Crowdsourcing with Endogenous Entry

Transcription

1 Crowdsourcng wth Endogenous Entry Arpta Ghosh Preston McAfee February 21, 212 Abstract We nvestgate the desgn of mechansms to ncentvze hgh qualty outcomes n crowdsourcng envronments wth strategc agents, when entry s an endogenous, strategc choce. Modelng endogenous entry n crowdsourcng markets s mportant because there s a nonzero cost to makng a contrbuton of any qualty whch can be avoded by not partcpatng, and ndeed many stes based on crowdsourced content do not have adequate partcpaton. We use a mechansm wth monotone, rankbased, rewards n a model where agents strategcally make partcpaton and qualty choces to capture a wde varety of crowdsourcng envronments, rangng from conventonal crowdsourcng contests wth monetary rewards such as TopCoder, to crowdsourced content as n onlne Q&A forums. We begn by explctly constructng the unque mxed-strategy equlbrum for such monotone rank-order mechansms, and use the partcpaton probablty and dstrbuton of qualtes from ths constructon to address the queston of desgnng ncentves for two knds of rewards that arse n the context of crowdsourcng. We frst show that for attenton rewards that arse n the crowdsourced content settng, the entre equlbrum dstrbuton and therefore every ncreasng statstc ncludng the maxmum and average qualty (accountng for partcpaton, mproves when the rewards for every rank but the last are as hgh as possble. In partcular, when the cost of producng the lowest possble qualty content s low, the optmal mechansm dsplays all but the poorest contrbuton. We next nvestgate how to allocate rewards n settngs where there s a fxed total reward that can be arbtrarly dstrbuted amongst partcpants, as n crowdsourcng contests. Unlke models wth exogenous entry, here the expected number of partcpants can be ncreased by subsdzng entry, whch could potentally mprove the expected value of the best contrbuton. However, we show that subsdzng entry does not mprove the expected qualty of the best contrbuton, although t may mprove the expected qualty of the average contrbuton. In fact, we show that free entry s domnated by taxng entry makng all entrants pay a small Yahoo! Research, Santa Clara, CA. arpta@yahoo-nc.com Yahoo! Research, Burbank, CA. mcafee@yahoo-nc.com 1

2 fee, whch s rebated to the wnner along wth whatever rewards were already assgned, can mprove the qualty of the best contrbuton over a wnner-take-all contest wth no taxes. 1 Introducton Crowdsourcng, where a problem or task s broadcast to a crowd of potental contrbutors for soluton, s a rapdly growng onlne phenomenon beng used n applcatons rangng from seekng solutons to challengng projects such as n Innocentve or TopCoder, all the way to crowdsourced content such as on onlne Q&A forums lke Y! Answers, StackOverflow or Quora. The two key ssues whch arse n the context of crowdsourcng are qualty s the obtaned soluton or set of contrbutons of hgh qualty? as well as partcpaton there s a nonzero effort or cost assocated wth makng a contrbuton of any qualty n a crowdsourcng envronment whch can be avoded by smply choosng to not partcpate, and ndeed many stes have too lttle content. In such a settng, the effort an agent decdes to exert wll depend on how many other agents are lkely to partcpate and how much effort they wll exert, snce the amount of effort necessary to obtan a partcular reward depends both on the number and strength of compettors an agent faces. Naturally, the level of effort agents choose, and therefore the qualty of the output created, depends on the ncentves offered to agents. How should rewards be desgned to ncentvze hgh effort, n a settng where entry s an endogenous, strategc choce? We are motvated by two dfferent knds of questons that arse n the context of desgnng rewards for crowdsourced content, dependng on the settng and the nature of the rewards. The frst s n the context of attenton rewards on usergenerated content (UGC based stes, such as onlne Q&A forums lke Quora or StackOverflow. Here, the mechansm desgner, or ste owner, has a choce about how many of the receved contrbutons to dsplay,.e., how to reward the contrbutons wth attenton he could choose to dsplay all contrbutons for a partcular task, or dsplay only the best few, suppressng some of the poorer contrbutons. What strategy mproves the qualty of the best contrbuton suppled? What about the average qualty of contrbutons? On the one hand, suppresson should cause qualty to rse, because the payoff to poor content falls; on the other hand, suppressng content also corresponds to decreasng the total reward pad out, whch could decrease qualty. Is t a good dea, n a game-theoretc sense, to dsplay all contrbutons? The second queston arses n settngs where there s some fxed total avalable reward whch can be dstrbuted arbtrarly amongst the agents. Ths happens, for example, n the settng of crowdsourcng contests wth monetary rewards, where the prncpal posng the challenge or task offers some fxed amount of money for obtanng the soluton to the challenge. Another nstance s systems whch reward agents wth vrtual ponts 1. (The dstncton between 1 If the value of ponts s determned only n proporton to the total number of ponts awarded (so that just doublng the number of ponts awarded for all tasks has no effect on 2

3 ths settng and attenton rewards s that t s not possble to take away attenton from the second poston and add t to the frst poston snce, to a frst approxmaton, attenton to the second spot comes from a subset of vewers provdng attenton to the frst; so attenton rewards cannot be arbtrarly redstrbuted across ranks. How can rewards be desgned to mprove the qualty of contrbutons n settngs wth arbtrarly redstrbutable rewards, when entry s endogenous? Our contrbutons. We use a mechansm wth monotone, rank-based rewards n a model wth contrbutors who strategcally choose both partcpaton and qualty to smultaneously capture a wde varety of crowdsourcng envronments, rangng from conventonal crowdsourcng contests wth monetary rewards such as TopCoder, to crowdsourced content such as n Q&A forums. We frst analyze the equlbra of such monotone rank-order mechansms, and explctly construct the unque mxed-strategy equlbrum for ths mechansm ( 3. We then use ths constructon, whch explctly gves us the equlbrum partcpaton probablty and dstrbuton of qualtes, to address the queston of how to desgn rewards for each of the two settngs prevously mentoned. We frst show ( 4 that for attenton rewards, the entre equlbrum dstrbuton and therefore every ncreasng statstc, ncludng the maxmum and average qualty (accountng for partcpaton mproves when the rewards for every rank but the last are as hgh as possble: f there are n potental contrbutors, then the optmal mechansm sets the attenton rewards for ranks 1 through n 1 to be the maxmum possble, whle the attenton to the nth rank s curtaled to the cost of producng the lowest possble qualty contrbuton (note here that k < n agents may partcpate, n whch case only the rewards for ranks 1,..., k are gven out. If ths cost s low, ths prescrbes, roughly speakng, dsplayng all but the poorest contrbuton. We next nvestgate redstrbuton of rewards ( 5. Unlke n models wth exogenous entry wth a fxed number of partcpants, t s possble here to ncrease the expected number of partcpants by subsdzng entry, for example, by provdng a small reward to all partcpants n addton to a large reward to the wnner. In models wth exogenous entry, more partcpants lead to hgher qualtes, suggestng that subsdzng entry may be productve n ths endogenous entry settng as well. Also, even f subsdzng entry (at the cost of payng less to the wnner were to reduce the equlbrum dstrbuton from whch each contrbutor chooses her qualty, the expected value of the maxmum qualty could nonetheless ncrease when the number of contrbutors ncreases, snce we have the maxmum of a larger number of random varables. However, we show that subsdzng entry does not mprove the expected value of the maxmum qualty, although t may mprove the expected value of the total contrbuton. In fact, we show that free entry (correspondng to a wnner-take-all contest s domnated by taxng entry - makng all entrants pay a small fee, whch s rebated ncentves, the total number of ponts avalable to reward agents wth s effectvely fxed as well. 3

4 to the wnner along wth whatever rewards were already assgned, can mprove the expected qualty of the outcome. Related work. There s a growng lterature on the optmal desgn of contests [13, 14, 12], as well as specfcally on the desgn of onlne crowdsourcng contests [1, 4, 3] and onlne procurement (e.g., [15]. The most relevant of these to our work are the followng. [13] nvestgates the optmal structure of rewards when the objectve s to maxmze the sum of qualtes of contrbutons, for concave, lnear and convex costs; [14] also consders the objectve of maxmzng the hghest qualty contrbuton. [1] studes the optmal desgn of crowdsourcng contests n a settng wth agents wth heterogeneous abltes and lnear costs, when the objectve s to maxmze the sum of the top k qualtes mnus the total reward pad out to agents. [3] study the desgn and approxmaton of optmal crowdsourcng contests modeled as all-pay auctons, agan for agents wth lnear costs, and nvestgate the extent of wasted effort compared to conventonal procurement. There s also a volumnous economcs lterature on contest desgn not focused on crowd-sourcng, see, for example, [2] and references theren. The key dfference between ths lterature and our work s endogenous entry all these papers assume some fxed number n of contestants who always partcpate (.e., the cost of producng the lowest possble qualty c( =, whereas whether to partcpate or not s an endogenous strategc choce n our model (.e., we allow for c( >. That endogenous partcpaton may matter s foreshadowed by the aucton lterature, whch s the bass for much of the modelng of crowdsourced content provson auctons wth endogenous entry are qute dfferent than auctons wth exogenous partcpaton. For nstance, whle postng monopoly reserve prces s always part of seller maxmzaton n aucton models wth exogenous partcpaton, a monopoly seller sets effcent reserve prces when partcpaton s endogenous [11]. Endogenous entry makes a substantal dfference n the crowd-sourcng models for much the same reason t s no longer possble to reduce the profts of the contrbutors, because those proft levels are determned by the cost of entry. Our results on the optmalty of taxng n 5 are foreshadowed by Taylor [16] and Fullerton and McAfee [5], both of whom show, albet n dfferent settngs, that free entry produces too much entry. An addtonal, though less mportant, dfference wth the lterature on onlne crowdsourcng contests [1, 3] s that we allow general cost functons rather than restrctng lnear cost functons. There s also a related lterature on models wth endogenous entry [11, 5, 16, 9, 7, 8], although largely outsde the specfc settng of contest desgn (wth the excepton of [16, 8]. [16] studes a settng wth agents who all have a common exogenous cost to partcpaton, and draw the qualty of ther output from some dstrbuton. An agent s only strategc choce s whether or not to enter n each perod of a possbly mult-perod game. In ths model, [16] fnds that restrctng entry wth taxaton s optmal. The key dfference from our work, of course, s that qualty s an endogenous choce n our model as opposed to an exogenous draw from a dstrbuton. [8] uses a very smlar agent model to that n [16], but nstead addresses the queston of mplementaton of optmal outcomes are contest structures where the hghest qualty contrbuton receves some hgh 4

5 prze and all other contrbutons receve some low prze adequate to mplement the optmal outcome achevable wth nonstrategc agents? We do not address the queston of mplementablty of optmal outcomes, but rather ask how to mprove equlbrum outcomes. [9, 7] address the queston of ncentvzng hgh-qualty user-generated content (UGC n a game-theoretc framework wth strategc agents and endogenous entry, a settng related to that of crowdsourced content. However, [9, 7] focus on the performance of mechansms n the lmt of dvergng rewards (as s the case wth attenton rewards n the context of very popular UGC stes such as Youtube or Slashdot, whle our results address the case of fnte, or bounded, rewards, as s relevant n much of crowdsourcng. 2 Model We model a general socal computng or crowdsourcng scenaro as a game wth rank-dependent rewards,.e., a rank-order mechansm wth reward a for producng the th best contrbuton, and focus on the effect of the reward structure on the qualtes of contrbutons produced by strategc agents n a sngle mcromarket, such as one crowdsourcng contest or a queston n a Q&A forum. There s a mcromarket wth a pool of n agents, each of whom s a potental partcpant n ths mcromarket. Each agent can choose whether to contrbute or not, as well as the qualty of the contrbuton she makes f she chooses to enter. Agents make the decson of whether to partcpate strategcally,.e., entry s endogenous, and each agent that chooses to partcpate then chooses her level of effort, modeled as the qualty q of the output she produces, strategcally to maxmze her utlty. We next descrbe the utlty of an agent. The cost, or effort, requred to producng a contrbuton of qualty q for each agent s c(q. We wll assume that c(q s some strctly ncreasng, contnuously dfferentable functon of q. Although we do not need ths assumpton, t wll be useful to magne that c( >,.e., there s a nonzero cost to producng a contrbuton, even one of the lowest possble qualty. Ths nonzero partcpaton cost models, for example, the cost of readng and understandng the task for whch contrbutons are beng solcted, whch can be avoded by smply choosng to not partcpate. Snce c( >, partcpaton always has a strctly postve cost, whereas not partcpatng at all ncurs zero cost and produces zero beneft, and therefore has a net utlty of. Homogenety. Note that our model of costs assumes homogenety amongst all potental contrbutors, correspondng to assumng that agents do not dffer n ther abltes, but smply n the amount of effort that they choose to put n. Whle there are ndeed settngs where potental contrbutors may dffer wdely n ther abltes, there are also settngs where t s effort, rather than ablty, whch domnates the qualty of the outcome produced (for example, wrtng a revew for a product on Amazon, or producng an artcle for a crowdsourced-content based ste such as Assocated Content whch requres exhaustvely researchng the topc rather than nherent expertse, fall n ths category. Also, n several settngs, such as specfc topcs or categores n Q&A 5

6 forums lke Quora or Stackoverflow, the set of potental contrbutors may be self-selected to have rather smlar abltes or expertse levels, and therefore have smlar costs to producng a partcular qualty. Whle the most complete model of the real world would allow for dfferences n both ablty and effort, we choose here to focus on strategc choce of effort,.e., to focus on the strategc queston faced by an agent of how lttle effort can I get away wth?, snce ths s a reasonable frst approxmaton n many settngs relevant to crowdsourcng. Mechansm G(a 1, a 2,..., a n. Once agents have made ther partcpaton and qualty choces, the mechansm observes the qualtes q produced by the agents who enter, and awards przes a to the partcpants n decreasng order of qualty. Specfcally, a mechansm G(a 1, a 2,..., a n awards a prze of value a to the entrant who produces the th hghest-qualty contrbuton. If more than one agent produces the same qualty, the mechansm breaks tes randomly amongst these agents to obtan a strct rank order, and assgns rewards accordng to ths order. No przes are awarded to agents who do not enter, and specfcally, f no agent partcpates, no prze s awarded. We note here that we assume that qualtes are perfectly observable, as n all the pror lterature on contest desgn and crowdsourcng contests [13, 3, 4, 1] snce each task n a crowdsourcng envronment s usually posed by some prncpal who can rank the contrbutons n decreasng order of qualty (such as the person postng the task n a crowdsourcng contest or the asker n an onlne Q&A forum, ths assumpton s reasonable, partcularly snce G(a 1, a 2,..., a n only uses the relatve ranks, and not the actual absolute values of the qualtes. We wll focus throughout on monotone mechansms, n whch hgher ranks receve hgher rewards, and not all rewards are equal. Defnton 2.1. Consder a mcromarket wth n agents. We say G(a 1, a 2,..., a n s a monotone mechansm f a 1 a 2... a n and at least one nequalty s strct,.e., a > a +1 for some 1 n 1. We say G(a 1, a 2,..., a n s monotone nonnegatve f G(a 1, a 2,..., a n s monotone and a n,.e., all rewards are nonnegatve. Soluton concept. We use the soluton concept of a symmetrc Nash equlbrum, snce agents payoff functons are symmetrc n the parameters of the game. In a symmetrc strategy, each contrbutor partcpates wth the same probablty and follows the same strategy of qualty choces condtonal on partcpatng. We wll denote a par of partcpaton probablty and CDF that consttute a symmetrc mxed strategy by (p, G(q. Defnton 2.2. A symmetrc mxed strategy equlbrum (p, G(q s a probablty p and a dstrbuton G over qualtes q such that when every agent enters wth probablty p, and chooses a qualty drawn from the CDF G(q condtonal on enterng, no agent can ncrease her expected utlty by devatng from ths strategy,.e., by changng ether the probablty wth whch she partcpates or the dstrbuton G from whch she draws a qualty. 6

7 3 Equlbrum Analyss We begn by analyzng the equlbra of the mechansm G(a 1, a 2,..., a n, whch we wll then use to compare outcomes n dfferent mechansms. We frst prove the followng smple lemma, whch elmnates the possblty of pure strategy equlbra n whch all partcpants choose the same qualty. Lemma 3.1. There exsts no symmetrc equlbrum n the game G(a 1, a 2,..., a n where all partcpants choose the same qualty q condtonal on enterng when the cost c(q s contnuous. Proof. Suppose there s a symmetrc equlbrum n whch all partcpants choose the same qualty q condtonal on enterng. If k agents enter (where k can be a random varable f partcpants randomzed over the choce of entry, the expected payoff to each agent that enters (where the expectaton s over random tebreakng s k =1 a /k. So the expected payoff to an agent who enters wth qualty q s the expectaton over all possble values of k, E[U(q, q ] = n Pr(k k=1 k =1 a k c(q. But note that enterng wth qualty q + ɛ s a proftable devaton f all other agents who enter choose qualty q: choosng q + ɛ gves ths agent a reward of a 1 for all values of k. Snce a > a +1 for some, a 1 > n a =1 n (and Pr(n > n a symmetrc equlbrum n whch all partcpants enter wth some postve probablty. Snce c(q s contnuous, there exsts a choce of ɛ such that E[U(q + ɛ, q ] > E[U(q, q ] whch consttutes a proftable devaton, contradctng the assumpton that there s a symmetrc equlbrum n whch all agents choose the same qualty. Snce there can exst no symmetrc pure strategy equlbra n whch all agents choose a sngle qualty q condtonal on enterng, we wll nvestgate symmetrc mxed strategy equlbra where all agents randomze over ther choce of qualty (condtonal on enterng usng the same dstrbuton. Frst, of course, we need to establsh the exstence of such symmetrc mxed strategy equlbra we wll do ths by explctly constructng such an equlbrum. The next theorem establshes some propertes that any symmetrc mxed strategy equlbrum to G(a 1, a 2,..., a n, f one exsts, must possess. We wll use these propertes to prove the exstence of a symmetrc mxed strategy equlbrum by constructng one n Theorem 3.2. Theorem 3.1. Let (p, G(q be any symmetrc mxed strategy equlbrum to G(a 1, a 2,..., a n. If the agents cost c(q s contnuous and strctly ncreasng n q, then G(q s contnuous,.e., contans no mass ponts, and has support on an nterval wth left endpont. 7

8 Proof. Let U(q = n =1 a P r( q, where P r( q s the probablty of beng ranked th when choosng qualty q, gven that the remanng agents partcpate wth probablty p and draw qualtes accordng to the dstrbuton G(q condtonal on partcpatng, and tes are broken at random. U(q s the beneft to ths agent from enterng wth qualty q when other agents play (p, G(q. The payoff to an agent who enters wth qualty q, when other agents play accordng to a mxed strategy (p, G(q n G(a 1, a 2,..., a n s π(q = U(q c(q = n a P r( q c(q. 1. G(q has no mass ponts: We frst show that G(q s contnuous on ts support,.e., t has no mass ponts. Suppose not; let q be a mass pont. Then, snce tes are broken randomly, note that lm U(q + ɛ > U(q, ɛ snce there s a postve probablty of a te at q = q, whch can be elmnated by choosng a slghtly hgher qualty. Ths mples that there s an ɛ such that the payoff from choosng q + ɛ s strctly greater than that at q, snce the cost functon c s contnuous n q: lm π(q + ɛ = lm(u(q + ɛ c(q + ɛ ɛ ɛ > U(q c(q. But ths means q cannot belong to the support of an equlbrum dstrbuton G, a contradcton. So G(q s contnuous on ts support. 2. q mn = : Let q mn denote the nfmum of the qualtes n the support of G. Snce G contans no mass ponts as shown above, G(q mn =. Suppose, for a contradcton, that q mn >. But then, an agent can proftably devate by choosng q mn ɛ nstead of q mn : snce G(q mn =, an agent choosng q mn wll be ranked lowest among all agents who enter anyway,.e., U(q mn ɛ = U(q mn. But c(q mn > c(q mn ɛ snce c(q s strctly ncreasng, so π(q mn ɛ > π(q mn, yeldng a proftable devaton, contradctng q mn belongng to the support of an equlbrum dstrbuton. Ths argument holds for any qualty strctly greater than the lowest qualty, whch s. Therefore, any equlbrum dstrbuton G must have q mn =. 3. Interval support: Fnally, we argue that the support of G(q must be an nterval (.e., the support contans no holes, or equvalently, G(q s strctly ncreasng between and q, where G( q = 1. Suppose not; then there must exst some q 1 < q 2 such that G(q 1 = G(q 2 (recall that we have already ruled out mass ponts, so specfcally, there can be no mass pont at q 2. But then U(q 1 = U(q 2, snce the qualty q affects the 8

9 probablty of beng ranked n any partcular poston only va G(q (see (1. So π(q 2 = U(q 2 c(q 2 < U(q 1 c(q 1 = π(q 1 snce c(q 1 < c(q 2, a contradcton to q 2 belongng to the support n equlbrum. Therefore, there can be no such holes n the support of G,.e., the support s an nterval. The probablty P r( q that an agent choosng qualty q has the th hghest qualty when the remanng n 1 agents play accordng to (p, G(q when G(q s contnuous, s the probablty that 1 other agents partcpate (each wth probablty p and choose qualty greater than q (each wth probablty 1 G(q, and the remanng n agents ether do not partcpate or choose qualty less than q,.e., ( n 1 P r( q = (p(1 G(q 1 (1 p(1 G(q n. 1 Note that ths expresson s vald only because G has no mass ponts, snce f there s a mass pont at q there s a postve probablty of more than one agent usng the same qualty, whch leads to tes that are broken randomly. Then, the beneft U = n =1 a P r( q under a contnuous CDF G s ( n 1 U = a +1 (p(1 G(q (1 p(1 G(q. (1 = Before proceedng wth the constructon of an equlbrum, we evaluate a dervatve whch wll be used repeatedly n our proofs n ths secton. Settng wrte Then, du dx = = = x(q = p(1 G(q, ( n 1 U(x = a +1 x (1 x. (2 ( n 1 a +1 x 1 (1 x = n 2 = (n 1 = a +2 ( n 2 n 2 x (1 x n 2 (n 1 ( n 1 a +1 (n 1 x (1 x n 2 = ( n 2 a +1 x (1 x n 2, so that n 2 du dx = (n 1 ( n 2 (a +2 a +1 x (1 x n 2. (3 = 9

10 For a monotone mechansm,.e., wth a a +1 and at least one strct nequalty, note that du dx > for x (, 1. Now we wll use Theorem 3.1 to construct, and therefore demonstrate the exstence of a symmetrc mxed strategy equlbrum (p, G(q n G(a 1, a 2,..., a n. Theorem 3.2 (Equlbrum Constructon. There exsts a symmetrc mxed strategy equlbrum (p, G(q to G(a 1, a 2,..., a n when a a +1 for all ; ths equlbrum s unque up to ncluson of the endponts of the support. Proof. We wll construct a canddate par (p, G(q for whch no agent can beneft by changng p and no agent wll want to devate from G(q; to fnsh the proof we verfy that p and G(q are ndeed a vald probablty and CDF, respectvely. Before constructng the equlbrum, we note that f a 1 c(,.e., the maxmum possble reward s less than the cost of producng the lowest possble qualty, no agent can derve nonnegatve utlty from partcpatng n G(a 1, a 2,..., a n rrespectve of the actons of other agents. In ths case, the only equlbrum s that no agents partcpate n G(a 1, a 2,..., a n (.e., p = ; the choce of G(q s meanngless, whch s not very nterestng. In what follows, therefore, we wll assume that a 1 > c(. Frst, from Theorem 3.1, we know that a mxed strategy equlbrum (p, G(q, f one exsts, has support on an nterval [, q] wth G( = and G( q = 1. Also, snce G(q s contnuous, the payoff at qualty q [, q] s π(q = U(p(1 G(q c(q, where U(x s the functon defned n (2. Usng the fact that belongs to the support and G( = (no mass ponts, we can wrte the payoff at as π( = U(p c( = ( n 1 a +1 (1 p p c(. = If p = 1, U(p = U(1 = a n 2. Therefore, f p = 1, π( = a n c(. But for p to be an equlbrum probablty of partcpaton, we must have π(. Therefore, p can be 1 only f a n c(,.e., f a n < c( then p < 1 n equlbrum. Conversely, f a n c(, we must have p = 1 n equlbrum. At p = 1, π( = a n c(. Snce U(p s a strctly decreasng functon of p on (, 1, π( > for any p < 1. But then f p < 1, an agent has an ncentve to devate and ncrease the probablty of partcpaton snce payoffs are strctly postve, contradctng the fact that p s an equlbrum partcpaton probablty. Therefore, p = 1 f and only f a n c(. 2 Ths corresponds to the fact that when all agents partcpate (p=1, an agent choosng qualty comes n last (recall that there s no mass pont at and gets beneft a n. 1

11 For (p, G(q to be an equlbrum, we must have equal payoffs throughout the support,.e., π(q = K for all q [, q]. Further, snce (p, G(q s a free entry equlbrum, no agent must have an ncentve to change her decson to partcpate. Ths means that f p < 1, we must have K =,.e., equlbrum payoffs must be zero unless p = 1. We now use ths together wth the prevous argument to construct our equlbrum. 1. a n < c(: If a n < c(, then set p to be the value that satsfes ( n 1 a +1 (1 p p = c(. (4 = Note that the left-hand sde s a contnuous, strctly decreasng functon of p on (, 1, takng value a 1 at p = and a n at p = 1. Therefore there s a unque soluton n (, 1 to ths equaton when c( satsfes a n < c( < a 1,.e., there s a unque soluton p whch s a vald probablty. The dstrbuton G(q s the soluton to ( n 1 a +1 (1 p(1 G(q (p(1 G(q = c(q. = for each q n [, q], where q s the unque soluton (snce c(q s strctly ncreasng to c( q = a 1. Note that the value of q s that whch solves G( q = 1 n the equaton above. 2. a n c(: If a n c(, set p = 1. The dstrbuton G(q has support on the nterval [, q], where and G(q s gven by the soluton to = c( q = a 1 a n + c(, ( n 1 a +1 (1 p(1 G(q (p(1 G(q = c(q + a n c( for each q n [, q]. Agan, note that q s obtaned by settng G( q = 1 n the equaton above. To verfy that our constructon s ndeed an equlbrum, note that no agent has an ncentve to devate and choose a dfferent p: when p < 1, π(q = so there s no beneft from ncreasng or decreasng p, and π(q for p = 1 so no agent wants to decrease partcpaton n ths case. Also, no agent wants 11

12 to devate from G(q condtonal on partcpatng: frst, π(q s equal for all q [, q], so an agent mght only want to devate by choosng qualty greater than q. But note that n both cases (p < 1 and p = 1, π(q = U( q c(q < π( q for any q > q, snce an agent choosng q s guaranteed to wn the maxmum possble reward anyway (recall that G has no mass ponts, specfcally at q. So no agent wants to devate from (p, G(q. We have already verfed that the value of p les between and 1,.e., t s a vald probablty. The last thng we need to verfy s that the dstrbuton G(q computed n both cases s ndeed a CDF (note that the clamed propertes of G, namely contnuty wth support on [, q] follow drectly from the contnuty of U(x n x and c(q, and by constructon. To show ths, we need to show that that G s ncreasng on (, q,.e., G q = ( U / q ( U G s nonnegatve on (, q. Now, observe that n ether case (a n c( or a n < c(, G(q can be wrtten as the soluton to U(p(1 G(q = c(q + max{a n c(, }, where p s determned approprately. Therefore, U(q = c(q+max{a n c(, } s a strctly ncreasng functon of q,.e., U q >. Also, wth x = p(1 G, U G = U x x G = p U x. Usng the expresson n (3, and the fact that a a +1 wth strct nequalty for some, U G > on (, q. So we have ( ( G U q = / p U > q x on (, q (recall that a 1 > c( by assumpton, so p >. By constructon, G( = and G( q = 1, so G(q s ncreasng and les n [, 1] for q [, q]. So G(q s a vald CDF. The followng two facts about the equlbrum are mmedate from the proof above. Corollary 3.1. For any rewards (a 1,..., a n such that a a +1 and at least one nequalty s strct, 1. The equlbrum partcpaton probablty p n G(a 1, a 2,..., a n s 1 f and only f a n c(. 2. The maxmum qualty q n the support of G s gven by c( q = a 1 max{a n c(, }. 12

13 4 Increasng Attenton Rewards We begn wth nvestgatng the desgn of ncentves n the context of attenton rewards. Such attenton rewards arse, for example, n stes that are based on user-generated content (UGC such as Q&A forums lke Quora or StackOverflow, or Amazon revews. In these settngs, there s some avalable amount of attenton reward for the top spot or answer (derved from all the vewers who read the contrbuton dsplayed frst, a smaller amount for the second spot (correspondng to the vewers that contnue on to the second, and so on,.e., some maxmum possble rewards A 1, A 2,..., A n that can be obtaned by always showng all avalable contrbutons for each poston 1,..., n. Attenton rewards have an unusual constrant when contrasted wth monetary or vrtual ponts rewards of the knd we dscuss n 5: the total avalable reward n =1 A cannot be arbtrarly redstrbuted amongst agents snce, to a frst approxmaton, attenton to the second spot comes from a subset of vewers provdng attenton to the frst. Thus, whle t s possble to freely ncrease or decrease each of the rewards a between and A (subject, of course, to the monotoncty constrant,.e., a a +1, t s not easy to take away reward from a 2 and redstrbute t to a 1. 3 Now, a ste featurng UGC could suppress some of the UGC, e.g. by only showng the top-ranked content, or reducng the promnence of lesser ranked content,.e., the ste could choose a < A by not always (or never dsplayng the th ranked contrbuton. Does ths strategy mprove the qualty of the best contrbuton suppled? On the one hand, equlbrum qualtes should rse, because the payoff to poor qualty falls. However, the payoff to supplyng any content also falls, so partcpaton falls as well. How do these two effects nteract? What f we were nterested n a dfferent metrc of performance, and not just n the best contrbuton for example, do the qualtes of the average contrbuton, or the qualty of the second best or thrd best contrbuton behave the same way as the qualty of the best contrbuton as a functon of a, or do they behave dfferently? Intutvely, t seems plausble that the soluton for maxmzng the qualty of the best contrbuton may dffer from what maxmzes the qualty of an average contrbuton, snce lower rewards for non-wnnng contrbutons should ncrease the ncentve to be best, but hgher rewards for non-wnnng contrbutons may ncrease the average qualty. The followng theorem says that the entre dstrbuton of equlbrum qualtes (accountng for the fact that agents partcpate probablstcally, and therefore every ncreasng statstc, mproves when the rewards for achevng any of the the frst through last-but-one ranks ncreases (ths unform mprovement s n contrast to the case wth redstrbuton, as we wll see n 5. Therefore, t s optmal to ncrease each of the a 1, a 2,..., a to the maxmum extent possble. However, the stuaton s somewhat more subtle for a n, the subsdy to the 3 We note that randomzng between dsplayng q 1 and q 2 n the frst and second spot can acheve the opposte redstrbuton, namely ncrease a 2 at the expense of a 1, but we do not consder ths here snce t adversely affects the user experence. The analyss n 5 addresses ths ssue. 13

14 contrbutor wth the lowest possble rank: f the current value of a n < c( then ncreasng a n mproves qualty, but f a n s farly large already,.e., a n > c( then a decrease n a n mproves qualty. Lemma 4.1. The dervatve of the probablty wth whch an agent chooses qualty greater than q n equlbrum wth respect to a, d(p(1 G da, s postve for = 1,..., n 1 for all q (, q. The dervatve wth respect to a n, d(p(1 G da n s postve when a n < c( but negatve for a n > c( for all q (, q. Proof. We have from the equlbrum constructon that H(p(1 G, a U(p(1 G(q c(q max{a n c(, } =, (5 where U(x s the beneft functon defned n (1. Dfferentatng (5 gves us We use the dervatve U x p(1 G: H x d(p(1 G da = H a H p(1 G. calculated n (3 for the denomnator, wth x = n 2 = (n 1 (a +2 a +1 < = ( n 2 x (1 x n 2 for x (, 1, snce a a +1 wth at least one strct nequalty. For a 1,..., a, ( H n 1 = (p(1 G 1 (1 p(1 G n > a 1 for q (, q. Therefore, d(p(1 G da > everywhere,.e., ncreasng the rewards for each of the frst through n 1th postons always mproves the equlbrum dstrbuton. For a n, when a n < c( or equvalently p < 1, U a n = (p(1 G >, on (, q, but when a n c( (so that p = 1, U a n = (p(1 G 1 <. Therefore, d(p(1 G da n > for a n < c( but d(p(1 G da n < for a n c(. Thus, the reward for the last poston behaves dfferently ncreasng a n untl t equals c( mproves equlbrum qualtes, but when a n c(, ncreasng a n further make the equlbrum qualtes worse. 14

15 Recall also from Corollary 3.1 that c( q = a 1 max{a n c(, }, so that the maxmum qualty n the support decreases lnearly wth a n when a n c(. Ths mmedately gves us the followng result. Theorem 4.1. Suppose each of the rewards a s constraned to le below some maxmum value A, a A, where A 1... A n. Then, the choce of rewards (a 1,..., a n that optmzes the equlbrum dstrbuton of qualtes, and therefore the expected value of any ncreasng functon of the contrbuted qualtes, s a = A, = 1,..., n 1; a n = mn(a n, c(. 5 Redstrbuton of Rewards We now address the queston of how to optmally redstrbute reward amongst agents to mprove equlbrum qualty. Ths queston arses n settngs where there s some total avalable reward that can be dstrbuted n any arbtrary way amongst agents, as n the case of crowdsourcng contests such as TopCoder, or even contests wth vrtual ponts, where ponts have value only relatve to the total number of ponts n the system, so that effectvely there s a fxed budget of avalable reward. We note that ths settng s the one that has been studed wdely n the contest desgn lterature n economcs, and n the growng lterature on the desgn of crowdsourcng contests, unlke the settng n 4; the key dfference, as dscussed n the secton on related work, s that our model allows for endogenous entry. Whch value of (a 1,..., a n leads to the best equlbrum outcome amongst all mechansms G(a 1, a 2,..., a n wth the same expected payout? What do we mean by best outcome,.e., what s the objectve to optmze? As we wll see, unlke n the prevous secton wth attenton rewards, not all ncreasng statstcs of the qualty dstrbuton need be optmzed by the same allocaton of rewards. We wll focus largely on the expected qualty of the best contrbuton, snce ths s the objectve of nterest n many settngs lke crowdsourcng contests wth an arbtrarly redstrbutable total reward, and fnally brefly address the expected total qualty, whch s potentally relevant n settngs lke Q&A forums such as Y! Answers. We frst wrte the budget constrant that says we are restrcted to redstrbutng rewards,.e., the total expected payout to contestants must reman the same. Snce entry s endogenous, the number of partcpants n equlbrum s a random varable when p < 1, so not all przes a are always pad out. The expected payment to the wnners n equlbrum s B = n j=1 ( n p j (1 p n j j j k=1 a k 15

16 ( snce the payment when j contrbutors enter, whch happens wth probablty n j p j (1 p n j where p s the equlbrum partcpaton probablty, s j k=1 a k. Rearrangng, we have n n ( n B = a k p j (1 p n j. (6 j k=1 j=k Note that when p = 1, B = n =1 a. Before dervng our results for the maxmum qualty, we state a couple of techncal lemmas. The proof of the frst proposton below s obtaned easly by ntegratng by parts. Proposton 5.1. For any k n, and p, n ( ( n n 1 p p j (1 p n j = n x k 1 (1 x n k dx. j k 1 j=k We ntroduce some notaton before our next lemma. Defnton 5.1 (B k (q, W(k. Consder n agents playng accordng to the symmetrc mxed strategy (p, G(q. We defne ( n 1 B k (q = (p(1 G(q k 1 (1 p(1 G(q n k ; k 1 B k (q s the probablty that an agent enterng and choosng qualty q s ranked at poston k. We also defne W(k = q B k (qpg (qdq. W(k s the probablty that a partcular one of the n agents who enter wth probablty p and choose qualty from the dstrbuton G(q s ranked n the kth poston. The proof of the followng proposton uses the dentty n Proposton 5.1: Proposton 5.2. For any ndex s n, ( n 1 (1 (1 p n p s 1 (1 p 1 s s 1 n j=s ( n p j (1 p n j. j The followng techncal lemma uses Proposton 5.2 above, and s central to the proof of the man lemma. Lemma 5.1. Suppose a n < c(,.e., p < 1, and we vary the reward a s for some rank s and change a 1 to keep the budget B unchanged. da 1 W(s B W(1. 16

17 Proof. j=s da 1 = B / B. B a s a 1 We frst evaluate the quantty B a s when p < 1: ( B n n n n ( = p (1 p n j n ( + a k jp j 1 (1 p n j (n jp (1 p n j 1 dp a s j=s k=1 j=k ( ( n n n = p (1 p n j n 1 + n p k 1 (1 p n k dp, k 1 whch, usng (4, gves us Now, note that dp = B a s = n j=s k=1 ( n p (1 p n j + nc( dp. (7 ( s 1 p s 1 (1 p n s (n 1 n 2 = (a +2 a +1 ( n 2 p (1 p n 2. Usng both of these, together wth the nequalty n Proposton 5.2 and the defnton of W(k, and rearrangng, gves the result. We now state and prove the man lemma n ths secton, whch wll mmedately gve us the theorem on maxmzng the expected qualty of the best contrbuton. Lemma 5.2. Suppose the cost functon c s such that c (q/c(q s nonncreasng n q. Then, for any monotone nonnegatve contests G(a 1, a 2,..., a n, redstrbutng reward away from the wnner to any lower rank s > 1 locally decreases the expected qualty of the best contrbuton n equlbrum,.e., deq max. Proof. The expected value of the hghest qualty contrbuton obtaned n an equlbrum of G(a 1, a 2,..., a n, countng the utlty from recevng no contrbutons as the same as from a zero qualty contrbuton, s Eq max = q 1 (1 p(1 G(q n dq. We are nterested n the effect of shftng reward from the wnner to some lower rank s on the expected hghest qualty contrbuton n equlbrum,.e., the 17

18 effect of changng a s when a 1 s adjusted so as to preserve the expected payout on Eq max : deq max d q = 1 (1 p(1 G(q n dq q { dp(1 G(q = np(1 p + pg(q + da } 1 dp(1 G(q dq. da 1 Recall the equlbrum condton from (5: H(p(1 G, a = U(p(1 G(q c(q max{a n c(, } =. Dfferentatng, we have Therefore, dp(1 G(q da k H p(1 G pg (q = c (q. (8 = H a H p(1 G B = B k(qpg (q c, (9 (q where B k (q s as n Defnton 5.1. Case 1: a n < c(, or p < 1. Usng the nequalty boundng da1 from Lemma 5.1, we have deq max q { np(1 p + pg(q B s(q W(s } pg W(1 B1(q (q c (q dq q { } (1 p + pg(q Bs(qpG (q = nw(s B1(qpG (q dq. c (q W(s W(1 Now, recall that each term multplyng (1 p+pg(q c (q densty: n ths dfference s a f k (q = B k(qpg (q W(k = B k(qpg (q q B k(qpg (qdq. Therefore, the term wthn the parentheses s the dfference between two denstes f s (q and f 1 (q, wth the property that the frst densty (correspondng to s puts less weght on hgher values of q (formally, t s easy to verfy that the two dstrbutons f 1 and f s satsfy the MLRP property f s > 1, whch mples frst order stochastc domnance. Therefore, the rght-hand sde s the dfference between the expected value of the functon (1 p+pg(q c (q computed wth respect to the denstes f s (q and f 1 (q. Therefore, deqmax s non-postve f (1 p+pg(q c (q s an ncreasng functon of q or equvalently that 18 c (q (1 p+pg(q

19 s decreasng. Recall that ths dervatve beng non-postve mples that ncreasng the reward a 1, whle decreasng a s n such a way as to hold the budget constant, mproves the expected hghest qualty. We now show that a suffcent condton for ths s that c (q c(q s non-ncreasng n q. The equlbrum condton gves us: 1 c(q ( n 1 j j= Usng ths, we have Snce c (q (1 p + pg(q = c (q c(q (p(1 G(q j (1 p(1 G(q n j 1 a j+1 = 1. ( n 1 j j= p(1 G(q (1 p(1 G(q j a j+1. ( (1 p(1 G(q j s decreasng n q for every j and each aj+1 by assumpton, we have that f c (q c(q s decreasng n q, then decreasng n q as well, completng the proof. c (q (1 p+pg(q Case 2: a n c(, or p = 1. In ths case, all agents always partcpate, and the budget constrant smplfes to B = n =1 a. Therefore, da 1 = 1. B Also, snce p = 1, we have deq max = d = = q q q 1 G(q n dq { dg(q ng(q + da 1 dg(q da 1 ng(q { dg(q dg(q da 1 } dq. } dq Usng (8 wth p = 1, and notng that H a k = B k (q for k = 1,..., n 1, we have as before dg(q = B k(qg (q da k c, (q for k = 1,..., n 1. We substtute ths to obtan for any s < n: deq max q G(q = c (nb s (qg (q nb 1 (qg (q dq. (q Now, nb k (qg (q, whch s equal to n ( k 1 (1 G(q k 1 G(q n k G (q s postve and ntegrates out to 1, so t s a densty. Moreover, t puts more weght s 19

20 on hgher q for lower k. Therefore, f G(q c (q s ncreasng, negatve snce s > 1. As before, we substtute to obtan ( n 1 a n c( + (1 G(q j G(q n j 1 a j+1 = c(q j j= c (q G(q = c (q c(q deq max wll be c( n 2 G(q + ( n 1 (1 G(q j j G(q j a j+1. Agan, the term wthn parentheses s decreasng n q, so f c (q c(q then G(q c (q j= s decreasng, s ncreasng, and the dervatve deqmax s nonpostve. Fnally, note that when s = n, dg(q = B k (q 1, so that dervatve can be mmedately seen to be negatve. Together, we have the result for the case a n c(. Ths lemma mmedately gves us the two man theorems. The frst result states that f we are restrcted to nonnegatve rewards a,.e., chargng for entry s not feasble, then a wnner-take-all contest maxmzes the expected qualty of the best contrbuton n equlbrum amongst all possble monotone, nonnegatve allocatons of the total budget amongst partcpants. Ths result agrees wth the results from the lterature on contest desgn and crowdsourcng contests whch do not model endogenous entry (eg [1, 3]. Theorem 5.1. Suppose the cost functon c s such that c (q/c(q s nonncreasng n q. Then, the expected qualty of the best contrbuton obtaned n equlbrum among all monotone nonnegatve contests G(a 1, a 2,..., a n that have the same total expected payout, s maxmzed by a wnner-take-all contest,.e., at (a 1,,...,. The second result states that free entry does not lead to the optmum level of qualty for the best contrbuton, and n fact restrctng entry by taxaton can mprove the maxmum equlbrum qualty. Theorem 5.2. Consder a wnner-take-all contest wth rewards (A,,...,, and suppose c( >. If the cost c s such that c (q/c(q s decreasng n q, then taxng entry locally mproves the expected qualty of the best contrbuton n equlbrum. Theorem 5.2 shows that t s advantageous to tax partcpants and use the proceeds to subsdze the best qualty result, n order to maxmze the best qualty result. In models where partcpaton s exogenous, the desrablty of a tax would not be surprsng, because some addtonal profts can be extracted wth no loss of partcpaton. In contrast, wth endogenous entry, a tax wll 2

21 drve down partcpaton, whch means we choose the maxmum from a smaller number of random varables, potentally leadng to a poorer outcome. It s therefore not surprsng that the theorem on the optmalty of a tax requres a condton on the cost functon, although we note that ths condton s satsfed by lnear costs whch are typcally used n the crowdsourcng contest desgn lterature [4, 1, 3], as well as exponental and other cost functons. What ths condton accomplshes s to nsure that the gan from mprovng the dstrbuton of qualty of partcpants wll domnate the loss of partcpaton from a small tax. Average or total qualty. What f we are nterested n the average, or total, qualty nstead of the maxmum qualty? The average qualty s smply the expected value of q drawn accordng to the CDF 1 p(1 G(q, where as before, we count nonpartcpaton, or no contrbuton, as producng the same utlty as a contrbuton wth qualty : Eq avg = q 1 (1 p(1 G(qdq = q p(1 G(qdq. The total qualty s n tmes ths average qualty. Here, unlke the case wth attenton rewards, we wll see that the mechansm that s best for maxmum qualty need not be the best for average qualty. We state the followng two theorems. Theorem 5.3. Suppose c (q = 1, and let denote the expected total qualty. Consder the wnner-take-all contest G(a,,...,. G(a,,...,. Then deqavg at Proof. Snce a n =, and we have assumed that partcpaton ncurs a nonzero cost,.e., c( >, we have p < 1. Then, the zero payoff equlbrum condton s c(q = a 1 (1 p(1 G(q. (Recall that we are consderng wnner-take-all contests. Dfferentatng, and usng the assumpton of lnear cost functons, we have c (q = a 1 (n 1(1 p(1 G(q n 2 pg (q. (1 We have deq avg = = d q p(1 G(qdq { dp(1 G(q + da 1 q B } dp(1 G(q dq. da 1 Usng (9 and settng a 2 =... = a n =, we have dp(1 G(q = ( s 1 (p(1 G(q s 1 (1 p(1 G(q n s a 1 (n 1(1 p(1 G(q n 2. 21

22 Usng the calculatons n the proof of Lemma 5.1, and agan usng a 2 =... = a n =, we get deq avg 1 q ( n 1 = (p(1 G(q s 1 (1 p(1 G(q 2 s a 1 (n 1 s 1 n ( n ( j=s j p j (1 p n j + n s 1 p s 1 (1 p n+1 s (1 p(1 G(qdq 1 (1 p n + n(1 pn (usng (1 q ( n 1 = ( (p(1 G(q s 1 (1 p(1 G(q n s pg (q s 1 n ( n ( j=s j p j (1 p n j + n s 1 p s 1 (1 p n+1 s (1 p(1 G(q pg (qdq 1 (1 p n + n(1 pn p ( n 1 = ( x s 1 (1 x n s = 1 n =, s 1 n ( n j=s j n j=s p j (1 p n j + n 1 (1 p n + n(1 pn ( n p j (1 p n j j n (1 pn n(1 + (1 pn n j=s ( s 1 p s 1 (1 p n+1 s n ( n j=s j p j (1 p n j + n (1 x dx 1 (1 p n + n(1 pn ( n p j (1 p n j (1 (1 p n j where the fnal nequalty follows from applyng Proposton 5.2. ( s 1 p s 1 (1 p n+1 s ( n 1 s 1 p s 1 (1 p 1 s Ths theorem says that for lnear cost functons, the equlbrum expected total qualty s ncreased by ncreasng the reward to the hghest rank at the expense of any lower rank at the wnner-take-all contest (a,,...,. Here, there s too much entry for the average or total qualty objectve as well, and taxaton, or chargng entrants a small fee that s rebated to the wnner, locally mproves total qualty. Next, we consder exponental cost functons, c(q = e kq (k > here, whether the average qualty mproves wth taxes or subsdes depends on the sze of the avalable reward B. Theorem 5.4. Suppose c (q/c(q = k, where k > s ndependent of q. Consder the wnner-take-all contest G(a,,...,. Then deqavg < for small enough B, whle deqavg > for large enough B. 1 (1 p n n 22

23 Proof. Usng the zero proft equlbrum condton and (1, we have k = c (q c(q = (n 1pG (q (1 p(1 G(q. We have deq avg = d q p(1 G(qdq 1 q (n 1pG ( (q n 1 = a 1 (n 1 (1 p(1 G(q ( (p(1 G(q s 1 (1 p(1 G(q 2 s s 1 n ( n ( j=s j p j (1 p n j + n s 1 p s 1 (1 p n+1 s (1 p(1 G(qdq 1 (1 p n + n(1 pn = 1 q ( n 1 ( (p(1 G(q s 1 (1 p(1 G(q 1 s pg (q a 1 s 1 n ( n ( j=s j p j (1 p n j + n s 1 p s 1 (1 p n+1 s pg (qdq 1 (1 p n + n(1 pn = 1 ( p (n n ( n ( 1 x s 1 (1 x 1 s j=s j p j (1 p n j + n s 1 p s 1 (1 p n+1 s dx a 1 s 1 1 (1 p n + n(1 pn ( = 1 p ( n ( n ( n 1 x s 1 (1 x 1 s j=s j p j (1 p n j + n s 1 p s 1 (1 p n+1 s dx p. a 1 s (1 pn Now let f(p = p ( n ( n n 1 x s 1 (1 x 1 s j=s j p j (1 p n j + n dx p s (1 pn ( s 1 p s 1 (1 p n+1 s, and observe that f(p s postve for large p and negatve for small p (recall s > 1. To see that f(p s negatve for small p, frst note that f( =. Also, f (p lm p p s 1 = ( n 1 s 1 ( n 1 = s 1 <. ( n s 1 n (s 1 Therefore f(p s negatve for small p, and deq avg < ( n s 1 (s 1 n 23

24 for p near zero, correspondng to a small B. For large p, p ( n 1 f(p x s 1 (1 x 1 s dx 1 n s 1 n 1, ( n 1 s 1 p s 1 (1 p n+1 s snce for s 2 (recall x < 1, p xs 1 (1 x 1 s dx p 1/2 ( x 1 x dx ln(1 p as p 1. Therefore, deq avg > when p s large, correspondng to large B, for exponental costs. Ths theorem says that for exponental costs, the effect on the expected average qualty of ncreasng a 1 whle decreasng a s to mantan the budget for any s > 1, depends on the value of B: when B s small, taxng entry mproves average qualty, but when B s large, the average qualty s ncreased by subsdzng entry. Recall that our results on the expected maxmum qualty do apply to exponental costs, and suggest that taxng entry s optmal for maxmzng the expected qualty of the best contrbuton. Thus, when the avalable reward B s large, the mechansms to maxmze the qualty of the best and average contrbutons need not be the same taxng entry mproves the best contrbuton s qualty, whereas subsdzng entry s what mproves the total qualty of contrbutons produced over a wnner-take-all contest for exponental cost functons and large B. 6 Acknowledgments. We are grateful to Matt Jackson for helpful remarks, and n partcular suggestng the dea of taxng entry to mprove qualty. References [1] N. Archak, A. Sundarajan, Optmal Desgn of Crowdsourcng Contests, Internatonal Conference on Informaton Sysytems (ICIS, 29. [2] M. R. Baye, D. Kovenock and C. G. de Vres, The all-pay aucton wth complete nformaton, Economc Theory Volume 8, Number 2, , [3] S. Chawla, J.D. Hartlne, B. Svan, Optmal Crowdsourcng Contests, ACM SODA 212. [4] D. DPalantno, M. Vojnovc, Crowdsourcng and all-pay auctons, 1th ACM conference on Electronc commerce, EC 9. [5] R. Fullerton, R.P. McAfee, Auctonng Entry nto Tournaments, Journal of Poltcal Economy (17:3, ,

25 [6] A. Glazer, R. Hassm, Optmal Contests, Economc Inqury, 26(1, , [7] A. Ghosh, P. Hummel, A Game-Theoretc Analyss of Rank-Order Mechansms for User-Generated Content, 12th ACM Conference on Electronc Commerce (EC, 211. [8] A. Ghosh, P. Hummel, Implementng Optmal Outcomes n Socal Computng: A Game-theoretc Approach, Workng paper, 211. [9] A. Ghosh, R.P. McAfee, Incentvzng Hgh-Qualty User Generated Content, 2th Internatonal World Wde Web Conference (WWW, 211. [1] V. Krshna, J. Morgan, The Wnner-Take-All Prncple n Small Tournaments, Advances n Appled Economcs Vol. 7 (M. Baye, ed. JAI Press, Stamford, [11] R.P. McAfee, J. McMllan, Auctons and Bddng, Journal of Economc Lterature, 25, no. 2, June 1987, [12] D.B. Mnor, Increasng Effort Through Softenng Incentves n Contests, Workng Paper, Unversty of Calforna, Berkeley, 211. [13] B. Moldovanu, A. Sela, The Optmal Allocaton of Przes n Contests, Amercan Economc Revew, 91(3, , 21. [14] B. Moldovanu, A. Sela, Contest archtecture, Journal of Economc Theory, 126(1:7-97, 26. [15] E. Snr and L. Htt. Costly Bddng n Onlne Markets for IT Servces, Management Scence, , 23. [16] C. Taylor, Dggng for golden carrots: An Analyss of Research Tournaments, Amercan Economc Revew, 85(4, ,