Uncertain contest success function

European Journal of Political Economy 33 (2014) 134–148

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Uncertain contest success function Martin Grossmann 1 Department of Business Administration, University of Zurich, Plattenstrasse 14, 8032 Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 16 February 2012 Received in revised form 6 November 2013 Accepted 18 November 2013 Available online 25 November 2013 JEL Classification: C72 D72 D81 Keywords: Contest Uncertainty Rent-seeking

a b s t r a c t In this article, contestants play with a certain probability in Contest A and with the complementary probability in Contest B. This situation is called contest uncertainty. In both contests, effort is additively distorted by a contest noise parameter which affects the sensitivity of the contest success function (CSF). In Contest A (B), this parameter is linearly added to (subtracted from) effort. We analyze the interaction of contest uncertainty and contest noise on contestant behavior and profit. For symmetric contestants, contest noise has an ambiguous effect on effort and profit. We show that more contest uncertainty can imply greater effort. Furthermore, an introduction of an infinitesimal degree of contest uncertainty can have a large impact on effort and profit. Based on the analysis, this article presents the contest organizer's incentive to manipulate the degree of uncertainty in the contest. For profit or effort maximization, the contest organizer should always eliminate any uncertainty. If contestants are asymmetric, more contest noise increases effort as well as competitive balance if both Contests A and B have the same probability of occurrence. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Contests are characterized by agents eliciting (financial) effort to win a contest prize. Examples include patent races, political campaigns, lobbying, sports contests, and arms races in military conflicts. In the contest literature, it is usually assumed that contestants know exactly the characteristics of the contest in which they are involved. That means that the contestants have full information regarding the technology and structure of the contest success function (CSF) and they are able to anticipate their probability of winning the contest, subject to their opponents' behavior. In this paper, we assume that two contestants are uncertain about which contest they are involved in. The contestants play with some probability of being in Contest A and with the complementary probability of being in Contest B. We denote this randomness as the contest uncertainty. In the model, the classical ratio form of the CSF is applied. In the CSF for both Contests A and B, a noise parameter is added linearly to the contestant effort where noise positively (negatively) affects effort in Contest A (B). Thus, the contestant effort is either strengthened or alleviated. We call this noise parameter contest noise. This paper analyzes the consequences of the contest uncertainty and noise on contestant behavior. According to the previous argumentation, the competition between the two contestants is interpreted as follows: there are two different contests, A and B, and the contestants are unsure about which contest they are involved in. However, the setting can be interpreted alternatively: there is only one contest in which contestants are involved, namely a proportional-prize contest, but contest noise distorts the CSF of this contest.2

1

Tel.: +41 44 634 5315; fax +41 44 634 5329. E-mail address: [email protected]. I am grateful to an anonymous referee who pointed out this interpretation. In a proportional-prize contest, the prize is shared by contestants in proportion to their investments (e.g., Long and Vousden, 1987). An experimental study of proportional-prize contests is provided by Cason et al. (2010); see also Dechenaux et al. (2012) for an extensive survey of the basic structures of contests and an excellent review of the corresponding experimental studies. 2

0176-2680/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejpoleco.2013.11.004

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

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This model can be applied in different situations. First, suppose that two athletes exercise with different levels of intensity for a competition. For the athletes it remains difficult to estimate how an additional hour of training will change their winning percentage in the competition. Second, two countries preparing for a war might be uncertain about the impact of additional military investment on their probability of winning in the conflict. Third, during their studies students may be uncertain about their future career choice and the impact of their investment in the job market. Depending on the particular sector and job area, they can be confronted with a highly professional human resources department. For instance, many well-respected consultancy and banking companies have sophisticated assessment centers trying to eliminate luck as much as possible from the application and selection process. On the other hand, an application to a smaller company or a start-up firm may be considered less professionally due to limited capacity, and therefore luck may be a more pronounced feature of the application and selection process. In this article, we will show that contest uncertainty as well as contest noise is crucial for contestants' behavior. In particular, when Contest B is more likely, then the efforts are greater and profits are lower for symmetric contestants. Moreover, greater contest noise has an ambiguous effect on efforts and profits. Furthermore, more contest uncertainty can increase efforts and the introduction of an infinitesimal element of contest uncertainty can have a large impact on contestant behavior. For asymmetric contestants, greater contest noise increases efforts as well as competitive balance if Contests A and B have the same probability. In addition, this article considers the implications for a contest designer. The next subsection briefly summarizes the results from the related literature and highlights the differences with this paper's approach.

1.1. Related literature Uncertainty has been introduced into the contest literature in different forms. In a seminal article, Lazear and Rosen (1981) analyze a rank-order tournament in which contestant effort is influenced by noise. They show that more noise leads to reduced efforts by risk-averse contestants.3 Nalebuff and Stiglitz (1983) also consider a rank-order tournament. They analyze the optimal design of contests with imperfect information. In their model, noise only affects contestant behavior in the case of risk aversion. One important branch of the contest literature focuses on uncertain contest prize valuations. The models used to explore this differ with respect to the information structure, the asymmetry and the distribution of the prize valuations, and the timing of contestant choice.4 Hurley and Shogren (1998b) analyze a contest with asymmetric information. They show that a contestant without information about the prize valuation of his/her opponent reduces effort as the contest becomes more risky. The opponent increases (decreases) his/her effort accordingly if he/she is – ex ante – the underdog (favorite) in the contest.5 Malueg and Yates (2004) consider a contest in which the players have private information concerning their valuation of the contest prize. However, the contestants' prize valuations can be correlated, and Malueg and Yates derive the critical conditions for which a Bayesian equilibrium exists. Wärneryd (2003) analyzes a two-player contest in which the contest prize is the same for both players but the specific value is uncertain. He shows that the expenditures are lower if only one player is uncertain about the prize, compared with a situation in which either both players are uncertain or neither of the two players is uncertain. Linster (1993) discusses the impact of uncertainty in a dynamic Stackelberg rent-seeking game in which the first player is unsure about the second player's type. Another interesting analysis is presented by Ludwig (2012), who provides a comparison of four contest settings. Contestant types can be privately or publicly observed and the game can be played simultaneously or sequentially.6 This article is also related to those by Baik and Shogren (1995) and Garakani and Gurtler (2011). Baik and Shogren (1995) analyze a contest in which the contestants are uncertain about their relative abilities.7 Thus, the contestants know their own ability but they do not know about their opponents' ability. They find that uncertainty increases or decreases effort depending on how relative abilities influence the marginal effect of effort on the CSF. Furthermore, they analyze the contestants' incentive to elicit effort by spying on their opponents' to learn about their ability. Garakani and Gurtler (2011), on the other hand, analyze the incentives for an omniscient contest organizer to deliver information to the contestants regarding their opponents' ability. The ratio form of the CSF is commonly applied in the contest literature. An interesting branch of the contest literature deals with the stochastic foundation of the ratio form. McFadden (1973) derives the logit form of the CSF on the basis of two assumptions: (i) effective effort depends linearly on individual effort and noise; and (ii) noise is distributed according to the extreme value distribution. Jia (2008) makes two different assumptions to obtain the ratio form of the CSF: (i) the contestants'

3 Sheremeta et al. (2009) and Sheremeta et al. (2012) analyze effort noise within the framework by Tullock (1980). In contrast to Lazear and Rosen (1981), noise is multiplicatively introduced (instead of additively) into their model. They conclude from their model that more noise decreases effort. 4 Nitzan (1994) presents alternative ways of modeling rent-seeking contests. He discusses risk aversion, uncertain prizes, and asymmetry in contests. Instead of incomplete information regarding the prize valuation, Fey (2008) and Ryvkin (2010) consider a contest in which players have private information regarding their effort costs. 5 See also Hurley and Shogren (1998a). 6 While the above-mentioned articles focus on an analysis of optimal behavior within a given uncertain environment, one could also ask: what is the optimal degree of risk? In a recent work, Kräkel (2008) discusses optimal contestant risk-taking behavior in asymmetric contests (see also Kräkel and Sliwka, 2004). He identifies two decisive effects for risk taking: (i) an effort effect (risk affects effort incentives), and (ii) a likelihood effect (risk affects the winning probabilities). 7 Kräkel (2012) also considers a model in which the contestants have different abilities. In this paper, however, there is no allowance for uncertainty. The author shows that the contestant with the lowest ability can overcompensate for the weakness by eliciting greater effort.

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effective effort is determined by effort, which is multiplicatively associated with a noise variable, and simultaneously (ii) this noise follows an inverse exponential distribution (see also Jia and Skaperdas, 2012; Jia et al., 2013). The main difference between the contest literature and the model presented in this article is as follows: while many of the above-mentioned authors analyze the consequences of individual effort distortions with full knowledge of the characteristics of the CSF, we introduce uncertainty on an aggregate level whereby players are uncertain about the characteristics of the CSF itself. Therefore, the source of uncertainty is different and requires a separate analysis, as presented in this paper. 1.2. Outline The paper is structured as follows. Section 2 presents the assumptions used in the model. After that, we present three possible degenerated contests in Section 3. Section 4 analyzes the nondegenerated contest. First, we consider the behavior of contestants with symmetric prize valuations in which the likelihood of Contests A and B can differ. After that, asymmetric prize valuations are introduced. However, we will assume that Contests A and B are subject to the same probability of occurrence. Finally, a short summary and conclusion are provided in Section 5. 2. Assumptions Two risk-neutral contestants i and j (i, j ∈ {1, 2} and i ≠ j) elicit nonnegative efforts xi and xj. Henceforth, we only present contestant i's setting due to the symmetry of contestant j. The contestants' relative effective efforts determine the probability of winning the contest. However, the contest itself is a random variable since the contestants are unsure of the type of contest in which they are playing. 2.1. Nondegenerated contest In a nondegenerated contest, the contestants expect with probability λ ∈ (0,1) participation in Contest A with the contest success function (CSF)
 A pi xi þ
; x j þ
¼

xi þ
xi þ
þ x j þ


and with the complementary probability 1 − λ participation in Contest B with the CSF
 B pi xi −
; x j −
¼

xi −
xi −
þ x j −


where
∈ ℝ+. Thus, the contest-specific parameter
is introduced linearly to contestants' effort in Contests A and B. Contestant i's contest prize valuation is Vi ∈ ℝ+. We call contestant i the favorite (underdog) when Vi N Vj (Vi b Vj). If the contestants are symmetric we define V ≡ Vi = Vj. The marginal effort costs are constant and normalized to 1. Contestant i maximizes the expected profit
 E½πi  ¼ P i xi ; x j ;
; λ V i −xi where




P i xi ; x j ;
; λ ≡

8 <λ

xi þ
xi −
þ ð1−λÞ ; xi þ
þ x j þ
xi −
þ x j −
: 0; else

ð1Þ

for

xi −
N0 ∧ x j −
N0

is called the nested CSF representing the randomness of the nondegenerated contest with λ ∈ (0, 1). Note that the nested CSF depends on whether xi −
and xj −
are positive.8 Contestants participate in the competition if the expected profit is nonnegative; thus, we assume an outside option of zero value. 2.2. Degenerated contest In a degenerated contest, λ ∈ {0,1} and/or
= 0. In these cases there is, de facto, no uncertainty. If λ ∈ {0,1} then the contestants know with probability 1 in which contest they are involved. On the other hand, if
= 0 but λ ∈ (0,1), then the contestants are unsure in which of the two contests they are taking part. However, since the two contests are identical for
= 0,

8 It makes sense to assume that contestant i has to elicit a larger effort than
in order to participate in the contest with Pi N 0 as
can be considered as the fixed cost of participation in Contest B. We will prove this result later. The stronger assumption is that the opponent simultaneously has to elicit a larger effort than
for Pi N 0. This simplification can be justified by arguing that no contest exists such that Pi = 0 if the opponent's effort is too low. In the absence of a competition, both contestants cannot win the prize.

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

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no “true” uncertainty is present in this case. Therefore, uncertainty is produced only if a positive value of
is interacting with an interior λ ∈ (0, 1). 2.3. Interpretation of parameters
and λ A value of λ closer to 0.5 implies a higher uncertainty in which contest the competitors are involved. The uncertainty of the nested contest is maximized for λ = 0.5.9 In this case, the contestants participate with probability 0.5 in Contests A and B respectively. Therefore, we call λ the contest uncertainty parameter. On the other hand, we call
the contest noise parameter, similar to Amegashie (2006).10 A larger
implies that the difference in the sensitivity of the individual CSFs increases – the sensitivity of the CSF in Contest A (B) decreases (increases) – and therefore contestants' uncertainty regarding the sensitivity of the realized CSF increases in the nondegenerated contest.11 Thus, if the contest is not degenerated, a larger value of
represents an additional source for contestants' uncertainty in the contest. On the other hand, contest noise
can be interpreted as follows: in a proportional-prize contest, the CSF pi(xi, xj) = xi ∕ (xi + xj) represents the share of the prize that contestant i receives. In this case, the introduction of
in the CSF represents noise in the proportional-prize contest. In this paper, we will discuss how the combination of the two elements
and λ affects uncertainty, and their impact on contestant behavior. In the following sections, we will show that a convex combination of two contests can imply a discontinuity in the convex combination of the results in the isolated (degenerated) contests. 3. Degenerated contest In this section, we first consider the three possibilities for a degenerated contest, i.e., λ = 1, λ = 0, and
= 0 separately. The next section then derives the results for a nondegenerated contest. The reason for this procedure is that the results of the degenerated contest are not necessarily a special case of the nondegenerated contest. 3.1. Case λ = 1 Both contestants expect Contest A with probability 1. Therefore, contestant i maximizes E[πi] = (xi +
)/(xi +
+ xj +
) Vi − xi subject to xi ⩾ 0. According to Amegashie (2006), the following effort xAi and expected profit E[πAi ] of contestant i result in equilibrium: h i Vi V j V 3i A A xi ¼
2 −
and E πi ¼
2 þ
Vi þ V j Vi þ V j 2

As shown by Amegashie (2006), contest noise has a negative impact on effort and a positive effect on expected profits. By this assumption, effort xi has to be nonnegative. Therefore, there is an upper bound on the noise parameter. In a symmetric contest, xAi ⩾ 0 if V/4 ⩾
. Note that the effective effort xi +
used in the CSF of Contest A is positive, too. However, since Contest B has zero weight in this degenerated contest, xi −
does not have to be positive. The expected profit is positive for all
⩾ 0. 3.2. Case λ = 0 Both contestants expect Contest B with probability 1. Therefore, contestant i maximizes E[πi] = (xi −
)/(xi −
+ xj −
) Vi − xi subject to xi N
. Then, the following effort xBi and expected profit E[πBi ] of contestant i result in equilibrium: h i Vi V j V 3i B B xi ¼
2 þ
and E πi ¼
2 −
Vi þ V j Vi þ V j 2

In contrast to the previous case, contest noise
has a positive impact on effort and a negative effect on expected profit for λ = 0. Note that in equilibrium, contestant i elicits an effort level which ensures that effort xi as well as effective effort xi −
are positive. However, contestant i only participates in Contest B if the expected profit is nonnegative, i.e., V3i /(Vi + Vj)2 ⩾
. For symmetric contestants the participation constraint is V/4 ⩾
.

9 For xi ≠ xj, it is easy to show that the conditional variance of the nested CSF V(Pi|ϵ,xi,xj) is maximized for λ = 0.5. A previous version of this paper based on the assumption that λ = 0.5. I would like to thank an anonymous referee who suggested generalizing the model assuming that λ ∈ (0,1). 10 Amegashie (2006) introduced the noise
in a degenerated contest with λ = 1 (see also Dasgupta and Nti, 1998). He shows that
can be interpreted as the sensitivity of the CSF. A larger value of
implies a lower sensitivity of the CSF regarding different effort levels chosen by contestants. This CSF was axiomatized by Rai and Sarin (2009). 11 It is easy to show that the conditional variance of the nested CSF V(Pi|ϵ,xi,xj) is increasing in
for xi ≠ xj.

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M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

It will be important later to understand the economic mechanism behind the result in Contest B in detail. Note that the optimization problem can be rewritten as follows: defining effective effort in Contest B as yi ≡ xi −
, contestant i maximizes E[πi] = yi/(yi + yj)Vi − yi −
with respect to effective effort yi. In this context,
can be interpreted as a fixed cost which has to be paid by contestant i in order to obtain positive effective effort. A larger value of
means a larger fixed cost. Then, it is not astonishing that, one-to-one,
increases effort xBi and decreases the expected profit E[πBi ] in equilibrium. As long as this fixed cost is not too large contestant i has an incentive to elicit effort strictly greater than
and he/she participates in the contest. 3.3. Case
= 0 Contestant i maximizes the expected profit E[πi] = xi/(xi + xj)Vi − xi with respect to xi independently of the value of λ if
= 0. In equilibrium, contestant i's optimal effort xCi and expected profit E[πCi ] are as follows12: V 2i V j C xi ¼
2 Vi þ V j

and

h i C E πi ¼


V 3i Vi þ V j

2 :

In all of the three degenerated cases, the contestant with the larger prize valuation has a larger probability of winning the contest. 4. Contest uncertainty and noise 4.1. Symmetric contestants In this subsection we assume that contestants are symmetric and the nested contest is not degenerated, i.e., λ ∈ (0,1) and
N 0. Then, the first-order condition of Eq. (1) for contestant i is derived as follows: xj þ
x j −
λ
2 V þ ð1−λÞ
2 V ¼ 1 xi þ x j þ 2
xi þ x j −2


ð2Þ

The analysis of contestant i's first-order condition (2) and a symmetrical first-order condition for contestant j leads to the following proposition: Proposition 1. For symmetric contestants with 0 b
⩽ V/4 and λ ∈ (0,1), (i) a unique equilibrium exists, in which

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V þ ðV þ 8
Þ2 −32
Vλ =8N
and  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (iii) each contestant has expected profit E½π ∗  ¼ 3V− ðV þ 8
Þ2 −32
Vλ =8N0. (ii) each contestant elicits positive effort x∗ ¼

Proof. See Appendix A. ■ This proposition demonstrates that a unique symmetric equilibrium always exists in a symmetric contest for all possible λ ∈ (0,1) as long as 0 b
⩽ V/4.13 In this equilibrium, contestants have the incentive to elicit larger efforts than
such that effective efforts used in Contests A and B are positive.14 As the next step, we derive the comparative statics of the results in Proposition 1 with respect to λ and
. We start with the effect of the parameter representing the contest uncertainty: Proposition 2. For symmetric contestants with 0 b
⩽ V/4 and λ ∈ (0,1), a larger λ unambiguously (i) decreases individual effort as well as aggregate effort; and (ii) increases expected profit.

12

For instance see Nti (1999), who discusses the comparative statics of this result extensively. The condition 0 b
⩽ V/4 guarantees positive profits for all λ ∈ (0,1) such that both contestants participate in the contest. 14 It is important to understand that the result x∗ N
is caused by incentives and not by the restriction in the nested CSF that effective efforts have to be positive. Thus, x∗ is not a corner solution to this restriction. However, we had to exclude a second solution for effort in equilibrium since this was smaller than
(see Appendix A). 13

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

Proof. See Appendix B.

139

■

The effect of the parameter representing the contest noise is as follows: ∗

Proposition 3. For symmetric contestants with 0 b
⩽ V/4, λ ∈ (0,1), and λ ≡ 0:5 þ 4
=V, a larger
∗

∗

(i) increases (decreases) individual effort as well as aggregate effort for λ b λ (λ N λ ); and ∗ ∗ (ii) increases (decreases) expected profit for λ N λ (λ b λ ).

Proof. See Appendix C.

■

From the results of the degenerated contests, Proposition 1, Proposition 2, and Proposition 3, the following four corollaries can immediately be derived: Corollary 1. For symmetric contestants, (i) expected individual and aggregate profits are maximized for λ = 1 and
= V/4; and (ii) individual and aggregate efforts are maximized for λ = 0 and
= V/4.

Corollary 1 shows that a contest organizer always eliminates any contest uncertainty if contestants' profits or efforts are maximized. However, the two objectives (profit and effort maximization) cannot be achieved together since profit maximization (effort maximization) requires λ = 1 (λ = 0). In either case, the contest organizer should maximize the contest noise
. According to result (i) of Corollary 1 contestant i has expected profit E[πi] = V/2 in equilibrium for λ = 1 and
= V/4.15 Note that in this degenerated contest there is no contest uncertainty. Contestants know for sure that
enters the CSF with a positive sign. Moreover, choosing
= V/4 is the maximum possible
guaranteeing nonnegative efforts. The intuition of this result is that a large
implies low effort incentives and therefore low effort costs such that profits are maximized. According to result (ii) of Corollary 1, effort maximization requires the elimination of any contest uncertainty and the maximization of contest noise, viz., λ = 0 and
= V/4. Then, the contestants know for sure that
enters the CSF with a negative sign. For λ = 0 and
= V/4, contestant i elicits effort xi = V/2 in equilibrium. 16 Choosing
= V/4 is the maximum possible
guaranteeing nonnegative profits. As discussed in the last section,
can be interpreted as a fixed cost a contestant has to pay in order to participate in the contest. We have shown that contestants participate in this contest if V/4 ⩾
and they always have an incentive to elicit a positive effective effort x i −
. Choosing the largest possible
(guaranteeing nonnegative profits) implies that the contestants' incentives for making efforts are maximized. Corollary 2. For symmetric contestants, greater contest uncertainty, i.e., a value of λ closer to 0.5, can result in increased effort. Suppose that λ N 0.5. A marginal decrease in λ implies a higher contest uncertainty. Nonetheless, individual and aggregate effort unambiguously increases as ∂x∗/∂λ b 0 in the nondegenerated contest. The reason for this result is that a lower value of λ increases the probability of Contest B where effort incentives are higher. Therefore, the underlying reason for this result is not a higher uncertainty per se, but rather a simultaneous effect of a higher λ on the sensitivity of the nested CSF. Corollary 3. For symmetric contestants, more contest noise, i.e., a larger value of
, unambiguously increases contestants' effort if λ = 0.5. In the degenerated Contest A (B), we know that effort is unambiguously decreasing (increasing) with contest noise. Therefore, one would intuitively expect that in a nondegenerated contest more contest noise should have a positive effect on effort if 0 b λ b 0.5, a negative effect on effort if 0.5 b λ b 1 and no influence on effort if λ = 0.5. However, efforts unambiguously ∗ increase for λ = 0.5 since λ N 0.5 according to Proposition 3. The intuition of this result is as follows: ceteris paribus, a higher
has a positive and negative effect on contestant i's marginal revenue of effort. However, the net effect is positive for λ = 0.5. Since 15

Note that expected profit E[πi] = V/2 in the degenerated contest with λ = 1 is larger than the expected profit in the nondegenerated contest since  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 V=2 N 3V− ðV þ 8
Þ −32
Vλ =8⇔V N− ðV þ 8
Þ −32
Vλ

where the term inside the root on the right-hand side is positive for all λ∈ (0,1). 16 Note that effort xi = V/2 in the degenerated contest with λ = 0 is larger than the largest possible effort in the nondegenerated contest if  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V=2 N V þ ðV þ 8
Þ2 −32
Vλ =8⇔8ðV þ 2
ÞðV−4
ÞN −32
Vλ: Note that the left-hand side 8(V + 2
)(V − 4
) ⩾ 0 since V ⩾ 4
and the right-hand side − 32
Vλ is negative in the nondegenerated contest with λ ∈ (0,1) and
N 0.

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M. Grossmann / European Journal of Political Economy 33 (2014) 134–148


does not affect the marginal costs, contestant i increases his/her optimal choice of effort. Formally, contestant i's marginal revenue of effort MRi is an increasing function in
for
N 0. The following derivation highlights this characteristic for symmetric prize valuations and λ = 0.5: ∂MRi −2 −2 ¼ −ð1=8Þðx þ
Þ V þ ð1=8Þðx−
Þ V N0: j ∂
xi ¼x j ≡ x;λ¼0:5 |{z} |{z} b0

N0

Note that the second term of the last derivative dominates the first term due to the additive characteristic of the CSF with respect to
. Therefore, contestant i and symmetrically contestant j play more aggressively with a larger
. We will prove in the next subsection that this result even holds for asymmetric contestants. Corollary 4. For symmetric contestants and V/8 b
⩽ V/4, (i) a larger
unambiguously increases effort; (ii) the introduction of an infinitesimal level of contest uncertainty can imply a jump discontinuity in effort and expected profit. We have seen in Corollary 3 that the effect of contest noise on effort is not canceled out for λ = 0.5. Nonetheless, one would expect that the effort incentives of Contest A should dominate effort incentives in contest B coercively on a higher level of λ, i.e., effort should be decreasing with contest noise for a large contest uncertainty. However, result (i) in Corollary 4 demonstrates that ∗ contestants unambiguously increase their effort if the contest noise is great, viz., V/8 b
⩽ V/4, such that λ N 1 (see also Proposition 3). In this case, the effort incentives of Contest B always dominate the effort incentives of Contest A, such that effort is unambiguously increasing with contest noise. For result (ii), suppose that the value of
is sufficiently large and λ = 1 such that contestants participate in Contest A with probability 1. A marginal reduction in λ means that contestants have to consider a small probability participating in Contest B. Ex ante, one might guess that efforts marginally increase and expected profits marginally decrease according to the results in the degenerated contests. However, the introduction of this small probability shifts efforts and profits discretely. Thus, the introduction of a small contest uncertainty has a large impact on contestant behavior. For
= V/4, for instance, contestant i's effort equals xAi = 0 and expected profit is E[πAi ] = V/2 in Contest A (where λ = 1). However, we obtain lim x∗ ¼ V=4 and λ→1

lim E½π∗  ¼ V=4 in the nondegenerated contest. The explanation of this discontinuity is as follows: for a sufficiently large contest

λ→1

noise an infinitesimal probability of a realization of Contest B means that contestants have an incentive to pay more than the fixed cost
.17 Therefore, effort discretely increases and consequently expected profit decreases. Note that this jump only happens around λ = 1 in the presence of a large contest noise. However, there is a discontinuity neither in the case of λ = 0 nor in the case of a small contest noise (
b V/8) combined with λ = 1. Fig. 1a and b shows the discontinuity for effort and expected profit around λ = 1 in the case of V = 10 and
= 2.5. If strong contest noise is present, then the discrete jump around λ = 1 provides an additional argument for the previous claim derived in Corollary 1 that a contest designer should eliminate any contest uncertainty to maximize profits. Even an infinitesimal uncertainty can have a large impact on contestant behavior and profit. The previous literature has predominantly suggested that more uncertainty leads to lower effort (e.g., Lazear and Rosen, 1981; Sheremeta et al., 2009).18 According to our model, we have to differentiate between the sources of uncertainty. Corollary 2 shows that a higher contest uncertainty can lead to more aggressive behavior by both contestants whenever a higher uncertainty is related to the sensitivity of the nested CSF. On the other hand, a larger contest noise can decrease or increase effort depending on the interaction of contest noise and contest uncertainty according to Proposition 3, Corollaries 3 and 4. 4.2. Asymmetric contestants In this subsection, we consider contestants with asymmetric prize valuations but with symmetrical weights for Contests A and B. For λ = 0.5, the first-order condition of Eq. (1) for contestant i is derived as follows: xj þ
x j −
0:5
2 V i þ 0:5
2 V i ¼ 1: xi þ x j þ 2
xi þ x j −2


ð3Þ

The analysis of contestant i's first-order condition (3) and the symmetrical first-order condition for contestant j leads to the following proposition: Proposition 4. Suppose that expected profits are positive in equilibrium; for asymmetric contestants with 10Vj N Vi N 0.1Vj and symmetric weights λ = 0.5,

17 18

On the other hand, contestants' effort is lower than
for V/8 b
⩽ V/4 in the degenerated Contest A. Sheremeta et al. (2009) show that contestants elicit a lower effort if noise is introduced multiplicatively in their model.

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

141

Fig. 1. a) Effort discontinuity; b) Profit discontinuity.

(i) a unique equilibrium exists, in which (ii) the favorite elicits a larger effort than the underdog. Proof. See Appendix D.

■

Part (i) of this proposition demonstrates that a unique equilibrium always exists for asymmetric contestants with symmetric probabilities for Contests A and B as long as contestants are not too asymmetric.19 It is intuitive that the favorite makes a greater effort than the underdog. As the next step, we derive the comparative statics of the results of Proposition 4 with respect to the contest noise. In order to do so, we first calculate the total differential for Eq. (3). Afterwards, Cramer's Rule is applied to derive the following proposition. Proposition 5. Suppose that expected profits are positive in equilibrium; for asymmetric contestants with 10Vj N Vi N 0.1Vj and symmetric weights λ = 0.5, a larger contest noise
increases the individual efforts in equilibrium. Proof. See Appendix E. Proposition 5 generalizes Corollary 3 of the previous subsection for asymmetric contestants. Thus, the previous result did not depend on the assumption regarding the symmetry of contestants. While Nti (1999) finds a non-monotonic effect for the return to scale parameter in a standard contest on effort, there is a negative effect of contest noise on effort in the degenerated contest

19 Note that 10Vj N Vi N 0.1Vj is a sufficient condition guaranteeing that contestants elicit larger efforts than ϵ in equilibrium. Simulations show that expected profits are positive if the asymmetry and the contest noise are not too great. These simulations and a complex condition (not presented in the paper) guaranteeing positive expected profits are available upon request.

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M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

Fig. 2. a) Contestant i’s effort; b) Contestant j’s effort.

with λ = 1 in Amegashie (2006). In our model, the convex combination of two similarly weighted contests, A and B, imply that more contest noise has an unambiguously positive effect. Thus, Contest B overcompensates for the effort incentives of Contest A. Next, we provide a simulation of the model in order to gain additional insights into contestant behavior. In particular, we analyze the effect of contest noise and asymmetry on competitive balance and expected profits. The expected competitive balance (CB) in the contest is derived and measured as follows: xi þ ∈ xi −∈ 1 xi þ ∈ þ x j þ ∈ 1 xi −∈−x j −∈ 1 xi þ ∈ 1 xi −∈ xi x j ¼ ∈ þ ¼ 2 þ ¼ CB ¼ xj þ ∈ x j −∈ 2 2 2 x j þ ∈ 2 x j −∈ x j −∈2 xi þ ∈ þ x j þ ∈ xi −∈−x j −∈ Note that the contest is balanced if CB is close to 1. CB is larger (lower) than 1 if contestant i (j) elicits larger efforts than contestant j (i). In the simulation, the parameters are set as follows: Vi ∈ [2,4], Vj = 3,
∈ {0.1,0.2,0.3}. Thus, we consider the

Fig. 3. Competitive balance.

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

143

Fig. 4. a) Contestant i’s expected profit; b) Contestant j’s expected profit.

effect of asymmetry regarding the contestants' prize valuation. For Vi N 3 (Vi b 3), contestant i (j) is the favorite. If Vi = 3, then the contestants have symmetric valuations. Fig. 2a and b replicates the following result by Nti (1999).20 The underdog's effort is increasing in his own prize valuation but decreasing in the favorite's prize valuation. The favorite's effort is increasing in his own prize valuation and in the valuation of the underdog. This is the reason why the effort of contestant i is an increasing function in Vi, and contestant j's effort is a concave function in Vi. Moreover, Fig. 2a and b confirms Proposition 5. More contest noise increases the effort of both contestants. In addition, the competitive balance increases unambiguously with greater contest noise (see Fig. 3). Thus, contest noise to some extent cancels out the differences between the asymmetric contestants. The reason for this result is as follows: more contest noise increases the effort incentives for both contestants. However, the increase is relatively larger for the underdog, such that the competitive balance increases. Furthermore, Fig. 4a and b shows that the expected profits decrease with greater contest noise. The rationale for this result is as follows: greater contest noise increases effort incentives. More effort implies larger costs, while the contest prize remains fixed. Therefore, the expected profits for both contestants decrease. To sum up, we obtain the following additional numerical result based on the previous simulation: Result. For asymmetric contestants and symmetric weights λ = 0.5, more contest noise increases the competitive balance and decreases the expected profits of both contestants.

5. Conclusion In this paper, we introduce uncertainty into a contest with respect to the parametrization of the CSF on an aggregate level. The contestants are unsure which CSF determines their winning probability. With some probability, contestants are involved in Contest A and with the complementary probability in Contest B with a different CSF. This uncertainty is called contest uncertainty.

20

See also Malueg and Yates (2005).

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M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

In our model, effort is additively distorted by a contest noise parameter. In Contest A (B), this parameter is linearly added to (subtracted from) effort. A larger contest noise increases the difference in the CSF sensitivities of the two contests, A and B. This paper complements the literature on contest theory by introducing uncertainty differently. Hitherto, this literature, in assessing the effect of uncertainty, has concentrated mainly on contestants' uncertainty regarding single variables like the valuation of the contest prize or the effectivity of their effort. The general insight obtained from this previous work is that greater uncertainty is related to decreased effort. We derive the following main results for symmetric contestants: a unique equilibrium exists if contest noise is not too large. An increase in the likelihood of Contest B increases effort and decreases profits since the incentives for effort are larger in Contest B. Contest noise has an ambiguous effect on effort and profit. This effect depends on the likelihood of Contests A and B and on the level of contest noise. An infinitesimal amount of uncertainty can have large effects on contestant behavior. Introducing uncertainty can discretely change contestant effort and profit if contest noise is relatively large. According to the model, the contest organizer, in maximizing effort or profit, should eliminate any contest uncertainty. A contest designer may have, at least in some types of contest, the power to manipulate the degree of uncertainty. In public procurement, for instance, a government can issue a project invitation to firms, and a government can decrease firms' uncertainty by defining – in advance – the procedure to be used for assessing the proposals submitted by firms in detail. By doing this, the government reduces the uncertainty for the firms regarding the CSF sensitivity. In rent-seeking contests, for instance, aggregate efforts in relation to the contest prize measure the degree of rent dissipation. According to our model, the combination of contest uncertainty and contest noise determines the size of rent dissipation. Rent dissipation is maximized if contest noise is large and contestants are certain of participating in Contest B with probability 1. In this case, the contest organizer should eliminate any contest uncertainty, but at the same time he/she should ensure a high sensitivity of the CSF by choosing a large contest noise parameter. In order to maximize profits, the contest organizer should provide contest A with probability 1 and he/she should choose a large contest noise, i.e., a low sensitivity of the CSF. If contestants are not too asymmetric, we conclude that a unique equilibrium exists. In this case, more contest noise increases efforts as well as competitive balance if Contests A and B both have the same probability of occurrence. This article shows that it is important to identify the underlying source of uncertainty in contests in order to derive the appropriate policy implications. An interesting topic for future research would be a thorough analysis of the interaction effects between the uncertain variables in the model (e.g., uncertain prize valuations) and the general uncertainty regarding the contest structure (i.e., the parametrization of the CSF). Acknowledgments This research was conducted while the author was a visiting researcher at the Northwestern University (Kellogg School of Management). The author thanks Rakesh Vohra and the Department of Managerial Economics and Decision Sciences for their support. Especially helpful comments were given by Andreas Hefti, Thành Nguyen, and Ron Siegel. I would like to thank two anonymous referees who significantly helped to improve the paper with their suggestions. Financial support was provided by the Stiefel-Zangger Foundation of the University of Zurich. Responsibility for any errors rests with the author. Appendix A. Proof of Proposition 1 In Part (1), we first derive the unique symmetric solution. Part (2) demonstrates that the second-order condition is satisfied for the symmetric solution. In Part (3), we show that additional asymmetric solutions do not exist. Part (4) derives the expected profit of contestants in equilibrium. Part (1). In equilibrium, efforts are symmetric such that xi = xj ≡ x. Thus, contestant i's first-order condition (2) can be written as follows: V λV ð1−λÞV þ ¼ 1⇔x1;2 ¼ 4ðx þ
Þ 4ðx−
Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðV þ 8
Þ −32
Vλ 8

:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV þ 8
Þ2 −32
Vλ N0 for λ ∈ (0,1) and therefore two real solutions exist.  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The first solution x1 ¼ V þ ðV þ 8
Þ2 −32
Vλ =8 is larger than
such that effective efforts are positive in the nested CSF:

It is easy to see that

ðV þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðV þ 8
Þ −32
VλÞ=8 N
⇔ 1 N λ

Thus, contestants have an incentive to elicit positive effective efforts.

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

145

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For the second solution, x2 ¼ V− ðV þ 8
Þ2 −32
Vλ =8, this condition is not satisfied since 21  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V− ðV þ 8
Þ −32
Vλ =8 N
⇔ 1 b λ: In this case, the assumptions regarding positive effective efforts in the nested contest are violated. Therefore, a unique and symmetric solution x∗ ≡ x1 N
exists in which both contestants elicit effort 

x ¼

 Vþ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðV þ 8
Þ −32
Vλ =8:

Part (2). The second-order condition of contestant i is as follows: xj þ
x j −
−2λ
3 V−2ð1−λÞ
3 Vb0 xi þ x j þ 2
xi þ x j −2
In the symmetric solution x∗ the second-order condition is satisfied since 



x þ
x −
λV ð1−λÞV −2λ   3 V−2ð1−λÞ   3 V b 0 ⇔ −   2 −   2 b 0 8 x þ
8 x −
4 x þ
4 x −
for λ ∈ (0,1).

Part (3). The first-order condition of contestant i with X ≡ xi + xj can be rewritten as follows: λ

xj þ
2

ðX þ 2
Þ

V þ ð1−λÞ

2

x j −
2

ðX−2
Þ

V ¼ 1

2

λVðx j þ
ÞðX−2
Þ þ ð1−λÞVðx j −
ÞðX þ 2
Þ

⇔ 2 2 ¼ ðX þ 2
Þ ðX−2
Þ :

Analogously for contestant j: 2

2

2

2

λV ðxi þ
ÞðX−2
Þ þ ð1−λÞV ðxi −
ÞðX þ 2
Þ ¼ ðX þ 2
Þ ðX−2
Þ : Combining the two first-order conditions we get:

 2 2 λ x j −xi ðX−2
Þ ¼ ð1−λÞ xi −x j ðX þ 2
Þ According to the last equation, it is easy to see by a proof of contradiction that xi ≠ xj is not possible in equilibrium. Therefore, asymmetric solutions do not exist.

Part (4). In equilibrium, contestant i's and j's expected profit E[π∗i ] = E[π∗j ] ≡ E[π∗] is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3V− ðV þ 8
Þ −32
Vλ 

x þ
x −
 : E π ¼λ V þ ð1−λÞ V−x ¼   8 2ðx þ
Þ 2ðx −
Þ Note that expected profit is positive if 3V N

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðV þ 8
Þ −32
Vλ ⇔ 8ðV þ 2
ÞðV−4
Þ þ 32
Vλ N0:

Thus, if
⩽ V/4 then contestants' profit is positive for all λ ∈ (0,1) and
N 0 and both contestants participate in the contest.

21

Note that the second solution x2 is even negative for 4
N V(2λ − 1).

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M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

Appendix B. Proof of Proposition 2 Part (i). The effect of λ on effort is: ∂x 2
V ¼ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0 ∂λ 2 ðV þ 8
Þ −32
Vλ Therefore, a larger λ decreases individual as well as aggregate effort for
N 0. Part (ii). The effect of λ on expected profit is: 

∂E½π  2
V ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N0: ∂λ 2 ðV þ 8
Þ −32
Vλ Therefore, a larger λ increases the expected profit for
N 0. Appendix C. Proof of Proposition 3 Part (i). The effect of
on effort is as follows: ∂x V þ 8
−2Vλ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T0 ∂
2 ðV þ 8
Þ −32
Vλ The critical value of λ for which the derivative is ∂x∗/∂
equals zero is defined as λ∗ ≡ 0.5 + 4
/V.22 Then, it is easy to see that





∂x  N 0⇔λbλ ∂


and

∂x  b0⇔λ N λ : ∂


Part (ii). The effect of
on expected profit is:  ∂E½π  V þ 8
−2Vλ ¼ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T0 ∂
2 ðV þ 8
Þ −32
Vλ ∗

The critical value of λ for which the derivative is ∂E[π∗]/∂ϵ equals zero is the same λ as above. Then, it is easy to see that ∂E½π   N 0⇔λ N λ ∂


and

∂E½π    b0⇔λbλ : ∂


Appendix D. Proof of Proposition 4 In Part (i), we first derive the unique solution and check the second-order condition. In Part (ii), we show that the favorite elicits greater effort than the underdog. Part (i). Contestant i's first-order condition (3) can be written as follows: xj þ
2

ðX þ 2
Þ

þ

x j −
2

ðX−2
Þ

¼

2 Vi

⇔x j ðX Þ ¼

2

2

2 ðX−2
Þ ðX þ 2
Þ 8X
2 þ V i ðX−2
Þ2 þ ðX þ 2
Þ2 ðX−2
Þ2 þ ðX þ 2
Þ2

ð4Þ

with X ≡ xi + xj. Therefore, we get an implicit solution xj(X) for xj. Replacing xi by X − xj in the numerator and replacing xi + xj by

22

∗

Note that λ N 0.5.

M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

147

X in the denominator, contestant j's first-order condition can be written as follows: X−x j þ
2

ðX þ 2
Þ

þ

X−x j −
ðX−2
Þ

2

¼

2 : Vj

Next, we replace xj in the last equation by the implicit solution (4). After some manipulation, we obtain 0¼X

2

Vi þ V j 2 Vi þ V j −X−4
: V iV j V iV j

The last equation shows that there are at most two solutions for X. Both solutions are real while one of those solutions is negative and the other one is positive. The positive solution is denoted by X∗∗ and defined as

X



¼

V iV j þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ffi V 2i V 2j þ 16 V i þ V j
2
 : 2 Vi þ V j

Using X∗∗ in the implicit Eq. (4) one can derive the unique optimal effort x∗∗ j by contestant j. Then, it is easy to derive the ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ optimal effort x∗∗ i by contestant i using the condition xi = X − xj . Note that effort xi N 0 and xj N 0. However, efforts should be larger than contest noise according to the nested CSF. ∗∗ One can show that Vi N Vj/10 is a sufficient condition for x∗∗ i N
. On the other hand, Vj N Vi/10 is a sufficient condition for xj N
. Thus, if the contestants' prize valuations are not too asymmetric a unique equilibrium exists. The second-order condition of contestant i is as follows: xj þ
x j −
−0:5
3 V i −0:5
3 V i b0: xi þ x j þ 2
xi þ x j −2
∗∗ It is easy to see that the second-order condition is satisfied for x∗∗ i N
and xj N
.

Part (ii). Combining contestant i's first-order condition (3) with contestant j's first-order condition, we obtain 0 B @


1 0 xj þ


1

x j −


xi þ
xi −
C B C Vj 2 þ
2 A=@
2 þ
2 A ¼ : Vi xi þ x j þ 2
xi þ x j −2
xi þ x j þ 2
xi þ x j −2


According to the last equation, it is easy to see that the favorite elicits greater effort than the underdog in equilibrium. Appendix E. Proof of Proposition 5 First, we slightly modify contestant i's first-order condition (3) as well as the symmetrical first-order condition of contestant j such that we get

i:


xj þ


x j −
2 2 þ
2 ¼ Vi xi þ x j þ 2
xi þ x j −2


ð5Þ

and

j:


xi þ
xi þ x j þ 2


2 þ


xi −
2 2 ¼ : Vj xi þ x j −2


Totally differentiating Eqs. (5) and (6), we get   # 2 x −3x −2
−x þ 3x −2
 3 3" x j þ
x j −
xi −x j xi −x j i j i j þ þ 3 3 7 7 6− 6 −2 A3 þ B3 A3 B3 B 6 A  7 dxi ¼ 6   7d
4 4 x j −xi x j −xi x j −3xi −2
−x j þ 3xi −2
5 xi þ
xi −
5 dx j þ −2 þ 3 − þ A3 B A3 B3 A3 B3 2

ð6Þ

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M. Grossmann / European Journal of Political Economy 33 (2014) 134–148

with A ≡ xi + xj + 2
and B ≡ xi + xj − 2
. Applying Cramer's Rule, we derive
2 3 dxi 8x j xi þ x j
þ 32xi
¼ 
N0  2 2 d
2 xi þ x j þ 4


and


2 3 dx j 8xi xi þ x j
þ 32x j
¼ 
N0  2 2 d
2 xi þ x j þ 4


for
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