{"title": "A Decomposition of Forecast Error in Prediction Markets", "book": "Advances in Neural Information Processing Systems", "page_first": 4371, "page_last": 4380, "abstract": "We analyze sources of error in prediction market forecasts in order to bound the difference between a security's price and the ground truth it estimates. We consider cost-function-based prediction markets in which an automated market maker adjusts security prices according to the history of trade. We decompose the forecasting error into three components: sampling error, arising because traders only possess noisy estimates of ground truth; market-maker bias, resulting from the use of a particular market maker (i.e., cost function) to facilitate trade; and convergence error, arising because, at any point in time, market prices may still be in flux. Our goal is to make explicit the tradeoffs between these error components, influenced by design decisions such as the functional form of the cost function and the amount of liquidity in the market. We consider a specific model in which traders have exponential utility and exponential-family beliefs representing noisy estimates of ground truth. In this setting, sampling error vanishes as the number of traders grows, but there is a tradeoff between the other two components. We provide both upper and lower bounds on market-maker bias and convergence error, and demonstrate via numerical simulations that these bounds are tight. Our results yield new insights into the question of how to set the market's liquidity parameter and into the forecasting benefits of enforcing coherent prices across securities.", "full_text": "A Decomposition of Forecast Error in\n\nPrediction Markets\n\nMiroslav Dud\u00edk\n\nMicrosoft Research, New York, NY\n\nmdudik@microsoft.com\n\nRyan Rogers\n\nUniversity of Pennsylvania, Philadelphia, PA\n\nrrogers386@gmail.com\n\nS\u00e9bastien Lahaie\n\nGoogle, New York, NY\nslahaie@google.com\n\nJennifer Wortman Vaughan\n\nMicrosoft Research, New York, NY\n\njenn@microsoft.com\n\nAbstract\n\nWe analyze sources of error in prediction market forecasts in order to bound\nthe difference between a security\u2019s price and the ground truth it estimates. We\nconsider cost-function-based prediction markets in which an automated market\nmaker adjusts security prices according to the history of trade. We decompose the\nforecasting error into three components: sampling error, arising because traders\nonly possess noisy estimates of ground truth; market-maker bias, resulting from\nthe use of a particular market maker (i.e., cost function) to facilitate trade; and\nconvergence error, arising because, at any point in time, market prices may still be\nin \ufb02ux. Our goal is to make explicit the tradeoffs between these error components,\nin\ufb02uenced by design decisions such as the functional form of the cost function\nand the amount of liquidity in the market. We consider a speci\ufb01c model in which\ntraders have exponential utility and exponential-family beliefs representing noisy\nestimates of ground truth. In this setting, sampling error vanishes as the number\nof traders grows, but there is a tradeoff between the other two components. We\nprovide both upper and lower bounds on market-maker bias and convergence error,\nand demonstrate via numerical simulations that these bounds are tight. Our results\nyield new insights into the question of how to set the market\u2019s liquidity parameter\nand into the forecasting bene\ufb01ts of enforcing coherent prices across securities.\n\n1\n\nIntroduction\n\nA prediction market is a marketplace in which participants can trade securities with payoffs that\ndepend on the outcomes of future events [19]. Consider the simple setting in which we are interested\nin predicting the outcome of a political election: whether the incumbent or challenger will win.\nA prediction market might issue a security that pays out $1 per share if the incumbent wins, and\n$0 otherwise. The market price p of this security should always lie between 0 and 1, and can be\nconstrued as an event probability. If a trader believes that the likelihood of the incumbent winning is\ngreater than p, she will buy shares with the expectation of making a pro\ufb01t. Market prices increase\nwhen there is more interest in buying and decrease when there is more interest in selling. By this\nprocess, the market aggregates traders\u2019 information into a consensus forecast, represented by the\nmarket price. With suf\ufb01cient activity, prediction markets are competitive with alternative forecasting\nmethods such as polls [4], but while there is a mature literature on sources of error and bias in polls,\nthe impact of prediction market structure on forecast accuracy is still an active area of research [17].\nWe consider prediction markets in which all trades occur through a centralized entity known as a\nmarket maker. Under this market structure, security prices are dictated by a \ufb01xed cost function and\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fthe current number of outstanding shares [6]. The basic conditions that a cost function should satisfy\nto correctly elicit beliefs, while bounding the market maker\u2019s loss, are now well-understood, chief\namong them being convexity [1]. Nonetheless, the class of allowable cost functions remains broad,\nand the literature so far provides little formal guidance on the speci\ufb01c form of cost function to use in\norder to achieve good forecast accuracy, including how to set the liquidity parameter which controls\nprice responsiveness to trade. In practice, the impact of the liquidity parameter is dif\ufb01cult to quantify\na priori, so implementations typically resort to calibrations based on market simulations [8, 18].\nPrior work also suggests that maintaining coherence among prices of logically related securities has\ninformational advantages [8], but there has been little work aimed at understanding why.\nThis paper provides a framework to quantify the impact of the choice of cost function on forecast\naccuracy. We introduce a decomposition of forecast error, in analogy with the bias-variance decom-\nposition familiar from statistics or the approximation-estimation-optimization decomposition for\nlarge-scale machine learning [5]. Our decomposition consists of three components. First, there is the\nsampling error resulting from the fact that the market consists of a \ufb01nite population of traders, each\nholding a noisy estimate of ground truth. Second, there is a market-maker bias which stems from the\nuse of a cost function to provide liquidity and induce trade. Third, there is convergence error due to\nthe fact that the market prices may not have fully converged to their equilibrium point.\nThe central contribution of this paper is a theoretical characterization of the market-maker bias and\nconvergence error, the two components of this decomposition that depend on market structure as\nde\ufb01ned by the form of the cost function and level of liquidity. We consider a tractable model of agent\nbehavior, originally studied by Abernethy et al. [2], in which traders have exponential utility functions\nand beliefs drawn from an exponential family. Under this model it is possible to characterize\nthe market\u2019s equilibrium prices in terms of the traders\u2019 belief and risk aversion parameters, and\nthereby quantify the discrepancy between current market prices and ground truth. To analyze market\nconvergence, we consider the trader dynamics introduced by Frongillo and Reid [9], under which\ntrading can be viewed as randomized block-coordinate descent on a suitable potential function.\nOur analysis is local in that the bounds depend on the market equilibrium prices. This allows us to\nexactly identify the main asymptotic terms of error. We demonstrate via numerical experiments that\nthese asymptotic bounds are accurate early on and therefore can be used to compare market designs.\nWe make the following speci\ufb01c contributions:\n1. We precisely de\ufb01ne the three components of the forecasting error.\n2. We show that the market-maker bias equals cb \u00b1 O(b2) as b \u2192 0, where b is the liquidity\n\nparameter, and c is an explicit constant that depends on the cost function and trader beliefs.\n\n3. We show that the convergence error decreases with the number of trades t as \u03b3t with \u03b3 = 1\u2212\u0398(b).\nWe provide explicit upper and lower bounds on \u03b3 that depend on the cost function and trader\nbeliefs. In the process, we prove a new local convergence bound for block-coordinate descent.\n4. We use our explicit formulas for bias and convergence error to compare two common cost\nfunctions: independent markets (IND), under which security prices vary independently, and\nthe logarithmic market scoring rule (LMSR) [10], which enforces logical relationships between\nsecurity prices. We show that at the same value of the market-maker bias, IND requires at least\nhalf-as-many and at most twice-as-many trades as LMSR to achieve the same convergence error.\nWe consider a speci\ufb01c utility model (exponential utility), but our bias and convergence analysis\nimmediately carry over if we assume that each trader is optimizing a risk measure (rather than an\nexponential utility function) similar to the setup of Frongillo and Reid [9]. Exponential utility was\nchosen because it was previously well studied and allowed us to focus on the analysis of the cost\nfunction and liquidity. The role of the liquidity parameter in trading off the bias and convergence error\nhas been informally recognized in the literature [7, 10, 13], but our precise de\ufb01nition of market-maker\nbias and explicit formulas for the bias and convergence error are novel. Abernethy et al. [2] provide\nresults that can be used to derive the bias for LMSR, but not for generic cost functions, so they do not\nenable comparison of biases of different costs. Frongillo and Reid [9] observe that the convergence\nerror can be locally bounded as \u03b3t, but they only provide an upper bound and do not show how \u03b3\nis related to the liquidity or cost function. Our analysis establishes both upper and lower bounds\non convergence and relates \u03b3 explicitly to the liquidity and cost function. This is necessary for a\n\n2\n\n\fmeaningful comparison of cost function families. Thus our framework provides the \ufb01rst meaningful\nway to compare the error tradeoffs inherent in different choices of cost functions and liquidity levels.\n\n2 Preliminaries\nWe use the notation [N ] to denote the set {1, . . . , N}. Given a convex function f : Rd \u2192 R \u222a {\u221e},\nits effective domain, denoted dom f, is the set of points where f is \ufb01nite. Whenever dom f is\nu \u2212 f (u)]. We\nnon-empty, the conjugate f\u2217 : Rd \u2192 R \u222a {\u221e} is de\ufb01ned by f\u2217(v) := supu\u2208Rd [v\nwrite (cid:107)\u00b7(cid:107) for the Euclidean norm. A centralized mathematical reference is provided in Appendix A.1\n\n(cid:124)\n\nCost-function-based market makers We study cost-function-based prediction markets [1]. Let\n\u2126 be a \ufb01nite set of mutually exclusive and exhaustive states of the world. A market administrator,\nknown as market maker, wishes to elicit information about the likelihood of various states \u03c9 \u2208 \u2126,\nand to that end offers to buy and sell any number of shares of K securities. Securities are associated\nwith coordinates of a payoff function \u03c6 : \u2126 \u2192 RK, where each share of the kth security is worth\n\u03c6k(\u03c9) in the event that the true state of the world is \u03c9 \u2208 \u2126. Traders arrive in the market sequentially\nand trade with the market maker. The market price is fully determined by a convex potential function\nC called the cost function. In particular, if the market maker has previously sold sk \u2208 R shares of\neach security k and a trader would like to purchase a bundle consisting of \u03b4k \u2208 R shares of each, the\ntrader is charged C(s + \u03b4\u03b4\u03b4) \u2212 C(s). The instantaneous price of security k is then \u2202C(s)/\u2202sk. Note\nthat negative values of \u03b4k are allowed and correspond to the trader (short) selling security k.\nLet M := conv{\u03c6(\u03c9) : \u03c9 \u2208 \u2126} be the convex hull of the set of payoff vectors. It is exactly the set\nof expectations E [\u03c6(\u03c9)] across all possible probability distributions over \u2126, which we call beliefs.\nWe refer to elements of M as coherent prices. Abernethy et al. [1] characterize the conditions that a\ncost function must satisfy in order to guarantee important properties such as bounded loss for the\nmarket maker and no possibility of arbitrage. To start, we assume only that C : RK \u2192 R is convex\nand differentiable and that M \u2286 dom C\u2217, which corresponds to the bounded loss property.\nExample 2.1 (Logarithmic Market Scoring Rule: LMSR [10]). Consider a complete market with a\nsingle security for each outcome worth $1 if that outcome occurs and $0 otherwise, i.e., \u2126 = [K] and\n\u03c6k(\u03c9) = 1{k = \u03c9} for all k. The LMSR cost function and instantaneous security prices are given by\n\n\u2202C(s)\n\n, \u2200k \u2208 [K].\n\n=\n\nand\n\nk=1 esk\n\nC(s) = log\n\nIts conjugate is the entropy function, C\u2217(\u00b5) = (cid:80)\n\n(1)\n\u2202sk\nk \u00b5k log \u00b5k + I{\u00b5 \u2208 \u2206K}, where \u2206K is the\nsimplex in RK and I{\u00b7} is the convex indicator, equal to zero if its argument is true and in\ufb01nity if\nfalse. Thus, in this case M = \u2206K = dom C\u2217.\nNotice that the LMSR security prices are coherent because they always sum to one. This prevents\narbitrage opportunities for traders. Our second running example does not have this property.\nExample 2.2 (Sum of Independent LMSRs: IND). Let \u2126 = [K] and \u03c6k(\u03c9) = 1{k = \u03c9} for all k.\nThe cost function and instantaneous security prices for the sum of independent LMSRs are given by\n\n(cid:96)=1 es(cid:96)\n\n(cid:16)(cid:80)K\n\n(cid:17)\n\nesk(cid:80)K\n\nC(s) =(cid:80)K\n\nesk\n\n\u2202C(s)\n\nC\u2217(\u00b5) =(cid:80)\n\n=\n\nand\n\nk=1 log (1 + esk )\n\n(2)\nk[\u00b5k log \u00b5k+(1\u2212\u00b5k) log(1\u2212\u00b5k)]+I{\u00b5 \u2208 [0, 1]K}, M = \u2206K, and dom C\u2217 = [0, 1]K.\nWhen choosing a cost function, one important consideration is liquidity, that is, how quickly prices\nchange in response to trades. Any cost function C can be viewed as a member of a parametric family\nof cost functions of the form Cb(s) := bC(s/b) across all b > 0. With larger values of b, larger trades\nare required to move market prices by some \ufb01xed amount, and the worst-case loss of the market\nmaker is larger; with smaller values, small purchases can result in big changes to the market price.\n\n1 + esk\n\n\u2202sk\n\n, \u2200k \u2208 [K],\n\nBasic model\nIn our analysis of error we assume that there exists an unknown true probability\ndistribution ptrue \u2208 \u2206|\u2126| over the outcome set \u2126. The true expected payoffs of the K market\nsecurities are then given by the vector \u00b5true := E\u03c9\u223cptrue [\u03c6(\u03c9)].\n\n1A longer version of this paper containing the appendix is available on arXiv and the authors\u2019 websites.\n\n3\n\n\fWe assume that there are N traders and that each trader i \u2208 [N ] has a private belief \u02dcpi over\noutcomes. We additionally assume that each trader i has a utility function ui : R \u2192 R for wealth\nand would like to maximize expected utility subject to her beliefs. For now we assume that ui\nis differentiable and concave, meaning that each trader is risk averse, though later we focus on\nexponential utility. The expected utility of trader i owning a security bundle ri \u2208 RK and cash ci is\nUi(ri, ci) := E\u03c9\u223c\u02dcpi\nis without loss of generality because we could incorporate any initial cash holdings into ui.\n\n(cid:1)(cid:3) . We assume that each trader begins with zero cash. This\n\n(cid:0)ci + \u03c6(\u03c9) \u00b7 ri\n\n(cid:2)ui\n\n3 A Decomposition of Error\n\nIn this section, we decompose the market\u2019s forecast error into three major components. The \ufb01rst is\nsampling error, which arises because traders have only noisy observations of the ground truth. The\nsecond is market-maker bias, which arises because the shape of the cost function impacts the traders\u2019\nwillingness to invest. Finally, convergence error arises due to the fact that at any particular point in\ntime the market prices may not have fully converged. To formalize our decomposition, we introduce\ntwo new notions of equilibrium.\nOur \ufb01rst notion of equilibrium, called a market-clearing equilibrium, does not assume the existence\nof a market maker, but rather assumes that traders trade only among themselves, and so no additional\nsecurities or cash are available beyond the traders\u2019 initial allocations. This equilibrium is described by\nsecurity prices \u00af\u00b5 \u2208 RK and allocations (\u00afri, \u00afci) of security bundles and cash to each trader i such that,\ngiven her allocation, no trader wants to buy or sell any bundle of securities at those prices. Trader\nbundles and cash are summarized as \u00afr = (\u00afri)i\u2208[N ] and \u00afc = (\u00afci)i\u2208[N ].\nDe\ufb01nition 3.1 (Market-clearing equilibrium). A triple (\u00afr, \u00afc, \u00af\u00b5) is a market-clearing equilibrium if\ni=1 \u00afci = 0, and for all i \u2208 [N ], 0 \u2208 argmax\u03b4\u2208RK Ui(\u00afri + \u03b4, \u00afci \u2212 \u03b4 \u00b7 \u00af\u00b5). We call\n\u00af\u00b5 market-clearing prices if there exist \u00afr and \u00afc such that (\u00afr, \u00afc, \u00af\u00b5) is a market-clearing equilibrium.\nSimilarly, we call \u00afr a market-clearing allocation if there exists a corresponding equilibrium.\n\n(cid:80)N\ni=1 \u00afri = 0,(cid:80)N\nThe requirements on(cid:80)N\n\ni=1 \u00afci guarantee that no additional securities or cash have\nbeen created. In other words, there exists some set of trades among traders that would lead to the\nmarket-clearing allocation, although the de\ufb01nition says nothing about how the equilibrium is reached.\nSince we rely on a market maker to orchestrate trade, our markets generally do not reach the market-\nclearing equilibrium. Instead, we introduce the notion of market-maker equilibrium. This equilibrium\nis again described by a set of security prices \u00b5(cid:63) and trader allocations (r(cid:63)\ni ), summarized as\n(r(cid:63), c(cid:63)), such that no trader wants to trade at these prices given her allocation. The difference is that\nwe now require r(cid:63) and c(cid:63) to be reachable via some sequence of trade with the market maker instead\nof via trade among only the traders, and \u00b5(cid:63) must be the market prices after such a sequence of trade.\nDe\ufb01nition 3.2 (Market-maker equilibrium). A triple (r(cid:63), c(cid:63), \u00b5(cid:63)) is a market-maker equilibrium\ni = Cb(0) \u2212 Cb(s(cid:63)),\n\u00b5(cid:63) = \u2207Cb(s(cid:63)), and for all i \u2208 [N ], 0 \u2208 argmax\u03b4\u2208RK Ui\ncall \u00b5(cid:63) market-maker equilibrium prices if there exist r(cid:63) and c(cid:63) such that (r(cid:63), c(cid:63), \u00b5(cid:63)) is a market-\nmaker equilibrium. Similarly, we call r(cid:63) a market-maker equilibrium allocation if there exists a\ncorresponding equilibrium. We sometimes write \u00b5(cid:63)(b; C) to show the dependence of \u00b5(cid:63) on C and b.\n\nfor cost function Cb if, for the market state s(cid:63) =(cid:80)N\n\ni \u2212 Cb(s(cid:63) + \u03b4) + Cb(s(cid:63))(cid:1). We\n\ni=1 r(cid:63)\n\ni , we have(cid:80)N\n(cid:0)r(cid:63)\n\ni + \u03b4, c(cid:63)\n\ni=1 c(cid:63)\n\ni , c(cid:63)\n\ni=1 \u00afri and(cid:80)N\n\nThe market-clearing prices \u00af\u00b5 and the market-maker equilibrium prices \u00b5(cid:63)(b; C) are not unique in\ngeneral, but are unique for the speci\ufb01c utility functions that we study in this paper.\nUsing these notions of equilibrium, we can formally de\ufb01ne our error components. Sampling error is\nthe difference between the true security values and the market-clearing equilibrium prices. The bias\nis the difference between the market-clearing equilibrium prices and the market-maker equilibrium\nprices. Finally, the convergence error is the difference between the market-maker equilibrium prices\nand the market prices \u00b5t(b; C) at a particular round t. Putting this together, we have that\n\n\u00b5true \u2212 \u00b5t(b; C) = \u00b5true \u2212 \u00af\u00b5\n\n+ \u00af\u00b5 \u2212 \u00b5(cid:63)(b; C)\n\n+ \u00b5(cid:63)(b; C) \u2212 \u00b5t(b; C)\n\n.\n\n(3)\n\n(cid:124)\n\n(cid:123)(cid:122)\n\n(cid:125)\n\nSampling Error\n\n(cid:123)(cid:122)\n\nBias\n\n(cid:125)\n\n(cid:124)\n\n(cid:123)(cid:122)\n\n(cid:125)\n\nConvergence Error\n\n(cid:124)\n\n4\n\n\f4 The Exponential Trader Model\n\nFor the remainder of the paper, we work with the exponential trader model introduced by Abernethy\net al. [2] in which traders have exponential utility functions and exponential-family beliefs. Under\nthis model, both the market-clearing prices and market-maker equilibrium prices are unique and can\nbe expressed cleanly in terms of potential functions [9], yielding a tractable analysis. The results of\nthis section are immediate consequences of prior work [2, 9], but our equilibrium concepts bring\nthem into a common framework.\nWe consider a speci\ufb01c exponential family [3] of probability distributions over \u2126 de\ufb01ned as p(\u03c9; \u03b8) =\ne\u03c6(\u03c9)\u00b7\u03b8\u2212T (\u03b8), where \u03b8 \u2208 RK is the natural parameter of the distribution, and T is the log partition\nunder p(\u00b7; \u03b8), and dom T \u2217 = conv{\u03c6(\u03c9) : \u03c9 \u2208 \u2126} = M.\nFollowing Abernethy et al. [2], we assume that each trader i has exponential-family beliefs with\nnatural parameter \u02dc\u03b8i. From the perspective of trader i, the expected payoffs of the K market securities\n\nfunction, T (\u03b8) := log(cid:0)(cid:80)\ncan then be expressed as the vector \u02dc\u00b5i with \u02dc\u00b5i,k :=(cid:80)\n\n\u03c9\u2208\u2126 e\u03c6(\u03c9)\u00b7\u03b8(cid:1). The gradient \u2207T (\u03b8) coincides with the expectation of \u03c6\n\n\u03c9\u2208\u2126 \u03c6k(\u03c9)p(\u03c9; \u02dc\u03b8i).\n\nAs in Abernethy et al. [2], we also assume that traders are risk averse with exponential utility for\nwealth, so the utility of trader i for wealth W is ui(W ) = \u2212(1/ai)e\u2212aiW , where ai is the the trader\u2019s\nrisk aversion coef\ufb01cient. We assume that the traders\u2019 risk aversion coef\ufb01cients are \ufb01xed.\nUsing the de\ufb01nitions of the expected utility Ui, the exponential family distribution p(\u00b7; \u02dc\u03b8i), the log\npartition function T , and the exponential utility ui, it is straightforward to show [2] that\n\nUi(ri, ci) = \u2212 1\nai\n\n\u03c9\u2208\u2126 e\u03c6(\u03c9)\u00b7( \u02dc\u03b8i\u2212airi) = \u2212 1\nai\n\neT ( \u02dc\u03b8i\u2212airi)\u2212T ( \u02dc\u03b8i)\u2212aici.\n\n(4)\n\ne\u2212T ( \u02dc\u03b8i)\u2212aici(cid:80)\n\nof traders\u2019 utilities is also locally maximized, as is(cid:80)N\nconditions require that(cid:80)N\nequilibrium must be a local minimum of(cid:80)N\n\nUnder this trader model, we can use the techniques of Frongillo and Reid [9] to construct potential\nfunctions which yield alternative characterizations of the equilibria as solutions of minimization\nT ( \u02dc\u03b8i + ais) for each\nproblems. Consider \ufb01rst a market-clearing equilibrium. De\ufb01ne Fi(s) := 1\ntrader i. From Eq. (4) we can observe that \u2212Fi(\u2212ri) + ci is a monotone transformation of trader i\u2019s\nai\nutility. Since each trader\u2019s utility is locally maximized at a market-clearing equilibrium, the sum\ni=1(\u2212Fi(\u2212ri) + ci). Since the equilibrium\ni=1 ci = 0, the security allocation associated with any market-clearing\ni=1 Fi(\u2212ri). This idea is formalized in the following\ntheorem. The proof follows from an analysis of the KKT conditions of the equilibrium. (See the\nappendix for all omitted proofs.)\nTheorem 4.1. Under the exponential trader model, a market-clearing equilibrium always exists and\nmarket-clearing prices are unique. Market-clearing allocations and prices are exactly the solutions\nof the following optimization problems:\n\n(cid:104)(cid:80)N\n\n(cid:105)\n\n\u00afr \u2208 argmin\n\nr:(cid:80)N\n\ni=1 ri=0\n\ni=1 Fi(\u2212ri)\n\n,\n\nrium is a local minimum of the function F (r) :=(cid:80)N\n\n(cid:104)(cid:80)N\n(cid:0)(cid:80)N\n\n(cid:105)\n\n(cid:1).\n(cid:105)\n\n\u00af\u00b5 = argmin\n\u00b5\u2208RK\n\ni=1 F \u2217\n\ni (\u00b5)\n\n.\n\n(5)\n\nUsing a similar argument, we can show that the allocation associated with any market-maker equilib-\n\ni=1 Fi(\u2212ri) + Cb\n\ni=1 ri\n\nTheorem 4.2. Under the exponential trader model, a market-maker equilibrium always exists and\nequilibrium prices are unique. Market-maker equilibrium allocations and prices are exactly the\nsolutions of the following optimization problems:\n\nr(cid:63) \u2208 argmin\n\nF (r) ,\n\nr\n\n\u00b5(cid:63) = argmin\n\u00b5\u2208RK\n\ni=1 F \u2217\n\ni (\u00b5) + bC\u2217(\u00b5)\n\n.\n\n(6)\n\n(cid:104)(cid:80)N\n\ncan be written as \u00af\u00b5 = E\u00af\u03b8 [\u03c6(\u03c9)], where \u00af\u03b8 := (cid:0)(cid:80)N\n\nSampling error We \ufb01nish this section with an analysis of the \ufb01rst component of error identi\ufb01ed in\nSection 3: the sampling error. We begin by deriving a more explicit form of market-clearing prices:\nTheorem 4.3. Under the exponential trader model, the unique market-clearing equilibrium prices\n\u02dc\u03b8i/ai\nweighted average belief and E\u00af\u03b8 is the expectation under p(\u00b7; \u00af\u03b8).\n\n(cid:1) is the risk-aversion-\n\n(cid:1)/(cid:0)(cid:80)N\n\ni=1 1/ai\n\ni=1\n\n5\n\n\fThe sampling error arises because the beliefs \u02dc\u03b8i are only noisy signals of the ground truth. From\nTheorem 4.3 we see that this error may be compounded by the weighting according to risk aversions,\nwhich can skew the prices. To obtain a concrete bound on the error term (cid:107)\u00b5true\u2212 \u00af\u00b5(cid:107), we need to make\nsome assumptions about risk aversion coef\ufb01cients, the true distribution of the outcome, and how this\ndistribution is related to trader beliefs. For instance, suppose risk aversion coef\ufb01cients are bounded\nboth from below and above, the true outcome is drawn from an exponential-family distribution with\nnatural parameter \u03b8true, and the beliefs \u02dc\u03b8i are independent samples with mean \u03b8true and a bounded\ncovariance matrix. Under these assumptions, one can show using standard concentration bounds\n\nthat with high probability, (cid:107)\u00b5true \u2212 \u00af\u00b5(cid:107) = O((cid:112)1/N ) as N \u2192 \u221e. In other words, market-clearing\n\nprices approach the ground truth as the number of traders increases. In Appendix B.4 we make\nthe dependence on risk aversion and belief noise more explicit. The analysis of other information\nstructures (e.g., biased or correlated beliefs) is beyond the scope of this paper; instead, we focus on\nthe two error components that depend on the market design.\n\n5 Market-maker Bias\n\ni are strongly convex on M (from properties of the log partition function).\n\nWe now analyze the market-maker bias\u2014the difference between the marker-maker equilibrium prices\n\u00b5(cid:63) and market-clearing prices \u00af\u00b5. We \ufb01rst state a global bound that depends on the liquidity b and cost\nfunction C, but not on trader beliefs, and show that \u00b5(cid:63) \u2192 \u00af\u00b5 with the rate O(b) as b \u2192 0. The proof\nbuilds on Theorems 4.1 and 4.2 and uses the facts that C\u2217 is bounded on M (by our assumptions on\nC), and conjugates F \u2217\nTheorem 5.1 (Global Bias Bound). Under the exponential trader model, for any C, there exists a\nconstant c such that (cid:107)\u00b5(cid:63)(b; C) \u2212 \u00af\u00b5(cid:107) \u2264 cb for all b \u2265 0.\nThis result makes use of strong convexity constants that are valid over the entire set M, which can\nbe overly conservative when \u00b5(cid:63) is close to \u00af\u00b5. Furthermore, it gives us only an upper bound, which\ncannot be used to compare different cost function families. In the rest of this section we pursue\na tighter local analysis, based on the properties of F \u2217\ni and C\u2217 at \u00af\u00b5. Our local analysis requires\nassumptions that go beyond convexity and differentiability of the cost function. We call the class of\nfunctions that satisfy these assumptions convex+ functions. (See Appendix A.3 for their complete\ntreatment and a more general de\ufb01nition than provided here.) These functions are related to functions\nof Legendre type (see Sec. 26 of Rockafellar [15]). Informally, they are smooth functions that are\nstrictly convex along directions in a certain space (the gradient space) and linear in orthogonal\ndirections. For cost functions, strict convexity means that prices change in response to arbitrarily\nsmall trades, while the linear directions correspond to bundles with constant payoffs, whose prices\nare therefore \ufb01xed.\nDe\ufb01nition 5.2. Let f : Rd \u2192 R be differentiable and convex. Its gradient space is the linear space\nparallel to the af\ufb01ne hull of its gradients, denoted as G(f ) := span{\u2207f (u) \u2212 \u2207f (u(cid:48)) : u, u(cid:48)\u2208 Rd}.\nDe\ufb01nition 5.3. We say that a convex function f : Rd \u2192 R is convex+ if it has continuous third\nderivatives and range(\u22072f (u)) = G(f ) for all u \u2208 Rd.\nIt can be checked that if P is a projection on G(f ) then there exists some a such that f (u) =\nu, so f is up to a linear term fully described by its values on G(f ). The condition on\n(cid:124)\nf (P u) + a\nthe range of the Hessian ensures that f is strictly convex over G(f ), so its gradient map is invertible\nover G(f ). This means that the Hessian can be expressed as a function of the gradient, i.e., there\nexists a matrix-valued function Hf such that \u22072f (u) = Hf (\u2207f (u)) (see Proposition A.8). The cost\nfunctions C for both the LMSR and the sum of independent LMSRs (IND) are convex+.\nExample 5.4 (LMSR as a convex+ function). For LMSR, the gradient space of C is parallel to\nthe simplex: G(C) = {u : 1\nu = 0}. The gradients of C are points in the relative interior of\nthe simplex. Given such a point \u00b5 = \u2207C(s), the corresponding Hessian is \u22072C(s) = HC(\u00b5) =\n(diagk\u2208[K] \u00b5k) \u2212 \u00b5\u00b5\n(cid:124), where diagk\u2208[K] \u00b5k denotes the diagonal matrix with values \u00b5k on the\ndiagonal. The null space of HC(\u00b5) is {c1 : c \u2208 R}, so C is linear in the all-ones direction (buying\none share of each security always has cost one), but strictly convex in directions from G(C).\nExample 5.5 (IND as a convex+ function). For IND, the gradient space is RK and the gradients are\nthe points in (0, 1)K. In this case, HC(\u00b5) = diagk[\u00b5k(1 \u2212 \u00b5k)]. This matrix has full rank.\nOur next theorem shows that for an appropriate vector u, which depends on \u00af\u00b5 and C, we have\n\u00b5(cid:63)(b; C) = \u00af\u00b5 + bu + \u03b5b, where (cid:107)\u03b5b(cid:107) = O(b2). Here, the O(\u00b7) is taken as b \u2192 0, so the error term\n\n(cid:124)\n\n6\n\n\f\u03b5b goes to zero faster than the term bu, which we call the asymptotic bias. Our analysis is local in\nthe sense that the constants hiding within O(\u00b7) may depend on \u00af\u00b5. This analysis fully uncovers the\nmain asymptotic term and therefore allows comparison of cost families. In our experiments, we show\nthat the asymptotic bias is an accurate estimate of the bias even for moderately large values of b.\nTheorem 5.6 (Local Bias Bound). Assume that the cost function C is convex+. Then\n\u00b5(cid:63)(b; C) = \u00af\u00b5 \u2212 b(\u00afa/N )HT ( \u00af\u00b5)\u2202C\u2217( \u00af\u00b5) + \u03b5b , where (cid:107)\u03b5b(cid:107) = O(b2).\n\nIn the expression above, \u00afa = N/((cid:80)N\n\ni=1 1/ai) is the harmonic mean of risk-aversion coef\ufb01cients and\n\nHT ( \u00af\u00b5)\u2202C\u2217( \u00af\u00b5) is guaranteed to consist of a single point even when \u2202C\u2217( \u00af\u00b5) is a set.\nThe theorem is proved by a careful application of Taylor\u2019s Theorem and crucially uses properties of\nconjugates of convex+ functions, which we derive in Appendix A.3. It gives us a formula to calculate\nthe asymptotic bias for any cost function for a particular value of \u00af\u00b5, or evaluate the worst-case bias\nagainst some set of possible market-clearing prices. It also constitutes an important step in comparing\ncost function families. To compare the convergence error of two costs C and C(cid:48) in the next section,\nwe require that their liquidities b and b(cid:48) be set so that they have (approximately) the same bias, i.e.,\n(cid:107)\u00b5(cid:63)(b(cid:48); C(cid:48)) \u2212 \u00af\u00b5(cid:107) \u2248 (cid:107)\u00b5(cid:63)(b; C) \u2212 \u00af\u00b5(cid:107). Theorem 5.6 tells us that this can be achieved by the linear\nrule b(cid:48) = b/\u03b7 where \u03b7 = (cid:107)HT ( \u00af\u00b5)\u2202C(cid:48)\u2217\n( \u00af\u00b5)(cid:107) /(cid:107)HT ( \u00af\u00b5)\u2202C\u2217( \u00af\u00b5)(cid:107). For C = LMSR and C(cid:48) = IND, we\nprove that the corresponding \u03b7 \u2208 [1, 2]. Equivalently, this means that for the same value of b the\nasymptotic bias of IND is at least as large as that of LMSR, but no more than twice as large:\nTheorem 5.7. For any \u00af\u00b5 there exists \u03b7 \u2208 [1, 2] such that for all b, (cid:107)\u00b5(cid:63)(b/\u03b7; IND) \u2212 \u00af\u00b5(cid:107) =\n(cid:107)\u00b5(cid:63)(b; LMSR)\u2212 \u00af\u00b5(cid:107)\u00b1O(b2). For this same \u03b7, also (cid:107)\u00b5(cid:63)(b; IND)\u2212 \u00af\u00b5(cid:107) = \u03b7(cid:107)\u00b5(cid:63)(b; LMSR)\u2212 \u00af\u00b5(cid:107)\u00b1O(b2).\nTheorem 5.6 also captures an intuitive relationship which can guide the market maker in adjusting the\nmarket liquidity b as the number of traders N and their risk aversion coef\ufb01cients ai vary. In particular,\nholding \u00af\u00b5 and the cost function \ufb01xed, we can maintain the same amount of bias by setting b \u221d N/\u00afa.\nNote that 1/ai plays the role of the budget of trader i in the sense that at \ufb01xed prices, the trader\ni(1/ai) corresponds to the total\namount of available cash among the traders in the market. Similarly, the market maker\u2019s worst-case\ni(1/ai) is natural.\n\nwill spend an amount of cash proportional to 1/ai. Thus N/\u00afa =(cid:80)\nloss, amounting to the market maker\u2019s cash, is proportional to b, so setting b \u221d(cid:80)\n\n6 Convergence Error\n\nWe now study the convergence error, namely the difference between the prices \u00b5t at round t and the\nmarket-maker equilibrium prices \u00b5(cid:63). To do so, we must posit a model of how the traders interact with\nthe market. Following Frongillo and Reid [9], we assume that in each round, a trader i \u2208 [N ], chosen\nuniformly at random, buys a bundle \u03b4 \u2208 RK that optimizes her utility given the current market state s\nand her existing security and cash allocations, ri and ci. The resulting updates of the allocation vector\ni=1 correspond to randomized block-coordinate descent on the potential function F (r) with\nr = (ri)N\nblocks ri (see Appendix D.1 and Frongillo and Reid [9]). We refer to this model as the all-security\n(trader) dynamics (ASD).2 We apply and extend the analysis of block-coordinate descent to this setting.\nWe focus on convex+ functions and conduct local convergence analysis around the minimizer of F .\nOur experiments demonstrate that the local analysis accurately estimates the convergence rate.\nLet r(cid:63) denote an arbitrary minimizer of F and let F (cid:63) be the minimum value of F . Also, let rt denote\nthe allocation vector and \u00b5t the market price vector after the tth trade. Instead of directly analyzing\nthe convergence error (cid:107)\u00b5t \u2212 \u00b5(cid:63)(cid:107), we bound the suboptimality F (rt) \u2212 F (cid:63) since (cid:107)\u00b5t \u2212 \u00b5(cid:63)(cid:107)2 =\n\u0398(F (rt) \u2212 F (cid:63)) for convex+ costs C under ASD (see Appendix D.7.1).\nConvex+ functions are locally strongly convex and have a Lipschitz-continuous gradient, so the\nstandard analysis of block-coordinate descent [9, 11] implies linear convergence, i.e., E [F (rt)] \u2212\nF (cid:63) \u2264 O(\u03b3t) for some \u03b3 < 1, where the expectation is under the randomness of the algorithm. We\nre\ufb01ne the standard analysis by (1) proving not only upper, but also lower bounds on the convergence\nrate, and (2) proving an explicit dependence of \u03b3 on the cost function C and the liquidity b. These\ntwo re\ufb01nements are crucial for comparison of cost families, as we demonstrate with the comparison\nof LMSR and IND. We begin by formally de\ufb01ning bounds on local convergence of any randomized\niterative algorithm that minimizes a function F (r) via a sequence of iterates rt.\n\n2In Appendix D, we also analyze the single-security (trader) dynamics (SSD), in which a randomly chosen\n\ntrader randomly picks a single security to trade, corresponding to randomized coordinate descent on F .\n\n7\n\n\fsuch that for some c > 0 and all t \u2265 t0, E(cid:2)F (rt)(cid:12)(cid:12) rt0(cid:3) \u2212 F (cid:63) \u2264 c\u03b3t\u2212t0\n\nDe\ufb01nition 6.1. We say that \u03b3high is an upper bound on the local convergence rate of an algorithm\nif, with probability 1 under the randomness of the algorithm, the algorithm reaches an iteration t0\nhigh . We say that \u03b3low is a lower\nbound on the local convergence rate if \u03b3high \u2265 \u03b3low holds for all upper bounds \u03b3high.\nTo state explicit bounds, we use the notation D := diagi\u2208[N ] ai and P := IN \u2212 11\n(cid:124)\n/N, where IN\nis the N \u00d7 N identity matrix and 1 is the all-ones vector. We write M + for the pseudoinverse of a\nmatrix M and \u03bbmin(M ) and \u03bbmax(M ) for its smallest and largest positive eigenvalues.\nTheorem 6.2 (Local Convergence Bound). Assume that C is convex+. Let HT := HT ( \u00af\u00b5) and\nHC := HC( \u00af\u00b5). For the all-securities dynamics, the local convergence rate is bounded between\n\nhigh = 1 \u2212 2b\n\u03b3ASD\nlow = 1 \u2212 2b\n\u03b3ASD\n\nN \u00b7 \u03bbmin(P DP ) \u00b7 \u03bbmin\nN \u00b7 \u03bbmax(P DP ) \u00b7 \u03bbmax\n\n(cid:0) H 1/2\n(cid:0) H 1/2\n\nT H +\nT H +\n\nC H 1/2\nC H 1/2\n\nT\n\nT\n\n(cid:1) + O(b2) ,\n(cid:1) \u2212 O(b2) .\n\nIn our proof, we \ufb01rst establish both lower and upper bounds on convergence of a generic block-\ncoordinate descent that extend the results of Nesterov [11]. We then analyze the behavior of the\nalgorithm for the speci\ufb01c structure of our objective to obtain explicit lower and upper bounds. Our\nbounds prove linear convergence with the rate \u03b3 = 1 \u2212 \u0398(b). Since the convergence gets worse as\nb \u2192 0, there is a trade-off with the bias, which decreases as b \u2192 0.\nTheorems 5.6 and 6.2 enable systematic quantitative comparisons of cost families. For simplicity,\nassume that N \u2265 2 and all risk aversions are a, so \u03bbmin(P DP ) = \u03bbmax(P DP ) = a. To compare\nconvergence rates of two costs C and C(cid:48), we need to control for bias. As discussed after Theorem 5.6,\ntheir biases are (asymptotically) equal if their liquidities are linearly related as b(cid:48) = b/\u03b7 for a suitable\n\u03b7. Theorem 6.2 then states that C(cid:48)\nb(cid:48) requires (asymptotically) at most a factor of \u03c1 as many trades as Cb\nto achieve the same convergence error, where \u03c1 := \u03b7 \u00b7 \u03bbmax(H 1/2\nC(cid:48)H 1/2\nT H +\nT ).\nb(cid:48), with \u03c1(cid:48) de\ufb01ned symmetrically to \u03c1.\nSimilarly, Cb requires at most a factor of \u03c1(cid:48) as many trades as C(cid:48)\nFor C = LMSR and C(cid:48) = IND, we can show that \u03c1 \u2264 2 and \u03c1(cid:48) \u2264 2, yielding the following result:\nTheorem 6.3. Assume that N \u2265 2 and all risk aversions are equal to a. Consider running LMSR with\nliquidity b and IND with liquidity b(cid:48) = b/\u03b7 such that their asymptotic biases are equal. Denote the\nIND and the respective market-maker equilibria\niterates of the two runs of the market as \u00b5t\nIND. Then, with probability 1, there exist t0 and t1 \u2265 t0 such that for all t \u2265 t1 and\nas \u00b5(cid:63)\nsuf\ufb01ciently small b\n\nT )/\u03bbmin(H 1/2\n\nLMSR and \u00b5(cid:63)\n\nLMSR and \u00b5t\n\nC H 1/2\n\nT H +\n\nIND\n\nLMSR\n\nLMSR\n\nEt0\n\nIND \u2212 \u00b5(cid:63)\n\nLMSR \u2212 \u00b5(cid:63)\n\n\u2212 \u00b5(cid:63)\nwhere \u03b5 = O(b) and Et0[\u00b7] = E[\u00b7 | rt0 ] conditions on the t0th iterate of a given run.\nThis result means that LMSR and IND are roughly equivalent (up to a factor of two) in terms of the\nnumber of trades required to achieve a given accuracy. This is somewhat surprising as this implies\nthat maintaining price coherence does not offer strong informational advantages (at least when traders\nare individually coherent, as assumed here). However, while there is little difference between the\ntwo costs in terms of accuracy, there is a difference in terms of the worst-case loss. For K securities,\nthe worst-case loss of LMSR with the liquidity b is b log K, and the worst-case loss of IND with the\nliquidity b(cid:48) is b(cid:48)K log 2. If liquidities are chosen as in Theorem 6.3, so that b(cid:48) is up to a factor-of-two\nsmaller than b, then the worst-case loss of IND is at least (bK/2) log 2, which is always worse than\nthe LMSR\u2019s loss of b log K, and the ratio of the two losses increases as K grows.\nWhen all risk aversion coef\ufb01cients are equal to some constant a, then the dependence of Theorem 6.2\non the number of traders N and their risk aversion is similar to the dependence in Theorem 5.6. For\ninstance, to guarantee that \u03b3 stays below a certain level for varying N and a requires b = \u2126(N/a).\n\n(cid:2)(cid:13)(cid:13)\u00b52t(1+\u03b5)\n\n(cid:13)(cid:13)2(cid:3) \u2264 Et0\n\n(cid:2)(cid:13)(cid:13)\u00b5t\n\n(cid:13)(cid:13)2(cid:3) \u2264 Et0\n\n(cid:2)(cid:13)(cid:13)\u00b5(t/2)(1\u2212\u03b5)\n\nLMSR\n\n(cid:13)(cid:13)2(cid:3) ,\n\n7 Numerical Experiments\n\nWe evaluate the tightness of our theoretical bounds via numerical simulation. We consider a complete\nmarket over K = 5 securities and simulate N = 10 traders with risk aversion coef\ufb01cients equal\nto 1. These values of N and K are large enough to demonstrate the tightness of our results, but\nsmall enough that simulations are tractable. While our theory comprehensively covers heterogeneous\n\n8\n\n\fFigure 1: (Left) The tradeoff between market-maker bias and convergence. Solid lines are for LMSR,\ndashed for IND, the color indicates the number of trades. (Center) Market-maker bias as a function\nof b. (Right) Convergence in the objective. Shading indicates 95% con\ufb01dence based on 20 trading\nsequences.\n\nrisk aversions and the dependence on the number of traders and securities, we have chosen to\nkeep these values \ufb01xed, so that we can more cleanly explore the impact of liquidity and number\nof trades. We consider the two most commonly studied cost functions: LMSR and IND. We \ufb01x the\nground-truth natural parameter \u03b8true and independently sample the belief \u02dc\u03b8i of each trader from\nNormal(\u03b8true, \u03c32IK), with \u03c3 = 5. We consider a single-peaked ground truth distribution with\nk = log \u03bd for k (cid:54)= 1, with \u03bd = 0.02. Trading is simulated\n1 = log(1 \u2212 \u03bd(K \u2212 1)) and \u03b8true\n\u03b8true\naccording to the all-security dynamics (ASD) as described at the start of Section 6. In Appendix E,\nwe show qualitatively similar results using a uniform ground truth distribution and single-security\ndynamics (SSD).\nWe \ufb01rst examine the tradeoff that arises between market-maker bias and convergence error as the\nliquidity parameter is adjusted. Fig. 1 (left) shows the combined bias and convergence error, (cid:107)\u00b5t\u2212 \u00af\u00b5(cid:107),\nas a function of liquidity and the number of trades t (indicated by the color of the line) for the two\ncost functions, averaged over twenty random trading sequences. The minimum point on each curve\ntells us the optimal value of the liquidity parameter b for the particular cost function and particular\nnumber of trades. When the market is run for a short time, larger values of b lead to lower error. On\nthe other hand, smaller values of b are preferable as the number of trades grows, with the combined\nerror approaching 0 for small b.\nIn Fig. 1 (center) we plot the bias (cid:107)\u00b5(cid:63)(b; C) \u2212 \u00af\u00b5(cid:107) as a function of b for both LMSR and IND. We\ncompare this with the theoretical approximation (cid:107)\u00b5(cid:63)(b; C) \u2212 \u00af\u00b5(cid:107) \u2248 b(\u00afa/N )(cid:107)HT ( \u00af\u00b5)\u2202C\u2217( \u00af\u00b5)(cid:107) from\nTheorem 5.6. Although Theorem 5.6 only gives an asymptotic guarantee as b \u2192 0, the approximation\nis fairly accurate even for moderate values of b. In agreement with Theorem 5.7, the bias of IND is\nhigher than that of LMSR at any \ufb01xed value of b, but by no more than a factor of two.\nIn Fig. 1 (right) we plot the log of \u02c6E[F (rt)] \u2212 F (cid:63) as a function of the number of trades t for our two\ncost functions and several liquidity levels. Even for small t the curves are close to linear, showing\nthat the local linear convergence rate kicks in essentially from the start of trade in our simulations.\nIn other words, there exist some \u02c6c and \u02c6\u03b3 such that, empirically, we have \u02c6E[F (rt)] \u2212 F (cid:63) \u2248 \u02c6c\u02c6\u03b3t, or\nequivalently, log(\u02c6E[F (rt)] \u2212 F (cid:63)) \u2248 log \u02c6c + t log \u02c6\u03b3. Plugging the belief values into Theorem 6.2, the\nslope of the curve for LMSR should be log10 \u02c6\u03b3 \u2248 \u22120.087b for suf\ufb01ciently small b, and the slope for\nIND should be between \u22120.088b and \u22120.164b. In Appendix E, we verify that this is the case.\n\n8 Conclusion\n\nOur theoretical framework provides a meaningful way to quantitatively evaluate the error tradeoffs\ninherent in different choices of cost functions and liquidity levels. We \ufb01nd, for example, that to\nmaintain a \ufb01xed amount of bias, one should set the liquidity parameter b proportional to a measure of\nthe amount of cash that traders are willing to spend. We also \ufb01nd that, although the LMSR maintains\ncoherent prices while IND does not, the two are equivalent up to a factor of two in terms of the\nnumber of trades required to reach any \ufb01xed accuracy, though LMSR has lower worst-case loss.\nWe have assumed that traders\u2019 beliefs are individually coherent. Experimental evidence suggests that\nLMSR might have additional informational advantages over IND when traders\u2019 beliefs are incoherent\nor each trader is informed about only a subset of events [12]. We touch on this in Appendix C.2, but\nleave a full exploration of the impact of different assumptions on trader beliefs to future work.\n\n9\n\nLiquidity Parameter b0.00.20.40.60.81.0Bias Plus Convergence Error 0.000.040.080.12 #Trades 100 200 5001000Liquidity Parameter b0.00.20.40.60.81.0Market\u2212Maker Bias0.000.040.08llllllllllllllllllllllll Actual Bias LMSRIND Asymptotic Bias LMSRIND\u22128\u22126\u22124\u221220025050075010001250Number of TradesLog10 of Suboptimality of F Liquidity b 0.010.030.050.07\fReferences\n[1] Jacob Abernethy, Yiling Chen, and Jennifer Wortman Vaughan. Ef\ufb01cient market making via\nconvex optimization, and a connection to online learning. ACM Transactions on Economics\nand Computation, 1(2):Article 12, 2013.\n\n[2] Jacob Abernethy, Sindhu Kutty, S\u00e9bastien Lahaie, and Rahul Sami. Information aggregation in\nexponential family markets. In Proceedings of the 15th ACM Conference on Economics and\nComputation (EC), 2014.\n\n[3] Ole Barndorff-Nielsen. 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