The optimal multi-stage contest

Transcription

1 MPRA Munich Persona RePEc Archive The optima muti-stage contest Fu, Qiang and Lu, Jingfeng UNSPECIFIED November 2006 Onine at MPRA Paper No. 946, posted 07. November 2007 / 0:25

2 The Optima Muti-Stage Contest Qiang Fu Nationa University of Singapore Jingfeng Lu Nationa University of Singapore Apri, 2006 Abstract This paper investigates the optima (effort-maximizing) structure of muti-stage sequentiaeimination contests with pooing competition in each stage. We aow the contest organizer to design the contest structure in two arms: contest sequence (the number of stages, and the number of remaining contestants in each stage), and prize aocation. First, we find that the optimaity of winner-take-a (singe fina winner, singe fina prize, no intermediate prizes) is independent of the contest sequence. Second, we show that the more compete the contest sequence is, the more efforts can be induced from the contestants. Therefore, the optima contest eiminates one contestant at each stage unti the finae, whie a singe winner takes over the entire prize purse. Our resuts not ony rationaize various forms of muti-stage contests conducted in reaity, such as the we-known Fox TV show American Ido, but aso shed ight on the design of interna organizationa hierarchy. JEL Nos: C7, D7 Introduction Situations in which economic agents expend costy and non-refundabe resources in order to win a imited number of prizes are fairy ubiquitous. For instance, high schoo students engage in academic efforts to compete for coege admissions. Firms participate in research We are especiay gratefu for the inspiring discussion with Kyung Hwan Baik. We thank Michae Baye, Ivan Png, Indrani Chakaraborty, Benny Modovanu, Lixin Ye, Hongjun Zhong and Jimmy Chan, and a seminar participants at Nationa University of Singapore, and Shanghai University of Finance and Economics for hepfu comments and suggestions. Qiang Fu: Department of Business Poicy, Nationa Univeristy of Singapore, Business Link, SINGA- PORE, 7592; Te: (65) , (65) , bizfq@nus.edu.sg. Jingfeng Lu: Department of Economics, Nationa University of Singapore, 0 Kent Ridge Crescent, SINGAPORE, 9260; Te: (65) , Fax: (65) , Emai: ecsjf@nus.edu.sg.

3 tournament to win a procurement contract of innovation. Cities obby the Internationa Oympic Committee in order to host Oympic Games. In modeing contests, the economic iterature conventionay assumes that the contest asts for ony one stage, in which each contestant competes against a others for a singe prize ( winner-take-a ). Many contests in reaity, however, ast for more than one stages, and require contestants to repeatedy exert efforts. In the fina stage, the finaists expend resources to compete for the prizes; whie in each preiminary stage, the remaining contestants compete for the tickets to the next round. In most of the contexts, no prize other than the tickets is awarded in the preiminary stages. Numerous rea word contest settings exhibit such a muti-phase sequentia competition structure. One such exampe is the eection of London to host the 202 summer Oympic Game. Whie 9 cities initiay submit appications, ony 5 of them (London, Madrid, Moscow, New York and Paris) are shortisted as the candidates for the fina eection. The 2005 DARPA (Defense Advanced Research Projects Agency) Grand Chaenge, a race among autonomous robots hed in the Mojave Desert aong a route of 32.2 mies, aso exempifies a sequentia competition setting. A tota of 43 teams were seected out of 95 appicants to participate in the NQE (Nationa Quaification Event), which cut the 43 teams down to 23 for the fina race. In the end, Staney, the Stanford Racing Team s autonomous robotic car, competed the course first and earned a $2 miion prize. In research tournaments, the procurement firms seect the few most attractive ideas from a arger poo of innovation proposas, and ony the seected are eigibe for the further race towards successfu innovation (see Fuerton and McAfee, 999). In many Asian countries, in contrast to the K-2 system in U.S., students have to take more than one major screening exams in order to be admitted into coeges. In recruiting new facuty members, economics departments usuay interview a arge group of candidates, but extend on-campus visit invitations to ony a sma number of them. Centra to the contest iterature is the inquiry how the design or the rue of the contest affects the tota efforts contestants expend. As argued by Gradstein and Konrad (999),... the contest structures are the outcome of a carefu design with the view of attaining a variety of objectives, one of which is maximization of efforts by contenders. The efforts of the contestants benefit the contest organizer in many occasions, such as professiona sports, We thank Ivan Png for aerting us of this exampe. 2

4 research tournament, as we as poitica rent-seeking. Foowing this thread of thinking, our paper addresses the issue of optima (effort-maximizing) contest design in a muti-phase sequentia Tuock contest setting with pooing competition in each stage. We recognize that the structure of a muti-stage contest consists of two aspects: the sequence (the number of stages, and the number of remaining contestants in each stage), and the aocation of prizes. Specificay, we answer the foowing questions. First, given the sequence of a muti-stage contest, and a fixed tota of prize purses, how does the contest organizer aocate the prize mass to the recipients in order to maximize the efforts? Does a winnertake-a (singe-winner and singe-prize) contest necessariy dominate a contest that awards intermediate prizes? Second, given the number of participants and the tota vaue of prizes, does a muti-stage contest, which sequentiay eiminates contestants, drive more efforts, as compared to a singe-stage simutaneous contest? What is the optima contest sequence? Third, does the sequence of the contest intertwine with the prize aocation in infuencing the efforts? We investigate the optima contest sequence and the optima prize aocation in a unified framework. We consider a muti-stage contest, in which N identica contestants compete for a fixed prize mass, instead of a given singe prize. Each stage-contest is a pooing contest, in which each remaining contestant competes against a other remaining contestants. In each preiminary stage, a contestant competes not ony for the tickets to the next stage, but aso for nonnegative intermediate prizes. In the fina stage, the remaining contestants compete for nonnegative fina prizes ony. We aow the contest organizer to maximize the tota efforts in two arms: choosing the optima contest sequence and aocating optimay the prize mass. We first show that the optima aocation of the prize mass is independent of the contest sequence structure, and a winner-take-a (singe-prize) contest dominates a other prize aocations. We then show that the more compete the contest sequence is, the more efforts the contestants expend. 2 As a consequence, the optima contest that maximizes the tota efforts is a (N )-stage Pyramid contest that eiminates one contestant at each stage, and a singe fina winner takes over the entire prize purse. To summarize, our study provides rationaes for (i) the muti-stage contest widey observed in reaity; and (ii) the winner-take-a principe commony assumed in modeing rent-seeking competition in a more 2 We wi define the concept of competeness in Section

5 genera setting. The Reation to the Literature Baye, Kovenock and de Vries (993) raise the question why do poiticians frequenty announce that they have narrowed down a set of potentia recipients of a prize to a sate of finaists? They examine the scenario, in which the poitician optimay shortists a set of finaists from a poo of rent seekers (who differ in their vauations for the fina prize) to induce more competition, whie the process of shortisting does not invove rent-seeking activities on the part of contestants. A handfu of papers, in contrast, have modeed the process of shortisting as the preiminary stages of a sequentia contest, which requires contestants to repeatedy expend efforts. For instance, Amegashie (999) considers a two-stage contest. In the first stage of the contest, contestants are divided into groups, and a singe winner stands out from each group to participate in the second-stage (fina) competition. Rosen (986) modes the organizationa hierarchy as a series of pairwise contests: empoyees compete for promotion aong the adder of hierarchy. He shows that a arger top prize increases the overa efforts. Harbaugh and Kumpp (2005), as we as Matros (2005), consider two-stage tournament contests that group contestants in preiminary stages. In contrast to Amegashie (999), they study the optima intertempora effort aocation of the contestants faced with resource constraints. Gradstein (998) aso contributes to this research agenda by comparing a simutaneous contest with a contest that consists of a series of pairwise subcontests. Rosen (986) considers a 2 N contestant N stage sequentia contest: in each stage, two of the remaining contestants are matched into head-to-head confrontation, and the winner survives for the next stage. Yet he does not justify the optimaity of the contest structure. In the sense of endogenizing the contest structure, our paper is more cosey reated to Gradstein and Konrad (999), as we as Modovanu and Sea (2006). Gradstein and Konrad (999) consider muti-stage imperfecty discriminatory contests that group identica contestants in preiminary stages. In contrast to Rosen (986) and Amegashie (999), they aow the contest organizer to fexiby design the contest structure as a matching scheme. They show that the muti-stage contest adopted by Rosen (986) may emerge as the optimum if the contest success function is ess discriminatory. In a perfecty discriminatory contest setting, 4

6 Modovanu and Sea (2006) study the optima contest architecture that specifies whether and how the contestants shoud be spit into subgroups in the first-stage competition. They find that the optimaity of contest architecture depends upon the properties of contestants cost function, as we as the objective of the contest organizer. A singe grand static contest maximizes the expected tota efforts for the case of inear cost of effort. If the effort cost function is convex, a designer can benefit from spitting the contestants into severa subgroups in the preiminary stage of the contest. Our paper differs from these papers in two regards. Firsty, we aow remaining contestants to compete against a others in each stage, instead of matching them into different groups. We consider each stage of competition in the sequentia eimination process as a mutipe-winner mutipe-oser contest (as first suggested by the semina paper of Cark and Riis (996)): the winners are first seected to proceed to the next stage, whie the rest of contestants are eiminated. 3 Amegashie (2000) compares the two foowing ways of shortisting in two-stage contests: pooing (contestants compete against a others in each stage) and grouping (contestants are divided into groups). He shows that the former generates a higher rent-dissipation rate. Fu and Lu (2005) aso provide theoretica evidence supporting that pooing competition generates higher rent-dissipation rate. These studies partiay justify why we adopt a pooing competition in each stage of the game. Our paper is aso reated to Fuerton and McAfee (999) in modeing shortisting. In a two-stage mode, they show that the optima research tournament requires competing companies to participate in an a-pay auction (as a screening scheme) to win the entry, whie ony a subset of firms (the most competitive firms) engage in innovation activities. Secondy, few of these papers aow for fexibe prize aocation in muti-stage contest settings. Most of papers in the contest iterature assume that the prize structure is exogenousy given, and is governed by the winner-take-a principe: a singe fina winner receives a singe indivisibe prize. Besides Rosen (986), the exceptions are Krishna and Morgan (998), Modovanu and Sea (200) and Matros (2005). Krishna and Morgan (998) justify the winner-take-a principe in sma tournaments. Modovanu and Sea (200) con- 3 Cark and Riis (998) suggest an interesting muti-stage contest with a different rue. In their paper, winners are first seected in each stage to receive the stage prizes, but then they have to exit from the contest; whie the osers proceed to the next stage and continue to exert efforts in order to compete for the remaining prizes. 5

7 sider the optima prize aocation in a one-stage perfecty discriminatory contest (a-pay auction). Matros (2005) aows the contest organizer to fexiby aocate his/her budget between a singe winner s prize and osers rewards to maximize the tota efforts in a two-stage contest, and shows that a winner-take-a contest dominates. Our paper proceeds as foows. Section 2 sets up the mode. In Section 3, we first derive the genera soution of equiibrium efforts in a muti-stage muti-prize contest. We then present our resuts on the optima contest structure. In addition, we discuss the impications and appications of these resuts. Section 4 provides a concuding remark. 2 The Mode N( 3) risk-neutra contestants are engaged in a muti-stage sequentia contest with nonnegative intermediate and fina prizes. The contest organizer has a tota budget of Γ 0 for prize aocation. For this moment, we fix the sequence of the contest, i.e. the number of stages and the numbers of survivors in each stage. Let L denote the number of stages in the contest, and N denote the number of contestants in stage {, 2,..., L}. In stage {, 2,..., L }, N contestants participate, and N + of them survive and proceed to the next stage. Ceary, we have N N. There are N nonnegative intermediate prizes W m,m {,...,N }, avaiabe for each stage {, 2,..., L }. AtstageL, N L contestants compete for N L nonnegative fina prizes WL m,m {,...,N L }. 4 The sequence of a given contest is therefore represented by a L-term non-increasing sequence {N, N 2,..., N L }. Ceary, we shoud have N = N N 2... N L. At a stages =, 2,...,L, the remaining contestants simutaneousy exert their nonnegative efforts e i,i=, 2,..., N. The tickets to the next stage and the stage prizes are aocated in a sequentia ottery process as modeed by Cark and Riis (996). Once a contestant is seected, he/she is immediatey removed from the poo, whie the rest of them are eigibe for the next draw. Define Ω m stage, where m {, 2,...,N }.Denote(e,..., ei probabiity that a contestant i Ω m 4 Some fina prizes are aowed to be zero. to be the set of remaining contestants up for the m th draw in,e i+,..., e N ) by e i. The conditiona is seected in the m th draw is then given by 6

8 p(e i, e i ; Ω m )=e i Á X j Ω m e j. () In the case that a contestants up for a draw make zero effort, we assume that the seected is randomy chosen from the poo. Moreover, we assume that if Ω m reduces to a singeton, then the ony contestant is automaticay seected for sure regardess of his effort. At stage {, 2,...,L}, the contestant seected in the m th draw is awarded the prize W m. In addition, at stage {, 2,..., L }, the contestants who are seected in the first N + draws proceed to the ( +) th stage, whie the other N N + contestants are eiminated. XN LX We define Γ to be the sum of prizes awarded in stage, andγ Γ to be m= W m the tota of the prizes in the whoe contest. Denote by V the conditiona (symmetric) equiibrium expected payoff of a representative contestant at stage. For convenience, we define V L+ =0. At stage {, 2,..., L}, a representative contestant i rationay chooses his/her effort e i to maximize his expected payoff NX + V i = [P m (e i, e i )(V + + W m )] + m= XN m=n + + = [P m (e i, e i )W m ] e i, (2) where P m (e i, e i ) is the probabiity that contestant i is seected in the m th draw. Note that P m (e i, e i )= X [Pr(Ω m )Pr(i Ω m )p(e i, e i ; Ω m )],wherepr(ω m ) is the probabiity that the Ω m remaining contestants up for the m-th draw are Ω m,andpr(i Ω m ) is the probabiity that contestant i beongs to Ω m. Since we consider the symmetric equiibrium, we assume a contestants other than i exert the same effort e 0 without oss of generaity. Under this simpification, P m (e i, e i )= (N )! e 0 (N m)! (Πm k= e i +(N ) k)e 0 e i +(N. (3) m)e 0 Denote by e the symmetric equiibrium effort. From (3), when e i = e,i=,..., N, P m(e i,e i ) e i is given by e i P m (e,...,e ) e i = ( m P ) N g N e. (4) 7

9 The first order condition for the interior equiibrium effort e is thus NX + [ P m(e,..., e ) (V + + W m )] + m= (4)and(5)eadto m= e i NX + [( m P N g )(V + + W m )] + XN m=n + + XN m=n + + [ P m(e,..., e ) e i [( m P W m ] =0. (5) N g )W m ] N e =0. (6) Proposition In a symmetric interior equiibrium of the contest in stage, each remaining contestant exerts an effort of e = P N+ m= m P [( N g )(V + + W m )] + XN m=n + + [( m P N g )W m ] N. (7) Proposition, which directy stems from the first order condition (6), gives the interior equiibrium individua effort of each remaining contestant in stage. It stricty increases with the expected future payoff V + as the coefficient of V + is P m N + P m= ( N + P N + N + g N 0. Note that the term m P g N g )= N g stricty increases with m, the index for the order of the draw. Thus, the equiibrium effort e aso increases with the vaue of earier prizes W m (the prizes awarded with earier draws), where m satisfies m P > 0. However,e N g decreases with the vaues of ater prizes W m,wherem satisfies m P < 0. N g Define Φ to be Φ P N + P m= [( m )(V N g ++W m )]+ XN m=n + + [( m P )W m N g ]. The soution for the symmetric equiibrium effort is vaid and satisfies (6) if and ony if Φ 0 hods for {W m } N m=. Otherwise a corner soution appies and the equiibrium effort woud be zero. Φ < 0 mayhappenwhensufficienty arge prizes are awarded for the atest draws. In this case, the contestants prefer not to make positive effort, but to wait for the atest prizes. As a resut, the first order condition for interior soution fais, and the corner soution of zero effort arises. Here and hereafter, we assume Φ 0, and restrict our attention to the (unique) symmetric interior equiibrium. In Section 3.2, we wi show in detai that there is no oss of generaity to focus ony on the prize aocations that ead to interior equiibrium. 8

10 Rearrange the terms in (6), we have N + P N e = N + V + V + Define the tota efforts E N + g N g + Γ XN (W m m= mx k= ). (8) N k + LX N e. In this context, we assume that the tota efforts = accrue to the benefit of the contest organizer. Thus, the contest organizer is to choose the optima sequences {N } L = the tota efforts E, subject to the budget constraint m and prize aocation {W m =,..., N ; =,...,L} to maximize Γ Γ 0. (9) 3 Resuts 3. Preiminary Resuts For convenience, we define Γ L+ =0.Inastage, {, 2,..., L}, every symmetric contestant has the same chance of winning each component of the tota stage-award N + V + + Γ (incuding N stage prizes and N + tickets to the next stage) in a symmetric equiibrium. Therefore, the conditiona equiibrium expected payoff of a representative contestant at stage is V =(N + V + + Γ )/N e,wheree is his/her equiibrium effortatstage. The tota of the N contestants equiibrium expected payoffs can then be written as N V = N + V + + Γ N e, (0) which impies N e = N + V + + Γ N V. () Lemma E = Γ NV. Proof. Summing up () over the L stages gives E = XL N e + N L e L = XL XL (N + V + N V )+ Γ + N L e L = = XL = N L V L NV + Γ + N L e L. (2) = 9

11 Thus, Note that in the ast stage of the contest, i.e. stage L, wehaven L V L = Γ L N L e L. Q.E.D. XL E = Γ L NV + = Γ = Γ NV. (3) Lemmaisfairyintuitive. V represents the conditiona expected payoff a representative contestant receives in the first stage. The contestant does not ony gain from the intermediate prizes awarded in this stage, but aso from the vaues of the tickets to future stages. As a consequence, V is the payoff every contestant expects from the whoe contest in the very beginning of the contest. Thus, NV represents the tota surpus a contestants receive in the contest, which, by the risk neutraity of the contestants, is equivaent to the difference betweenthetotaprizemassγ and the tota efforts E. From (8), the tota equiibrium efforts in each stage are determined by the conditiona expected payoffs of a representative contestant in next stage, and the stage prizes awarded inthecurrentstage. Using(8)and(),wecanwriteinthefoowingLemmacontestants payoffs in the current stage (V ) in terms of the future expected payoff (V + ), and the intermediate prizes of the current stage prizes. N + P Lemma 2 N V = V + N + g N g + XN m= (W m mx k= N ), for {, 2,..., L}. k+ Using Lemma 2, we are abe to recursivey sove for V as in Lemma 3, which states that V can be written as a function of the prizes in the current and a future stages. Lemma 3 V = LX = {( Π j= N j )(Π j= N j+ P N j+ g N j g X (W m ) N m= mx k= )}. N k+ Combining Lemmas and 3, we have the foowing resut. Proposition 2 A N-person L-stage sequentia-eimination contest, with the sequence structure {N } L =,prizeaocation{w m m =,..., N ; =,...,L} and a tota prize purse Γ, induces the tota equiibrium efforts of E = Γ N LX = {(Π j= )(Π j= N j N j+ P N j+ g N j g ) N X m= (W m mx k= )}. (4) N k + 0

12 3.2 The Optima Prize Aocation Lemma 3 shows that given the sequence of the contest, a contestant s equiibrium surpus V competey depends on the aocation of the prize purses. If the contest organizer intends to maximize the tota efforts E, howmanyfina winners shoud be aowed for? Does it pay to create intermediate prizes that are awarded in earier stages? Next, we address how the LX contest organizer optimay aocates his/her tota budget Γ 0 over the N possibe prizes. By Lemma, the optima prize aocation that maximizes the tota efforts must minimize the equiibrium surpus V. Thus, we can focus on the minimization of V to sove the origina effort-maximization probem. Before we proceed, we first carify why there is no oss of generaity if we focus on prize aocations that ead to symmetric interior equiibrium. Reca (6). An interior equiibrium requires Φ P N + P m= [( m )(V XN N g + +W m )]+ [( m P )W m N g ] 0. Suppose m=n + + that the prize aocation makes Φ < 0, which eads to the corner soution, i.e. e =0.First, note that if we aocate the entire Γ to the first draw prize, it must foow that Φ 0. Second, Φ is continuous in {W m } N m=. Thus, the contest organizer can aways shift the prize mass Γ from ater prizes to earier prizes within the stage, and reach an aocation that deivers exacty Φ =0. The first order condition (6) is then reinstated and it renders an interior equiibrium with e =0. Thus, by appropriatey shifting the prize mass from ater prizes to earier prizes, we can aways appy condition (6) and obtain an interior equiibrium, which generates the same outcome as the corner soution equiibrium. Note from (0) that this adjustment of prizes does not ater V as we. As a resut, it does not affect the tota efforts in any other stage of the contest. This means that we can ignore the possibiity of a corner soution, and assume Φ to be nonnegative without oss of generaity. = Theorem ( Winner-take-a ) Given a contest sequence {N } L =, the optima contest prize aocation that maximizes the tota efforts E requires the entire prize purse to be aocatedtothefirst prize in the fina stage, i.e. W L = Γ 0. Proof. The proof consists of three steps. Define for brevity D ( Π j= N j )(Π N j+ P j= N j+ g N j g X (W m ) N m= mx k= ).FromLemma N k+

13 3, we can then write V as the sum of the L separate terms D, {, 2,...,L}, i.e., Step One: W m V = LX D. (5) = =0,ifm>, {, 2,..., L}. For any fixed Γ, {, 2,..., L}, D is minimized if the entire prize purse in stage is mx aocated to the first prize W because increases with m. Thus for the optima prize aocation, we must have W m Step Two: W purse Γ Γ 0. k= N k+ =0, if m>, {, 2,..., L}. =0,for {, 2,..., L }, andw L = Γ, givenafixed tota prize Step one impies that V can be reduced to the foowing form for the optima prize aocation V = LX {( Π j= )(Πj= N j = N j+ P N j+ g N j g )W }. (6) N Next,weshowthatfor {, 2,..., L },ifwemovetheprizemassw to W+, it further reduces V. For this purpose, we need ony to compare the coefficients of W and W+.Thus, we compare (Π j= N j )(Π N j+ P N j+ g j= ) N j g N with ( Π + j= N j+ P N j )(Π N j+ g j= ) N j g N +. Ignoring the common eements, we ony need to compare N to ( N + ) 2 N + P N + g N g < ( N + ) 2 (N + N + N )= N, because, N + g N g < N + N N + P N + g N g. Obviousy,( N + ) 2 if g>0. It foows that the weight on W is stricty greater than that on W+, which impies that shifting the mass of W to W + stricty reduces V. Thus in order to minimize V, a prize mass aocated to prizes in an earier stage ( <L) shoud be reaocated to W L. Given the tota amount of prizes Γ, wethushavew L = Γ. Thus, we show that a winner-take-a contest maximizes the tota efforts for any given contest sequence. Step Three: Γ = Γ 0, i.e., the contest organizer uses up the entire budget on prizes. Based on steps one and two, Proposition 2 eads to that E can be reduced to the foowing form for the optima prize aocation E = Γ[ N( Π L j= )(Π L j= N j N j+ P N j+ g N j g ) N L ]. (7) 2

14 N L Note that N( Π L j= N j )(Π L N j+ P N j+ g j= ) N j g N L = N Π L NL 2 j= [ N j+ P N j+ g ] < N N j N j Π L N j+ g NL 2 j= N j =. ThusE stricty increases with Γ. Therefore, the entire budget Γ 0 shoud be aocated to W L. Q.E.D. Theorem estabishes that if the contestant organizer has the fexibiity to aocate a fixed prize mass, a muti-stage contest that maximizes the tota efforts must combine a the resource into a singe fina prize and reward it to a singe fina winner, regardess of the sequence of the contest. Our resuts therefore provide a rationae for the commony assumed winner-take-a principe in modeing rent-seeking competition. Cark and Riis (996) show that contestants expend more efforts if the contest is governed by a winner-take-a rue than they do if the number of positive prizes exceeds one. Theorem confirmsthisinsight in the context of muti-stage eimination contests. Lemma3showsthatV is a weighted sum of the prizes awarded in a stages. The weights on prizes exhibit two interesting features. On one hand, within a singe stage, an earier prize has a smaer weight. This is due to the fact that a higher probabiity of winning an earier prize demands higher efforts from a contestant. As a resut, within a stage, the contest organizer can increase the efforts by aocating the entire stage purse Γ to the firstdraw prize, i.e. W = Γ. 5 Therefore, we can focus on the contest structure that aocates the prize purse ony to the first-draw prize of each stage. On the other hand, between any two stages, a first-draw prize awarded in a ater stage has a smaer weight. The intuition is that contestants have to (repeatedy) exert their further efforts to win a prize awarded in a ater stage. In other words, a prize at a higher rank of the adder demands more efforts from a contestant. To induce the highest subsequent efforts, a resources shoud then be aocated as one singe prize at the ast stage. Aggregating both the within and between effects eads to the optimaity of winner-take-a in our muti-stage setting. 3.3 The Optima Contest Sequence: Pyramid Contest Having estabished the winner-take-a principe as the optima prize aocation rue in any muti-stage contest, it remains to ask what is the optima sequence of the contest (the number of stages and the number of remaining contestants in each stage). Next, we study 5 This effect is consistent with the winner-take-a principe in Modovanu and Sea (200). 3

15 how the contest organizer optimay chooses the sequence {N } L = to maximize the tota efforts E. By Theorem, the aocation of prize mass is independent of the sequence of the contest, we therefore simpy restrict our attention to the case with a singe prize W L = Γ 0. Rewriting Lemma 3 by setting a prizes other than W L to be zero eads to V =(Π L j= )(Π L j= N j N j+ P N j+ g N j g ) Γ 0 N L. (8) First of a, for mathematica convenience, we consider ony the contest sequences with N L =for the optima sequence without oss of generaity. By the optimaity of winnertake-a principe in prize aocation, a singe winner survives the ast stage of the contest and takes over the entire prize purse. Thus, a L stage contest with N L > is equivaent to a hypothetica (L +) stage contest represented by the sequence {{N } L =, }, i.e.n L+ =. In the hypothetica (L+) stage contest, one contestant is seected at stage L to enter stage L+, but does not receive prize in stage L. But the ast man standing automaticay wins Γ 0 in stage L + without exerting effort. Consequenty, we have e L+ =0and V L = Γ 0,and equation (8) sti appies. We thus consider in the foowing anaysis ony contest sequences with N L =. Second, we assume that the contest sequence {N } L = is stricty decreasing. Equation (6) impies that if N = N +, then eiminating stage does not affect V. Based on the above two resuts, we can search for the optima contest sequence by considering ony the sequences {N } L =, where N = N>N 2 >...>N L =, without oss of generaity. In a L stage contest with a sequence of {N } L = where N L =, suppose there exists J<Lsuch that N J N J+ >. We imagine to insert an additiona stage between stage J and stage J +,inwhichm {N J+ +,...,N J } contestants seected from the N J contestants at the J-th stage compete for the N J+ tickets to stage J +. Does adding this additiona stage necessariy induce more tota efforts from the contestants? The effect of this additiona stage on the tota efforts in the contest is not readiy seen: athough the additiona stage M creates a new source of efforts Me M, the impact of this additiona stage on the contestants effort entries in previous stages is ambiguous. First, from (8), N J e J may either decrease or increase, athough N J e J tends to be reduced as the vaue of the ticket to the next stage is ower (V M =(N J+ V J+ Me M )/M < V J+ ). This is due to the fact that in (8), N J e J is not a monotonic function of the number of survivors to 4

16 the next stage. 6 For the same reason, the impact of the additiona stage on the stage payoff V J is aso indefinite. Second, the additiona stage s impact on the efforts in stages prior to stage J, N j e j,j<j,is aso ambiguous. From (8), N j e j,j<j, woud change in the same direction as V j+, which in turn changes in the same direction as V J. Let E({N }) denote the set composed of a the integers in the sequence {N }. Definition Acontestsequence{Ñ} is more compete than {N } if and ony if E({N }) E({Ñ }). We show in the foowing theorem that any additiona stage aways increases the tota efforts, regardess of the existing contest structure. Theorem 2 The more compete the contest sequence is, the higher the tota efforts are induced. Proof. Denote by E 0 the tota efforts in the origina contest {N },whiebye M the tota efforts in the new contest after one additiona stage M is inserted. We ony need to show E M >E 0. Denote by V e the equiibrium expected payoff that the N contestants anticipate at the first stage of the contest after the additiona stage is inserted. By Lemma, we simpy need to show ev <V. Under the optima prize aocation characterized in Theorem, Lemma 2 eads to that V = V J+ JQ N + P N + g ( ). (9) JQ = N g N Simiary, ev = V J J+ Q [ ( Q M J = N = P = N + N + g )](M P N g 6 N + P The foowing exampe shows that the component N + N + P N +.AssumeN =5. Then N + N + g N g N + P N N + =3;N + + g N g =.28 when N + =4. NJ+ M g N J g )( P N + g N g =.35 when N + =2;N + N J+ g ). (20) M g is not a monotonic function of N + P N + g N g =.57 when 5

17 To estabish that ev <V, we need to show N ( J+ P M Since M g N J g is decreasing in g, M P M g N J g M < N J+ P N J+ M g N J g N J+ g M g.thus, P )(M N J+ M g ) < P N J g N J+ g N J g. < < NJ+ M ( P N J+ g )(M P M g N J+ P N J+ g ( N J+ N J+ P M g N J g ) NJ+ M g )( P M g N J g ) N J+ g N J g. (2) The ast step foows Chebyshev Sum Inequaity. Q.E.D. Theorem 2 is important. It estabishes that an additiona stage of competition aways reduces a representative contestant s expected payoff V,soastoincreasethetotaefforts. A contest sequence is not optima, as ong as the difference between any two successive terms in the sequence {N } L = exceeds one. The tota efforts of the contest can be increased if additiona stages can be inserted, regardess of its existing structure. Thus, the optima contest sequence is represented by a N term stricty decreasing arithmetic sequence {N N = N +, =, 2,..., N.}. The ast term N N =represents the unique fina winner. In other words, the contest asts for N stage, and one contestant is eiminated in each stage. 7 We name it as a compete-sequence Pyramid contest. Theorem 3 In a setting with N contestants and a prize budget Γ 0,theeffort-maximizing sequentia contest with pooing competition in each stage asts for N stages, whie eiminating one contestant each stage, and a singe fina winner takes over the entire prize purse of Γ 0. Theorem 3 naturay stems from Theorem and Theorem 2. Thus, we concude that the optima contest must be organized as a winner-take-a compete-sequence Pyramid contest. Theorem 4 The optimay designed N-person contest with the tota prize purse of Γ 0, i.e. 7 In stage N, two remaining contestants compete for one fina prize. 6

18 the winner-take-a Pyramid contest, induces a tota equiibrium effort of E = Γ 0 { Q P N [ N (N ) g ] (N +) g = Q (N +) N = }. (22) Theorem 4 expicity shows the equiibrium tota efforts in the optimay designed N- person contest. The resut directy arises from Lemma and (8), and the fact that the sequence structure of the optima contest is represented by a compete sequence of integers from N to. 3.4 Discussion So far we have shown that a contest drives more efforts, provided that the contestants have to survive a onger ine of shots before they win the fina prize. Our resuts therefore provide a rationae for the widey observed muti-phase sequentia competition in reaity. Baye, Kovenock, and de Vries (993) argue that the contest organizer (poiticians) may strategicay shortist a subset of finaists, according to their vauations, to participate in rent-seeking competitions (for instance, IOC seects potentia candidate cities up for the eection of hosting cities of Oympic Games), in order to increase the rent-seeking revenue. We suggest an aternative view to this thread of thinking: shortisting benefits the contest organizer even if the contestants are identica. Our paper provides usefu insights to contests design. We show that the optima contest that generates the most efforts is a winner-take-a compete-sequence Pyramid contest. The famous Fox TV show American Ido echoes our resuts pretty we. The show is basicay a singing contest. Tweve contestants are picked out from thousands and thousands of hopefu superstars. These tweve winners, however, are not met with immediate success. The remaining part of the contest proceeds exacty in the form of a Pyramid contest which we have estabished to be optima. The series of the shows then ast for eeven weeks. In each week every remaining contestant makes his/her performance. After each show, one of the remaining contestants is voted off by viewers, whie the others proceed to the next stage. The procedure repeats unti the finae, in which two survivors compete face to face, and one of them becomes the new American Ido. We see that the organization 7

19 structure of American Ido turns out to coincide with the effort-maximizing contest we have estabished. Our resuts provide insights to contests design even when the organizer is faced with constraints on contest structure. For instance, a contest may have to be conducted with a fixed number of stages, whie the contest organizer has ony the fexibiity to choose the number of contestants who survive each stage. Proposition 2 has fuy characterized the maxima tota efforts that resut from any given contest sequence structure. The restricted optima contest sequence can thus be identified through direct comparison across finite possibiities. Rosen (986) and Gradstein and Konrad (999) draw the anaogy between the interna hierarchy of an organization and a muti-stage contest. As argued by Gradstein and Konrad (999): One can interpret the organization s hierarchy in a steady state as consisting of a series of contests among the individua members of each eve of the hierarchy for the promotion to an upper eve. Our resuts have two important impications on the design of interna organizationa structure. Firsty, we show, by Theorem 2, that a hierarchica structuredoesincreasethetotaefforts agents expend inside the organization. This resut provides an aternative rationae for organizationa hierarchy. Secondy, our resuts shed ight on the design of interna incentive (compensation) scheme. We show that in the optima contest, the tota of prize purses shoud be combined into a singe fina prize, and no intermediate award shoud be given away. In the context of organizationa hierarchy, the winner-take-a principe may not be feasibe. Yet our resuts do not ose its appea in this aspect. Reca Lemma 3, which gives a representative contestant s expected payoffs interms of the prize structure. We see that V is in fact a weighted sum over the prizes, and the weights associated with the prizes diminish as the eve of the prizes ascends. It impies that more generous purses for top-ranking prizes (wage, or other benefits) maintain the incentive in career (Rosen (986)) and increase the overa efforts. Our resuts confirm the insights suggested by Rosen (986): contestants who succeed in attaining high ranks in eimination career adders rest on their aures in attempting to cimb higher, uness top-ranking prizes are given a disproportionate weight in the purse. 8

20 4 Concuding Remarks This paper studies the optima contest structure in a muti-phase sequentia competition setting. We aow the contest organizer to design the optima contest in two arms: the contest sequence and the aocation of a fixed tota of prize purses. We show that the contest organizer must aocate the entire prize mass to a singe fina prize, regardess of the contest sequence. We further show that additiona stages aways increase the tota efforts. This resut provides important insights for the design of muti-stage contests. Therefore we concude that the optima contest must be a winner-take-a compete-sequence Pyramid contest that eiminates one contestant in each stage unti the finae. This paper concerns itsef with one aspect of contests design: the maximization of tota efforts, and eaves tremendous room for future extensions. First of a, we have not considered the cost of organizing the contest. The contest organizer may be concerned about the additiona costs that coud arise from additiona stages, which shoud be taken into account in future research on the optima design of muti-stage contests. Secondy, we do not consider the heterogeneity in abiities or preferences among contestants. One interesting extension is to aow for contestants with differing types. We beieve that extensions in this direction wi not vary the main themes of our resuts. Nevertheess, it is sti interesting to investigate whether a stronger contestant is more ikey to win the fina prize in a muti-stage contest than he/she does in a singe-stage one. In that sense, a mode with asymmetric payers may shed ight on the screening effect of the muti-stage contest. In addition, our study considers a contest success function with inear impact of effort. Perhaps, another chaenging extension woud be to aow for other forms of contest technoogies. Finay, our mode assumes that contestants efforts affect ony the outcome of the sub-contest in the current stage. One may extend this mode by aowing for accumuatabe efforts, in which case efforts made in the current stage can be carried over into future stages and continue to infuence contestants ikeihoods of winning. 9

21 References [] Amegashie, J.A., The Design of Rent-Seeking Competitions: Committees, Preiminary and Fina Contest, Pubic Choice, 999, 99, [2] Amegashie, J.A., Some Resuts on Rent-Seeking Contests with Shortisting, Pubic Choice, 2000, 05, [3] Baye, M.R., Kovenock, D., and de Vries, C.G., Rigging the Lobbying Process: An Appication of A-Pay Auction, American Economic Review, 993, 86, [4] Cark, D.J., and Riis, C., A Mutipe-Winner Nested Rent-Seeking Contest, Pubic Choice, 996, 87, [5] Cark, D.J., and Riis, C., Infuence and the Discretionary Aocation of Severa Prizes, European Journa of Poitica Economy, 998, 4, [6] Fu, Q., and Lu, J., Repicating Contests: Revisited, working paper, 2005, Nationa University of Singapore [7] Fuerton, R.L., and McAfee, P.R., Auctioning Entry into Tournaments, Journa of Poitica Economy, 999, 07, [8] Gradstein, M., Optima Contest Design: Voume and Timing of Rent-Seeking in Contests, European Journa of Poitica Economy, 998, 4, [9] Gradstein, M., and Konrad, K.A., Orchestrating Rent Seeking Contests, Economic Journa, 999, 09, [0] Harbaugh, R., and Kumpp, T., Eary Round Upsets and Championship Bowouts, Economic Inquiry, 2005, 43, [] Krishna, V., and Morgan, J., The Winner-Take-A Principes in Sma Tournaments, in Advances in Appied Microeconomics, M. Baye, ed., JAI Press, Stamford, 998. [2] Matros, A., Eimination Tournaments where Payers Have Fixed Resources, 2005, working paper, University of Pittsburgh. 20

22 [3] Modovanu, B., and Sea, A., The Optima Aocation of Prizes in Contests, American Economic Review, 200, 9, [4] Modovanu, B., and Sea, A., Contest Architecture, Journa of Economic Theory, 2006, 26, [5] Rosen, S., Prizes and Incentives in Eimination Tournaments, American Economic Review, 986, 76,