seeker’s profit approaches
vmax.
d) When α ≤ cvmax
, the salvage value is lower than the cost of participation even for a solver
with a submission of the highest possible quality (vmax). Therefore, when the number of potential
solvers is large (n→∞), all solvers participating in the tournament receive negative profit, i.e.,
limn→∞ π(vi) < 0 for vi ∈ [vmin, vmax). As a result, limn→∞ k(n,A∗) = 0 and limn→∞Π(A∗) = 0.
The reward corresponding to the participation threshold vl, A(vl), is increasing and convex in
the number of potential solvers n. The expected quality of the winning submission for a given
Clemons, Pac and Savin Designing Innovation Tournaments 7
threshold vl, E[maxi=1,...,k(vl) vi
]= vmax − vmax−vmin
k(n,A(vl))+1, is increasing and concave in the number of
potential solvers, n. Therefore, for every threshold vl < vmax, there exists a finite n(α) such that
A(vl) exceeds the expected quality of the winning submission. The optimal expected profit of the
seeker, Π(A∗), is increasing in α and R, which implies that the threshold n(α) is increasing in R
and α.
¤