Crowdsourcing Contests with Private Types and Evaluation Uncertainty

Transcription

1 Crowdsourcing Contests with Privte Types nd Evlution Uncertinty Konstntinos I. Stours University of Virgini, Chrlottesville, Virgini, November, 2017 Abstrct. We introduce generl model of crowdsourcing contests in which the performnce of n gent is driven by his privtely known bility type dverse selection, his costly effort choice nd it is further ffected by evlution uncertinty or luck morl hzrd. We show tht when the mrginl cost of effort is sufficiently lrge nd the solvers types, s well s, the evlution uncertinty re independent nd identiclly distributed respectively, there exists unique Byes Nsh equilibrium effort. This equilibrium effort is in symmetric nd pure strtegies nd involves inctive types. Our model includes s specil cses the Tullock contest Tullock, 1980, the ll-py uction model with privte informtion of Moldovnu nd Sel 2001, nd the symmetric effort with evlution uncertinty model of Lzer nd Rosen 1981 tht hve been studied seprtely thus fr. Our results suggest tht severl comprtive sttics results of the ll-py uctions with privte informtion re robust in the presence of noisy feedbck. Key words: crowdsourcing, contests, existence, uniqueness, pure-strtegy Byes-Nsh equilibrium, privte informtion, morl hzrd 1. Introduction Despite the prevlence of crowdsourcing contests in prctice the outcome of which is bsed on privtely known expertise level bility of ech solver, his costly choice of effort nd noisy feedbck by the contest orgnizer seeker, bsic issues such s the existence nd uniqueness of n equilibrium hve been ddressed only prtilly. Indeed, previous work focused on purely dverse selection models of the ll-py contest see for exmple Moldovnu nd Sel 2001 nd the models discussed in Chpter 3 of Vojnović, 2016, or on purely morl hzrd models see Lzer nd Rosen 1981 nd Tullock 1980, nd the models discussed in Chpter 4 of Vojnović, To our knowledge, little in generl is known bout the uniqueness of n equilibrium of generl crowdsourcing contest model tht combines both dverse selection nd morl hzrd. In this pper, we estblish condition tht is sufficient for the uniqueness of the equilibrium of generl crowdsourcing contest model. Previous work by Chwl nd Hrtline 2013 hs shown tht the equilibrium of purely dverse selection ll-py contest with independent nd identiclly distributed IID vlutions with liner effort cost is unique nd involves pure, symmetric nd strictly monotone strtegies. The result of Chwl nd Hrtline is robust in the 1

2 2 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty ddition of noise ffecting the rnkings of the solvers: the unique equilibrium is shown to be in pure, symmetric nd wekly monotone strtegies. We consider crowdsourcing contest model in which the cost of effort is convex function nd solvers effort re perturbed by n IID noise with continuous nd tomless density. In prticulr, equilibrium existence, but not uniqueness, is gurnteed to lie in wekly monotone pure strtegies by the seminl work of Athey For our setting the result of Athey 2001 cn be strengthened to show the existence of symmetric equilibrium. As long s the mrginl cost of effort is sufficiently lrge, we estblish tht the equilibrium is unique nd my entil inctive types Wärneryd, 2003; Levin nd Smith, 1994; Wsser, 2013; Stours, Generlly, the equilibrium existence ensures tht the model is consistent, while uniqueness strengthens the nlysis of the optiml design of such contest. 2. The model A firm seeker, she seeks to solve problem through the use of crowdsourcing contest. The seeker hs fixed budget r nd elicits solutions from popultion of n solvers who could potentilly prticipte in the contest. One of the benefits of crowdsourcing is tht the seeker cn tp into pool of tlent from outside solvers. However, these solvers exhibit significnt heterogeneity in terms of their expertise. Ech solver i is privtely informed bout his own bility type i. Abilities re independently drwn from commonly known identicl distribution F C 1 with n tomless density f with support [, ], where 0 < < <. Upon entry ech solver i simultneously chooses his effort e i i 0 t cost 1 ce i i i. In the specil cse where the cost of effort function is the identity, the solvers hve qusi-liner utility s it is stndrd in the mechnism design literture. We ssume tht c is twice continuously differentible with c e > 0 nd c e 0 for ll e > 0, nd with c 0 = 0 nd c 0 > 0. Further, we normlize solvers outside option nd entry fee to zero. A solver with bility i who chooses effort e i i submits solution with performnce level X i given by X i = e i i + E i, solvers performnce where E i is rndom vrible tht models seeker s subjective tste, luck, or simply noise in observing solver i s effort perfectly. Tht is, the bsolute performnce of solver is not known to him. We ssume tht the noise terms re identiclly nd independently distributed from common knowledge nd tomless distribution G C 2 with zero men, support R nd density g. Further, ssume tht the limits lim x g x nd lim x + g x exist. An exmple of distribution tht stisfies these conditions is the Stndrd Norml distribution. Observe tht when the vrince of noise is zero, we obtin the observble effort model with privte types 1 This is without loss of generlity since the bility type i is known constnt to solver i.

3 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty 3 of Moldovnu nd Sel Similrly, when the vrince of solvers bility is zero, solvers performnce is equivlent to the symmetric output model with morl hzrd of Lzer nd Rosen To motivte solvers to prticipte nd exert effort, the seeker lloctes her budget r to the solver with the highest performnce. Tht is, we ssume tht the winner of the contest is determined ccording to Winner-Tkes-All WTA lloction. Tht is, the utility of solver i is either r ce i i if his performnce rnks 1st out of n, or ce i i otherwise. Note tht ties re events of mesure zero. Using order sttistics rguments, our nlysis cn be extended 2 to ccount for wekly decresing rewrd lloction in which r j r j+1 for j = 1,..., N 1. In summry, the timing of the sttic gme is s follows. The seeker nnounces her budget. Ech solver privtely lerns his bility. The solvers simultneously choose effort levels ccording to their bility types. Ech submitted solution is ffected by seeker s subjective tste. Finlly, the solution with the highest performnce is llocted the seeker s budget. 3. Anlysis Our sttic gme hs two min fetures. First, the effort choice of solver is ffected by his privtely known bility, s well s, by the effort choices of the other solvers. Yet, ech solver hs only distributionl informtion on the bilities of the other solvers. Second, deterministic increse in effort of solver is ffected by noise which is outside the control of the solver. Tht is, given n ordered sequence of solvers efforts, the ddition of noise cn entirely flip the rnk-order of efforts with positive probbility. The conventionl pproch in ll-py uctions to estblishing the existence of unique symmetric equilibrium effort entils three steps. First, specific form of the equilibrium is conjectured. Second, the best response of n rbitrry solver i is found given the conjectured equilibrium strtegy of the other solvers. Third, it is shown tht the best response of solver i is unique nd identicl to the proposed equilibrium strtegies of the rest solvers. However, s we show below, this conventionl pproch hs only limited utility in our setting. The resons re twofold. First, the co-existence of privte informtion nd morl hzrd mke the derivtion of symmetric equilibrium intrctble in closed-form. Hence, it is problemtic to estblish the equilibrium uniqueness. Second, with the exception of very specil cses, the second-order condition of the expected utility of solver to be strictly negtive t the proposed equilibrium my limit the region for which unique equilibrium exists. It remins chllenging to chrcterize properties of solvers effort such s monotonicity with respect to their type nd comprtive sttics without closed-form of proposed equilibrium. 2 Our nlysis remins vlid for ny given rewrd lloction chosen by the seeker. The relted mechnism design question of the optiml rewrd lloction shll be ttempted by the uthors in future reserch.

4 4 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty As result, we rely on generl tools developed by Athey 2001 to estblish the existence of n equilibrium in non-decresing, pure nd symmetric strtegies. Then, we show tht the sufficient conditions of Theorem 4 of Mson nd Vlentinyi 2010 re stisfied nd hence the equilibrium is unique. Theorem 1 Equilibrium existence nd uniqueness. There exists Byes-Nsh equilibrium effort in non-decresing, pure nd symmetric strtegies. For sufficiently lrge c 0 the equilibrium effort is unique. The first prt of Theorem 1 shows tht pure equilibrium effort exists under our frmework 2 nd it is wekly incresing in the bility types of the solvers. We prove tht solvers ction spce is bounded nd tht their expected utilities stisfy specific single crossing property. Then, by the fixed point theorem of Athey 2001 our gme of incomplete informtion with continuum ctions nd continuous expected utilities must hve n equilibrium in pure nd nondecresing strtegies. In our setting, the result of Athey 2001 cn be extended to estblish the existence of symmetric equilibrium in pure nd non-decresing strtegies. The second prt of Theorem 1 estblishes sufficient condition for equilibrium uniqueness. In prticulr, we show tht for sufficiently lrge mrginl cost of effort, ech solver s mrginl interim utility is strictly incresing in his bility type, nd tht ech solver s type hs lrger impct on his mrginl interim utility thn the efforts of his opponents. This implies tht given solver s type, the best response correspondence is contrction; hence, there is t most one equilibrium Mson nd Vlentinyi, Tken together, the two prts of Theorem 1 imply the existence nd uniqueness of n equilibrium effort in crowdsourcing contests with privte types nd evlution uncertinty tht stisfy our modeling ssumptions 2. Importntly, Theorem 1 estblishes condition tht rules out the existence of mixed or symmetric equilibri in brod clss of noisy contests ll-py uctions with convex costs pyments. Chwl nd Hrtline 2013 present structurlly reveling rgument for the non-existence of symmetric equilibri in uction gmes where the rnk-order of the bid of solver is deterministic function of his rnk reltively to the rnk of the bids of the other solvers. In our nottion, Chwl nd Hrtline 2013 show tht when the cost of effort function is the identity function, then symmetric equilibri with pure continuous nd strictly incresing strtegies cnnot be in Byes-Nsh equilibrium. We note tht the proofs of Chwl nd Hrtline 2013 continue to hold in the bsence of evlution uncertinty when cost of effort is convex, or equivlently when solvers re rnked ccording to concve function of effort. Further, in deterministic contests with IID types ny mixed equilibrium is pyoff equivlent to pure equilibrium s it ensures equl probbility to chieve certin rnk-order for lmost ll solver

5 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty 5 types. Tht is, the pure nd symmetric structure of the equilibri of deterministic contests with IID types is robust in the ddition of noisy feedbck. Strtegies in Byes-Nsh equilibrium in deterministic contests re lwys strictly monotone in solvers types Chwl nd Hrtline, In contrst, if the performnce rnk-order of solver is ffected by perturbtion of his effort, the equilibrium effort is wekly monotone in generl. Proposition 1. Assume tht the sufficient condition for equilibrium uniqueness of Theorem 1 holds. There exists unique solver with bility min [, ] such tht e i = 0 for ll i [, min ], nd e i > 0 nd strictly incresing for ll i min, ]. Proposition 1 shows tht noisy contests with unique equilibrium induce screening of types. It is known tht deterministic contests with strictly positive entry fees or outside options operte s mechnism tht restricts the entry of solvers of sufficiently low bility see for exmple Moldovnu nd Sel, Surprisingly, in noisy contests this holds even when the vlue of solvers outside option is normlized to zero. The existence of screening of types implies tht the solvers re priori uncertin bout the number of other solvers tht would exceed the screening threshold min nd will exert strictly positive effort see for exmple Levin nd Smith 1994, Moldovnu nd Sel 2001 nd Stours et l Tht is, in noisy contests with unique equilibrium, the number of prticipting solvers follows the Binomil distribution with popultion prmeter n nd entry probbility p = 1 F min. We note tht the presence of inctive types, or tht the equilibrium effort in noisy contests is zero for mss of types is lso observed by Wärneryd 2003 in common-vlue setting. Our ssumption of continuous types is sufficient condition for solvers to ply wekly monotone strtegies in symmetric equilibrium. When solvers types re discrete, we cn recover this structurl property when the noise hs sufficiently smll support. To see tht, suppose tht there re n = 2 solvers with bility types either 0 Low with probbility p L or 1 High otherwise. Also, ssume tht the cost of effort is the liner function c e = 3 e nd let R = 1. Then, in equilibrium e H 1 3 nd e L = 0. Suppose tht when solver exerts effort, he is impcted by noise with support 0.1, 0.1. Since rnkings in bility re preserved when rnked by costly effort nd noise, the unique pure symmetric equilibrium is e L = e H = 0. Next, we relte our generl model 2 to other well known models of the contest literture. Corollry 1. i Assume tht solvers bility is constnt lmost surely, but solvers effort is ffected by noise. Then, solvers expected utility is equivlent to the expected utility of Lzer nd Rosen 1981 with WTA rewrd lloction.

6 6 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty ii Assume tht the noise is zero lmost surely for ll solvers i, but solvers bility is privte informtion. Then, solvers expected utility is equivlent to the expected utility of Moldovnu nd Sel 2001 with WTA rewrd lloction. A detiled proof is unnecessry. As the vrince of bility V [A] 0, the gent types become more homogeneous ex-nte nd the equilibrium effort tends to the effort predicted by the symmetric model of Lzer nd Rosen Similrly, s the vrince of noise V [E i ] 0, min nd rnkings in bility re preserved by rnkings in effort in equilibrium s in Moldovnu nd Sel We note tht winner-tkes-ll contest with bility homogeneous solvers nd noise following n extreme vlue distribution is equivlent stndrd Tullock contest. Luce nd Suppes 1965 were the first to show tht the extreme vlue distribution leds to the discrete choice model of logit. McFdden 1974 completed the equivlence by showing lso the converse: the logit formul for the choice probbility must imply tht the unobserved utility is distributed ccording to the extreme vlue distribution. To illustrte, consider contests with contest success function of Tullock For discrimintory power r > 0, contest success function e r i n j=1 er j nd cost of effort c e = e Tullock contest cn be represented s contest with bility homogeneous solvers nd noise from the Gumbel distribution. Following McFdden 1974 the probbility of winning the contest is P [ e i E i < e i E i] where Ei n i=1 re IID with CDF F ε = exp ε r. By simple trnsformtion, this is equivlent to contest with noise Ẽi = ln E i following the Gumbel distribution F ε = F exp ε nd cost of effort c e = e 0 xdx = e2 2. However, lthough solving the respective first-order conditions of these contests would led to the sme equilibrium effort, the stndrd Tullock contest nd its ssocited noisy contest re not pyoff equivlent. In prticulr, the equilibrium existence nd uniqueness of the stndrd Tullock contest holds for smller rnge of discrimintory powers r n n 1, compred to its noisy contest counterprt. 4. Concluding remrks While this pper hs focused on the existence nd uniqueness of the equilibrium effort of crowdsourcing contest with privte IID types nd IID noise, our techniques cn be used to estblish the uniqueness of the equilibrium effort in contest models with purely privte informtion or morl hzrd. Future reserch to be ttempted by the uthors shll study whether noisy feedbck benefits the objective of the contest seeker, s well s, the optiml design of such noisy contest. 5. Acknowledgments We thnk Louis Lester Chen nd Ionnis Pnges for fruitful discussions.

7 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty 7 References S. Athey. Single crossing properties nd the existence of pure strtegy equilibri in gmes of incomplete informtion. Econometric, 694: , S. Chwl nd J. D. Hrtline. Auctions with unique equilibri. In Proceedings of the fourteenth ACM conference on Electronic Commerce, pges ACM, O. Kdn. Discrete prices nd competition in deler mrket. Working Pper, Wshington University in St. Louis, E. Lzer nd S. Rosen. Rnk-order tournments s optimum lbor contrcts. Journl of Politicl Economy, 895: , D. Levin nd J. L. Smith. Equilibrium in uctions with entry. The Americn Economic Review, pges , R. D. Luce nd P. Suppes. Preference, utility, nd subjective probbility. Hndbook of mthemticl psychology, 3: , R. Mson nd A. Vlentinyi. The existence nd uniqueness of monotone pure strtegy equilibrium in byesin gmes. Working Pper, University of Mnchester, D. McFdden. The mesurement of urbn trvel demnd. Journl of public economics, 34: , B. Moldovnu nd A. Sel. The optiml lloction of prizes in contests. Americn Economic Review, pges , K. I. Stours. Incentive design of on-demnd mrketplces for service nd innovtion. INSEAD Ph.D. Disserttion, K. I. Stours, J. Hutchison-Krupt, nd R. O. Cho. Motivting prticiption nd effort in innovtion contests. Working Pper, University of Virgini, G. Tullock. Efficient rent seeking. In J. Buchnn, R. Tollison, nd G. Tullock, editors, Towrd theory of the rent-seeking society, chpter 3, pges Texs A&M University Press, College Sttion, M. Vojnović. Contest Theory. Cmbridge University Press, K. Wärneryd. Informtion in conflicts. Journl of Economic Theory, 1101: , C. Wsser. Incomplete informtion in rent-seeking contests. Economic Theory, 531: , 2013.

8 8 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty Appendix A. Proofs All proofs of the sttements re given in the order of their ppernce in the min text. Any dditionl results re stted nd proved here. Lemm 1. Consider continuous rndom vrible with support R nd continuous probbility density function g tht is strictly positive on R. Further, ssume tht the limits lim x g x nd lim x + g x exist. Then, g is uniformly bounded. Proof of Lemm 1. For ny rbitrry x R fix n ε > 0 nd consider the bounded intervl [x ε, x + ε]. By continuity of g there would exist constnt k > 0 such tht g x k. Assume now tht lim x + g x exists nd tht lim x + g x = +. This implies tht s x + the corresponding cumultive distribution function G x +, which is contrdiction. Proof of Theorem 1. Existence. Let U i i, e i, e i := E [ U i i, e i, e i]. Note tht U i i, e i, e i r ce i nd tht by choosing effort e i = 0 solver i cn gurntee himself i non-negtive expected utility. Therefore, effort levels e i > c 1 i r re clerly dominted for solver i. Tht is, we cn restrict the effort choice of solver i to the bounded intervl [ 0, c 1 i r ]. We compre the performnce of solver i ginst n rbitrry solver j i. If ech solver j i employs strtegy e j A j, the expected utility of solver i who hs privtely known bility i nd exerts effort e i is E [ U i i, e i, e ] i = r P [X i rnked 1st out of n] c e i i = r j i = r = r = r j i P [ e i + Z i > e j A j + Z j ] c e i i P [ Z j < e i e j A j + Z i ] c e i i R j i R P [ ] Z j < e i e j A j + x dg x c e i G e i e j j + x i df j dg x c e i j i i From the uniform bound 2 U i e i i i, e i, e i = c e i > c 0 > 0 the Single Crossing Condition 2 i 2 for gmes of incomplete informtion SCC in Athey 2001 is stisfied. Our model is lso consistent with ssumption A1 in Athey Further, the ction spce of ech solver is bounded nd the expected utility U i i, e i, e i is continuous in ei, e i for ll i. Hence, by Corollry 2.1 of Athey 2001 there exists n equilibrium in pure nd non-decresing strtegies e i i.

9 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty 9 Next, we show existence of symmetric equilibrium in pure nd non-decresing strtegies. Theorem 1 in Appendix B of Kdn 2002 extends Corollry 2.1 of Athey 2001 nd shows existence of n equilibrium in symmetric, pure nd non-decresing strtegies for gmes of incomplete informtion when the ction spce is finite nd types re independently drwn from the sme distribution IID. Then, Theorem 2 of Athey 2001 implies the existence of symmetric equilibrium for gmes of incomplete informtion with continuum of ctions nd IID types. Thus, we conclude the existence of n equilibrium in symmetric, pure nd non-decresing strtegies for our setting. Uniqueness. Due to independence, t symmetric equilibrium the expected utility of solver i becomes E [ U i i, e i, e ] i = r R n 1 G e i e + x df dg x c e i i We now provide sufficient condition for the equilibrium to be unique. By our existence result bove, this would gurntee tht the unique equilibrium would be in symmetric, pure, nd non-decresing strtegies. We focus on symmetric pure strtegies nd pply the Theorem 4 of Mson nd Vlentinyi We show tht their ssumptions U1, U2, U3, D1 nd D2 hold in our model. U1: By the Men Vlue Theorem nd the uniform bound 2 U i e i i i, e i, e i > c 0 > 0 we 2 hve tht U1 holds with δ := c 0 nd the Eucliden metric d 2 T e i, e j := e i e j. U2: We first show tht U i e i i, e i, e i is uniformly bounded. Note tht the first term on the RHS of U i i, e i, e n 2 i = r n 1 G e i e + x df e i R g e i e + x df dg x c e i i is positive nd it is mximized t e i = c 1 i r nd e = 0 for ll [, ]. Set L i, e i := c e i i nd We hve tht U i, e i := r R n 1 n 2 G c 1 r + x df g c 1 r + x df L i, e i U i e i i, e i, e i < U i, e i dg x c e i i

10 10 Stours: Crowdsourcing Contests with Privte Types nd Evlution Uncertinty Then, by the Men Vlue Theorem U2 holds with ω := mx i, e i {L i, e i, U i, e i } > 0 U3: By the Men Vlue Theorem sufficient condition for U3 to hold is U i U i mx i, e i, e i min i, e i, e i κ e i,e i e i e i,e i e i for ll i Tht is, U3 holds with κ := r n 1 G c 1 r + x df n 2 g c 1 r + x df dg x > 0 R D1 As noted on p.28 in Mson nd Vlentinyi 2010 the ssumption tht solver types re independent implies tht D1 holds with ι := 0. D2 By Lemm 1 g is uniformly bounded, i.e. g x ν for ll x R. Then, D2 holds for ν := mx x R g x. there exists constnt ν > 0 such tht By Theorem 4 in Mson nd Vlentinyi 2010, if δ > ι ω + ν κ, or equivlently if c 0 > mx g x r n 1 G 2 x R R c 1 r + x n 2 df g c 1 r + x df dg x there exists unique Byes Nsh equilibrium. Proof of Proposition 1. The first-order derivtive E[U i i, e i, e ] e i is equl to n 2 r n 1 G e i e + x df g e i e + x df R dg x c e i i 1 Assume tht the sufficient condition for equilibrium uniqueness of Theorem 1 holds. Then, E [U i i, e i, e ] is strictly concve in e i. Tht is, 1 is strictly decresing in e i. The first-order condition FOC of solver i s mximiztion problem is E[U i i, e i, e ] ei e i 0 =e i with equlity for bility i where e i > 0. Hence, if e i = e i > 0, we must hve tht e j = e j > e i for ll j > i nd j i. Thus, unique solver with bility min [, ] must exist such tht e i = 0 for ll i [, min ], nd e i > 0 nd strictly incresing for ll i min, ].