Combinatorial Prediction Markets for Event Hierarchies

Transcription

1 Combinatorial rdiction Markts for Evnt Hirarchis Mingyu Guo Duk Univrsity Dpartmnt of Computr Scinc Durham, NC, USA David M. nnock Yahoo! Rsarch 111 W. 40th St. 17th Floor Nw York, NY ABSTRACT W study combinatorial prdiction markts whr agnts bt on th sum of valus at any tr nod in a hirarchy of vnts, for xampl th sum of pag viws among all th childrn within a wb subdomain. W propos thr xprssiv btting languags that sm natural, and analyz th complxity of pricing using Hanson s logarithmic markt scoring rul (LMSR) markt makr. Sum of arbitrary subst (SAS) allows agnts to bt on th wightd sum of an arbitrary subst of valus. Sum with varying wights (SVW) allows agnts to st thir own wights in thir bts but rstricts thm to only bt on substs that corrspond to tr nods in a fixd hirarchy. W show that LMSR pricing is N-hard for both SAS and SVW. Sum with prdfind wights (SW) also rstricts bts to nods in a hirarchy, but using prdfind wights. W driv a polynomial tim pricing algorithm for SW. W discuss th algorithm s gnralization to othr btting contxts, including btting on maximum/minimum and btting on th product of binary valus. Finally, w dscrib a prototyp w built to prdict wb sit pag viws and discuss th implmntation issus that aros. Catgoris and Subjct Dscriptors J.4 [Computr Applications]: Social and Bhavioral Scincs Economics Gnral Trms Economics, Thory Kywords Combinatorial prdiction markts, logarithmic markt scoring rul markt makr, computational complxity 1. INTRODUCTION rdiction markts ar powrful mchanisms for liciting probability stimats of futur vnts. Th markts assssmnt can b rmarkably accurat [12, 13]. Th Iowa Elctronic Markts (IEM), a ral-mony basd prdiction markt maintaind by th This matrial is basd on work don primarily whil Guo was visiting Yahoo! Rsarch. Whil at Duk, Guo is supportd undr NSF IIS Cit as: Combinatorial rdiction Markts for for Evnt Evnt Hirarchis, Mingyu Mingyu Guo, Guo and David David M. M. nnock, nnock, roc. roc. of of 8th 8th Int. Int. Conf. Conf. on on Autonomous Autonomous Agnts and Multiagnt Systms (AAMAS 2009), Dckr, Sichman, Agnts and Multiagnt Systms (AAMAS 2009), Dckr, Sichman, Sirra and Castlfranchi (ds.), May, 10 15, 2009, Budapst, Hungary, pp. Sirra and Castlfranchi (ds.), May, 10 15, 2009, Budapst, Hungary, pp. XXX-XXX. Copyright c 2009, Intrnational Foundation for Autonomous Agnts and Multiagnt Copyright Systms 2009, Intrnational ( Foundation All rights for Autonomous rsrvd. Agnts and Multiagnt Systms ( All rights rsrvd. Univrsity of Iowa, significantly outprforms th traditional polls ovr th past fiv US prsidntial lctions [1, 5, 6, 10]. Th Forsight Exchang ( and Hollywood Stock Exchang ( two play-mony basd prdiction markts, hav bn succssful in prdicting th probabilitis of various vnts on a larg rang of topics, such as unrsolvd scintific qustions and Oscar award winnrs [11]. In a typical prdiction markt, agnts trad scuritis with ach othr or with a cntral markt makr. An xampl scurity would b $1 on Duk to win th 2009 NCAA mn s basktball championship. Such a scurity pays off $1 if Duk indd wins th titl, and $0 othrwis. If th markt pric for this scurity is $0.2, thn it mans that th consnsus stimatd probability of Duk winning th titl is 20 prcnt at th tim of th quot. Th markt pric changs along with th trading activitis: mor dmand rsults in highr pric and vis vrsa. From Las Vgas to Wall Strt, narly all oprating prdiction markts ar singl dimnsional. That is, scuritis of diffrnt kinds ar bing tradd in sparat markts, vn if thy may b logically rlatd. For xampl, th pric of on scurity of Duk winning th 2009 NCAA mn s basktball championship should b rlatd to th pric of anothr scurity of Duk gtting into th Final Four. Whn rlatd scuritis ar handld sparatly, stimat discrpancis and undsirabl arbitrag opportunitis aris. To captur th undrlying rlationship among diffrnt scuritis, w nd a combinatorial markt in which all allowabl scuritis ar bing tradd, and th markt must maintain a consistnt st of prics for all th scuritis. Lt us considr a combinatorial prdiction markt on th statby-stat rsults of th US prsidntial lction. Th outcom spac for such a combinatorial markt is xtrmly larg (2 51, considring District of Columbia). Low liquidity bcoms a problm, as th agnts attntion gts dividd among xponntially many outcoms. Gnralization of standard doubl auctions may simply fail to find any trads [7, 3]. A bttr ida is to implmnt a markt makr that is willing to buy or sll any scurity at any tim. Whn an agnt coms in, sh asks for a quot of th scurity of hr intrst. If th quotd pric is lowr than th pric in th agnt s mind, th agnt can start buying th scurity, until th pric grows to a point that is clos to th agnt s stimation. On th othr hand, if th pric is considrd too high, th agnt can start slling th scurity (quivalnt to buying th ngation of th vnt). Whn th markt gts stabl, th markt prics rflct th agnts consnsus probability stimations. In this papr, w will b focusing on a spcific typ of combinatorial markt makr Hanson s logarithmic markt scoring rul markt makr (LMSR) [8, 9]. A prdiction markt basd on LMSR rquirs only boundd subsidy and it has in som sns infinit 201

2 AAMAS th Intrnational Confrnc on Autonomous Agnts and Multiagnt Systms May, 2009 Budapst, Hungary liquidity. LMSR is bcoming th standard markt makr for combinatorial stting, and it has bn th subjct of intrst in a numbr of rsarch paprs [4]. In principl, agnts should b allowd to bt on (buy/sll scuritis of) any vnt (subst of outcoms). Howvr, whn th outcom spac is larg, pricing (computing th xact pric quot ) may tak xponntial tim for som scuritis. Thrfor, as a trad off, w can rstrict th st of vnts that ar allowd to b bt on, usually through rstricting th btting languag. Svral paprs xamin th balanc btwn xprssivnss and computational complxity [7, 4, 2, 3]. This lin of rsarch was initiatd by Fortnow t al. [7], followd by Chn t al. [3], in which th authors study btting languags on Boolan combinatorics and prmutations for markt claring problms. In Chn t al. [2], th authors analyz th computational complxity of LMSR pricing for prmutations and Boolan combinatorics. Th authors show that for subst btting, pair btting, and btting on conjunctions and disjunctions, pricing for LMSR markt makr is #-hard. Th work closst to our own is that of [4]. Th authors study a spcial cas of Boolan combinatorics in which th agnts bt on how far a tam gos in a singl-limination tournamnt. Thy propos a polynomial-tim algorithm for th problm of LMSR pricing in th tournamnt contxt. Th authors also show that th pricing problm is N-hard for som mor gnral btting languags. In this papr, w will follow this lin of rsarch. W study combinatorial prdiction markts in which th agnts can bt on th wightd sum of valus that ar associatd with futur vnts. Blow w giv two dtaild xampl application contxts that involv btting on sum. W will b rfrring to ths two xampl contxts throughout th papr. Btting on pag viws: Th total pag viws or imprssion of a subdomain of a wb sit (.g. is a subdomain of is th numbr of visits to this subdomain (for a givn priod of tim). ag viws is a standard mtric for Intrnt advrtising as it capturs quantity of advrtismnts that can b supplid to th advrtisrs. If w can prdict th pag viws of a subdomain for th coming month, that is, if w can prdict th quantity of supply of th coming month, thn w can st bttr prics for advrtismnts. Traditionally, th prdiction has bn solly basd on machin larning algorithms. rdiction markt improvs upon th traditional approach by adding an xtra twaking/corrcting stag to th prdicting procss: W initializ th markt according to th bst algorithm availabl, thn lav it for th invisibl hand to figur out th rights and wrongs. (W hav implmntd a prototyp to prdict wb sit pag viws. Mor dtails ar in Sction 8.) A subdomain is calld a laf subdomain, if it contains no child subdomains undr it. For xampl, is not a laf subdomain bcaus it contains th child subdomain Th total pag viws of a nonlaf subdomain is th sum of th pag viws of its child subdomains. For xampl, a subdomain about NCAA contains a list of child subdomains: NCAA hompag, NCAA basktball, NCAA football, tc. Btting on th pag viws of a non-laf subdomain is ssntially btting on th sum of th pag viws of all its child subdomains. A natural bt (scurity) in this contxt would b th pag viws of subdomain x is btwn v 1 and v 2 (for th coming month) th sum of pag viws of all th child subdomains of x is btwn v 1 and v 2 (for th coming month). Btting on lctoral vot count: 1 In th US prsidntial lc- 1 For simplicity, w assum all stats ar winnr-taks-all and w tion, if a party wins th popular vot of a stat, thn it wins all th lctoral vots of that stat (winnr-taks-all). W can us a binary variabl to dnot th lction rsult of a stat: it taks valu 1 if th Dmocrats win, and it taks valu 0 if th Rpublicans win. Th total numbr of lctoral vots won by th Dmocrats is thn th wightd sum of all th binary variabls, whr th wights ar th numbr of lctoral vots of diffrnt stats (.g. Ohio has 20 lctoral vots its wight is 20). Btting on th numbr of lctoral vots won by th Dmocrats from a st of stats is ssntially btting on th wightd sum of th binary variabls rprsnting thos stats. A natural bt (scurity) in this contxt would b th total numbr of lctoral vots won by th Dmocrats is btwn v 1 and v 2 th wightd sum of stats won by th Dmocrats is btwn v 1 and v 2. Our papr is organizd as follows: In Sction 2, w rviw th prliminaris of Hanson s logarithmic markt scoring rul markt makr. In Sction 3, w propos thr xprssiv btting languags that sm natural. Th first btting languag (SAS) allows agnts to bt on th wightd sum of an arbitrary subst of valus. Th scond btting languag (SVW) allows agnts to st thir own wights in thir bts but rstricts substs to form a hirarchy. In Sction 4 and Sction 5, w show that LMSR pricing is N-hard for both SAS and SVW. Th third btting languag (SW) allows th agnts to bt on th wightd sum of slctd substs of valus, whr th wights ar prdfind and substs form a hirarchy. W driv a polynomial tim pricing algorithm for SW in Sction 6. In Sction 7, w discuss th algorithm s gnralization to othr btting contxts, including btting on maximum/minimum and btting on th product of binary valus. Finally, in Sction 8, w dscrib a prototyp w built to prdict wb sit pag viws and discuss th implmntation issus that aros. 2. LOGARITHMIC MARKET SCORING RULE MARKET MAKER (LMSR) Logarithmic markt scoring ruls [8, 9] ar squntial vrsions of logarithmic scoring ruls. Scoring ruls map probability distributions and rsults of futur vnts into amounts of rward. Logarithmic scoring ruls ar propr in th sns that whn facing such ruls, risk-nutral agnts will rval thir tru subjctiv probability distributions of th futur vnts to maximiz thir xpctd rward. Logarithmic markt scoring ruls can b intrprtd as follows: Th markt starts with som initial distribution ovr th outcom spac. Whn an agnt coms in, sh can modify th currnt markt distribution at hr will. Hr rward is thn th rward, undr a spcific logarithmic scoring rul, for th modifid distribution, minus th rward for th distribution bfor modification. At any tim, for any agnt, sinc th rward for th distribution bfor modification is byond th agnt s control, ssntially, th agnt can only focus on maximizing th rward for th modifid distribution. That is, th agnts always fac a (propr) logarithmic scoring rul. Thrfor, it is a dominant stratgy for a myopic agnt to rval hr tru blifs undr LMSR. LMSR is usually implmntd as a markt makr. That is, instad of asking th agnts to dirctly modify th markt distribution, thr is a markt makr that is in charg of maintaining a consistnt st of prics (probabilitis) for all th allowabl scuritis, and th agnts modify th markt distribution through buying or slling scuritis. For xampl, slling scuritis of an outcom is quivalnt to marking down th probability of that outcom in ignor all third partis. 202

3 Mingyu Guo, David M. nnock Combinatorial rdiction Markts for Evnt Hirarchis th markt distribution. Obviously, trading scuritis is mor natural than playing with distribution ovr an outcom spac that is usually xponntial in siz. A gnric LMSR offrs scuritis corrsponding to all outcoms. A scurity on outcom ω pays off $1 if ω happns, and $0 othrwis. At any momnt, th markt makr kps track of a vctor q (q ω) ω Ω, which indicats th numbr of outstanding shars of all outcoms. That is, th numbr of (activ) scuritis covring outcom ω is dnotd by q ω. Ω is th st of all outcoms. Th instantanous pric for scurity ω undr LMSR is p ω(q) qω b is a positiv paramtr. Whn b is small, purchasing or shorting a fw scuritis can significantly chang th markt distribution. Whn b is larg, th ffct of buying or slling a fw scuritis is lss noticabl, maning th ffctiv liquidity of th markt is larg. Suppos an agnt wants to purchas/short on scurity of ω. Th currnt outstanding shars ar dnotd by q, and aftr purchasing/shorting ω, th st of outstanding shars bcom ˆq. Thn th cost of th transaction quals th intgral of th instantanous pric following any path from q to ˆq. Th cost can b writtn as C(ˆq) C(q), whr function C is a cost function with th following form: C(q) b log X qτ Function C has anothr maning. At any momnt, th worstcas subsidy rquird to run th markt makr is at most C(q). If th markt starts with 0 shars on all outcoms (which is an usual assumption), thn th worst-cas subsidy is b log Ω. In most cass, it is natural to only bt on compound scuritis on collctions of outcoms. For xampl, th compound scurity It will rain on xactly on day in th nxt wk is a collction of 7 scuritis on singl outcoms: It will rain on day x only, for all choics of x. A compound scurity s instantanous pric is just th sum of th instantanous prics of all th outcoms covrd by th compound scurity. 3. BETTING LANGUAGES In this papr, w considr combinatorial prdiction markts in which th final outcoms can b rprsntd as tupls of valus. Spcifically, w considr outcom spac Ω whos lmnts ar ω (x 1,x 2,...,x n) whr x i {0, 1,...,N} for all i. 2 It is asy to s that th siz of th outcom spac is (N +1) n. For btting on pag viws, n is th numbr of laf subdomains, and N is th uppr bound on th pag viws of th laf subdomains. For btting on lctoral vot count, n is th numbr of US stats, and N 1(rcall that th rsult of a stat is dnotd by a binary variabl). W propos thr xprssiv btting languags that sm natural for btting on sum. Thy offr diffrnt lvls of xprssivnss, 2 It is without loss of gnrality to rstrict th valus of th x i to intgrs from 0 to N, as long as th x i tak thir valus from a finit st of rational numbrs. For xampl, if x 1 s valu is ithr 1/7 or 2/3, thn(21x 1 3) s valu is ithr 0 or 11, which is in {0, 1,...,N} for N 11. W can simply bt on th valus of th x i aftr crtain linar transformation. and fac diffrnt lvls of computational difficulty. W first introduc th SAS btting languag, which is th most gnral on among th thr. Sum of arbitrary subst (SAS) btting on sum of arbitrary subst of th x i. A scurity undr SAS has th following form: v 1 X i S c ix i v 2 whr S {0, 1,...,n}, v 1, v 2 and th c i ar all nonngativ intgrs. 3 W also allow bts that spcify only on nd of th tripl inquality. In Sction 4, w will show that LMSR pricing is N-hard vn for a rstrictd vrsion of SAS in which all th c i ar rquird to b constant 1 (unit wights). Allowing btting on arbitrary substs maks LMSR pricing computationally infasibl. Thrfor, w nd to sacrific som xprssivnss. A natural stp is to mak crtain rstrictions on th substs that ar allowd to b bt on. Actually, this is not ncssarily a bad thing bcaus chancs ar our intrsts ar focusd on slctd substs anyway. W notic that natural vnts somtims form vnt hirarchis vntsofintr- sts (what w ar intrstd in btting on) corrspond to nods of a tr, and th vnt corrsponding to a non-laf nod is dtrmind by its child nods. For both xampl contxts mntiond in th introduction, w s such hirarchis. Evnt hirarchy for btting on pag viws: Th following tr dscribs a typical subdomain hirarchy. Sports NCAA Hompag Basktball Football... Th pag viws of NCAA is th sum of th pag viws of its child subdomains. A bt on th pag viws of NCAA is a bt on th sum of th pag viws of its child subdomains, that is, th sum of th pag viws of all th laf subdomains whos ancstor is NCAA. Evnt hirarchy for btting on lctoral vot count: In th US prsidntial lction, th stat-by-stat rsults, as wll as th ovrall lction rsult, form th following hirarchy: Ovrall Elction Rsult Alabama Alaska Arizona Arkansas... Th ovrall lction rsult dpnds on its childrn. Th numbr of lctoral vots rcivd by th Dmocrats in th lction is th wightd sum of th rsults of all stats. (Rcall that w us binary variabls to dnot th rsult of a stat: it taks valu 1 if th Dmocrats win, and 0 othrwis.) Th tr structur of an vnt hirarchy dtrmins which substs ar allowd to b bt on (ths substs corrspond to th tr nods). For xampl, for th tr blow, w ar allowd to bt on (th wightd sum of) th following substs: {x 1}, {x 2}, {x 3}, {x 4}, {x 5}, {x 1,x 2,x 3}, {x 4,x 5}, {x 1,x 2,x 3,x 4,x 5}. 3 Again, it is without loss of gnrality to rstrict th valus of v 1, v 2 and th c i to intgrs, as long as v 1, v 2 and th c i ar rational numbrs. For xampl, a bt on 2/5 1/5x 1 +1/2x 2 3/7 can simply b rwrittn as 28 14x 1 +35x

4 AAMAS th Intrnational Confrnc on Autonomous Agnts and Multiagnt Systms May, 2009 Budapst, Hungary x 1 x 2 x 3 x 4 x 5 Now w ar rady to introduc SVW and SW. Sum with varying wights (SVW) btting on th wightd sum of slctd substs of th x i. Only substs corrsponding to tr nods ar allowd to b bt on. Th agnts can st thir own wights in thir bts. A scurity undr SVW has th following form: v 1 X i S c ix i v 2 whr S corrsponds to a tr nod, v 1, v 2 and th c i ar intgrs spcifid by th agnts. v 1 and v 2 ar nonngativ. Th c i ar positiv. 4 W also allow bts that spcify only on nd of th tripl inquality. In Sction 5, w will show that LMSR pricing is N-hard for SVW for any vnt hirarchy (tr structur). Sum with prdfind wights (SW) btting on th wightd sum of slctd substs of th x i. Only substs corrsponding to tr nods ar allowd to b bt on. Th wights ar prdfind. A scurity undr SW has th following form: v 1 X c ix i v 2 i S whr S corrsponds to a tr nod, v 1, v 2 ar nonngativ intgrs spcifid by th agnts, and th c i ar prdfind constant intgrs. W also allow bts that spcify only on nd of th tripl inquality. In Sction 6, w driv a polynomial tim pricing algorithm for th SW btting languag. 4. COMLEXITY OF SAS In this sction, w analyz th complxity of pricing using LMSR for th SAS btting languag. W show that LMSR pricing is Nhard vn for a rstrictd vrsion of SAS in which all th wights arrquirdtob1 (unit wights). CLAIM 1. LMSR pricing for th SAS btting languag is Nhard. ROOF. Rcall that a scurity undr SAS has th following form: v 1 i S cixi v2,whrs is an arbitrary subst of {0,...,n}, v 1, v 2 and th c i ar nonngativ intgrs. In this proof, w will only nd to considr scuritis with th following form: v 1 i S xi v2. That is, w only considr scuritis in which all th c i ar qual to constant 1. W will show that vn if agnts only bt on ths rstrictd scuritis, pricing using LMSR is still N-hard. Lt us considr an arbitrary 3-SAT xprssion with n v binary variabls z 1,z 2,...,z nv and n c clauss. E.g. (z 1 z 2 z nv ) (z 5 z 6 z nv )... (z 1 z 1 z 7) {z } n c W will show that LMSR pricing for SAS involvs solving th satisfiability problm of th abov 3-SAT xprssion. Rcall that th outcom spac Ω is th st of all n tupls ω (x 1,x 2,...,x n), whrx i {0, 1,...,N} for all i. Ltus considr a LMSR markt markr for which n (2n v +2)n c.for prsntation purpos, w rnam th x i so that th outcoms ar now n tupls as follows: 4 If th c i ar allowd to b zros, thn it rducs to th cas of btting on arbitrary substs. (z 11, z 11,z 21, z 21,...,z nv1, z nv1,u 1,v 1, {z } 2n v+2 z 12, z 12,z 22, z 22,...,z nv2, z nv2,u 2,v 2, {z } 2n v+2. z 1nc, z 1nc,z 2nc, z 2nc,...,z nvnc, z nvnc,u nc,v nc ) {z } 2n v+2 (altogthr n c rows) z ij, z ij, u j and v j ar in {0, 1,...,N} for i from 1 to n v and j from 1 to n c. Basically, w want to link th valu of z ij to th valu of z i in th j-th claus of th 3-SAT xprssion undr considration. (For ach i, w nd to mak sur that th valus of th z ij ar th sam ovr diffrnt j,sincz i s valu should b th sam in all clauss.) W want to link th valu of z ij to th logical ngativ of z i.th u j and v j ar auxiliary variabls. Suppos th following scuritis hav bn purchasd. (W assum that thr wr no outstanding scuritis whn th markt startd. That is, th following scuritis ar th only outstanding scuritis.) 1. scuritis on 0 z ij 1 for all i from 1 to n v and all j from 1 to n c 2. scuritis on 0 z ij 1 for all i from 1 to n v and all j from 1 to n c 3. scuritis on 0 u j 1 for all j from 1 to n c 4. scuritis on 0 v j 1 for all j from 1 to n c 5. scuritis on n c j1 zij 0 for all i from 1 to nv; scuritis on n c j1 zij nc for all i from 1 to nv 6. scuritis on z ij + z ij 1for all i from 1 to n v and all j from 1 to n c 7. scuritis on 0 w j + u j + v j 3 for all j, whr w j is th sum of thr slctd variabls among z ij and z ij that corrspond to th thr litrals in th j-th claus of th 3-SAT xprssion undr considration. For xampl, if th j-th claus of th 3-SAT xprssion is (z 1 z 2 z 3),thn w j z 1j + z 2j + z 3j 8. Q scuritis on w j + u j + v j 3for all j, whrw j is dfind th sam as abov (n c + 1)(2n v +2)n c log(n)b Q (2n v +2)n c log(n)b If thr xists a satisfactory assignmnt of th 3-SAT xprssion, thn thr xists on outcom that satisfis all th abov groups of scuritis 5. (Lt th z i b any satisfactory assignmnt. Th following outcom satisfis all th groups: z ij z i for all i and j; z ij z i for all i and j; Ifw j 1,thnu j v j 1;Ifw j 2, thn u j 1and v j 0;Ifw j 3,thnu j v j 0.) If thr xists on outcom that satisfis all th abov groups of scuritis, thn it corrsponds to a satisfactory assignmnt of th 3-SAT xprssion. On satisfactory assignmnt is simply z i z ij for arbitrary j. (All th variabls ar binary according to th first four groups of bts. For spcific i, th valus of th z ij ar th sam 5 W say an outcom satisfis th fifth group of scuritis if it satisfis half of thm (n v out of 2n v). 204

5 Mingyu Guo, David M. nnock Combinatorial rdiction Markts for Evnt Hirarchis ovr all j (ithr all 0 or all 1) according to th fifth group of bts. Th valu of z ij and z ij ar diffrnt according to th sixth group of bts. That is, z i corrsponds to z ij for all j. All th clauss of th 3-SAT ar satisfid by th z i according to th ighth group of scuritis, sinc w j + u j + v j 3implis w j 1.) That is, thr xists on outcom that satisfis all th abov groups of scuritis if and only if th 3-SAT xprssion has a satisfactory assignmnt. Considr th pricing problm of th following scurity: Xn c j1 (w j + u j + v j)3n c This scurity is allowd by th SAS btting languag, sinc it is th sum of a subst of variabls with unit wights. From now on, w rfr to this scurity as th objctiv scurity. W first assum that th 3-SAT xprssion is satisfiabl. Thr xists at last on outcom that satisfis all th abov groups of scuritis, and it must b covrd by th objctiv scurity according to th ighth group of xisting scuritis. Rcall that th instantanous pric for outcom ω is p ω(q),whrq qω ω is th numbr of outstanding shars for outcom ω. Th numbr of outstanding shars for an outcom that satisfis all th xisting groups of scuritis is (2n vn c +2n c + n v + n vn c + n c)+qn c. Th instantanous pric for such an outcom is ( (3nvnc+3nc+nv)+Qnc) Sinc a compound scurity s instantanous pric is th sum of th instantanous prics of all th outcoms covrd by th compound scurity, w hav that th pric of th objctiv scurity is gratr than or qual to th abov xprssion. If an outcom is not covrd by th objctiv scurity, thn it maks w j + u j + v j 3for at last on j. Th numbr of outstanding shars for such an outcom is at most (2n vn c +2n c + n v + n vn c + n c)+q(n c 1). Thrfor, th instantanous pric for such an outcom is at most ( (3nvnc+3nc+nv)+Q(nc 1)) Thr ar at most N n 1N (2nv+2)nc 1 such outcoms (N n is th siz of th outcom spac, and thr is at last on outcom that is covrd by th objctiv scurity according to our assumption). So th sum of th instantanous prics of all outcoms that ar not covrd by th objctiv scurity is at most < (N (2nv+2)nc ( (3nvnc+3nc+nv)+Q(nc 1)) 1) log(n)+( (3nvnc+3nc+nv)+Q(nc 1)) (2nv+2)nc (3nvnc+3nc+nv)+Qnc) ( That is, th pric of th objctiv scurity is gratr than th total pric of all th outcoms that ar not covrd by it. Hnc, if th 3- SAT xprssion is satisfiabl, thn th pric of th objctiv scurity is gratr than 1 2. Now w assum that th 3-SAT xprssion is not satisfiabl. Thr dos not xist an outcom that satisfis all th xisting groups of scuritis. If an outcom is covrd by th objctiv scurity and satisfis th first svn groups of xisting scuritis, thn it also satisfis th ighth group of scuritis, which is contrary to th fact that th 3-SAT xprssion is not satisfiabl. That is, all outcoms covrd by th objctiv scurity must violat som of th first svn groups of scuritis. Th numbr of outstanding shars for any outcom that is covrd by th objctiv scurity is at most (2n vn c +2n c + n v + n vn c + n c 1) +Qn c. Thrfor, th instantanous pric for such an outcom is at most ( (3nvnc+3nc+nv 1)+Qnc) Thr ar at most N n N (2nv+2)nc such outcoms. So th sum of th instantanous prics of all th outcoms that ar covrd by th objctiv scurity is at most N (2nv+2)nc ( (3nvnc+3nc+nv 1)+Qnc) log(n)+( (3nvnc+3nc+nv 1)+Qnc) (2nv+2)nc (3nvnc+3nc+nv 1)+Q(nc+1)) ( Now considr an outcom that corrsponds to an arbitrary assignmnt of th 3-SAT xprssion. (E.g. z ij z i and z ij z i for all i and j; u j 0and v j 0for all j.) Th outcom satisfis th first svn groups of xisting scuritis, and dos not satisfy th objctiv scurity. Its instantanous pric is at last (3nvnc+3nc+nv) (3nvnc+3nc+nv 1)+Q(nc+1)) ( That is, if th 3-SAT xprssion is not satisfiabl, thn th pric of th objctiv scurity is lss than or qual to 1. 2 If thr xists a LMSR pricing algorithm for SAS that taks only (n) tim, thn thr xists an algorithm that solvs any 3- SAT satisfiability problm with n v variabls in (n) tim. Sinc n (2n v +2)n c (2n v v +2)`2n, 3 th algorithm solvs any 3- SAT satisfiability problm with n v variabls in (n v) tim. This is impossibl unlss N. Thrfor, LMSR pricing for th SAS btting languag is N-hard. 5. COMLEXITY OF SVW In this sction, w analyz th complxity of pricing using LMSR for th SVW btting languag. Th proof of Claim 1 tlls us that if agnts ar allowd to bt on arbitrary substs of th x i, thn th pricing problm is N-hard vn if w rquir unit wights. SVW rstricts th st of substs that ar allowd to b bt on (only substs corrsponding to tr nods ar ligibl), but on th othr hand, it still allows agnts to st thir own wights in th thir bts. It turns out that th rsult is still ngativ: LMSR pricing is N-hard for SVW for any vnt hirarchy. CLAIM 2. LMSR pricing for th SVW btting languag is Nhard for any vnt hirarchy. ROOF. Rcall that a scurity undr SVW has th following form: v 1 i S cixi v2, whrs corrsponds to a tr nod, 205

6 AAMAS th Intrnational Confrnc on Autonomous Agnts and Multiagnt Systms May, 2009 Budapst, Hungary v 1, v 2 and th c i ar intgrs spcifid by th agnts. v 1 and v 2 ar nonngativ. Th c i ar positiv. For any vnt hirarchy, th following substs ar always allowdtobbton:{x i} for i 1, 2,...,n,and{x 1,x 2,...,x n} (thy corrspond to th lafs and th root). W will construct our proof basd on scuritis only on ths substs. Thrfor, our rsult applis to any vnt hirarchy. Suppos th following scuritis hav bn purchasd. (W assum that thr wr no outstanding scuritis whn th markt startd. That is, th following scuritis ar th only outstanding scuritis.) n log(n)b scuritis on x i 1(c ix i c i)foralli from 1 to n. n log(n)b scuritis on x i 0(c ix i 0)foralli from 1 to n. Considr th pricing of th following two scuritis: n i1 cixi I (scurity A) and n i1 cixi 0(scurity B). I is a spcific positiv intgr. If thr xists a subst of th c i that sum to xactly I, thn thr xists on outcom (t 1,t 2,...,t n) whos numbr of outstanding shars is n 2 log(n)b, by stting t i 1 if c i is in th subst of numbrs that sum to I, andt i 0 othrwis. This outcom is covrd by scurity A. For scurity B, th only outcom it covrs is (0, 0,...,0), whos numbr of outstanding shars is also n 2 log(n)b. Thrfor, th pric of scurity A is at last as grat as th pric of scurity B, if thr xists a subst of th c i that sum xactly to I. If thr dos not xist a subst of th c i that sum to I, thnall outcoms covrd by scurity A hav at most n(n 1) log(n)b outstanding shars. Thr ar at most N n 1 such outcoms. Th instantanous pric of scurity A is thn at most (N n n(n 1) log(n) 1) < N n n(n 1) log(n) n 2 log(n) W notic that th right-hand sid of th inquality is xactly th instantanous pric of scurity B. That is, th pric of scurity A is lss than th pric of scurity B, if thr dos not xist a subst of th c i that sum xactly to I. Thrfor, LMSR pricing for th SVW btting languag is at last as difficult as th Subst-sum problm with n positiv intgrs, which is N-complt. 6. A OLYNOMIAL-TIME RICING ALGORITHM FOR SW Th SW btting languag allows agnts to bt on th wightd sum of slctd substs of th x i. Ths substs corrspond to vnts that form a tr. Th wights ar prdfind. For xampl, for th following vnt hirarchy (valus in th parnthsis ar th prdfind wights of th nods): r r L x 1(3) x 2(1) x 3(5) x 4(1) r R r RR x 5(7) x 6(2) Th agnts ar allowd to bt on th valus of x 1,x 2,...,x 6, r L (3x 1 +1x 2 +5x 3), r RR (7x 5 +2x 6), r R (1x 4 +7x 5 +2x 6), and r (3x 1 +1x 2 +5x 3 +1x 4 +7x 5 +2x 6). Bfor introducing our algorithm, w first propos th following lmmas. LEMMA 1. Lt y b a random variabl associatd with any vnt. Th distribution of y is charactrizd by th outstanding scuritis in th markt. Lt v b an arbitrary constant. If (y v) p, thn aftr introducing on xtra scurity on y v, w hav (y v) p 1 p 1 +(1 p). On way to intrprt th abov claim is that, aftr introducing on xtra copy of scurity y v, th probability of y v is first magnifid by a factor of 1, thn th distribution vctor of y is normalizd (multiplid by som valu so that th sum of all th lmnts is back to 1). ROOF. W us q (q τ ) to indicat th numbr of outstanding shars of all outcoms bfor introducing th xtra scurity. W hav p τ {τ yv} τ {τ yv} τ {τ yv} + τ {τ y v} Aftr introducing th xtra scurity, w hav (y v) +1) (qτ τ {τ yv} τ {τ yv} (qτ +1) + τ {τ y v} p 1 p 1 +(1 p) LEMMA 2. Lt y, z b two random variabls that rprsnt th valus of two arbitrary vnts. Th distribution of y is charactrizd by th outstanding scuritis in th markt. Lt v y, v z b two arbitrary constants. Lt p (y v y), p (y v y z v z) and p (y v y z v z). 1. If w introduc M copis of scurity z v z into th markt, thn w hav lim M (y v y)p. 2. If w introduc ngativ M copis of scurity z v z (both M copis of z > v z and M copis of z < v z) into th markt, thn w hav lim M (y v y)p. ROOF. Du to spac constraint, w will only prsnt th proof of statmnt 2. W us q (q τ ) to indicat th numbr of outstanding shars of all outcoms. W hav p (y v y z v z) (y v y z v z)/ (z v z) τ {τ yv y z v z} / τ {τ yv y z v z} τ {τ z v z} τ {τ z v z} Aftr shorting M copis of scurity z v z,whav (y v y) quals τ {τ yv y zv z} (qτ M) + τ {τ yv y z v z} τ {τ zv z} (qτ M) + τ {τ z v z} 206

7 Mingyu Guo, David M. nnock Combinatorial rdiction Markts for Evnt Hirarchis As M gos to infinity, (qτ M) gos to 0 for any τ. So (y v y) τ {τ yv y z v z} p τ {τ z v z} LEMMA 3. Th outcom spac consists of tupls of n coordinats. If th n coordinats can b sparatd into k groups, and no outstanding scurity mntions coordinats of diffrnt groups, thn coordinats of diffrnt groups ar indpndnt th markt can b intrprtd as k sparat markts. On simpl xampl suffics to illustrat th ida bhind th abov claim. Lt th outcom spac b {(x 1,x 2,x 3) x 1 is th numbr of stats won by th Dmocrats in th lction; x 2 is th numbr of stats won by th Rpublicans in th lction; x 3 is tomorrow s tmpratur}. Whn th markt starts (with no outstanding scuritis), th x i ar pairwis indpndnt (proprty of LMSR). Now suppos th scuritis ar dividd into two groups. On group of scuritis ar on lction. Thy hav som ffct on th distribution of x 1 or x 2 or both. Th othr group of scuritis ar on tmpratur. Thy hav som ffct on th distribution of x 3. With ths two groups of scuritis, (x 1,x 2) and x 3 ar still indpndnt. A dtaild proof is omittd du to spac constraint. Now w ar rady to introduc th LMSR pricing algorithm for SW. Th algorithm taks as input th st of outstanding scuritis and an objctiv scurity of th following form: r v whr r is an vnt (a tr nod) and v is a constant intgr. Th algorithm outputs th instantanous pric (probability) of th objctiv scurity. (Th pric of a scurity on a rang,.g. v 1 r v 2, can b computd as v 2 vv 1 (r v).) Th tr nods ar random variabls that tak intgr valus from 0 to CnN,whrC is th maximal wight (constant). W will us array of siz CnN +1as th data structur for storing distribution of a random variabl. Th algorithm is basd on th following routin dist(r): itcomputs th markt distribution of th random variabl corrsponding to tr nod r, considring only outstanding scuritis on r and r s offspring (ignoring all othr outstanding scuritis). Outlin of th algorithm Lt r 0 b th root of th tr. To comput th pric of scurity r 0 v, w simply run dist(r 0) (no scuritis ignord). To comput th pric of scurity r v whr r r 0,wfirst run dist(r 0) to gt th distribution of r 0. Thn w rcomput dist(r 0), considring an xtra infinit copis of scurity r v. By Lmma 2, dist(r 0) rturns th distribution of r 0 conditional on r v. Thn w rcomput dist(r 0), considring an xtra ngativ infinit copis of scurity r v. By Lmma 2, w gt th distribution of r 0 conditional on r v. Sinc (r 0 v 0) (r 0 v 0 r v) (r v)+ (r 0 v 0 r v)(1 (r v)) for any v 0, w can solv for th valu of (r v) basd on th computd distributions. Outlin of th routin dist(r) r is not a laf nod: Rcall that whn computing dist(r),war considring only scuritis on r and r s offspring. W furthr ignor all scuritis on r. Th rmaining scuritis ar sparatd into a fw groups, with ach group corrsponding to a branch of r s offspring. Lt r 1,r 2,...,r k b r s childrn. According to Lmma 3, th valus of r s childrn ar indpndnt, and th distribution of r i is just dist(r i). W comput dist(r i) for all i. Thn w comput th distribution of r by aggrgating all dist(r i). Wfirst comput th distribution of s 2 r 1 + r 2 by aggrgating th distribution of r 1 and r 2. W thn comput th distribution of s 3 r 1 + r 2 + r 3 by aggrgating th distribution of s 2 and r 3. W ar don in k 1 stps. Th tim complxity of ach stp is at most th squar of th siz of th distribution vctor, which is polynomial in n and N. Thrfor, th whol aggrgation procss is polynomial tim. Now w hav th distribution of r. Howvr, this is th distribution that considrs only scuritis on r s offspring. To gt dist(r), w nd to add back in all scuritis on r. According to Lmma 1, w only nd to magnify th probability of r v by a factor of xv (x v is th numbr of scuritis on r v: scuritis having th form of v 1 r v 2, with v 1 v v 2), and thn normaliz th distribution vctor. 6 r is a laf nod: For laf nod r, computing dist(r) is much asir. For any possibl valu v, th probability of r v is proportional to xv,whrx v is th numbr of scuritis on r v. Complxity of th algorithm W only nd to show that th routin dist(r) is polynomial tim. dist(r) is a rcursiv routin, but it visits any nod at most onc. Th numbr of nods is polynomial in n (ach non-laf nod has at last two childrn). Th non-rcursiv part of dist(r) taks polynomial tim (in n and N). Thrfor, dist(r) is polynomial tim in n and N, so is our algorithm. 7. OTHER BETTING CONTEXTS So far w hav bn only focusing on vnt hirarchis basd on wightd sum. In principl, th algorithm w proposd in th prvious sction can b applid to any btting contxt, as long as th vnts of intrst form a tr structur. Howvr, for som btting contxts, th algorithm may not b polynomial tim. A sufficint condition for th algorithm to b polynomial tim is that Th siz of th st of all possibl valus ovr all tr nods is polynomial of n and N. Lt r b an arbitrary non-laf tr nod. Lt r 1,r 2,...,r k b r s childrn. r can b writtn as r 1 r 2... r k,whr is an associativ binary oprator (.g. addition, multiplication). That is, r ((((r 1 r 2) r 3) r 4)... r k ). Th oprator may b diffrnt for diffrnt tr nods. 7 In this sction, w giv two xampl contxts basd on oprators othr than sum btting on maximum/minimum and btting on th product of binary valus. Both xampl contxts satisfy th abov sufficint condition. Hnc for both contxts, our algorithm can b applid (polynomial tim). Btting on maximum or minimum: Lt us considr th following scnario. An lctronic gam company wants to prdict th arlist possibl rlas dat of its nxt gnration gam. Th gam s componnts ar organizd as follows: Gam Graphics Sound Ntwork Charactrs Background W may run a combinatorial prdiction markt that allows popl to bt on th maximum numbr of days it taks from today to finish a componnt. A bt (scurity) would b lik Background can b 6 If x v,thn (r v) 1.Ifx v, (r v) 0. 7 A mor gnral vrsion of this condition is that r can b writtn as f((((r 1 1 r 2) 2 r 3)... k 1 r k ),whrth i ar arbitrary binary oprators and f is an arbitrary function. (W assum that th oprators and th function can b valuatd in polynomial tim.) 207

8 AAMAS th Intrnational Confrnc on Autonomous Agnts and Multiagnt Systms May, 2009 Budapst, Hungary finishd in 60 days, or Th whol gam can b finishd in 100 days. This is an vnt hirarchy basd on th maximum oprator x y max(x, y): th numbr of days it taks to finish a (nonlaf) componnt is th maximum of th numbr of days it taks to finish any child componnt. Btting on th product of binary valus: Lt us considr a slightly modifid scnario. Th company wants to prdict whthr th gam can b rlasd bfor som dadlin. W may run a combinatorial prdiction markt in which popl bt on whthr a componnt can b finishd on tim. W us a binary valu to dnot whthr a componnt can b finishd bfor th dadlin. A bt (scurity) would b lik Background can b finishd bfor th dadlin, or Th whol gam can not b finishd bfor th dadlin. This is an vnt hirarchy basd on th product of binary valus: a (non-laf) componnt can b finishd on tim if and only if all its child componnts can b finishd bfor th dadlin. 8. BETTING ON AGE VIEWS: IMLEMENTATION ISSUES Basd on th algorithm proposd in Sction 6, w implmntd a prototyp to prdict wb sit pag viws. Th prdiction markt w implmntd is truly combinatorial with xponntial-siz stat spac, yt th prics of th allowabl scuritis can b computd in ral tim. Th prototyp taks th form of a multi-usr wb application. Blow w brifly discuss a fw issus w ncountrd during th implmntation. Th pag viws of a subdomain can b hug (in th magnitud of billions). Our algorithm can not handl such a larg N from a practical point of viw. Lt L and U b th lowr bound and uppr bound on th pag viws of th subdomains. Rcall that a bt taks th form of v 1 r v 2. Instad of allowing v 1 and v 2 to b arbitrary intgrs in {0, 1,...,U}, w rquir thm to b takn from {L + iδ i 0, 1,...,(U L)/Δ}. Larg (small) Δ lads to fastr (slowr) computation and lowr (highr) prcision. Evn though our aim is to build a combinatorial prdiction markt whr w can bt on th pag viws of all subdomains, w choos to compromis on this ida of having a singl markt whn th pag viws of a subdomain is significantly lss than that of its siblings (judging from historical data). W simply ignor such subdomains or run sparat markts for thm, bcaus activitis on ths subdomains do not affct th markt distribution (of thir siblings and ancstors) in a noticabl way, and by rmoving thm w spd up th computation. Evn whn spd is not a concrn, somtims it is bnficial to sparat th markt. Lt r b a nod dp down th tr. In our algorithm, to comput th pric of r v, w comput thr distributions: th distribution of r 0 (root), th distribution of r 0 conditional on r v, and th distribution of r 0 conditional on r v. Thn w solv for th valu of (r v) basdonthfact that (r 0 v 0) (r 0 v 0 r v) (r v)+ (r 0 v 0 r v)(1 (r v)) for any v 0. Howvr, if th thr distributions ar clos(whichisliklytobthcasifr is dp down th tr), thn th solution basd on th abov quation may contain a significant numrical rror (division by a valu that is clos to 0). A bttr ida is to prtnd that th root is only a fw lvls abov r. Thatis,w only considr a branch of th markt whn daling with nods dp down th tr. 9. CONCLUSION W studid combinatorial prdiction markts whr agnts bt on th sum of valus at any nod in a hirarchy of vnts, for xampl th sum of pag viws among all th childrn within a wb subdomain. W proposd thr xprssiv btting languags that sm natural, and analyzd th complxity of pricing using Hanson s logarithmic markt scoring rul (LMSR) markt makr. Sum of arbitrary subst (SAS) allows agnts to bt on th wightd sum of an arbitrary subst of valus. Sum with varying wights (SVW) allows agnts to st thir own wights in thir bts but only allows bts on nods in a hirarchy. W showd that LMSR pricing is Nhard for both SAS and SVW. Sum with prdfind wights (SW) allows agnts to bt on th wightd sum of substs corrsponding to nods in a hirarchy, whr th wights ar prdfind. W drivd a polynomial tim pricing algorithm for SW. W discussd th algorithm s gnralization to othr btting contxts, including btting on max/min and btting on th product of binary valus. Finally, w dscribd a prototyp w built to prdict wb sit pag viws and discussd th implmntation issus that aros. 10. REFERENCES [1] J. Brg, F. Nlson, and T. Ritz. rdiction markt accuracy in th long run. Intrnational Journal of Forcasting, 24: , [2] Y. Chn, L. Fortnow, N. Lambrt, D. M. nnock, and J. Wortman. Complxity of combinatorial markt makrs. In rocdings of th ACM Confrnc on Elctronic Commrc (EC), Chicago, IL, USA, [3] Y. Chn, L. Fortnow, E. Nikolova, and D. M. nnock. Btting on prmutations. In rocdings of th ACM Confrnc on Elctronic Commrc (EC), pags , San Digo, CA, USA, [4] Y. Chn, S. Gol, and D. M. nnock. ricing combinatorial markts for tournamnts. 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