Prize allocation and incentives in team contests

Transcription

1 Prize allocatio ad icetives i team cotests Beoit S Y Crutze Erasmus Uiversity Sabie Flamad Rovira i irgili Uiversity November 9, 07 Nicolas Sahuguet HEC Motréal ad CEPR Abstract We study a cotest betwee teams that compete for multiple idivisible prizes. Team output is a CES fuctio of all the team members efforts. We use a geeralized Tullock cotest success fuctio to allocate prizes betwee teams. We study how differet itra-team prize allocatio rules impact team output. We cosider a egalitaria rule that gives all members the same chace of receivig a prize, ad a list rule that sets ex-ate the order i which members receive a prize. The covexity of the cost of effort fuctio ad the complemetarity of idividual efforts determie which rule maximizes team output ad success. Our results speak to may reald world situatios, such as electios, cotests for the allocatio of local public goods ad the iteral orgaizatio of firms. Departmet of Ecoomics, Erasmus School of Ecoomics, The Netherlads. crutze@ese.eur.l. Departmet of Ecoomics ad CREIP, Rovira i irgili Uiversity, Spai. sabie.flamad@urv.cat. Departmet of Applied Ecoomics, HEC Motréal, Caada. icolas.sahuguet@hec.ca.

2 Itroductio I may ecoomic ad political settigs, cotests are used to allocate idivisible prizes betwee teams of idividuals. Each team competes agaist other teams to wi as may prizes as possible. Prizes are allocated to teams o the basis of their output. The prizes wo are the allocated betwee team members. To maximize its output, a team eeds to provide the right icetives to its members. The desig of a efficiet itra-team prize allocatio mechaism thus requires to take ito accout both iter-team ad itra-team strategic cosideratios. Our leadig example comes from parliametary electios. Political parties are best described as teams of idividual cadidates (see Dows (957)). I electios uder proportioal represetatio i multiple-seat electoral districts, political parties propose to voters a list of politicias who compete as a team to wi as may legislative seats as possible. Legislative seats are allocated to parties proportioally to their vote shares. These vote shares deped i tur o the list s electoral output that team members geerate durig the electoral campaig. Withi parties, each legislative seat wo by a party is a idividual prize that politicias o the list strive to wi. Uder closedlist proportioal represetatio the electoral rule used i, for istace, Argetia, Germay (partially), Icelad, Israel ad Spai voters caot vote for idividual cadidates, but oly for the etire party list ad seats are allocated to politicias strictly followig the order of the party list. Our model allows us to uderstad the coditios uder which a ordered list of cadidates is the optimal itra-team prize allocatio rule. This iterplay betwee itra-team icetives ad iter-team competitio is also relevat i other cotexts. For istace, cosider two departmets withi a firm. These departmets ca be viewed as teams. The firm s CEO wats to desig the hirig ad promotio policy to provide icetives to their various employees. As performace typically correlates positively with the eed to expad the size of a departmet, the umber of positios ad promotios goig to each departmet will be proportioal to the departmet s relative performace. The, employees i a departmet will exert effort to boost their departmet performace. Each of them hopes to the be give oe of the Eve i flexible proportioal represetatio systems, i which voters ca also cast a vote for their favourite idividual cadidate(s) o the party electoral list, the vast majority of legislative seats are allocated i the order of the electoral list. For example, i the last electio of March 07 i The Netherlads, which uses PR with party lists but also the eed for voters to express such a preferetial vote, a full oe hudred percet of the oe hudred ad fifty members of Parliamet were elected followig the order of their parties electoral lists. These party lists also serve other purposes, such as selectio ad geder/sex/origi balace. I this paper, we focus purely o the icetive effects of these lists.

3 prizes their departmet wo. I this cotext, what itra-departmetal prize allocatio rule should the firm use to maximize the outputs of the teams? Oce agai, to aswer this questio, we eed to study the iterplay betwee icetives withi departmets ad competitio betwee departmets. As a last example, cosider a cetral govermet that eeds to choose where to build a ew hospital, school or military base. As the local public good geerates both direct ad idirect jobs ad ecoomic activity for the regio i which they are located, each regio i the coutry would like to see the public good built o their territory. The differet regios thus lobby or compete agaist each other to secure the local public good. Yet, oce a regio has wo this cotest, the exact locatio of the local public good must still be decided amog the regio s differet cities or muicipalities. The regios ca thus be described as teams ad, oce agai, to pi dow which itra-team allocatio rule maximises team output, we eed to study the iterplay betwee icetives withi regios ad competitio betwee regios. I lie with the above examples, we develop a model of a cotest betwee teams of idetical idividuals who compete, via the provisio of costly idividual effort, for multiple idivisible prizes. 3 The team s productio fuctio aggregates the efforts of all team members ad is assumed to be a costat elasticity of substitutio techology. Idividual efforts ca thus be substitutes or complemets. We also parametrize the cost fuctio of team members to allow for differet degrees of covexity. The umber of prizes a team wis is proportioal to team output. To model the iter-team part of the cotest, we propose a ovel allocatio mechaism for the prizes. We exted the classical Tullock (980) imperfectly discrimiatig cotest success fuctio to the case of may prizes. Each prize is wo by a team with a probability equal to the ratio of its ow output over the sum of all the teams outputs. The umber of prizes wo by a team thus follows a biomial distributio. I the mai body of the paper, to model the itra-team allocatio of prizes, we focus o the case of o observable (or o-cotractible) efforts. The, the allocatio mechaism caot deped directly o the efforts of idividual members ad specifies the probability that a give team member wis a prize as a fuctio of the total umber of prizes wo by his team. We compare two itrateam allocatio rules: a list rule that allocates prizes accordig to a pre-specified list as uder closed-list proportioal represetatio ad a egalitaria allocatio rule that treats all members equally by radomly distributig the prizes wo. The list rule is thus biased ad discrimiatory as 3 I a compaio paper, we cosider the case of teams of agets with heterogeeous abilities, which impacts their cost of effort fuctio ad productivity, which i tur impacts team output. 3

4 it gives a ex-ate advatage to some members over others, eve though all members are idetical. We show that which itra-team allocatio rule is associated to highest team output depeds o the complemetarity (substitutability) of the idividual efforts i the team productio techology ad the covexity of the idividual cost fuctio. I particular, the egalitaria rule domiates the list rule whe idividual efforts are complemets ad the margial cost fuctio is covex. Whe efforts are substitutes ad the margial cost of effort is cocave, the list rule domiates the egalitaria oe. We the tur to the questio of fidig the optimal mechaism amog all possible itra-team allocatio rules. We first restrict attetio to mootoic rules, uder which the idividual probability of wiig a prize must be (weakly) icreasig i the umber of prizes wo by the team. We show that, depedig o the same coditio o complemetarity of effort ad covexity of cost, the optimal mootoic rule is either the list or the egalitaria allocatio rule. This justifies our decisio to focus o these two rules to start with. To the best of our kowledge, our fidigs offer the first theoretical justificatio for the rule that specifies, for electios uder proportioal represetatio i multiple-seat districts, that the seats a party has wo are to be distributed to the cadidates o the list i the order of the list. Ideed, our fidigs imply that this itra-team allocatio rule is the optimal icetive provisio mechaism whe efforts are substitutable eough ad the cost of effort is ot too covex. This cotrasts sharply with the typical egative views about the icetive effects of closed-list proportioal represetatio both i the ecoomics ad i the political sciece literature (see for istace, Persso ad Tabellii (000, 003), Persso, Tabellii ad Trebbi (003) ad Tavits (007)). Eve though the list allocatio rule is associated with the free-ridig ad demotivatig effects the political ecoomy literature has focused o, we show that, i our model, it is the best allocatio rule if the team s electoral output exhibits eough substitutability (ad/or the cost of effort is ot too covex), as low effort provisio by the first ad last politicias o a party list are more tha compesated by high effort by the cadidates i the media positios of the list. We close our aalysis of optimal rules by otig that, ituitively, removig the mootoicity costrait opes the way to itra-team allocatio rules that ca provide eve better icetives tha the list rule. Ideed, by removig the mootoicity costrait, we allow the rule desiger to allocate icetives more freely across the differet team members. thaks to the o-egativity costrait o effort provisio. I particular, givig o-mootoic icetives to some team member frees icetive tokes that ca be redistributed to other members, ad the extra effort geerated by 4

5 such redistributio ca over-compesate the drop i effort by the members who are give egative icetives. Further, because of the additioal flexibility that removig the mootoicity costrait geerates, we show that the egalitaria rule is the optimal rule for a smaller set of parameter values. We the tur to various extesios. First, we allow idividual efforts to be fully cotractible: the allocatio of prizes ca deped o the effort exerted by team members. This implies that each team member ca be put exactly o their participatio costrait. I this case, the egalitaria rule is always the optimal mootoic rule. The, we exted the model to the case of more tha two teams ad to the case of a biased iter-team cotest, whe oe team has a advatage over the other teams. The rest of the paper is structured as follows. The ext sectio reviews the relevat literature. Sectio 3 presets the model. Sectio 4 solves for the equilibrium uder the egalitaria ad list allocatio rules ad compares idividual effort choices ad team output uder the two itra-team allocatio rules. I Sectio 5, we discuss optimal mechaisms. Sectio 6 cosiders extesios. The last sectio cocludes ad offers aveues for further research. Related Literature Our paper first cotributes to the literature o cotests. It offers a bridge betwee two differet strads of the literature: team cotests ad cotests for multiple prizes. I team cotests, several teams compete i order to wi oe prize, which may be of a public or private ature, or a mix of both. The focus of this strad of the literature is o the sharig rule that determies how to split the sigle available (private part of the) prize across the wiig team s members, so as to maximize team output. Importat cotributios iclude the semial work of Nitza (99) ad the cotributios of Lee (995), Esteba ad Ray (00), Ueda (00), Baik ad Lee (00), Nitza ad Ueda (0), Baik ad Lee (0) ad Balart et al. (05). 4 Our cotributio is to exted the literature o team cotests to the case i which teams compete for may idivisible prizes. Turig to the literature o multiple prizes, our cotributio is to exted it to the case i which it is teams, as opposed to idividuals, which compete for these prizes. 5 I particular, our Biomial-Tullock mechaism offers a ovel way of allocatig the prizes amog the differet teams competig i the cotest. Oe classical predecessor to our iter-team allocatio mechaism is the probabilistic votig mechaism developed by Eelow ad Hiich (977) ad Lidbeck ad Weibull 4 For a recet survey o sharig rules i collective ret seekig, see Flamad ad Troumpouis (05). 5 For a recet survey of the cotest literature o cotests with multiple prizes, see Sisak (009). 5

6 (987) ad used recetly by Galasso ad Naiccii (06) i their aalysis of cadidate selectio issues uder closed-list proportioal represetatio. I that model, the probability of wiig a extra prize is idepedet of the umber of prizes a team has already wo, whereas i our model the teams probability of wiig a extra prize decreases with the umber of prizes it already wo, a feature that we believe is desirable as it is more realistic. I the touramet literature, some cotributios, such as that of Nalebuff ad Stiglitz (983), also cosider the case of multiple prizes. Oe major advatage of our mechaism is its aalytical tractability that allows for closed-form solutios. Our paper cotributes to the literature o icetives i teams, ad i particular to the literature that liks icetives ad discrimiatio or o-equal treatmet of ex-ate idetical team members. Witer (004) aalyzes whether agets who are idetical i their qualificatios should receive asymmetric rewards to improve icetives ad efficiecy. Witer (004) relies o a O-Rig techology where all agets must succeed i their task for the team to be successful. Also, Witer (004) models effort as a biary choice ad thus caot discuss the role of the covexity of the cost fuctio. I cotrast, as we rely o a stadard CES productio fuctio, we have a more cotiuous way to parametrize complemetarity. We also let efforts be a cotiuous variable which allows us to aalyze the effect of the covexity of the cost fuctio. Bose et al. (00) also study the uequal treatmet of idetical agets i teams. I their model, complemetarities i team productio lead to higher output whe effort decisios are take sequetially as opposed to simultaeously. Thus, they fid that, withi each team, it is optimal to treat differetly the two team members i the presece of strategic complemetarities.our team output fuctio is a CES, ad all team members choose their efforts simultaeously. I that case, complemetarities call for equal treatmet. I most applicatios, the use of a sequetial mechaism does ot appear realistic. Closer to our basic model setup is Ray et al. (007) who also use a CES fuctio to model team productio. Yet, their model focuses o oe team oly ad there is a cotiuous, fully divisible prize to be wo. I accordace to our results, they fid that uequal sharig rules are efficiet whe efforts are substitutes. Our fidigs suggest that their result exteds to the case of team cotests for multiple idivisible prizes. Our aalysis also goes further tha Ray et al.(007), as we derive the optimal mootoic ad o-mootoic allocatio rules uder cotractible ad o-cotractible effort, whe efforts are complemets as well as substitutes. Give that our setup coicides with the icetive provisio versio of a electoral competitio game uder closed-list proportioal represetatio i multiple-seat districts, we also cotribute to 6

7 furtherig the research o the icetive effects of electoral rules; see for example Myerso (993 ad 999), Persso ad Tabellii (000, 003), Persso Tabellii ad Trebbi (003), Tavits (007), Buisseret ad Prato (07) ad Crutze ad Sahuguet (07). 6 I this literature, proportioal represetatio, especially with closed-list, has received quite some bad press. 7 This is maily because of the fact that, as we also show i our theory, icetives to work hard are bluted for the team members at the top ad bottom of the list. Yet, our results show that, as log as idividual efforts are ot too complemetary ad/or the cost of effort is ot too covex, the attributio of seats followig the order of a pre-defied list is the optimal mechaism parties should use uder closedlist proportioal represetatio. Buisseret ad Prato (07) ad Crutze ad Sahuguet (07) both offer some more specific cautioary tales about the ucoditioal validity of the bad press closed-list proportioal represetatio has bee subject to. For example, Crutze ad Sahuguet (07) show that the cadidates idividual icetives to exert effort ca be stroger uder closedlist proportioal represetatio tha uder plurality rule, cotrary to what the curret wisdom i comparative politics is, oce we let party leaders attribute the differet slots o the party list i a competitive fashio. Buisseret ad Prato (07) poit out that party lists uder proportioal represetatio ca lead to a better aligmet of the idividual cadidates objectives with parties ad the electorate at large. 3 Model 3. Idividual ad team efforts Two teams are competig i a cotest for idetical prizes. Each prize has value. Each team is composed of idetical members who ca at most wi oe prize. Member i of team j exerts costly effort e ij 0 to improve his team s chaces of wiig prizes. Team members are idetical i the sese that they all have the same effort productivity, that we ormalize to uity, ad they all face the same cost of effort fuctio, which is icreasig ad covex: c(e ij ) = e β ij /β, with β >. () 6 See also Kuicova ad Rose-Ackerma (005), Chag ad Golde (007), Schleiter ad ozaya (04) ad Rafler (06). 7 With the exceptio of Myerso (993 ad 999), Buisseret ad Prato (07) ad Crutze ad Sahuguet (07). 7

8 Team j s aggregate output is deoted by E j. We assume that the productio fuctio aggregatig idividual efforts exhibits costat elasticity of substitutio: ] E j = (e ij ) σ σ, with σ <. () i= 3. Allocatio of prizes betwee teams The allocatio of the prizes betwee teams depeds o the aggregate output of each team. We assume that the allocatio fuctio is what we call hereafter a Biomial-Tullock imperfectly discrimiatig cotest success fuctio. This techology is a atural geeralizatio of the Tullock (980) cotest success fuctio to multiple prizes. As i a Tullock cotest, the probability that team j wis a give prize is give by the ratio-form cotest success fuctio p j = E j E +E. Thus, prizes are awarded to team j usig idepedet draws from a Beroulli distributio with parameter p j. The probability that team j wis k prizes follows a biomial distributio ad is give by: ( ) P j (k) = Ck Ej k ( ) E +E E k j E +E. (3) 3.3 Allocatio of prizes withi teams Idividual efforts are ot cotractible. The allocatio of prizes caot deped directly o these efforts 8, but oly o the umber of prizes wo by the team. A allocatio rule specifies, for each umber of prizes wo by the team, the probability that a give team member wis a prize. Eve though idividual effort does ot eter directly ito the prize allocatio mechaism, team members still icrease their chaces of gettig a prize by exertig more effort. We cotrast two allocatio rules: the egalitaria allocatio rule which treats all team members equally, ad the list allocatio rule which treats team members differetly as it gives priority to some team members i the allocatio of prizes. We focus o these rules for two reasos. First they are atural rules. Secod, they tur out to be the optimal mootoic rules, as we show i Sectio 5. Uder the egalitaria rule, for ay umber k of prizes wo by a team, all group members have the same probability k/ of wiig a prize. The, member i i team j chooses his level of effort to solve: Max eij k= P j (k) k eβ ij β 8 I Sectio 6., we relax this assumptio ad aalyze the case where idividual effort is perfectly cotractible ad prize allocatio ca deped directly o idividual efforts. ] (4) 8

9 Uder the list allocatio rule, as uder closed-list proportioal represetatio, the team members are ordered o a list, which determies the order of the allocatio of prizes wo by the team. Thus, the member i mth positio o the list wis a prize if their team wis at least m prizes. The list order is idepedet of effort decisios ad based o effort-idepedet characteristics 9 of the differet team members. This allocatio rule treats similar members i differet ways ad is thus biased ad discrimiatory. Team member i mth positio o the list i team j solves ] Max eij P j (k) eβ mj β k=m (5) Notice that the summatio goes from m to ad ot from to, as the team member i mth positio o the list oly gets a prize whe his team wis at least m prizes. 4 Equilibrium efforts Uder the egalitaria rule, member i i team j chooses his level of effort to solve equatio (4) above. Usig the formula for the expectatio of the biomial distributio, we ca rewrite the objective fuctio (4) as: β ij E j e E + E β Thus, the problem uder the egalitaria rule is equivalet to a team cotest over oe fully divisible prize of idividual value with a egalitaria itra-team sharig rule, as i Nitza (99). Propositio : Uder the egalitaria allocatio rule, i a symmetric Nash-Equilibrium: Team output E E Idividual effort e E is give by: is give by : E E = (6) ( ) β β+σ β( σ) (7) 4 ( e E = 4EE σ σ ) ( β /β ( ) β β = 4) (8) Proof. See appedix. 9 Examples of such characteristics iclude seiority or age. 9

10 Idividual effort icreases i the value of a prize ad decreases with the umber of prizes., while team output icreases i both ad. Uder the list allocatio rule, team member i positio m o the list i team j maximizes: We the have: k=m P j (k) eβ mj β = k=m C k ( Ej E + E ) k ( E ) k j eβ mj E + E β Propositio : Uder the list allocatio rule, i a symmetric Nash-Equilibrium, team output (9) E L is give by: EL = k= kc k ( ) ] σ β+σ β+σ β( σ) ( Idividual effort e m of the team member i the mth positio o the list is give by: e m = EL σ mcm ( ) ] + β+σ 4 ) β (0) () Proof. See appedix. The mai differece betwee the two allocatio rules lies i the way they treat the differet team members. Uder the egalitaria rule all team members receive the same icetives ad exert the same effort i equilibrium. Uder the list allocatio rule, icetives ad idividual equilibrium efforts vary with the positio o the list. I particular the members located aroud the media list positio face the largest margial beefit of effort, as these positios are associated to the steepest slope of the biomial distributio. To the cotrary, the first ad last members o the list have little icetive to exert effort, as the slope of the biomial distributio is essetially flat. Ideed, as we illustrate i Figure (for the case of a cotest for 30 prizes), the shape of the distributio of the equilibrium wiig probabilities associated with the differet positios o the list, represeted by the blue bars (scaled so as to have the two curves of roughly equal size), implies that the distributio of idividual equilibrium efforts is bell-shaped. This heterogeeity i icetives is the mai reaso why closed-list proportioal represetatio i multiple-seat districts has received so much bad press i the political ecoomy ad the political sciece literature. 0

11 Figure : Biomial-Tullock probabilities ad effort provisio 4. Compariso of the allocatio rules We ow compare team output ad team success uder the two allocatio rules. We show that the parameter β, which represets the covexity of the cost fuctio, ad parameter σ, which parametrizes the degree of complemetarity i the team output productio fuctio, play a cetral role. Ideed, we have: Propositio 3: The list allocatio rule leads to higher team output tha the egalitaria rule if ad oly if β < σ. The two allocatio rules yield the same team output if ad oly if β = σ. Proof. See appedix. The ituitio behid this result is as follows. As we saw above, idividual icetives are uiform uder the egalitaria rule ad bell-shaped uder the list rule. Whe will the bell-shaped distributio of idividual efforts yield higher team output? Suppose that the cost of effort fuctio is close to beig liear (β is close to ). If idividual efforts are highly substitutable (σ close to 0), team output is equal to the sum of efforts, ad this sum is what matters, ot so much the level of the differet idividual efforts. Thus, iducig differeces i effort ca be optimal. Whe efforts are complemetary (σ > /), iducig differeces i idividual effort is suboptimal as the decisio of the low effort providers depresses team output. What about the covexity of the idividual cost of effort fuctio? Suppose for simplicity that σ = 0. Whe this fuctio is very covex (β > ), the

12 margial cost is also covex. The, asymmetric icetives are bad for team performace. Ideed, startig from equal margial beefits of effort, icreasig the margial beefit of oe team member ad decreasig the beefit of aother oe will have a positive effect if the margial cost icreases more slowly for the idividual with stroger icetives tha for the oe with weaker icetives. Yet, whe the margial cost of effort is covex, this is simply ot possible. 4. Teams choose their allocatio rule idepedetly The result applies to a symmetric cotest betwee teams that (are required to) use the same itrateam prize allocatio rule. Yet, i reality teams choose strategically their allocatio rule. We thus add a stage to our game to check how our results are affected i this exteded game. I the first stage, each team chooses a allocatio rule to maximize team success. I the secod stage, after observig the allocatio rule chose by both teams, idividual team members choose their effort. Solvig for the (pure-strategy) subgame perfect equilibrium, we fid that the coditio drivig the teams choice of the allocatio rule is the same as i Propositio 3 above: Propositio 4: I the pure-strategy subgame perfect equilibrium of the two-stage game, the list allocatio rule is chose if ad oly if β < σ. Proof. See appedix. This result shows that to maximize team output ad thus the expected umber of prizes wo by the team, the optimal allocatio rule is still pied dow by coditio β < σ. Thus, give β ad σ, both teams have a domiat strategy (strictly domiat whe β σ) i the first stage of the game. To wrap up, propositios 3 ad 4 show that the covexity of the margial cost ad the complemetarity of the team productio fuctio drive the choice of allocatio rule. Whe the margial cost of effort is covex, givig powerful icetives to a few idividuals is ot productive, as eve these idividual are ot goig to choose to exert much effort. With covex margial costs, it is thus more efficiet to give all team members the same icetives ad treat them i a equal, symmetric way. Whe the margial cost is ot too covex or eve cocave, it is efficiet to provide powerful icetives to few idividuals who will exert very high levels of effort. I that case, a rule that treats some members differetly is optimal. The degree of complemetarity plays a similar role. Whe efforts are substitutes, there is o cost i gettig very differet effort levels withi the team. Whe efforts are complemetary, it is better to iduce similar efforts ad oce agai the egalitaria rule becomes the optimal rule.

13 Goig back to our leadig example, proportioal represetatio i multiple-seat districts, the above fidigs suggest that whe efforts are ot too complemetary withi each party ad/or the idividual cost of effort is ot too covex, the use of closed lists ca be optimal. I particular, it gives better icetives tha ay arragemet which treats all politicias who are ruig for electio i a ex-ate fair ad egalitaria way. Thus, propositios 3 ad 4 provide a icetive argumet for the use of closed lists over a egalitaria rule uder proportioal represetatio. I the ext sectio, we go further ad show that closed lists are the optimal icetive mechaism (whe efforts are substitutes ad the cost of effort is ot too covex) whe we impose a mootoicity costrait. 5 Optimal allocatio rule A allocatio rule ca be represeted as a ( + ) matrix of weights λ ik ]. Etry λ ik correspods to the probability that team member i gets a prize whe the team has wo k prizes. Probabilities eed to be betwee 0 ad, ad the umber of prizes distributed caot be larger tha the umber of prizes wo by the team. These feasibility costraits thus require 0 λ ik ad i= λ ik = k. The egalitaria allocatio rule ca be represeted as a matrix i which each colum has equal etries λ ik = k/. The list allocatio rule ca be represeted as a matrix with λ ik = 0 if i > k ad λ ik = if i k. For istace, with three prizes, the matrices correspodig to the egalitaria rule ad the list 0 /3 /3 0 allocatio rule are 0 /3 /3 ad 0 0 respectively. 0 /3 / To simplify the expositio, we oly preset the case of perfect substitutes (σ = 0). The extesio to the case of complemets is straightforward, albeit algebraically more tedious. Member i of team j maximizes k= λ ikp j (k) eβ ij. I a Nash equilibrium, as equilibrium effort caot be egative, the first order coditio implies that the optimal effort of team member i is give by: max k= E e ij E i λ (E +E ) ik Ck kp k j At a symmetric Nash equilibrium, the above boils dow to: β ] ( P j ) k ( k) Pj k ( P j ) k ] β, 0. () 3

14 e i = max k= E e i 4E λ ikck ( ) β (k )] Simplifyig ad forgettig for ow that effort caot be egative, we get: E = + i ] λ ik Ck (k ) k= β β β, 0. (3). (4) Deotig ik = λ i( k) λ ik ad exploitig the fact that λ i0 = 0, we ca rewrite team output as: E = i + / k=0 Thus, team output is maximized whe i ik Ck ( k) β β β / k=0 ikc k ( k) ] β. (5) is maximized. The costraits take two forms. First, there is the prize budget costrait: 0 i ik. Secod, idividual probabilities must be betwee 0 ad, implyig that: ik. 5. Optimal mootoic rules We first derive the optimal mechaism uder a mootoicity costrait, which requires that λ ik λ i,k, that is, that ik 0 for all k ad i. This costrait imposes that the probability that a idividual wis a prize is (weakly) icreasig i the umber of prizes wo by the team. Solvig for the optimal allocatio rule yields the followig: Propositio 5: Whe β, the egalitaria allocatio rule is the optimal mootoic allocatio rule. Whe β, the list allocatio rule is the optimal mootoic allocatio rule. Proof. See appedix. The ituitio behid the result is simple. The itra-team prize allocatio rule determies idividual icetives ad effort choices. Whe β, it is optimal to equalize icetives across team members, while whe β, it is optimal to make icetives as strog as possible for some idividuals. The maximizatio problem is similar to that of the optimal allocatio of risk. With risk-averse idividuals, the allocatio is as egalitaria as possible, while with risk-lovig agets, it is optimal to make the allocatio as uequal as possible. Thus, as already aticipated, the aalysis above proves that the use of closed-lists uder proportioal represetatio i multiple-seat electoral districts is the optimal icetive mechaism to 4

15 use (provided coditio β < σ is met). This fidig also offers a ovel iterpretatio to two importat ad related puzzles i the empirical literature i political ecoomy: the lack of empirically sigificat icetive costs associated with the use of closed-lists uder proportioal represetatio see for example Persso, Tabellii ad Trebbi (003) ad the fact that some of the coutries that are most praised for their electoral system s capacity to hold their politicias o their toes have as their electoral rule proportioal represetatio with closed-lists. The above fidigs suggest that this electoral rule ca actually be the optimal oe to use i electios i multiple-seat districts if the goal is to maximize the party list s total electoral output. Applyig our fidigs to the optimal orgaizatio of firms, oe testable predictio is that departmets i which members are highly complemetary for team productio should treat their members i a more egalitaria fashio tha departmets i which idividual efforts are more substitutable. Oe casual observatio about the orgaizatio of firms i the techologically most advaced sectors, such as that of software applicatios for mobile techologies, is that at least some of these firms are much more horizotal, much less hierarchical tha the firms i older, more mature sectors. Whereas this observatio falls very short of beig a empirical test of our fidigs, it is certaily cosistet with them: as the efforts of coders workig o the same project i a software compay are very complemetary, we should expect the way their departmet is orgaised to be such that all workers face equal treatmet. Takig this observatio to the data would certaily be a very iterestig exercise. 5. Optimal o-mootoic rules A o-mootoic allocatio rule ca give egative icetives to some team members. Negative icetives appear whe a idividual faces a higher probability of gettig a prize whe his team gets fewer prizes. Yet, effort caot be egative. Also, egative icetives free up icetive tokes that ca be redistributed to other team members. The combiatio of the zero lower boud o effort ad the possibility of redistributig icetives withi the team may the geerate a higher team output tha what was possible whe the mootoicity costrait was bidig. We illustrate how this redistributio of icetives ca ideed geerate higher team output with the help of two examples. As the optimal rule ca be o-mootoic, the member labels should ot be give ay rakig iterpretatio aymore. A team member with a lower umber is ot ecessarily treated more favorably by the itra-team allocatio rule. 5

16 Example Optimal rules with four prizes I that case, the optimal allocatio rule maximizes ( ) 4 i= max + (λ i3 λ i ) β, 0. Whe β =, the optimal allocatio rule maximizes 4 i= max ( + (λ i3 λ i ), 0). Uder the list allocatio rule ad the egalitaria allocatio rule, 4 i= (λ i3 λ i ) =. This meas that 4 i= ( + (λ i3 λ i )] = 8. Oce we remove the mootoicity costrait, we ca set (λ i3 λ i ) to be egative, geeratig egative icetives for team member i who, as a cosequece, chooses zero effort. Yet, this also frees icetive tokes that ca be strategically redistributed to the most resposive team member(s). 0 0 Exploitig this redistributio, it is easy to check that a optimal rule is The optimal allocatio rule is o-mootoic i the sese that team member wis a prize if the team wis oe prize but does ot wi a prize if the team wis three prizes. With that rule, 4 i= max ( + (λ i3 λ i ), 0) = 9 > 8. Also, observe that members to 4 are all treated equally (as the weights i colums ad 4 do ot matter for icetives): the etries i colums ad 3 are the same for these players. This rule clearly remais optimal whe β give that it creates eve more uequal icetives tha the list allocatio rule. For β >, there is a trade-off betwee relyig o the egalitaria allocatio rule ad a optimal o-mootoic rule. To pi dow the value of β below which it is optimal to depart from the egalitaria rule, oe eed oly compare 4 i= ( + /) β = 4 β ad 3 (+ ) β = 3 3 β, where the effort of oly three team members matter for the secod team s output as, uder the o-mootoic rule above, member exerts zero effort. The, simple algebra implies that the egalitaria rule is suboptimal wheever 4 β < 3 3 β β < l(3/)+l(4/3) l(4/3).4. Example Optimal rule with five prizes With more prizes, the optimal o-mootoic rule ca take differet forms depedig o the value β takes o. The optimal allocatio rule maximizes 4 i= max (λ i4 λ i ) + 0 (λ i3 λ i ), 0] β. 0 This is ot the oly optimal rule, as optimality puts o costraits o the value the differet λ i ca take o. 6

17 As before, whe β the fuctio above is a covex fuctio of λ i4 λ i ad λ i3 λ i, implyig that it is optimal to make icetives as heterogeeous as possible. Also, as all colums of the rule matrix but the first ad last eter i the equatio of team effort, the optimal o-mootoic rule is this time uique. It is easy to check that it is give by Thus, whe β, it is optimal to treat differetly members ad ad treat equally members 3 to 5. Whe β > idetifyig which itra-team allocatio rule is optimal is a bit more ivolved. Usig umerical aalysis to pi dow the critical values for β, it appears that there are two cases to cosider: β 3 ad β (, 3). Whe β 3, the egalitaria rule is optimal: the covexity of the idividual cost of effort is too strog for the beefits of geeratig egative icetives to compesate for their costs. Whe β (, 3), the optimal rule gives equal icetives to four team members ad egative icetives to /4 3/4 oe: 0 0 /4 3/ /4 3/4 0 0 /4 3/4 The last example shows that, uder o-mootoicity ad with a sufficiet umber of prizes, the space of parameter values for β ca be partitioed more fiely to fie tue icetives tha whe the allocatio rules are costraied to be mootoic. 6 Extesios 6. Cotractible efforts Assume efforts are ow fully observable or cotractible. What are the optimal allocatio rules i this case? Because efforts ca be cotracted upo, ay optimal allocatio rule will deped ot oly o the umber of prizes wo by the team, but also o the actual effort decisio of each team member. This meas that the participatio costrait of team members ow becomes cetral. 7

18 The egalitaria rule the specifies that each team member who exerts effort e e has a equal chace of gettig oe of the prizes wo by the team. The list rule still assigs to each team member a rak i the list, but ow also specifies a miimal effort level associated with each rak. If a team member exerts the specified effort (or more), they get a prize if the team wis more prizes tha their rak. If they exert less effort or if the team wis less prizes tha their rak, they get o prize. The, we have the followig propositio: Propositio 6: Whe efforts are cotractible, the egalitaria allocatio rule leads to higher team output tha the list allocatio rule for all values of the parameters β ad σ. Proof. See appedix. Whe efforts are observable, the level of effort is determied by the participatio costrait of team members. The cotract ca impose a effort level that drives each team member s utility to his outside optio. The, the cost of effort eters directly i the participatio costrait. Whe effort is ot observable, it is the margial cost of effort that is cetral to ay member s effort choice. Comparig team outputs across the two itra-team allocatio rules, we fid that the egalitaria rules geerates higher team output tha the list rule if ad oly if β σ, which is always satisfied. Thus, whe effort is cotractible, it is always optimal to treat every team member i the same way. Ufair allocatio rules are iefficiet allocatio rules. This last fidig suggests that the imperfect cotractibility of effort is a ecessary coditio for the optimality of discrimiatory rules regardig icetives. 6. More tha two teams How are our fidigs modified if we let K > teams compete? This extesio is atural for the political ecoomy applicatio as it is commo to see more tha two parties competig i electios uder a proportioal system. Surprisigly perhaps, the mai result of Propositio 3 exteds to this set-up. We still get that closed-lists may be optimal uder proportioal represetatio whe there are more tha two parties i the electio. With more tha two teams, the distributio of efforts withi each team becomes asymmetric ad right-skewed. Ideed, i ay symmetric equilibrium, each team member expects his team to wi /K prizes. Thus, depedig o how may teams eter the competitio, the relevat itermediate list members, those who exert highest effort, are aroud positio /K o their team list. Figure below illustrates this for K = (the blue bars) ad K = 4 (the red bars) (with =, β =, σ = 0 ad = 30). 8

19 K= K= m Figure : Effort distributios with two ad four parties Despite the fact that efforts are ow distributed asymmetrically, we still have that: Propositio 7: With K > teams, the list allocatio rule leads to higher team output tha the egalitaria rule if ad oly if β < σ. The two allocatio rules yield the same team output if ad oly if β = σ. Proof. See appedix. 6.3 Asymmetric cotests (oe team with a advatage) Ofte, iter-team competitios are biased i favour of some team. I electoral cotests, for example, some parties ejoy a ex-ate ideological advatage over some of their competitors. I cotests for the allocatio of local public goods such as schools, hospitals or military bases, the socioecoomic coditios of the differet regios may make some regios (dis-)advataged i the competitio. I firms, the CEO may favour oe departmet over the others. Agai somewhat surprisigly, coditio still determies the rakig of effort betwee allocatio rules. Ideed, cosider a cotest betwee two teams that is biased i favor of team, say. Let the probability that team wis a prize give efforts E ad E be give by λe / (λe + E ) with λ >. The distributio of the umber of prizes is ow: P (k) = Ck ad we have: ( λe ) k ( ) k λe +E λe λe +E, 9

20 Propositio 8: I a biased cotest betwee two teams (λ > ), the list allocatio rule leads to higher team output tha the egalitaria allocatio rule if ad oly if β < σ. The two allocatio rules yield the same team output if ad oly if β = σ. Proof. See appedix. 6.4 Some other possible extesios We discuss quickly some other extesios. Icreasig the size of the teams. We have assumed i the model that the umber of members i each team is equal to the umber of prizes that a team ca wi. We ca easily adapt the model to aalyze how the umber of team members ifluece total effort uder the two allocatio rules of iterest. Whe the team uses the egalitaria allocatio rule, it is easy to show that icreasig the umber of team members leads to higher team output whe coditio is satisfied, whereas the opposite is true whe coditio is ot satisfied. Of course, with a list allocatio rule, icreasig the umber of team members has o beefit. Additioal team members have o chace of gettig a prize ad would thus exert zero effort. Our fidigs also allow us to exted the literature o the group size paradox (see for example Esteba ad Ray (00) ad the cotributios that follow). If that literature focuses o the case of a sigle prize, we offer some thoughts about the optimal way to orgaize a team whe prizes are multiple ad idivisible ad the umber of team members is higher tha the umber of potetial prizes. Our results prove that the optimal itra-team prize allocatio rule is crucial to uderstad the coditios such that larger groups are more efficiet at geeratig team output. Ideed, whe the list allocatio rule is optimal, icreasig the size of the team has o effect o idividual effort decisios ad thus team output. It thus makes sese to thik about the group size paradox i our setup oly whe the egalitaria allocatio rule is optimal. Broader objective fuctio for team members. We ca also exted the model to aalyze the cosequeces of lettig team members care about, for example, the total umber of prizes their team wis. The presece of this type of prefereces seems especially relevat for political applicatios: the typical political cadidate does ot care oly about themselves gettig elected, but also about the overall performace of their party. There are several ways to model this type of prefereces. For istace, we could add a compoet to the utility fuctio of the idividual that depeds o the umber of prizes wo by the team. We could also add a beefit if the team wis a majority of the prizes (as wiig a majority of legislative seats implies cotrol of govermet 0

21 ad thus a discrete jump i utility for the political party that achieves such a result). The algebra the becomes less trasparet, but the same coditio o β ad σ is still drivig the compariso of team outputs. Allowig for aggregate oise. We ca also modify the cotest techology to allow for aggregate oise i the iter-team distributio of prizes. For example, we could make use of or γ + ( γ) E i E +E E r i E r +Er where r ad γ parametrize the resposiveess of success to effort. It is easy to show that it is still true that the rakig betwee the egalitaria allocatio rule ad the list allocatio rules follows coditio. This fidig mirrors Balart, Chowdhury ad Troumpouis (05) which proves that the above two ways of modellig aggregate oise yield the same icetives for cotest participats. 7 Coclusio We developed a model to study the impact of itra-team prize allocatio rules i team cotests whe prizes are idivisible. We showed that the covexity of the margial cost fuctio ad the degree of effort complemetarity drive which type of allocatio rule is best for icetives. Whe the margial cost of effort is covex or efforts are complemets, the egalitaria allocatio rule, that treat all team members the same way, domiates. Whe the margial cost of effort is cocave or efforts are substitutes, it is optimal to treat team members asymmetrically. I particular, the list allocatio rule is optimal. These results hold i a cotext i which idividual efforts are ot cotractible ad i which the allocatio of prizes oly depeds o the total umber of prizes wo by the team. Whe efforts are cotractible, the optimal mechaism is oe that liks directly idividual effort provisio to the prospect of obtaiig a prize, ad i that case the egalitaria rule domiates, always. We also cosidered several extesios, such as havig more tha two teams, havig a asymmetric cotest or itroducig a luck compoet i the cotest betwee teams, to show that our results are robust to such extesios. We believe our model is very ameable to further extesios ad applicatios. For example, i a compaio paper, we aalyze a similar game but allow members to differ i ability, which impacts their cost of effort fuctio, ad productivity, which impacts team output via the CES productio fuctio. These additioal sources of heterogeeity allow us to aswer several additioal iterestig questios, such as uderstadig how to treat the most ad least productive team

22 members ad piig dow the eviromets i which heterogeeous teams are most useful. I political ecoomy, Crutze ad Sahuguet (07) adapt the preset model to compare icetives uder differet electoral rules, ad i particular British-style first-past-the-post ad Israeli-style proportioal represetatio. Fially, the stark predictios of our model should be empirically testable, as some ecoomic sectors exhibit much stroger complemetarity across workers tha others. Carryig out such empirical tests is thus aother iterestig aveue for further research. 8 Refereces Baik, K. H. ad S. Lee (00). Strategic groups ad ret dissipatio, Ecoomic Iquiry, 39, Balart, P., S. M. Chowdhury, ad O. Troumpouis (05). Likig idividual ad collective cotests through oise level ad sharig rules, Ecoomics Letters, 55, Balart, P., S. Flamad, ad O. Troumpouis (06). Strategic choice of sharig rules i collective cotests, Social Choice ad Welfare, 46(), Bose, Arup, Debashis Pal, ad David EM Sappigto.(00) Asymmetric treatmet of idetical agets i teams. Europea Ecoomic Review, 54.7: Buisseret, Peter ad Carlo Prato, (07). Electoral Accoutability i Multi-Member Districts. mimeo Uiversity of Chicago School of Public Policy. Chag, Eric CC, ad Miriam A. Golde, (007). Electoral systems, district magitude ad corruptio. British Joural of Political Sciece, 37(), Crutze, Beoit S. Y. ad Nicolas Sahuguet, (07). Electoral icetives: the iteractio betwee cadidate selectio ad electoral rules, Erasmus School of Ecoomics, Erasmus Uiversiteit Rotterdam. Dows, Athoy (957). A Ecoomic Theory of Democracy. Pearso Ed. Eelow ad Hiich (977). XXX. Which paper is this? Esteba, Joa ad Debraj Ray, (00). Collective actio ad the group size paradox, America Political Sciece Review, 95, Flamad, S. ad O. Troumpouis (05): Prize-sharig rules i collective ret seekig, i Compaio to the Political Ecoomy of Ret Seekig, ed. by R. D. Cogleto ad A. L. Hillma, A very iterestig recet cotributio that has a simlar flavour is Cubel ad Sachez-Pages (07), i which itra-group icome iequality is mapped ito the groups capacity to defed themselves from exteral threats.

23 Edward Elgar Publishig, 9-. Galasso, icezo ad Tommaso Naicii, (05). So closed: Political selectio i proportioal systems. Europea Joural of Political Ecoomy, 40, Korad, Kai A., (009). Strategy ad Dyamics i Cotests, Oxford UK: Oxford Uiversity Press. Kuicova, Jaa, ad Susa Rose-Ackerma.(005) Electoral rules ad costitutioal structures as costraits o corruptio. British Joural of Political Sciece, 35(4), Lee, S. (995). Edogeous sharig rules i collective-group ret-seekig, Public Choice, 85, Lidbeck, Assar, ad Jörge W. Weibull. (987) Balaced-budget redistributio as the outcome of political competitio. Public Choice, 5(3), Myerso, Roger, (993). Effectiveess of Electoral Systems for Reducig Govermet Corruptio: A Game-Theoretic Aalysis. Games ad Ecoomic Behavior 5: 8-3. Myerso, Roger B. (999). Theoretical comparisos of electoral systems. Europea Ecoomic Review, 43(4), Nalebuff, Barry J. ad Joseph E. Stiglitz, (983). Prizes ad icetives: towards a geeral theory of compesatio ad competitio, The Bell Joural of Ecoomics, -43. Nitza, Shmuel, (99). Collective ret dissipatio, Ecoomic Joural, 0, Nitza, Shmuel ad K. Ueda, (0). Prize sharig i collective cotests, Europea Ecoomic Review, 55, Persso, Torste ad Guido Tabellii. (000). Political Ecoomics: Explaiig Ecoomic Policy. The MIT Press. Persso, Torste ad Guido Tabellii. (003). The Ecoomic Effect of Costitutios. The MIT Press. Persso, Torste, Guido Tabellii ad Fracesco Trebbi. (003). Electoral rules ad Corruptio. The Joural of the Europea Ecoomic Associatio : Ray, Debraj, Jea-Marie Balad ad Olivier Dagelie (007). Iequality ad iefficiecy i joit projects, The Ecoomic Joural, 7, Schleiter, Petra, ad Alisa M. ozaya, (04). Party system competitiveess ad corruptio. Party Politics, 0(5), Sisak, Daa, (009). Multiple-prize cotests: The optimal allocatio of prizes, Joural of Ecoomic Surveys, 3,

24 Tavits, M., (007). Clarity of resposibility ad corruptio, America Joural of Political Sciece, 5(), pp.8-9. Tullock, G. (980): Efficiet Ret Seekig, i Toward a Theory of the Ret-Seekig Society, ed. by J. M. Buchaa, R. D. Tolliso, ad G. Tullock, College Statio, TX: Texas A&M Uiversity Press, 97-, reprited i Roger D. Cogleto, Arye L. Ueda, K. (00). Oligopolizatio i collective ret-seekig, Social Choice ad Welfare, 9, Witer Eyal (004), Icetives ad Discrimiatio, America Ecoomic Review, 94, Appedix: proofs Proof of Propositio We start by provig a useful lemma. ( ) Lemma : E j Ej σ e ij = e ij. Proof of Lemma Usig the defiitio of E j, E j = E j e ij = i= (e ij) σ] σ, we get: ] σ ( σ) (e ij) σ (e ij ) σ σ = (e ij ) σ i= i= (e ij ) σ ] σ σ = ( Ej e ij ) σ. The first-order coditio associated to maximisig (6) is: E e i E (E + E ) eβ i = 0. At a symmetric Nash equilibrium, e ij = e kj for ay i ad k ad j, thus E (E +E ) = /4E. We thus get: Thus: 4E σ σ e β = 0. e = ( ) 4E σ β σ ( Ej e ij ) σ = σ σ ad 4