A Unified Framework for Dynamic Pari-Mutuel Information Market Design

Transcription

1 A Unfed Framework for Dynamc Par-Mutuel Informaton Market Desgn Shpra Agrawal Stanford Unversty Stanford, Calforna Zzhuo Wang Stanford Unversty Stanford, Calforna [Extended Abstract] Erck Delage HEC Montréal Montréal, Canada Ynyu Ye Stanford Unversty Stanford, Calforna Mark Peters Stanford Unversty Stanford, Calforna ABSTRACT Recently, concdng wth and perhaps drvng the ncreased popularty of predcton markets, several novel par-mutuel mechansms have been developed such as the logarthmc market scorng rule (LMSR), the cost-functon formulaton of market makers, and the sequental convex parmutuel mechansm (SCPM). In ths work, we present a unfed convex optmzaton framework whch connects these seemngly unrelated models for centrally organzng contngent clams markets. The exstng mechansms can be expressed n our unfed framework usng classc utlty functons. We also show that ths framework s equvalent to a convex rsk mnmzaton model for the market maker. Ths facltates a better understandng of the rsk atttudes adopted by varous mechansms. The utlty framework also leads to easy mplementaton snce we can now fnd the useful cost functon of a market maker n polynomal tme through the soluton of a smple convex optmzaton problem. In addton to unfyng and explanng the exstng mechansms, we use the generalzed framework to derve necessary and suffcent condtons for many desrable propertes of a predcton market mechansm such as proper scorng, truthful bddng (n a myopc sense), effcent computaton, controllable rsk-measure, and guarantees on the worst-case loss. As a result, we develop the frst proper, truthful, rsk controlled, loss-bounded (n number of states) mechansm; none of the prevously proposed mechansms possessed all A full verson of ths paper s avalable as A Unfed Framework for Dynamc Par-Mutuel Informaton Market Desgn at Ths author s contrbuton was done whle completng hs PhD at Stanford Unversty. 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 dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. EC 09, July 6 10, 2009, Stanford, Calforna, USA. Copyrght 2009 ACM /09/07...$5.00. these propertes smultaneously. Thus, our work could provde an effectve tool for desgnng new market mechansms. Categores and Subject Descrptors J.4 [Computer Applcatons]: Socal and Behavoral Scences Economcs General Terms Economcs, Theory Keywords Predcton Markets, convex optmzaton, rsk measures, unfed framework 1. INTRODUCTION Contngent clam markets are organzed for a varety of purposes. Predcton markets are created to aggregate nformaton about a partcular event. Fnancal markets nvolvng contngent clams allow traders to hedge ther exposure to certan event outcomes. Bettng markets are desgned for entertanment purposes. The partcpants n these markets trade securtes whch wll pay a fxed amount f a certan event occurs. Some examples of these events would be the wnner of the World Seres, the value of the latest consumer prce ndex or the release date of Wndows Vsta. Predcton markets have grown n popularty as research nto the accuracy of ther predctons has shown that they effectvely aggregate nformaton from the tradng populaton. One of the longest-runnng predcton markets s the Iowa Electronc Market whch allows real money bettng on varous electons. Studes by Berg and her coauthors ([4], [2] and [3]) have shown that the nformaton generated by these markets often serves as a better predcton of actual outcomes than pollng data. Google has run nternal predcton markets over a varety of events and Cowgll et al. [7] have shown that ther predctons also perform qute well. Despte the potental value created by these markets, there can be some dffcultes wth ther ntroducton and development. Frst, many nascent markets suffer from lqudty problems. Occasonally these problems stem from the choce

2 of mechansm used to operate the market. Organzng markets as a contnuous double aucton (lke the NASDAQ stock market) s a popular opton and usually performs well. However, n thn markets, Bossaerts et al. [5] have demonstrated that some problems surface whch nhbt the growth of lqudty. To overcome ths stuaton, the market organzer could ntroduce an automated market maker whch wll centrally nteract wth the traders. Ths mechansm wll have some rules for prcng shares. The market organzer must determne these rules wth one key queston beng hs own tolerance for rsk. Recently, there has been a surge n research of these automated market makers. New market-makng mechansms based on par-mutuel prncples have recently been developed by Hanson [9], Pennock et al. [13] and Peters et al. [14]. These market makers allow contngent clams n nascent market to be mmedately prced accordng to rules of the mechansm. The mechansms are par-mutuel n the sense that the wnners are generally pad out by the stakes of the losers. The clams beng traded are commtments to pay out a fxed amount f a partcular event occurs n the future. The mechansm developed by Hanson has been shown to perform well n smulated markets [15] and has been adopted by many onlne predcton markets. However, the orgns of these new mechansms dffer. Peters et al. [14] developed ther mechansm by creatng a sequental verson of a call aucton problem whch s solved by convex optmzaton. Ther Sequental Convex Par-mutuel Mechansm (SCPM) uses an optmzaton problem to determne when to accept orders and how to prce accepted orders. On the other hand, Hanson s mechansm s derved from scorng rules. Scorng rules are functons often used to compare dstrbutons. In partcular, Hanson uses the logarthmc scorng rule to determne how much to charge a trader for a new order. Hs mechansm s called the logarthmc market scorng rule (LMSR). Usng a smlar approach as Hanson, t s possble to create market makers for other scorng rules (MSR). In contrast to the SCPM, the MSR model doesn t drectly provde an optmzaton problem from the market organzer s standpont. Recently, there has been some nterest n comparng and unfyng these mechansms for predcton markets. Chen and Pennock [6] gve an equvalent cost functon based formulaton for the MSR market makers, and relate them to utlty-based 1 market makers. Peters et al. [15] emprcally compare the performances of varous market mechansms. In ths work, we provde strong theoretcal foundaton for unfyng exstng market makers lke the SCPM, the MSR, cost functon and utlty-based markets under a sngle convex optmzaton framework. Our model not only ads n comparng varous mechansms, but also provdes ntutve understandng of the behavor of the market organzer n these seemngly dfferent mechansms. Specfcally, our man contrbutons are as follows: A unfyng framework: We propose a generalzed verson of the SCPM as a unfed convex optmzaton framework for market makers. The framework subsumes exstng models of predcton market desgn. In partcular, any market scorng rule, cost functon or 1 Utlty functon s defned n [6] n a dfferent context and should not to be confused wth the utlty functon defned n ths paper. utlty-based market (of [6]) can be expressed nto ths framework by varyng the choce of a concave utlty functon. Utlty/cost functon to scorng rule: It was shown n [6] that any proper scorng-rule mples a cost functon defned by certan condtons. However, t had been unknown what type of cost functons would mply proper scorng-rules. We establshed ths drecton of the relaton by showng that every monotone, onto, and concave utlty functon (easy to check) nduces a cost functon and mplctly a proper rule. Moreover, the cost functon can be constructed by a statc convex optmzaton problem n polynomal tme. Intutve rsk-based nterpreteton: We establsh the equvalence of predcton market mechansms to convex rsk mnmzaton model for the market maker. The rsk atttude of the market maker explans the desgn choces of varous mechansms. For popular mechansms lke the LMSR, the mplct rsk functon turns out to characterze precsely how much the market maker s prepared to nvest n order to learn a dstrbuton p whch s dfferent from hs pror belef. New mechansm desgn: Our framework ads new mechansm desgn by provdng easy-to-check necessary and suffcent condtons for a predcton market mechansm to be myopcally truthful, proper, loss-bounded, and rsk measured. As a result, we derve the FIRST loss-bounded, rsk-measured, and proper predcton market mechansm that uses a quadratc utlty functon (Quad-SCPM). The orgnal quadratc rule was nether monotone nor rsk measured, and the loss of LMSR was not bounded n the number of states. Ths opens the possblty to score problems wth large number of states. The rest of the paper s organzed as follows. To begn, Secton 2 wll provde some background on the mechansms whch we wll be studyng. In Secton 3, we propose our new framework based on a generalzaton of the SCPM, and demonstrate ts propertes of truthful prcng scheme, cost functon based formulaton, and guarantees on worst-case loss bounds. In Secton 4, we show that the SCPM framework s equvalent to the MSR market makers. Secton 5 wll further explan how the market organzer s decson problem n our unfed framework s actually equvalent to a convex rsk mnmzaton problem. In Secton 6, we dscuss the mplcatons of our work on new mechansm desgn. 2. BACKGROUND In ths secton, we wll provde some background on the key mechansms for predcton markets dscussed n ths work the market scorng rule mechansms (MSR), costfuncton and utlty-based formulaton of markets, and the Sequental Convex Par-mutuel mechansm (SCPM). Let ω represent a dscrete or dscretzed random event to be predcted, wth N mutually exclusve and exhaustve outcomes. We consder a contngent clams market where clams are of the form Pays $1 f the outcome state s. A new trader arrves and submts an order whch essentally specfes the clams over each outcome state that the trader desres to buy. The market maker then decdes what prce

3 to charge for the new order. Varous mechansms treat a new order n the followng seemngly dfferent manners. 2.1 Market Scorng Rules Let r = (r 1, r 2,..., r N) represents a probablty estmate for the random event ω. A scorng rule s a sequence of scorng functons, S = S 1( r), S 2( r),..., S N( r), such that a score S ( r) s assgned to r f outcome of the random varable ω s realzed. A proper scorng rule [16] s a scorng rule that motvates truthful reportng. Based on proper scorng rules, Hanson[9] developed a market scorng rule (MSR) mechansm. In the MSR market, the market maker wth a proper scorng rule S begns by settng an ntal probablty estmate, r 0. Every trader can change the current probablty estmate to a new estmate of hs choce as long as he agrees to pay the market maker the scorng rule payment assocated wth the current probablty estmate and receve the scorng rule payment assocated wth the new estmate. Some examples of market scorng rules are the logarthmc market scorng rule (LMSR): S ( r) = b log(r ) (b > 0) and the quadratc market scorng rule: S ( r) = 2br b j r 2 j (b > 0). where b s a constant reflectng the rsk atttude of the market makers. Hanson s MSR has many favorable characterstcs. It s desgned as a par-mutuel mechansm whch bounds the rsk of the market organzer. It functons as an automated market maker n the sense that t s always able to calculate prces for new orders. Crucally, the LMSR s also known to elct truthful bds from the market traders. 2.2 Cost Functon and Utlty-Based Market Makers Recently, Chen & Pennock [6] proposed a cost functon based mplementaton of market makers. Let the vector q R N represents the number of clams on each state currently held by the traders. In the cost functon formulaton, the total cost of all the orders q s calculated va some cost functon C( q). A trader submts an order characterzed by the vector a R N where a reflects the number of clams over state that the trader desres. The market organzer wll charge the new trader C( q + a) C( q) for hs order. At any tme n the market, the gong prce of a clam for state, p ( q), equals C/ q. The prce s the cost per share for purchasng an nfntesmal quantty of securty. Chen & Pennock [6] show that any scorng rule has an equvalent cost functon formulaton. For example, below are the specfc cost and prcng functons for LMSR: ( C( q) = b log j/b) j eq and p ( q) = eq /b. j eq j /b For general market scorng rules, they proposed three equatons that the cost functon C should satsfy so that the cost functon based market maker s equvalent to the market based on a gven scorng rule S: S ( p) = q C( q) + K p = 1 p = C q. (1) We wll use ths formulaton later to prove equvalence of MSR and SCPM mechansms. In [6], the authors also proposed a utlty-based market maker. Ths market maker formulates a utlty functon over the proft generated from the market and reles on subjectve probablty to mantan hs rsk poston, measured n terms of expected utlty, whle runnng the market. For a specal class of utlty functons (the HARA utlty functons), they show that the utlty-based market maker s equvalent to weghted pseudosphercal scorng rule market makers. 2.3 Sequental Convex Par-Mutuel Mechansm The SCPM was desgned to requre traders to submt orders whch nclude three elements: a lmt prce (π), a lmt quantty (l) and a vector ( a) that represents whch states the order should contan. The components of the vector a wll contan ether a 1 (f a clam over the specfed state s desred) or a 0 (f t s not desred). The lmt prce refers to the maxmum amount that the trader wshes to pay for one share. The lmt quantty represents the maxmum number of shares that the trader s wllng to buy. The market maker decdes the actual number of shares x to be granted to a new order, and the prce to be charged for the order. The market maker solves the followng optmzaton problem for makng ths decson: maxmze πx z + θ log(s ) x,z, s subject to ax + s + q = z e (2) 0 x l, where parameters q stands for the numbers of shares currently held by the traders pror to the new order (π, l, a) arrves, and e represents the vector of all ones. Each tme a new order arrves, the optmzaton problem (2) s solved and the state prces are defned to be the optmal dual varables correspondng to the frst set of constrants denoted as p. The trader s then charged accordng to the nner product of the fnal prce and the order flled ( p T a). Ths optmzaton problem, wthout the utlty functon θ log(s), has the followng nterpretaton for the market maker: besdes x, decson varable z represents the maxmum number of accumulated shares, ncludng s representng the contngent numbers of surplus shares would be kept by the market maker, over all states; and (πx z) n the objectve represents the proft that would be made from the new order. Thus, market organzer ams at maxmzng hs worst-case proft. As we establsh later n ths paper, addng the utlty functon θ log(s) enhances the rsk takng ablty of the market maker. 3. THE UNIFYING FRAMEWORK In ths paper, we demonstrate that a generalzed form of SCPM provdes a unfyng framework for par-mutuel market mechansms. We propose the followng convex optmzaton model wth a concave contnuous utlty functon u( s): 2 2 Indeed, there s another techncal condton on u( ) that s requred for ths model to be feasble and bounded, that s, q 0, t, u(t e q) T e = 1, where u( ) denotes the (sub-)gradent functon.

4 maxmze x,z, s πx z + u( s) subject to ax + s + q = z e (3) 0 x l. Note that the utlty functon u( s) = θ log(s) used n the orgnal SCPM model of Peters et al. [15] s a specal case of (3). From here on, SCPM refers to the above generalzed SCPM model. When requred, we dsambguate by referrng to the orgnal model wth u( s) = θ log(s) as Log-SCPM. The optmzaton model (3) has exactly the same meanng for the market maker as the Log-SCPM, and nherts many desrable propertes lke ntutve nterpretaton, convex formulaton, global optmalty, Lagrange dualty, polynomal computatonal complexty, etc., as n the orgnal Log-SCPM model. Next, we demonstrate some new desrable propertes of the new SCPM framework ncludng truthfulness of the prcng scheme, effcent cost functon based scorng rules, and easly computable guarantees on worst-case loss. 3.1 Truthful Prcng Scheme In ths secton, we desgn a truthful prcng scheme for the generalzed SCPM framework. We shall assume lmted msreports, that s, we assume a bdder may only le about hs valuaton π of an order. There are no msreports of the arrval tme of the bdder n the market, and a bdder s only allowed to bd once. Such truthfulness s also known as myopc truthfulness [15]. The orgnal SCPM model does not provde ncentves for the traders to bd truthfully [14]. On the other hand, market scorng rules such as LMSR ensure truthful bddng. We show that ths dfference n ncentves s attrbuted to a dfference between the mplementaton of SCPM and MSR prcng scheme. In the SCPM model, the market organzer wll typcally charge the trader for an accepted number of shares based on the fnal prce calculated by the mechansm. However, n the market scorng rules such as LMSR, the trader s actually charged by a cost functon whch s equvalent to the ntegral of the prcng functon over the number of shares accepted. Thus, as the prce ncreases whle the order s flled, the trader s charged the nstantaneous prce for each nfntesmally small porton of hs order that s flled. We show that our general SCPM framework equally admts truthfulness f the market maker charges traders accordng to the ntegral of the prce functon over nfntesmally small accepted shares. More specfcally, the new chargng scheme solves the optmzaton problem for nfntesmally small orders, thus computng ncremental prces. Defne p(ǫ) to be the dual prces correspondng to the followng optmzaton problem (wth l = ǫ): maxmze x,z, s πx z + u( s) subject to ax + s + q = z e (4) 0 x ǫ. The SCPM model (3) can be equvalently vewed as gradually ncreasng ǫ untl the prce become more than the bd offered, or the trader reaches hs lmt l. Let x be the optmal soluton to (3),.e. the largest ǫ such that the last constrant n (4) s tght at optmalty. Now, nstead of chargng the trader the fnal prce p( x) T a as n the conventonal SCPM, we charge an accepted order by the followng formula: ( x 0 p(ǫ)dǫ ) T a. (5) Below, we establsh the ntegrablty of the prcng functon and some mportant propertes of the new prcng mechansm: Lemma 3.1. The followng propertes are satsfed: 1. The prce vector sums up to 1 for every ǫ 2. The prce s non-negatve f the utlty functon u( ) s non-decreasng 3. The prce s consstent,.e., the market organzer wll accept another nfntesmal order f π s greater than the nstantaneous prce of the order and vce versa 4. The nstantaneous prce p(ǫ) T a s non-decreasng n ǫ 5. The prce functon s ntegrable. Proof. The proof of ths theorem s referred to [1]. The above propertes of the prcng scheme lead to the followng strong result about the truthfulness of ths scheme: Theorem 3.2. Irrespectve of the choce of utlty functon u( ), the optmal bddng strategy n the SCPM s myopcally truthful when the traders are charged accordng to prcng scheme (5). Proof. Assume that the trader has a valuaton of γ for hs desred order a wth a quantty lmt of l. The trader s proft can be expressed as: R(x) = γx ( x 0 p(ǫ)dǫ)t a. The trader seeks to maxmze R by choosng a proper π. From the prevous lemma, we know that p(ǫ) T a s non-decreasng n ǫ. Thus, t s easy to see that an optmal strategy when p(0) T a γ s to not place an order. Thus, we can assume that p(0) T a < γ. Under ths assumpton, the optmal x l (that maxmzes trader s proft R(x)) s gven by the followng optmalty condtons: (x l) ( γ p(x) T a ) = 0 γ p(x) T a. On the other hand, the market organzer wll accept a bd f and only f p(ǫ) T a s less than π. Hence, settng π = γ gves the optmal x, and thus optmal proft for the trader. Ths s equvalent to bddng truthfully snce the trader wll essentally bd such that the nstantaneous prce s drven ether to ther valuaton or as close to ther valuaton as possble before httng the lmt quantty constrant. Therefore, we have shown that the truthfulness of ths mechansm does not depend on the partcular choce of utlty functon but rather on the mplementaton of the chargng method. By revsng the chargng method to an ncremental one, we can create a truthful mplementaton for the general SCPM. As we show n the next secton, the ntegral n the expresson (5) does not need to be explctly calculated n order to compute t; gven x, we can compute the total charge effcently usng a convex cost functon formulaton. 3.2 Cost Functon of the Market Maker Secton 2.2 dscussed a cost functon based mplementaton for market makers, ntroduced by Chen and Pennock [6]. In ths secton, we derve a convex cost functon for the SCPM, whch wll reduce the problem of computng the ntegral (5) to a smple convex optmzaton problem.

5 Theorem 3.3. Let q be the number of shares on each state held by the traders n the SCPM market, and a new order (π, l, a) s accepted up to level x and s charged accordng to prcng scheme (5). Then, the charge s ( x T p(ǫ)dǫ) a = C( q + a x) C( q), 0 where C( q) s a convex cost functon defned by C( q) = mn t t u(te q), (6) and has the property e T C( q) = 1, q 0. Proof. Note that p(ǫ) s the optmal dual soluton assocated wth the optmzaton problem: maxmze z, s πǫ z + u( s) subject to aǫ + s + q = z e. Let V ( q, ǫ) denote the optmal objectve value of the above problem. Then, max V ( q, ǫ) = z, s πǫ z + u( s) subject to aǫ + q + s = z e } = πǫ mn z u(z e q aǫ) z { = πǫ C( q + aǫ). Next, usng local senstvty analyss results [10], the optmal dual varables p(ǫ) s such that V ( q, ǫ) C( q + aǫ) p(ǫ) = =. q q Thus, we can conclude the statement presented n our theorem by performng the ntegraton x 0 p(ǫ) T adǫ = = x 0 x 0 qc( q + aǫ) T adǫ dc( q + aǫ) dǫ dǫ = C( q + a x) C( q). Fnally, we can verfy that C( q) s a convex functon of q snce t s the mnmum over t of a functon that s jontly convex n both t and q. Note that not only s the cost functon convex, but t can also be smply computed for any gven q by solvng a sngle varable convex optmzaton problem. Ths s n contrast to the market scorng rules, where computng the cost functon s non-trval and requres solvng a set of dfferental equatons (refer equaton (1) n Secton 2.2). Remark 3.4. Another way to derve our truthful prcng scheme s to relate our optmzaton model to the generalzed VCG mechansm. Note that our decson model (3) s of the form x = arg max πx + f(x), x where f(x) = max z z + u(z e q ax),.e. f s a functon that depends only on allocaton x and s constant wth respect to the bd π. Such allocaton method s known as an affne maxmzer, a truthful prcng scheme for such mechansm s gven by generalzed VCG prce: f(0) f(x ) (Proposton 9.31 [12]). By smple computaton, ths s equal to C( q + a x) C( q) as defned n (6). Thus, our optmzaton model naturally establshes the connecton of predcton market mechansms and ther cost functon formulatons to the popular VCG mechansm. Remark 3.5. Observe that the cost functon defned n (6) always satsfes no arbtrage condton C( q + t e) = t + C( q). Thus, our model s naturally arbtrage free, regardless of the choce of concave utlty functon. 3.3 Worst-case Loss for the Market Maker An nterestng consequence of the cost functon representng the SCPM developed n the prevous secton, s that the worst-case loss can be formulated as a convex optmzaton problem. Theorem 3.6. Assumng the market starts wth 0 shares ntally, then the worst-case loss for the market maker usng the SCPM mechansm s gven by B + C(0) where B = max{max s u( s) s } and C( ) s the cost functon defned by (6). Proof. Let the number of shares held by the traders at tme t be q t. By Theorem 3.3, assumng we started wth 0 shares ntally, the total money collected at tme t s C( q t ) C(0). On the other hand, f state scenaro occurs, the market maker needs to pay amount q t, Thus, we can fnd for each state scenaro, the worst-case loss by solvng the optmzaton problem : max q 0 {q C( q) + C(0)} = max{q t + u(t e q)} + C(0) t, q 0 = max{u( s) s} + C(0). s Then, we take the maxmum among all scenaros to conclude the proof. As a corollary of the above theorem, we have that: Corollary 3.7. Computng the worst-case loss bound for the SCPM s a convex optmzaton problem. Furthermore, a necessary and suffcent condton to guarantee a bounded loss s that u( s) s s bounded from above. Below, we llustrate the applcaton of the above theorem through some examples. Detaled proofs for these examples are avalable n [1]. ( Example 3.8. For u( s) = b log /b) e s, we can compute B = 0, C(0) = b log(n), gvng worst-case loss as b log(n). ( ) Example 3.9. For u( s) = et s 1 N 4b st I e et s, we N can derve B = b ( ) 1 1 N, C(0) = 0 gvng bound (N 1)b. N Later n Secton 4, we demonstrate that the above two choces of utlty functons are equvalent to the LMSR and the quadratc market scorng rules respectvely. Observe that the derved bounds match those known n the lterature for these scorng rules [6].

6 Example For the Log-SCPM, u( s) = θ log(s), we can show that B s unbounded by usng s 1 = 1, s = α, and lettng α. Actually, B lm α θ1 log(1) + 1 θ log(α) 1 = and C(0) = θ θ log ( θ ). Thus, the worst-case loss s unbounded n ths case. Example For u( s) = mn s, the worst-case loss s 0; snce for all values of ( s, t): u( s) s and t = u(t e). Observe that here the utlty of the surplus proft s s equal to the mnmum or worst-case proft. Ths represents extreme rsk averseness of the market organzer. The last example provded a glmpse of how the utlty functon relates to the rsk averseness of the market maker. In Secton 5, we wll further buld ths ntuton for the behavor of the organzer by recastng our optmzaton framework as a rsk mnmzaton problem. 4. RELATION TO EXISTING MECHANISMS The market scorng rules (MSR) form a large class of popular par-mutuel mechansms. In ths secton, we demonstrate a strong equvalence between the SCPM and the MSR markets. Partcularly, we show that: Theorem 4.1. Any proper market scorng rule wth cost functon C( ) of (1) can be formulated as an SCPM model (3) wth the concave utlty functon u( s) = C( s), and the two models are equvalent n terms of the orders accepted and the prce charged for a submtted order. Theorem 4.2. The SCPM wth any utlty functon u( ) gves an mplct proper scorng rule, as long as the utlty functon has the property that ts dervatve spans the smplex { r : e T r = 1, r 0}, that s, for all vectors r n the smplex: u( s) = r, s. (7) Thus, the SCPM framework subsumes the class of proper scorng rule mechansms. Moreover, a proper scorng rule based market can be created by smply choosng a utlty functon u( ) that satsfes condton (7). As we shall demonstrate later n ths secton, ths condton s not dffcult to satsfy or valdate, thus provdng a useful tool to desgn market mechansms that correspond to a proper scorng rule. We frst prove the above theorems for general MSR based market, and then llustrate wth specfc examples of the LMSR and the quadratc market scorng rules. 4.1 Equvalent SCPM for MSR (Theorem 4.1) To establsh Theorem 4.1, we use the cost functon formulaton of market scorng rules dscussed n [6], and brefly explaned n Secton 2.2. We frst establsh the followng mportant propertes of the cost functon for any proper scorng rule: Lemma 4.3. The cost functon C( ) for any proper scorng rule has followng propertes: 1. C( q) s a convex functon of q 2. For any vector q and scalar d, t holds that: C( q+d e) = d + C( q) Proof. The proof s referred to [1]. Note that n [6], the second property above was treated as an assumpton based on the prncple of no arbtrage. Here, we show that t can actually be derved from the propertes of cost functon formulaton tself. Now, we are ready to prove Theorem 4.1: Proof. Usng the result n part 1 of Lemma 4.3, clearly the proposed utlty functon u( s) = C( s) s concave. In lght of part 2 of Lemma 4.3, we have C( s + z e) = C( s) + z. Therefore, the proposed utlty functon u( s) = C( s) = z C(z e s). Incorporatng u( s) = z C(z e s) n the SCPM model (3), we get the followng optmzaton problem: maxmze x,z, s subject to πx C(z e s) z e ax s = q 0 x l Snce, C( ) was proven to be convex n part 1 of Lemma 4.3, ths s a convex optmzaton problem wth KKT condtons: p T a + y π, x ( p T a + y π) = 0, p = C( q+ ax) q, y (l x) = 0, y 0, 0 x l. Thus, x s ncreased untl x = l, or the prce p T a = C( q + ax) T a becomes greater than the bd prce π. In partcular, under contnuous chargng scheme, where a seres of optmzaton problems are solved wth small l = ǫ, ths corresponds to chargng the ncreasng nstantaneous prce p(ǫ) T a = C( q + aǫ) T a untl t s greater than the prce offered by the bdder. For nfntesmally small ǫ, lm ǫ 0 p(ǫ) = C q. Thus, the orders accepted and prce charged become equvalent to the market scorng rule wth cost functon C( q). 4.2 Properness of the SCPM (Theorem 4.2) In ths secton, we prove that the SCPM mechansm s proper, that s, t mplctly corresponds to scorng the reported belefs wth a proper scorng rule, as long as the utlty functon satsfes the spannng condton (7). Proof. By defnton, a scorng rule S( ) s proper f and only f gven any outcome dstrbuton r, an optmal strategy of a selfsh trader s to report belef r, that s: r arg max p r S ( p) (8) In cost functon based markets lke the SCPM, the traders do not drectly report a belef p. Instead, they buy shares q payng a prce that equals to the dfference of the cost functon, and thus ndrectly reportng the belef as the resultng prce vector p. For these markets, accordng to [6] an mplct scorng rule s defned n the followng manner : S ( p) = q C( q) + K where p = C q (also refer to equaton (1) n Secton 2.2). Therefore, the properness condton n terms of q s represented as: p := C( q ) = r, (9) where q arg max r (q C( q)) (10) q 0

7 for all dstrbutons r. Intutvely, snce the traders receve $1 for each share on the actual outcome, the proft of traders for outcome state s q C( q). Thus, the properness condton ensures that an optmal strategy for selfsh traders s to buy orders q so that the resultng prce vector s equal to ther actual belef. Now, the optmalty condtons for (10) are: r C( q ) q + η = 0, η 0, q 0, η q = 0,. Thus, condton (9) s satsfed f there exsts a postve optmal soluton to (10). As derved n Theorem 3.3, the cost functon of the SCPM mechansm s gven by (6). Therefore, the optmzaton problem (10) s equvalent to max q 0 r(q mnt{t u(t e q)}) max q 0,t r(q t + u(t e q)) max q 0,t r T ( q t e) + u(t e q) max s: s=t e q, q 0 u( s) r T s max s u( s) r T s. As long as there exsts an optmal soluton s to the above problem, we can set t as a large postve value and set q = t e s > 0. Thus, the condton (7), that s, u( s) spans the smplex, ensures the properness. It s easy to see that ths s also a necessary condton. Ths proves Theorem 4.2. A concern however s that the prce vector p that maxmzes trader s expected proft may not be unque. Ths could be ether because there are multple sub-gradents of the cost functon C( q) at optmal q resultng n multple prce vectors p, or because there are multple optmal q and they all result n dfferent correspondng prce vectors C( q ). Ths s typcally undesrable snce n ths case, ether buyng the orders q assocated wth the true belef r s not the only optmal strategy for the traders, or even n the case that the traders acqure q, the market maker s stll unable to recover the true belef. Ths stuaton s avoded by the concept of strctly proper scorng rules. A scorng rule s called strctly proper f the only optmal strategy for a trader s to honestly report hs belef [16]. In terms of our market mechansm, t means that r s the unque soluton to (8). Snce u( ) s concave, t s easy to see that a suffcent condton to ensure strct properness n the SCPM s that u( ) s a smooth functon, that s, u( ) s contnuous (over the smplex). Next, we llustrate the equvalence between the SCPM and proper scorng rules usng some popular mechansms as examples. 4.3 Examples Example 4.4. The LMSR market maker s equvalent to the SCPM framework wth utlty functon u( s) = C( s) = b log ( e s /b ). Ths scorng rule s known to be strctly proper [9]. Note that our condton for properness s satsfed as well snce u( ) s smooth and [ e s ] /b u( s) = e s /b whch clearly spans the smplex. Example 4.5. A market maker usng quadratc scorng rule s equvalent to the SCPM framework ( wth ) utlty functon u( s) = C( s) = et s 1 N 4b st I e et s. Ths scorng rule s known to be strctly proper [6]. Our condton for N properness s satsfed snce u( ) s smooth and [ ] 1 s s u( s) = + N 2b where s = e T s/n. Thus, for any r n smplex, we can set s = 2br to get u( s) = r Example 4.6. For the Log-SCPM [15], u( s) = θ log(s), u( s) = θ /s, whch clearly spans the nteror of the smplex for any postve θ. Also, u( ) s smooth, thus ths mechansm s strctly proper. Example 4.7. Consder a lnear utlty functon u( s) = c T s. Ths mechansm s not proper, snce the dervatve of the functon s a constant vector, and does not span the smplex. So far, we establshed the SCPM as a general market mechansm that ncludes all scorng rules whch are proper and possess common desred propertes. The cost functon formulatons of the two mechansms were used to establsh the equvalence. A natural queston now would be: what s the dfference among the dfferent utlty objectve functons adopted n the SCPM? Next, we wll show that they represent dfferent rsk measures for the market maker. Thus, when usng ths model to accept ncomng orders the market maker s actually takng ratonal decsons wth respect to a specfc rsk atttude defned n terms of u( ). In [6], the authors propose to model the market maker s rsk atttude usng expected utlty. However, the expressveness of ther model s lmted as t reles on the formulaton of the market maker s subjectve probabltes. 3 In what follows, we wll show that the SCPM framework adopts a dstrbuton-free representaton of rsk whch s more realstc because market makers are typcally lttle nformed about the probablty of each outcome. It s also the case that the SCPM strctly subsumes the model proposed n [6]: many concave utlty functons ncludng a pecewse quadratc utlty functon cannot be replcated n the utltybased framework of [6]. A deeper comparson between the two frameworks s avalable n the full verson [1] of ths paper. 5. RISKS FOR THE MARKET MAKER Each tme he or she s offered an order, the market maker must consder the rsks nvolved n acceptng t. Ths s due to the fact that the monetary return generated from the market depends on the actual state outcome. In the earler par-mutuel market ntroduced n [11], ths rsk was effectvely handled n terms of maxmzng the worst-case return generated by the market relatve to the set of outcomes (.e., u( s) = mn s, refer Example 3.11). Unfortunately, ths rsk atttude s somewhat lmtng as t wll create a market whch s lkely to accept very few orders and extract lttle nformaton. In what follows, we consder 3 Actually, expected utlty maxmzaton s known to be nconsstent wth practce (see Ellsberg paradox) where such probabltes often cannot be properly defned.

8 the return generated by the market to be a random varable Z and demonstrate that, when acceptng orders based on the SCPM wth a non-decreasng utlty functon, the market maker effectvely takes ratonal decsons wth respect to a rsk atttude. We use dualty theory to gan new nsghts about how ths atttude relates to the concept of pror belef about the true probablty of outcomes. 5.1 The SCPM Markets and the Convex Rsk Measures In a fntely dscrete probablty space (Ω, F), the set of random varables Z can be descrbed as the set of functons Z : Ω R. A convex rsk measure on the set Z s defned as follows: Defnton 5.1. When the random outcome Z represents a return, a rsk measure s a functon ρ : Z R that descrbes one s atttude towards rsk as : random return Z s preferred to Z f ρ(z) ρ(z ). Furthermore, a rsk measure s called convex f t satsfes the followng: Convexty : ρ(λz +(1 λ)z ) λρ(z)+(1 λ)ρ(z ), Z, Z Z, and λ [0, 1] Monotoncty : If Z, Z Z and Z Z then ρ(z) ρ(z ) Translaton Equvarance : If α R and Z Z, then ρ(z + α) = ρ(z) α Convex rsk measures are ntutvely appealng. Frst, even n a context where the decson maker does not know the probablty of occurrence for the dfferent outcomes, t s stll possble to descrbe a rsk functon ρ(z). The three propertes of convex rsk measures are also natural ones to expect from such a functon. Convexty states that dversfyng the returns leads to lower rsks. Monotoncty states that f the returns are reduced for all outcomes then the rsk s hgher. And fnally, translaton nvarance states that f a fxed ncome s added to random returns then t s rrelevant wether ths fxed ncome s receved before or after the random return s realzed. We refer the reader to [8] for a deeper study of convex rsk measures. Next, we formulate the SCPM model for predcton markets as a convex rsk mnmzaton problem. In context of predcton markets, the random return Z wll represent the revenue for the market organzer, whch depends on the actual outcome of the random event n queston. Let q represent the total orders held by the traders, and c represent the total money collected so far from the traders n the market. Snce the market organzer has to pay $1 for each accepted order that matches the outcome, hs revenue for outcome state s c q. When a new trader enters wth a bd of π, based on the number of accepted orders x, the total revenue for state s gven by (c q +πx a x). The rsk mnmzaton model seeks to choose the number of accepted orders x to mnmze the rsk on total revenue. Below, we formally show that the SCPM model s equvalent to a convex rsk mnmzaton model. Theorem 5.2. Let Ω = {ω 1, ω 2,..., ω m}, Z R m be the vector representaton of Z such that Z = Z(ω ), and Z x (ω ) = c q + πx a x. Then, gven that u( ) s non-decreasng, the SCPM optmzaton model (3) s equvalent n terms of set of optmal solutons for x to the rsk mnmzaton model mnmze x subject to ρ(z x ) 0 x l wth convex rsk measure ρ(z) = mn t{t u( Z + t e)}. Proof. The equvalence can be obtaned by frst elmnatng s n (3), and then performng a smple change of varable t = z πx c: max z {πx z + u (z e ax q)} = max t { t + u (t e ( + (πx)} + c) e ax q) c} = mn t {t u Z x + t e c = ρ(z x ) c. Snce maxmzng ρ(z x ) c over x s equvalent to mnmzng ρ(z x ) n terms of optmal soluton set, the equvalence follows drectly. It remans to show that, when u( ) s concave and nondecreasng, the proposed measure satsfes the three propertes (convexty, monotoncty, and translatonal equvarance) of a convex rsk measure. The convexty and the monotoncty follow drectly from concavty and monotoncty of u( ). We refer the reader to [1] for more detals on ths part of the proof. Remark 5.3. More mportantly, Theorem 5.2 essentally shows that any convex rsk measure ρ(z) can potentally be used to create a verson of the SCPM market whch accepts orders accordng the rsk atttude descrbed by ρ(z). Ths s acheved by smply choosng the utlty functon u( s) = ρ(y s ) where Y s : Ω R s a random varable defned as Y s (ω ) = s. Such a constructed u( ) s necessarly concave and ncreasng. 5.2 Rsk Atttude Through the Belef Space We just showed that the SCPM actually represents a rsk mnmzaton problem for the market maker when u( ) s non-decreasng. In fact, we can get more nsghts about the specfc rsk atttude by studyng the dual representaton of rsk measure ρ(z): ρ(z) = mn p { p p 0, E p[z] + L( p), p =1} where L( p) = max s u( s) p T s, and E p [Z] = p Z. We refer the reader to [8] for more detals on the equvalence of ths representaton. Note that ρ(z) s evaluated by consderng the worst dstrbuton p n terms of tradng off between reducng expected return and reducng the penalty L( p). In terms of the SCPM, ths representaton equvalence leads to the concluson that orders are accepted accordng to: { } max 0 x l mn p { p p 0, p =1} p Z x + L( p) In ths form, t becomes clearer how L( p) encodes the ntents of the market maker and relates t to hs belef about the true dstrbuton of outcomes. For nstance, we know that the frst order s accepted only f { p p p 0, } p = 1, p Z x (L( p) L(ˆp)),.

9 where ˆp = arg mn p { p p 0, p =1} L( p). For any gven p, the penalty L( p) L(ˆp) therefore reflects how much the market maker s wllng to lose n terms of expected returns n the case that the true dstrbuton of outcomes ends up beng p. It s also the case that after acceptng q orders, the dstrbuton descrbed by p = arg mn p p T (c e q) + L( p) s actually the vector of dual prces computed n the SCPM. In other words, the prce vector n the SCPM market reflects the dstrbuton that s beng consdered as the outcome dstrbuton by the market organzer n order to determne hs expected return. Ths confrms the nterpretaton of prces as a belef consensus on outcome dstrbuton generated from the market. As we wll see next, the functon L( p) wll typcally be chosen so that L( p) L(ˆp) s large f p s far from ˆp, and ˆp wll reflect a pror belef of the market organzer. That s, the market organzer s wllng to lose on the expected return n order to learn a dstrbuton p that s very dfferent from hs pror belef. Ths s n accordance wth the fact that the market s beng organzed as a predcton market rather than a pure fnancal market, and one of the goals of market organzer s to learn belefs even f at some rsk to the generated returns. We make the above nterpretatons clearer through the followng examples. Example 5.4. The utlty functon u( s) = mn {s } corresponds to cost L( p) = mn {s } p T s = 0 for all p. That s, the market organzer s purely maxmzng hs worst-case return. ( Example 5.5. For the LMSR, u( s) = b log /b) e s, whch s equvalent to usng L( p) as the Kullback-Lebler dvergence of p from unform dstrbuton L KL ( p; U). Ths s mnmzed at ˆp = U reflectng a unform pror. The correspondng rsk measure s also known as the Entropc Rsk Measure and ts level of tolerance s measured by b. Example 5.6. The Log-SCPM uses u( s) = θ log(s), whch s equvalent to choosng the penalty functon to be the negatve log-lkelhood of p beng the true dstrbuton gven a set of observatons descrbed by the vector θ. More specfcally, L LL ( p ; θ) = log ( pθ ) +K, whch s mnmzed at ˆp = θ. θ and tolerance to rsk s measured through θ These examples llustrate how the rsk mnmzaton representaton provdes nsghts on how to choose u( ). In the case of the Mn-SCPM (Example 5.4), the assocated penalty functon leads to a market where trades that mght generate a loss for the market maker are necessarly rejected. Hence, the traders have no ncentve for sharng ther belef. On the other hand, both the LMSR and the Log-SCPM are mechansm that wll accept orders leadng to negatve expected returns under a dstrbuton p, as long as ths dstrbuton s far enough from ˆp. Effectvely, a trader wth a belef that dffers from ˆp wll have hs order accepted gven that he submts t early enough. In practce, choosng between the LMSR and the Log-SCPM nvolves determnng whether the Kullback Lebler dvergence or a lkelhood measure better characterzes the market maker s commtment to learnng the true dstrbuton. 6. DESIGN OF NEW MECHANISMS In the prevous sectons, we used the SCPM framework to establsh that varous mechansms actually dffer only n terms of the choce of utlty functon. More mportantly, we establshed necessary and suffcent condtons on the utlty functons n order to acheve varous propertes desred n a predcton market mechansm. We frst showed that for any concave functon, a generalzed VCG lke prcng mechansm ensures truthful bddng (n a myopc sense). Also, the prcng mechansm always leads to prces that can be computed effcently by solvng a smple convex optmzaton problem. We derved exact bounds for worst-case loss obtaned from the market n ths framework. We also dscussed condtons on the utlty functon to ensure properness of the mechansm nlne wth the defnton of a proper scorng rule. Further, we showed that when selectng a non-decreasng utlty functon, the market maker s mplctly defnng hs rsk atttude wth respect to the potental revenues obtaned from the market. More nterestngly, we have shown that a rsk measure can be explctly chosen to desgn a market maker wth a partcular rsk atttude. These propertes take a decsve role when selectng the most effectve market mechansm for a gven applcaton. Table 1 summarzes our conclusons for popular mechansms: the LMSR, the orgnal SCPM (Log-SCPM), the rskless Market (Mn-SCPM), and the market based on the quadratc scorng rule. We also use these observatons to desgn two new mechansms: Exponental-SCPM and Quad-SCPM (descrbed below n Example 6.1 and 6.2). Exponental-SCPM provdes propertes smlar to LMSR whle usng a classc separable utlty functon. Quad-SCPM mproves upon the quadratc scorng rule. Although ths rule s known to be myopcally truthful, n practce a market maker that uses ths rule needs to explctly restrct prces to be between [0,1] at all tmes. Quad-SCPM modfes the utlty functon of quadratc scorng rule to provde non-negatve prces and a convex rsk-measure, whle retanng all the attractve propertes of the scorng rule, that s, strct properness, and a worst case loss that s bounded n the number of states (N). Ths opens the possblty to score problems wth large number of states. More detals about the dervaton of these propertes are avalable n [1]. Example 6.1. Consder the Exponental-SCPM obtaned from usng the followng utlty: ( ) u( s) = b 1 1 e s /b. N Ths utlty functon s concave, non-decreasng and separable. It has a bounded worst-case loss equal to b log(n). And, the functon L( p) s the Kullback-Lebler dvergence of p from unform dstrbuton L KL ( p ; U). Usng the prcng scheme descrbed n Secton 3.1, t leads to a predcton market that s myopcally truthful. Also, the correspondng scorng rule s strctly proper. And, the orders can be prced usng convex ( cost functon C( q) = b log 1 N eq /b). Example 6.2. Consder the Quad-SCPM obtaned from usng the followng utlty: { 1 u( s) = max θ v s N T v 1 } 4b vt v

10 Table 1: A summary of propertes of varous market mechansms Worst Convex u( s) Truthful L( p) Cost Rsk Measure Properness LMSR b log( e s /b ) Yes b log(n) Yes bl KL ( p U) Strctly Proper Log-SCPM b log(s) Yes Yes bl LL ( p U) Strctly Proper Mn-SCPM mn s Yes 0 Yes 0 Proper 1 Quad. Scorng Rule N et s 1 4b st (I 1 N e et ) s Yes b ( ) 1 1 No - Strctly Proper N Exponetal-SCPM b(1 1 N e s /b ) Yes b log (N) Yes bl KL ( p U) Strctly Proper 1 Quad-SCPM max v s N et v 1 4b vt v Yes b ( ) 1 1 Yes b p U 2 N 2 Strctly Proper for some θ such that θ = 1. Ths utlty functon s nondecreasng and concave whch ensures that resultng prces are non-negatve and sum to 1. It has bounded worst-case loss gven by b( θ mn θ ). And, the dstance from the pror ˆp = θ s measured by L( p) = b p θ 2 2, that s the 2-norm dstance. The resultng predcton market s myopcally truthful and leads to orders that can be prced usng the cost functon: { C( q) = mn t θ T v + 1 } t, v t e q 4b vt v. whch requres solvng a quadratc program. Also, the correspondng scorng rule s strctly proper. Ths market s closely related to the market assocated wth the quadratc scorng rule, snce C( q) reduces to the popular quadratc cost functon when the constrant v t e q s replaced wth v = t e q. However, the slght modfcaton ensures that prces are non-negatve, and has an ntutve nterpretaton n terms of dstance to the pror belef. 7. REFERENCES [1] S. Agrawal, E. Delage, M. Peters, Z. Wang, and Y. Ye. A unfed framework for dynamc par-mutuel nformaton market desgn, [2] J. Berg, R. Forsythe, F. Nelson, and T. Retz. Results from a Dozen Years of Electon Futures Markets Research, volume 1 of Handbook of Expermental Economcs Results, chapter 80, pages Elsever, [3] J. E. Berg, F. D. Nelson, and T. A. Retz. Predcton market accuracy n the long run. Internatonal Journal of Forecastng, 24(2): , [4] J. E. Berg and T. A. Retz. The Iowa Predcton Markets: Stylzed Facts and Open Issues. Informaton Markets: A New Way of Mak ng Decsons n the Publc and Prvate Sectors [5] P. Bossaerts, L. Fne, and J. Ledyard. Inducng lqudty n thn fnancal markets through combned-value tradng mechansms. European Economc Revew, 46(9): , October [6] Y. Chen and D. Pennock. A utlty framework for bounded-loss market makers. In Proceedngs of Uncertanty n Artfcal Intellgence, pages 49 56, [7] B. Cowgll, J. Wolfers, and E. Ztzewtz. Usng predcton markets to track nformaton flows: Evdence from google. Workng paper, [8] H. Föllmer and A. Sched. Convex measures of rsk and tradng constrants. Fnance and Stochastcs, 6(4): , [9] R. Hanson. Combnatoral nformaton market desgn. Informaton Systems Fronters, 5(1): , [10] H. Hnd. A tutoral on convex optmzaton II: Dualty and nteror pont methods. In Proceedngs of Amercan Control Conference, Mnneapols, Mnnesota, USA, [11] J. Lange and N. Economdes. A parmutuel market mcrostructure for contngent clams. European Fnancal Management, 11(1):25 49, [12] N. Nsan, T. Roughgarden, E. Tardos, and V. V. Vazran. Algorthmc Game Theory. Cambrdge Unversty Press, New York, NY, USA, [13] D. Pennock, Y. Chen, and M. Dooley. Dynamc par-mutuel market. Workng paper, [14] M. Peters. Convex Mechansms for Par-Mutuel Markets. Doctoral thess, Stanford Unversty, [15] M. Peters, A. M.-C. So, and Y. Ye. Par-mutuel markets: Mechansms and performance. In Proceedngs of Thrd Internatonal Workshop on Internet and Network Economcs, pages 82 95, [16] R. L. Wnkler. Scorng rules and the evaluaton of probablty assessors. Journal of the Amercan Statstcal Assocaton, 64(327): , 1969.