Betting on Permutations

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1 Bettng on Permutatons ABSTRACT Ylng Chen Yahoo! Research 45 W. 18th St. 6th Floor New York, NY Evdoka Nkolova CS & AI Laoratory Massachusetts Insttute of Technology Camrdge, MA We consder a permutaton ettng scenaro, where people wager on the fnal orderng of n canddates: for example, the outcome of a horse race. We examne the auctoneer prolem of rsklessly matchng up wagers or, equvalently, fndng artrage opportuntes among the proposed wagers. Requrng dders to explctly lst the orderngs that they d lke to et on s oth unnatural and ntractale, ecause the numer of orderngs s n! and the numer of susets of orderngs s 2 n!. We propose two expressve ettng languages that seem natural for dders, and examne the computatonal complexty of the auctoneer prolem n each case. Suset ettng allows traders to et ether that a canddate wll end up ranked among some suset of postons n the fnal orderng, for example, horse A wll fnsh n postons 4, 9, or 13-21, or that a poston wll e taken y some suset of canddates, for example horse A, B, or D wll fnsh n poston 2. For suset ettng, we show that the auctoneer prolem can e solved n polynomal tme f orders are dvsle. Par ettng allows traders to et on whether one canddate wll end up ranked hgher than another canddate, for example horse A wll eat horse B. We prove that the auctoneer prolem ecomes NP-hard for par ettng. We dentfy a suffcent condton for the exstence of a par ettng match that can e verfed n polynomal tme. We also show that a natural greedy algorthm gves a poor approxmaton for ndvsle orders. Categores and Suject Descrptors J.4 [Computer Applcatons]: Socal and Behavoral Scences Economcs Part of ths work was done whle the author was at Yahoo! Research. Supported n part y Amercan Foundaton for Bulgara Fellowshp. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstruted for proft or commercal advantage and that copes ear ths notce and the full ctaton on the frst page. To copy otherwse, to repulsh, to post on servers or to redstrute to lsts, requres pror specfc permsson and/or a fee. EC 07, June 13 16, 2007, San Dego, Calforna, USA. Copyrght 2007 ACM /07/ $5.00. Lance Fortnow Department of Computer Scence Unversty of Chcago Chcago, IL General Terms Economcs, Theory Keywords Davd M. Pennock Yahoo! Research 45 W. 18th St. 6th Floor New York, NY Predcton market, expressve ettng, order matchng, computatonal complexty 1. INTRODUCTION Buyng or sellng a fnancal securty n effect s a wager on the securty s value. For example, uyng a stock s a et that the stock s value s greater than ts current prce. Each trader evaluates hs expected proft to decde the quantty to uy or sell accordng to hs own nformaton and sujectve proalty assessment. The collectve nteracton of all ets leads to an equlrum that reflects an aggregaton of all the traders nformaton and elefs. In practce, ths aggregate market assessment of the securty s value s often more accurate than other forecasts relyng on experts, polls, or statstcal nference [16, 17, 5, 2, 15]. Consder uyng a securty at prce ffty-two cents, that pays $1 f and only f a Democrat wns the 2008 US Presdental electon. The transacton s a commtment to accept a ffty-two cent loss f a Democrat does not wn n return for a forty-eght cent proft f a Democrat does wn. In ths case of an event-contngent securty, the prce the market s value of the securty corresponds drectly to the estmated proalty of the event. Almost all exstng fnancal and ettng exchanges par up lateral tradng partners. For example, one trader wllng to accept an x dollar loss f a Democrat does not wn n return for a y dollar proft f a Democrat wns s matched up wth a second trader wllng to accept the opposte. However n many scenaros, even f no lateral agreements exst among traders, multlateral agreements may e possle. For example, f one trader ets that the Democratc canddate wll receve more votes than the Repulcan canddate, a second trader ets that the Repulcan canddate wll receve more votes than the Lertaran canddate, and a thrd trader ets that the Lertaran canddate wll receve more votes than the Democratc canddate, then, dependng on the odds they each offer, there may e a three-way agreeale match even though no two-way matches exst. We propose an exchange where traders have consderale flexlty to naturally and succnctly express ther wagers, and examne the computatonal complexty of the aucton-

2 eer s resultng matchng prolem of dentfyng lateral and multlateral agreements. In partcular, we focus on a settng where traders et on the outcome of a competton among n canddates. For example, suppose that there are n canddates n an electon (or n horses n a race, etc.) and thus n! possle orderngs of canddates after the fnal vote tally. Traders may lke to et on artrary propertes of the fnal orderng, for example canddate D wll wn, canddate D wll fnsh n ether frst place or last place, canddate D wll defeat canddate R, canddates D and R wll oth defeat canddate L, etc. The goal of the exchange s to search among all the offers to fnd two or more that together form an agreeale match. As we shall see, the matchng prolem can e set up as a lnear or nteger program, dependng on whether orders are dvsle or ndvsle, respectvely. Attemptng to reduce the prolem to a lateral matchng prolem y explctly creatng n! securtes, one for each possle fnal orderng, s oth cumersome for the traders and computatonally nfeasle even for modest szed n. Moreover, traders attenton would e spread among n! ndependent choces, makng the lkelhood of two traders convergng at the same tme and place seem remote. There s a tradeoff etween the expressveness of the ddng language and the computatonal complexty of the matchng prolem. We want to offer traders the most expressve ddng language possle whle mantanng computatonal feaslty. We explore two ddng languages that seem natural from a trader perspectve. Suset ettng, descred n Secton 3.2, allows traders to et on whch postons n the rankng a canddate wll fall, for example canddate D wll fnsh n poston 1, 3-5, or 10. Symetrcally, traders can also et on whch canddates wll fall n a partcular poston. In Secton 4, we derve a polynomal-tme algorthm for matchng (dvsle) suset ets. The key to the result s showng that the exponentally g lnear program has a correspondng separaton prolem that reduces to maxmum weghted partte matchng and consequently we can solve t n tme polynomal n the numer of orders. Par ettng, descred n Secton 3.3, allows traders to et on the fnal rankng of any two canddates, for example canddate D wll defeat canddate R. In Secton 5, we show that optmal matchng of (dvsle or ndvsle) par ets s NP-hard, va a reducton from the unweghted mnmum feedack arc set prolem. We also provde a polynomallyverfale suffcent condton for the exstence of a parettng match and show that a greedy algorthm offers poor approxmaton for ndvsle par ets. 2. BACKGROUND AND RELATED WORK We consder permutaton ettng, or ettng on the outcome of a competton among n canddates. The fnal outcome or state s S s an ordnal rankng of the n canddates. For example, the canddates could e horses n a race and the outcome the lst of horses n ncreasng order of ther fnshng tmes. The state space S contans all n! mutually exclusve and exhaustve permutatons of canddates. In a typcal horse race, people et on propertes of the outcome lke horse A wll wn, horse A wll show, or fnsh n ether frst or second place, or horses A and B wll fnsh n frst and second place, respectvely. In practce at the racetrack, each of these dfferent types of ets are processed n separate pools or groups. In other words, all the wn ets are processed together, and all the show ets are processed together, ut the two types of ets do not mx. Ths separaton can hurt lqudty and nformaton aggregaton. For example, even though horse A s heavly favored to wn, that may not drectly oost the horse s odds to show. Instead, we descre a central exchange where all ets on the outcome are processed together, thus aggregatng lqudty and ensurng that nformatonal nference happens automatcally. Ideally, we d lke to allow traders to et on any property of the fnal orderng they lke, stated n exactly the language they prefer. In practce, allowng too flexle a language creates a computatonal urden for the auctoneer attemptng to match wllng traders. We explore the tradeoff etween the expressveness of the ddng language and the computatonal complexty of the matchng prolem. We consder a framework where people propose to uy securtes that pay $1 f and only f some property of the fnal orderng s true. Traders state the prce they are wllng to pay per share and the numer of shares they would lke to purchase. (Sell orders may not e explctly needed, snce uyng the negaton of an event s equvalent to sellng the event.) A dvsle order permts the trader to receve fewer shares than requested, as long as the prce constrant s met; an ndvsle order s an all-or-nothng order. The descrpton of ets n terms of prces and shares s wthout loss of generalty: we can also allow ets to e descred n terms of odds, payoff vectors, or any of the dverse array of approaches practced n fnancal and gamlng crcles. In prncple, we can do everythng we want y explctly offerng n! securtes, one for every state s S (or n fact any set of n! lnearly ndependent securtes). Ths s the so-called complete Arrow-Dereu securtes market [1] for our settng. In practce, traders do not want to deal wth low-level specfcaton of complete orderngs: people thnk more naturally n terms of hgh-level propertes of orderngs. Moreover, operatng n! securtes s nfeasle n practce from a computatonal pont of vew as n grows. A very smple ddng language mght allow traders to et only on who wns the competton, as s done n the wn pool at racetracks. The correspondng matchng prolem s polynomal, however the language s not very expressve. A trader who eleves that A wll defeat B, ut that nether wll wn outrght cannot usefully mpart hs nformaton to the market. The prce space of the market reveals the collectve estmates of wn proaltes ut nothng else. Our goal s to fnd languages that are as expressve and ntutve as possle and reveal as much nformaton as possle, whle mantanng computatonal feaslty. Our work s n drect analogy to work y Fortnow et. al. [6]. Whereas we explore permutaton comnatorcs, Fortnow et. al. explore Boolean comnatorcs. The authors consder a state space of the 2 n possle outcomes of n nary varales. Traders express ets n Boolean logc. The authors show that dvsle matchng s co-np-complete and ndvsle matchng s Σ p 2 -complete. Hanson [9] descres a market scorng rule mechansm whch can allow ettng on comnatoral numer of outcomes. The market starts wth a jont proalty dstruton across all outcomes. It works lke a sequental verson of a scorng rule. Any trader can change the proalty dstruton as long as he agrees to pay the most recent trader accordng to the scorng rule. The market maker pays the

3 last trader. Hence, he ears rsk and may ncur loss. Market scorng rule mechansms have a nce property that the worst-case loss of the market maker s ounded. However, the computatonal aspects on how to operate the mechansm have not een fully explored. Our mechansms have an auctoneer who does not ear any rsk and only matches orders. Research on ddng languages and wnner determnaton n comnatoral auctons [4, 14, 18] consders smlar computatonal challenges n fndng an allocaton of tems to dders that maxmzes the auctoneer s revenue. Comnatoral auctons allow dders to place dstnct values on undles of goods rather than just on ndvdual goods. Uncertanty and rsk are typcally not consdered and the central auctoneer prolem s to maxmze socal welfare. Our mechansms allow traders to construct ets for an event wth n! outcomes. Uncertanty and rsk are consdered and the auctoneer prolem s to explore artrage opportuntes and rsklessly match up wagers. 3. PERMUTATION BETTING In ths secton, we defne the matchng and optmal matchng prolems that an auctoneer needs to solve n a general permutaton ettng market. We then llustrate the prolem defntons n the context of the suset-ettng and parettng markets. 3.1 Securtes, Orders and Matchng Prolems Consder an event wth n competng canddates where the outcome (state) s a rankng of the n canddates. The ddng language of a market offerng securtes n the future outcomes determnes the type and numer of securtes avalale and drectly affects what nformaton can e aggregated aout the outcome. A fully expressve ddng language can capture any possle nformaton that traders may have aout the fnal rankng; a less expressve language lmts the type of nformaton that can e aggregated though t may enale a more effcent soluton to the matchng prolem. For any ddng language and numer of securtes n a permutaton ettng market, we can succnctly represent the prolem of the auctoneer to rsklessly match offers as follows. Consder an ndex set of ets or orders O whch traders sumt to the auctoneer. Each order O s a trple (, q, φ ), where denotes how much the trader s wllng to pay for a unt share of securty φ and q s the numer of shares of the securty he wants to purchase at prce. Naturally, (0, 1) snce a unt of the securty pays off at most $1 when the event s realzed. Snce order s defned for a sngle securty φ, we wll omt the securty varale whenever t s clear from the context. The auctoneer can accept or reject each order, or n a dvsle world accept a fracton of the order. Let x e the fracton of order O accepted. In the ndvsle verson of the market x = 0 or 1 whle n the dvsle verson x [0, 1]. Further let I (s) e the ndcator varale for whether order s wnnng n state s, that s I (s) = 1 f the order s pad ack $1 n state s and I (s) = 0 otherwse. There are two possle prolems that the auctoneer may want to solve. The smpler one s to fnd a suset of orders that can e matched rsk-free, namely a suset of orders whch accepted together gve a nonnegatve proft to the auctoneer n every possle outcome. We call ths prolem the exstence of a match or sometmes smply, the matchng prolem. The more complex prolem s for the auctoneer to fnd the optmal match wth respect to some crteron such as proft, tradng volume, etc. Defnton 1 (Exstence of match, ndvsle orders). Gven a set of orders O, does there exst a set of x {0, 1}, O, wth at least one x = 1 such that ( I (s))q x 0, s S? (1) Smlarly we can defne the exstence of a match wth dvsle orders. Defnton 2 (Exstence of match, dvsle orders). Gven a set of orders O, does there exst a set of x [0,1], O, wth at least one x > 0 such that ( I (s))q x 0, s S? (2) The exstence of a match s a decson prolem. It only returns whether trade can occur at no rsk to the auctoneer. In addton to the rsk-free requrement, the auctoneer can optmze some crteron n determnng the orders to accept. Some reasonale ojectves nclude maxmzng the total tradng volume n the market or the worst-case proft of the auctoneer. The followng optmal matchng prolems are defned for an auctoneer who maxmzes hs worst-case proft. Defnton 3 (Optmal match, ndvsle orders). Gven a set of orders O, choose x {0, 1} such that the followng mxed nteger programmng prolem acheves ts optmalty max x,c s.t. c (3) P ` I (s) q x c, s S x {0,1}, O. Defnton 4 (Optmal match, dvsle orders). Gven a set of orders O, choose x [0, 1] such that the followng lnear programmng prolem acheves ts optmalty max x,c s.t. c (4) P ` I (s) q x c, s S 0 x 1, O. The varale c s the worst-case proft for the auctoneer. Note that, strctly speakng, the optmal matchng prolems do not requre to solve the optmzaton prolems (3) and (4), ecause only the optmal set of orders are needed. The optmal worst-case proft may reman unknown. 3.2 Suset Bettng A suset ettng market allows two dfferent types of ets. Traders can et on a suset of postons a canddate may end up at, or they can et on a suset of canddates that wll occupy a partcular poston. A securty α Φ where Φ s a suset of postons pays off $1 f canddate α stands at a poston that s an element of Φ and t pays $0 otherwse. For example, securty α {2, 4} pays $1 when canddate α s ranked second or fourth. Smlarly, a securty Ψ j where

4 Ψ s a suset of canddates pays off $1 f any of the canddates n the set Ψ ranks at poston j. For nstance, securty {α, γ} 2 pays off $1 when ether canddate α or canddate γ s ranked second. The auctoneer n a suset ettng market faces a nontrval matchng prolem, that s to determne whch orders to accept among all sumtted orders O. Note that although there are only n canddates and n possle postons, the numer of avalale securtes to et on s exponental snce a trader may et on any of the 2 n susets of canddates or postons. Wth ths, t s not mmedately clear whether one can even fnd a tradng partner or a match for trade to occur, or that the auctoneer can solve the matchng prolem n polynomal tme. In the next secton, we wll show that somewhat surprsngly there s an elegant polynomal soluton to oth the matchng and optmal matchng prolems, ased on classc comnatoral prolems. When an order s accepted, the correspondng trader pays the sumtted order prce to the auctoneer and the auctoneer pays the wnnng orders $1 per share after the outcome s revealed. The auctoneer has to carefully choose whch orders and what fractons of them to accept so as to e guaranteed a nonnegatve proft n any future state. The followng example llustrates the matchng prolem for ndvsle orders n the suset-ettng market. Example 1. Suppose n = 3. Ojects α, β, and γ compete for postons 1, 2, and 3 n a competton. The auctoneer receves the followng 4 orders: (1) uy 1 share α {1} at prce $0.6; (2) uy 1 share β {1, 2} at prce $0.7; (3) uy 1 share γ {1, 3} at prce $0.8; and (4) uy 1 share β {3} at prce $0.7. There are 6 possle states of orderng: αβγ, αγβ, βαγ, βγα, γαβ,and γβα. The correspondng statedependent proft of the auctoneer for each order can e calculated as the followng vectors, c 1 = ( 0.4, 0.4, 0.6, 0.6, 0.6, 0.6) c 2 c 3 = ( 0.3, 0.7, 0.3, 0.3, 0.7, 0.3) = ( 0.2, 0.8, 0.2, 0.8, 0.2, 0.2) c 4 = ( 0.7, 0.3, 0.7, 0.7, 0.3, 0.7). 6 columns correspond to the 6 future states. For ndvsle orders, the auctoneer can ether accept orders (2) and (4) and otan proft vector c = (0.4, 0.4, 0.4, 0.4, 0.4, 0.4), or accept orders (2), (3), and (4) and has proft across state c = (0.2, 1.2, 0.2, 1.2, 0.2, 0.2). 3.3 Par Bettng A par ettng market allows traders to et on whether one canddate wll rank hgher than another canddate, n an outcome whch s a permutaton of n canddates. A securty α > β pays off $1 f canddate α s ranked hgher than canddate β and $0 otherwse. There are a total of N(N 1) dfferent securtes offered n the market, each correspondng to an ordered par of canddates. Traders place orders of the form uy q shares of α > β at prce per share no greater than. n general should e etween 0 and 1. Agan the order can e ether ndvsle or dvsle and the auctoneer needs to decde what fracton x of each order to accept so as not to ncur any loss, wth x {0, 1} for ndvsle and x [0, 1] for dvsle orders. A.5 B Fgure 1: Every cycle has negatve worst-case proft of 0.02 (for the cycles of length 4) or less (for the cycles of length 6), however acceptng all edges n full gves a postve worst-case proft of The same defntons for exstence of a match and optmal match from Secton 3.1 apply. The orders n the par-ettng market have a natural nterpretaton as a graph, where the canddates are nodes n the graph and each order whch ranks a par of canddates α > β s represented y a drected edge e = (α, β) wth prce e and weght q e. Wth ths nterpretaton, t s temptng to assume that a necessary condton for a match s to have a cycle n the graph wth a nonnegatve worst-case proft. Assumng q e = 1 for all e, ths s a cycle C wth a total of C edges such that the worst-case proft for the auctoneer s e ( C 1) 0, e C snce n the worst-case state the auctoneer needs to pay $,1 to every order n the cycle except one. However, the example n Fgure 1 shows that ths s not the case: we may have a set of orders n whch every sngle cycle has a negatve worst-case proft, and yet there s a postve worstcase match overall. The edge laels n the fgure are the prces e; oth the optmal dvsle and ndvsle soluton n ths case accept all orders n full, x e = COMPLEITY OF SUBSET BETTING The matchng prolems of the auctoneer n any permutaton market, ncludng the suset ettng market have n! constrants. Brute-force methods would take exponental tme to solve. However, gven the specal form of the securtes n the suset ettng market, we can show that the matchng prolems for dvsle orders can e solved n polynomal tme. Theorem 1. The exstence of a match and the optmal match prolems wth dvsle orders n a suset ettng market can oth e solved n polynomal tme. Proof. Consder the lnear programmng prolem (4) for fndng an optmal match. Ths lnear program has O D C E F

5 varales, one varale x for each order and the proft varale c. It also has exponentally many constrants. However, we can solve the program n tme polynomal n the numer of orders O y usng the ellpsod algorthm, as long as we can effcently solve ts correspondng separaton prolem n polynomal tme [7, 8]. The separaton prolem for a lnear program takes as nput a vector of varale values and returns f the vector s feasle, or otherwse t returns a volated constrant. For gven values of the varales, a volated constrant n Eq. (4) asks whether there s a state or permutaton s n whch the proft s less than c, and can e rewrtten as I (s)q x < q x c s S. (5) Thus t suffces to show how to fnd effcently a state s satsfyng the aove nequalty (5) or verfy that the opposte nequalty holds for all states s. We wll show that the separaton prolem can e reduced to the maxmum weghted partte matchng 1 prolem [3]. The left hand sde n Eq. (5) s the total money that the auctoneer needs to pay ack to the wnnng traders n state s. The frst term on the rght hand sde s the total money collected y the auctoneer and t s fxed for a gven soluton vector of x s and c. A weghted partte graph can e constructed etween the set of canddates and the set of postons. For every order of the form α Φ there are edges from canddate node α to every poston node n Φ. For orders of the form Ψ j there are edges from each canddate n Ψ to poston j. For each order we put weght q x on each of these edges. All mult-edges wth the same end ponts are then replaced wth a sngle edge that carres the total weght of the mult-edge. Every state s then corresponds to a perfect matchng n the partte graph. In addton, the auctoneer pays out to the wnners the sum of all edge weghts n the perfect matchng snce every canddate can only stand n one poston and every poston s taken y one canddate. Thus, the auctoneer s worst-cast state and payment are the soluton to the maxmum weghted partte matchng prolem, whch has known polynomal-tme algorthms [12, 13]. Hence, the separaton prolem can e solved n polynomal tme. Naturally, f the optmal soluton to (4) gves a worst-case proft of c > 0, there exsts a matchng. Thus, the matchng prolem can e solved n polynomal tme also. 5. COMPLEITY OF PAIR BETTING In ths secton we show that a slght change of the ddng language may rng aout a dramatc change n the complexty of the optmal matchng prolem of the auctoneer. In partcular, we show that fndng the optmal match n the par ettng market s NP-hard for oth dvsle and ndvsle orders. We then dentfy a polynomally-verfale suffcent condton for decdng the exstence of a match. The hardness results are surprsng especally n lght of the oservaton that a par ettng market has a seemngly more restrctve ddng language whch only offers n(n 1) securtes. In contrast, the suset ettng market enales traders to et on an exponental numer of securtes and 1 The noton of perfect matchng n a partte graph, whch we use only n ths proof, should not e confused wth the noton of matchng ets whch we use throughout the paper. yet had a polynomal tme soluton for fndng the optmal match. Our hope s that the comparson of the complextes of the suset and par ettng markets would offer nsght nto what makes a ddng language expressve whle at the same tme enalng an effcent matchng soluton. In all analyss that follows, we assume that traders sumt unt orders n par ettng markets, that s q = 1. A set of orders O receved y the auctoneer n a par ettng market wth unt orders can e represented y a drected graph, G(V, E), where the vertex set V contans canddates that traders et on. An edge e E, denoted (α, β, e), represents an order to uy 1 share of the securty α > β at prce e. All edges have equal weght of 1. We adopt the followng notatons throughout the paper: G(V, E): orgnal equally weghted drected graph for the set of unt orders O. e: prce of the order for edge e. G (V, E ); a weghted drected graph of accepted orders for optmal matchng, where edge weght x e s the quantty of order e accepted y the auctoneer. x e = 1 for ndvsle orders and 0 < x e 1 for dvsle orders. H(V, E): a generc weghted drected graph of accepted orders. k(h): soluton to the unweghted mnmum feedack arc set prolem on graph H. k(h) s the mnmum numer of edges to remove so that H ecomes acyclc. l(h): soluton to the weghted mnmum feedack arc set prolem on graph H. l(h) s the mnmum total weghts for the set of edges whch, when removed, leave H acyclc. c(h): worst-case proft of the auctoneer f he accepts all orders n graph H. ǫ: a suffcently small postve real numer. Where not stated, ǫ < 1/(2 E ) for a graph H(V,E). In other cases, the value s determned n context. A feedack arc set of a drected graph s a set of edges whch, when removed from the graph, leave a drected acyclc graph (DAG). Unweghted mnmum feedack arc set prolem s to fnd a feedack arc set wth the mnmum cardnalty, whle weghted mnmum feedack arc set prolem seeks to fnd a feedack arc set wth the mnmum total edge weght. Both unweghted and weghted mnmum feedack arc set prolems have een shown to e NP-complete [10]. We wll use ths result n our complexty analyss on par ettng markets. 5.1 Optmal Indvsle Matchng The auctoneer s optmal ndvsle matchng prolem s ntroduced n Defnton 3 of Secton 3. Assumng unt orders and consderng the order graph G(V, E), we restate the auctoneer s optmal matchng prolem n a par ettng market as pckng a suset of edges to accept such that worst-case proft s maxmzed n the followng optmzaton

6 prolem, max x e,c s.t. c (6) P e `e I e(s) x e c, s S x e {0, 1}, e E. Wthout lose of generalty, we assume that there are no mult-edges n the order graph G. We show that the optmal matchng prolem for ndvsle orders s NP-hard va a reducton from the unweghted mnmum feedack arc set prolem. The latter takes as nput a drected graph, and asks what s the mnmum numer of edges to delete from the graph so as to e left wth a DAG. Our hardness proof s ased on the followng lemmas. Lemma 2. Suppose the auctoneer accepts all edges n an equally weghted drected graph H(V,E) wth edge prce e = (1 ǫ) and edge weght x e = 1. Then the worst-case proft s equal to k(h) ǫ E, where k(h) s the soluton to the unweghted mnmum feedack arc prolem on H. Proof. If the order of an edge gets $1 payoff at the end of the market we call the edge a wnnng edge, otherwse t s called a losng edge. For any state s, all wnnng edges necessarly form a DAG. Conversely, for every DAG there s a state n whch the DAG edges are wnners (though the remanng edges n G are not necessarly losers). Suppose that n state s there are w s wnnng edges and l s = E w s losng edges. Then, l s s the cardnalty of a feedack arc set that conssts of all losng edges n state s. The edges that reman after deletng the mnmum feedack arc set form the maxmum DAG for the graph H. Consder the state s max n whch all edges of the maxmum DAG are wnners. Ths gves the maxmum numer of wnnng edges w max. All other edges are necessarly losers n the state s max, snce any edge whch s not n the max DAG must form a cycle together wth some of the DAG edges. The numer of losng edges n state s max s the cardnalty of the mnmum feedack arc set of H, that s E w max = k(h). The proft of the auctoneer n a state s s proft(s) = e w e E = (1 ǫ) E w (1 ǫ) E w max, where equalty holds when s = s max. Thus, the worst-case proft s acheved at state s max, proft(s max) = ( E w max) ǫ E = k(h) ǫ E. Consder the graph of accepted orders for optmal matchng, G (V, E ), whch conssts of the optmal suset of edges E to e accepted y the auctoneer, that s edges wth x e = 1 n the soluton of the optmzaton prolem (6). We have the followng lemma. Lemma 3. If the edge prces are e = (1 ǫ), then the optmal ndvsle soluton graph G has the same unweghted mnmum feedack arc set sze as the graph of all orders G, that s k(g ) = k(g). Furthermore, G s the smallest such sugraph of G,.e., t s the sugraph of G wth the smallest numer of edges, that has the same sze of unweghted mnmum feedack arc set as G. Proof. G s a sugraph of G, hence the mnmum numer of edges to reak cycles n G s no more than that n G, namely k(g ) k(g). Suppose k(g ) < k(g). Snce oth k(g ) and k(g) are ntegers, for any ǫ < 1 we have that k(g ) ǫ E < E k(g) ǫ E. Hence y Lemma 2, the auctoneer has a hgher worst-case proft y acceptng G than acceptng G, whch contradcts the optmalty of G. Fnally, the worst-case proft k(g) ǫ E s maxmzed when E s mnmzed. Hence, G s the smallest sugraph of G such that k(g ) = k(g). The aove two lemmas prove that the maxmum worstcase proft n the optmal ndvsle matchng s drectly related to the sze of the mnmum feedack arc set. Thus computng each automatcally gves the other, hence computng the maxmum worst-case proft n the ndvsle par ettng prolem s NP-hard. Theorem 4. Computng the maxmum worst-case proft n ndvsle par ettng s NP-hard. Proof. By Lemma 3, the maxmum worst-case proft whch s the optmum to the mxed nteger programmng prolem (6), s k(g) ǫ E, where E s the numer of accepted edges. Snce k(g) s nteger and ǫ E ǫ E < 1, solvng (6) wll automatcally gve us the cardnalty of the mnmum feedack arc set of G, k(g). Because the mnmum feedack arc set prolem s NP-complete [10], computng the maxmum worst-case proft s NP-hard. Theorem 4 states that solvng the optmzaton prolem s hard, ecause even f the optmal set of orders are provded computng the optmal worst-case proft from acceptng those orders s NP-hard. However, t does not mply whether the optmal matchng prolem,.e. fndng the optmal set of orders to accept, s NP-hard. It s possle to e ale to determne whch edges n a graph partcpatng n the optmal match, yet unale to compute the correspondng worst-case proft. Next, we prove that the ndvsle optmal matchng prolem s actually NP-hard. We wll use the followng short fact repeatedly. Lemma 5 (Edge removal lemma). Gven a weghted graph H(V, E), removng a sngle edge e wth weght x e from the graph decreases the weghted mnmum feedack arc set soluton l(h) y no more than x e and reduces the unweghted mnmum feedack arc set soluton k(h) y no more than 1. Proof. Suppose the weghted mnmum feedack arc set for the graph H {e} s F. Then F {e} s a feedack arc set for H, and has total edge weght l(h {e})+x e. Because l(h) s the soluton to the weghted mnmum feedack arc set prolem on H, we have l(h) l(h {e})+x e, mplyng that l(h {e}) l(h) x e. Smlarly, suppose the unweghted mnmum feedack arc set for the graph H {e} s F. Then F {e} s a feedack arc set for H, and has set cardnalty k(h {e})+1. Because k(h) s the soluton to the unweghted mnmum feedack arc set prolem on H, we have k(h) k(h {e}) + 1, gvng that k(h {e}) k(h) 1. Theorem 6. Fndng the optmal match n ndvsle par ettng s NP-hard.

7 Proof. We reduce from the unweghted mnmum feedack arc set prolem agan, although wth a slghtly more complex polynomal transformaton nvolvng multple calls to the optmal match oracle. Consder an nstance graph G of the mnmum feedack arc set prolem. We are nterested n computng k(g), the sze of the mnmum feedack arc set of G. Suppose we have an oracle whch solves the optmal matchng prolem. Denote y optmal match(g ) the output of the optmal matchng oracle on graph G wth prces e = (1 ǫ) on all ts edges. By Lemma 3, on nput G, the oracle optmal match returns the sugraph of G wth the smallest numer of edges, that has the same sze of mnmum feedack arc set as G. The followng procedure fnds k(g) y usng polynomally many calls to the optmal match oracle on a sequence of sugraphs of G. set G := G teratons := 0 whle (G has nonempty edge set) reset G := optmal match(g ) f (G has nonempty edge set) ncrement teratons y 1 reset G y removng any edge e end f end whle return (teratons) Ths procedure removes edges from the orgnal graph G layer y layer untl the graph s empty, whle at the same tme computng the mnmum feedack arc set sze k(g) of the orgnal graph as the numer of teratons. In each teraton, we start wth a graph G and replace t wth the smallest sugraph G that has the same k(g ). At ths stage, removng an addtonal edge e necessarly results n k(g {e}) = k(g ) 1, ecause k(g {e}) < k(g ) y the optmalty of G, and k(g {e}) k(g ) 1 y the edgeremoval lemma. Therefore, n each teraton the cardnalty of the mnmum feedack arc set gets reduced exactly y 1. Hence the numer of teratons s equal to k(g). Note that ths procedure gves a polynomal transformaton from the optmal matchng prolem to the unweghted mnmum feedack arc set prolem, whch calls the optmal matchng oracle exactly k(g) E tmes, where E s the numer of edges of G. Hence the optmal matchng prolem s NP-hard. 5.2 Optmal Dvsle Matchng When orders are dvsle, the auctoneer s optmal matchng prolem s descred n Defnton 4 of Secton 3. Assumng unt orders and consderng the order graph G(V, E), we restate the auctoneer s optmal matchng prolem for dvsle orders as choosng quantty of orders to accept, x e [0, 1], such that worst-case proft s maxmzed n the followng lnear programmng prolem, max x e,c s.t. c (7) P e `e I e(s) x e c, s S x e [0, 1], e E. We stll assume that there are no mult-edges n the order graph G. When orders are dvsle, the auctoneer can e etter off y acceptng partal orders. Example 2 shows a stuaton when acceptng partal orders generates hgher worst-case proft than the optmal ndvsle soluton. Example 2. We show that the lnear program (7) sometmes has a non-nteger optmal soluton. A B Fgure 2: An order graph. Letters on edges represent order prces. Consder the graph n Fgure 2. There are a total of fve cycles n the graph: three four-edge cycles ABCD, ABEF, CDEF, and two sx-edge cycles ABCDEF and ABEFCD. Suppose each edge has prce such that 4 3 > 0 and 6 5 < 0, namely (.75,.80), for example =.78. Wth ths, the optmal ndvsle soluton conssts of at most one four-edge cycle, wth worst case proft (4 3). On the other hand, takng 1 fracton of each of the three four-edge cycles 2 would yeld hgher worst-case proft of 3 (4 3). 2 Despte the potental proft ncrease for acceptng dvsle orders, the auctoneer s optmal matchng prolem remans to e NP-hard for dvsle orders, whch s presented elow va several lemmas and theorems. Lemma 7. Suppose the auctoneer accept orders descred y a weghted drected graph H(V,E) wth edge weght x e to e the quantty accepted for edge order e. The worst-case proft for the auctoneer s c(h) = e E( e 1)x e + l(h). (8) Proof. For any state s, the wnnng edges form a DAG. Thus, the worst-case proft for the auctoneer acheves at the state(s) when the total quantty of losng orders s mnmzed. The mnmum total quantty of losng orders s the soluton to weghted mnmal feedack arc set prolem on H, that s l(h). Consder the graph of accepted orders for optmal dvsle matchng, G (V, E ), whch conssts of the optmal suset of edges E to e accepted y the auctoneer, wth edge weght x e > 0 gettng from the optmal soluton of the lnear program (7). We have the followng lemmas. D C E F

8 Lemma 8. l(g ) k(g ) k(g). Proof. l(g ) s the soluton of the weghted mnmum feedack arc set prolem on G, whle k(g ) s the soluton of the unweghted mnmum feedack arc set prolem on G. When all edge weghts n G are 1, l(g ) = k(g ). When x e s are less than 1, l(g ) can e less than or equal to k(g ). Snce G s a sugraph of G ut possly wth dfferent edge weghts, k(g ) k(g). Hence, we have the aove relaton. Lemma 9. There exsts some ǫ such that when all edge prces e s are (1 ǫ), l(g ) = k(g). Proof. From lemma 8, l(g ) k(g). We know that the auctoneer s worst-case proft when acceptng G s c(g ) = ( e 1)x e + l(g ) = l(g ) ǫ x e. e E e E When he accepts the orgnal order graph G n full, hs worstcase proft s c(g) = e E( e 1) + k(g) = k(g) ǫ E. Suppose l(g ) < k(g). If E P e E xe = 0, t means that G s G. Hence, l(g ) = k(g) regardless of ǫ, whch P contradcts wth the assumpton l(g ) < k(g). If E e E x e > 0, then when ǫ < k(g) l(g ) E P e E x e, c(g) s strctly greater than c(g ), contradctng wth the optmalty of c(g ). Because x e s are less than 1, l(g ) > k(g) s mpossle. Thus, l(g ) = k(g). Theorem 10. Fndng the optmal worst-case proft n dvsle par ettng s NP-hard. Proof. Gven the optmal set of partal orders to accept for G when edge weghts are (1 ǫ), f we can calculate the optmal worst-case proft, y lemma 9 we can solve the unweghted mnmum feedack arc set prolem on G, whch s NP-hard. Hence, fndng the optmal worst-case proft s NP-hard. Theorem 10 states that solvng the lnear program (7) s NP-hard. Smlarly to the ndvsle case, we stll need to prove that just fndng the optmal dvsle match s hard, as opposed to eng ale to compute the optmal worstcase proft. Snce n the dvsle case the edges do not necessarly have unt weghts, the proof n Theorem 6 does not apply drectly. However, wth an addtonal property of the dvsle case, we can augment the procedure from the ndvsle hardness proof to compute the unweghted mnmum feedack arc set sze k(g) here as well. Frst, note that the optmal dvsle sugraph G of a graph G s the weghted sugraph wth mnmum weghted feedack arc set sze l(g ) = k(g) and smallest sum of edge weghts P e E xe, snce ts correspondng worst case proft s `k(g) ǫ P e E x e accordng to lemmas 7 and 9. Lemma 11. Suppose graph H satsfes l(h) = k(h) and we remove edge e from t wth weght x e < 1. Then, k(h {e}) = k(h). Proof. Assume the contrary, namely k(h {e}) < k(h). Then y Lemma 5, k(h {e}) = k(h) 1. Snce removng a sngle edge cannot reduce the mnmum feedack arc set y more than the edge weght, l(h) x e l(h {e}). (9) On the other hand H {e} H so we have, l(h {e}) k(h {e}) = k(h) 1 = l(h) 1. (10) Comnng (9) and (10), we get x e 1. The contradcton arses. Therefore, removng any edge wth less than unt weght from an optmal dvsle graph does not change k(h), the mnmal feedack arc set sze of the unweghted verson of the graph. We now can augment the procedure for the ndvsle case n Theorem 6, to prove hardness of the dvsle verson, as follows. Theorem 12. Fndng the optmal match n dvsle par ettng s NP-hard. Proof. We reduce from the unweghted mnmum feedack arc set prolem for graph G. Suppose we have an oracle for the optmal dvsle prolem called optmal dvsle match, whch on nput graph H computes edge weghts x e (0,1] for the optmal sugraph H of H, satsfyng l(h ) = k(h). The followng procedure outputs k(g). set G := G teratons := 0 whle (G has nonempty edge set) reset G := optmal dvsle match(g ) whle (G has edges wth weght < 1) remove an edge wth weght < 1 from G reset G y settng all edge weghts to 1 reset G := optmal dvsle match(g ) end whle f (G has nonempty edge set) ncrement teratons y 1 reset G y removng any edge e end f end whle return (teratons) As n the proof of the correspondng Theorem 6 for the ndvsle case, we compute k(g) y teratvely removng edges and recomputng the optmal dvsle soluton on the remanng sugraph, untl all edges are deleted. In each teraton of the outer whle loop, the mnmum feedack arc set s reduced y 1, thus the numer of teratons s equal to k(g). It remans to verfy that each teraton reduces k(g) y exactly 1. Startng from a graph at the egnnng of an teraton, we compute ts optmal dvsle sugraph. We then keep removng one non-unt weght edge at a tme and recomputng the optmal dvsle sugraph, untl the latter contans only edges wth unt weght. By Lemma 11 throughout the teraton so far the mnmum feedack arc set of the correspondng unweghted graph remans unchanged. Once the oracle returns a graph G wth unt edge weghts, removng any edge would reduce the mnmum feedack arc set: otherwse G s not optmal snce G {e} would have

9 the same mnmum feedack arc set ut smaller total edge weght. By Lemma 5 removng a sngle edge cannot reduce the mnmum feedack arc set y more than one, thus as all edges have unt weght, k(g ) gets reduced y exactly one. k(g) s equal to the returned value from the procedure. Hence, the optmal matchng prolem for dvsle orders s NP-hard. 5.3 Exstence of a Match Knowng that the optmal matchng prolem s NP-hard for oth ndvsle and dvsle orders n par ettng, we check whether the auctoneer can dentfy the exstence of a match wth ease. Lemma 13 states a suffcent condton for the matchng prolem wth oth ndvsle and dvsle orders. Lemma 13. A suffcent condton for the exstence of a match for par ettng s that there exsts a cycle C n G such that, e C 1, (11) e C where C s the numer of edges n the cycle C. Proof. The left-hand sde of the nequalty (11) represents the total payment that the auctoneer receves y acceptng every unt orders n the cycle C n full. Because the drecton of an edge represents predcted orderng of the two connected nodes n the fnal rankng, formng a cycle meanng that there s some logcal contradcton on the predcted orderngs of canddates. Hence, whchever state s realzed, not all of the edges n the cycle can e wnnng edges. The worst-case for the auctoneer corresponds to a state where every edge n the cycle gets pad y $ 1 except one, wth C 1 e the maxmum payment to traders. Hence, f nequalty (11) s satsfed, the auctoneer has non-negatve worst-case proft y acceptng the orders n the cycle. path(w, v) s mnmzed. Comparng the shortest cycle found for every vertex, we then can determne the shortest overall cycle for the graph H. Because the short path prolem can e solved n polynomal tme [3], we can fnd the soluton to our prolem n polynomal tme. If the worst-case proft for the optmal cycle s non-negatve, we know that there exsts a match n G. However, the condton n lemma 13 s not a necessary condton for the exstence of a match. Even f all sngle cycles n the order graph have negatve worst-case proft, the auctoneer may accept multple nterweavng cycles to have postve worstcase proft. Fgure 1 exhts such a stuaton. If the optmal ndvsle match conssts only of edge dsjont cycles, a natural greedy algorthm can fnd the cycle that gves the hghest worst-case proft, remove ts edges from the graph, and proceed untl no more cycles exst. However, we show that such greedy algorthm can gve a very poor approxmaton. n It can e shown that dentfyng such a non-negatve worstcase proft cycle n an order graph G can e acheved n polynomal tme. Lemma 14. It takes polynomal tme to fnd a cycle n an order graph G(V, E) that has the hghest worst-case proft, that s! max C C e ( C 1) e C where C s the set of all cycles n G. Proof. Because e ( C 1) = ( e 1) + 1 = 1 e), e C e C(1 e C fndng the cycle that gves the hghest worst-case proft n the orgnal order graph G s equvalent to fndng the shortest cycle n a converted graph H(V, E), where H s acheved y settng the weght for edge e n G to e (1 e). Fndng the shortest cycle n graph H can e done wthn polynomal tme y resortng to the shortest path prolem. For any vertex v n V, we consder every neghor vertex w such that (v, w) E. We then fnd the shortest path from w to v, denoted as path(w,v). The shortest cycle that passes vertex v s found y choosng the w such that e (v,w) +, Fgure 3: Graph wth n vertces and n + n edges on whch the greedy algorthm fnds only two cycles, the dotted cycle n the center and the unque remanng cycle. The laels n the faces gve the numer of edges n the correspondng cycle. Lemma 15. The greedy algorthm gves at most an O( n)- approxmaton to the maxmum numer of dsjont cycles. Proof. Consder the graph n Fgure 3 consstng of a cycle wth n edges, each of whch partcpates n another (otherwse dsjont) cycle wth edges. Suppose all edge weghts are (1 ǫ). The maxmum numer of dsjont cycles s clearly n, takng all cycles wth length. Because smaller cycles gves hgher worst-case proft, the greedy algorthm would frst select the cycle of length n, after whch there would e only one remanng cycle of length n. Thus the total numer of cycles selected y greedy s 2 and the approxmaton factor n ths case s n/2. In lght of Lemma 15, one may expect that greedy algorthms would gve n-approxmatons at est. Approxma-

10 ton algorthms for fndng the maxmum numer of edgedsjont cycles have een consdered y Krvelevch, Nutov and Yuster [11, 19]. Indeed, for the case of drected graphs, the authors show that a greedy algorthm gves a n-approxmaton [11]. When the optmal match does not consst of edge-dsjont cycles as n the example of Fgure 3, greedy algorthm tryng to fndng optmal sngle cycles fals ovously. 6. CONCLUSION We consder a permutaton ettng scenaro, where traders wager on the fnal orderng of n canddates. Whle t s unnatural and ntractale to allow traders to et drectly on the n! dfferent fnal orderngs, we propose two expressve ettng languages, suset ettng and par ettng. In a suset ettng market, traders can et ether on a suset of postons that a canddate stands or on a suset of canddates who occupy a specfc poston n the fnal orderng. Par ettng allows traders et on whether one gven canddate ranks hgher than another gven canddate. We examne the auctoneer prolem of matchng orders wthout ncurrng rsk. We fnd that n a suset ettng market an auctoneer can fnd the optmal set and quantty of orders to accept such that hs worst-case proft s maxmzed n polynomal tme f orders are dvsle. The complexty changes dramatcally for par ettng. We prove that the optmal matchng prolem for the auctoneer s NP-hard for par ettng wth oth ndvsle and dvsle orders va reductons to the mnmum feedack arc set prolem. We dentfy a suffcent condton for the exstence of a match, whch can e verfed n polynomal tme. A natural greedy algorthm has een shown to gve poor approxmaton for ndvsle par ettng. Interestng open questons for our permutaton ettng nclude the computatonal complexty of optmal ndvsle matchng for suset ettng and the necessary condton for the exstence of a match n par ettng markets. We are nterested n further explorng etter approxmaton algorthms for par ettng markets. 7. ACKNOWLEDGMENTS We thank Rav Kumar, Yshay Mansour, Amn Saer, Andrew Tomkns, John Tomln, and memers of Yahoo! Research for valuale nsghts and dscussons. 8. REFERENCES [1] K. J. Arrow. The role of securtes n the optmal allocaton of rsk-earng. Revew of Economc Studes, 31(2):91 96, [2] J. E. Berg, R. Forsythe, F. D. Nelson, and T. A. Retz. Results from a dozen years of electon futures markets research. In C. A. Plott and V. Smth, edtors, Handook of Expermental Economc Results (forthcomng) [3] T. H. Cormen, C. E. Leserson, R. L. Rvest, and C. Sten. Introducton to Algorthms (Second Edton). MIT Press and McGraw-Hll, [4] P. Cramton, Y. Shoham, and R. Stenerg. Comnatoral Auctons. MIT Press, Camrdge, MA, [5] R. Forsythe, T. A. Retz, and T. W. Ross. Wshes, expectatons, and actons: A survey on prce formaton n electon stock markets. Journal of Economc Behavor and Organzaton, 39:83 110, [6] L. Fortnow, J. Klan, D. M. Pennock, and M. P. Wellman. Bettng oolean-style: A framework for tradng n securtes ased on logcal formulas. Decson Support Systems, 39(1):87 104, [7] M. Grötschel, L. Lovász, and A. Schrjver. The ellpsod method and ts consequences n comnatoral optmzaton. Comnatorca, 1(2): , [8] M. Grötschel, L. Lovász, and A. Schrjver. Geometrc Algorthms and Comnatoral Optmzaton. Sprnger-Verlag, Berln Hedelerg, [9] R. D. Hanson. Comnatoral nformaton market desgn. Informaton Systems Fronters, 5(1): , [10] R. M. Karp. Reduclty among comnatoral prolems. In Complexty of computer computatons (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heghts, N.Y.), pages Plenum, New York, [11] M. Krvelevch, Z. Nutov, and R. Yuster. Approxmaton algorthms for cycle packng prolems. Proceedngs of the sxteenth annual ACM-SIAM symposum on Dscrete algorthms, pages , [12] H. W. Kuhn. The hungaran method for the assgnment prolem. Naval Research Logstc Quarterly, 2:83 97, [13] J. Munkres. Algorthms for the assgnment and transportaton prolems. Journal of the Socety of Industral and Appled Mathematcs, 5(1):32 38, [14] N. Nsan. Bddng and allocaton n comnatoral auctons. In Proceedngs of the 2nd ACM Conference on Electronc Commerce (EC 00), Mnneapols, MN, [15] D. M. Pennock, S. Lawrence, C. L. Gles, and F. A. Nelsen. The real power of artfcal markets. Scence, 291: , Feruary [16] C. Plott and S. Sunder. Effcency of expermental securty markets wth nsder nformaton: An applcaton of ratonal expectatons models. Journal of Poltcal Economy, 90:663 98, [17] C. Plott and S. Sunder. Ratonal expectatons and the aggregaton of dverse nformaton n laoratory securty markets. Econometrca, 56: , [18] T. Sandholm. Algorthm for optmal wnner determnaton n comnatoral auctons. Artfcal Intellgence, 135:1 54, [19] R. Yuster and Z. Nutov. Packng drected cycles effcently. Proceedngs of the 29th Internatonal Symposum on Mathematcal Foundatons of Computer Scence (MFCS), 2004.