Equilibrium in Prediction Markets with Buyers and Sellers
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1 Equlbrum n Predcton Markets wth Buyers and Sellers Shpra Agrawal Nmrod Megddo Benamn Armbruster Abstract Predcton markets wth buyers and sellers of contracts on multple outcomes are shown to have unque equlbrum prces, whch can be computed n polynomal tme. 1 Introducton Predcton markets are ncreasngly seen as effcent belef-aggregaton devces. In these markets, traders buy and sell contracts on possble outcomes of some experment, e.g., sports competton or electon. A contract wll pay $1 f the actual outcome satsfes the condtons specfed n the contract. Thus, f the subectve belefs of the traders gve rse to unque market prces, then these prces may be nterpreted as the market-generated probabltes of outcomes. Predcton markets have grown n popularty as there s ncreasng emprcal evdence that predctons from these markets are at least as accurate, and often outperform tradtonal polls [2]. However, the exstng theory does not suffce to support ths practce. A market wth multple rsk-neutral buyers and one organzer that acts as a seller ( parmutuel bettng markets ) was analyzed by Esenberg and Gale [3]. They showed that the equlbrum prce of a contract s equal to the (budget-weghted) fracton of traders who beleve that the respectve outcome s most lkely. However, for a general exchange market, such an analyss exsts only for the smpler case of bnary outcomes [4, 5]. Here, the general predcton market wth buyers and sellers s reduced to parmutuel bettng markets. Thus, predcton markets have unque equlbrum prces, whch reflect market belef n the same sense as proven by Esenberg and Gale for parmutuel bettng. The results are also extended to traders wth concave utlty functons. 2 Equlbrum prces n predcton markets Let M = 1,..., m} be the set of all the outcomes of an experment, e.g., sports competton or electon. Let N = 1,..., n} the set of traders who buy and sell bettng contracts. For M, a sngle contract C enttles the buyer to receve from the seller one dollar when the outcome s. It s mportant to note that n our model the contracts can be traded n fractons. Ths assumpton s necessary for provng exstence of an equlbrum lke n many other economc models. A buyer must pay the current market prce of a contract. A seller of a set of contracts must depost the worst-case amount he may have to pay on the contracts he sells. Denote the market prces by π 1,..., π m. If trader buys b C -contracts, he must pay π b. If sells s C -contracts, he receves π s, but must depost max s }. Department of computer Scence, Stanford Unversty, Stanford, CA emal: shpra@stanford.edu IBM Almaden Research Center, San Jose, CA emal: megddo@almaden.bm.com Department of Industral Engneerng and Management Scences, Northwestern Unversty, Evanston, IL emal: armbruster@northwetern.edu. 1
2 Each has an ntal budget m > 0 and a subectve probablty dstrbuton p = (p 1,..., p m ),.e, p = 1 and p 0, M. We frst assume the traders are rsk-neutral,.e, trader s utlty s equal to hs subectve expectaton of the amount of money he wll have after the experment has been performed and the oblgatons are have been settled. A generalzaton to rsk-averse traders s dscussed n Secton 5. Thus, trader wshes to solve the followng problem: (P ) Maxmze subect to (p π ) (b s ) π (b s ) + maxs } m ( N) b, s 0 ( M). We call π 1,..., π m equlbrum prces f there exst b s and s s that solve the ndvdual optmzaton problems (P ), such that the market clears,.e., for every M: b = s. (1) For prces to represent probabltes, they must sum to 1. Indeed, Proposton 1. If π 1,..., π n are equlbrum prces, then M π = 1. Proof. If sells one C -contract, then from s pont of vew, hs fnal wealth wll be m + π 1 wth probablty p and m + π wth probablty 1 p. On the other hand, f buys one C k -contract for each k, then from s pont of vew, hs fnal wealth wll be m k π k wth probablty p and m k π k + 1 wth probablty 1 p. If k M π k < 1, then both m + π 1 < m k π k and m + π < m k π k + 1. It follows that would not sell C. Ths concluson holds for every N and M and, therefore, n ths case there would be no sellng and no buyng. However, ths can be an equlbrum only f p = π for every N and M, whch mples π = 1. Suppose M π > 1. If sells x C -contracts for each M, then receves x M π, whch s more than the requred depost x, so every value of x s feasble. The fnal wealth of s equal to m +x M π x, whch can be arbtrarly large, and hence not n equlbrum. Note that the proof of Proposton 1 holds for nonlnear utlty functons. The dea of the proof s also nstrumental n provng equvalence of equlbrum prces n predcton markets to equlbrum prces n parmutuel bettng markets. The latter was studed n [3]. 2
3 3 Equvalence to Parmutuel Bettng In parmutuel bettng, each trader acts only as a buyer. Thus, the optmzaton problem for trader s: Maxmze subect to (p π ) b π b m (2) b 0 ( M). Prces π 1,..., π m are parmutuel bettng equlbrum prces(pbep) f there exst b s that solve the ndvdual optmzaton problems (2), respectvely, so that under every outcome, the buyers are exactly pad off by the total money collected,.e., for every, b = π k b k (3),k Proposton 2. If π 1,..., π m are PBEP, then π = 1. Proof. By defnton, at equlbrum, m = π b = π b = π m. Ths mples the clam. Proposton 3. Prces π 1,..., π m are equlbrum prces n the predcton market f and only f they are PBEP n the correspondng parmutuel bettng market. Proof. Let b } be optmal purchases, satsfyng (1), made by traders at equlbrum prces π 1,..., π m n the parmutuel bettng market. Defne We show that the b s and s s satsfy the condtons requred for b = b f < m 0 f = m s = b m f < m 0 f = m. π 1,..., π m to be equlbrum prces n the predcton market. Frst, trader s wealth under every outcome s the same n both formulatons. Trader s wealth under < m s: m k π b k + b s = m k<m π b k + b b m = m k π b k + b, and under = m, m k π b k + b s = m k<m π b k = m k π b k + b. Second, the amounts of money spent n both optmzaton problems are equal as well; m m 1 π k (b k s k) + max s k = k π k (b k b m ) + b m = m 1 ( m 1 π k bk + 1 π k ) bm = m π k b k. Therefore, each N s ndfferent between the purchases b,..., b m and the transactons b 1,..., b m and s 1,..., s m. 3
4 Furthermore, the b s and s s satsfy (1) because, for = m, b m = s m = 0, and for every < m, (b s ) = (b b m ) = b b m = 0. For the converse, suppose π (π 1,..., π m ) s a vector equlbrum prces n the predcton market. Denote A M : π > p }. Frst, we show that there exst b s and s s that satsfy the equlbrum condtons wth respect to π, such that for all A, s = s, and for all A, s = 0. Suppose, on the contrary, that for some k A, s k < max A s. By ncreasng s k and b k smultaneously, the obectve value and the left-hand sde of (2) reman unchanged. Therefore, we can smultaneously ncrease s k and b k untl s k = s, whle mantanng (1). Smlarly, f s > 0 for some A, then necessarly b s (refer to (2)), so we can smultaneously decrease s k and b k untl s k = 0. Therefore, w.l.o.g., s = 0 f A, and s = s f A. For every N, defne replacements b = b + s / A b A. We show that, from the pont of vew of trader, the probablty dstrbuton over hs fnal wealth s unchanged n the replacement. In the orgnal predcton market, wth probablty A p, the net proft from sales was s A π s, and wth probablty A p, t was s A π. Now, f buys s C -contracts for every A, then wth probablty A p the net proft from these purchases s s A π, and wth probablty A p t s s s A π. Snce π = 1, the net profts under these two scenaros are equal. The total costs for each of these sets of contracts are also equal: π b + s s π b + s π b. A π = / A π = Furthermore, the b s satsfy (3) snce for every M, b = b + s = s + s = s. : / A : A : / A On the other hand, for every, π b =, π s = s. 4 Equlbrum prces reflect market belef Esenberg and Gale [3] presented a concave maxmzaton problem for computng an equlbrum n parmutuel bettng, where the prces are obtaned as Karush-Kuhn-Tucker (KKT) multplers, and proved exstence and unqueness of equlbrum prces. By our equvalence mappng, we obtan followng result for predcton market equlbrum: Theorem 1. There exsts a unque vector of equlbrum prces for predcton markets as defned by (2), (1). The equlbrum prces can be computed n polynomal tme. Furthermore, the equlbrum prce of C s equal to the fracton of money spent on C. 4
5 Proof. Exstence and unqueness follow from [3] and Proposton 3. Polynomal-tme computablty also follows from the concave-maxmzaton formulaton [3]. Denote by m the amount of money spent by on C at equlbrum. Thus, b = m /π. By (3), π = m. To get a precse relaton between equlbrum prces and traders belefs, n lne wth the results of [4, 5], consder a market wth a nonatomc contnuum of traders. Each trader s represented by a probablty dstrbuton p = (p 1,..., p m ) over a fxed M. The set of all such p s a smplex. Consder a nonatomc measure µ : R +, so that µ(p) s the densty of money (.e., budget) per unt volume at p. W.l.o.g, assume µ(p) dp = 1. (4) π = (π 1,..., π m ) are equlbrum prces f there exsts a vector β = (β 1,..., β m ) of measures β : R +, M, where β (p) represents the densty of C -contracts bought per unt volume at p ; two condtons must hold: m Indvdual optmalty. Consder a gven vector π = (π 1,..., π m ) and a vector of belefs p = (p 1,..., p m ). The optmzaton problem of traders n the neghborhood of p s to choose the values of β 1 (p),..., β m (p) so as to Maxmze (p π ) β (p) (5) subect to M π β (p) µ(p) (6) M β (p) 0 ( M). Here, the cost per unt volume s constraned by the budget per unt volume, and the obectve s to maxmze the net proft per unt volume. Balance constrant. β (p) dp = 1 ( M). (7) The ndvdual-optmalty condton mples that, n equlbrum, p < max π k M p k π k β (p) = 0. (8) For every 0 < π and M, denote ( (π) p : ( k M) k p > p )} k π π k. (9) Theorem 2. If µ s nonatomc, then, n equlbrum, for every M, µ(p) dp = π. (10) (π) Thus, at equlbrum, for every M, the fracton of the total budget that comes from traders who prefer C s equal to π. 5
6 Proof. Denote 0 (π) = (π). M Snce the Lesbegue-measure of the set \ 0 (π) s zero, β (p) dp = Therefore, by (7) (9), for every M, 1 = β (p) dp = k M k (π) 0(π) β (p) dp. β (p) dp = (π) β (p) dp. (11) On the other hand, by (6), n equlbrum, π k β k (p) = µ(p). (12) k M It follows from (12) and (11) that for every, µ(p) dp = π k β k (p) dp = π (π) k M (π) (π) β (p) dp = π. In the case of two outcomes, 1 ([π, 1 π]) and 2 ([π, 1 π]) are exactly the set of p s such that π < p and π p, respectvely. Thus, n ths case, Theorem 9 proves that the equlbrum prce π s the (budget-weghted) π-tle of traders belefs, consstent wth the observatons n [4, 5] for ths specal case. 5 Extenson to nonlnear utltes So far we consdered rsk-neutral traders,.e., the utlty equals expected proft. Our observatons can be extended to nonlnear utltes, for example, when traders are rsk averse,.e., the utltes are concave functons of money (see [1] for detals). Here s a sketch of the maor deas nvolved. Note that the proof of Proposton 3 holds wthout the assumpton of lnear utlty functons. Therefore, we can reduce our problem to consderng equlbrum n the parmutuel bettng markets. The latter are equvalent to Fsher market wth nonlnear utltes. Thus, results on exstence and unqueness of equlbrum for those markets can be appled. References [1] S. Agrawal, N. Megddo, and B. Armbruster, Equlbrum n Predcton Markets wth Buyers and Sellers, Research Report RJ10453, IBM Almaden Research Center, [2] J. E. Berg, F. D. Nelson, and T. A. Retz, Predcton market accuracy n the long run, Internat. J. Forecastng 24 (2008) [3] E. Esenberg and D. Gale, Consensus of Subectve Probabltes: The Par-Mutuel Method, Annals Math. Stat. 30 (1959) [4] C. F. Mansk, Interpretng the Predctons of Predcton Markets, Economcs Letters 91 (2006) [5] J. Wolfers and E. Ztzewtz, Predcton Markets n Theory and Practce, n: The New Palgrave Dctonary of Economcs, 2nd Edton, S. N. Durlauf and L. E. Blume, Eds. (2008). 6
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