Betting on the Real Line

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1 Betting on the Rea Line Xi Gao 1, Yiing Chen 1,, and David M. Pennock 2 1 Harvard University, 2 Yahoo! Research, Abstract. We study the probem of designing prediction markets for random variabes with continuous or countaby infinite outcomes on the rea ine. Our interva betting anguages aow traders to bet on any interva of their choice. Both the ca market mechanism and two automated market maker mechanisms, ogarithmic market scoring rue (LMSR) and dynamic parimutue markets (DPM), are generaized to hande interva bets on continuous or countaby infinite outcomes. We examine probems associated with operating these markets. We show that the auctioneer s order matching probem for interva bets can be soved in poynomia time for ca markets. DPM can be generaized to dea with interva bets on both countaby infinite and continuous outcomes and remains to have bounded oss. However, in a continuous-outcome DPM, a trader may incur oss even if the true outcome is within her betting interva. The LMSR market maker suffers from unbounded oss for both countaby infinite and continuous outcomes. Key words: Prediction Markets, Combinatoria Prediction Markets, Expressive Betting 1 Introduction Prediction markets are specuative markets created for forecasting random variabes. In practice, they have been shown to provide remarkaby accurate probabiistic forecasts [1, 2]. Existing prediction markets mainy focus on providing an aggregated probabiity mass function for a random variabe with finite outcomes or discretized to have finite outcomes. For exampe, to predict the future printer saes eve, the vaue of which ies on the positive rea ine, Hewett-Packard s saes prediction markets partition the range of the saes eve into about 10 excusive intervas, each having an assigned Arrow-Debreu security that pays off $1 if and ony if the future saes eve fas into the corresponding interva [3]. The price of each security represents the market probabiity that the saes eve is within the corresponding interva. The set of prices provides a probabiity mass function for the discretized random variabe. However, many random variabes of interest have continuous or countaby infinite outcome spaces. For exampe, the carbon dioxide emission eve in a certain period of time can be thought of as a continuous random variabe on the positive rea ine; the printer saes eve can be treated as a random variabe with Part of this work was done whie Yiing Chen was at Yahoo! Research.

2 countaby infinite outcomes, taking positive integer vaues. Discretizing such random variabes into finite outcomes can potentiay hurt information aggregation, as market participants may have information that can not be easiy expressed with the ex-ante specified discretization. It is desirabe to provide more expressive betting anguages so that market participants can express their information more accuratey and preferaby in the same way they possess it. In this paper, we design and study prediction market mechanisms for predicting random variabes with continuous or countaby infinite outcomes on the rea ine. We provide betting anguages that aow market participants to bet on any interva of their choice and create the security on the fy. We generaize both the ca market mechanism and two automated market maker mechanisms, ogarithmic market scoring rues (LMSR) [4] and dynamic parimutue markets (DPM) [5, 6], to hande interva bets on continuous or countaby infinite outcomes, and examine probems associated with operating these markets. We show that the auctioneer s order matching probem can be soved in poynomia time for ca markets. DPM can be generaized to dea with interva bets on both countaby infinite and continuous outcomes and remains to have bounded oss. However, in a continuous-outcome DPM, a trader may incur oss even if the interva she bets on incudes the true outcome. The LMSR market maker suffers from unbounded oss for both countaby infinite and continuous outcomes. Due to space constraints, the Appendix is omitted and avaiabe upon request. Reated Work. Our work is situated in the broad framework of designing combinatoria prediction market mechanisms that provide more expressiveness to market participants. Various betting anguages for permutation combinatorics have been studied for ca markets, incuding subset betting and pair betting [7], singeton betting [8], and proportiona betting [9]. Fortnow et a. [10] anayzed betting on Booean combinatorics in ca markets. For LMSR market makers, Chen et a. [11] showed that computing the contract price is #P-hard for subset betting, pair betting, and Booean formuas of two events. In a tournament setting, pricing in LMSR becomes tractabe for some restricted Booean betting anguages [12]. Yoopick is a combinatoria prediction market impementation of LMSR that aows traders to bet on point spreads of their choice for sporting events [13]. It is impemented as a LMSR with a arge number of finite outcomes. Agrawa et a. [14] proposed Quad-SCPM, which is a market maker mechanism that has the same worst-case oss as a quadratic scoring rue market maker. Quad-SCPM may be used for interva bets on countaby infinite outcomes since its worst-case oss does not increase with the size of the outcome space. 2 Background In this section, we briefy introduce three market mechanisms that have been used by prediction markets to predict random variabes with finite outcomes. 2.1 Ca Markets A ca market is an auctioneer mechanism, where the auctioneer (market institution) riskessy matches received orders. In a ca market, participants submit

3 buy or se orders for individua contracts. A orders are assembed at one time in order to determine a market cearing price at which demand equas suppy. Buy orders whose bid prices are higher than the cearing price and se orders whose ask prices are ower than the cearing price are accepted. A transactions occur at the market cearing price. Most ca markets are biatera matching buy and se orders of the same contract. For muti-outcome events, ca markets can be mutiatera aowing participants to submit orders on different contracts and performing goba order matching [15 18]. 2.2 Logarithmic Market Scoring Rues Let v be a random variabe with N mutuay excusive and exhaustive outcomes. A ogarithmic market scoring rue (LMSR) [4, 19] is an automated market maker that subsidizes trading to predict the ikeihood of each outcome. An LMSR market maker offers n contracts, each corresponding to one outcome and paying $1 if the outcome is reaized [4, 20]. Let q i be the tota quantity of contract i hed by a traders combined, and et q be the vector of a quantities hed. The market maker utiizes a cost function C(q) = b og N j=1 eqj/b that records the tota amount of money traders have spent. A trader that wants to buy any bunde of contracts such that the tota number of outstanding shares changes from q to q must pay C( q) C(q) doars. Negative quantities encode se orders and negative payments encode sae proceeds earned by the trader. At any time, the instantaneous price of contract i is p i (q) = eq i /b N j=1 eq j /b, representing the cost per share of purchasing an infinitesima quantity. An LMSR is buit upon the ogarithmic scoring rue, s i (r) = b og(r i ). It is known that if the market maker starts the market with a uniform distribution its worst-case oss is bounded by b og N. 2.3 Dynamic Parimutue Markets A dynamic parimutue market (DPM) [5, 6] is a dynamic-cost variant of a parimutue market. From a trader s perspective, DPM acts as a market maker in a simiar way to LMSR. There are N securities offered in the market, each corresponding to an outcome of the random variabe. The cost function of the market maker, which captures the tota money wagered in the market, is C(q) = κ N j=1 q2 j, whie the instantaneous price for contract i is p i(q) = κq i N j=1 q2 j where κ is a free parameter. Unike in LMSR, the contract payoff in DPM is not a fixed $1. If outcome i happens and the quantity vector at the end of the market = κ j (qf is q f, the payoff per share of the winning security is o i = C(q f ) j )2. q f i q f i A nice property of DPM is that if a trader wagers on the correct outcome, she is guaranteed to have non-negative profit, because p i is aways ess than or equa to κ and o i is aways greater than or equa to κ. Because the price functions are not we-defined when q = 0, the market maker must begin with a non-zero quantity vector q 0. Hence, the market maker s oss is bounded by C(q 0 ).,

4 3 Ca Markets for Interva Betting For a random variabe X that has continuous or countaby infinite outcomes on the rea ine, we consider the betting anguage that aows traders to bet on any interva (, u) of their choice on the rea ine and create a security for the interva on the fy. The security pays off $1 per share when the betting interva contains the reaized vaue of X. For countaby infinite outcomes, the interva is interpreted as a set of outcomes that ie within the interva. Suppose that the range of X is (L, U) where L R { } and U R {+ }. Traders submit buy orders. Each order i O is defined by (b i, q i, i, u i ), where b i denotes the bid price for a unit share of the security on interva ( i, u i ), and q i denotes the number of shares of the security to purchase at price b i. We note i L and u i U. Given a set of orders O submitted to the auctioneer, the auctioneer needs to decide which orders can be riskessy accepted. We consider the auctioneer s probem of finding an optima match to maximize its worst-case profit given a set of orders O. We first define the state space S to be the partition of the range of X formed by orders O. For any order i O, ( i, u i ) defines 2 boundary points of the partition. Let A = ( i O i ) {L} be the set of eft ends of a intervas in O and the eft end of the range of X, and B = ( i O u i ) {U} be the set of right ends of a intervas in O and the right end of the range of X. We rank a eements of A and B in order of increasing vaues, and denote the i-th eement as e i. Ceary, e 1 = L and e A + B = U. We formay define the state space S as foows. Definition 1. Let s i S be the i-th eement of the state space S for a 1 i ( A + B 1). If e i = e i+1, then s i = {e i }. Otherwise, s i = (e i, e i+1 ] if both e i A and e i+1 A; s i = (e i, e i+1 ) if e i A and e i+1 B; s i = [e i, e i+1 ] if e i B and e i+1 A; and s i = [e i, e i+1 ) if e i B and e i+1 B. Because S = A + B 1, A O +1, and B O +1, we have S 2 O +1. With the definition of states given orders O, we formuate the auctioneer s optima match probem as a inear program, anaogous to the one used for permutation betting [7]. Definition 2 (Optima Match). Given a set of order O, choose x i [0, 1] such that the foowing inear program is optimized. max x i,c c (1) s.t. i (b i I i (s))q i x i c, s S 0 x i 1, i O I i (s) is the indicator variabe for whether order i is winning in state s. I i (s) = 1 if the order gets a payoff of $1 in s and I i (s) = 0 otherwise. The variabe c represents the worst-case profit for the auctioneer, and x i [0, 1] represents the fraction of order i O that is accepted. As the number of structura constraints is at most 2 O +1 and the number of variabes is O, (1) can be soved efficienty. We state it in the foowing theorem.

5 Theorem 3. For ca markets, the auctioneer s optima order matching probem for interva betting on countaby infinite and continuous outcomes can be soved in poynomia time. Thus, if the optima soution to (1) generates positive worst-case profit c, the auctioneer accepts orders according to the soution. Otherwise, when c 0, the auctioneer rejects a orders. When there are few traders in the market, finding a counterpart to trade in a ca market may be hard and the market may suffer from the thin market probem. Aowing traders to bet on different intervas further exacerbates the probem by dividing traders attention among a arge number of subsets of securities, making the ikeihood of finding a muti-atera match even more remote. In addition, ca markets are zero-sum games and hence are chaenged by the no-trade theorem [21]. In the next two sections, we examine market maker mechanisms, which not ony provide infinite iquidity but aso subsidize trading, for interva betting. 4 Dynamic Parimutue Markets for Interva Betting For interva betting in DPMs, traders aso create a security on the fy by choosing an interva (, u). However, the payoff of the security is not fixed to be $1. Instead, each share of the security whose interva contains the reaized vaue of the random variabe entites its hoder to an equa share of the tota money in the market. We generaize DPM to aow for (but not imited to) interva betting on countaby infinite and continuous outcomes. The probem that we consider is whether these mechanisms sti ensure the bounded oss of the market maker. 4.1 Infinite-Outcome DPM j=1 q2 j We generaize DPM to aow for countaby infinite outcomes, and ca the resuting mechanism infinite-outcome DPM. In an infinite-outcome DPM, the underying forecast variabe can have countaby infinite mutuay excusive and exhaustive outcomes. Each state security corresponds to one potentia outcome. An interva bet often incudes a set of state securities. The market maker keeps track of the quantity vector of outstanding state securities, sti denoted as q, which is a vector of dimension. The cost and price functions for the infiniteoutcome DPM are C I (q) = κ j=1 q2 j, and pi i (q) = κq i per winning security if outcome i happens is o I i = κ j=1 (qf j )2 q f i.. The payoff The oss of the market maker in an infinite-outcome DPM is sti her cost to initiate the market. The market maker needs to choose an initia quantity vector q 0 such that her oss C I (q 0 ) is finite. In practice, an infinite-outcome DPM market maker can start with a quantity vector that has ony finite positive eements and a others are zeros, or use an infinite converging series. Whenever a trader purchases a state security whose current price is zero or that has not

6 been purchased before, the market maker begins to track quantity and cacuate price for that security. Hence, infinite-outcome DPM can be operated as a finiteoutcome DPM that can add new state securities as needed. The market maker does not need to record quantities and cacuate prices for a infinite outcomes, but ony for those having outstanding shares. Infinite-outcome DPM maintains the desirabe price-payoff reationship of DPM the payoff of a security is aways greater than or equa to κ and its price is aways ess than or equa to κ. 4.2 Continuous-Outcome DPM We then generaize DPM to aow for continuous outcomes, and ca the resuting mechanism continuous-outcome DPM. The cost and price functions of a + continuous-outcome DPM are C = κ q(y)2 d y and p(x) = κq(x) + q(y)2 d y. A trader can buy δ shares of an interva (, u). The market maker then increases q(x) by δ for a x (, u). The trader s payment equas the change in vaue of the cost function. However, stricty speaking, function p(x) does not represent price, but is better interpreted as a density function. The instantaneous price for buying infinitey sma amounts of the security for interva (, u) is p (,u) = p(x)d x = κ q(x)d x + q(y)2 d y. If the reaized vaue of the random variabe is x, each share of a security on any interva that contains x has payoff C = κ + o(x ) = shares for securities whose interva contains y at the cose of the market. A continuous-outcome DPM market maker can choose an initia quantity distribution q 0 (x) such that her oss is finite. However, the desirabe price-payoff reationship that hods for the origina DPM no onger hods for continuousoutcome DPM. A trader who bets on the correct outcome may sti ose money. Theorem 4 states the price-payoff reationship for continuous-outcome DPM. Proof of the theorem is provided in Appendix A. qf (y) 2 d y, where q f (y) is the number of outstanding Theorem 4. The price per share for buying a security on interva (, u) is aways ess than or equa to κ u. If traders can bet on any non-empty open interva, the payoff per share is bounded beow by 0. If traders coud bet ony on open intervas of size at east z, the payoff per share is bounded beow by κ 2z 2. 5 Logarithmic Market Scoring Rue for Interva Betting For LMSR, we define the same interva betting anguage as in ca markets. A trader can create a security by specifying an interva (, u) to bet on. If the reaized vaue of X fas into the interva, the security pays off $1 per share. We generaize LMSR to aow countaby infinite and continuous outcomes and study whether the market maker sti has bounded oss. LMSR for finite outcomes can be extended to accommodate interva betting on countaby infinite outcomes simpy by changing the summations in the price

7 and cost functions to incude a countaby infinite outcomes. However, as the LMSR market maker s worst-case oss is b og N, the market maker s worst-case oss is unbounded as N approaches. We generaize LMSR to accommodate continuous outcome spaces. A ogarithmic scoring rue for a continuous random variabe is s(r(x)) = b og(r(x)) where x is the reaized vaue for the random variabe and r(x) is the reported probabiity density function for the random variabe evauated at x. Using an equation system simiar to the one proposed by Chen and Pennock [20], we derive the corresponding price and cost functions for the continuous ogarithmic scoring rue: C = b og( + eq(y)/b e d y), and p(x) = q(x)/b +. Here, p(x) eq(y)/b d y does not represent price, but is best interpreted as a density function. The instantaneous price for buying infinitey sma amounts of the security for interva (, u) is p(x)d x. If the interva (, u) contains the reaized vaue, one share of the security entites its hoder $1 payoff. However, the worst-case oss is sti unbounded for a continuous LMSR market maker even with the restriction on the size of aowabe intervas, as shown by Theorem 5. Proof of the theorem is presented in Appendix B. Theorem 5. A continuous ogarithmic market scoring rue market maker has unbounded worst-case oss, with or without the restriction that traders can bet ony on intervas of size at east z. 6 Concusion and Future Directions We study interva betting on random variabes with continuous or countaby infinite outcomes for ca markets, DPM, and LMSR. We show that the auctioneer s order matching probem in ca markets can be soved in poynomia time for interva bets. DPM can be generaized to hande both countaby infinite and continuous outcomes. Unfortunatey, in a continuous-outcome DPM, a trader may incur oss even if her betting interva contains the true outcome. LMSR market makers, however, suffer from unbounded oss for both countaby infinite and continuous outcomes. One important future direction is to design automated market maker mechanisms with desirabe properties, especiay bounded oss, when handing continuous outcome spaces. In particuar, it may be fruitfu to expore interva bets with variabe payoffs for outcomes within the interva. The interva contracts for ca markets and LMSR give the same payoff as ong as the outcome fas within the specified interva. Impicity, this assumes that a trader s prediction of the random variabe is a uniform distribution over the given interva. Aternativey, it woud be interesting to aow for the trader s probabiity distribution of the random variabe to take other shapes over the given interva, and hence to aow payoffs to vary correspondingy for outcomes within the interva. References 1. Berg, J.E., Forsythe, R., Neson, F.D., Rietz, T.A.: Resuts from a dozen years of eection futures markets research. In Pott, C.A., Smith, V., eds.: Handbook of Experimenta Economic Resuts. (2001)

8 2. Wofers, J., Zitzewitz, E.: Prediction markets. Journa of Economic Perspective 18(2) (2004) Chen, K.Y., Pott, C.R.: Information aggregation mechanisms: Concept, design and impementation for a saes forecasting probem. Working paper No. 1131, Caifornia Institute of Technoogy (2002) 4. Hanson, R.D.: Combinatoria information market design. Information Systems Frontiers 5(1) (2003) Pennock, D.M.: A dynamic pari-mutue market for hedging, wagering, and information aggregation. In: ACM Conference on Eectronic Commerce (EC). (2004) 6. Mangod, B., Dooey, M., Dornfest, R., Fake, G.W., Hoffman, H., Kasturi, T., Pennock, D.M.: The tech buzz game. IEEE Computer 38(7) (2005) Chen, Y., Fortnow, L., Nikoova, E., Pennock, D.M.: Betting on permutations. In: ACM Conference on Eectronic Commerce (EC). (2007) Ghodsi, M., Mahini, H., Mirrokni, V.S., ZadiMoghaddam, M.: Permutation betting markets: singeton betting with extra information. In: ACM Conference on Eectronic Commerce (EC). (2008) Agrawa, S., Wang, Z., Ye, Y.: Parimutue betting on permutations. In: Lecture Notes in Computer Science, The Internationa Workshop on Internet and Network Economics (WINE). (2008) 10. Fortnow, L., Kiian, J., Pennock, D.M., Weman, M.P.: Betting booean-stye: A framework for trading in securities based on ogica formuas. Decision Support Systems 39(1) (2004) Chen, Y., Fortnow, L., Lambert, N., Pennock, D.M., Wortman, J.: Compexity of combinatoria market makers. In: ACM Conference on Eectronic Commerce (EC). (2008) Chen, Y., Goe, S., Pennock, D.M.: Pricing combinatoria markets for tournaments. In: ACM Symposium on Theory of Computing (STOC). (2008) Goe, S., Pennock, D., Reeves, D.M., Yu, C.: Yoopick: A combinatoria sports prediction market. In: AAAI. (2008) Agrawa, S., Deage, E., Peters, M., Wang, Z., Ye, Y.: A unified framework for dynamic pari-mutue information market design. In: ACM Conference on Eectronic Commerce (EC). (2009) Bossaerts, P., Fine, L., Ledyard, J.: Inducing iquidity in thin financia markets through combined-vaue trading mechanisms. European Economic Review (46) (2002) Lange, J., Economides, N.: A parimutue market microstructure for contingent caims trading. NYU Schoo of Business Discussion Paper No. EC (2007) 17. Baron, K., Lange, J.: Parimutue Appications in Finance: New Markets for New Risks. Pagrave Macmian (2005) 18. Peters, M., So, A.M.C., Ye, Y.: Pari-mutue markets: Mechanisms and performance. In: Lecture Notes in Computer Science, The Internationa Workshop on Internet and Network Economics (WINE). (2007) Hanson, R.D.: Logarithmic market scoring rues for moduar combinatoria information aggregation. Journa of Prediction Markets 1(1) (2007) Chen, Y., Pennock, D.M.: A utiity framework for bounded-oss market makers. In: Conference on Uncertainty in Artificia Inteigence (UAI 2007). (2007) Migrom, P., Stokey, N.L.: Information, trade and common knowedge. Journa of Economic Theory 26(1) (1982) 17 27

9 A Proof of Theorem 4 Proof. By Cauchy-Schwarz inequaity, we have Hence, ( 2 q(x) d x) q(x) 2 d x q(x) d x (u ) 1 2 d x = (u ) The price of buying a security on interva (, u) is then p (,u) = κ q(x)d x + q(y)2 d y κ q(x)d x q(y) 2 d y q(x) 2 d x. q(x) 2 d x. (2) κ u. Suppose the smaest partition that incudes the true outcome x at the end of the market is (c, d). q f (x) = for a x (c, d). Then, payoff per share is o(x ) = κ + qf (y) 2 d y κ d c qf (y) 2 d y = κ 2 (d c) = κ d c. (3) If traders can bet on any non-empty interva, (d c) in (3) may approach 0. Thus, payoff per share o(x ) > 0. If the market maker restricts that traders can ony bet on intervas no smaer than z, we consider the interva (d z, c + z) for the case of (d c) < z. Because (c, d) is the smaest partition that contains x, =. (4) d c Any time when q(x) d x is increased by a, it must be the case that c c d z q(x) d x + c+z q(x) d x is increased at east by a d d c (z (d c)), because the smaest interva size is z. Hence, c d z q f (x) d x + c+z d q f (x) d x (z (d c)). d c

10 Thus, c+z d z q f c (x) d x qf (x) d x (z (d c)) + d c = z d c c c q f (x) d x q f (x) d x. (5) Payoff per share is o(x ) = κ + qf (y) 2 d y + κ(d c) = qf (y) 2 d y c+z κ(d c) d z qf (y) 2 d y κ(d c) c+z d z qf (y) d y (c + z) (d z) κ(d c) z d d c c qf (y) d y (c + z) (d z) κz = 2z (d c) > κ 2z. 2 The second equaity comes from (4). The fourth inequaity is a resut of appying (2). Appying (5), we get the fifth inequaity. Letting (d c) 0, we obtain the ast inequaity. B Proof of Theorem 5 Proof. Let x be the reaized vaue of the random variabe. Let p 0 (x ) and p f (x ) be the initia and fina price density for x. Then, the market maker s oss is b og p f (x ) b og p 0 (x ).

11 Suppose the smaest partition that incudes the true outcome x at the end of the market is (c, d). If traders can bet on any non-empty intervas, p f (x ) = = e qf (x )/b + eqf (y)/b d y eqf (x )/b c eqf (y)/b d y e qf (x )/b (d c) e qf (x )/b = 1 d c. 1 As (d c) approaches 0, d c approaches. Hence, the worst-cast oss is not bounded because b og p f (x ) <. If the smaest interva that traders can bet on is of size z, consider the situation that (d c) < z and traders buying equa shares of intervas (d z, d) and (c, c + z). Then, p f (x ) = e qf (x )/b + eqf (y)/b d y e qf (x )/b c d z eqf (y)/b d y + c eqf (y)/b d y + c+z d e qf (x )/b = 2(z (d c)) e qf (x )/2b + (d c) e qf (x )/b 1 = 2(z (d c)) e qf (x )/2b + (d c). e qf (y)/b d y When and (d c) 0, the p f (x ) approaches to 0 according to the above expression. Hence, the worst-case oss of the market maker is not bounded because b og p f (x ) <.

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