Information Aggregation in Exponential Family Markets

X Information Aggregation in Exponential F amily Marke ts JA COB A BERNETHY , University of Michigan, Ann Arbor SINDHU KUTTY , University of Michigan , Ann Arbor S ´ EBASTIEN LAHAIE , Microsoft Research, New Y ork City RAHUL SAMI , Google India W e co nsider the design of predic tion market mechanisms known as automated market makers . W e show that w e can design these mechanisms via the mold of exponential family distributions , a popular and well- studied probability distribution template used in statis t ics . W e give a full development of this relationsh i p and explore a range of benefits. W e dra w connections betw een the information aggregation of market prices and th e belief aggregation of learning agents th at rely on exponentia l family distributions. W e develop a very natural analysis of th e market behavior as well as the price equilibrium under th e a s sumption that the traders exhibit risk aversion according t o exponential util i t y . W e also consider s i mi l a r aspects under alternative models , s uch as when traders are bu dget constrained. Categories and Su bject Descriptors: J .4 [ Social and Behavio ral Sciences ]: Economics; I.2.6 [ Artificial Intell igence ]: Learning General T erms: Algorithms, Economics Additional Key W ords an d Phrases: logarithmic score, exponential family , maximum entropy , risk aversion, budget constraints 1. INTRODUCTION Prediction markets are aggre gation mechanisms that allow market prices to be inter- preted as pr e dictive probabilities on an eve nt. Each trade r in the market is assumed to have some p rivate information that he u ses to make a pr e diction on the outcome of the e vent. Trade rs are allow ed to report their beliefs by buying and selling securi- ties whose ultimate payof f de pends on the future outcome. This will affe ct the state of the market, thus u pdating the pre d ictive p r o babilities for the ev e nt. Further , since the trades are do ne sequentially , the trader is allowed to observe all past trades in the market and u pdate his private info rmation based o n this information. In this sense the market p rices, w hich are in effect the prices at which the marginal trader is willing to buy or sell the available securities, can be interpreted as an agg regate “consensus probability forecast” of the eve nt in question. Much of the work on prediction market design has focused heavily o n structural prop- erties of the mechanism: incen tive compatibility , the market maker loss, the available liquidity , the fluctuations of the prices as a f unction of the trading volume, to name P ermission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided th at copies are not made or distributed for profit or commercial advan tage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others tha n ACM must b e honored. Abstracting with credit is per- mitted. T o copy otherwise, to republish, to post on servers , to redistribute to lists, or to us e an y component of this work in other works requires prior specific permission and /or a fee. P ermissions may be requested from Publications Dept., ACM, Inc., 2 P enn Plaza, S u ite 701, New Y ork, NY 10121-0701 USA, fax + 1 (212) 869-0481, or permissions@acm.org . c  2014 ACM 0000-0000 /2014/02-ARTX $15.00 DOI: http://dx. doi.org/10.1145/0000000.0000000 EC’14 , June 8 –12, 2014, St anford University , Palo Alto, C A, USA, V ol. X, No. X, Article X, Publication da te: F ebruary 2014. X:2 Aber nethy et al. a few . Absent fr om much of the literature is a co rresponding semantics o f the market behavior or the observ ed prices. That is , how can we interpre t the equilibrium mar- ket state when w e have a numbe r of traders with dive rse beliefs on the unde rlying state of the world? In what sen se is the market an agg r egation mechanism? D o price changes re late to ou r usual Bayesian notion o f info rmation inco rporation via po sterior updating? In the present work we sho w that a number of classical statistical to o ls can be le ver- aged to d esign a p rediction market f ramework in the mold of exponentia l family dis- tributions ; we show that this statistical framewo rk leads to a number o f attractive properties and inter pretations. Common c o ncepts in statistics—including entro py max - imization , log lo ss , and bayesia n in f erence —relate to natural aspects of our class o f mechanisms . In particular , the ce ntral objects in our market f ramework can be inte r - preted via concepts used to d e fine exponen tial families: — the market’s pay off functio n co r responds to the sufficient sta tistics of the distribution; — the ve c tor of outstanding sh ares in the market correspo n ds to the natura l parame ter vector of the distribution; — the market price s correspond to mean parameters ; — the market’s co st function correspo n ds to the d istribution’ s log-p artition fun ction. W e begin in Section 2 w ith a discussion o f scoring rule s based on expone ntial family distributions , and we show how the framew ork leads to a variety of scoring rules for continuous outcome spaces. W e turn our attention to market de sign in Sec tio n 3 and give a full description of o u r pro posed me chanisms . In addition to showing the syn- tactic relationship betwe en expon e ntial families and prediction markets, we exp lore a number of rich semantic imp lications as w e ll. In particular , we show that ou r form u - lation allows us to analyze the evolution of the market under various m odels of trader behavior: — Trader be havior varies de p ending o n how they assimilate inf ormation; f or e x ample, should w e consider our agents as Bayesians or frequen tists. In Section 4 we consider traders that u se a c o njugate p rior to u pdate the ir belief s , and w e study h ow their trades would affect the market state. — In Se c tion 5 we consider r i sk-averse age nts that optimize their bets acco rding to ex - ponential utility . In this case we can characterize pre cisely h ow a single trader in- teracts with the marke t, as well as the equilibrium reached give n multiple trade rs; this result is achieved via a p otential game argumen t. The ev e ntual market state is a weighted combination of traders’ beliefs and the initial state; the we ig hts are proportion al to risk aversion parameters. — In Section 6 we consider budget-limi ted tra ders who are constrained in ho w m u ch they influence the market. W e analyze the market un der these circumstances; we are able to show that traders w ith goo d information c an ex pect to profit and their influence over the market state increases over time whe reas malicious traders have limited impact on the market. Related W ork. The notion of an expon ential family distribution is fundame ntal to this paper . F or com p rehensive introductions to these distributions, see [Barn dorff-N ielsen 1978; W ainwright and Jordan 2008]. Ex ponential families are intimately tied to the notions of log loss and en tro py , but can be g eneralized to other types o f co n vex losses and information, as show n by Gr ¨ unwald and Dawid [2004], w ho also make a co nnec- tion to scoring rules. EC’14 , June 8–12, 2014, Stanford University , Palo Alto , CA, USA, V ol. X, No. X, Article X, Publi cation date: February 201 4. Informati on Aggregati on in Exponentia l F amily Markets X:3 Scoring rules are a me asure of pre d iction accuracy , and we are concerne d here with sco r in g r u les fo r statistic expectations, typically over infinite outcome spaces. Such rules have been characterized by Savage [1971] ; see also [Frongillo 2013; Lambert et al. 2008]. O ur rules are of co urse special cases o f this characterization, but it appears the range of ele gant scoring r ules that arise from expone ntial families has not be en appr eciated. Indee d, Gn eiting and Raftery [2007] observe that specifi c instances of scoring rules f or con tinuous outcomes are lacking, and surv e y variou s possibilities . In a seminal paper , Hanson [2003] showe d ho w to form a prediction market based on a sequentially-shared sco r in g r ule, and specifically pro posed the logarithmic market scoring rule (LMSR) based on lo g loss for finite outcome space s [ Hanson 2007]. The markets we introduce are direct gene ralizations of the LMSR to continuou s ou tco mes, but take the form of c o st-function based markets as introduced by Chen and P enno ck [2007]. Gao et al. [ 2009] and Che n e t al. [ 2013] also co nsider extend ing various market makers to infinite outcome spaces. Prediction markets are kno wn to p erform well in practice [P ennock and Sami 2007; P ennock e t al. 2001]. Howev er , a sound theo ry fo r in te rpreting trader behavior and market prices is an on going area o f study [W olf e rs and Zitzewitz 2006]. At one ex- treme, agents are assumed my opic and risk-neutral, imply ing they move the market state to their be lief [C h en and V aug han 2010] . At the o ther e xtreme, agents are strate- gic and the market fully inco r p orates all in f ormation [ Ostrovsky 2012]. W e are not aware of any wor ks that conside r risk-averse agents within c ost-function based markets. Howeve r , risk aversion is a fundamen tal comp onent of mathemati- cal fin ance and portfolio optimization, an d there are close connections betwe en the notion of a cost function and that of a convex risk measure [F ¨ ollmer and Schied 2002; F ¨ ollmer and Knispel 2011 ]. In d eed, they arise f rom the same axioms as noted by Othm an and Sandholm [2011]. W e see the potential to draw more on the mathe - matical finance literature to take into acco unt risk aversion, as prediction markets are simply single-pe riod financial markets [F ¨ ollmer and Schied 2004, P art I]. W e also note that co nnections betwe en Machine Learning and market mechanisms have been explore d in [Storkey 2011]. 2. GENERALIZED LOG SCORING RULES W e con sider a measurable space con sisting o f a set of outco mes X toge the r with a σ - algebra F . An age nt or e x pert has a belief over potential outcome s taking the form of a probability measure absolutely co ntinuous with respe ct to a base me asure ν . 1 Throughou t w e represen t the belief as the correspo nding density p with re spe ct to ν . Let P denote the set of all such pr o bability densities . W e are intere sted in eliciting info rmation about the agent’s belief, in particular expec- tation info r mation. Let φ : X → R d be a v ector -valued random variable or statistic , where d is finite. The aim is to elicit µ = E p [ φ ( x )] where x is the rand o m outcome. A scoring rule is a dev ice fo r this purpose. Let M =  µ ∈ R d : E p [ φ ( x )] = µ, fo r some p ∈ P  1 Recall th at a measure P is abs olutely contin u ous with respect to ν if P ( A ) = 0 for every A ∈ F for wh ich ν ( A ) = 0 . In essence the base measure ν restricts the support of P . In our examples ν will ty pically be a restriction of t he Lebesgue measure for continuous outcomes or the counti n g measure for discrete outcomes . EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:4 Aber nethy et al. be the set of realizable statistic e xpectations. A scoring rule S : M × X → R ∪ {−∞} pays the age nt S ( ˆ µ, x ) according to how well its report ˆ µ ∈ M ag rees with the even tual outcome x ∈ X . The follo wing d e finition is due to Lambert et al. [ 2008]. Definition 2.1 . A sco ring rule S is pro p er for statis tic φ if fo r each µ ∈ M an d p ∈ P with expe cted statistic µ , w e have fo r all ˆ µ 6 = µ E p [ S ( µ, x )] ≥ E p [ S ( ˆ µ, x )] . (1) Given a prope r scoring ru le S any affin e transform ation ˜ S ( µ, x ) = aS ( µ, x ) + b ( x ) of the rule, w ith a > 0 and b an arbitrary real-v alued f unction of the outco mes, again yie ld s a proper scoring rule terme d equ i valent [Dawid 1998 ; Gneiting and Raftery 2007] . Throughou t we will implicitly apply such af fine transformations to obtain the clear - est version of the scoring rule. W e will also focus on scoring rules where ine quality (1) is strict to avoid trivial cases such as con stant sco r ing rules. Classically , scorin g rules take in the e ntire density p rather than just some statistic, and ince n tive co mpatibility must h old o v er all of P . When the o u tcome space is large or infinite, it is no t feasible to d irectly comm unicate p , so the definition allows for summary information of the belief. Note that Definition 2.1 places o nly m ild in f ormation requiremen ts on the part of the agent to ensure truthful re porting. Be cause condition ( 1) h o lds for all p co nsistent with expe ctation µ , it is e n ough for the agent to simply kno w the latter and not the complete de nsity to be prop erly incentivize d . Howe v er , the agent must also ag r ee w ith the support of the de nsity as implicitly define d by base me asure ν . When the outcome space is fi n ite we re cover classical scoring rules by using the statis- tic φ : X → { 0 , 1 } X that maps an outcome x to a unit vector with a 1 in the compo nent correspond ing to x . The ex pectation o f φ is then exactly the pr o bability mass function. 2.1. Proper Scoring fr om Maximum Entropy Our starting point for de sign ing prope r scoring rules is the classic logarithmic scor- ing rule for eliciting pro babilities in the case o f fin ite outcomes. This rule is simply S ( p, x ) = log p ( x ) , namely we take the log likelihood o f the reported density at the eventual outcome. T o generalize the rule to ex p ected statistics rathe r than full densi- ties , we co nsider a subset of densities D ⊆ P . I f there is a bijec tio n betwe en the sets D and M , then we say that M parame trizes D and write p ( · ; µ ) for the de nsity mapping to µ . Given such a family parametrized by the relev ant statistics , the g eneralized log scoring rule is then S ( µ, x ) = lo g p ( x ; µ ) . (2) Even though the log score is only app lie d to densities fro m D , according to De fini- tion 2.1 it must wo rk over all densities in P . It turns out this is possible if D is cho- sen appropriately , drawing on a we ll-known duality betwee n maximum likelihood and maximum entropy [Gr ¨ unwald and Dawid 2004]. EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:5 Exponent ial F amilie s. W e let p ( x ; µ ) be the maximum entropy distribution with ex- pected statist ic µ . Specifically , it is the solution to the fo llowing program: 2 min p ∈P F ( p ) s . t . E p [ φ ( x )] = µ, (3) where the objective function is the ne gative e ntropy of the distribution, name ly F ( p ) = Z x ∈X p ( x ) log p ( x ) dν ( x ) . Note that the explicit set of constraints in (3) are linear , where as the obje ctive is con- vex. W e le t G : M → R be the o ptimal v alue fun c tion of (3), meaning G ( µ ) is the negative entropy of the maximum entrop y distribution with expecte d statis tics µ . It is well-know n that solutions to ( 3) are expon e ntial family distributions, whose den - sities with respect to ν take the form p ( x ; θ ) = exp( h θ, φ ( x ) i − T ( θ )) . (4) The de nsity is stated here in terms of its na tural parametrization θ ∈ R d , where θ arises as the Lagrang e multiplier associated with the linear constraints in (3). The term T ( θ ) essentially arises as the multiplier for the normalization constraint (the density must integrate to 1), and so e nsures that (4) is normalized: T ( θ ) = log Z X exp h θ, φ ( x ) i dν ( x ) . (5) The function T is known as the log-pa r ti tion or cumulant f unction cor r esponding to the ex p onential family . Its d omain is Θ = { θ ∈ R d : T ( θ ) < + ∞} , called the natural parameter space. The expone ntial family is regula r if Θ is open—almost all ex ponential families of intere st, and all tho se we consider in this w ork, are regular . The family is minimal if there is no α ∈ Θ such that h α, φ ( x ) i is a con stant over X ( ν -almost everyw here); m in im ality is a prope r ty of the associated statistic φ , usually called the sufficient statistic in the literature. The fo llowing prop osition collects the relev ant results on reg ular exp onential families; proofs may be fo und in W ainwright and Jordan [2008, Prop. 3.1–3.2, Thm. 3.3–3.4] and see also Banerje e et al. [2005a, Lem. 1, Thm. 2]. A co nvex function T is o f Legendre type if it is prope r , closed, strictly convex and diff erentiable on the interior of its domain, and lim θ → ¯ θ ||∇ T ( θ ) || = + ∞ whe n ¯ θ lies on the boundary of the domain. P R O P O S I T I O N 2.1. Consider a regul a r exponen tial family w ith minimal s ufficient statistic . The fol lowing properties hold: ( 1 ) T and G are o f Legendre type , and T = G ∗ (equivalently G = T ∗ ). ( 2 ) The gradi ent ma p ∇ T is o ne-to-on e and onto the interior of M . Its inverse is ∇ G which is one-to-one and onto the interi or of Θ . ( 3 ) The exponen ti al family distribution with natural parameter θ ∈ Θ ha s ex pected statistic µ = E p [ φ ( x )] = ∇ T ( θ ) . ( 4 ) The max imum entrop y distribution w i th expected statistic µ is the expon ential fa m - ily distributio n with natura l par a meter θ = ∇ G ( µ ) . 2 W e assume that t he minimum is finite and achieved for all µ ∈ M . S ome care is needed to ensure this holds for specific statistics an d outcome spaces. F or example, taking outcomes to be the real n u mbers , there is no maximum entropy distribution with a given mean µ (one can take dens ities tending towards the uniform distribution over the reals), but there is always a solution if we constrain both the mean and varian ce. EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:6 Aber nethy et al. In the above T ∗ denotes the c o nvex conjugate o f T , which he re can be ev aluated as T ∗ ( µ ) = sup θ ∈ Θ h θ, µ i − T ( θ ) . Similarly , G ∗ ( θ ) = sup µ ∈M h θ, µ i − G ( µ ) . Proper Log Scori ng. W e are now in a position to analyze the log sco ring rule unde r ex - ponential family distributions. From our discussion so f ar , w e have that an exp onential family density can be parametrized either by the natural p arameter θ , or by the me an parameter µ , an d that the tw o are re lated by the inv ertible gradient map µ = ∇ T ( θ ) . W e w ill write p ( x ; θ ) or p ( x ; µ ) given the parametrization used. The follo wing observation is crucial. Let ˜ p ∈ P be a density (n ot n ecessarily fro m an expone ntial f amily) with e xpected statistic µ , let p ( · ; µ ) be the e xponen tial family with the same ex pected statistic, and let ˆ µ ∈ M be an alternative report. The n from (4) note E ˜ p [log p ( x ; ˆ µ )] = E p ( · ; µ ) [log p ( x ; ˆ µ )] = h ˆ θ , µ i − T ( ˆ θ ) , (6) where ˆ θ = ∇ G ( ˆ µ ) is the natural parame ter for the expon ential f amily with statistic ˆ µ . W e see from this that the e xpected log score only depen ds on the exp ectation µ of the underlying density , not the f ull density , wh ich is how w e can achieve prop er scoring according to Definition 2.1. T H E O R E M 2.2. Consider the logarithmi c scoring rule S ( µ, x ) = lo g p ( x ; µ ) defi ned over a set o f densities D para metrized by M . The scoring rule is p r oper i f and o n ly if D is the exponen tial family with statistic φ . P R O O F . Let µ, ˆ µ ∈ M be the age nt’s true belief and an alternative report, and let p ∈ P be a density co nsistent with µ . Let θ = ∇ G ( µ ) and ˆ θ = ∇ G ( ˆ µ ) , and note that µ = ∇ T ( θ ) . W e have E p [log p ( x ; µ )] − E p [log p ( x ; ˆ µ )] = h θ , µ i − T ( θ ) − h ˆ θ, µ i + T ( ˆ θ ) = T ( ˆ θ ) − T ( θ ) − h ˆ θ − θ , µ i = T ( ˆ θ ) − T ( θ ) − h ˆ θ − θ , ∇ T ( θ ) i . (7) The latter is po sitive by the strict convexity of T , which show s that the log score is proper . F or the c o nverse, assume the de fined log score is pro per . By the Savage char- acterization of p r oper scoring ru les for ex pectations (see Gneiting and Raf tery [2007, Thm. 1] and Savage [1971]), we must have S ( µ, x ) = G ( µ ) − h∇ G ( µ ) , µ − φ ( x ) i for some strictly co n vex function G . Let T = G ∗ , so that ∇ G = ∇ T − 1 , and let θ = ∇ G ( µ ) . Then the above can be written as log p ( x ; µ ) = G ( µ ) − h∇ G ( µ ) , µ − φ ( x ) i = h θ , µ i − T ( θ ) − h θ , µ − φ ( x ) i = h θ , φ ( x ) i − T ( θ ) , which sho ws that p ( x ; µ ) takes the form of an exp onential f am ily . As fu rther intuition for the result, no te that (7) is the definition of the ‘Bre gman diver- gence’ with respect to strictly co nvex function T , written D T . Therefor e we have E p [log p ( x ; µ )] − E p [log p ( x ; ˆ µ )] = D T ( ˆ θ , θ ) = D G ( µ, ˆ µ ) , where the last equality is a w ell-known identity relating the Bregman dive rgence s o f T and T ∗ = G . The equation states that the age nt’s regre t fr om misreporting its mean parameter does not dep end on the full density p , only the mean µ . EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:7 2.2. Examples: Moments over the Real Lin e Theorem 2.2 le ads to a straightforward pro cedure f o r constructing sco re rules for ex- pectations. De fine the re le vant statistic, and consider the maximum entropy (equiva- lently , e xpone ntial family) distribution consistent with the agent’s r eported mean µ . The scoring rule compensates the agent according to the log likelihoo d of the even tual outcome acco rding to this distribution. The inter pretation is that the agent is only pro - viding partial inf ormation about the underly ing density , so the p rincipal first in f ers a full density accor d ing to the principle of maximum entropy , and then sco r es the age nt using the usual log score. An adv an tage of this g e neralization of the log score is that, for many domains (multi- dimensional included) and e xpectations of interest, it le ads to nove l closed-form scor- ing rules. B y examining the log densities of various expo nential families, w e can fo r instance obtain scoring rules for several differe n t combinations of the arithmetic, g eo- metric, an d harmonic me ans , as well as higher orde r moments. The following examp les illustrate the construction. Example 2.3 . As base measure we take the Lebesgue restricted to [0 , + ∞ ) , and we consider the statist ic φ ( x ) = x so that we are simply e liciting the mean. The maximum entropy distribution with a given me an µ is the expo nential distribution, an d taking its lo g density gives the scoring rule S ( µ, x ) = − x µ − log µ. (8) W e stress that although this rule is d e rived f r om the ex p onential distribution, The- orem 2.2 implies that it elicits the mean of an y distribution supported on the non- negative reals (e.g., P areto, lognor mal). Indee d, it is easy to see that the e xpected score (8) dep ends only o n the mean of the ag ent’s belief be cause it is linear in x . As a gene ralization of this example, the maximum entro py distribution for the k -th mo- ment φ ( x ) = x k with respect to the same base me asure is the W eibull distribution. T aking its lo g density leads to the follow ing equivalent scoring rule: S ( µ, x ) = ( k − 1) log x − k log µ − Γ  1 + 1 k  k  x µ  k , (9) where Γ de notes the gamma function (the extension of the f actorial to the re als). W e have n ot f ound either scoring rule (8) o r ( 9) in the literature. Example 2.4 . As a base measure we take the Lebe sgu e ov er the real number s R . W e are interested in eliciting the mean µ and varianc e σ 2 , so as a statistic we take φ ( x ) = ( x, x 2 ) for which E p [ φ ( x )] = ( µ, µ 2 + σ 2 ) . The max entropy distribution for a given mean and variance is the Gaussian, w h ose log density gives the scoring rule S (( µ, σ 2 ) , x ) = − ( x − µ ) 2 σ 2 − log σ 2 . (10) Again, we stress that this scor ing rule elicits the mean and variance of any density over the re al numbers, n ot just tho se of a normal d istribution. The construction e as- ily gen eralizes to a multi-dimen sion al outcome space by taking the log density of the multivariate normal: S (( µ, Σ ) , x ) = − ( x − µ ) ′ Σ − 1 ( x − µ ) − log | Σ | . (11) Here the statistics being elicited are the mean vector µ and the covariance matrix Σ . These scoring rules have been studied by Dawid and Sebastiani [1999] as rules that EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:8 Aber nethy et al. only d e pend on the mean and v arianc e of the reported de nsity . The y note that these rules are weakly pro per ( because they do not distinguish be tw e en d ensities with the same first and second mome nts), but do no t make the p oint that knowledge of the full density is not nec e ssary o n the part of the agent. In the above, Example 2.4 illustrates an impor tant point about parame trizations of the elicited expe ctations . The variance σ 2 cannot be w ritten as E [ φ ( x )] for any φ , because the mean µ e nters the definition of σ 2 but is no t available when φ is defined (ind eed it is elicited in tandem with the variance) . 3 Instead o n e must use the first two uncentered moments E [ x ] and E [ x 2 ] . These are in bije ction with µ and σ 2 , so the resulting scoring rule can be re-wr itten in terms o f the latter . Therefo re, it is po ssible to e licit no t ju st expectations but also bijective transformations o f ex pectations. 3. EXPONENTIAL F AMIL Y MARKETS In a single-ag e nt setting, a scoring rule is used to elicit the agent’s belief . In a multi- agent setting, a prediction market can be used to a ggregate the beliefs of the agents. In his seminal pap e r Hanson [2003] introduce d the idea of a market scoring rule, which inherits the appe aling elicitation and aggregation prop e rties of both in o rder to per- form well in thin or thick marke ts. In this section , we adapt the g eneralized log scor- ing ru le to a market scorin g rule which leads to markets with simple closed-form cost functions for many statistics of interest. 3.1. Prediction Market In a prediction market an agent’s exp e cted belie f µ is elicited indirectly through the purchase and sale of contingent claim securities. Under this appro ach, each comp o - nent i of the statistic φ is inte r preted as the p ayof f function o f a security; that is, a single share of security i pays off φ i ( x ) whe n x ∈ X occurs . Thus if the portfolio of shares held by the age nt is δ ∈ R d , w here entry δ i correspond s to the number of shares of security i , the n the payoff to the age nt when x occu rs is ev aluated by taking the inner product h δ, φ ( x ) i . As a concre te examp le, recall that in the classic fi nite-outcome case the statistic h as a com p onent for each outcome x such that φ x ( x ′ ) = 1 if x ′ = x and 0 otherwise. Therefore the corre sponding security pays 1 dollar if ou tco me x occurs. (These are known as Arrow- Debreu secur ities.) In Example 2.3 the one-dime nsional statistic is φ ( x ) = x , correspo nding to a security w hose p ayof f is linear in the o utcome x ∈ R + . (This amounts to a future s con tract.) The standard wa y to imp lement a prediction marke t in the literature, due to Chen and P enno ck [ 2007], is via a centralized market maker . The market m aker maintains a conv ex, dif ferentiable cost functio n C : R d → ( −∞ , + ∞ ] , where C ( θ ) records the re venue collected when the vector of o utstanding share s is θ . The cost to an agent of purchasing po r tf o lio δ under a market state of θ is C ( θ + δ ) − C ( θ ) , and therefore the instantaneous prices of the securities are given by the gradien t ∇ C ( θ ) . A risk-ne u tral agent will choose to acquire shares up to the poin t whe re, fo r e ach share, expe cted payoff equals marginal price. F ormally , if the age nt acquires po rtfolio 3 This is an intuitiv e but far from formal explanation for the fact that the dimen sion of t h e message space, or elicitation complexity , for eliciting the variance is at least 2 [Lambert et al. 2008]. EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:9 δ , movin g the m arke t state vector to θ ′ = θ + δ , then we must have E p [ φ ( x )] = ∇ C ( θ ′ ) . (12) In this way , by its choice of δ , the agen t reve als that its exp ected belief is µ = ∇ C ( θ ′ ) . W e stress that this o bservation re lies on the assumptions that 1) the ag ent is risk- neutral, 2) the age nt does no t incorp orate the market’s inform ation into its own be- liefs, and 3) the agent is not budget constrained . W e will examin e relaxations of each assumption in later sections. 3.2. Information-Theo retic Interp retation In the re mainder of this paper we fo cus on the following cost fun c tio n , which arises from the “generalized” logarithmic market scoring rule (LMSR): C ( θ ) = log Z x ∈X exp h θ , φ ( x ) i dν ( x ) . ( 13) This is of course exactly the log-partition fun c tion (5) fo r the expo nential family with sufficient statistic φ , and w e rec o ver the classic LMSR using o utcome indicator v ectors as statistics . Because an agent would neve r select a portfo lio with infinite co st, the effective domain (i.e., the possible vecto rs of outstanding shares) of C is Θ = { θ ∈ R d : C ( θ ) < + ∞} , which give s an econo mic interpretation to the natural parameter space of an expo n ential family . The correspond e nce be tween the c o st function ( 13) and the log-partition f unction (5) suggests the follow ing interp retation. The market maker maintains an expone ntial family distribution ov er the state space X parametrize d by share vectors that lie in Θ . When an agen t buys shares, it move s the distribution’s natural parameter so that the market pr ices matches its beliefs, o r in other words the market’s mean parametrization matches the ag e nt’s expe ctation. There is a we ll-known duality be tween scoring rules and co st-function based mar- kets [Abernethy et al. 2013; Hanson 2003]. T o see this in our con te xt, re call from (6): E ˜ p [log p ( x ; ˆ µ )] = h ˆ θ , µ i − T ( ˆ θ ) where ˜ p is the agent’s be lief and ˆ µ the age n t’ s repo rt. The expe cted log sco r e from reporting ˆ µ is exactly the same as the e xpected payoff from buying portfo lio of shares ˆ θ = ∇ C ( ˆ µ ) (assuming an initial marke t state of 0), as h ˆ θ, µ i is the expected reven ue and T ( ˆ θ ) is the cost. As in Section 2 this reasoning relies on the assumption of risk- neutrality , not on any specific form for the agent’s belie f. The agent’s exp e cted p rofit from moving the share vector from θ to θ ′ is h θ ′ − θ , µ i − C ( θ ′ ) + C ( θ ) = C ( θ ) − C ( θ ′ ) − h θ − θ ′ , ∇ C ( θ ) i = D C ( θ, θ ′ ) = D C ∗ ( µ ′ , µ ) , recalling ( 7). Now Baner j ee e t al. [2005b ] have observed (among others) that the Kullback-Leibler d iv ergence betwe en two expo nential family d istributions is the Breg - man diverg ence, with re pect to the log-partition fun ction, between their natural pa- rameters. The agen t’ s exp ected pro fi t is therefo re the KL diverg ence between the mar- ket’s imp lied ex pectation and the expon ential family correspon ding to the agen t’ s ex- pectation, a well-known prope rty fro m the class ical LMSR [ Hanson 2007]. EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:10 Aber nethy et al. 3.3. Examples: Real Line and the Sphere Let u s no w re visit our scoring rule e xamples f rom Se c tion 2 in the context of prediction markets . The rele vant entities now are the payoff function, the effective domain o f shares , and the cost fun ction. Example 3.1 . W e con sider outcome s o ver the positive reals R + and set up a m arke t for the e x pected ou tcome, consisting o f a single security that pays off φ ( x ) = x . The lo g partition function of the expo nential distribution leads to the follow in g cost function : C ( θ ) = − log( − θ ) . The ef fective domain is Θ = { θ ∈ R : θ < 0 } . This me an s the marke t must start with a negative number of o u tstanding shares f or the security , and the number of shares must stay negative. The market maker need not explicitly enforce this, because by the Legendre proper ty of C the co st ten ds to + ∞ as the outstanding shares appro ach the boundary , wh ich is straightforward to see in this e xample. Example 3.2 . W e consider outcome s o v er the real line R an d set up a market with securities correspon ding to the first two unce ntered mome nts (i.e, agents are betting on the return and volatility). The securities are defined by the payoffs φ ( x ) = ( x, x 2 ) . The log partition function o f the normal distribution, under its natural p aram e trization, leads to the following cost f u nction: C ( θ ) = − θ 2 1 4 θ 2 − 1 2 log( − 2 θ 2 ) . The effec tive do main is Θ = { ( θ 1 , θ 2 ) ∈ R 2 : θ 2 < 0 } . Again, we have here an instance where it is not possible for the number of outstanding shares of the sec o nd security to exce ed 0. However , an arbitrary amount o f the securities can be sold short, which correspond s to increasing the v ariance of the market’s estimate. Example 3.3 . As another e xample let the outcome space be the d -dimensional unit sphere. This setting was co nsidered by Abernethy et al. [ 2013] w ho prov ide a c o st fu nc- tion implicitly defi ned through a variational characterization. The maximum entropy approach lead s to another alternative. W e have a security for each of the d dime nsions, and security i simply pays off φ i ( x ) = x i , whe re x ∈ R d is the unit-norm outcom e. The maximum entrop y distribution over the sphere with such sufficient statistics is the von Mises-Fis her distribution. The log partition fu nction correspond s to C ( θ ) = I d 2 − 1 ( || θ || ) −  d 2 − 1  log || θ || , where I r refers to the modifi e d Be ssel function o f first kind and order r ; see Banerje e et al. [2005b] fo r an ex planation of these quantities . The ef fective do main of θ is the positive orthant in R d . The m ean parame trization of the v on Mises-Fisher distribution g ives a generalized log scorin g rule for the expe cted outcome componen ts , but it is unwieldy and invo lves several special functions. 4. BA YESIAN TRADERS WITH LINEAR UTILI TY In the standard model of cost-function based p rediction markets, a sequence o f my- opic, risk-neutral ag e nts arrive and trade in the market [Ch en and P ennock 2007; Chen and V au g han 2010 ]. As we saw in Section 3.1, such a trader move s the prices EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:11 to its own ex pectation µ . Ho w ever , this means that the market does not pe rform me an - ingful aggre gation of ag ents’ be liefs, as the final price s are simply the fi nal age nt’s expectation. In this sec tio n we e xamine the agg regation be havior of the marke t whe n agen ts are Bayesian and take into account the cu rrent market state when forming their beliefs. This require s mo re structure to their be liefs. F or this section and the re m ainder of the paper , we will assume that age nts have e x ponentia l family beliefs . The expon ential families framew o rk is we ll- suited to reasoning about Bayesian up- dates. As before let the data distribution be given by p ( x ; θ ) = exp( h θ , φ ( x ) i − T ( θ )) where T is the log partition fun ction and φ are the sufficient statistics . Instead of di- rect beliefs about the data distribution the agent maintains a conju g ate prior ov er the parameters θ . Every expo nential family admits a co njugate prior of the form p ( θ ; b 0 ) = exp( h nν, θ i + nT ( θ ) − ψ ( ν, n )) . Note that this is also an e xponen tial family with natural parameter b 0 = ( nν, n ) whe re ν ∈ R d and n is a positive integer . The sufficient statistic maps θ to ( θ , T ( θ )) , and the log partition function ψ is defi n ed as the n ormalizer as usual. F or a complete treatment of e xponen tial families con jugate priors, see for instance Barnd orff-N ie lsen [1978]. Now Diacon is and Ylvisaker [1979, Thm . 2] and Jewell [1974] have shown that E θ ∼ b 0 E x ∼ θ [ φ ( x )] = ν, (14) meaning that ν = nν / n is the posterio r mean. Thus, it is he lpful to think o f the prior as being based on a ‘ph antom’ sample o f size n and mean ν . Sup p ose n o w that the age nt observes an e mpirical sample with mean ˆ µ and size m . By a standard der iv ation [see Diaconis and Ylvisaker 1979], the posterior co njugate prior parame te rs become nν ← nν + m ˆ µ and n ← n + m , and the posterior expectation (14) evaluates to nν + m ˆ µ n + m . (15) Thus the posterior mean is a conve x combination o f the prior and empirical means, and their relative weights depen d on the phantom and empirical sample sizes. Consider Bayesian age nts maintaining an ex ponential family co njugate prior o ver the data mode l’s natural paramete rs (equivalen tly , the expected security payoffs). Each agent has access to a private sample of the data of size m with me an statistic ˆ µ . If n agents have arrived before to trade, then the cu rrent market price s µ corr espond to the phantom sample, and the phantom sample size is n m . After fo r ming the poste- rior (15) with the se substitutions , the (risk-neutral) age nt purchases shares δ to mo ve the current market share vecto r to ∇ C ( θ + δ ) = nν + ˆ µ n + 1 . As a result, the fin al market price s under this behavior are a simple average o f the agent’s mean parameters and the initial market prices. W e note that to facilitate such belief updating, the market should p ost the number of trades since initialization. 5. RISK-A VERSE TRADERS WITH EXPONENTIAL UTILITY In this section w e relax the standard assumption that age nts in the market are risk- neutral. W e show that w ith sufficient extra structure to the agen ts’ be liefs and utilities , EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:12 Aber nethy et al. the market perfor m s a clean agg r e gation of the age nts’ beliefs via a simple we ig hted average. Assume that the agen t has an expo nential utility func tio n fo r wealth w : U a ( w ) = − 1 a exp( − aw ) . (16) Here a co ntrols the risk aversion: the agent’s aversion g rows as a increases, and as a tends to 0 we approach line ar utility ( r isk-neutrality). Specifically , a is the Arrow-Pratt coefficie nt of absolute risk aversion, and e xpone ntial utilities of the form (16) are the unique utilities that exhibit constant absolute risk aversion [V arian 1992, Ch ap. 11]. If we alth is d istributed acco rding to a pro bability measure P , then the certain ty equiv- alent of a rando m amoun t of wealth is define d as C E ( w ) = U − 1 a ( E P [ U a ( w )] ) . Suppose as befor e that the ag e nt’s belief ove r outcome s takes the f orm of a den- sity p with respect to base m easure ν . There is a close relationship betwee n the log-partition function and the ce rtainty e quivalent und er expo nential utility [see Ben-T al and T eboulle 2007]. L E M M A 5.1 . The certainty equiva l ent of the a g ent’s expe cted p r ofit, with expon ential utility , when acquirin g shares δ under a market state of θ is log a − T p ( − aδ ) − aC ( θ + δ ) + aC ( θ ) , (17) where T p is the lo g par ti tion fu nction (5) wi th a base meas ure of p dν . Furthermore, if the agent’s beli ef is a n expon ential fami ly with natural pa rameter ˆ θ , we have T p ( δ ) = T ( ˆ θ + δ ) − T ( ˆ θ ) , where T is the usual lo g pa rtition function with base measure ν . P R O O F . Explicitly , the certainty equivale n t o f the profit is C E ( h δ, φ ( x ) i − [ C ( θ + δ ) − C ( θ )] ) = − log Z X 1 a exp ( h− aδ, φ ( x ) i + a [ C ( θ + δ ) − C ( θ )]) p ( x ) dν ( x ) = log a − a [ C ( θ + δ ) − C ( θ )] − lo g Z X exp h− aδ, φ ( x ) i p ( x ) dν ( x ) = log a − a [ C ( θ + δ ) − C ( θ )] − T p ( − aδ ) . F or the second part of the result, we have T p ( δ ) = log Z X exp h δ, φ ( x ) i p ( x ; ˆ θ ) dν ( x ) = log Z X exp( h δ + ˆ θ, φ ( x ) i − T ( ˆ θ )) dν ( x ) = T ( ˆ θ + δ ) − T ( ˆ θ ) + log Z X exp( h δ + ˆ θ, φ ( x ) i − T ( ˆ θ + δ )) dν ( x ) = T ( ˆ θ + δ ) − T ( ˆ θ ) + log Z X p ( x ; ˆ θ + δ ) dν ( x ) = T ( ˆ θ + δ ) − T ( ˆ θ ) , where the last line follows f rom the fact that den sity p ( x ; ˆ θ + δ ) integ r ate s to 1. Recall that for the ge neralized LMSR, the co st function C is ex actly the log parti- tion function T . W e are therefo re lead to the f ollowing un derstanding of a risk-averse agent’s behavior in such a market. EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:13 T H E O R E M 5.2. Supp o se a n agen t h as exp onential utility with c o efficient a an d ex- ponential fam i ly belief s with na tural parameter ˆ θ . In the g eneralized LMSR ma rket with current market state θ , the agent’s optim al trade δ moves the state vector to θ + δ = 1 1 + a ˆ θ + a 1 + a θ. (18) P R O O F . The agent’s optimal trade maximizes its ex pected utility , or e qu ivalently the certainty equivalent. From Lemma 5.1 and that T = C , the agen t max im ize s log a − T ( ˆ θ − aδ ) + T ( ˆ θ ) − aT ( θ + δ ) + aT ( θ ) . This objective is strictly concave, fro m the strict convexity o f T . The optimum is there- fore characterized by the fir st-order c ondition ∇ T ( ˆ θ − aδ ) = ∇ T ( θ + δ ) . As the gradi- ent map ∇ T is o ne-to-o ne, this is solve d by equating the argume nts , which leads to δ = ( ˆ θ − θ ) / (1 + a ) and (18). Note that as, a te n ds to 0, we approach risk neutrality and the agen t moves the share vector all the way to its p rivate estimate ˆ θ . As a grows larger ( the agent grows more risk averse) the agent m akes smaller trades to re d uce it e x posure, and the final state stays closer to the cur rent state θ . Update (18) im p lies that, und er the conditions of the theorem, a market that receive s a sequence of m y opic traders aggregates their natural parameters in the form of an exp onentially we ighted moving average. The final m arke t estimates (i.e., prices) are obtained by applying ∇ T to this average. Liqui dity Adjustment . In practice the centralized market maker allow s itself some con- trol o ver the liquidi ty in the market, w hich capture s h ow re sponsive prices are to trades. T o adjust liquidity we consider the parametrized cost C λ ( θ ) = 1 λ C ( λθ ) . He re λ is co nstrued as the in verse liquidity , or price responsivene ss. A larger setting of λ means fewer shares need to be bou ght to reach the same prices. 4 In the context o f the gen eralized LMSR we write T rather than C where T is the log partition function, with liquidity-adjusted ve rsion T λ . Let ˆ µ be the age nt’s me an belief w ith correspo nding natural parameter ˆ θ = ∇ T − 1 ( ˆ µ ) . Recall that a risk-neutral agent moves the share vector so that the price s match its mean parame ter . Therefo re, define the tar get shares as ˜ θ = ∇ T − 1 λ ( ˆ µ ) . The target shares ˆ θ and natural parameter ˜ θ are re lated by ∇ T ( ˆ θ ) = ∇ T λ ( ˜ θ ) = ˆ µ . In add ition it is straightforward to check that ∇ T λ ( ˜ θ ) = ∇ T ( λ ˜ θ ) so we have ˜ θ = ˆ θ/ λ. (19) Higher price re sponsiveness means f ewer shares must be bou ght to make the mar- ket pr ice s match the agent’s exp ectation, so the natural parameter is scaled dow n accordingly . With a liquidity adju stment the analysis of Theo rem 5.2 can be e xtended and yields the following result, wher e as befor e θ and µ are the market’s o utstanding shares and prices respective ly . C O R O L L A R Y 5.3 . U nder the c o nditions of Theorem 5.2 and an inverse liquid ity o f λ , the agent’s optimal trade δ moves the state vector to θ + δ = λ λ + a ˜ θ + a λ + a θ = λ λ + a ∇ T − 1 λ ( ˆ µ ) + a λ + a ∇ T − 1 λ ( µ ) . (20) 4 The liquidity adjustment t o the cost fu nction t akes the same form as th e risk-aversion adjustment to the exponential utility in (16). In convex anal y sis, this transformation is known a s the perspective func- tion [Hiriart-Urruty and Lemar ´ echal 2 0 0 0, p. 90]. EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:14 Aber nethy et al. According to (20), as λ gro ws large the ag ent m oves the market state closer to the target shares , rather than its true natural parame ter . Note that the targ e t shares themselve s depend on λ by (19), but the update can be directly w ritten in terms o f the age nt’s beliefs as in the right-h and side o f (20). 5.1. Repeated T rading and the Effective Belief In prev ious sections we have analyze d trader behavior assuming it is his first entry into the market. W e now pose the question: how will a trader reason abo ut a p ossible future inv estment when the trade r h olds an ex isting po rtfolio? In the context of a trader po ssessing an exp onential family be lief toge ther with expo nential utility , we show that we can exp licitly analyze ho w an agent incorpo rates an existing portfo lio. The key conclusion is that a trader will reason about a future investment simply as though he had updated his be lie f and h ad no prior investment. Suppose an expone ntial utility agent has expo nential f amily belief p arame trized by natural parameter ˆ θ . Based o n this belief, let δ 1 be the v e ctor o f shares the age nt has purchased on first entry in the market. On a subsequen t en try into this market with market state θ ′ , his optimal purchase δ ∗ 2 is given by the solution of arg max δ 2 E x ∼ p ( x ; ˆ θ ) U [ h δ 1 + δ 2 , φ ( x ) i − C ( δ 1 + θ ) + C ( θ ) − C ( δ 2 + θ ′ ) + C ( θ ′ )] . Then if θ ′′ = ˆ θ − aδ 1 is the e ffective belief, the trade r’s o ptimal purchase is giv en by δ 2 = ( θ ′′ − θ ′ ) / (1 + a ) , movin g the share ve ctor to θ ′ + δ 2 = 1 1+ a θ ′′ + a 1+ a θ ′ , which is a convex combination of the ef fective belief and the current market state. T H E O R E M 5.4. Suppose an ex ponentia l utility maximizing trader with utility pa- rameter a who ha s belie f ˆ θ m akes a p u rchase δ in a market. On subsequently re-entering the market, he wi l l behave identically to an e xponentia l util ity maximizing trader with belief ˆ θ − aδ a nd no prior exp osure in the market. Theorem 5.4 implies that financial exposure can be equiv alently und e rstood as chang- ing the privately held beliefs. 5.2. Equilibrium Market State for Exponential Utility Agents W e have shown that eve ry expo nential-utility maximizing trader picks the share vec- tor δ so that the e ventual market state c an be re presented as a conve x com bination of the current market state and the natural parame ter of his ( expone ntial family) be lief distribution. In this section we will compu te the equilibrium state in an e xpone n tial family market with multiple such traders. W e draw a well-known result f rom g ame theory re garding the class o f potential game s . W e say a f unction f ( ~ x ) is at a l o cal optim um if changing any coordinate o f ~ x d oes not increase the value of f . T H E O R E M 5.5 ( M O N D E R E R A N D S H A P L E Y [ 1 9 9 6 ] ) . Let U i ( ~ δ ) be the util i ty func- tion of the i th trader given s tra tegies ~ δ = ( δ 1 , . . . , δ i , . . . , δ n ) . If there exists a potential function Φ( ~ δ ) su ch that U i ( ~ δ ) − U i ( ~ δ − i , δ ′ i ) = Φ( ~ δ ) − Φ( ~ δ − i , δ ′ i ) then ~ δ i s a Nash equilibriu m if and only if Φ( ~ δ ) i s at a local optim um. EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:15 In the e xponen tial family market, the cost function C is identical to the log partition function T de fined in ( 5). Let ~ δ be the matrix of share ve c to rs purchased by every trader in the market at equilibrium. Let θ be the initial market state, ˆ θ i the natural parameter of trader i ’s belief distribution and a i his risk aversion parameter . Define a potential function as Φ( ~ δ ) = T ( θ + P i δ i ) + P i 1 a i T ( ˆ θ i − a i δ i ) . Rather than wo rking dire ctly with the u tilities of ev ery trader , we will wor k w ith the log of their utility values. 5 Now the log- utility of trade r i is U i ( ~ δ ) = − T ( θ + X j δ j ) + T ( θ + X j 6 = i δ j ) − 1 a i T ( ˆ θ i − a i δ i ) + 1 a i T ( ˆ θ i ) . W e can n ow apply Theore m 5.5, henc e the equilibrium state is obtained by jointly maximizing Φ( ~ δ ) for each δ i : ∇ δ i Φ( ~ δ ) = ∇ T   θ + n X j =1 δ i   − ∇ T ( ˆ θ i − a i δ i ) = 0 . This leads to the follow ing ex pression f o r the final market state. θ + n X j =1 δ j = θ + P n i =1  ˆ θ i a i  1 + P n i =1 1 a i W e see that the equilibrium state is a conve x combination of the in itial m arke t state and all agent beliefs, w ith the latter w e ighted according to risk toleran c e. 6. BUDGET -L IMITED A GGREGA TION In this section, we consider the e volution of the market state whe n traders are budg e t- limited. W e assume that the traders trade in multiple instances of the m arket. As before, the market price is interpreted as a probability density over the outcome space and the share vector as the natural parameter of an e xpone n tial family distribution. Consistent with the connection s d r awn in Section 2 and through out, we measure the error in prediction using the standard log loss. W e show that traders with f aulty inf ormation can only impose a limited amount o f ad- ditional loss to the market’s p rediction. Further , since inf ormative traders exper ience an expecte d incre ase in budget, they w ill eventually be unco nstrained and allowed to carry out unrestricted trade s. T aken toge ther , this means that w hile the market suf- fers limited damag e fr om ill-informe d traders, it is also able to make use of all the information from informative traders in the long run . Budget-lim ited trades. Let α be the bud get of a trader in the market. Suppose that with infinite budge t, the trader w ould h ave move d the market state fro m θ to ˆ θ , where ˆ θ repre sents his true belie f. Now suppo se further that α < C ( ˆ θ ) − C ( θ ) ; that is, the trader’s budget does not allow for purchasing enough shares to move the market state to his be lief . In this case, we want to budg et-limit the trade r’s infl uence on the market state. 5 It is important to note that th e potent ial function analysis still applies for any monotonically increasing transformation of the traders’ utility functions. EC’14 , J une 8–12 , 2014, Stanford University , Palo Alto, CA, USA , V ol. X, No. X, Article X, Publication date: F ebruary 2014 . X:16 Aber nethy et al. Let the current market state be given by θ and let the final marke t state be θ ′ = λ ˆ θ + (1 − λ ) θ w here λ = min  1 , α C ( ˆ θ ) − C ( θ )  . The cost to the trade r to mov e the market state from θ to θ ′ is at mo st his budg et α and is called his budg et-limited trad e . Limited Dama ge. W e w ill now quantify the error in pre diction that the market maker might have to endure as a re sult of ill-inf ormed en tities e ntering the market. W e as- sume that these e ntities trade in multiple instances of the marke t; thus the e xposure of the market maker is over sev eral rou nds. The log loss function f o r θ shares he ld is defined as L ( θ , x ) = − log p ( x ; θ ) = C ( θ ) − h θ φ ( x ) i . L E M M A 6.1 . The loss induced o n the ma rket by an u ninformative trader is bounded by his initial budget. P R O O F . F irst co n sider the change in budget of a trader i ove r multiple round s of the prediction market. Let his budget at ro unds t and t − 1 be α t i and α t − 1 i respectively . The change in budget for trader i mov in g the m arke t state from θ to θ ′ with outcome x t is α t i − α t − 1 i = C ( θ ) − C ( θ ′ ) − ( θ − θ ′ ) T φ ( x t ) = L ( θ , x t ) − L ( θ ′ , x t ) = ∆ t i Where ∆ t i is called the myop ic im p act o f a trader i in round t . Thus, the myopic impact captures incremental g ain in pre d iction due to the trader in a round and is e qual to the change in his budget in that roun d. Since the market evolve s so that the budge t of any trade r nev er falls belo w zer o, the total my opic impact in T rou n ds caused d ue to trader i is ∆ i := P T t =1 ∆ t i = P T t =1 ( α t i − α t − 1 i ) = α T i − α 0 i ≥ − α 0 i . An interesting aspe ct of Lemma 6.1 is that the log lo ss can be quantified in the same units as the traders’ budge ts. Budget of In form ative T raders. W e now characterize the e xpected change in budg et fo r an informative trader . L E M M A 6.2 . Let θ be the cur r ent market state . Suppose that an in formative trader with belief distribution parametrized by ˆ θ moves the market state to the budge t- limited state θ ′ = λ ˆ θ + (1 − λ ) θ . Then, the expectatio n (over the trader’s belief) of the trader’s profit is strictly positive w henever his budget i s positive an d his bel ief di ffers fro m the previous market po sition θ . P R O O F . Let the cost function C be equal to the log partition function T of the belief distribution. The payof f is give n by the sufficie nt statistics φ ( x ) . Then, the trade r ’s expected net payoff is given by E x ∼ P ˆ θ [ C ( θ ) − C ( θ ′ ) − ( θ − θ ′ ) φ ( x )] = T ( θ ) − θ ∇ T ( ˆ θ ) − ( T ( θ ′ ) − θ ′ ∇ T ( ˆ θ )) = D T ( θ, ˆ θ ) − D T ( θ ′ , ˆ θ ) ≥ λD T ( θ, ˆ θ ) ≥ 0 where D T ( · , · ) is the Bregman diver gence based o n T . The second to last inequality holds since D T ( θ ′ , ˆ θ ) is conve x in θ ′ and we have: D T ( θ ′ , ˆ θ ) = D T  λ ˆ θ + (1 − λ ) θ, ˆ θ  ≤ λD T ( ˆ θ, ˆ θ ) + (1 − λ ) D T ( θ, ˆ θ ) = (1 − λ ) D T ( θ, ˆ θ ) A trader who adjusts the market state may expect p ositive profit ≥ λD T ( θ, ˆ θ ) . EC’14 , June 8–12, 2014, Stanford University , Palo Alto, CA, USA, V ol. X, No. X, Article X, Publication date: F ebruary 2014. Informati on Aggregati on in Exponentia l F amily Markets X:17 W e note one important aspec t of Lemma 6.2: the exp ectation is taken with re spect to each trader’s belief at the time of trade, rather than with re spect to the true distribu- tion. This is nee ded because w e have made no assumptions abo ut the optimality o f the traders’ belief updating proc edure. If we assume that the traders’ belief formation is optimal, then this growth result will exte nd to the true distribution as well. Given a continuou s de nsity the probability a trader will fo r m exactly the same beliefs as the cu r rent market position is 0 , and thus, each trader will have po sitive expe cted profit on almo st all sequ e nces o f o bserved samples and beliefs. This result suggests that, ev entually , e very info rmative trader will have the ability to influe n ce the m arke t state in accordance with his belief s , without being budget limited. Notice that Lemma 6.2 only required that the market state to which the trader m oves be rep r e sentable as a co nvex co mbination of the current market state and his belief . This means that the re sult holds for exp onential utility traders aiming to max imize their utility by Theor em 5.2. In this case, the trader who mov e s the market state can expect his pro fit to be positive and at least 1 a D T ( θ, ˆ θ ) w here a is the ex p onential utility parameter . 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