Strategy and incentive in contest and tournament

STRATEGY AND INCENTIVE IN CONTEST AND TOURNAMENT LIU XUYUAN (B.A., 2008, Xi'an Jiaotong University, M.A.,2010, York University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2014 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. _____________ Liu Xuyuan 10 Oct 2014 To my parents and my husband Ma Ting, for their love, support, and encouragement Acknowledgements I have bene…ted greatly from the guidance and support of many people over the past four years. Without their love and help, this thesis would not have been possible. On this occasion, I would like to express my gratitude toward them. In the …rst place, I am particularly indebted to my main supervisor Prof. Qiang Fu for his support and help in the past few years. He showed great kindness and patience to me, and guided me through each step of research. My work has bene…tted enormously from his comments and critique. Moreover, he is a great life mentor, and always gives invaluable suggestions on academic and non-academic matters. I am always feeling lucky and honorable to be supervised by him. I would also like to sincerely thank my co-supervisor, Prof. Jingfeng Lu, for his supervision and support in various ways. Prof. Lu’s comprehensive knowledge and incisive insight on contest theory as well as his uncompromising and prudent attitude toward research and insistence on quality works have deeply in‡uenced me and will de…nitely bene…t my future study. I would like to thank my committee members, Prof. Parimal Bag, Prof. Ko Chiu Yu, who spent their valuable time providing me with insightful feedback i Acknowledgements ii and support. All of them are very encouraging, patient, gracious, and helpful to my research. The same level of appreciation also goes to Prof. Songfa Zhong and Prof. Roy Chen, who provided me with the knowledge of behavioral economics. I also wish to thank Prof. Jie Gong for her sel‡ess teaching skill sharing. I would also like to thank Prof. Shin-Hwan Chiang and Prof. Xianghong Li for academic guidance during my master program in York University. Thanks are also due to my friends and colleagues at the department of Economics for their thoughtful suggestions and comments, especially to Dr. Li Li, Dr. Zhang Shen, Dr. Wan Jing, Dr. Miao Bing, Dr. Hong Bei, Dr. Li Jingping, Mun Lai Yoke, Jiang Wei, Jiang Yushi, Sun Yifei, Shen Bo, Yang Guangpu, Zeng Ting, Cai Xiqian, Lu Yunfeng, Jiangtao Li and many others. Finally, to my parents and my husband, all I can say is that it is your unconditional love that gives me the courage and strength to face the challenges and di¢ culties in pursuing my dreams. Thanks for your acceptance and endless support to the choices I make all the time. Contents Acknowledgements i Summary v List of Tables viii List of Figures ix Chapter 1. The E¤ort-Maximizing Contest with Heterogeneous Prizes 1.1. Introduction 1.2. The Analysis 1.3. Concluding Remarks 15 Chapter 2. Optimal Prize Rationing In all pay auction With Incomplete Information 17 2.1. Introduction 17 2.2. Literature Review 20 2.3. Model Setup 21 2.4. Prize Rationing 24 2.5. Concluding Remarks 36 Chapter 3. R&D Contests with Imperfect Quality Signals 3.1. Introduction 38 38 iii CONTENTS iv 3.2. Relation to Literature 45 3.3. Setup and Preliminaries 48 3.4. Analysis in A Stylized Setting 53 3.5. Discussion 67 3.6. Conclusion 78 Chapter 4. Pro rata or Fixed : The Price Strategy of Crowdsourcing Contest Intermediaries 80 4.1. Introduction 80 4.2. Related Literature 84 4.3. Setup and Preliminaries 85 4.4. Price Strategy Analysis 87 4.5. Discussion 96 4.6. Conclusion 99 References 101 Appendix . A: Proofs of Chapter 105 A1: Properties of Gamma and Beta Functions 105 A2: Proofs 105 Appendix . B: Proofs of Chapter 126 B1: Proofs 126 B2: Comparative Static Summary 147 Appendix . C: Proofs of Chapter 149 Summary My dissertation contains four essays on strategy and incentive in contest and tournament. The …rst chapter studies optimal contest design in environments where the organizer commits to allocate a given set of heterogeneous prizes and each contestant wins one and only one prize. Contestants’ e¤ort e¢ ciencies are their private information and they are ex ante symmetric. We …nd that under the regularity condition of increasing virtual e¤ort e¢ ciency, a grand all pay auction where all contestants compete together for all available prizes maximizes the total expected e¤ort, where higher e¤ort wins higher prize. Chapter investigates optimal prize rationing rule in all pay auctions with incomplete information. The contest organizer has a set of …xed indivisible prizes, such as a number of certi…cates, which he can utilize to incentivize the agents to exert productive e¤ort. Each agent at most wins one prize. We study the optimal number of prizes the organizer should grant in order to induce higher e¤ort from agents. The analysis shows that under the regularity condition of increasing virtual e¤ort e¢ ciency, marginal contribution of extra prize decreases and it is never optimal for the organizer to award a prize beyond the point where marginal revenue turns negative. Moreover, we found that for a family of Beta distributions, the optimal number of prizes v SUMMARY vi weakly increases with the expansion of contestant pool and the improvement of contestant quality no matter organizer concerns with expected total e¤ort or expected highest e¤ort. In addition, compared to expected total e¤ort maximization, expected highest e¤ort maximization requires a smaller set of prizes to be awarded. Chapter is about R&D contests with imperfect quality signals. A buyer searches for an innovative product and invites two …rms to participate in a contest. A …rm’s bid has two components: the intrinsic quality of the product and the price. The two …rms can be heterogeneous, in that one bears a higher marginal cost in producing higher quality. Firms simultaneously commit to their R&D e¤orts to improve their products’ quality and submit their price o¤ers. The buyer inspects …rms’submissions and awards the contract to the …rm that provides the highest perceived buyer surplus. The buyer is unable to precisely observe the true quality of a …rm’s product. Instead, she receives a noisy signal of the actual quality o¤ered by each …rm. Due to the noise, she may award the contract to a …rm that submits a less competitive bid. With a nontrivial noisy term in her quality evaluation, a pure-strategy equilibrium exists in the game. Our analysis depicts the main properties of the equilibrium and characterizes …rms’responses to the noise in the quality-evaluation process. We show that the noise exercises substantial impact on …rms’behavior in structuring their bids, i.e., the trade-o¤ between high quality and low price. We compare the ex ante expected surplus in this game to that of a benchmark model in which quality can be perfectly observed. We …nd that a SUMMARY vii nonexpert buyer can, paradoxically, obtain higher expected surplus in spite of possible errors it may commit ex post. Chapter focuses on the behavioral strategy of contest intermediaries. Leveraging the wisdom of crowds in the form of a contest is not rare among modern entrepreneurs. The rise in popularity of contest intermediaries based on easy access to a large pool of the talented around the world and greatly reduced risk for entrepreneurs. We examined the behavioral strategy of a monopoly intermediary who is a pro…t maximizer. We found that given an entrepreneur’s …xed crowdsourcing budget, the intermediary will invariantly favor a …xed pricing scheme, whether entrepreneur sets her quality standard exogenously or endogenously. a) If s2 = 0, …rm will bid (x1 = 12 ; p1 = a) optimally. Given s1 = a, 4c …rm will deviate to s2 = 2a to get positive pro…t 2a. Accordingly, s2 = can never be an equilibrium. b) If a > s2 > 0, …rm still wants to deviate to s2 = to minimize the loss, which is contrary with part a). Therefore, a > s2 > can not be an equilibrium. c) If a < s2 < , given s1 = s2 +a , …rm will switch to s2 = s1 +a+" > to get positive payo¤, which means a < s2 < is also not an equilibrium. By the same token, we can prove the case s1 a. Therefore, the pure s2 = equilibrium strategies can not exist when < a < . 8c Proof of Proposition 3.12. Proof. We now verify s1 = a, s2 = is an equilibrium when 8c a . If s2 = 0, …rm chooses x1 and p1 to maximize its expected utility as follows: M ax p1 fp1; x1 g s1 + a 2a x21 = (x1 st: x1 > ; a s1 ) s1 + a 2a s1 a. From …rst order condition, …rm will bid x1 = pro…t a s1 = ( 16a )s2 2a + ( 8a )s + . 16 As x21 , 8c s1 +a , 4a a and the corresponding , 16a2 2a and a, …rm will choose to bid s1 = a. Given s1 = a, …rm maximizes 145 its expected utility as following: M ax p2 fp2; x2 g s2 2a cx22 = (x2 st: x2 > ; a Similarly, …rm will bid x2 = )s2 . 2a Since 8c a , 16a2 c s2 4ac 2a s2 ) s2 s2 2a cx22 , a. optimally, then her pro…t is and a s2 = ( 16a1 c a, …rm will choose to bid s2 = 0. Accordingly, s1 = a; s2 = is an equilibrium when 146 8c a . B2: Comparative Static Summary Table 3.1: Noise E¤ect Variable Case @ p < @a Case when < c < c 0, and 18 < a < a @ p < @a p when when c @ p @a c 0, or a a q win @q win > @a x1 @x1 < @a @x1 < @a x2 @x2 > @a @x2 > @a @p1 @a p ( when c 0, 1+c 12c or a @q win < @a 1) q 1+c2 c2 12 @p1 @a p when < c < 23 ( 1) p1 @p1 > @a 0, and 12 < q a @p1 > @a < 1+c 12c 1+c2 c2 when 0, 1+c 12c + @p2 > @a @p2 > @a s1 @s1 < @a @s1 < @a and 12 s2 @s2 @a q , or a > 1+c 12c 12 q 1+c 12c 2+3c c2 c2 when c 1+c2 c2 when < c < 0, 12 when a > 1+c 12c q 0, 1+c2 + 12 c2 p2 @s2 @a a q @s2 @a @s2 @a 2+3c c2 c2 147 , when c > , when < c < Table 3.2: E¢ ciency Di¤erential E¤ect Variable p Case @ p < @c 0;when < c < @ p @c 0;when c Case 2a (18a 1)(4a 1) 2a (18a 1)(4a 1) when < c < @q win @c 0, and p q win @ p @c 0;when c 2a (18a 1)(4a 1) 2a (18a 1)(4a 1) 14 13 18c+17c2 24 @q win @c 0;when c 6a 72a2 30a+1 14 13 @q win @c > 0;when c > 6a 72a2 30a+1 > 0, or a > p 0;when < c < 1+5c 24c @c @x2 @c p1 @p1 < @c @p1 @c p2 @p2 > @c @p2 @c s1 @s1 < @c @s1 @c s2 @s2 > @c @s2 @c 148 APPENDIX C: Proofs of Chapter Proof of Lemma 4.1 Proof. When there are N prizes, suppose a monotone bidding strategy exists. Then the payo¤ function = N X for each contestant is vi F N i (e (e))[1 F (e (e))]i i=1 = N X1 vN F N i i (e (e))[1 F (e (e))] + vN (1 i=1 = N X1 N X1 F N i (e (e))[1 F (e (e))]i ) i=1 (vi vN )F N i (e (e))[1 F (e (e))]i . i=1 Adopting the methodology of Moldovanu and Sela (2006) leads to the following equilibrium bidding strategy in an all-pay auction with incomplete information: e( ) = N X1 s=1 (vs vN ) Z tdFsN (t) = e( ) = (vs vN ) s=1 As N X N X (vs vN ) s=1 149 Z tdFsN (t), Z tdFsN (t), which implies that e( ) = N X vs s=1 = N X vs s=1 = N X vs s=1 = N X s=1 vs Z Z Z Z tdFsN (t) N Z X vN tdFsN (t) s=1 N X1 Z tdFsN (t)(t) tdFsN (t) vN f vN f tdFsN (t) + s=1 N X1 Z tdFsN (t)(t) + s=1 Z Z tdFNN (t)g td(1 N X1 FsN (t))g s=1 tdFsN (t). Proof of Lemma 4.2 Proof. The intermediary will choose pex f to maximize the expected payo¤ as follows, = Mexaxpex f (1 pex f pf = pex f (1 = pex f (1 F N ( (pex f ))) ( )N ) (N s0 N 1)(I pex f ) ). The necessary condition implies that the …rst-order condition should be satis…ed, which means (N s0 N I = 0, 1)(I pex f ) 150 and pf ex p =I N s0 I(N N 1) . Proof of Lemma 4.3 Proof. For the sake of convenience, we …rst denote I 1+w NN ( )N )v= I 1+wq and ( )N = N ( N )N = q, as v = N q. N The intermediary will choose w and s to maximize expected pro…t as follows: pex r w = M ax w,s Z e( )F N ( )N dF ( ) + ws [1 ( )N ]. This objective function of the intermediary is equivalent to pex r = M ax w, Iw N 2N [( ) + wq 2N ( )2N ]d + Iw N N ( ) [1 + wq N s20 (1 + wq)2 Iw ] + q[1 I2 + wq N Iw [q = M ax w,q + wq 2(N 1) N N ( )N ] q]. The necessary condition implies the satisfaction of …rst-order conditions: d p dq d p dw (C.1) =0 =0 . Also, the commission payment cannot exceed the entrepreneur’s budget. Solving (C.1) and (4.4) together, we get w = (N 1)s0 I (N 151 p N (N 1)s20 1)s30 I , s = vq = Iq = s0 . 1+w q Proof of Lemma 4.4 Proof. Each contestant will choose an e¤ort level e to maximize expected payo¤ as follows: M ax vF N e e (e (e)) . Meanwhile, there exists a critical type of contestant e( ) = 0; otherwise, e( ) r, such that if < r, 0. The optimal strategy for player should satisfy the …rst-order condition, which means 1) F N e ( ) = v(N ( ). Combining with the boundary condition, we get e( ) = e( r ) + = v rF Z N vydF N r ( r) + v 152 Z (y) ydF N r (y). Proof of Lemma 4.5 Proof. The platform will charge pen f to maximize expected pro…t, as follows: = Men axpen f (1 pen f F N ( r (pen f ))) pf r = Men axpen f (1 I pf pen f ). The satisfaction of the …rst-order condition indicates that p pfen = I rI. Proof of Lemma 4.6 N ( N Proof. De…ne )N = q; then the expression for the prize can be simpli- …ed as v= Meanwhile, as ( )N = N q N I w Nr . + wq and ( r )N = r V , when the intermediary maximizes expected pro…t pen r = M ax w w,s Z e( )F N ( )N dF ( ) + ws [1 ( )N ]. r This is equivalent to pex r = M axw w,p Z r r N I w Nr [ + N N + wq N ]N 153 N r I w Nr Nq d +w( + q)(1 ). N + wq N The necessary conditions indicates that d p dq d p dw (C.2) =0 =0 . Combining (4.4) and (C.2), we get w = p I Ir r ; s = r. Proof of Lemma 4.7 Proof. We prove this by contradiction. Suppose r < s0 , then the prize left p for contestants is v1 = r I. However, if entrepreneur sets r = s0 , then p v2 = s0 I.In other words, v2 > v1 . If r < s0 , the expected entrepreneur’s pro…t will be f1 =t Z [ N 1p r + r I N N N N ]N d I[1 F N ( r )]. If r = s0 , the expected pro…t the entrepreneur can earn is f2 =t Z [ r N 1p + s0 I N N N N ]N Clearly, f1 as [1 F N ( r )] > [1 < F N ( )]. 154 f2 , d I[1 F N ( )]. Therefore, for each r < s0 , the entrepreneur’s pro…t will be dominated by choosing s0 . In other words, r s0 : Proof of Lemma 4.9 Proof. In this case, the entrepreneur’s objective function is equivalent to N 1p N r + rI ]N N d r N N r p p r(r + N r rI + I N I) p M ax . r 2N I M ax = Z [ Also, r > s0 , so the …rst-order condition yields p + 3N + + 3N I; s0 g. r = maxf 9(1 + N )2 Proof of Lemma 4.10 Proof. If an entrepreneur holds a contest independently, she will maximize pro…t as follows, M ax r where v = p r I and r Z r N r + v N N [ N ]N N = ( rv ) N . Then the …rst-order condition demonstrates that r (1 + N ) = 1, v 155 d , which implies that r = p r I . 1+N As p + 3N + + 3N r = maxf I, s0 g, 9(1 + N )2 p p p +4 1+3N I I s , ) s and r = s0 , accordingly, r r . if 5+3N9(1+N 0 )2 1+N p p p p +4 1+3N 2 +4 1+3N 5+3N +4 1+3N However, if 5+3N9(1+N > s0 , r = 5+3N9(1+N I, r = I. )2 )2 3(1+N )2 Meanwhile, p p p + 3N + + 3N + 3N + + 3N = 3(1 + N )2 9(1 + N )2 = p < 1, p + 3N + + 3N which means r r . Proof of Lemma 4.11 Proof. Since s0 = (1 when B( ) > 0, )V A( ) + V B( ), A( ), the entrepreneur will never use the second prize, so if the contestants’ e¤ort level will uniformly decrease with the second prize. Now let us consider the situation when B( ) > A( ), which can be 156 written as follows: B( ) A( ) = (N e where F (e) = N . N 1) Z Z tF N (t)[N N F (t)]dF (t) FN = eF N (t)[N ( )[N N F (t)]dF (t) N F ( )], versa. Moreover, the critical type of contestant varies with d d = (1 )(N N , N Clearly, when B( ) > A( ), F ( ) 1) [F ( )]N and vice as A( ) B( ) f ( ) + (N 1) [F ( )]N [(1 N )F ( ) + N 2]f ( ) And the …rst order condition is d d p =V Z 0( ) (1 A( )gN F N ( )dF ( ) )A( ) + B( )gN [F ( )]N f(1 =V[ fB( ) Z 0( ) fB( ) )(N Suppose F (e) = A( )gN F N 1) [F ( )]N N N f ( 0) ( )dF ( ) d d s0 N [F ( )]N A( ) B( ) f ( ) + (N 1) [F ( )]N . We will discuss the sign of 157 d d p f ( 0) [(1 N )F ( ) + N using two cases. 2]f ( ) ]. . F (e), Case 1: if F ( ) Z = = 0( Z 0( Z 0( = NF N NF N ( ) ( ) ) Z Z NF N ) e + e ) A( )gN F N fB( ) e 0( NF ( ) N 0N Z Z ( ) ) eN F ( ) Z ( )dF ( ) (N 1)t[F (t)]N [(N 2) N F (t)]dF (t)dF ( ) (N 1)t[F (t)]N [(N 2) N F (t)]dF (t)dF ( ) (N 1)t[F (t)]N [(N 2) N F (t)]dF (t)dF ( ) 2) N F (t)]dF (t)dF ( ) Z 1)[F (t)]N (N [(N 2F 2N +1 ( )N + F 2N ( )(2N 2F ( )(2N 1) F 2( 0) 1) 2F 2N +1 ( )N + F 2N ( )(2N 2F ( )(2N 1) 1) . and B( ) A( ) = 0( ) (N 1)t[F (t)]N N (N [F ( )] F (e), Case 2: if F ( ) Z Z fB( ) eN F ( ) A( )gN F N [(N 2) N F ( )). ( )dF ( ) 2F 2N +1 ( )N + F 2N ( )(2N 2F ( )(2N 1) and 158 N F (t)]dF (t) 1) , B( ) e A( ) Z (N = e[F ( )]N 1)t[F (t)]N (N [(N 2) N F (t)]dF (t) N F ( )). Therefore, in case 1, d d p F 2( 0) 0N f +[F ( )]2N 2F 2N +1 ( )N + F 2N ( )(2N 2F ( )(2N 1) (N N F ( )) (1 (1 )(N 1) )F ( ) + (1 F ( ))F ( )(N 1)F ( ) + (N 1)[N (N 1) g. 1)F ( )] Meanwhile, we de…ne g(F ( ); ) = as dg(F ( d ); ) (1 (1 )(N )F ( ) + (1 F ( ))F ( )(N 1)F ( ) + (N 1)[N (N 0, max g(F ( ), ) = g(F ( ), 21 ) and ) = g( NN , 12 ) = 2(N 1) (N 2)N < .Then N dg(F ( ); ) dF ( ) 1) , 1)F ( )] 0, max g(F ( ), we can further narrow the range of 159 d d p by applying the above results: d d p 0N f F 2( 0) +[F ( )]2N 2(2N (N N F ( )) 1) N f + 2F 2N [F ( )]2N 2(2N 2F 2N +1 ( )N + F 2N ( )(2N 2F ( )(2N 1) (N +[F ( )]2N (N k(F ( ); N ) = 1+2F 2N ( )N F 2N when F ( ) = 19 37N +18N , 9N (2N 1) N N g F 2N ( )N N F ( )) 1) N f + 2F 2N ( )N 1) 2(2N F 2N ( )(2N 1) + 1)g ( )(2N 1) N F ( )) 20g. De…ne ( )(2N 1)+[F ( )]2N k gets the maximum value k (N ) , and (N N F ( )) 20, dk (N ) dN < 0. Therefore, the maximum value of k (N ) is obtained when k = 3. As k (3) < 0, d d p 0, the same proof applies as for the case 2. When N = 2, B( ) < A( ) and two prizes will never be optimal. 160 [...]... bidder has a single unit demand,5 Ando (2004) demonstrate that in an all pay auction setting with homogenous prizes and private e¤ort e¢ ciency, a grand contest dominates any equally divided contest consisting of identical subcontests in terms of expected total e¤ort The dominance result of Ando (2004) requires a regular condition of increasing virtual e¤ort e¢ ciency.6 Fu and Lu (2009) investigate whether... constraint is Medical/Doctor certi…cation exams and Bar exams which aim at controlling professional standard of candidates as well as improving their quali…cation in the related industry Despite lacking direct monetary incentives in such certi…cation exam, the improved future prospects from gaining the certi…cation is the key motivation for increasing candidate’ e¤orts s Even though the funding pressure... and private e¤ort e¢ ciency, a grand contest dominates any equally divided contest consisting of identical subcontests in terms of total e¤ort The dominance result of Ando (2004) requires a regular condition of increasing virtual e¤ort e¢ ciency Fu and Lu (2009) investigate whether an arbitrary division of a grand contest can induce higher total e¤ort in a nested Tullock contest framework with a given... (1998), and Glazer and Hassin (1988) focus on multi-prize all 20 pay auction with heterogeneous prizes under the institution of complete information In contrast, Moldovanu and Sela (2001, 2006) examine an incomplete information contest and establish the unique symmetric (increasing) pure strategy bidding equilibrium for an all pay auction with heterogeneous prizes Among theses studies, the underline assumption... must be in v, and every element of v must go to one and only one vm When M = 1, the split contest degenerates to the grand contest Proposition 1.1 immediately implies the following two results Corollary 1.1 (Superadditive E¤ort) Suppose the Increasing Virtual E¤ ort E¢ ciency condition holds Within an institution of all pay auction, a grand contest induces more expected total e¤ort than any split contest. .. grand contest can induce higher total e¤ort in a nested Tullock contest framework with a given set of heterogeneous prizes In the analysis of Fu and Lu (2009), contestants’e¤ort e¢ ciency instead is public information They …nd that as long as the impact function is log concave, a grand contest generates more e¤ort than any set of subcontests The optimality of a grand all pay auction established in. .. provides incentive for productive e¤ort, contest has generated great interest among academic researchers Most of the large economic literature on contests has focused on the issue of optimal contest design given the …xed prizes budget or prizes set, including the work Barut and Kovenock (1998), Glazer and Hassin (1988), and Moldovanu and Sela (2001, 2006) However, there are many real world cases where a contest. .. the grand prize splits into identical single prizes for identical subcontests Speci…cally, they …nd that as long as the cost function is linear or concave, the dominance of the grand contest always holds regardless of the distributions of e¤ort e¢ ciency One should note that in Moldovanu and Sela (2006) the prizes di¤er across the grand and divided contests, which di¤erentiates their study from Ando... unlimited indivisible prizes set and each contestant wins at most one prize For tractability, we simply assume that the demand of whole market is nearly perfectly elastic and awarding more prizes will not alter the market value of these prizes We …nd that under the regularity conditions, marginal contribution of extra prize is always diminishing and organizer should stop at a point where marginal bene…t... total expected e¤ort inducible when e¤ort is contractable.3 Speci…cally, as we have multiple prizes that are certainly to be distributed and each contestant wins one and only one prize, an arbitrary prize allocation outcome in our setting has to be speci…ed appropriately by a one-to-one matching between prizes and contestants.4 We …nd that under a regularity condition of increasing virtual e¤ort e¢ . Chapter 4 149 Summary My dissertation contains four essays on strategy and incentive in contest and tournament. The …rst chapter studies optimal contest design in environments where the organizer commits. setting with homogenous prizes and private e¤ort e¢ ciency, a grand contest dominates any equally divided contest consisting of identical subcontests in terms of expected total e¤ort. The dominance. whole proving process. 6 Moldovanu and Sela (2006) reveal the dominance of an all pay auction with a single grand prize, if in divided contests the grand prize splits into identical single prizes