Crowdsourcing with Tullock Contests: A New Perspective

Transcription

1 5 IEEE Conferene on Computer Communiations (INFOCOM) Crowdsouring with Tullok Contests: A New Perspetive Tie Luo, Salil S. Kanhere, Hwee-Pink Tan, Fan Wu, Hongyi Wu Institute for Infoomm Researh, A*STAR, Singapore Shool of Computer Siene and Engineering, The University of New South Wales, Australia Shanghai Key Laboratory of Salable Computing and Systems, Shanghai Jiao Tong University, China Center for Advaned Computer Studies, University of Louisiana at Lafayette, USA luot@ir.a-star.edu.sg, salilk@unsw.edu.au, hptan@ir.a-star.edu.sg, fwu@s.sjtu.edu.n, wu@as.louisiana.edu Abstrat Inentive mehanisms for rowdsouring have been extensively studied under the framework of all-pay autions. Along a distint line, this paper proposes to use Tullok ontests as an alternative tool to design inentive mehanisms for rowdsouring. We are inspired by the onduiveness of Tullok ontests to attrating user entry (yet not neessarily a higher revenue) in other domains. In this paper, we explore a new dimension in optimal Tullok ontest design, by superseding the ontest prize whih is fixed in onventional Tullok ontests with a prize funtion that is dependent on the (unknown) winner s ontribution, in order to maximize the rowdsourer s utility. We show that this approah leads to attrative pratial advantages: (a) it is well-suited for rapid prototyping in fully distributed web agents and smartphone apps; (b) it overomes the disinentive to partiipate aused by players antagonism to an inreasing number of rivals. Furthermore, we optimize onventional, fixed-prize Tullok ontests to onstrut the most superior benhmark to ompare against our mehanism. Through extensive evaluations, we show that our mehanism signifiantly outperforms the optimal benhmark, by over three folds on the rowdsourer s utility um profit and up to nine folds on the players soial welfare. I. INTRODUCTION Crowdsouring represents a new problem-solving model that eliits solutions, ideas, data, et. referred to as ontributions from an undefined, generally large group of people. Classi examples inlude Amazon Mehani Turk, Yahoo! Answers, GalaxyZoo.org, TopCoder.om, et. Reently, a new variant of rowdsouring alled partiipatory sensing emerged as a new data-olletion model, whih eliits sensor data ontributed from user-owned mobile devies suh as smartphones. Examples inlude GreenGPS [], LiveCompare [], ContriSense:Bus [3] and Waze.om, to name a few. Pivotal to the viability of all suh rowdsouring systems, is whether there is enough inentive to attrat suffiient partiipation. A large body of prior work [4] [] has been dediated to designing inentive mehanisms for suh senarios, where eah inentive mehanism essentially determines some reward aording to users ontributions. The ommonly adopted approah turns out to be autions, where eah bidder tenders a bid (e.g., planned sensing duration [4] or desired payment [5]) to the rowdsourer, who will then hoose the highest or lowest bidder(s) as the winner(s) to give out some reward. A widely used form among these autions is all-pay autions [6] [], whih niely aptures the senario where eah bid represents some irreversible effort. In other words, effort has to be sunk at the time of bidding, for example working out a solution to a problem, or sensing and sending data through a smartphone. A distintive harateristi of autions in general, is that they are perfetly disriminating []: the best (highest or lowest) bidder wins with probability one while the others lose for sure. Thus in all-pay autions, due to the inevitable sunk ost (every bidder has to pay for his bid regardless of whether he wins the aution or not), all bidders substantially shade (i.e., derease) their bids for fear of loss [3]. Furthermore, if a bidder believes that there exists some other bidder who will bid higher than him, he will hoose not to bid at all. Indeed, as Franke et al. [4] pointed out, all-pay autions are so disriminative that, under a omplete-information setting, only two (strongest) players will enter the aution in equilibria and only one of them will have a positive expeted payoff. Clearly, this is not desirable in many rowdsouring ampaigns that favor a large partiipant pool (yet not neessarily higher revenue) for the sake of more population diversity (e.g., LiveCompare []) and/or larger geographi overage (e.g., GreenGPS [] and Waze). Taking a radially different approah, this paper proposes to use Tullok ontests as an alternative framework to design inentive mehanisms for rowdsouring. Fathered by Tullok s seminal work [5], Tullok ontests represent a distint ontest regime that is imperfetly disriminating: every player has a stritly positive probability to win (determined by a ontest suess funtion) as long as he bids. This harateristi makes Tullok ontests highly onduive to attrating user entry, espeially weak players [4], whih is of high pratial interest beause weak players often onstitute the majority of potential partiipants in rowdsouring. This explains why, in reality, we see lotteries the simplest form of Tullok ontests muh more often, and usually engaging a larger number of partiipants, than all-pay autions. On the other hand, Tullok ontests are not neessarily superior to all-pay autions in terms of revenue, or total user ontribution. In fat, there is no onlusive theory as to whih ontest regime revenue-dominates the other in general [4], [6], [7], and an experimental study also shows that revenue in all-pay autions may be independent of the number of partiipants at some stable state [8]. Intuitively, the reason is that (a) the fiere ompetition indued by all-pay autions effiaiously inentivizes (a small number of) strong players to exert high effort, while (b) the mild ompetition in Tullok ontests attrats a medium amount of ontributions from more (albeit weaker) players. Therefore, In an asymptoti limiting ase, Tullok ontests subsume all-pay autions. However, they are generally lassified as two different ontest regimes. A strong or weak player in this paper refers to a player with a strong or weak type; type, as a term in Bayesian games and mehanism design, refers to the private information held by a player, suh as his valuation of the autioned item or his prodution ost. Therefore, a strong player means a player with high valuation or low ost, and vie versa /5/$3. 5 IEEE 55

2 5 IEEE Conferene on Computer Communiations (INFOCOM) all-pay autions are advantageous in eliiting the highest-quality ontribution from the strongest players, suh as seleting the best performer for a ompetition, while Tullok ontests are superior in attrating more users and hene are benefiial to population diversity and geographi overage, suh as in lifestyle [] and transport mobile apps [], [3]. Thus Tullok ontests are omplementary to all-pay autions. We note that these two frameworks have been ompared in terms of their respetive benefits in other domains suh as fundraising [6], lobbying [7], and general ontests [9]. We find that the omparison results therein an apply to rowdsouring in priniple. Therefore, the rest of the paper will fous, within the regime of Tullok ontests, on optimizing this framework for rowdsouring. The objetive, as is most ommon, is to maximize the rowdsourer s revenue. To this end, we explore a new dimension in the spae of Tullok ontest design, by superseding the ontest prize whih is fixed in onventional Tullok ontests with a prize funtion that is dependent on the (unknown) winner s ontribution. The rationale is to reate a two-tier inentive to improve the effiay of Tullok ontests: the first tier, as exists in onventional ontests as well, is for a player to win a prize by ompeting with and outdoing other players; on top of this, the seond-tier inentive is for eah player to outdo himself in order to amplify the prize. Logially, this approah also leads to a hange of the rowdsourer s objetive: maximizing revenue beomes maximizing profit revenue minus the (non-fixed) ost (prize) whih is also his utility. To the best of our knowledge, this paper is the first that introdues prize as a funtion into optimal Tullok ontest design, a subjet that is being pursued sine the 99 s [] following Tullok s seminal work [5] in 98. Ultimately, the new perspetive in the title of this paper has dual interpretations: (a) a new alternative mehanism-design framework for rowdsouring, and (b) a novel dimension of optimal Tullok ontest design. To find an appropriate benhmark for a new mehanism designed as suh to ompare against, we need a fixed-prize Tullok ontest. However, even this onventional and seemingly simple ase turns out to be hallenging a general analytial solution to its equilibria does not exist and only numerial ones are available in the literature [] [3]. Furthermore, we go one signifiant step beyond prior art, by not only solving equilibria of suh onventional ontests, but also optimizing the ontests by finding the best equilibrium in terms of the same (utility-maximizing) objetive. This allows us to ompare our proposed mehanism with the best possible benhmark. Extensive performane evaluations reveal that our mehanism outstrips the optimal benhmark by a remarkable margin: a 5% inrease in the rowdsourer s utility (profit) and a 83% improvement in the players aggregate utility (soial welfare). The improved performane ahieved with these two typially ompeting metris reflets a highly desirable win-win situation. A. Related Work Even in their simplest form, Tullok ontests are analytially more hallenging to takle than most lassi autions. This is partiularly true in the inomplete-information setting 3 whih is a more realisti setting for rowdsouring. Speifially, the equilibria of most lassi autions with omplete information, or with inomplete information and symmetri players, an be solved in losed form; but Tullok ontests with inomplete information is generally intratable in analytial means, even in the simplest form (lottery) [4]. This an be attributed to the double unertainty: in autions with inomplete information, the unertainty about other players types is the only soure of unertainty; but in Tullok ontests, the imperfetly disriminating nature or more speifially the probabilisti winner seletion (unlike in autions the highest bid guarantees winning) reates another soure of unertainty. As a onsequene, the literature on Tullok ontests exlusively deals with the omplete-information setting or restritive versions of the inomplete-information setting (e.g., only two disrete types for two players [5]). It was not until 8 that Fey [] first proved the existene of a (symmetri) equilibrium for a lottery with inomplete information. However, the model is limited to two players and uniform distribution, and the equilibrium strategy is only numerially haraterized. A subsequent breakthrough was made by Ryvkin [], who extended Fey s model [] by allowing for more than two players, arbitrary ontinuous distributions, and a more general ontest suess funtion. He also proved the existene of equilibria (leaving uniqueness as future work), following the spirit of []. Still, the equilibrium strategy was only numerially omputed, due to the limited analytial tratability of Tullok ontests. So far, the only known analytial solution to equilibria with a ontinuous type distribution, is due to Ewerhart [6] who onstruted a rather speial distribution to obtain a losed-form expression. However, the distribution is rather omplex and not generalizable, and the model is still limited to a two-player lottery only. In this paper, under a general rowdsouring model with inomplete information, we derive the optimal prize funtion that maximizes the rowdsourer s utility um profit. Surprisingly, our solution of the unique Bayesian Nash equilibrium (a) an be expressed in a simple and losed form in general ases, and (b) is agnosti to the number of players. These are in stark ontrast to prior art, and in pratial terms, imply that our mehanism (a) an be easily implemented in web agents and smartphone apps that at in a fully distributed manner, and (b) overomes the disinentive to partiipate aused by player s antagonism to an inreasing number of rivals. Along the line of optimal Tullok ontest design, two general diretions have been pursued in prior work. One stream of researh explores whether the prize should be alloated to a single winner or all the players in a hierarhial manner. For example, [] applies a rank-dependent expeted utility model to a lottery in whih the prize was divided into, aording to ranks, a few large prizes and a large number of small prizes. However, [7] proves that it is optimal to give the entire prize to a single winner in a symmetri equilibrium. The other diretion fouses on whether and how to bias players in suh a way that indues the maximum revenue. For instane, [4] allows the rowdsourer to assign different weights (preferenes) to players in a disriminative manner for revenue maximization. [9] proves 3 Briefly speaking, with omplete information all the players are informed of all the others types, while with inomplete information eah player only knows his own type. Setion II explains this in more detail. 56

3 5 IEEE Conferene on Computer Communiations (INFOCOM) that a biased lottery (like [4]) ahieves the same revenue as a biased all-pay aution, when both are fully optimized. In the fair ase, [9] proves that an optimal lottery is always superior to an optimal all-pay aution. Our proposed approah represents a new dimension in the design spae of Tullok ontests. Provisioning ontest prize as a funtion (of the unknown winner s ontribution) sets this work apart from all prior work on Tullok ontests in whih prizes are fixed and known ex ante. B. Contributions The main ontributions of this paper are summarized below: ) This work is the first attempt in the rowdsouring literature that uses Tullok ontests as a new framework to design inentive mehanisms. ) We explore a new dimension of optimal Tullok ontest design by provisioning the prize as a funtion. We demonstrate the simpliity of our approah whih makes it partiularly wellsuited for rapid prototyping in fully distributed web agents and smartphone apps. We also show that our approah overomes the disinentive aused by players antagonism to an inreasing number of rivals. 3) As a byprodut of this work, we onstrut an optimal fixedprize Tullok ontest as the benhmark for omparison and outline a step-by-step algorithm for it. This benhmark preisely falls in line with standard Tullok ontests on whih extensive studies are based. Therefore, the benhmark, its onstruting algorithm and the assoiated performane analysis, are highly instrutive for future researh on Tullok ontests. 4) Our last ontribution, whih is not mentioned above, is that we introdue a new parameter the rowdsourer s valuation of user ontribution into the ontest model, and show that it has an exponential positive effet on the performane of both our and onventional mehanisms. In pratial terms, this means that a rowdsourer an arue higher payoff by improving his business proesses via a better utilization of the rowdsoured ontributions. The rest of this paper proeeds as follows. Setion II presents our model with our proposed mehanism, and Setion III analyzes the model to derive the optimal Tullok ontest. The optimal benhmark is then onstruted in Setion IV. Following that, an extensive performane evaluation is provided in Setion V whih demonstrates key results and offers intuition as well. Finally, Setion VI onludes. II. CONTEST MODEL Sitting at the ore of a Tullok ontest framework is a ontest suess funtion (CSF) whih speifies the probability that a player i =,,...,n who exerts (or bids ) effort b i wins the ontest. Our model assumes a very general form of CSF: Pr(b i )= g(b i ) P n j= g(b j) whih generalizes the lassi Tullok CSF, b r i / P n j= br j where r>, as well as the most-studied form of r =(also known as lottery). In (), g( ) is a nonnegative, stritly inreasing funtion that satisfies g() = and onverts player i s effort b i into his ontribution i. For mathematial onveniene, we assume that g( ) is twie differentiable and onave, whih aptures the ommon phenomenon of diminishing marginal return when () exerting effort. When b i =for all i, i.e., no one exerts effort, we assume Pr(b i )=, i.e., no one will win any prize. 4 Another ruial omponent of our ontest model is a prize funtion V ( w ) that we speifially introdue in the ontest, whih is a (monetary) prize of a ommon value that is dependent on the (unknown) winner s ontribution w. Aordingly, a player i will reeive an expeted inome of Pr(b i )V ( i ). This funtion V ( ) is ommon knowledge to all players (e.g., via announement by the rowdsourer). Eah player is haraterized by his type his marginal ost of exerting effort denoted by i [, ], where <<. That is, exerting effort b i will inur a ost of i b i to player i. Thus, if the effort profile of all the players is b := (b,b,...,b n ), the payoff of player i an be expressed in the following quasi-linear form: Pr(b i )V ( i ) i b i. Sine i = g(b i ), a player s (ex post) payoff given the ontribution strategy profile of all the players := (,,..., n ), is ũ i ( i, ) = i P n j= j V ( i ) h( i ) i () where h := g is the inverse funtion of g( ). We note that g exists beause g( ) is stritly monotone. As rowdsouring typially involves an undefined group of people, we assume the interim stage whih orresponds to an inomplete-information setting: eah player i is informed of his own type i but not of others, yet it is ommon knowledge that all the i are independently drawn from a ontinuum [, ] aording to a.d.f. F () or p.d.f. f() =F (). 5 On the ontrary, the ex ante stage orresponds to a no-information setting where players do not know anyone s type inluding their own, and the ex post stage orresponds to a omplete-information setting where all the players types are ommon knowledge. The rowdsourer ollets revenue from the aggregate ontribution of all the players, and bears the ost of paying for the (variable) prize. The profit um utility of the rowdsourer is thus nx = i V ( w ) (3) i= where is the rowdsourer s valuation of per unit user ontribution. This parameter does not appear in prior work where a unity value is always impliitly assumed. However, expliitly modeling this parameter not only homogenizes the dimension of the expression (3), but also allows us to investigate the impat of on key metris suh as the player strategy, prize, profit, and soial welfare, whih turns out (f. Setion V-D) to be an interesting ombination of both linearity and nonlinearity. 4 An alternative and more ommonly adopted pratie in the literature, is assuming Pr(b i )=/n. However, the prizes therein are all fixed, and hene if b i = for all i, a player j will have inentive to deviate by exerting an infinitesimal effort >to inrease his payoff by ( n )v j O( ) where v j is his valuation of the prize. Therefore, an all-zero-bid equilibrium does not exist. In our ase, however, the prize is a funtion V ( ) of ontribution and V ( ) an be so small that players lose the inentive to deviate from all-zero bids. Therefore, we impose Pr(b i )=if b i =, 8i to reinstate this inentive. Indeed, we will show later in Proposition that our mehanism ensures that a player will reeive stritly positive payoff if he exerts non-zero effort. 5 In pratie, suh a distribution F () an be obtained (and published) by the rowdsourer based on histori data, or when histori data is not available assume uniform distribution as widely used in the Bayesian game literature (our model onstitutes a Bayesian game with private information being player types and ommon prior being the type distribution). 57

4 5 IEEE Conferene on Computer Communiations (INFOCOM) III. PROPOSED OPTIMAL TULLOCK CONTEST The solution onept of a game with inomplete information is a pure strategy Bayesian Nash equilibrium, in whih eah player plays a strategy that maximizes his expeted utility given his belief about other players types and that other players also play their respetive equilibrium strategies. Formally, it is a strategy profile BNE =( BNE, BNE,..., n BNE ) that satisfies u i ( i, i BNE ; BNE i ) u i ( i, i ; BNE i ), 8 i, 8i, where u i is the expeted utility of player i, defined as u i ( i, i ):= i [ũ i ( i, )] (4) where =( i, i ), and ũ i is defined in (). Definition (4) an be expanded as follows. In a Bayesian Nash equilibrium, eah player s strategy i is a funtion of his own type i and the ommon prior, i.e., eah player s belief about all the other players types. As our setting is symmetri, in that the prior is a ommon distribution F ( ) for all the players, 6 we fous on symmetri equilibria in whih any player i s equilibrium strategy is speified by a funtion :[, ]! + as i = ( i ), 8i. Therefore, definition (4) an be rewritten based on (), as u(, ) =p( )V ( ) h( ) (5) for an arbitrary type and strategy, where Z p( ) := n + P n j= ( j ) ny j= df ( j ) (6) in whih :=[, ]. Thus, for a partiular player i, his expeted utility is u i = u( i, i ) whih an be omputed from (5). Proposition (Existene and Uniqueness of Equilibrium). Our Tullok ontest model admits a unique, monotone dereasing, pure-strategy Bayesian Nash equilibrium. Due to spae onstraint, we defer all the proofs of this paper to [8]. Heneforth, we will exlusively deal with the equilibrium, and thereby drop the supersript BNE for brevity. Lemma (Equilibrium Contribution Strategy). Given an arbitrary prize funtion V ( ), the (symmetri) equilibrium strategy ( ) of our Tullok ontest, as in = (), is impliitly given by p( )V ( ) h( ) = Z We remark on the following: h( ( )) d, 8 [, ]. (7) Equation (7) has an intuitive interpretation: player i s expeted utility, as represented by the l.h.s., is determined by his ost advantage relative to the highest-ost player, modulated by his ontribution level. There is no losed-form solution to (7) for an arbitrary V ( ). In fat, even the fixed-prize ase (as in onventional ontests) does not have a losed-form solution in general either [4]. However, ounter-intuitively, our approah of using a non-fixed prize (as 6 In an asymmetri setting, player types follow their respetive and generally different distributions F i ( i ), whih is ommon knowledge. Not only is solving suh asymmetri equilibria still an open problem with no analytial solution in general [4], but this setting also makes all players onymous and thus may engender privay onerns in pratie. a funtion) whih seems to be introduing more omplexity leads to a losed-form equilibrium solution in most general ases. We remark on this tratability following the next theorem. Theorem (Optimal Prize Funtion, Strategy, and Maximum Profit). The optimal prize funtion that maximizes the rowdsourer s utility (profit) in our Tullok ontest is given by " Z # w. V ( w )= ( w )h( w ) h( )d ( ) p( w ) (8) where = () and ( ) is the inverse funtion of the equilibrium strategy ( ) whih, as in = (), is given by h ( ) = + F () f(), 8 [, ]. (9) The indued maximum profit of the rowdsourer is Z apple = n () h( ()) + F () [h( ()) h( ())] df (). () f() We will illustrate how to put Theorem to use, in Setion V with a ase study. Here we remark on the following: The strategy or (), as of the highest-ost or weakest player, is always in all-pay autions under standard assumptions. However, in Tullok ontests, this is not neessarily the ase (unless = ), whih will be evidened in Setion V. The reason is that any Tullok ontestant has a positive winning probability as long as he exerts nonzero effort, whereas all-pay autions perfetly disriminate the weakest bidders who have no hane to win. Analytial tratability and Pratial impliation: An interesting and somewhat surprising observation is that, while funtionizing the ontest prize would, intuitively, seem to introdue omplexity to onventional, fixed-prize ontests, the equilibrium strategy (9) turns out to be muh simpler ompared to the fixed-prize ase (f. () in Setion IV). In fat, for most and ommon funtions h( ), it (9) an be expressed in losed form. This onvenient analytial tratability is in stark ontrast to prior art (see [] [3] and a survey [4]) where equilibria do not have analytial solutions in general and an only resort to numerial methods. Theoretially, this lends us a lot of onveniene in subsequent tehnial treatments and other possible future extensions. Pratially, this fosters the appliation of our mehanism via easily-implementable software deployed in web agents and smartphone apps that at on eah user s behalf to determine his ontribution strategy in a fully distributed manner. 7 Agnostiism of strategy and Pratial impliation: Another surprising observation is that the equilibrium strategy (9) is agnosti to n. This is ounter-intuitive and in diret ontrast with prior findings (again see [] [4]; also f. ()) where players are antagonisti to an inreasing number of rivals: when the number of players inreases, eah individual player s hane 7 Note that the optimal prize funtion (8) and maximum profit () are omputed on the other hand by a entralized, and omputationally powerful server, and the omputation takes the already-solved ( ) (9) as input, unlike in () ( ) is unknown. In fat, in many ases suh as demonstrated in Setion V, the omputation in our mehanism is fairly straightforward. 58

5 5 IEEE Conferene on Computer Communiations (INFOCOM) of winning will be diluted, and hene if the prize is fixed, eah player will have to expet a lower utility, resulting in a disinentive to partiipate. However, now that players an stay agnosti to the partiipant pool size, 8 they need not worry about an inreasing number of rivals, whih ertainly strengthens the motivation to partiipate or stay in the ampaign. In more pratial terms, partiipants would even not be averse to spreading the awareness and publiity of a ampaign, whih helps further expand the partiipant pool. Now we state an important ondition pertaining to general inentive mehanisms: individual rationality (IR) [9]. It means that any partiipating player should reeive in equilibrium a nonnegative expeted utility, or in other words, eah player should be better off or at least remain neutral by partiipating. In the following, we prove that our mehanism possesses a stronger version of IR. Proposition (Strit Individual Rationality). Our Tullok ontest with the prize funtion given by Theorem satisfies strit individual rationality (SIR), where all the players reeive stritly positive expeted utility, exept that a player of type whih happens with probability zero expets a surplus of zero (and hene is indifferent in partiipating). Another often-disussed property in mehanism design is inentive ompatibility (IC) [9] or truthfulness, whih means that all players report their types truthfully. This property is tehnially irrelevant to our mehanism beause, unlike some other mehanisms suh as [4], [5] whih determine workers wages based on worker-reported types (osts or desired payments), our mehanism determines players reward based on observable user ontributions rather than unobservable (and private) player types (osts). On the other hand, those other mehanisms an satisfy IR trivially by paying a wage no less than a worker s reported ost or payment, provided that IC is satisfied; but in our ase, satisfying IR requires a arefully designed prize funtion (8) (whih is demonstrated by the proof of Proposition ). IV. OPTIMAL FIXED-PRIZE TULLOCK CONTESTS This setion onstruts a onventional, i.e., fixed-prize, Tullok ontest for the sake of omparison with our mehanism (later in Setion V). 9 However, we augment the prior art of dealing with onventional ontests [] [3], from solving the equilibrium to optimizing (as well as solving) the equilibrium, in order to reate the most superior benhmark to hallenge our mehanism. Speifially, we set to find an optimal fixed prize V that maximizes the rowdsourer s utility through a partiular equilibrium, and suh a solution needs to be found for every possible value of, the valuation of ontribution. Formally, the problem is formulated as max V where " n # X Z = i V = n ()df () V. () i= The equilibrium strategy = () is given by Proposition 3. 8 The agnostiism is ahieved by the prize funtion (8) whih absorbs n and thereby isolates the player strategy from this number. 9 It is sound and fair to ompare our mehanism with a onventional, fixedprize ounterpart, as our prize funtion is an extension of the onventional (single) fixed prize. This is also in aordane with [7] whih proves that it is optimal to alloate a single (fixed) prize as ompared to multiple (fixed) prizes. Proposition 3. In a Tullok ontest with fixed prize V, the equilibrium strategy = () is impliitly determined by Z P n n j= ( j ) Y n [ ()+ P n df ( j )=h ( ). () j= ( j )] V Unfortunately, () does not have an analytial solution. The speial ase of V =was numerially takled by [] [3]. In our ase, V is not given and we need to find the optimal V that maximizes (). In the meantime, ontains () () whih is analytially intratable. Moreover, the optimal solution V must not be a value but a funtion of. This problem is tehnially hallenging. Our solution was inspired by solving the Fredholm equations using a numerial method desribed in [3] (Chap. 5). Consider a two-player ase for simpliity. The first key idea in our solution is to transform the integral in () into a quadrature sum, by supposing that V is given: mx (t j )f(t j ) j [ ()+ (t j )] + R m() =h ( ()), (3) V j= j= where t j,j =,,...,m, are the quadrature points distributed in, j is determined by the hosen quadrature sheme (e.g., Gaussian), and R m () is the residual error due to transforming the original integral into the quadrature sum. In our ase, a uniform quadrature sheme suffies and thereby j = m := ( )/m. In addition, sine the integrand is atomless ( (t j )=for all j is not an equilibrium beause a player will have inentive to deviate by an infinitesimal amount to gain positive utility), the integral an be losely approximated by the quadrature sum for a suffiiently large m, in whih ase R m () an be safely ignored. The seond key idea is to note that, sine (3) holds for all [, ], it must also hold for all the i that equal to the quadrature points t j,j =,,...,m. Thus, (3) is further transformed into a system of m nonlinear equations: mx ( j )f( j ) m [ ( i )+ ( j )] h ( ( i )) i =,i=,..., m. V j= (4) This system an be solved using the Matlab funtion fsolve. Similarly but on a muh simpler sale, the profit () an be approximated by ˆ (, V, ()) = n m m X j= ( j )f( j ) V (5) where ( i ) m i= are the solutions to the system (4). The entire solution is outlined in a self-explanatory fashion by the psuedo-ode in Algorithm. In the atual implementation, we also added a testing ondition to ensure the range [V, V ] to be large enough to inlude the peak point of ˆ (whih an be shown to be onave in V ); we also improved the effiieny by adding a stopping ondition to terminate the inner loop faster. These are peripheral and hene omitted in Algorithm. V. PERFORMANCE EVALUATION In this setion, we ompare for the same rowdsouring ampaign a Tullok ontest that employs our design against a Tullok ontest that adopts a fixed prize, eteris paribus. The former uses the optimal prize funtion determined by Theorem, 59

6 5 IEEE Conferene on Computer Communiations (INFOCOM) Algorithm : Optimizing fixed-prize Tullok ontest Input: bounds,,,,v, V ; m; step sizes, Output: V :! + (a vetor of optimal prizes) :! + (a vetor of maximum profits) :! m + (a matrix of whih eah row is an m-element equilibrium strategy profile orresponding to a ) m ( )/m; {,+ m,+ m,...,+(m ) m}; 3 V {V,V +,V +,...,V }; 4 Create a V m strategy matrix A ; 5 for to (step size: ) do 6 ˆ ; 7 for i to V do 8 reate F m (, m, V [i]) = aording to Eq. (4); 9 := ( j j =,,...,m) fsolve(f m ); A [i] where A [i] is the i-th row of A ; ˆ [i] ˆ (, V [i], ) alulated using Eq. (5); end 3 () max i (ˆ ); 4 V () V [arg max i (ˆ )]; 5 () 6 end A [arg max i (ˆ )]; whih we refer to as Tullok-OPF, and the latter uses the optimal fixed prize determined by Algorithm, whih we refer to as OptBenhmark. We stress that this benhmark is not designed to favor our proposed mehanism, but honestly follows the wellestablished standard model and, in fat, is fully optimized. The performane metris we evaluate are: (a) the equilibrium ontribution strategy of players, whih orresponds to revenue; (b) the prize that the rowdsourer provisions, whih orresponds to ost; () the expeted utility of the rowdsourer, whih orresponds to profit; (d) the soial welfare of the ampaign, whih is the aggregate utility of all the players at equilibrium and we denote by U := ommunity by partiipating in the rowdsouring ampaign. The rationale is that a ompany is typially profit-driven and thus onerned about the metris (a) (), while a government ageny or a non-profit organization may fous more on the metri (d). The primary senario in this setion onsists of two players whose marginal ontribution osts are independently drawn from [ P n i= u i]; it measures the total surplus of the a uniform distribution F () =, [, ]. Player effort b is onverted to user ontribution aording to = g(b) = p b, and hene h( ) :=g ( ) =. Suh a setup is ommon in the literature suh as [], []. This senario is then extend to a n-player setting. A. Tullok-OPF: Analytial Results Our mehanism an be solved analytially, and the solving proess also illustrates how to put Theorem to use. First, the equilibrium ontribution strategy an be obtained via (9) as = () = 4. (6) Hene where = p( ) = ( ) =/(4 )+/, and the numerator of (8) equals ( 4 + Z )+ 4 d = + = Z () =/6. Using (6), the denominator of (8) equals Z d = d (4 4 log(4 + ) = ) log 4 +. Therefore, the optimal prize funtion (8) is obtained as V ( w )= w + w 4 4 w log 6 w+ w+. (7) Finally, the indued maximum profit is obtained via (): Z apple = 4 (4 ) +( ) 36 (4 ) d Z = log 3 d = + (8) B. OptBenhmark: Numerial Results OptBenhmark an only be numerially solved, using Algorithm, of whih the parameters are speified by Table I. Table I: Parameters for Algorithm (OptBenhmark) Parameter V V m Value Fig. reveals the trajetory of finding the duple (V, ), i.e., the optimal prize and maximum profit, by evaluating over a range of possible prizes, for eah =,, 3. The optimal duple is found at the peak of eah urve, and the urves learly demonstrate the onavity of profit versus prize, whih onfirms the existene and uniqueness of the optimum. Furthermore, how the optimal duple (V, ) is affeted by is examined by Fig.. Interestingly, prize V oinides with profit for all the s. This indiates that the revenue of OptBenhmark is double of the ost (prize). This also implies that, if in Fig. we draw all the other trajetories (in addition to the in-situ three), the peak points of all the trajetories will all fall onto the same straight line y = x. Another observation in Fig. is that both prize and profit are onvexly inreasing in. We will revisit this nonlinear behavior together with other subsequent observations in Setion V-D. C. Comparison Given that both Tullok-OPF and OptBenhmark are solved by now, we proeed to ompare them with respet to the four metris mentioned earlier. Crowdsouring Revenue: Fig. 3a examines the equilibrium ontribution strategy of players as a funtion of player type, for eah =,, 3. We remark on three observations. First, in all the ases (3 urves), the strategy is monotone dereasing in type, whih onforms to our Proposition (whih subsumes V ( ) being onstant). The onvex trend is also onsistent with the literature: for example, we verified a speial ase of OptBenhmark with h( ) =, =, [.,.], whih is the same as [], and the result (not reprodued here) exatly mathed []. Seond, for any, Tullok-OPF eliits signifiantly higher ontribution 5

7 5 IEEE Conferene on Computer Communiations (INFOCOM) Profit y=x (.775,.77).4 (.35,.343).3. OptBenhmark (ν =). (.75,.85) OptBenhmark (ν =) OptBenhmark (ν =3) Candidate Prize Figure : OptBenhmark: Trajetory of finding the duple of optimal prize and maximum profit, annotated at the peak of eah urve OptBenhmark: Optimal Prize OptBenhmark: Maximum Profit ν: Valuation of per unit Contribution Figure : OptBenhmark: Impat of on prize and profit. Soial Welfare OptBenhmark Tullok OPF ν: Valuation of per unit Contribution Figure 4: Comparison of Tullok-OPF against OptBenhmark: Soial Welfare. ξ: Contribution Strategy (BNE) OptBenhmark (ν =) Tullok OPF (ν =) OptBenhmark (ν =) Tullok OPF (ν =) OptBenhmark (ν =3) Tullok OPF (ν =3) : Marginal Contribution Cost (a) Revenue: Equilibrium ontribution strategy. Out of the m= quadrature points, only are shown for better visibility. Optimal Prize Tullok OPF (ν =) Tullok OPF (ν =) Tullok OPF (ν =3) OptBenhmark (ν =3) OptBenhmark (ν =) OptBenhmark (ν =).5.5 ξ w : Winner s Contribution (b) Cost: Prize. Note that the support of Tullok- OPF is [/6,/]. Figure 3: Comparison of Tullok-OPF against OptBenhmark. Maximum Profit OptBenhmark Tullok OPF ν: Valuation of per unit Contribution () Profit. than OptBenhmark for every possible type, by about 5% for high-ost (weak) players and 4% for low-ost (strong) players. Third, in both mehanisms, the highest-ost or weakest player makes stritly positive ontribution, i.e., = () >, indiating a sheer ontrast between Tullok ontests and autions where =; here we reall the first remark below Theorem. Crowdsouring Cost: Fig. 3b ompares the optimal prize funtion V ( w ) in Tullok-OPF against the optimal fixed prize V in OptBenhmark. Note that the support of V ( w ) is [, ] = [/6,/] whih follows from (6). While it is intuitive that eah of the three V ( w ) urves inreases in winner s ontribution, it is interesting to note that eah urve fits a straight line very well, whih suggests that the omputation of (7) ould be remarkably simplified in pratie via linear approximation. While this advantage should not be overstated as to how it generalizes, it hints at a possible line of future work. Moreover and noteworthily, the diagram shows that the prize offered by Tullok-OPF is generally well above OptBenhmark. This raises an important question that whether this muh higher ost to be borne by the rowdsourer will eventually pay off, whih is answered next. Crowdsouring Profit: Fig. 3 evaluates the maximum profit that a rowdsourer garners from the two mehanisms. The salient observation is that Tullok-OPF outperforms OptBenhmark by a large profit margin, whih strongly orroborates our proposal of using an optimized prize funtion to supersede the onventional, fixed prize in Tullok ontests. For a loser examination, we ollate the results into Table II and alulate the ratio between eah pair of profits for all the s. It is interesting to note that the ratio almost remains onstant, at about This ould be explained by the nature of optimization whih has pushed the profit to the limit in both mehanisms. In addition, a rigorous analysis as well as investigating to what extent this result an generalize may be worth future exploring. Soial Welfare: In Tullok-OPF, this an be analytially obtained. To solve for U = [ P n i= u i], rather than using the definition (5), we leverage on Lemma whih lends us muh more onveniene: U = = n = " n # X u i = n i= Z Z Z 8 Z Z d df () (4 ) log 3 d = 3 8 h( ( )) d df () (9) One the other hand, the soial welfare in OptBenhmark has to resort to numerial methods. To do so, we take the output of Algorithm, and denote the row () orresponding to eah 5

8 5 IEEE Conferene on Computer Communiations (INFOCOM) Table II: Profit Comparison and Ratios OptBenhmark Tullok-OPF Ratio by a strategy profile. Thus, we an ompute the soial welfare as 3 X m X m U = n m 4f( i m A5, () i= j=i whih an be understood by rewriting (9) as U = n Z f() Z j d d and noting that Lemma applies, without hange, to fixed-prize ases where V ( ) is a onstant. The results are presented in Fig. 4. The observation is exiting: Tullok-OPF outstrips OptBehmark in an even more striking manner as ompared to profit in Fig. 3. The detailed data are ollated in Table III, whih indiates a remarkable improvement of folds. On the other hand, the nearly onstant ratio in Table II is not dupliated here. It may be puzzling as to why the rowdsourer an reap higher profit while, at the same time, users altogether also gain higher surplus. This onstitutes a win-win situation whih is highly desirable but typially hard to attain. To explain this, we note the following rationales. First, generally speaking, rowdsouring is not a zero-sum game like the stok market; rather, it involves a wealth reation proess in whih users exert effort to reate something valuable that we have abstrated as ontribution. Seond, there exists a value asymmetry between players and the rowdsourer, where the rowdsourer typially values ontribution higher than players do. This value asymmetry is a ommon phenomenon in reality: for example in the worldwide emerging Smart City and Smart Nation initiatives nowadays, itizen-generated data suh as ambient noise and GPS traes olleted by smartphones do not usually bear muh value to the phone owners but an be very valuable to a rowdsourer suh as a noise ontrol bureau or a transport ompany (like Waze). Thus, it makes perfet business sense for a rowdsourer to provision ertain attrative reward to inentivize more user ontribution whih in turn bears even more value to the rowdsourer. D. Impat of Crowdsourer s Valuation Reall that we have introdued a new parameter,, to our model in Setion II (f. (3)). Herein, we investigate how it affets the various performane indiators. This effet is expliitly examined with respet to the optimal fixed prize (Fig. ), maximum profit (Fig. 3), and soial welfare (Fig. 4), where it is learly shown that the impat of on these three metris is nonlinear (onvex). Furthermore, this effet is also impliitly examined with respet to the equilibrium ontribution strategy (Fig. 3a) and optimal prize funtion (Fig. 3b) (in both diagrams one needs to ompare aross different urves orresponding to different s), and we see that the impat of on these two metris is approximately linear. The main message onveyed here is that, if a rowdsourer inreases his valuation of user ontribution, e.g. by improving his business proesses to better exploit user ontribution, his profit and the players soial welfare will both inrease faster. This is an interesting finding unovered due to our introdution of. Moreover, this observation applies to both Tullok-OPF and the onventional ase. Now we dive in deeper to explain the orrelation between the above nonlinearity and linearity. On the one hand, the ase of OptBenhmark an be niely explained: (a) the nonlinear profit (Fig. 3) results from the linear revenue (ontribution; Fig. 3a) and nonlinear ost (prize; Fig. ), and (b) the nonlinear soial welfare (Fig. 4) follows from the nonlinear prize (player s gain; Fig. ) and the linear strategy (player s ost; Fig. 3a). On the other hand, the ase of Tullok-OPF is not that straightforward, sine both revenue (ontribution) and ost (prize) seem to be linear aross different s. In fat, an overlooked fat was that the prize funtions in Fig. 3b are shifted horizontally, and thus one should ompare prizes aross s for the same winner rather than for the same amount of ontribution. To do this, a simple way is to ompare the maximum (or the minimum) winner ontribution w aross different urves, as it an uniquely identify a partiular winner. This reveals that the impat of on prize is atually nonlinear. Combined with the linearity on revenue (Fig. 3a), this explains why the profit and soial welfare in Tullok-OPF are nonlinear in. E. n-player Case In this setion, we extend our investigation to n players. We onjeture that the above omparison results will ontinue to hold in the n-player ase, and hene we will not repeat the same omparisons. In fat, as Ryvkin [] pointed out, it is omputationally infeasible to numerially solve fixed-prize Tullok ontests for an arbitrary large number of players beause of the urse of dimensionality. Therefore, this setion fouses on Tullok-OPF. In partiular, we are interested in how the omposition of a partiipant pool, i.e., the distribution of the player types, affets the two key metris, profit and soial welfare. We onsider two partiipant pools: Population- draws player types from the same distribution as above, i.e., F () =, [, ], while Population- draws from another distribution G() = 4, [.5,.5]. Thus, Population- is more diverse or is more unertain in player types than Population-, while they statistially share the same mean value (.5). Following from Theorem, the equilibrium strategy in 4 Population- is G = n,g = n Z.5 apple 4 = n.5 Z.5.5, and thus the maximum profit is (4 ) +( ) 8 (4 ) d log 3 d = + n For Population-, we leverage (8) as a shortut to obtain log 3 n,f = + n. () 8 7 5

9 5 IEEE Conferene on Computer Communiations (INFOCOM) Table III: Soial Welfare Comparison and Ratios OptBenhmark Tullok-OPF Ratio Soial welfare an be solved by referring to (9): Z Z log 3 d d = (4 ) 6 Un,G = n log 3 Un,F = 6 4 n. Thus it immediately follows from the above that n, 36 n,g > n,f, U n,g >U n,f, 8n, 8 >. This tells that Population- is superior to Population- in terms of both profit and soial welfare. An insight that may be drawn from this set of results is that population diversity or unertainty is benefiial to Tullok-ontest based rowdsouring for both rowdsourers and partiipants. VI. CONCLUSION To reap, this work has presented a first attempt to use Tullok ontests as a new framework to design inentive mehanisms for rowdsouring. Furthermore, we have explored a novel dimension in the spae of optimal Tullok ontest design, by superseding the onventional, fixed prize by an optimal prize funtion for utility maximization. In stark ontrast to prior art, we have obtained an analytial solution to the unique Bayesian equilibrium, and found that the equilibrium is robust to an inreasing number of rivals. As our model employs a very general ontest suess funtion and assumes inomplete information, the mehanism and results would fit a wide range of pratial rowdsouring appliations. For example, WiFiSout [3] is a mobile app that aims to profile the performane of itywide WiFi aess points by eliiting personal experiene on WiFi usage from smartphone users. Similarly, OpenSignal [3] aims to onstrut itywide 3G and 4G LTE ell overage maps through rowdsouring too. The superiority of our design has been demonstrated through extensive evaluations by omparing against a fully-optimized benhmark. Construting this optimal benhmark signifiantly extends prior art whih only solves onventional, fixed-prize Tullok ontests. This benhmark would be highly relevant to a wider researh ommunity for future performane evaluations. 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