Designing Incentives for Online Question and Answer Forums


 Letitia Lucas
 3 years ago
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1 Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA Yilig Che School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA David C. Parkes School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA ABSTRACT I this paper, we provide a simple gametheoretic model of a olie questio ad aswer forum. We focus o factual questios i which user resposes aggregate while a questio remais ope. Each user has a uique piece of iformatio ad ca decide whe to report this iformatio. The asker prefers to receive iformatio sooer rather tha later, ad will stop the process whe satisfied with the cumulative value of the posted iformatio. We cosider two distict cases: a complemets case, i which each successive piece of iformatio is worth more to the asker tha the previous oe; ad a substitutes case, i which each successive piece of iformatio is worth less tha the previous oe. A bestaswer scorig rule is adopted to model Yahoo! Aswers, ad is effective for substitutes iformatio, where it isolates a equilibrium i which all users respod i the first roud. But we fid that this rule is ieffective for complemets iformatio, isolatig istead a equilibrium i which all users respod i the fial roud. I addressig this, we demostrate that a approvalvotig scorig rule ad a proportioalshare scorig rule ca eable the most efficiet equilibrium with complemets iformatio, uder certai coditios, by providig icetives for early respoders as well as the user who submits the fial aswer. Categories ad Subject Descriptors H.5.3 [Iformatio Iterfaces ad Presetatio (e.g. HCI)]: Group ad Orgaizatioal Iterfaces; J.4 [Social ad Behavioral Scieces]: Ecoomics Geeral Terms Desig, Ecoomics, Theory. INTRODUCTION Yahoo! Aswers is a questio ad aswer forum where users ca post questios or aswer questios o wide va Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. EC 09, July 6 0, 2009, Staford, Califoria, USA. Copyright 2009 ACM /09/07...$5.00. riety of topics. Yahoo! Aswers has 25 categories ragig from Computers & Iteret to Travel to Family & Relatioships to Health. Users may post discussio questios, factual questios or polls. I Yahoo! Aswers, people do ot exchage moey for aswers to questios. Participatio i Yahoo! Aswers is ecouraged through a elaborate poit system, ad with leaderboards ad topcotributor desigatios to ecourage participats to accumulate more poits. I this paper, we provide a gametheoretic model of behavior for olie questio ad aswer forums such as Yahoo! Aswers. We focus o modelig the oeshot questio ad aswer game. Additioally, we focus o modelig factual questios, such as What are the mai causes of the curret housig crisis?, rather tha discussio questios (e.g. What is your favorite movie of all time? ). Factual questios have bee demostrated to have a higher archival value tha discussio questios [5]. Our iterest is i uderstadig the structure of the equilibria i a model that captures some qualitative features of these eviromets, ad especially i cosiderig the effect of alterate scorig rules o the quality of these equilibria for the asker. I the model that we propose, each user has a uique piece of iformatio that is relevat to a questio ad ca decide whe to report this iformatio. As iformatio is reported it is aggregated ito the resposes, so that the value to the asker mootoically improves while the questio remais ope. I the case that multiple pieces of iformatio are simultaeously revealed i the fial roud, we assume that the asker is able to combie the iformatio. I cosiderig the iteractios betwee the iformatio, we cosider two distict cases: a complemets case i which each successive piece of iformatio is worth more to the asker tha the previous oe; ad a substitutes case i which each successive piece of iformatio is worth less. The asker prefers to receive iformatio sooer rather tha later, ad will stop the process whe satisfied with the cumulative value of the posted iformatio. We first aalyze the equilibrium for a bestaswer scorig rule, that is desiged to model the curret Yahoo! Aswers eviromet. Upo stoppig the process, the asker assigs oe poit to the best aswer amog all resposes. As we assume that every user has the ability to combie her ow iformatio with iformatio that has already bee revealed, uder the bestaswer scorig rule the asker assigs the poit to the most recet aswer (breakig ties at radom i the case that multiple aswers are received i the most recet roud). We fid that this scorig rule is ef
2 fective i isolatig the efficiet equilibrium i the case of substitutes iformatio, i which all iformatio is posted i the very first roud. O the other had, the bestaswer rule is ieffective for complemets iformatio, where it istead isolates the least efficiet equilibrium i which every user posts iformatio i the very last roud. I addressig this problem, we cosider two alterative scorig rules. The first is a approvalvotig scorig rule, i which the asker assigs oe poit to each of the best k > aswers. I our settig, this meas that the asker assigs oe poit to each of the most recet set of k aswers (with ties broke at radom if more tha k aswers were received i the most recet roud, or more tha k k users i the peultimate roud with k i the most recet roud, ad so o). With this scorig rule, we fid that it is ow possible to have the most efficiet equilibrium outcome for complemets iformatio, with certai restrictios o the asker s valuatio fuctio. This scorig rule also retais the efficiet equilibrium i the case of substitutes iformatio. However, the dowside of this rule is that it also retais the least efficiet equilibrium for complemets iformatio ad itroduces the least efficiet equilibrium for substitutes iformatio, agai uder certai restrictios o the asker s valuatio fuctio. A iterestig feature of this scorig rule is the tuable parameter k, which represets the tradeoff betwee the beefit of this scorig rule for the case of complemets iformatio ad the disadvatage for the case of substitutes iformatio. The secod scorig rule we propose is the proportioalshare scorig rule, i which the asker assigs some share of the available poits i proportio to the margial value cotributed by a user i the roud i which the user submits iformatio. With this scorig rule, we fid that it is ow possible to support the efficiet equilibrium outcome (i.e., the equilibrium i which all aswers are received i the first roud) for complemets iformatio, agai with certai restrictios o the asker s valuatio fuctio. This scorig rule also retais the efficiet equilibrium as the uique outcome i the case of substitutes iformatio. O the other had, while the efficiet equilibrium is uique across all poolig equilibrium i which users respod i the same roud (for certai restrictios o the valuatio fuctio), we are uable to rule out separatig equilibrium i which users respod i differet rouds for complemets valuatios uder the proportioalshare rule. The approvalvotig ad proportioalshare rules differ i the iformatioal requiremets that they place o the asker. Approval votig does ot require ay additioal iformatio beyod that required i the bestaswer rule, i.e. just a sigal as to whe the asker is happy with the most recet aswer(s). O the other had, the proportioalshare rule requires the asker, upo stoppig the process, to associate a umerical score with the total iformatio value as the aswers aggregate across rouds. It is a iterestig ope questio to uderstad whether scorig rules ca be desiged that preclude iefficiet equilibrium for complemets valuatios without requirig this additioal iformatio from the asker. 2. YAHOO! ANSWERS AND Q&A FORUMS I Yahoo! Aswers, users maitai a tally of poits ad are pealized for askig questios, rewarded for loggig oto the system each day, rewarded for aswerig a questio ad Activity first time user log ito the system oce per day ask a questio aswer a questio have your aswer chose as best aswer pick a best aswer for your questio Poits +00 poits + poit 5 poits +2 poits +0 poits +3 poits Table : The Poits Scheme i Yahoo! Aswers rewarded heavily for havig aswers selected as a best aswer. The complete poit system is give i Table. Based o the umber of poits a user accumulates, the user receives a level desigatio, with there beig seve levels i total. The higher the level, the greater privileges a user will get i terms of the umber of questios she ca ask per day, the umber of questios she ca aswer per day, etc. All users have a profile where the umber of poits the user has, the level, ad the percetage of best aswers is clearly displayed. Perhaps this is the most visible iformatio displayed about a user. I additio to the poit system, leaderboards ad top cotributor desigatios ecourage users to accumulate more poits. For each category ad subcategory, the top cotributor is displayed at the top of the page. Likewise, for each category ad subcategory, there is a leaderboard of the top te users. It has bee show that poits are a factor i motivatig users to participate i poitsbased questio ad aswer forums [4]. Note, though, that poits are ot eeded to ask questios because users ca always create ew idetities ad obtai 00 ew poits! While we believe that this is the first gametheoretic ivestigatio of questio ad aswer forums, there do exist a umber of existig empirical studies, there have bee a umber of empirical studies devoted to uderstadig how users participate i such forums [4, 9, ]. Nam et al. [4] aim to uderstad the uderlyig motivatios for why users participate i questio ad aswer forums. They study the Naver KowledgeiN (KiN) system, the largest questio ad aswer commuity i South Korea. They give a survey to 26 users of KiN ad fid that altruism alog with selfish reasos (e.g. poits, promotig persoal busiesses, maitaiig persoal kowledge, etc.) are top motivatios for participatig i KiN. These authors also show that the average umber of aswers to a questio icreases as the expected reward icreases. Ideed, others observe that higher rewards lead to icreased participatio. Yag et al. [9] fid that higher moetary rewards for a solutio to a task o Taskc, a popular webbased kowledgesharig market i Chia, attract more views for the task ad icreased task participatio. Harper et al. [6] compare differet questio ad aswer forums ad fid that moetary rewards ad icreased rewards lead to higher quality aswers. Perhaps i support of the fact that users ca build upo previous aswers ad receive credit for the aggregated iformatio, Nam et al. [4] preset a iterestig result: I the C++ forum, the exttolast questio is chose as a best aswer 5% of the time ad the last questio 69% of the time [4]. 2 These authors get virtually idetical results for KiN allows a asker to icrease the reward for a questio by up to 00 additioal poits. 2 KiN allows for selectig more tha oe best aswer.
3 the Java forum ad similar results for the Siger forum. Makig the case that users are behavig strategically, Yag et al. [9] also poit out that users lear to select tasks where they are competig agaist few oppoets, to icrease their chaces of wiig. Users also, over time, select tasks with higher expected rewards. Yag et al. [9] also study user behavior over time ad fid that most users become iactive after oly a few submissios. They also fid that there is a very small core of successful users who maage to icrease their wi percetage over time. This core group accouts for 20% of the wiig solutios o Taskc. Nam et al. [4] also observe a heavy tailed distributio i terms of user cotributio ad that may users drop out after a few cotributios. This is a geeral theme across may websites with user geerated cotet [8]. I a attempt to improve the quality of iformatio o Yahoo! Aswers, Harper et al. [5] focus o distiguishig factual questios i Yahoo! Aswers from discussio questios. Harper et al. poit out that factual questios have a higher archival value tha discussio questios, which ideed is our motivatio for focusig o factual questios. Harper et al. the use a umber of classifiers to group questios as either factual questios or discussio questios. Adamic et al. [] also observe the differece betwee factual forums ad discussio forums ad fid that factual forums ted to have a smaller umber of resposes, while each respose is relatively log, whereas discussio forums teds to have more resposes that are shorter i legth. I additio, these authors fid that discussio forums ted to have a greater amout of overlap betwee users who ask questios ad users who ask questios tha factual forums. Other research i questio ad aswer forums has focused o determiig experts i such commuities [20,, 2], while others have observed the redudacy i such systems ad attempt to retrieve sematically similar questios [0]. May studies otice the varyig quality of user geerated cotet, ad aim to retrieve high quality cotet [2, 4] ad improve cotet quality [8]. Yet other studies try to predict certai properties of Q&A forums (e.g., the likelihood a aswer will be chose as a best aswer [] or asker satisfactio [3]). Olie questio ad aswer forums fit ito the larger realm of peer productio systems [3], which is a term used to refer to decetralized system of users that cotribute to a system to achieve a global goal, without receivig moetary compesatio for their work. Examples of peer productio systems iclude Wikipedia, YouTube, ad huma computatio systems [6, 7]. Prior work has preseted a gametheoretic aalysis of huma computatio systems, specifically the PhotoSlap game [7] ad the ESP game [9]. There have also bee a umber of empirical studies aalyzig user cotributio to various peer productio systems. It has bee show that there exists strog regularities amog a wide rage of peer productio systems [8], amely power laws i terms of cotributio to such systems. Pouwelse et al. [5] also provide a ice empirical survey of the growth of may peer productio systems over the past decade. 3. OUR MODEL We focus o modelig how users participate i aswerig a sigle questio posted by asker. We assume that the questio is o a particular topic that has pieces of disjoit iformatio. Deote I = {I, I 2,..., I } as the iformatio space of the questio. There are users i the system that ca potetially aswer the questio. Each user i {,..., } possesses a uique piece of iformatio I i. Eve though iformatio is private, the fact that everyoe possesses a piece of iformatio out of total pieces is commo kowledge for all users ad the asker. The questioaswerig process has T rouds, uless the asker closes the questio earlier. The users each make a decisio about which roud to participate i, ad are able to observe the resposes by other users before respodig. Whe participatig i roud t >, every user has the ability to combie her ow piece of iformatio with all other pieces of iformatio that have bee revealed i previous aswers, ad submit a itegrated aswer. Each user, however, ca oly aswer the questio oce. This restrictio is also preset i the Yahoo! Aswers forum. At the ed of each roud, the asker decides whether or ot to close the questio. We assume that users seek to maximize their expected score i aswerig the questio ad are thus selfish ad motivated solely by poits, ot altruism. Because there is o cost i our model to aswerig a questio, ad because users seek to maximize their score, we ca restrict attetio to strategies i which a user will always submit a aswer to the questio. Moreover, a user will always choose to submit a aswer that aggregates her iformatio with previous iformatio because this ca oly icrease her score uder all scorig rules that we cosider. Whe multiple ew aswers are received i the same roud, we assume that the asker is herself able to combie the iformatio i these resposes. The asker is modeled with a valuatio fuctio, that maps sequeces of iformatio from iformatio space I ito a real umber represetig the asker s value for the cumulative iformatio associated with the sequece of iformatio. Let σ be a permutatio of I ad σ i(i) be the ith elemet of σ. The asker s valuatio fuctio v satisfies the followig properties: P: v(i 0 ) = 0, whe I 0 = ; P2: v(i, I 2,..., I j) = v(σ (I), σ 2(I),..., σ j(i)) for all j ad all permutatios σ of I; P3: v(σ (I), σ 2 (I),..., σ j (I)) < v(σ (I), σ 2 (I),..., σ j+ (I)) for all j < ad all permutatios σ of I. From ow o, we use v(i) = v(i, I 2,..., I i) to deote the asker s valuatio for ay i pieces of iformatio, ot just those from agets {,..., i}. This is possible due to property P2 which requires that all iformatio is equivalet i the sese that ay i pieces of iformatio geerate the same value. To defie the stoppig rule, we model the asker as drawig a threshold value, θ U[0, ], uiformly at radom betwee 0 ad the value of receivig all iformatio i the iformatio space I of the questio. Oce the asker s valuatio for items received exceeds θ, the asker is satisfied ad closes the questio, ad awards poits to oe or more users accordig to some specific scorig rule. The distributio o this threshold value is commo kowledge to all users but oly the askers kows the actual threshold value. That the questio is closed oce the threshold is exceeded models the ituitio that the asker prefers to receive a aswer sooer rather tha later.
4 Give that a earlier aswer is preferred to a later aswer ad that more iformatio is better tha less iformatio, we ca idetify the efficiet outcome as that i which every user respods i the very first roud. This ca be thought of as a poolig equilibrium, with every user coordiatig o a particular roud i which to submit iformatio. Apart from a poolig equilibrium, we ca also cosider a separatig equilibrium, i which pieces of iformatio are received i distict rouds. The least efficiet outcome is that i which each user waits util the very last roud to respod. Note that i all equilibria, the asker will certaily receive all iformatio by the last roud because it is costless for users to submit a aswer. Therefore, a poolig equilibrium i which all users coordiate o the last roud is the least efficiet of all outcomes. Depedig o the ature of the questio, the pieces of iformatio related to the questio may be complemets or substitutes. For example, suppose the asker posts the questio: What should I do for a oeday visit to Bosto? The two pieces of iformatio, walk alog the Freedom Trial ad have luch at Quicy Market (which is o the Freedom Trial) are complemets, because the value of kowig both pieces of iformatio for the asker is higher tha the sum of the values of oly kowig a sigle piece of iformatio. However, if the asker posts the questio: Where should I have luch i Times Square?, the aswers Becco ad Kodama are substitutes for the asker, sice the asker must choose betwee the two. To model the differet ature of these two questios ad the associated iformatio, we cosider the case where the value of each successive piece of iformatio received by the asker is of greater value tha the previous oe (the complemets case) ad the case where the value of each successive pierce of iformatio received by the asker is less tha the value of the previous oe (the substitutes case). I the followig defiitios, let δ j = v(j) v(j ). Defiitio 3.. I the complemets case, the valuatio fuctio must satisfy δ j < δ j+ for all j <. The substitutes case is defied aalogously. Defiitio 3.2. I the substitutes case, the valuatio fuctio must satisfy δ j > δ j+ for all j <. Uder these cofiguratios, we provide the probability for the asker to close a questio at time t. Let b(t) deote the amout of iformatio that the asker has at the ed of each time roud, where t T. Remark 3.. The probability of stoppig after each roud t, coditioal o that the questio has ot bee closed at roud t, is v(b(t)) for both the complemets ad the substitutes cases. Furthermore, i the complemets case, if b(t) = i, the v(b(t)) < i for all i. Likewise, for the substitutes case, if b(t) = i, the v(b(t)) > i for all i. Proof. Coditioal o that the questio has ot bee closed, the probability of stoppig at step t equals P [θ v(b(t))]. Sice θ follows a uiform distributio, we have P [θ v(b(t))] = v(b(t)) v( ) = v(b(t)). We must have v(b(t)) = i j= δ j ad = j= δ j. For the complemets case, by Defiitio 3., δ i < δ i+. We have, v(b(t)) i j= δj = j= δ = j + j=i+ < δ j ij= δ j = i + ( i)δ i. iδ i For the substitutes case, by Defiitio 3.2, δ i > δ i+. Hece the iequality i the above expressio is reversed for the substitutes case. Remark 3.2. If T = 2, the probability of stoppig i the first roud is p where p = v(i) ad i is the umber of items received i the first roud. The probability of stoppig i the secod roud is p = v(i). Proof. By Remark 3., the probability of stoppig at the first roud is p = v(i), ad the probability of stoppig at the secod roud, coditioal o ot stoppig at the first roud, is =. Hece, the ucoditioal probability of stoppig at the secod roud is ( p) = p. Remark 3.3. If T 2, the probability of stoppig i the first roud is p where p = v(i) ad i is the umber of items received i the first roud. The probability of stoppig i the secod roud is p where p = v(i+j) v(i) ad j is the umber of items received i the secod roud, ad so o. More geerally, the probability of stoppig i roud k is q where q = v(b(k)) v(b(k )), where b(k) is the set of iformatio available at the ed of roud k. Proof. The probability that the questio was ot closed at roud k is P [θ > v(b(k ))]. The coditioal o the fact that the questio was ot closed at roud k, the probability of stoppig at roud k is P [θ < v(b(k)) θ > v(b(k ))]. Hece, the ucoditioal probability of stoppig at roud k is the product of the two probabilities, which equals P [v(b(k )) < θ < v(b(k))] = v(b(k)) v(b(k )). 4. ANALYSIS OF BESTANSWER RULE The bestaswer rule models the scorig method curretly used by Yahoo! Aswers. I Yahoo! Aswers, upo closig the questio, the asker ca select oe aswer as the best aswer ad the associated user is the awarded some fixed umber of poits. Without loss of geerality, we ormalize the umber of poits awarded to. 3 Whe the asker closes the questio because the value has reached the threshold, the asker selects the user that aswered i the most recet roud as the wier. Whe there are multiple aswers provided i the fial roud, the asker uiformly picks oe of them as the best aswer. O oe had we see that users would prefer to wait so that the value of the aswer that they submit is maximized sice their iformatio will be aggregated with earlier aswers. But o the other had, waitig too log could result i a missed opportuity because the questio may be closed i a earlier stage. 3 This is without loss of geerality because we model oly a sigle game, ad thus the relative weight of poits for beig selected as the best aswer vs. askig a questio ad so forth is irrelevat i our aalysis.
5 This tradeoff betwee waitig to form better aswers ad participatig before the questio is closed is the key strategic cosideratio facig users. I the rest of this sectio, we explore this tradeoff ad perform equilibrium aalysis for both complemets ad substitutes cases. We use the otio of a active roud i our aalysis. A roud is active if at least oe user participates i that roud, otherwise it is iactive. We will establish a clea separatio for the complemets ad substitutes cases: the uique Nash equilibrium profile for complemets valuatios is the most iefficiet outcome, while the uique Nash equilibrium profile for substitutes valuatios is the most efficiet outcome. 4. Complemets Case For the complemets case, the asker s valuatio of havig a collectio of several pieces of iformatio is higher tha the sum of her valuatios for idividual pieces. The beefit of waitig to form a better aswer is therefore relatively high. The followig results show that the oly Nash equilibrium for the complemets case uder the best aswer rule, is that all users aswer the questio i the fial roud, just before the questio will defiitely close. This is the least efficiet equilibrium, because the asker must wait to get a aswer util the last possible roud. Lemma 4.. Cosider ay strategy profile that ivolves all users playig i the same roud. The oly oe of these strategy profiles that forms a purestrategy Nash equilibrium is the oe i which all users play last, for ay valuatio fuctio satisfyig the complemets coditio ad uder the bestaswer rule. Proof. Ay strategy profile that ivolves all users playig i the same roud, yields a expected payoff of to each user, because with probability their aswer is selected as the best aswer. Let the active roud be t. Whe t < T, a user ca deviate by participatig i roud t +. The probability that the questio is closed at the ed of roud t is p <, due to Remark 3.. The deviatig user ears a expected payoff p >. Thus, all users playig at roud t < T ca ot be a Nash equilibrium. Fially, cosider the strategy profile cosistig of all users participatig i the T th roud. If user i deviates by goig earlier, his expected payoff equals the probability that the questio is closed before roud T, which i this case is v() < accordig to Remark 3.. Theorem 4.2. For ay valuatio fuctio satisfyig the complemets coditio, the uique purestrategy Nash equilibrium uder the bestaswer rule is the least efficiet strategy profile, i which all users participate i the last roud. Proof. Lemma 4. idicates if a strategy profile forms a equilibrium ad it is ot the strategy profile that ivolves all users playig i the last roud, the the strategy profile must have more tha oe active roud. Cosider the first active roud, call this roud t. Suppose i users have played i the t th roud. The probability that the questio is closed at the ed of the t th roud is give by p = v(i). The expected payoff of a user who plays i roud t is p/i. Cosider the expected payoff of a user from roud t who deviates to the ext active roud, call this roud t. Suppose other j users have played i roud t. The probability that the questio is closed at the ed of roud t uder this deviatio is p = v(i+j) v(i ) accordig to Remark 3.3. The expected payoff of the deviatig user is ow p /(j + ). It is easy to see that p /(j +) > p/i uder the complemets coditio. Thus ay strategy profile that has more tha oe active roud caot be a equilibrium. 4.2 Substitutes Case For the substitutes case, the asker s valuatio of havig a collectio of several pieces of iformatio is lower tha the sum of her valuatios for idividual pieces. The beefit of waitig to form a better aswer i this case is therefore relatively low. I cotrast to the complemets case, the oly Nash equilibrium for the substitutes case uder the best aswer scorig rule, is that all users aswer the questio i the very first roud. This is the most efficiet equilibrium, because the asker will get all the aswers without waitig. Lemma 4.3. Cosider ay strategy profile that ivolves all users playig i the same roud. The oly oe of these strategy profiles that forms a purestrategy Nash equilibrium is the oe i which all users play first, for ay valuatio fuctio that satisfies the substitutes coditio ad uder the bestaswer rule. Proof. Ay strategy profile that ivolves all users playig i the same roud, yields a expected payoff of to each user. Let the active roud be t. Whe t >, a user ca deviate by participatig i roud t. The probability that the questio is closed at the ed of roud t is p >, due to Remark 3.. The deviatig user ears a expected payoff p >. Fially, cosider the strategy profile cosistig of all users participatig i the st roud. If a user deviates by goig later, the probability that the questio is closed after the first roud is greater tha accordig to Remark 3. ad the user s expected payoff is less tha. Theorem 4.4. For ay valuatio fuctio satisfyig the substitutes coditio, the uique purestrategy Nash equilibrium uder the bestaswer rule is the most efficiet strategy profile, i which all users participate i the first roud. Proof. Lemma 4.3 idicates that if a strategy profile is i equilibrium ad it is ot the strategy profile that ivolves all users playig i the first roud, the the strategy profile must have more tha oe active roud. Cosider the last active roud, call this roud t. Suppose i users have played i the t th roud. This meas that i users played earlier. The expected payoff for a user who played i the last roud is ( p)/i, where p = v( i) is the probability that the questio is closed before the last roud was reached. Cosider the value of p. We kow from Remarks 3., 3.2, ad 3.3 that the probability of stoppig before the last roud is reached is greater tha i, so the expected payoff of participatig i the last active roud must be less tha. A user who participates by goig i the last roud ca deviate by playig i the first active roud. Assume that there are j users who play i the first active roud, icludig the user who deviated. The probability that the questio is closed at the ed of the first roud is v(j) > j. So the expected payoff of participatig i the first roud is greater tha. 5. ANALYSIS OF APPROVAL VOTING Uder the bestaswer rule, the uique equilibrium for the complemets case is all users goig last, which is iefficiet.
6 It is possible that by chagig the desig of the scorig rule, we ca iduce a useful chage i the behavior of users ad i particular, eable a more efficiet equilibrium. I this sectio, we cosider a approvalvotig scheme ad aalyze the equilibrium play of users uder this rule. Uder the proposed approvalvotig scheme, the asker assigs oe poit to each of k > users, where k <. The umber of wiers, k, is a desig parameter. Note that if k =, this reduces to the bestaswer rule of Yahoo! Aswers. The Naver KowledgeiN forum, i compariso, does allow for askers to select more tha oe best aswer. I approval votig, the wiers are the k most recet users to aswer before the questio is closed, with ties broke uiformly at radom. I the special case i which the questio is closed ad more tha k users respod i the most recet roud, the a subset of k wiers is selected uiformly at radom. Similarly, whe k < k users respod i the most recet roud the each receive oe poit ad a subset of k k users that respoded i the previous roud are also selected as wiers, ad so forth. We cosider the approvalvotig scheme because it is simple ad also because it seems possible that allowig users that respoded earlier to receive poits will facilitate additioal equilibrium, by lesseig the icetive i the complemets case for every user to wait to the very last momet to respod to the questio. Remark 5.. For ay valuatio fuctio satisfyig the complemets coditio, ad for ay strategy profile cosistig of all users playig i the same roud, a user caot profitably deviate by playig i a earlier roud uder the approvalvotig rule, for ay k >. Proof. If all users play i the same roud, their expected payoff is k. If a user deviates by goig earlier, she receives a payoff of oe uit oly if the questio is closed after the first active roud, which occurs with probability < for ay valuatio fuctio satisfyig the complemets coditio. Remark 5.2. For ay valuatio fuctio that satisfies the coditio v( ) k, ad for ay strategy profile cosistig of all users playig i the same roud, a user ca ot profitably deviate by goig i a later roud, uder the approvalvotig rule for k > wiers. Proof. If all users play i the same roud, their expected payoff is k. If a user deviates by goig later, she receives a payoff of oe uit oly if the questio is ot closed after the first active roud. The questio is closed after the first active roud with probability p = v( ). The user who deviates, gets a expected payoff of p. We eed p k or i other words, p k. Remark 5.3. For ay valuatio fuctio satisfyig the substitutes coditio, ad for ay strategy profile cosistig of all users playig i the same roud, a user caot profitably deviate by playig i a later roud uder the approvalvotig rule, for ay k >. Proof. If all users play i the same roud, their expected payoff is k. If a user deviates by goig later, she receives a payoff of oe uit oly if the questio is ot closed after the first active roud, which occurs with probability less tha for ay valuatio fuctio satisfyig the substitutes coditio. Remark 5.4. For ay valuatio fuctio that satisfies the coditio v() k, ad for ay strategy profile cosistig of all users playig i the same roud, a user caot profitably deviate by playig i a earlier roud, uder the approvalvotig rule for k > wiers. Proof. If all users play i the same roud, their expected payoff is k. If a user deviates by goig earlier, she receives a payoff of oe uit oly if the questio is closed after the first active roud, which occurs with probability v(). This deviatio is ot profitable if ad oly if v() k. Lemmas 5. ad 5.2 are two techical lemmas that together show that ay strategy profile i which there are at least two active rouds caot be a Nash equilibrium for ay valuatio fuctio that satisfies the complemets iformatio criterio. The proof of these lemmas will appear i the full versio of the paper. Lemma 5.. Ay strategy profile that has at least two active rouds, where at least k users participate i the last active roud, caot be a purestrategy Nash equilibrium for ay valuatio fuctio satisfyig the complemets coditio, uder the approvalvotig rule for k > wiers. Lemma 5.2. Ay strategy profile that has at least two active rouds, where less tha k users participate i the last active roud, caot be a purestrategy Nash equilibrium for ay valuatio fuctio satisfyig the complemets coditio, uder the approvalvotig rule for k > wiers. Theorem 5.3. For ay valuatio fuctio that satisfies the complemets coditio ad v( ) k, all users playig i the same roud is a purestrategy Nash equilibrium, for ay roud, uder the approvalvotig rule for k > wiers. Moreover, these are the oly purestrategy Nash equilibria. For ay valuatio fuctio that satisfies the complemets coditio ad v( ) < k, the oly purestrategy Nash equilibria is for all users to play i the last roud. Proof. From Lemmas 5. ad 5.2, we kow that ay strategy profile that has two or more active rouds caot be a Nash equilibrium. Therefore ay equilibrium must have oly oe active roud. We kow from Remark 5., that if all users are goig i the last roud, a user caot profitably deviate for ay valuatio fuctio that satisfies the complemets coditio. Therefore, this strategy profile is a Nash equilibrium for ay valuatio fuctio that satisfies the complemets coditio. We kow from Remark 5.2, that whe v( ) < k ad all users are playig i the same roud, a user ca profitably deviate by goig later. Thus, whe v( ) < k, ay strategy profile where all users participate i a roud that is ot the last roud caot be a Nash equilibrium. We kow from Remark 5.2, that whe v( ) k ad all users are playig i the same roud, a user caot profitably deviate by goig later. We kow from Remark 5., that if all users are playig i the same roud, a user caot profitably deviate by goig earlier. Thus, whe v( ) > k, ay strategy profile where all users participate i a sigle roud is a Nash equilibrium. Similar to the case of complemets iformatio, Lemmas 5.4 ad 5.5 are two techical lemmas that together show
7 that ay strategy profile i which there are at least two active rouds caot be a purestrategy Nash equilibrium for ay substitutes valuatio. The proof of these lemmas will appear i the full versio of the paper. Lemma 5.4. Ay strategy profile that has at least two active rouds, where at least k users participate i the exttolast active roud, caot be a purestrategy Nash equilibrium for ay valuatio fuctio satisfyig the substitutes coditio, uder the approvalvotig rule for k > wiers. Lemma 5.5. Ay strategy profile that has at least two active rouds, where less tha k users participate i the exttolast active roud, caot be a purestrategy Nash equilibrium for ay valuatio fuctio satisfyig the substitutes coditio, uder the approvalvotig rule for k > wiers. Theorem 5.6. For ay asker valuatio fuctio satisfyig the substitutes coditio ad v() k, all users playig i the same roud is a purestrategy Nash equilibrium, for ay roud, uder the approvalvotig rule for k > wiers. Moreover, these are the oly purestrategy Nash equilibria. For ay valuatio fuctio that satisfies the substitutes coditio ad v() > k, the oly purestrategy Nash equilibrium is for all users to play i the first roud. Proof. From Lemmas 5.4 ad 5.5, we kow that ay strategy profile that has two or more active rouds caot be a Nash equilibrium. Therefore ay equilibrium must have oly oe active roud. We kow from Remark 5.3, that if all users are goig i the first roud, a user caot profitably deviate for ay valuatio fuctio that satisfies the substitutes coditio. Therefore, this strategy profile is a Nash equilibrium for ay valuatio fuctio that satisfies the complemets coditio. We kow from Remark 5.4, that whe v() > k ad all users are playig i the same roud, a user ca profitably deviate by goig earlier. Thus, whe v() > k, ay strategy profile where all users participate i a roud that is ot the first roud caot be a Nash equilibrium. We kow from Remark 5.4, that whe v() k ad all users are playig i the same roud, a user caot profitably deviate by goig earlier. We kow from Remark 5.3, that if all users are playig i the same roud, a user caot profitably deviate by goig later. Thus, whe v( ), ay strategy profile where all users participate i a sigle roud is a Nash equilibrium. > k I cotrast to the bestaswer rule, the approvalvotig rule ca eable the most efficiet equilibrium outcome for the case of complemetary iformatio. However, it is ot possible to isolate this as the oly equilibrium. For substitutes, we see that approvalvotig ca sometimes isolate the most efficiet equilibrium (as was the case for the bestaswer rule.) The umber of wiers, k >, is a tuable parameter i the approvalvotig rule that chages the equilibrium structure. The larger k is, the more likely it is to eable the most efficiet equilibrium for the complemets case, however, the larger k is, the more likely it is to itroduce the least efficiet equilibrium for the substitutes case. I what follows, we itroduce some special cases of complemets ad substitutes i order to study this tradeoff i a little more detail. We first tur to complemets valuatios. Defiitio 5.7. Valuatio fuctio v satisfies additive complemets if ad oly if, i additio to satisfyig Defiitio 3., v satisfies δ j+ = δ j + c with c > 0 for all j <. Defiitio 5.8. Valuatio fuctio satisfies multiplicative complemets if ad oly if i additio to satisfyig Defiitio 3., v satisfies δ j+ = δ j c with c > for all j <. Corollary 5.9 is a very positive result, that we eable the efficiet equilibrium for ay additive complemets valuatios ad ay value of, i other words, ay additive complemets valuatio ad pair. Corollary 5.9. For ay valuatio fuctio satisfyig the additive complemets coditio, all users playig i the same roud is a purestrategy Nash equilibrium for ay roud, uder the approvalvotig rule for k > wiers. These are the oly Nash equilibria of the game. I cotrast to the previous result, we fid that the umber of wiers k must be relatively large, i order to eable the most efficiet equilibrium. Corollary 5.0. For ay valuatio fuctio satisfyig the multiplicative complemets coditio, all users playig i the same roud is a purestrategy Nash equilibrium for ay roud, uder the approvalvotig rule for k c (c ) c wiers. These are the oly Nash equilibria of the game. If k < c (c ), the all users playig i the last roud is c a uique purestrategy Nash equilibrium. Defiitio 5.. Valuatio fuctio v satisfies additive substitutes if ad oly if, i additio to satisfyig Defiitio 3.2, v satisfies δ j+ = δ j +c with c < 0 for all j <. Defiitio 5.2. Valuatio fuctio v satisfies multiplicative substitutes if ad oly if i additio to satisfyig Defiitio 3.2, v satisfies δ j+ = δ j c with c < for all j <. Recall that for the case of substitutes iformatio, the relevat questio is to uderstad whe it is possible to isolate the efficiet equilibrium from amogst the poolig equilibrium (as i the bestaswer scorig rule). For additive substitutes we see that this is ot possible for k 2: Corollary 5.3. For ay valuatio fuctio satisfyig the additive substitutes coditio, all users playig i the same roud, for ay roud, is a purestrategy Nash equilibria for ay k > wiers i the approvalvotig rule. These are the oly Nash equilibria of the game. Oe would eed to resort to the bestaswer rule (equivaletly, approvalvotig with k = ), to isolate the efficiet equilibrium i this additive substitutes case. Ad, i pickig a value of k there is a clear tradeoff to make betwee hadlig additive substitutes ad additive complemets. O the other had, we obtai positive results for multiplicative substitutes valuatios: Corollary 5.4. For ay valuatio fuctio satisfyig the multiplicative substitutes coditio, all users playig i the first roud is a uique purestrategy Nash equilibrium uder the approvalvotig rule with k < ( c). Otherwise, c all users playig i the same roud, for ay roud, is a purestrategy Nash equilibrium for k ( c), ad these are the c oly Nash equilibria of the game.
8 c =.0 c = 2 c = 0 = 5 k 2 k 3 k 5 = 0 k 2 k 6 k 9 = 50 k 2 k 25 k 45 Table 2: Necessary coditio o the umber of wiers, k >, i approvalvotig with multiplicative complemets i order to eable the most efficiet equilibrium. c = 0.99 c = 0.5 c = 0. = 5 k = 2 k 4 = 0 k 5 k 9 = 50 k 25 k 45 Table 3: Necessary coditio o the umber of wiers, k >, i approvalvotig with multiplicative substitutes i order to eable the most efficiet equilibrium as the uique equilibrium. Etry idicates that this is ot possible. I Tables 2 ad 3 we illustrate the requiremets for multiplicative complemets ad multiplicative substitutes. Oe ca ifer the followig kid of difficulty with the approval votig rule: the requiremet o k to allow for the most efficiet equilibrium for the case of complemets valuatios teds to be at odds with the requiremet o k to isolate the most efficiet equilibrium as the uique equilibrium for the case of substitutes valuatios. Uder the proportioalshare rule, proposed i the ext sectio, we do ot have this problem sice i the case of substitutes valuatios, the most efficiet outcome is always a uique equilibrium. However, we caot eable the most efficiet equilibrium outcome for all complemets valuatios with the proportioal share rule ad so either rule domiates the other. 6. ANALYSIS OF PROPORTIONALSHARE SCORING RULE I this sectio, we cosider the proportioalshare scorig rule, ad aalyze the equilibrium behavior of users. I the proportioalshare scorig rule, the asker is give a fixed umber of poits that she ca distribute. Without loss of geerality, we ormalize the total umber of poits to distribute to so that each user that participates gets some fractio of a poit. We assume that the asker distributes this poit accordig to her valuatio fuctio. More specifically, suppose the questio closes after C T active rouds, collects k pieces of iformatio i total, ad at each active roud t C there are t participats. I the proportioalshare scorig rule, the asker distributes v(b()) equally amog the v(k) users participated i the active roud, ad, similarly, distributes v(b(t)) v(b(t )) v(k) to the t users that participated i active roud t >, where v(b(t)) deotes the value of the items received at the ed of roud t. I additio to beig a atural scorig rule, with each user receivig credit i proportio to the margial value cotributed to the system i the period i which his or her aswer is provided, we are iterested i this rule because we wat to explore whether or ot it ca remove the iefficiet equilibrium i the complemets case. While the approvalvotig rule was successful i itroducig the efficiet equilibrium (uder certai coditios o the asker valuatio), it was uable to isolate this as the oly equilibrium. The proportioalshare scorig rule is desiged to provide more credit to early respoders tha the approvalvotig rule i order to mitigate this problem. We first preset a lemma o the behavior of users whe there is oly oe active roud. Lemma 6.. For ay strategy profile i which all users v() play i the same roud, ad if, a user caot profitably deviate by goig i a earlier roud uder the proportioalshare rule. For ay strategy profile i which all users play i the same roud, ad if v( ), a user caot profitably deviate by goig i a later roud. Proof. Cosider the strategy profile cosistig of all users goig i the same roud. The expected payoff of each user is. The expected payoff of a user who deviates by playig i a later roud is ( p) ( p), where p = v( ). I order for this deviatio ot to be profitable, we eed ( p) 2, or equivaletly, p. The expected payoff of a user who deviates by playig i a earlier roud is p+( p) p, where p = v(). I order for this deviatio ot to be profitable, we eed p + ( p) p, or equivaletly, p. Applyig Lemma 6., we get the followig theorem, which characterizes the equilibrium structure whe we restrict our attetio to poolig equilibrium. v() Theorem 6.2. Cosider the proportioalsharig rule ad poolig equilibria where all users participate i a sigle active roud. If v( ) v() ad >, the strategy profile cosistig of all users goig i the first roud is a uique purestrategy poolig Nash equilibrium. If v() v( ) ad <, the strategy profile cosistig of all users goig i the last roud is a uique purestrategy poolig Nash equilibrium. If v() ad v( ), the ay strategy profile i which there is oly oe active roud ca be a purestrategy poolig Nash equilibrium. Fially if > ad v( ) <, there is o purestrategy poolig Nash equilibrium. We summarize the results of Theorem 6.2 i Table 4. Although Theorem 6.2 completely characterizes the equilibrium structure whe we restrict attetio to poolig equilibrium, we are uable to rule out the possibility of separatig equilibria. We kow that separatig equilibria do exist, however, they appear to hold for a very arrow rage of valuatio fuctios. Theorem 6.2 gives us a partial characterizatio of the equilibrium structure uder special cases of complemets valuatios. We will retur to this below. For ow we retur to substitutes valuatios ad see that we retai the same property as for the bestaswer rule ad isolate the efficiet equilibrium as the oly purestrategy Nash equilibrium.
9 v( ) v( ) < v() > v() all go first ay poolig o poolig all go last Table 4: Summary of partial equilibrium characterizatio results for the proportioalsharig rule. (Theorem 6.2) Theorem 6.3. For ay valuatio fuctio that satisfies the substitutes coditio, the oly purestrategy Nash equilibrium uder the proportioalshare scorig rule is the strategy profile cosistig of all users goig first. Proof. Cosider ay strategy profile i which there are at least two active rouds. Suppose that j users play i the last active roud. The expected payoff of a user who participates i the last active roud is ( p) ( p), where j p = v( j). Cosider the expected payoff of a user i the last active roud who deviates by goig i the first active roud. Suppose that i users participate i the first active roud, icludig the user who deviated. His expected payoff is at least p, where i p = v(i). For ay valuatio fuctio that satisfies the substitutes coditio, we kow that p i p, so p > ( p) ( p). Thus ay strategy profile i j i j which there are at least two active rouds caot be a Nash equilibrium. Cosider ay strategy profile i which there is oly oe active roud. Lemma 6. tells us that if all users are goig i the first roud, o user has icetive to deviate if ad oly ifthe valuatio fuctio satisfies the coditio:, which is always satisfied by ay valuatio v( ) fuctio that satisfies the substitutes coditio. Lemma 6. also tells us that if all users are goig i the same roud, that is ot the first roud, o user has icetive to deviate by goig earlier if ad oly if the valuatio fuctio satisfies v() the coditio:. However, this coditio is ever satisfied by ay valuatio fuctio that satisfies the substitutes coditio, therefore ay strategy profile i which all users play i the same roud, that is ot the first roud, caot be a Nash equilibrium. Ulike the case of the approvalvotig rule, it is ot always possible to achieve the most efficiet equilibrium for complemets valuatios. O the other had, we do ot eed to worry about more iefficiet equilibria beig itroduced for the case of substitutes iformatio. To better uderstad the coditio o positive results for complemets valuatios, we ca agai cosider the special case of additive complemets ad multiplicative complemets. Note that i this case there is o desig parameter, ad thus o explicit tradeoff that eeds to be made betwee good performace across differet valuatio models. For additive complemets, we see that the value of c eeds to be quite small with respect to δ to eable a good equilibrium. We see that give a fixed additive complemets valuatio, it may ot be possible to have the most efficiet equilibrium outcome, depedig o the value of. I other words, we caot eable the most efficiet equilibrium outcome for all valuatio ad pairs, whe the valuatio satisfies additive complemets. Corollary 6.4. Cosider the proportioalshare rule ad 4. If the valuatio fuctio of the asker satisfies additive complemets with c < 2 δ, the most efficiet outcome ( ) of all users goig i the first roud is a purestrategy Nash equilibrium. Moreover, this is the oly strategy profile that has oe active roud that is a purestrategy Nash equilibrium (rulig out other poolig equilibrium). The followig corollary tells us that we ca eable the most efficiet equilibrium outcome for the case of multiplicative complemets, however, we ca oly do so whe the value of c ad are both relatively small. Agai we see that, depedig o the valuatio fuctio ad pair, it may ot be possible to have the most efficiet equilibrium outcome. This fact is also illustrated i Table 5. Corollary 6.5. Cosider the proportioalshare rule. If the valuatio fuctio of the asker satisfies multiplicative complemets with c c < c ad c (c ) c, the most efficiet outcome of all users goig i the first roud is a purestrategy Nash equilibrium. Moreover, this is the oly strategy profile that has oe active roud that is a Nash equilibrium (rulig out other poolig equilibrium). We preset i Table 5 some example values of ad c ad whether the all goig first equilibrium ca be eabled for the multiplicative complemets case. c =.0 c = 2 c = 0 = 5 yes o o = 0 yes o o = 50 yes o o Table 5: Examples o whether the most efficiet equilibrium ca be eabled for the case of multiplicative complemets. 7. CONCLUSIONS We believe that appropriate icetive desig ca help to improve the iformatio quality i questio ad aswer forums. I studyig this, we have itroduced a simple, gametheoretic model of a questio ad aswer forum such as Yahoo! Aswers. We aalyze the bestaswer scorig rule, which models that of Yahoo! Aswers, ad show that it is effective with iformatio items that are substitutes but eables oly the least efficiet outcome, i which every user plays i the very last roud, i the case of complemets. I cosiderig the effect of differet scorig rules o the equilibrium structure of the game, we have idetified two scorig rules that lead to efficiecyimprovig chages i the equilibrium. Specifically, the approvalvotig rule ca eable the most efficiet equilibrium for ay complemets valuatio ad pair, with a appropriately chose value of k (the umber of wiers). O the other had, for ay substitutes valuatio ad pair, the approvalvotig rule ca itroduce the least efficiet outcome as a equilibrium, depedig o the value of k. The tuable parameter, k, eables a tradeoff betwee the beefit of this scorig rule for the case of complemets iformatio ad the disadvatage of itroducig this scorig rule for the case of substitutes iformatio. The proportioalshare rule, i compariso, ever itroduces a less efficiet equilibrium for the case of substitutes
10 valuatios. Moreover, for certai valuatios it is possible with the proportioalshare rule to isolate the efficiet equilibrium while rulig out all other poolig equilibrium for complemets valuatios. O the other had, there are some complemets valuatios ad pairs for which the proportioalshare rule does ot allow the efficiet equilibrium ad we are ot, i geeral, able to rule out additioal separatig equilibria (i which users respod i differet rouds) for the case of complemets valuatios. Take altogether, while the approvalvotig rule (for a small eough k) ad the proportioalscore rules seem to have more desirable properties tha the bestaswer rule, we do ot yet have a clear orderig betwee our two ew rules. I cosiderig the appropriate rule, oe must also remember that the approvalvotig rule requires less iformatio from the asker to allocate poits while the proportioalshare rule requires the asker to allocate a value to each aswer (or to each set of aswers, whe multiple aswers are received i the same roud) i a sequece of aswers. Clearly there are a lot of aveues for future work. I additio to characterizig the complete equilibrium structure (icludig split equilibria) for complemets valuatios i the proportioalshare rule, oe could study variatios o our simple model, such as aswerers that have differet valued pieces of iformatio (from the asker side) ad aswerers that have overlappig iformatio. Such extesios would remove symmetry of the aswerers ad move us towards a richer model. It would also be iterestig to icorporate the fact that some users are partially motivated by altruistic reasos ito our model. Aother directio for future work would be to model the cost to the asker of combiig iformatio provided by multiple users i the same roud, leadig to the idetificatio of scorig rules that promote build equilibrium where the user resposes are optimally sequeced ad build of each other. Fially, aother extesio is to cosider iformatio cascade effects, wherei oe user s respose triggers aother user to recall a ew piece of iformatio that would have ot bee available if ot triggered by the first user. Ackowledgemets We would like to thak the aoymous EC reviewers. The work of the first author is supported i part by a AT&T Labs Fellowship ad a NSF Graduate Research Fellowship. 8. REFERENCES [] L. A. Adamic, J. Zhag, E. Bakshy, ad M. S. Ackerma. Everyoe kows somethig: Examiig kowledge sharig o Yahoo aswers. I Proc. 7th Itl. Cof. o the World Wide Web (WWW), [2] E. Agichtei, C. Castillo, D. Doato, A. Giois, ad G. Mishe. Fidig highquality cotet i social media. I Proc. st ACM Itl. Cof. o Web Search ad Data Miig (WSDM), [3] Y. Bekler. Coase s Pegui, or, Liux ad the Nature of the firm. The Yale Law Joural, 2, [4] J. Bia, Y. Liu, E. Agichtei, ad H. Zha. Fidig the right facts i the crowd: Factoid questio ad aswerig over social media. I Proc. 7th Itl. Cof. o the World Wide Web (WWW), [5] F. M. Harper, D. Moy, ad J. A. Kosta. Facts or frieds? Distiguishig iformatioal ad coversatioal questios i social Q&A sites. I Proc. SIGCHI Cof. o Huma Factors i Computig Systems (CHI), [6] F. M. Harper, D. Raba, S. Rafaeli, ad J. A. Kosta. Predictors of aswer quality i olie q&a sites. I Proc. SIGCHI Cof. o Huma Factors i Computig Systems (CHI), [7] C.J. Ho, T.H. Chag, ad J. Y. je Hsu. Photoslap: A multiplayer olie game for sematic aotatio. I Proc. 22d Cof. o Artificial Itelligece (AAAI), [8] G. Hsieh ad S. Couts. mimir: A marketbased realtime questio ad aswer service. I Proc. SIGCHI Cof. o Huma Factors i Computig Systems (CHI), [9] S. Jai ad D. C. Parkes. A gametheoretic aalysis of games with a purpose. I Proc. 4th Itl. Workshop o Iteret ad Network Ecoomics (WINE), [0] J. Jeo, W. B. Croft, ad J. H. Lee. Fidig similar questios i large questio ad aswer archives. I Proc. ACM Cof. o Iformatio ad Kowledge Maagemet (CIKM), [] P. Jurczyk ad E. Agichtei. Discoverig authorities i questio aswer commuities by usig lik aalysis. I Proc. ACM Cof. o Iformatio ad Kowledge Maagemet (CIKM), [2] P. Jurczyk ad E. Agichtei. Hits o questio aswer portals: exploratio of lik aalysis for author rakig. I Proc. 30th ACM SIGIR Cof., [3] Y. Liu, J. Bia, ad E. Agichtei. Predictig iformatio seeker satisfactio i commuity questio aswerig. I Proc. 3st ACM SIGIR Cof., [4] K. K. Nam, M. S. Ackerma, ad L. A. Adamic. Questios i, Kowledge in?: A study of Naver s questio aswerig commuity. I Proc. SIGCHI Cof. o Huma Factors i Computig Systems (CHI), [5] J. Pouwelse, P. Garbacki, D. Epema, ad H. Sips. Pirates ad Samaritas: A decade of measuremets o peer productio ad their implicatios for et eutrality ad copyright. Telecommuicatios Policy Joural, 32:70 72, [6] L. vo Ah ad L. Dabbish. Labelig images with a computer game. I Proc. SIGCHI Cof. o Huma Factors i Computig Systems (CHI), [7] L. vo Ah ad L. Dabbish. Desigig games with a purpose. Commuicatios of the ACM, 5(8):58 67, [8] D. M. Wilkiso. Strog regularities i olie peer productio. I Proc. 9th ACM Cof. o Electroic Commerce (EC), [9] J. Yag, L. A. Adamic, ad M. S. Ackerma. Crowdsourcig ad kowledge sharig: Strategic user behavior o Taskc. I Proc. 9th ACM Cof. o Electroic Commerce (EC), [20] J. Zhag, M. Ackerma, ad L. Adamic. Expertise etworks i olie commuities: Structure ad algorithms. I Proc. 6th Itl. Cof. o the World Wide Web (WWW), 2007.
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