{"title": "Interpreting prediction markets: a stochastic approach", "book": "Advances in Neural Information Processing Systems", "page_first": 3266, "page_last": 3274, "abstract": "We strengthen recent connections between prediction markets and learning by showing that a natural class of market makers can be understood as performing stochastic mirror descent when trader demands are sequentially drawn from a fixed distribution. This provides new insights into how market prices (and price paths) may be interpreted as a summary of the market's belief distribution by relating them to the optimization problem being solved. In particular, we show that the stationary point of the stochastic process of prices generated by the market is equal to the market's Walrasian equilibrium of classic market analysis. Together, these results suggest how traditional market making mechanisms might be replaced with general purpose learning algorithms while still retaining guarantees about their behaviour.", "full_text": "Interpreting prediction markets: a stochastic\n\napproach\n\nRafael M. Frongillo\n\nComputer Science Divison\n\nUniversity of California, Berkeley\n\nraf@cs.berkeley.edu\n\nNicol\u00b4as Della Penna\n\nResearch School of Computer Science\nThe Australian National University\n\nme@nikete.com\n\nMark D. Reid\n\nResearch School of Computer Science\n\nThe Australian National University & NICTA\n\nmark.reid@anu.edu.au\n\nAbstract\n\nWe strengthen recent connections between prediction markets and learn-\ning by showing that a natural class of market makers can be understood\nas performing stochastic mirror descent when trader demands are sequen-\ntially drawn from a \ufb01xed distribution. This provides new insights into how\nmarket prices (and price paths) may be interpreted as a summary of the\nmarket\u2019s belief distribution by relating them to the optimization problem\nbeing solved. In particular, we show that under certain conditions the sta-\ntionary point of the stochastic process of prices generated by the market\nis equal to the market\u2019s Walrasian equilibrium of classic market analysis.\nTogether, these results suggest how traditional market making mechanisms\nmight be replaced with general purpose learning algorithms while still re-\ntaining guarantees about their behaviour.\n\n1 Introduction and literature review\n\nThis paper is part of an ongoing line of research, spanning several authors, into formal\nconnections between markets and machine learning. In [5] an equivalence is shown between\nthe theoretically popular prediction market makers based on sequences of proper scoring\nrules and follow the regularised leader, a form of no-regret online learning. By modelling\nthe traders that demand the assets the market maker is o\ufb00ering we are able to extend\nthe equivalence to stochastic mirror decent. The dynamics of wealth transfer is studied\nin [3], for a sequence of markets between agents that behave as Kelly bettors (i.e. have log\nutilities), and an equivalence to stochastic gradient decent is analysed. More broadly, [9, 2]\nhave analysed how a wide range of machine learning models can be implemented in terms\nof market equilibria.\n\nThe literature on the interpretation of prediction market prices [7, 11] has had the goal of\nrelating the equilibrium prices to the distribution of the beliefs of traders. More recent work\n[8] has looked at a stochastic model, and studied the behavior of simple agents sequentially\ninteracting with the market. We continue this latter path of research, motivated by the\nobservation that the equilibrium price may be a poor predictor of the behavior in a volitile\nprediction market. As such, we seek a more detailed understanding of the market than the\nequilibrium point \u2013 we would like to know what the \u201cstationary distribution\u201d of the price\nis, as time goes to in\ufb01nity.\n\n1\n\n\fAs is standard in the literature, we assume a \ufb01xed (product) distribution over traders\u2019 beliefs\nand wealth. Our model features an automated market maker, following the framework of [1]\nis becoming a standard framework in the \ufb01eld.\n\nWe obtain two results. First, we prove that under certain conditions the stationary point\nof our stochastic process de\ufb01ned by the market maker and a belief distribution of traders\nconverges to the Walrasian equilibrium of the market as the market liquidity increases. This\nresult, stated in Theorem 1, is general in the sense that only technical convergence conditions\nare placed on the demand functions of the traders \u2013 as such, we believe it is a generalisation\nof the stochastic result of [8] to cases where agents are are not limited to linear demands,\nand leave this precise connection to future work.\n\nSecond, we show in Corollary 1 that when traders are Kelly bettors, the resulting stochastic\nmarket process is equivalent to stochastic mirror descent; see e.g.\n[6]. This result adds to\nthe growing literature which relates prediction markets, and automated market makers in\ngeneral, to online learning; see e.g. [1], [5], [3] .\n\nThis connection to mirror descent seems to suggest that the prices in a prediction market\nat any given time may be meaningless, as the \ufb01nal point in stochastic mirror descent often\nhas poor convergence guarantees. However, standard results suggest that a prudent way\nto form a \u201cconsensus estimate\u201d from a prediction market is to average the prices. The\naverage price, assuming our market model is reasonable, is provably close to the stationary\nprice.\nIn Section 5 we give a natural example that exhibits this behavior. Beyond this,\nhowever, Theorem 2 gives us insight into the relationship between the market liquidity and\n\u221a\nthe convergence of prices; in particular it suggests that we should increase liquidity at a rate\nof\n\nt if we wish the price to settle down at the right rate.\n\n2 Model\n\nOur market model will follow the automated market maker framework of [1]. We will equip\nour market maker with a strictly convex function C : Rn \u2192 R which is twice continuously\ndi\ufb00erentiable. For brevity we will write \u03d5 := \u2207C. The outcome space is \u2126, and the contracts\nare determined by a payo\ufb00 function \u03c6 : \u2126 \u2192 Rn such that \u03a0 := \u03d5(Rn) = ConvHull(\u03c6(\u2126)).\nThat is, the derivative space \u03a0 of C (the \u201cinstantaneous prices\u201d) must be the convex hull\nof the payo\ufb00s.\nA trader purchasing shares at the current prices \u03c0 \u2208 Rn pays C(\u03d5\u22121(\u03c0) + r) \u2212 C(\u03d5\u22121(\u03c0))\nfor the bundle of contracts r \u2208 Rn. Note that our dependence solely on \u03c0 limits our model\nslightly, since in general the share space (domain of C) may contain more information than\nthe current prices (cf. [1]). The bundle r is determined by an agent\u2019s demand function\nd(C, \u03c0) which speci\ufb01es the bundle to buy given the price \u03c0 and the cost function C.\n\nOur market dynamics are the following. The market maker posts the current price \u03c0t, and\nat each time t = 1 . . . T , a trader is chosen with demand function d drawn i.i.d. from some\ndemand distribution D. Intuitively, these demands are parameterized by latent variables\nsuch as the agent\u2019s belief p \u2208 \u2206\u2126 and total wealth W . The price is then updated to\n\n(1)\nAfter update T , the outcome is revealed and payout \u03c6(\u03c9)i is given for each contract i \u2208\n{1, . . . , n}.\n\n\u03c0t+1 = \u03d5(\u03d5\u22121(\u03c0t) + d(C, \u03c0t)).\n\n3 Stationarity and equilibrium\n\nWe \ufb01rst would like to relate our stochastic model (1) to the standard notion of market\nequilibrium from the Economics literature, which we call the Walrasian equilibrium to avoid\nconfusion. Here prices are \ufb01xed, and the equilibrium price is one that clears the market,\nmeaning that the sum of the demands r is 0 \u2208 Rn. In fact, we will show that the stationary\npoint of our process approaches the Walrasian equilibrium point as the liquidity of the\nmarket approaches in\ufb01nity.\n\n2\n\n\ffunction C(s) = b ln(cid:80)\n\ni esi/b), we de\ufb01ne\n\nFirst, we must add a liquidity parameter to our market. Following the LMSR (the cost\n\nCb(s) := b C(s/b).\n\n(2)\n\nThis transformation of a convex function is called a perspective function and is known to\npreserve convexity [4]. Observe that \u03d5b(s) := \u2207Cb(s) = \u2207C(s/b) = \u03d5(s/b), meaning that\nthe price under Cb at s is the same as the price under C at s/b. As with the LMSR, we\ncall b the liquidity parameter ; this terminology is justi\ufb01ed by noting that one de\ufb01nition of\nliquidity, 1/\u03bbmax\u22072Cb(s) = b/\u03bbmax\u22072C(s/b) (cf. [1]). In the following, we will consider the\nlimit as b \u2192 \u221e.\nSecond, in order to connect to the Walrasian equilibrium, we need a notion of a \ufb01xed-price\nif a trader has demand d(C,\u00b7) given C, what would the same trader\u2019s\ndemand function:\ndemand be under a market where prices are \ufb01xed and do not \u201cchange\u201d during a trade? For\nthe sake of generality, we restrict our allowable demand functions to the ones for which the\nlimit\n\n(3)\nexists; this demand d(F,\u00b7) will be the corresponding \ufb01xed-price demand for d. We now de\ufb01ne\nthe Walrasion equilibrium point \u03c0\u2217, which is simply the price at which the market clears\nwhen traders have demands distributed by D. Formally, this is the following condition:1\n\nb\u2192\u221e d(Cb, \u03c0)\n\nd(F, \u03c0) := lim\n\nd(F, \u03c0\u2217) dD(d) = 0\n\n(4)\n\n(cid:90)\n\nD\n\nNote that 0 \u2208 Rn; the demand for each contract should be balanced.\nThe stationary point of our stochastic process, on the other hand, is the price \u03c0s\nthe expected price \ufb02uctuation is 0. Formally, we have\n\nb for which\n\nE\nd\u223cD[\u2206(\u03c0s\n\nb , d(Cb, \u03c0s\n\nb ))] = 0,\n\n(5)\nwhere \u2206(\u03c0, d) := \u03d5(\u03d5\u22121(\u03c0) + d) \u2212 \u03c0 is the price \ufb02uctuation. We now consider the limit of\nour stochastic process as the market liquidity approaches \u221e.\nTheorem 1. Let C be a strictly convex and \u03b1-smooth2 cost function, and assume that\n\u2202b d(Cb, \u03c0) = o(1/b) uniformly in \u03c0 and all d \u2208 D. If furthermore the limit (3) is uniform\n\u2202\nin \u03c0 and d, then limb\u2192\u221e \u03c0s\nProof. Note that by the stationarity condition (5) we may de\ufb01ne \u03c0\u2217 and \u03c0s\nof the following \u201cexcess demand\u201d functions, respectively:\n\nb to be the roots\n\nb = \u03c0\u2217.\n\n(cid:90)\n\nD\n\nZ(\u03c0) :=\n\nd(F, \u03c0) dD(d),\n\nb (\u03c0) := b E\nZ s\n\nd\u223cD[\u2206(\u03c0, d(Cb, \u03c0))],\nb does not limit to the zero function.\n\nwhere we scale the latter by b so that Z s\nLet s = \u03d5\u22121(\u03c0) be the current share vector. Then we have\n\nlim\nb\u2192\u221e b\u2206(\u03c0, d(Cb, \u03c0)) = lim\n\nb\u2192\u221e b(cid:0)\u03d5(cid:0)\u03d5\u22121(\u03c0) + d(Cb, \u03c0)/b(cid:1) \u2212 \u03c0(cid:1)\n\u03d5(cid:0)s + a d(C1/a, \u03c0)(cid:1) \u2212 \u03c0\n\u2207\u03d5(cid:0)s + a d(C1/a, \u03c0)(cid:1)(cid:0)d(C1/a, \u03c0) + a \u2202\nb d(Cb, \u03c0)(cid:1)(cid:0)d(Cb, \u03c0) + 1\nb\u2192\u221e\u2207\u03d5(cid:0)s + 1\nb\u2192\u221e\u22072C(s) d(Cb, \u03c0) = \u22072C(s) d(F, \u03c0),\n\na\n\nb\n\n= lim\na\u21920\n= lim\na\u21920\n= lim\n\n= lim\n\n\u2202a d(C1/a, \u03c0)(cid:1)\n\u2202b d(Cb, \u03c0)(\u2212b2)(cid:1)\n\n\u2202\n\nclass of demands for which uniqueness is satis\ufb01ed.\n\n1Here and throughout we ignore technical issues of uniqueness. One may simply restrict to the\n2C is \u03b1-smooth if \u03bbmax\u22072C \u2264 \u03b1\n\n3\n\n\fwhere we apply L\u2019Hopital\u2019s rule for the third equality. Crucially, the above limit is uniform\nwith respect to both d \u2208 D and \u03c0 \u2208 \u03a0; uniformity in d is by assumption, and uniformity in\n\u03c0 follows from \u03b1-smoothness of C, since C is dominated by a quadratic. Since the limit is\nuniform with respect to D, we now have\n\n(cid:21)\n\nb\u2192\u221e Z s\nlim\n\nb (\u03c0) = lim\n\nlim\nb\u2192\u221e b\u2206(\u03c0, d(Cb, \u03c0))\n\n(cid:20)\nd\u223cD[\u2206(\u03c0, d(Cb, \u03c0))] = E\nd\u223cD\nd\u223cD[d(F, \u03c0)] = \u22072C(s) Z(\u03c0).\n\nb\u2192\u221e b E\n= \u22072C(s) E\n\nAs \u22072C(s) is positive de\ufb01nite by assumption on C, we can conclude that limb\u2192\u221e Z s\nb and\nZ share the same zeroes. Since Z has compact domain and is assumed continuous with a\nunique zero \u03c0\u2217, for each \u0001 \u2208 (0, \u0001max) there must be some \u03b4 > 0 s.t. |Z(\u03c0)| > \u0001 for all \u03c0 s.t.\n(cid:107)\u03c0 \u2212 \u03c0\u2217(cid:107) > \u03b4 (otherwise there would be a sequence of \u03c0n \u2192 \u03c0(cid:48) s.t. f (\u03c0(cid:48)) = 0 but \u03c0(cid:48) (cid:54)= \u03c0\u2217).\nBy uniform convergence there must be a B > 0 s.t. for all b > B we have (cid:107)Z s\nb \u2212 Z(cid:107)\u221e < \u0001/2.\nIn particular, for \u03c0 s.t. (cid:107)\u03c0 \u2212 \u03c0\u2217(cid:107) > \u03b4, |Z s\nb must\nbe within \u03b4 of \u03c0\u2217. Hence limb\u2192\u221e \u03c0s\n\nb (\u03c0)| > \u0001/2. Thus, the corresponding zeros \u03c0s\n\nb = \u03c0\u2217.3\n\n3.1 Utility-based demands\n\nMaximum Expected Utility (MEU) demand functions are a particular kind of demand func-\ntion derived by assuming a trader has some belief p \u2208 \u2206n over the outcomes in \u2126, some\nwealth W \u2265 0, and a monotonically increasing utility function of money u : R \u2192 R. If such\na trader buys a bundle r of contracts from a market maker with cost function C and price \u03c0,\nher wealth after \u03c9 occurs is \u03a5\u03c9(C, W, \u03c0, r) := W +\u03c6(\u03c9)\u00b7r\u2212[C(\u03d5\u22121(\u03c0)+r)\u2212C(\u03d5\u22121(\u03c0))]. We\nensure traders do not go into debt by requiring that traders only make demands such that\nthis \ufb01nal wealth is nonnegative: \u2200\u03c9 \u03a5\u03c9(C, \u03c0, r) \u2265 0. The set of debt-free bundles for wealth\nW and market C at price \u03c0 is denoted S(C, W, \u03c0) := {r \u2208 Rn : min\u03c9 \u03a5\u03c9(C, W, \u03c0, r) \u2265 0}.\nA continuous MEU demand function du\nW,p(C, \u03c0) is then just the demand that maximizes a\ntrader\u2019s expected utility subject to the debt-free constraint. That is,\n\ndu\nW,p(C, \u03c0) := argmax\nr\u2208S(C,W,\u03c0)\n\nE\n\u03c9\u223cp\n\n[u (\u03a5\u03c9(C, W, \u03c0, r))] .\n\n(6)\n\nWe also de\ufb01ne a \ufb01xed-price MEU demand function du\nsimilarly, where\n\u03a5\u03c9(F, W, \u03c0, r) := W + \u03c6(\u03c9)\u00b7 r\u2212 \u03c0 \u00b7 r and S(F, W, \u03c0) := {r \u2208 Rn : min\u03c9 \u03a5\u03c9(F, W, \u03c0, r) \u2265 0}\nare the \ufb01xed price analogues to the continuously priced versions above. Using the notation\nbS := {b r | r \u2208 S}, the following relationships between the continuous and \ufb01xed price ver-\nsions of \u03a5, SW , and the expected utility are a consequence of the convexity of C. Their main\npurpose is to highlight the relationship between wealth and liquidity in MEU demands. In\nparticular, they show that scaling up of liquidity is equivalent to a scaling down of wealth\nand that the continuously priced constraints and wealth functions monotonically approach\nthe \ufb01xed priced versions.\n\nW,p(F, \u03c0)\n\nLemma 1. For any strictly convex cost function C, wealth W > 0, price \u03c0, demand\nr, and liquidity parameter b > 0 the following properties hold:\n1. \u03a5\u03c9(Cb, W, \u03c0, r) =\n3. S(C, W, \u03c0) is convex for all\nb \u03a5\u03c9(C, W/b, \u03c0, r/b);\n2. S(Cb, W, \u03c0) = b S(C, W/b, \u03c0);\nC;\n5. For monotone utilities\nu, E\u03c9\u223cp [u (\u03a5\u03c9(F, W, \u03c0, r))] \u2265 E\u03c9\u223cp [u (\u03a5\u03c9(C, W, \u03c0, r))].\n\n4. S(C, W, \u03c0) \u2286 S(Cb, W, \u03c0) \u2286 S(F, W, \u03c0) for all b \u2265 1.\n\nProof. Property (1) follows from a simple computation:\n\n\u03a5\u03c9(Cb, W, \u03c0, r) = W + \u03c6(\u03c9) \u00b7 r \u2212 b C(\u03d5\u22121(\u03c0) + r/b) + b C(\u03d5\u22121(\u03c0))\n\n= b(cid:0)W/b + \u03c6(\u03c9) \u00b7 (r/b) \u2212 C(\u03d5\u22121(\u03c0) + r/b) + C(\u03d5\u22121(\u03c0))(cid:1) ,\n\nwhich equals b \u03a5\u03c9(C, W/b, \u03c0, r/b) by de\ufb01nition. We now can see property (2) as well:\n\nS(Cb, W, \u03c0) = {r : min\n\n\u03c9\n\nb \u03a5\u03c9(C, W/b, \u03c0, r/b) \u2265 0} = {b r : min\n\n\u03c9\n\n\u03a5\u03c9(C, W/b, \u03c0, r) \u2265 0}.\n\n3We thank Avraham Ruderman for a helpful discussion regarding this proof.\n\n4\n\n\fconvexity we have for f := fC,s,\u03c9 we have f (r/b) = f(cid:0) 1\n\nFor (3), de\ufb01ne fC,s,\u03c9(r) = C(s + r)\u2212 C(s)\u2212 \u03c6(\u03c9)\u00b7 r, which is the ex-post cost of purchasing\nbundle r. As C is convex, and fC,s,\u03c9 is a shifted and translated version of C plus a linear\nterm, fC,s,\u03c9 is convex also. The constraint \u03a5\u03c9(C, W, \u03c0, r) \u2265 0 then translates to fC,s,\u03c9(r) \u2264\nW , and thus the set of r which satisfy the constraint is convex as a sublevel set of a convex\nfunction. Now S(C, W, \u03c0) is convex as an intersection of convex sets, proving (3).\nFor (4) suppose r satis\ufb01es fC,s,\u03c9(r) \u2264 W . Note that fC,s,\u03c9(0) = 0 always. Then by\nb 0 \u2264 W/b,\nwhich implies S(C, W, \u03c0) \u2286 S(Cb, W, \u03c0) when considering (3). To complete (4) note that\nfC,s,\u03c9 dominates fF,s : r (cid:55)\u2192 (\u03d5(s)\u2212 \u03c6(\u03c9))\u00b7 r by convexity of C: C(s + r)\u2212 C(s) \u2265 \u2207C(s)\u00b7 r.\nFinally, proof of (5) is obtained by noting that the convexity of C means that C(\u03d5\u22121(\u03c0) +\nr) \u2212 C(\u03d5\u22121(\u03c0)) \u2265 \u2207C(\u03d5\u22121(\u03c0)) \u00b7 r = \u03c0 \u00b7 r and exploting the monotonicty of u.\n\nb 0(cid:1) \u2264 1\n\nb f (r) + b\u22121\n\nb r + b\u22121\n\nLemma 1 shows us that MEU demands have a lot of structure, and in particular, properties\n(4) and (5) suggest that they may satisfy the conditions of Theorem 1; we leave this as an\nopen question for future work. Another interesting aspect of Lemma 1 is the relationship\nbetween markets with cost function Cb and wealths W and markets with cost function C\nand wealths W/b \u2013 indeed, properties (1) and (2) suggest that the liquidity limit should\nin some sense be equivalent to a wealth limit, in that increasing liquidity by a factor b\nshould yield similar dynamics to decreasing the wealths by b. This would relate our model\nto that of [8], where the authors essentially show a wealth-limit version of Theorem 1 for a\nbinary-outcome market where traders have linear utilities (a special case of (6)). We leave\nthis precise connection for future work.\n\n4 Market making as mirror descent\n\nWe now explore the surprising relationship between our stochastic price update and standard\nstochastic optimization techniques. In particular, we will relate our model to a stochastic\nmirror descent of the form\n\n{\u03b7 x \u00b7 \u2207F (xt; \u03be) + DR(x, xt)},\n\nx\u2208R\n\nxt+1 = argmin\n\n(7)\nwhere at each step \u03be \u223c \u039e are i.i.d. and R is some strictly convex function. We will refer to\nan algorithm of the form (7) a stochastic mirror descent of f (x) := E\u03be\u223c\u039e[F (x; \u03be)].\nTheorem 2. If for all d \u2208 D we have some F (\u00b7 ; d) : Rn \u2192 Rn such that d(R\u2217, \u03c0) =\n\u2212\u2207F (\u03c0; d), then the stochastic update of our model (1) is exactly a stochastic mirror descent\nof f (\u03c0) = Ed\u223cD[F (\u03c0; d)].\n\nProof. By standard arguments, the mirror descent update (7) can be rewritten as\n\nxt+1 = \u2207R\u2217(\u2207R(xt) \u2212 \u2207F (xt; \u03be)),\n\nwhere R\u2217 is the conjugate dual of R. Take R = C\u2217, and let \u03be = d \u223c D. By assumption,\nwe have \u2207F (x; d) = \u2212d(R\u2217, x) = \u2212d(C, x) for all d. As \u2207R\u2217 = \u2207C = \u03d5, we have \u03d5\u22121 =\n\n(\u2207R\u2217)\u22121 = \u2207R by duality, and thus our update becomes xt+1 = \u03d5(cid:0)\u03d5\u22121(xt) + d(C, xt)(cid:1),\n\nwhich exactly matches the stochastic update of our model (1).\n\n(cid:18)\n\nAs an example, consider Kelly betters, which correspond to \ufb01xed-price demands d(C, \u03c0) :=\ndlog\nW,p(F, \u03c0) with utility u(x) = log x as de\ufb01ned in (3). A simple calculation shows that our\nupdate becomes\n\np \u2212 \u03c0\n1 \u2212 \u03c0\nwhere W and p are drawn (independently) from P and W.\nCorollary 1. The stochastic update for \ufb01xed-price Kelly betters (8) is exactly a stochas-\ntic mirror descent of f (\u03c0) = W \u00b7 KL(p, \u03c0), where p and W are the means of P and W,\nrespectively.\n\n\u03d5\u22121(\u03c0t) +\n\n\u03c0t+1 = \u03d5\n\nW\n\u03c0\n\n(cid:19)\n\n,\n\n(8)\n\n5\n\n\fProof. We take F (x; dlog\n\nW,p) = W \u00b7 (KL(p, x) + H(p)). Then\n\n\u2207F (x; dlog\n\nW,p) = W\n\np \u2212 1\n1 \u2212 x\n\n+\n\n= \u2212 W\nx\n\np \u2212 x\n1 \u2212 x\n\n= \u2212dlog\n\nW,p(F, x).\n\n(cid:18)\u2212p\n\nx\n\n(cid:19)\n\nHence, by Theorem 2 our update is a stochastic mirror descent of:\n\nf (x) := E[F (x; dlog\n\nW,p)] = E[W p log x + W (1 \u2212 p) log(1 \u2212 x)] = W \u00b7 (KL(p, x) + H(p)) ,\nwhich of course is equivalent to W \u00b7 KL(p, x) as the entropy term does not depend on x.\n\nNote that while this last result is quite compelling, we have mixed \ufb01xed-price demands with\na continuous-price market model \u2013 see Section 3.1. One could interpret this combination as\na model in which the market maker can only adjust the prices after a trade, according to a\n\ufb01xed convex cost function C. This of course di\ufb00ers from the standard model, which adjusts\nthe price continuously during a trade.\n\n4.1 Leveraging existing learning results\n\nTheorem 2 not only identi\ufb01es a fascinating connection between machine learning and our\nstochastic prediction market model, but it also allows us to use powerful existing techniques\nto make broad conclusions about the behavior of our model. Consider the following result:\nProposition 1 ([6]). If (cid:107)\u2207F (\u03c0; p)(cid:107)2 \u2264 G2 for all p, \u03c0, and R is \u03c3-strongly convex, then\nwith probability 1 \u2212 \u03b4,\n\n(cid:18) D2\n\n\u03b7T\n\n(cid:19)(cid:32)\n\n(cid:114)\n\n(cid:33)\n\n1\n\u03b4\n\nf (\u03c0T ) \u2264 min\n\n\u03c0\n\nf (\u03c0) +\n\n+\n\nG2\u03b7\n2\u03c3\n\n1 + 4\n\nlog\n\n.\n\nIn our context, Proposition 1 says that the average\nof the prices will be a very good estimate of the min-\nimizer of f , which as suggested by happens to be the\nunderlying mean belief p of the traders! Moreover, as\nthe Kelly demands are linear in both p and W , it is\neasy to see from Theorem 1 that p is also the station-\nary point and the Walrasian equilibrium point (the\nlatter was also shown by [11]). On the other hand, as\nwe demonstrate next, it is not hard to come up with\nan example where the instantaneous price \u03c0t is quite\nfar from the equilibrium at any given time period.\n\nBefore moving to our empirical work, we make one\n\ufb01nal point. The above relationship between our\nstochastic market model and mirror descent sheds\nlight on an important question: how might an auto-\nmated market maker adjust the liquidity so that the\nmarket actually converges to the mean of the traders\u2019 beliefs? The learning parameter \u03b7\ncan be thought of as the inverse of the liquidity, and as such, Proposition 1 suggests that\nt may cause the mean price to converge to the mean belief\nincreasing the liquidity as\n(assuming a \ufb01xed underlying belief distribution).\n\nFigure 1: Price movement for Kelly\nbetters with binomial(q = 0.6, n = 6,\n\u03b1 = 0.5) beliefs in the LMSR market\nwith liquidity b = 10.\n\n\u221a\n\n5 Empirical work\n\nExample: biased coin Consider a classic Bayesian setting where a coin has unknown\nbias Pr[heads] = q, and traders have a prior \u03b2(\u03b1, \u03b1) over q (i.e., traders are \u03b1-con\ufb01dent that\nthe coin is fair). Now suppose each trader independently observes n \ufb02ips from the coin, and\nupdates her belief; upon seeing k heads, a trader would have posterior \u03b2(\u03b1 + k, \u03b1 + n \u2212 k).\nWhen presented with a prediction market with contracts for a single toss of the coin, where\nand contract 0 pays $1 for tails and contract 1 pays $1 for heads, a trader would purchase\n\n6\n\n05001000150020000.500.550.600.650.70Trade numberPrice of contract 1PriceAvg priceAvg belief\fFigure 2: Mean square loss of average and instantaneous prices relative to the mean belief\nof 0.26 over 20 simulations for State 9 for b = 1 (left), b = 3 (middle), and b = 10 (right).\nBars show standard deviation.\ncontracts as if according to the mean of their posterior. Hence, the belief distribution P of\n\n(cid:1)qk(1 \u2212 q)n\u2212k to belief p = (\u03b1 + k)/(2\u03b1 + n), yielding\n\nthe market assigns weight P(p) =(cid:0)n\n\na biased mean belief of (\u03b1 + nq)/(2\u03b1 + n).\n\nk\n\nWe show a typical simulation of this market in Figure 1, where traders behave as Kelly\nbetters in the \ufb01xed-price LMSR. Clearly, after almost every trade, the market price is\nquite far from the equilibrium/stationary point, and hence the classical supply and demand\nanalysis of this market yields a poor description of the actual behavior, and in particular, of\nthe predictive quality of the price at any given time. However, the mean price is consistently\nclose to the mean belief of the traders, which in turn is quite close to the true parameter q.\n\nElection Survey Data We now compare the quality of the running average price versus\nthe instantaneous price as a predictor of the mean belief of a market. We do so by simulating\na market maker interacting with traders with unit wealth, log utility, and beliefs drawn from\na \ufb01xed distribution. The belief distributions are derived from the Princeton election survey\ndata[10]. For each of the 50 US states, participants in the survey were asked to estimate\nthe probability that one of two possible candidates were going to win that state.4 We use\nthese 50 sets of estimates as 50 di\ufb00erent empirical distributions from which to draw trader\nbeliefs.\n\nmarket liquidity parameter b to de\ufb01ne the LMSR cost function C(s) = b ln(cid:80)\n\nA simulation is con\ufb01gured by choosing one of the 50 empirical belief distributions S, a\ni esi/b, and an\ninitial market position vector of (0, 0) \u2013 that is, no contracts for either outcome. A con\ufb01gured\nsimulation is run for T trades. At each trade, a belief p is drawn from S uniformly and\nwith replacement. This belief is used to determine the demand of the trader relative to the\ncurrent market pricing. The trader purchase a bundle of contracts according to its demand\nand the market moves its position and price accordingly. The complete price path \u03c0t for\nt = 1, . . . , T of the market is recorded as well as a running average price \u00af\u03c0t := 1\ni=1 \u03c0t for\nt\nt = 1 . . . , T . For each of the 50 empirical belief distributions we con\ufb01gured 9 markets with\nb \u2208 {1, 2, 3, 5, 10, 15, 20, 30, 50} and ran 20 independent simulations of T = 100 trades. We\npresent a portion of the results for the empirical distributions for states 9 and 11. States 9\nand 11 have, respectively, sample sizes of 2,717 and 2,709; means 0.26 and 0.9; and variances\n0.04 and 0.02. These are chosen as being representative of the rest of the simulation results:\nState 9 with mean o\ufb00-center and a spread of beliefs (high uncertainty) and State 11 with\nhighly concentrated beliefs around a single outcome (low uncertainty).\n\n(cid:80)t\n\nThe results are summarised in Figures 2, 3, and 4. The \ufb01rst show the square loss of the\naverage and instaneous prices relative to the mean belief for high uncertainty State 9 for\nb = 1, 3, 10. Clearly, the average price is a much more reliable estimator of the mean belief\nfor low liquidity (b = 1) and is only outperformed by the instaneous price for higher liquidity\n(b = 10), but then only early in trading. Similar plots for State 11 are shown in Figure 3\nwhere the advantage of using the average price is signi\ufb01cantly diminished.\n\n4The original dataset contains conjunctions of wins as well as conditional statements but we\n\nonly use the single variable results of the survey.\n\n7\n\n0204060801000.000.020.040.060.080.10Square loss of price to mean belief for State 9TradesLossb = 1InstantAveraged0204060801000.000.020.040.060.080.10Square loss of price to mean belief for State 9TradesLossb = 3InstantAveraged0204060801000.000.020.040.060.080.10Square loss of price to mean belief for State 9TradesLossb = 10InstantAveraged\fFigure 3: Mean square loss of average and instantaneous prices relative to the mean belief\nof 0.9 over 20 simulations for State 11 for b = 1 (left), b = 3 (middle), and b = 10 (right).\nBars show standard deviation.\n\nFigure 4 shows the improvement the average price has over the instantaneous price in square\nloss relative to the mean belief for all liquidity settings and highlights that average prices\nwork better in low liquidity settings, consistent with the theory. Similar trends were observed\nfor all the other States, depending on whether they had high uncertainty \u2013 in which case\naverage price was a much better estimator \u2013 or low uncertainty \u2013 in which case instanteous\nprice was better.\n\nFigure 4: An overview of the results for States 9 (left) and 11 (right). For each trade\nand choice of b, the vertical value shows the improvement of the average price over the\ninstantaneous price as measure by square loss relative to the mean.\n\n6 Conclusion and future work\n\nAs noted in Section 3.1, there are several open questions with regard to maximum expected\nutility demands and Theorem 1, as well as the relationship between trader wealth and market\nliquidity. It would also be interesting to have a application of Theorem 2 to a continuous-\nprice model, which yields a natural minimization as in Corollary 1. The equivalence to\nmirror decent stablished in Theorem 2 may also lead to a better understanding of the\noptimal manner in which a automated prediction market ought to increase liquidity so as\nto maximise e\ufb03ciency.\n\nAcknowledgments\n\nThis work was supported by the Australian Research Council (ARC). NICTA is funded\nby the Australian Government as represented by the Department of Broadband, Commu-\nnications and the Digital Economy and the ARC through the ICT Centre of Excellence\nprogram. The \ufb01rst author was partially supported by NSF grant CC-0964033 and by a\nGoogle University Research Award.\n\n8\n\n0204060801000.000.020.040.060.080.10Square loss of price to mean belief for State 11TradesLossb = 1InstantAveraged0204060801000.000.020.040.060.080.10Square loss of price to mean belief for State 11TradesLossb = 3InstantAveraged0204060801000.000.020.040.060.080.10Square loss of price to mean belief for State 11TradesLossb = 10InstantAveragedTrades20406080100b1020304050Loss Difference-0.020.000.020.040.06Improvement of Average over Instant Prices for State 9Trades20406080100b1020304050Loss Difference-0.08-0.06-0.04-0.020.000.02Improvement of Average over Instant Prices for State 11\fReferences\n\n[1] J. Abernethy, Y. Chen, and J.W. Vaughan. An optimization-based framework for\nautomated market-making. In Proceedings of the 11th ACM conference on Electronic\nCommerce (EC\u201911), 2011.\n\n[2] A. Barbu and N. Lay. 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International Foundation for Autonomous Agents\nand Multiagent Systems, 2010.\n\n[9] A. Storkey. Machine learning markets. AISTATS, 2012.\n\n[10] G. Wang, S.R. Kulkarni, H.V. Poor, and D.N. Osherson. Aggregating large sets of\nprobabilistic forecasts by weighted coherent adjustment. Decision Analysis, 8(2):128,\n2011.\n\n[11] J. Wolfers and E. Zitzewitz.\n\nInterpreting prediction market prices as probabilities.\n\nTechnical report, National Bureau of Economic Research, 2006.\n\n9\n\n\f", "award": [], "sourceid": 1510, "authors": [{"given_name": "Rafael", "family_name": "Frongillo", "institution": ""}, {"given_name": "Nichol\u00e1s", "family_name": "Della Penna", "institution": ""}, {"given_name": "Mark", "family_name": "Reid", "institution": ""}]}