Putting Your Money Where Your Mouth Is A Betting Platform for Better Prediction. Abstract

Transcription

1 Revew of Network Eonoms Vol.6, Issue June 007 Puttng Your Money Where Your Mouth Is A Bettng Platform for Better Predton FANG FANG College of Busness Admnstraton, Calforna State Unversty at San Maros MAXWELL STINCHCOMBE Department of Eonoms, the Unversty of Texas at Austn ANDREW WHINSTON Department of IROM, the Unversty of Texas at Austn Abstrat We desgned a platform wth a bettng mehansm for eltng ostly, dspersed nformaton of dfferent qualty. Our obetve s to elt both dspersed nformaton and the preson of the nformaton so as to effently weght dspersed nformaton to produe relable foreasts. After dedng to nur the osts to aqure nformaton, partpants wll report ther foreasts and dede the sze of ther bets to ndate preson. Our mehansm selets those wth relatvely prese nformaton at lower osts. We also dsuss the mplementaton ssues of our mehansm and the mplaton to fulfl the task of rowd sourng. Introduton Aggregatng dspersed nformaton an produe aurate predtons. These an be extremely useful for deson makng n a world full of unertantes. The development of the Internet provdes us wth a ommunaton base to ollet hghly dspersed nformaton. Therefore, how to buld a relable platform upon ths ommunaton base to entralze and ntegrate dspersed nformaton beomes an aademally hallengng task. In ths paper, we study the possblty of reatng a sustanable busness model by desgnng an effent nformaton eltaton and aggregaton mehansm to form an aurate foreast n a ost-effetve fashon. Contat Author. Fang Fang. College of Busness Admnstraton, Calforna State Unversty at San Maros, 333 S. Twn Oaks Valley Road, San Maros, CA 9096, U.S.A. E-mal: fangfang@susm.edu The paper has been presented at the Workshop of Eonoms of E-ntermedaton, Pars, Frane, 005 and reeved valuable feedbak from the audene. The authors also thank Jaques Crémer, Svetlana Boyarhenko, Alfredo DTllo, Takash Hayash, Dale Stahl, and Mhael Whnston for many nsghtful omments. The authors take responsblty for all errors. 4

2 Revew of Network Eonoms Vol.6, Issue June 007 When desgnng the relable mehansm to produe foreasts, we not only need to elt the dspersed nformaton truthfully from soures but also to onsder mportant fators suh as the relablty of eah pee of nformaton and the ost to ndue the truthful nformaton. Dfferenes n the qualty of ndvduals nformaton arse from dfferent experenes, dfferent bakgrounds, and other forms of dfferental aess to, and ompetene wth, nformaton soures. Further, these dfferenes are themselves varable aross tme and tops, meanng that who s better nformed vares. Identfyng those who have more relable nformaton n a spef predton task s mportant for dedng how muh weght to put on eah person s opnon. However, eah person prvately holds suh nformaton and may not have an nentve to reveal t. People who know they have more aurate nformaton are generally wllng to bet more money on t. They are wllng to put ther money where ther mouth s. We explot ths regularty to nfer the preson of people s nformaton. Our mehansm generates relable predtons by effently auray-weghtng eah person s nformaton based on the nferred preson. The other fator we look at s the ost for eah person to aqure and share nformaton. There s ommerally useful nformaton that s free but we study the nformaton that people need to spend resoures, money, and/or tme on to aqure and assmlate. In addton, opportunty osts are another type of ost that people nur to obtan nformaton. People wll only spend and share the resoures f they beleve t to be worthwhle. Costs, as well as relablty, are varable and unobservable by others. For ths reason, our seond desgn task s to motvate those agents wth low ost aess to more relable nformaton to spend ther resoures aqurng and assmlatng the nformaton. The early researh on nformaton eltaton mehansm desgn manly fouses on eltng a sngle agent s foreast of a future event (for example, Savage, 97. More reent work on eltng the foreasts from several agents has assumed that the qualty of the foreasts s equal and there s no ost of nformaton aquston (for example, Chen et al, 00 and 004. Weghtng observatons from dfferent soures aordng to ther relablty has a long hstory n the statsts lterature under the assumpton that the relablty s known. Based on our knowledge, we are the frst to dsuss nformaton aggregaton when the relablty s only prvately known. In the lterature on optmal aggregaton (for example, Clemen and Wnkler, 993, past performane s suggested as one possble ndator of relablty. However, suh prate an only be adopted to foreastng onventonal tops wthn a small group of pre-seleted experts. In reent busness applatons, the range of foreastng tops has beome wder, and hstoral data ould be unavalable or unrelable. In ths researh, we propose to systematally weght opnons based on the amount of money the partpants are wllng to put on what they say. Our dea of aggregatng dspersed nformaton from a large number of people falls nto the ategory of utlzng the wsdom of rowds (Surowesk, 004. The man dea s to rely upon the dversty and produtvty of the rowd to resolve unonventonal tasks. Spefally, Surowesk (004 lsted three types of problems to whh the rowds may provde a better soluton than that provded by a small group of seleted experts. The In other words, the agents are born wth the nformaton. The only deson s whether to share. In Surowesk (004, the rowd s referred to as a large group of people who are not verfed to be knowledgeable to solve the entre task. However, as argued by Surowesk, the aggregate effet of the rowd an be outstandng n many of the applatons. 5

3 Revew of Network Eonoms Vol.6, Issue June 007 foreastng problem fts n the ategory of ognton problems. The dsperson of partpants an nrease the dmenson of problem solvng and brng n more perspetves to dentfy the best soluton. In order to nsure the relablty of a random unknown rowd s nputs, we need to desgn proper nentve algnment mehansms to dsover the good rowds and motvate ther ontrbuton. Whle advoatng the utlzaton of the rowd s wsdom, our paper dsusses a theoretal framework to better motvate and manage the rowds. Our foreastng problem has a spef task of predtng a future value. 3 Example applatons are the demand foreast of a ertan produt, the predton of the number of seats n the U.S. Congress ouped by the Demorat Party after the next eleton, the overall orange produton n the next year n Calforna, and many more. It s also worth notng that our mehansm does not suggest that the experts opnons are not mportant. In fat, the experts are among the rowds. They are, most of the tme, the better rowd. Ther analyses and opnons are valuable soures of the rowds wsdom. However, utlzng only the seleted experts s subet to rsk. For example, the seleton proess may be based due to the organzer s lmted abltes to aess all the anddates and to udge the best qualfed group. In addton, the experts opnons may stll have a very small hane of beng wrong, even when they are the most qualfed group. Sometmes, they fal to (or are unable to observe a few relevant fators. Or they an rely too muh upon tradtonal ways of thnkng to resolve unonventonal problems. The rowd serves as a dspersed fore to proess and evaluate the experts opnons and provde more perspetves. Members of the rowd an therefore orret possble based opnons from experts, further nreasng the relablty of the foreast. There are eletron predton markets smulatng stok markets where the market pres reflet the aggregaton of dspersed nformaton. Suh mehansms also provde a method of puttng your money where your mouth s. Eletron predton markets have reently prolferated and generated a great deal of publ nterest. 4 Empral researh has olleted evdene to prove the predtablty of suh markets n the feld of poltal eletons. Forsythe, et al (99 provded empral results on the 988 presdental eleton and show that the market worked extremely well, domnatng opnon polls. Varous researhers have desgned many types of markets for dfferent predton tasks and report promsng but dfferent levels of predtablty. For example, Forsythe et al (995 reported that the market provdes a very good foreast of the popular vote shares n 993 Canadan federal eleton but a less aurate result n predtng the splt of seats among partes. Wolfers and Ztzewtz (003 provded a omprehensve revew of the researh work on eletron predton markets and dsussed expltly the ssue of dfferent desgns for dfferent foreastng tasks. However, t s not fully understood how relable 3 There are two dfferent types of problems n the lterature on foreastng: probablty foreastng and value foreastng. Probablty foreastng fouses on predtng the probablty of a future event, for example, Hllary Clnton has a 37.8% hane to beome the frst female presdent of the Unted States. In ontrast, value foreastng attempts to predt the value of a random varable whose value wll be realzed n the future, for example, the Demorat Party wll gan 5.6% of the votes n the next ongressonal eleton. 4 Examples of atvely tradng markets are Iowa Eletron Markets ( Hollywood Stok Exhange ( Tradesports ( and Intrade ( 6

4 Revew of Network Eonoms Vol.6, Issue June 007 suh markets are, nor what determnes ther relablty. 5 Lews-Bek (005 surveyed the exstng statstal models for produng eleton foreasts, ompared the outomes wth those produed by eletron predton markets, and argued that foreast produton was a omplated feld and t was stll too early to determne that the eletron predton market s a good enough mehansm. The theoretal foundaton for stok market pres provdng good predtons are the Effent Market Hypothess (EMH; Fama, 970 and the Ratonal Expetatons Equlbrum (REE; Grossman and Stgltz, 976 frameworks. There are two related theoretal questons on how and whether the market pre an effently aggregate prvate nformaton. The frst s a oneptual problem ponted out by Grossman and Stgltz (980. If the market pre aggregated all avalable nformaton, then no ratonal person would have an nentve to take part n the market sne ther prvate nformaton s almost ertanly less aurate than the aggregated nformaton of everyone else. Further, f market partpants shall pay a small ost to aqure ther prvate nformaton, then the only equlbrum s that no one wll pay the ost, therefore, the market annot reveal any of suh nformaton. That s, f the market somehow an perform ts nformaton aggregaton funton, t does not provde ustfable motvaton for partpants to ollet nformaton. More generally, Mlgrom and Stokey (98 pont out that when someone else s wllng to make a bet aganst you, t reveals nformaton makng you less wllng to bet aganst hm. Your ntal best estmate of the odds s based on your prvate nformaton. Knowng that someone else, on the bass of hs own prvate nformaton, s wllng to bet aganst you means that you know that hs prvate nformaton ontradts yours. Ths should derease your wllngness to bet. To resolve these oneptual problems, outsde subsdes must be provded to guarantee that the agents orret nentves to trade, espeally when people nur osts for aqurng nformaton. There s a thrd theoretal queston, a type of Luas rtque (Luas, 976. If a predton market beomes relable and ths relablty hanges poly or polts, ths may reate strateg nentves to manpulate the market. If the strateg nentves are strong enough, they ould offset any monetary losses nurred by the manpulators. Ths paper seeks a more relable nentve mehansm that an provde theoretally ustfable outomes. We propose a novel bettng mehansm n whh a prnpal who needs the foreast wll resort to a group of agents who may have some nformaton about the subet. In our mehansm, the predton s relable n the sense that the prnpal an estmate the auray of the predton from all the bets submtted. We ondut our analyss wth a one-prnpal-mult-agent framework to resolve the followng questons: ( How do we gve agents nentves to truthfully reveal both ther prvate nformaton and the qualty of ther nformaton? ( What do these nentves ost? (3 How do we trade off the nentve osts and the benefts of the nformaton we gather? 5 Reent falures n the Iowa Eletron Markets were ts predton that John Kerry s hane of wnnng the 004 Demorat Cauuses was less than 35%, and ts short-lastng predton, apparently reatng to some small number of polls, that John Kerry s hane of wnnng the eleton was well over 60%. 7

5 Revew of Network Eonoms Vol.6, Issue June 007 In our proposed bettng mehansm, the prnpal asks eah agent to submt a report as a prvate foreast and plae money as a bet on the report. After the unertanty s realzed, the agents an get rewards aordng to how lose ther reports are to the atual realzaton of the event and how muh money they bet on the reports. Wth suh a mehansm, the agents annot smply say that they are experts. They an only demonstrate ther expertse by havng the onfdene to plae bets on ther reports. We present a famly of reward funtons wth two rual propertes. Frst, ondtonal on dedng to nur the ost to gather nformaton, eah agent s domnant strategy s to report hs or her true foreast. Seond, agents bet an amount that s monotonally nreasng n the preson of ther nformaton. We dsuss the hoe of the optmal reward funton n two dfferent setups: ( a smultaneous bettng mehansm, where the prnpal ommts to the same reward funton for all the agents and agents bet wthout observng the atvtes of the other agents; ( a sequental bettng mehansm, where the prnpal dynamally adusts the reward funton for the next agent after observng eah bet and the report submtted by the prevous agents. Wth both mehansms, the agents nur the osts of nformaton aquston and bet only when ther expeted gan overs the osts. Therefore, the prnpal s hoe of the reward funton s essentally a hoe of her wllngness to pay for eah pee of nformaton. If the wllngness to pay s hgh, then there ould be more agents partpatng n the market, whh wll generate a more aurate predton. The tradeoff s hgher payments by the prnpal. The rest of the paper s organzed as follows. We present the foreastng model n a one-prnpal-mult-agent framework and dsuss the mehansm desgn task n seton. Seton 3 analyzes a benhmark ase where the prnpal an verfy the relablty of the agents nformaton soures. We prove the exstene of an optmal set of agents wth relatvely lower osts and hgher preson from ths analyss and dsuss the property of the seleton rteron. Seton 4 presents a bettng mehansm wth a famly of parameterzed reward funtons assumng that the agents preson s not avalable. Seton 5 extends the smultaneous bettng mehansm to a sequental one and shows how the sequental bettng an mprove the ost-effetveness for the prnpal. Seton 6 onludes wth a dsusson of the mplementaton of a bettng market and the mplatons of the rowd s wsdom. Model and analyss For deson purposes, a prnpal needs a foreast of a random varable to be realzed n the future, say the next quarter sales of a new produt. Aess to nformaton useful for makng the foreast s dspersed among many people. At a ost, people an aqure and proess ther nformaton. Both the qualty of the nformaton and the ost of aqurng t vary amongst the people nvolved. The prnpal s task s to motvate a set of agents wth relatvely lower ost to obtan more prese nformaton and to report truthfully both ther foreast and the relablty of ther foreast. The prnpal an then use the relablty nformaton to weght the agents reports and form the aggregated predton. Throughout our modellng of the future random varable and the dspersed nformaton, we assume normal dstrbutons and quadrat utlty funtons. By ontrast, we assume general dstrbutons for the osts and preson of the nformaton. We ould replae the 8

6 Revew of Network Eonoms Vol.6, Issue June 007 normal-quadrat formulaton wth more general dstrbutons and utlty funtons, for example, replang lower varane wth Blakwell nformaton mprovements, however, we strongly suspet that the ost of ths generalty would outwegh any gan. For example, we would be oblged to establsh the exstene and bas theoretal propertes of an optmal weghtng sheme for foreasts nstead of usng the explt weghtng sheme that arses n the normal-quadrat ase. Ths sheme has the same bas propertes that would arse n a more general settng.. The foreastng model A rsk neutral prnpal (for example, a frm wants to foreast a future outome, X ~ N s,. The prnpal s payoff from generatng represented by a random varable ( 0 0 a foreast ˆX s g ( X ˆ, x v p ( = ˆX x, where x s the realzaton of X, v s the value to the prnpal when the foreast s extremely prese, and p ( Xˆ x s a quadrat penalty term for mstakes n the foreast. To obtan a good predton, the prnpal resorts to N rsk-neutral agents. Eah agent I {,,, N} an aess a prvate ndependent nformaton soure at a ost > 0. 6 If agent dedes not to aqure the nformaton, she has the same belef about the random varable, X ~ N( s 0, 0, as the prnpal. If she does aqure nformaton from her prvate soure, she observes a prvate sgnal s = x+ ε. 7 Here ε ~ N ( 0, s the random error of the foreast. The varane of the error, as well as ts reproal, measures ex ante the relablty of the sgnal s. The larger the value of, the smaller the varane of and the more relable s s as an ndator of the future value x. In the rest of the paper, we refer to as the preson of agent s nformaton for =,,, N. We also all the reproal of the prnpal s pror varane 0 the preson of the pror belef. We assume that { X, ε,, εn, (,,, ( N, N } s an ndependent olleton of random varables/vetors. Before aqurng nformaton soures, the agents prvately know ther osts and preson, (,, before they dede whether to aqure the sgnal s. The prnpal needs to selet from among the agents those wth low osts and hgh presons n order to produe the foreast n a ost-effetve fashon. Therefore, the par ( s the, 6 In prate, an agent may have already obtaned the nformaton. Then we an assume that the agent s ost s 0. Our ost assumpton also nludes the opportunty ost for the agent to be wllng to reveal hs nformaton. An agent wll aqure and share nformaton only when the ost s ompensated. 7 In ths paper, we use the smple form of sgnal s = x+ ε. However, n the real world, the sgnal an agent observes may be of a more omplated form. For example, s = α x+ β + ε, where the oeffent α represents the relevane of the sgnal and β represents a onstant shft (or bas. However, we an transform ths s to s = x+. The ost term s an aggregate onept representng how muh the prnpal should pay to ndue the agents to aqure nformaton, nludng both the agents atual ost to aqure and proess the sgnal and ther opportunty ost. ε 9

7 Revew of Network Eonoms Vol.6, Issue June 007 agents ex ante type the prnpal wants to dentfy, whh s the goal of our mehansm desgn. The pars of (, are drawn from a ommonly known dstrbuton, Q, (, wth, and I I F ( beng the umulatve dstrbuton funton of the margnal dstrbuton of the H( beng the umulatve dstrbuton funton of the margnal dstrbuton of the. We assume throughout that both dstrbutons have fnte expetatons.. The optmal aggregaton problem If a subset of agents, S I, aqures ther prvate sgnals and report both s and truthfully, Lemma desrbes the optmal way to aggregate those sgnals. Lemma. When agents S truthfully report s and to the prnpal, the optmal ˆ 0s0+ S s X S =. 0 + predtor that maxmzes the prnpal s expeted payoff s ( S 0 S 8 payoff. = + s the predton preson, and S v p s the prnpal s expeted Lemma defnes the optmal way to aggregate dspersed nformaton, f avalable. Based on ths result, the optmal predtor s a weghted average of all the dspersed sgnals. Besdes eah agent s sgnal s, the preson s mportant n dedng the optmal weght of s. In addton, the total sum of presons ( S s the only parameter n the prnpal s ex ante valuaton funton. Therefore, the prnpal who maxmzes the ex ante expeted payoff s onerned about how to nrease the aggregated preson. p The prnpal s expeted gan v S, whh ndates that the ( S ( S s onave n prnpal s nentve to nrease the auray dereases as the aggregated predton gets more prese. The prnpal s less wllng to ompensate an agent s ost when aumulatng more sgnals. In addton, t ould be too ostly for the prnpal to ompensate everyone. A maor task for the prnpal s to fgure out, from an effeny pont of vew, whh people should be tapped for nformaton and whh should not. Another aspet of the prnpal s problem omes from the agents lak of nentve to report the sgnals and the preson truthfully. The reasons ould be varous. For example, a salesperson may want to send a report lower than her estmate beause she knows the foreast wll be used to generate her sales quota. Sometmes, people may exaggerate the preson beause they want to be onsdered experts. Even when an agent has no other nentve to le about the sgnals, he or she may stll not be wllng to tell the truth, whh s only a weakly domnatng strategy wthout a proper ompensaton sheme. Ths makes the report unrelable. In our setup, our agents ould also pretend to obtan the nformaton. To establsh a relable way to enourage agents to atually aqure nformaton and report truthfully, t s mportant to desgn a relable nentve algnment mehansm. 8 See the Appendx for all the proofs of theorems n ths paper. 0

8 Revew of Network Eonoms Vol.6, Issue June Mehansm desgn proedure We summarze the tasks of our mehansm desgn problem as follows: ( Eah agent wll have the rght nentve to report both s and truthfully. ( The agent who does not pay the ost to ollet the nformaton wll not have an nentve to lam that she dd. (3 The mehansm effently selets those wth relatvely low ost and relatvely hgh preson. To dsentangle the tasks, we ondut our analyss n the followng steps: Frst, we assume that the preson of eah agent s nformaton soure s verfable. We then look for a funton ( so that only those agents wth osts ( wll aqure nformaton. Our mehansm wll, for eah, have a maxmal ost that the agents are wllng to nur. Ths hooses the random set of agents who wll gather nformaton. We, and examne ts bas propertes. show that there s an optmal funton, Spefally, we gve ondtons for obtanng a non-dereasng funton that has the potental to be mplemented n a bettng mehansm. Seond, we desgn a bettng mehansm wth a parameterzed reward funton that ndues the agents wth suffently low osts relatve to ther preson to aqure nformaton, and to gve both ther true predton and the preson. The more onfdent about her knowledge someone s, the more she s wllng to bet on t. We gve the ondtons when the optmal funton an be mplemented so that the agents wth osts lower than ( wll plae a postve bet n the market. We provde a dret revelaton mehansm where the sze of an agent s bet equals the preson of her nformaton. Thrd, we show how the prnpal an properly weght the agents nformaton and apply the bettng mehansm n two dfferent settngs. In the frst settng, tmelness s mportant. The struture of rewards s announed and all those who hoose to aess ther nformaton do so and submt ther bets and foreasts. In the seond settng, some delay s aeptable and the agents are hosen one at a tme. They are asked f they are wllng to nur ther nformatonal osts and make a foreast and a bet. Sne the reward struture an be adusted for later agents, ths an lead to a savngs for the prnpal. 3 The optmal set when presons are verfable In ths seton, we assume that the prnpal an verfy whether agents have reported ( s, truthfully. The ssue left to be determned s whh agents should aqure, so that the agents wth nformaton. The dea s to fnd an optmal reward funton, preson and ost lower than ( the wll be wllng to aqure nformaton. We all the rerutng funton n the sense that the funton determnes the set of partpatng agents. In the followng setons, we wll then relax the assumpton of

9 Revew of Network Eonoms Vol.6, Issue June 007 and dsuss when we an reprodue suh an optmal verfable, as well as ndue agents to report ther s truthfully. We wll restrt our searh for the optmal rerutng funtons to the set of nondereasng funtons. Intutvely, hgher preson soures are worth more to the prnpal. In addton, we wll look for methods to mplement the optmal rerutng funton n seton 4 and the set of mplementable rerutng funtons needs to be non-dereasng. Otherwse, hgh preson agents wll have an nentve to pretend that they are the low preson ones. Fnally, under ertan assumptons on the ont dstrbuton of preson and osts dsussed later, the set of non-dereasng ontrats s ompat.. That s, an agent wth Pk an arbtrary non-dereasng rerutng funton wll get pad preson f she aqures the nformaton and shares t wth the prnpal. It an be shown that the set of agents who wll ollet nformaton s S = :0. The prnpal s deson s to hoose an optmal reward funton { } ( from the set of non-dereasng funtonal forms { : for } N N N Let = (, ( = = =, ω = (,, and ω = ( ω = C.. Q denotes the ont dstrbuton of ( N ω +. We use dq ( ω to ndate ntegraton wth respet to the margnal of Q on N the th omponent of ( +. Let Ψ be the prnpal s overall payoff, whh s affeted by the atual group of agents the prnpal faes (, and by the reward funton the prnpal adopted to selet the agents (. The payoff Ψ ( ((,, s then omposed of two parts, the expeted gan from the predton E g X ˆ ( S, X s, p Ψ = v S S and the ost to generate the predton Ψ. In the above equaton of S Ψ, ˆX S and S are the optmal predtor and the overall preson of the predtor defned n Lemma. Therefore, the prnpal s expeted payoff Ψ s p Ψ ( ((,, =Ψ S Ψ = v S as ( S ( S takes values n the nterval [ ( Ψ = p v s nreasng, onave, and takes values n the nterval 0,. p v, v 0 +. The funton beng nreasng means that reevng reports from more people nreases preson. The onavty means that, on average, the return to reevng reports from more people s dereasng. The set S s unquely determned gven the utoff (. The prnpal s problem s to hoose the optmal ( so as to optmze the expeted rewards,

10 Revew of Network Eonoms Vol.6, Issue June 007 Ψ = max EΨ,, C ( (( p = v dq ( (, ( dq(, S. ( S N N ( + ( + 3. Exstene of optmal rerutng funton An optmal rerutng funton exsts under ertan general assumptons of the agents dstrbuton Q. For example, f there s an atom part of Q, then the prnpal s reward funton s not ontnuous on, and an optmal urve may not be well-defned. Assumpton. The ondtonal dstrbuton funton H ( > 0, H H( for any > 0. has the property that, for any Proposton (Exstene. If Assumpton holds, then the problem max C EΨ( (,(, has a soluton. That s, there exsts an nreasng optmal rerutng funton (. In summary, there s no loss n restrtng attenton to those rerutng funtons C ( p that are bounded above by p beause the maxmal beneft n redung unertanty to 0 s bounded above by p. Indators of the subgraphs of suh funtons form an L - ompat set and domnated onvergene shows that the expeted payoffs are ontnuous. 3. A speal ase The searh for an optmal urve s not an easy task. Smulaton methods ould be used. The optmal urve depends on both how muh the prnpal desres an aurate foreast (that s, p and the dstrbuton of agents type Q. It s not lkely to fnd an analytal soluton of ( for a general Q. One relatvely easy speal ase has preson and osts ndependent, and unformly dstrbuted on [ 0, ] and [ 0, ]. Equvalently, Q s the unform dstrbuton on N ([ 0, ] [ 0, ]. In ths ase, we an study the frst-order dervatve of Equaton ( and show that t s unquely satsfed by a rerutng funton. The unqueness arses from a ontraton mappng and the latte struture of the set of non-dereasng funtons makes the analyss of the optmal ontrat transparent. The densty of eah agent s preson and ost s q(, =, the ondtonal H umulatve dstrbuton funton of osts gven preson s margnal of s preson s subgraph of where ( = { }, and the ϕ = C =, : + denote the. Takng frst order dervatves on both sdes of Equaton (, we obtan (. Let ( ( p M = + = ( 3

11 Revew of Network Eonoms Vol.6, Issue June 007 { } dq ( C C ( = 0 + (, (, M,. (3 N ( + 4 Bettng mehansm desgn In seton 3, we provded ondtons where an optmal funton exsts and s nondereasng. We also examned a speal ase when Q s unformly dstrbuted and studed the frst-order ondton of. In ths seton, we dsuss when we an mplement the funton n a bettng mehansm where we need to elt ( s, for the agent S(. We now assume that the prnpal an nether observe nor verfy an agent s sgnal, preson, and ost. He annot fore an agent to ollet the nformaton f the agent does not want to. Suh an assumpton fts the senaro of olletng the rowd s wsdom. A proper nentve mehansm needs to be desgned to elt all the relevant nformaton from agents. To aheve a relable foreast, the truthful eltaton of ( s, has to be guaranteed. Meanwhle, the ompensaton should be made to those agents wth suffently low osts relatve to ther preson to motvate them to aqure nformaton. Inspred by the folk sayng, Puttng Your Money where Your Mouth Is, we desgn a bettng mehansm that asks agents to report ther nformaton and to determne an amount of money as a bet. The agents are rewarded after the unertanty s realzed. The reward depends on how muh they bet and how lose ther report s to the realzaton of the future state. The reward s desgned so that an agent wth the hgher preson wll fnd t optmal to plae a larger bet whle revealng hs true foreast. Ths way, the agents preson s revealed and the weghts on the agents nformaton an then be determned. In the rest of ths seton, we frst show the exstene of a dret revelaton bettng mehansm desgn that an ndue the optmal ( when t s strtly nreasng, dfferentable and not too onave. That s, an agent wth preson an expet to earn (. The prnpal an aheve the same expeted payoff as f he an verfy eah agent s preson desrbed n seton 3. We then dsuss some potental strateges of the agents and ther mpat on the prnpal s payoff. In seton 5, we extend the bettng mehansm to a dynam settng and show how the prnpal an mprove the foreast. 4. The bas bettng mehansm To elt ( s,, we ask eah agent to send a report r as hs prvate foreast and to plae an amount of money B as a bet to sgnal hs preson. The agents are rewarded after the future state X = x s realzed. To provde the agents orret and quantfable nentves to report truthfully ( s,, the payoff they reeve wll depend on how lose ther reports are to the true realzaton of x (that s, the agents atual performane and how muh they bet as a sgnal of ther expeted performane. Formally, the reward s a funton f ( B, r, x. Gven ths reward struture, agent who has already aqured the sgnal solves max E[ π s, ] = E f ( B, r, x B s, (4 r, B 4

12 Revew of Network Eonoms Vol.6, Issue June 007 ( for hs optmal bettng strategy B ( s,, r ( s,. ( s fully revealng f t s nvertble for I. If, n addton, the strategy satsfes that B ( s, = and r ( s, = E [ X s, ], we say that the strategy s dretly revealng. A bettng mehansm that an mplement the dretly revealng strategy s alled a dret revelaton bettng mehansm. Defnton (Revelaton. We say that a bettng strategy B ( s,, r ( s, Gven the bettng mehansm, t ours that an agent s expeted reward s senstve to the form of the reward funton. The prnpal s expeted payoff by mplementng the bettng mehansm s optmzed when eah agent wth preson s rewarded ( f he bets. Defnton. We say that a reward funton f mplements ( f E π B, r s, = for [ 0, +. The reward funtons that mplement may not be unque. In ths paper, we examne f r, B, x = g B g B r x, where the lass of reward funtons wth the form g ( B and g ( B are strtly postve funtons. Our hoe of the reward funton s a modfed quadrat loss funton, whh ensures that the agents report truthfully ther prvate foreast of X as long as g ( s postve. The funtons g ( B and g ( B need to be desgned to ndue the agents to bet more when they beleve ther preson s hgher. Proposton (Bettng funton. When ( ( s non-dereasng and onave, and ( ( ( < + for +, a reward funton f unquely exsts wthn the above lass of funtons that satsfes the followng two ondtons: the reward funton f mplements ( ; the agents optmal bettng strategy (, In addton, f r, B, x = g B g B r x, where ( g B = B + B + B + B, and ( g B = B + B. B r s dretly revealng. 5

13 Revew of Network Eonoms Vol.6, Issue June 007 Proposton shows that the prnpal an aheve the same expeted payoff usng the bettng mehansm as f he an verfy the preson f the optmal rerutng funton s onave enough. In the speal ase of seton 3., the ondton < +, s unformly dstrbuted. Therefore, the prnpal an atually mplement holds when ( an optmal bettng sheme to aheve ost-effetve nformaton eltaton and aggregaton. 4. Bas bettng mehansm dsusson To mplement, the prnpal an all for a smultaneous bet to ollet nformaton from the agents. That s, a prnpal wll announe a predton task and a reward funton as defned n Proposton to let the agents bet n a short tme span. Suh smultaneous bettng allows the prnpal to ollet relevant nformaton as qukly as possble beause the agents, one havng deded to partpate, wll aess ther nformaton soure smultaneously. People wll bet beause they want to make money from ther prvate nformaton soures. Anonymty, as well as aountablty, s mportant n suh a bettng system desgn. Anonymty elmnates people s reputaton onern, whh has been dentfed n lterature as one potental soure of reportng bas. For example, Ferderer and College (004 reported that reputaton-drven experts may herd when they are onerned that ther foreasts ould be bad. However, people need to be held aountable n our desgn beause our reward funton does not satsfy the lmted lablty ondton. That s, the reward may beome negatve f the report dffers too muh from x, even though the overall expeted reward s always non-negatve (that s, 0. Therefore, the prnpal needs to have the ablty to punsh the bad bettors. There are yet ssues n suh an open bettng envronment. Below we dsuss three potental ssues: multple denttes, agents talk, and possble ex post regret. 4.. Aquston of multple bettng denttes In ths paper, we assume that people s nformaton s obtaned from ndependent nformaton soures. Ths assumpton an be establshed n general f we an assume that one person only bets one. However, ths s dffult when the number of agents s large and the agents are anonymous. Even though arefully desgned verfaton proesses, suh as requrng one redt ard per bet an restrt the hanes of multple bettng, t s mpossble to ompletely prevent. Theorem (Repeated Bettng Strategy. If an agent wth preson manages to bet. twe, her expeted payoff wll be Theorem shows that an agent an expet to double her payoffs by bettng twe. Suh strateg behavor has two negatve mpats on the prnpal. Frst, the prnpal pays extra money for a pee of useless nformaton. Seond, ths redundant pee of nformaton 6

14 Revew of Network Eonoms Vol.6, Issue June 007 affets the weghts assgned to aggregate all the nformaton, whh redues the effeny of the fnal predtor. 9 To redue suh a potental problem, the prnpal an revew all the bets and weghts arefully to detet potental repeated bettng. Modern tehnologes provde some methodologes to dentfy ssues suh as repeated bettng to some degree. However, ths s beyond the sope of ths paper. 4.. Agents talk and sgnal dependeny Our analyss assumes that agents obtan ndependent sgnals. Eah agent aesses ndependent nformaton soures and an ontrbute to the aggregate foreast wth an ndependent pont of vew, mprovng the foreastng qualty. If two agents (for example, agent a and b share ther sgnals, the ndependene assumpton s volated. The two have the same prvate foreasts and wll produe the same reports and bet the same amount of money. That s: (, r = r = E X s s Ba = Bb = a + b a b a b Ths has the same effet to the prnpal as f one agent wth preson a + b bets twe, whh has been dsussed n seton 4... Thngs get worse f more agents an talk together. In addton, the prnpal ould pay for overlappng sgnals f the nformaton soures are dependent, whh has a smlar effet as n ths ase. One possble soluton s for the prnpal to ask the agents to submt ustfatons for ther reports and bets. If the prnpal observes smlar ustfatons, t s an ndaton that the agents are obtanng smlar sgnals. The prnpal, therefore, an adust the weght n the aggregaton to restore effent weghtng. If two reports are exatly the same, the prnpal an refuse the bet from one to avod payng for the same nformaton repeatedly. However, verfyng eah agent s report sgnfantly nreases the prnpal s workload, espeally when the potental number of bettors s large. It loses the attraton of utlzng the rowd s wsdom, whh suggests that the maor omputaton load s aomplshed by deentralzed agents. Another downsde of requestng a report s the nreasng ost to ndue agents to bet. An agent wll bet only when the expeted reward overs both the ost of nformaton aquston and the ost of wrtng a report. Ths auses a shft n the optmal reward funton and an adustment of the reward funton f. Essentally, t makes the prnpal less wllng to selet agents, sne they are, n general, more ostly. A better soluton for ths problem wll be dsussed n seton Analyss of ex-post neffeny The prnpal pks an optmal reward funton based on the pror dstrbuton of the agent s preson and osts (that s, Q. Suh a reward funton s optmal n the ex ante sense. The prnpal s ex post payoff depends on the atual dstrbuton of the agents 9 Followng Greene (003, we all a predtor effent when t has the least varane among all the lnear unbased estmators. Based on ths rteron, our predtor gven n Lemma s effent even when we relax the normalty assumptons on X and ε for =,,, N. 7

15 Revew of Network Eonoms Vol.6, Issue June 007 preson and osts. In ths sense, the prnpal may regret payng too muh to the agents or not gettng enough nformaton. The followng two examples demonstrate two extreme ases where the prnpal overpad and underpad, respetvely. Example (Overpad. Assume that there are two agents (ndexed as =,. Both have a aess to aurate nformaton soures. That s, = = +. In addton, a a max,. { } <. In ths ase, both agents wll bet and expet to be rewarded However, learnng from ether agent {, }, the prnpal an obtan an aurate foreast already. The margnal beneft of learnng from the other agent 0 s a a a p 0 + p 0 + whle payng to agent. The prnpal would regret havng allowed the seond agent to bet beause the seond pee of nformaton has no value to hm. Example (Underpad. Assume that p =00. The optmal rerutng urve = + and the urrent preson = 0. The prnpal s expeted punshment term ( ˆ (, = p E p X X s = 0. After the foreast has been produed, the prnpal S fnds out that an agent wth = 6 and ost = dd not partpate beause 6 a 7 a a = < =. If the prnpal norporates ths agent s nformaton, the expeted punshment term wll be redued to p + a = 6.5 a, whh only osts the prnpal a =. In ths sense, the prnpal ddn t reward the agents enough. In other words, the pror belef of the agents types, Q, s too optmst. In our spef ontext of nformaton aggregaton, the prnpal s nentve for payng for a new pee of nformaton dereases as the urrent preson nreases. At some pont, the prnpal may want to stop aeptng bets f he sees that the preson s aurate enough. However, he annot do t n the smultaneous bettng envronment. In seton 5, we adust the bettng mehansm to a dynam settng, where only one agent an submt a bet at a tme. As we wll show, suh sequental bettng an help redue the problems dsussed n seton Extenson: Bettng dynams 5. Dynam bettng nreases effeny In ths seton, we extend the bettng to a dynam settng. The agents are arranged n a random queue to bet sequentally. For example, the prnpal an label the agents based on the order of ther arrval. The prnpal allows only one agent to bet at a tme. At tme t the t t prnpal promses agent t a reward funton ft( Bt, rt, x = g( Bt g( Bt( rt x, where ( r, B s the agent t s report and bet. The prnpal observes ( r, B to update hs belef t t t t 8

16 Revew of Network Eonoms Vol.6, Issue June 007 on X. The prnpal wll then adust the reward funton to f ( B, r, x t+ t+ t+ and allows the next agent t + to bet. Wth the dynam bettng setup, the prnpal has the ablty to adust the reward aordng to the observaton of how the agents nformaton s revealed. Therefore, the prnpal an observe whether the pror estmaton of Q s too optmst or pessmst. The prnpal an dede to stop at any tme one he beleves the foreast s aurate enough. ( t t Denote V as the prnpal s expeted gan from the bettng n perods t to N, whh s ontngent on the predton preson the prnpal has olleted from the t t prevous states = + s the rerutng funton the prnpal pks at tme. 0 = (, where ( = E f B, r, X B s, ( ( ( = E g, B g B r X B s g ( B = E g ( B B, 0 + s (5 The valuaton V perod, + t t V t+ t t ( t ( dq( t, s omposed of two parts, the expeted valuaton from the next t, mnus the expeted ost to pay agent t n ths perod. Denote as the ndator funton whether agent t plae a postve bet. To determne ( t the optmal rerutng funton, the prnpal not only needs to onsder the potental ontrbuton of the agent n the urrent perod, but also the future rounds of bettng. The problem an be modelled wth the followng dynam programmng formulaton (6: t ( + ( t t- =max + t t ( δ Q (, t t V t V t t d t t + t t t t (6 where δ s the dsount fator for the prnpal to delay the deson one more perod and the termnal ondton for the dynam programmng problem s N+ N ( = N V v p. (7 0 Defne the prnpal s expeted gan n the sequental settng as EΠ seq V ( = 0 and the expeted gan n the smultaneous settng as EΠsmul EΨ, as n equaton (. When δ =, the prnpal does not worry about makng a deson early or the tme nterval t s short. For example, the agents an gather nformaton from soures qukly. The prnpal an then dynamally adust the reward based on the ex post observaton of the agents atual types. The followng proposton proves that the prnpal gans from the ablty to dynamally adust the reward. 9

17 Revew of Network Eonoms Vol.6, Issue June 007 Proposton 3. When, N > E ( δ Π = > EΠ always holds. seq smul 5. Dynam bettng wth publ learnng Wth dynam bettng, the prnpal an hoose to release to the other agents what the prevous agents have reported and bet. The agents an therefore learn from the prevous agents reports and update ther belefs before bettng. Therefore, at tme t, an agent t s belef of X wll be where t E X s,, s,,, s, s ( ( ( t t = t s the reward funton when the agents an learn from the past. In addton, the preson of agent t s foreast mproves to t. Compared to the equaton (5, the reward funton hanges to: = t ( funton n ( = (,, (,,(,,, (, g ( B = E g ( B B. E f B r X B s s s (8 When the prnpal shares prevous bets and reports wth agents, the prnpal allows the agents to observe the up-to-date nformaton. Eah agent (exept for = has more aurate nformaton before she bets n suh a publ learnng proess. The agent s reward s less rsky. In addton, sne the prnpal has the ablty to adust the reward funton, she ould do t n eah round to exlude those agents who do not have any new pee of nformaton to bet. Therefore, the prnpal an effetvely elmnate the ssue of repeated bettng desrbed n seton 4... In seton 4.., we onsdered the ase that two agents agree to share ther foreast. Wth suh a publ learnng proess, the prnpal an effetvely elmnate suh nentves. Ths s beause f two agents share ther sgnals, only the frst agent wll gan from bettng by submttng a more prese foreast (learned from the seond agent. After the bettng, both sgnals beome avalable to all and the seond agent wll not have any new nformaton to share n her round of bettng. Therefore, the seond agent s expeted gan wll be zero. Meanwhle, the frst agent may tell the seond agent what her sgnal s. But ths sgnal wll be shared by the prnpal anyway. Therefore, the seond agent wll make the same deson when she bets and the prnpal s payoff remans unaffeted. The prnpal an allevate the problem of orrelated sgnals by sharng the nformaton wth all the agents. The ntuton s that the prnpal wll share the nformaton that has been aggregated from the past agents. The new agent wll then only gan from utlzng the ndependent omponent of her sgnal, as shown n Proposton 4. Proposton 4. The followng two ases yeld the same expeted payoff to the prnpal: 30

18 Revew of Network Eonoms Vol.6, Issue June 007 (3 Two agents and ( < have aess to orrelated nformaton soures. That s, s x ε = + + ε where ~ N ( 0, = + and s x γε ( γ ε and ~ N ( 0, ε. (4 Two agents and ( < have aess to ndependent nformaton soures. That s, s = x+ ε and s x ε = + where ε ~ N ( 0, and ~ N ( 0, In addton, f γ =, the agent n ase ( wll not bet. ε. Therefore, the later agents wll only seek and make money from ndependent nformaton soures. Ths mproves the ost-effetveness of the bettng mehansm by avodng payng repeatedly for the same set of nformaton. It also motvates agents to seek unonventonal soures, whh s the full utlzaton of the dversty of the rowds. In the smultaneous bettng setup, we dsussed that one possble soluton to redue the dependeny problem s for the prnpal to determne whether there are orrelated nformaton soures, whh mposes too muh burden on the prnpal to verfy the ndependene f the number of agents s large. In the settng wth dynam bettng and nformaton sharng, the agents self-reommend themselves to the bettng market only when they have a new ndependent pee of nformaton. The prnpal an save sgnfant amounts of workload by dentfyng the orrelaton among the reports submtted by the agents. It an also be shown that the dynam bettng mehansm redues the problem of ex post neffeny desrbed n seton Sne the prnpal has the ablty to adust the reward funton eah tme, she an redue the rerutng urve to a very low level one havng olleted nformaton prese enough. Therefore, the prnpal avods payng too muh as n Example. If the prnpal ollets too lttle nformaton, she an nrease the reward n the next round to mprove the stuaton as n Example. Though there are sgnfant benefts, a settng wth dynam bettng and nformaton sharng s not always strtly preferred. It omes at the ost that the prnpal needs to wat longer to ollet all the nformaton, espeally sne the agents need to observe the new rerutng urve to dede whether to nur the ost to ollet the nformaton, whh sometmes ould takes a long tme. In addton, the prnpal may not want to share wth the new agents the bets and reports submtted n the past rounds. Ths s true espeally when the prnpal s onerned about the serey of the fnal foreast. 6 Conludng remarks In ths paper, we propose a theoretal framework to aggregate dspersed nformaton. Spefally, we propose a mehansm that elts agent s prvate nformaton, as well as ts qualty, so as to effently weght the nformaton. The mpled statstal effeny gan yelds more relable foreasts. Our mehansm provdes a self-seleton proess where the agents an trade off between the ost to aess ther nformaton and the qualty of nformaton they feel they an provde. Suh a self-seleton proess releves the prnpal s burden of evaluatng the potental agents and faltates aess to a wder range of potental agents. We also dsuss several varatons of the market desgn for the prnpal s dfferent predton needs. 3

19 Revew of Network Eonoms Vol.6, Issue June Rsk neutralty and rsk averson Our model s the frst to expltly ndue the preson of the foreast from an agent. To faltate the demonstraton, we assume that the prnpal and all the agents are rsk neutral. If the agents are rsk averse, they wll bet less, and/or bet only when ther preson s hgher. If the agents rsk-averson an be learnt or estmated, the prnpal an orrespondngly adust the way she updates the nformaton based on the bets. However, f the agents rsk-averson s also prvate nformaton, a more omplated mehansm s requred. In Chen et al (00, a two-round market mehansm s ntrodued wth the frst round spefally desgned to estmate the agents rsk-averson. Our dsusson therefore, fouses on the seond round desgn, where the mehansm uses the agents reports and bets to elt ther prvate sgnals and preson. 6. Potental applatons Our mehansm desgn for predton produton s valuable espeally n the urrent busness world where nformaton s dspersed and t s beomng nreasngly rual to get more aurate predtons. The dea s to let ndependent agents self-selet whether to partpate based on ther preson and osts. It has the potental to save the prnpal a large amount of tme and resoures to evaluate the qualfaton of the experts. Our bettng mehansms an be appled to many busness deson senaros. One applaton s n the area of supply han demand foreast. Guo et al (006 dsussed the mportane of foreastng a maro level fator (for example, a well-defned ndex for a spef ndustry that affets all retalers loal demand. The authors, therefore, propose to use ertan market mehansms to ndue relable foreasts of the maro fator. Our bettng mehansm s a possble soluton to generate suh a predton. The supply han members an ontly organze a bettng market (as the prnpal and aept bets from agents all over the world as a bass for aggregatng the nformaton. A seond applaton an be the foreast of the sales of a new produt. Hewlett- Pakard has deployed an nternal market to predt the sales of ts prnters (Chen and Plott, 00. The nternal market has a desgn smlar to the stok market (also the Iowa eletron market, where partpants trade shares ndatng ther prvate predtons. Ther nternal market utlzes market pres to onvey the aggregated predtons, whh are publly observable by all the traders. The market s run nternally among a small group of partpants n order to avod releasng senstve busness nformaton outsde the ompany. A bettng market desrbed n seton 4 allows the prnpal to ollet nformaton from people outsde the ompany beause the prnpal an aggregate all the bets and reports wthout releasng them to the other traders. In ths way, Hewlett-Pakard s able to norporate valuable nformaton from ts ustomers and other related soures. In addton, the bettng mehansm an also be mplemented n applatons suh as sportng event foreasts, move box-offe foreasts, the pre of a future IPO, and many others. 6.3 Extenson: Parallel methods to ndue rowd s wsdom Our bettng mehansm attempts to ollet dspersed nformaton n an ad ho fashon. The Internet plays a rual role n aggregatng suh dspersed nformaton. As the Internet onnets the world, t hanges sgnfantly how people do busness. It espeally faltates unpreedented produtvty n developng the proets of wkpeda, blogsphere, 3

20 Revew of Network Eonoms Vol.6, Issue June 007 and PP network applatons suh as Bt-torrent and Skype. How to better organze and utlze the power of the rowd has also stmulated a great deal of researh work. In our mehansm, we fous on the Internet s ablty to brng agents together to solve a one-tme unonventonal problem. After the predtons are aggregated, the agents an walk away. Therefore, the prnpal needs to provde explt reward funtons so as to motvate the agents n ths one-tme nteraton. Parallel researh has studed long-tme nteratons between the prnpal and agents. In suh a setup, a reputaton mehansm may play a ertan role n estmatng the agents relablty. (See Dellaroas, 005, for dsusson on reputaton mehansm desgn and ommunty buldng. For example, the agents wll exert effort to obtan the best foreast, takng aount of ther reputaton. In addton, the prnpal an look at the agents long-term performane to dede the relablty of the report. A reputaton mehansm ould play an mportant role n motvatng agents to ontrbute to onlne ommunty buldng and resoure sharng (for example, ebay, wkpeda, and Skype. In omparson, our mehansm apples to the foreastng problems on unonventonal subets, where hstoral data/reputaton s not avalable to evaluate the agents performane. 7 Referenes Ba, S, Y. Stallaert, and A. B. Whnston (00 Researh Commentary: Introdung a Thrd Dmenson n Informaton Systems Desgn The Case for Inentve Algnment, Informaton Systems Researh, : Chen, K. Y., L. R. Fne and B. A. Huberman (00 Foreastng Unertan Events wth Small Groups, Pro. ACM EC 0 Conf. ACM, Tampa, FL. Chen, K. Y., L. R. Fne and B. A. Huberman (004 Elmnatng Publ Knowledge Bases n Informaton-Aggregaton Mehansms, Management Sene, 50: Chen, K. Y., and C. R. Plott (00 Informaton Aggregaton Mehansms: Conept, Desgn and Implementaton for a Sales Foreastng Problem, CIT Soal Sene Workng Paper. Clemen, R., and R. Wnkler (993 Aggregatng Pont Estmates: A Flexble Modellng Approah, Management Sene, 39: DeGroot, M. H. (970 Optmal Statst Deson. MGraw-Hll: New York. Dellaros, C. (005 Reputaton Mehansm Desgn n Onlne Tradng Envronments wth Pure Moral Hazard, Informaton Systems Researh, 6: Fama, E. F. (970 Effent Captal Markets: A Revew of Theory and Empral Work, The Journal of Fnane, 5: Fang, F., M. B. Stnhombe, and A. B. Whnston (006 Optmal Aggregaton of Informaton Soures wth Dfferng Qualty, CREC Workng Paper, Center for Researh on E-Commere, The Unversty of Texas at Austn. 33