{"title": "Market Scoring Rules Act As Opinion Pools For Risk-Averse Agents", "book": "Advances in Neural Information Processing Systems", "page_first": 2359, "page_last": 2367, "abstract": "A market scoring rule (MSR) \u2013 a popular tool for designing algorithmic prediction markets \u2013 is an incentive-compatible mechanism for the aggregation of probabilistic beliefs from myopic risk-neutral agents. In this paper, we add to a growing body of research aimed at understanding the precise manner in which the price process induced by a MSR incorporates private information from agents who deviate from the assumption of risk-neutrality. We first establish that, for a myopic trading agent with a risk-averse utility function, a MSR satisfying mild regularity conditions elicits the agent\u2019s risk-neutral probability conditional on the latest market state rather than her true subjective probability. Hence, we show that a MSR under these conditions effectively behaves like a more traditional method of belief aggregation, namely an opinion pool, for agents\u2019 true probabilities. In particular, the logarithmic market scoring rule acts as a logarithmic pool for constant absolute risk aversion utility agents, and as a linear pool for an atypical budget-constrained agent utility with decreasing absolute risk aversion. We also point out the interpretation of a market maker under these conditions as a Bayesian learner even when agent beliefs are static.", "full_text": "Market Scoring Rules Act As Opinion Pools For\n\nRisk-Averse Agents\n\nMithun Chakraborty, Sanmay Das\n\nDepartment of Computer Science and Engineering\n\nWashington University in St. Louis\n\n{mithunchakraborty,sanmay}@wustl.edu\n\nSt. Louis, MO 63130\n\nAbstract\n\nA market scoring rule (MSR) \u2013 a popular tool for designing algorithmic prediction\nmarkets \u2013 is an incentive-compatible mechanism for the aggregation of probabilis-\ntic beliefs from myopic risk-neutral agents. In this paper, we add to a growing\nbody of research aimed at understanding the precise manner in which the price\nprocess induced by a MSR incorporates private information from agents who de-\nviate from the assumption of risk-neutrality. We \ufb01rst establish that, for a myopic\ntrading agent with a risk-averse utility function, a MSR satisfying mild regular-\nity conditions elicits the agent\u2019s risk-neutral probability conditional on the latest\nmarket state rather than her true subjective probability. Hence, we show that a\nMSR under these conditions effectively behaves like a more traditional method of\nbelief aggregation, namely an opinion pool, for agents\u2019 true probabilities. In par-\nticular, the logarithmic market scoring rule acts as a logarithmic pool for constant\nabsolute risk aversion utility agents, and as a linear pool for an atypical budget-\nconstrained agent utility with decreasing absolute risk aversion. We also point out\nthe interpretation of a market maker under these conditions as a Bayesian learner\neven when agent beliefs are static.\n\n1\n\nIntroduction\n\nHow should we combine opinions (or beliefs) about hidden truths (or uncertain future events) fur-\nnished by several individuals with potentially diverse information sets into a single group judgment\nfor decision or policy-making purposes? This has been a fundamental question across disciplines\nfor a long time (Surowiecki [2005]). One simple, principled approach towards achieving this end is\nthe opinion pool (OP) which directly solicits inputs from informants in the form of probabilities (or\ndistributions) and then maps this vector of inputs to a single probability (or distribution) based on\ncertain axioms (Genest and Zidek [1986]). However, this technique abstracts away from the issue\nof providing proper incentives to a sel\ufb01sh-rational agent to reveal her private information honestly.\nFinancial markets approach the problem differently, offering \ufb01nancial incentives for traders to sup-\nply their information about valuations and aggregating this information into informative prices. A\nprediction market is a relatively novel tool that builds upon this idea, offering trade in a \ufb01nancial se-\ncurity whose \ufb01nal monetary worth is tied to the future revelation of some currently unknown ground\ntruth. Hanson [2003] introduced a family of algorithms for designing automated prediction markets\ncalled the market scoring rule (MSR) of which the Logarithmic Market Scoring Rule (LMSR) is\narguably the most widely used and well-studied. A MSR effectively acts as a cost function-based\nmarket maker always willing to take the other side of a trade with any willing buyer or seller, and\nre-adjusting its quoted price after every transaction.\nOne of the most attractive properties of a MSR is its incentive-compatibility for a myopic risk-neutral\ntrader. But this also means that, every time a MSR trades with such an agent, the updated market\n\n1\n\n\fprice is reset to the subjective probability of that agent; the market mechanism itself does not play an\nactive role in unifying pieces of information gleaned from the entire trading history into its current\nprice. Ostrovsky [2012] and Iyer et al. [2014] have shown that, with differentially informed Bayesian\nrisk-neutral and risk-averse agents respectively, trading repeatedly, \u201cinformation gets aggregated\u201d in\na MSR-based market in a perfect Bayesian equilibrium. However, if agent beliefs themselves do not\nconverge, can the price process emerging out of their interaction with a MSR still be viewed as an\naggeragator of information in some sense? Intuitively, even if an agent does not revise her belief\nbased on her inference about her peers\u2019 information from market history, her conservative attitude\ntowards risk should compel her to trade in such a way as to move the market price not all the way\nto her private belief but to some function of her belief and the most recent price; thus, the evolving\nprice should always retain some memory of all agents\u2019 information sequentially injected into the\nmarket. Therefore, the assumption of belief-updating agents may not be indispensable for providing\ntheoretical guarantees on how the market incorporates agent beliefs. A few attempts in this vein\ncan be found in the literature, typically embedded in a broader context (Sethi and Vaughan [2015],\nAbernethy et al. [2014]), but there have been few general results; see Section 1.1 for a review.\nIn this paper, we develop a new uni\ufb01ed understanding of the information aggregation characteristics\nof a market with risk-averse agents mediated by a MSR, with no regard to how the agents\u2019 beliefs are\nformed. In fact, we demonstrate an equivalence between such MSR-mediated markets and opinion\npools. We do so by \ufb01rst proving, in Section 3, that for any MSR interacting with myopic risk-averse\ntraders, the revised instantaneous price after every trade equals the latest trader\u2019s risk-neutral proba-\nbility conditional on the preceding market state. We then show that this price update rule satis\ufb01es an\naxiomatic characterization of opinion pooling functions from the literature, establishing the equiva-\nlence. In Sections 3.1, and 3.2, we focus on a speci\ufb01c MSR, the commonly used logarithmic variety\n(LMSR). We demonstrate that a LMSR-mediated market with agents having constant absolute risk\naversion (CARA) utilities is equivalent to a logarithmic opinion pool, and that a LMSR-mediated\nmarket with budget-constrained agents having a speci\ufb01c concave utility with decreasing absolute\nrisk aversion is equivalent to a linear opinion pool. We also demonstrate how the agents\u2019 utility\nfunction parameters acquire additional signi\ufb01cance with respect to this pooling operation, and that\nin these two scenarios the market maker can be interpreted as a Bayesian learning algorithm even\nif agents never update beliefs. Our results are reminiscent of similar \ufb01ndings about competitive\nequilibrium prices in markets with rational, risk-averse agents (Pennock [1999], Beygelzimer et al.\n[2012], Millin et al. [2012] etc.), but those models require that agents learn from prices and also\nabstract away from any consideration of microstructure and the dynamics of actual price formation\n(how the agents would reach the equilibrium is left open). By contrast, our results do not presuppose\nany kind of generative model for agent signals, and also do not involve an equilibrium analysis \u2013\nhence they can be used as tools to analyze the convergence characteristics of the market price in\nnon-equilibrium situations with potentially \ufb01xed-belief or irrational agents.\n\n1.1 Related Work\n\nGiven the plethora of experimental and empirical evidence that prediction markets are at least as ef-\nfective as more traditional means of belief aggregation (Wolfers and Zitzewitz [2004], Cowgill and\nZitzewitz [2013]), there has been considerable work on understanding how such a market formu-\nlates its own consensus belief from individual signals. An important line of research (Beygelzimer\net al. [2012], Millin et al. [2012], Hu and Storkey [2014], Storkey et al. [2015]) has focused on a\ncompetitive equilibrium analysis of prediction markets under various trader models, and found an\nequivalence between the market\u2019s equilibrium price and the outcome of an opinion pool with the\nsame agents. Seminal work in this \ufb01eld was done by Pennock [1999] who showed that linear and\nlogarithmic opinion pools arise as special cases of the equilibrium of his intuitive model of securities\nmarkets when all agents have generalized logarithmic and negative exponential utilities respectively.\nUnlike these analyses that abstract away from the microstructure, Ostrovsky [2012] and Iyer et al.\n[2014] show that certain market structures (including market scoring rules) satisfying mild condi-\ntions perform \u201cinformation aggregation\u201d (i.e. the market\u2019s belief measure converges in probability\nto the ground truth) for repeatedly trading and learning agents with risk-neutral and risk-averse util-\nities respectively. Our contribution, while drawing inspiration from these sources, differs in that\nwe delve into the characteristics of the evolution of the price rather than the properties of prices\nin equilibrium, and examine the manner in which the microstructure induces aggregation even if\nthe agents are not Bayesian. While there has also been signi\ufb01cant work on market properties in\n\n2\n\n\fcontinuous double auctions or markets mediated by sophisticated market-making algorithms (e.g.\nCliff and Bruten [1997], Farmer et al. [2005], Brahma et al. [2012] and references therein) when the\nagents are \u201czero intelligence\u201d or derivatives thereof (and therefore de\ufb01nitely not Bayesian), this line\nof literature has not looked at market scoring rules in detail, and analytical results have been rare.\nIn recent years, the literature focusing on the market scoring rule (or, equivalently, the cost function-\nbased market maker) family has grown substantially. Chen and Vaughan [2010] and Frongillo et al.\n[2012] have uncovered isomorphisms between this type of market structure and well-known ma-\nchine learning algorithms. We, on the other hand, are concerned with the similarities between price\nevolution in MSR-mediated markets and opinion pooling methods (see e.g. Garg et al. [2004]). Our\nwork comes close to that of Sethi and Vaughan [2015] who show analytically that the price sequence\nof a cost function-based market maker with budget-limited risk-averse traders is \u201cconvergent under\ngeneral conditions\u201d, and by simulation that the limiting price of LMSR with multi-shot but myopic\nlogarithmic utility agents is approximately a linear opinion pool of agent beliefs. Abernethy et al.\n[2014] show that a risk-averse exponential utility agent with an exponential family belief distribu-\ntion updates the state vector of a generalization of LMSR that they propose to a convex combination\nof the current market state vector and the natural parameter vector of the agent\u2019s own belief distri-\nbution (see their Theorem 5.2, Corollary 5.3) \u2013 this reduces to a logarithmic opinion pool (LogOP)\nfor classical LMSR. The LMSR-LogOP connection was also noted by Pennock and Xia [2011] (in\ntheir Theorem 1) but with respect to an arti\ufb01cial probability distribution based on an agent\u2019s ob-\nserved trade that the authors de\ufb01ned instead of considering traders\u2019 belief structure or strategies. We\nshow how results of this type arise as special cases of a more general MSR-OP equivalence that we\nestablish in this paper.\n\n2 Model and de\ufb01nitions\n\nConsider a decision-maker or principal interested in the \u201copinions\u201d / \u201dbeliefs\u201d / \u201cforecasts\u201d of a\ngroup of n agents about an extraneous random binary event X \u2208 {0, 1}, expressed in the form of\npoint probabilities \u03c0i \u2208 (0, 1), i = 1, 2, . . . , n, i.e. \u03c0i is agent i\u2019s subjective probability Pr(X = 1).\nX can represent a proposition such as \u201cA Republican will win the next U.S. presidential election\u201d\nor \u201cThe favorite will beat the underdog by more than a pre-determined point spread in a game of\nfootball\u201d or \u201cThe next Avengers movie will hit a certain box of\ufb01ce target in its opening week.\u201d In this\nsection, we brie\ufb02y describe two approaches towards the aggregation of such private beliefs: (1) the\nopinion pool, which disregards the problem of incentivizing truthful reports, and focuses simply on\nunifying multiple probabilistic reports on a topic, and (2) the market scoring rule, an incentive-based\nmechanism for extracting honest beliefs from sel\ufb01sh-rational agents.\n\n2.1 Opinion Pool (OP)\nThis family of methods takes as input the vector of probabilistic reports pi, i = 1, 2,\u00b7\u00b7\u00b7 , n submitted\nby n agents, also called experts in this context, and computes an aggregate or consensus operator\n\n(cid:98)p = f (p1, p2,\u00b7\u00b7\u00b7 , pn) \u2208 [0, 1]. Garg et al. [2004] identi\ufb01ed three desiderata for an opinion pool\n\n(other criteria are also recognized in the literature, but the following are the most basic and natural):\n\n1. Unanimity: If all experts agree, the aggregate also agrees with them.\n2. Boundedness: The aggregate is bounded by the extremes of the inputs.\n3. Monotonicity:\n\nIf one expert changes her opinion in a particular direction while all other\n\nexperts\u2019 opinions remain unaltered, then the aggregate changes in the same direction.\n\nDe\ufb01nition 1. We call (cid:98)p = f (p1, p2,\u00b7\u00b7\u00b7 , pn) a valid opinion pool for n probabilistic reports if it\n\npossesses properties 1, 2, and 3 listed above.\n\nIt is easy to derive the following result for recursively de\ufb01ned pooling functions that will prove\nuseful for establishing an equivalence between market scoring rules and opinion pools. The proof is\nin Section 1 of the Supplementary Material.\nLemma 1. For a two-outcome scenario, if f2(r1, r2) and fn\u22121(q1, q2, . . . , qn\u22121) are valid opinion\npools for two probabilistic reports r1, r2 and n\u22121 probabilistic reports q1, q2, . . . , qn\u22121 respectively,\nthen f (p1, p2, . . . , pn) = f2(fn\u22121(p1, p2, . . . , pn\u22121), pn) is also a valid opinion pool for n reports.\n\n3\n\n\fLinOP(p1, p2,\u00b7\u00b7\u00b7 , pn)=(cid:80)n\nLogOP(p1, p2,\u00b7\u00b7\u00b7 , pn)=(cid:81)n\n\ni=1 \u03c9lin\ni=1 p\u03c9log\n\ni pi,\n\n(cid:46)(cid:20)(cid:81)n\ni +(cid:81)n\ni \u2265 0 \u2200i = 1, 2, . . . , n,(cid:80)n\n\ni=1 p\u03c9log\n\n, \u03c9log\n\ni\n\ni\n\ni\n\ni=1(1 \u2212 pi)\u03c9log\n\ni\n\n(cid:21)\ni = 1,(cid:80)n\n\n,\n\nTwo popular opinion pooling methods are the Linear Opinion Pool (LinOP) and the Logarithmic\nOpinion Pool (LogOP) which are essentially a weighted average (or convex combination) and a\nrenormalized weighted geometric mean of the experts\u2019 probability reports respectively.\n\nfor a two-outcome scenario, where \u03c9lin\n\ni\n\ni=1 \u03c9lin\n\ni=1 \u03c9log\n\ni = 1.\n\n2.2 Market Scoring Rule (MSR)\nIn general, a scoring rule is a function of two variables s(p, x) \u2208 R \u222a {\u2212\u221e,\u221e}, where p is an\nagent\u2019s probabilistic prediction (density or mass function) about an uncertain event, x is the realized\nor revealed outcome of that event after the prediction has been made, and the resulting value of s\nis the agent\u2019s ex post compensation for prediction. For a binary event X, a scoring rule can just be\nrepresented by the pair (s1(p), s0(p)) which is the vector of agent compensations for {X = 1} and\n{X = 0} respectively, p \u2208 [0, 1] being the agent\u2019s reported probability of {X = 1} which may or\nmay not be equal to her true subjective probability, say, \u03c0 = Pr(X = 1). A scoring rule is de\ufb01ned\nto be strictly proper if it is incentive-compatible for a risk-neutral agent, i.e. an agent maximizes\nher subjective expectation of her ex post compensation by reporting her true subjective probability:\n\u03c0 = arg maxp\u2208[0,1] [\u03c0s1(p) + (1 \u2212 \u03c0)s0(p)], \u2200\u03c0 \u2208 [0, 1].\nIn addition, a two-outcome scoring rule is regular if sj(\u00b7) is real-valued except possibly that s0(1)\nor s1(0) is \u2212\u221e; any regular strictly proper scoring rule can written in the following form (Gneiting\nand Raftery [2007]):\n\nj \u2208 {0, 1}, p \u2208 [0, 1],\n\nsj(p) = G(p) + G(cid:48)(p)(j \u2212 p),\n\n(1)\nG : [0, 1] \u2192 R is a strictly convex function with G(cid:48)(\u00b7) as a sub-gradient which is real-valued expect\npossibly that \u2212G(cid:48)(0) or G(cid:48)(1) is \u221e; if G(\u00b7) is differentiable in (0, 1), G(cid:48)(\u00b7) is simply its derivative.\nA classic example of a regular strictly proper scoring rule is the logarithmic scoring rule:\n\ns0(p) = b ln(1 \u2212 p), where b > 0 is a free parameter.\n\n(2)\nHanson [2003] introduced an extension of a scoring rule wherein the principal initiates the process\nof information elicitation by making a baseline report p0, and then elicits publicly declared reports\npi sequentially from n agents; the ex post compensation cx(pi, pi\u22121) received by agent i from the\nprincipal, where x is the realized outcome of event X, is the difference between the scores assigned\nto the reports made by herself and her predecessor:\n\ns1(p) = b ln p;\n\nx \u2208 {0, 1}.\n\ncx(pi, pi\u22121) (cid:44) sx(pi) \u2212 sx(pi\u22121),\n\n(3)\nIf each agent acts non-collusively, risk-neutrally, and myopically (as if her current interaction with\nthe principal is her last), then the incentive compatibility property of a strictly proper score still holds\nfor the sequential version. Moreover, it is easy to show that the principal\u2019s worst-case payout (loss)\nis bounded regardless of agent behavior. In particular, for the binary-outcome logarithmic score, the\nloss bound for p0 = 1/2 is b ln 2; b can be referred to as the principal\u2019s loss parameter.\nA sequentially shared strictly proper scoring rule of the above form can also be interpreted as a\ncost function-based prediction market mechanism offering trade in an Arrow-Debreu (i.e. (0, 1)-\nvalued) security written on the event X, hence the name \u201cmarket scoring rule\u201d. The cost function\nis a strictly convex function of the total outstanding quantity of the security that determines all\nexecution costs; its \ufb01rst derivative (the cost per share of buying or the proceeds per share from\nselling an in\ufb01nitesimal quantity of the security) is called the market\u2019s \u201cinstantaneous price\u201d, and can\nbe interpreted as the market maker\u2019s current risk-neutral probability (Chen and Pennock [2007]) for\n{X = 1}, the starting price being equal to the principal\u2019s baseline report p0. Trading occurs in\ndiscrete episodes 1, 2, . . . , n, in each of which an agent orders a quantity of the security to buy or\nsell given the market\u2019s cost function and the (publicly displayed) instantaneous price. Since there is\na one-to-one correspondence between agent i\u2019s order size and pi, the market\u2019s revised instantaneous\nprice after trading with agent i, an agent\u2019s \u201caction\u201d or trading decision in this setting is identical to\nmaking a probability report by selecting a pi \u2208 [0, 1]. If agent i is risk-neutral, then pi is, by design,\nher subjective probability \u03c0i (see Hanson [2003], Chen and Pennock [2007] for further details).\n\n4\n\n\fDe\ufb01nition 2. We call a market scoring rule well-behaved if the underlying scoring rule is regular\nand strictly proper, and the associated convex function G(\u00b7) (as in (1)) is continuous and thrice-\ndifferentiable, with 0 < G(cid:48)(cid:48)(p) < \u221e and |G(cid:48)(cid:48)(cid:48)(p)| < \u221e for 0 < p < 1.\n\n3 MSR behavior with risk-averse myopic agents\n\nWe \ufb01rst present general results on the connection between sequential trading in a MSR-mediated\nmarket with risk-averse agents and opinion pooling, and then give a more detailed picture for two\nrepresentative utility functions without and with budget constraints respectively. Please refer to\nSection 2 of the Supplementary Material for detailed proofs of all results in this section.\nSuppose that, in addition to a belief \u03c0i = Pr(X = 1), each agent i has a continuous utility function\nof wealth ui(c), where c \u2208 [cmin\n,\u221e] denotes her (ex post) wealth, i.e. her net compensation from\n\u2208 [\u2212\u221e, 0] is her minimum\nthe market mechanism after the realization of X de\ufb01ned in (3), and cmin\nacceptable wealth (a negative value suggests tolerance of debt); ui(\u00b7) satis\ufb01es the usual criteria of\ni(c) > 0 except possibly that u(cid:48)\nnon-satiation i.e. u(cid:48)\ni (c) < 0\ni (\u221e) = 0, through out its domain (Mas-Colell et al. [1995]); in other words\nexcept possibly that u(cid:48)(cid:48)\nui(\u00b7) is strictly increasing and strictly concave. Additionally, we require its \ufb01rst two derivatives to\n) = \u2212\u221e. Note\nbe \ufb01nite and continuous on [cmin\non the agent\u2019s wealth, we can account for any starting\nthat, by choosing a \ufb01nite lower bound cmin\nwealth or budget constraint that effectively restricts the agent\u2019s action space.\nLemma 2. If |cmin\n| < \u221e, then there exist lower and upper bounds, pmin\n\u2208\n[pi\u22121, 1] respectively, on the feasible values of the price pi to which agent i can drive the market\nregardless of her belief \u03c0i, where pmin\ni + s0(pi\u22121)).\n\ni(\u221e) = 0, and risk aversion, i.e. u(cid:48)(cid:48)\n\n,\u221e] except that we tolerate u(cid:48)\n\n1 (cmin\n\ni + s1(pi\u22121)) and pmax\n\n\u2208 [0, pi\u22121] and pmax\n\ni = s\u22121\n\ni = s\u22121\n\n0 (cmin\n\ni(cmin\n\ni\n\n) = \u221e, u(cid:48)(cid:48)\n\ni (cmin\n\ni\n\ni\n\ni\n\ni\n\ni\n\ni\n\ni\n\ni\n\nSince the latest price pi\u22121 can be viewed as the market\u2019s current \u201cstate\u201d from myopic agent i\u2019s\nperspective, the agent\u2019s \ufb01nal utility depends not only on her own action pi and the extraneously\ndetermined outcome x but also on the current market state pi\u22121 she encounters, her rational action\nbeing given by pi = arg maxp\u2208[0,1] [\u03c0iui(c1(p, pi\u22121)) + (1 \u2212 \u03c0i)ui(c0(pi, pi\u22121))]. This leads us\nto the main result of this section.\nTheorem 1. If a well-behaved market scoring rule for an Arrow-Debreu security with a starting\ninstantaneous price p0 \u2208 (0, 1) trades with a sequence of n myopic agents with subjective probabil-\nities \u03c01, . . . , \u03c0n \u2208 (0, 1) and risk-averse utility functions of wealth u1(\u00b7), . . . , un(\u00b7) as above, then\nthe updated market price pi after every trading episode i \u2208 {1, 2, . . . , n} is equivalent to a valid\nopinion pool for the market\u2019s initial baseline report p0 and the subjective probabilities \u03c01, \u03c02, . . . , \u03c0i\nof all agents who have traded up to (and including) that episode.\n\nProof sketch.\nFor every trading epsiode i, by setting the \ufb01rst derivative of agent i\u2019s expected\nutility to zero, and analyzing the resulting equation, we can arrive at the following lemmas.\nLemma 3. Under the conditions of Theorem 1, if pi\u22121 \u2208 (0, 1), then the revised price pi after agent\ni trades is the unique solution in (0, 1) to the \ufb01xed-point equation:\n\npi =\n\n\u03c0iu(cid:48)\n\ni(c1(pi, pi\u22121))\ni(c1(pi, pi\u22121)) + (1 \u2212 \u03c0i)u(cid:48)\n\n\u03c0iu(cid:48)\n\ni(c0(pi, pi\u22121))\n\n.\n\n(4)\n\nSince p0 \u2208 (0, 1), and \u03c0i \u2208 (0, 1) \u2200i, pi is also con\ufb01ned to (0, 1) \u2200i, by induction.\nLemma 4. The implicit function pi(pi\u22121, \u03c0i) described by (4) has the following properties:\n\n1. pi = \u03c0i (or pi\u22121) if and only if \u03c0i = pi\u22121.\n2. 0 < min{pi\u22121, \u03c0i} < pi < max{pi\u22121, \u03c0i} < 1 whenever \u03c0i (cid:54)= pi\u22121, 0 < \u03c0i, pi\u22121 < 1.\n3. For any given pi\u22121 (resp. \u03c0i), pi is a strictly increasing function of \u03c0i (resp. pi\u22121).\n\nEvidently, properties 1, 2, and 3 above correspond to axioms of unanimity, boundedness, and mono-\ntonicity respectively, de\ufb01ned in Section 2. Hence, pi(pi\u22121, \u03c0i) is a valid opinion pooling function for\npi\u22121, \u03c0i. Finally, since (4) de\ufb01nes the opinion pool pi recursively in terms of pi\u22121 \u2200i = 1, 2, . . . , n,\n(cid:3)\nwe can invoke Lemma 1 to obtain the desired result.\n\n5\n\n\fThere are several points worth noting about this result. First, since the updated market price pi is\nalso equivalent to agent i\u2019s action (Section 2.2), the R.H.S. of (4) is agent i\u2019s risk-neutral probability\n(Pennock [1999]) of {X = 1}, given her utility function, her action, and the current market state.\nThus, Lemma 3 is a natural extension of the elicitation properties of a MSR. MSRs, by design, elicit\nsubjective probabilities from risk-neutral agents in an incentive compatible manner; we show that, in\ngeneral, they elicit risk-neutral probabilities when they interact with risk-averse agents. Lemma 3 is\nalso consistent with the observation of Pennock [1999] that, for all belief elicitation schemes based\non monetary incentives, an external observer can only assess a participant\u2019s risk-neutral probability\nuniquely; she cannot discern the participant\u2019s belief and utility separately. Second, observe that\nthis pooling operation is accomplished by a MSR even without direct revelation. Finally, notice\nthe presence of the market maker\u2019s own initial baseline p0 as a component in the \ufb01nal aggregate;\nhowever, for the examples we study below, the impact of p0 diminishes with the participation of\nmore and more informed agents, and we conjecture that this is a generic property.\nIn general, the exact form of this pooling function is determined by the complex interaction between\nthe MSR and agent utility, and a closed form of pi from (4) might not be attainable in many cases.\nHowever, given a paticular MSR, we can venture to identify agent utility functions which give rise\nto well-known opinion pools. Hence, for the rest of this paper, we focus on the logarithmic market\nscoring rule (LMSR), one of the most popular tools for implementing real-world prediction markets.\n\n3.1 LMSR as LogOP for constant absolute risk aversion (CARA) utility\nTheorem 2. If myopic agent i, having a subjective belief \u03c0i \u2208 (0, 1) and a risk-averse utility func-\ntion satisfying our criteria, trades with a LMSR market with parameter b and current instantaneous\nprice pi\u22121, then the market\u2019s updated price pi is identical to a logarithmic opinion pool between the\ncurrent price and the agent\u2019s subjective belief, i.e.\n\ni\u22121 + (1 \u2212 \u03c0i)\u03b1i(1 \u2212 pi\u22121)1\u2212\u03b1i(cid:3) , \u03b1i \u2208 (0, 1),\n\npi = \u03c0\u03b1i\n\ni p1\u2212\u03b1i\ni\u22121\n\n(cid:14)(cid:2)\u03c0\u03b1i\n\ni p1\u2212\u03b1i\n\nif and only if agent i\u2019s utility function is of the form\n\nui(c) = \u03c4i (1 \u2212 exp (\u2212c/\u03c4i)) ,\n\nc \u2208 R \u222a {\u2212\u221e,\u221e},\n\nconstant \u03c4i \u2208 (0,\u221e),\n\nthe aggregation weight being given by \u03b1i = \u03c4i/b\n\n1+\u03c4i/b .\n\n(5)\n\n(6)\n\nThe proof is in Section 2.1 of the Supplementary Material. Note that (6) is a standard formulation\nof the CARA (or negative exponential) utility function with risk tolerance \u03c4i; smaller the value of\n\u03c4i, higher is agent i\u2019s aversion to risk. The unbounded domain of ui(\u00b7) indicates a lack of budget\nconstraints; risk aversion comes about from the fact that the range of the function is bounded above\n(by its risk tolerance \u03c4i) but not bounded below.\nMoreover, the LogOP equation (5) can alternatively be expressed as a linear update in terms of\nlog-odds ratios, another popular means of formulating one\u2019s belief about a binary event:\n\nl(p) = ln(cid:0) p\n\n1\u2212p\n\n(cid:1) \u2208 [\u2212\u221e,\u221e]\n\nfor p \u2208 [0, 1].\n\nl(pi) = \u03b1il(\u03c0i) + (1 \u2212 \u03b1i)l(pi\u22121),\n\n(7)\nAggregation weight and risk tolerance: Since \u03b1i is an increasing function of an agent\u2019s risk\ntolerance relative to the market\u2019s loss parameter (the latter being, in a way, a measure of how much\nrisk the market maker is willing to take), identity (7) implies that the higher an agent\u2019s risk tolerance,\nthe larger is the contribution of her belief towards the changed market price, which agrees with\nintuition. Also note the interesting manner in which the market\u2019s loss parameter effectively scales\ndown an agent\u2019s risk tolerance, enhancing the inertia factor (1 \u2212 \u03b1i) of the price process.\nBayesian interpretation: The Bayesian interpretation of LogOP in general is well-known (Bordley\n[1982]); we restate it here in a form that is more appropriate for our prediction market setting. We\ncan recast (5) as pi = pi\u22121\n. This shows\nthat, over the ith trading episode \u2200i, the LMSR-CARA agent market environment is equivalent\nto a Bayesian learner performing inference on the point estimate of the probability of the forecast\nevent X, starting with the common-knowledge prior Pr(X = 1) = pi\u22121, and having direct access\nto \u03c0i (which corresponds to the \u201cobservation\u201d for the inference problem), the likelihood function\n\nassociated with this observation being L (X = x|\u03c0i) \u221d(cid:12)(cid:12)(cid:12) 1\u2212x\u2212\u03c0i\n\n(cid:16) 1\u2212\u03c0i\n(cid:12)(cid:12)(cid:12)\u03b1i, x \u2208 {0, 1}.\n\n(cid:17)\u03b1i(cid:14)(cid:104)\n\n+ (1 \u2212 pi\u22121)\n\n(cid:16) \u03c0i\n\n(cid:17)\u03b1i(cid:105)\n\n(cid:16) \u03c0i\n\npi\u22121\n\n(cid:17)\u03b1i\n\n1\u2212x\u2212pi\u22121\n\npi\u22121\n\npi\u22121\n\n1\u2212pi\u22121\n\n6\n\n\fj = \u03b1j\n\nj+1/(cid:101)\u03b1n\n\n0 =(cid:81)n\n\n0 l(p0)+(cid:80)n\ni=1(cid:101)\u03b1n\ni l(\u03c0i). This is a LogOP where(cid:101)\u03b1n\n(cid:81)n\ni=j+1 (1 \u2212 \u03b1i), j = 1, . . . , n \u2212 1, and(cid:101)\u03b1n\n\nSequence of one-shot traders:\nIf all n agents in the system have CARA utilities with potentially\ndifferent risk tolerances, and trade with LMSR myopically only once each in the order 1, . . . , n, then\nl(pn) =(cid:101)\u03b1n\nthe \u201c\ufb01nal\u201d market log-odds ratio after these n trades, on unfolding the recursion in (7), is given by\ni=1(1\u2212\u03b1i) determines the inertia\n(cid:101)\u03b1n\nof the market\u2019s initial price, which diminishes as more and more traders interact with the market, and\nbelief;(cid:101)\u03b1n\nj , j \u2265 1 quanti\ufb01es the degree to which an individual trader impacts the \ufb01nal (aggregate) market\nn = \u03b1n. Interestingly, the weight of an\nagent\u2019s belief depends not only on her own risk tolerance but also on those of all agents succeeding\nher in the trading sequence (lower weight for a more risk tolerant successor, ceteris paribus), and is\nindependent of her predecessors\u2019 utility parameters. This is sensible since, by the design of a MSR,\ntrader i\u2019s belief-dependent action in\ufb02uences the action of each of (rational) traders i + 1, i + 2, . . .\nso that the action of each of these successors, in turn, has a role to play in determining the market\nimpact of trader i\u2019s belief. In particular, if \u03c4j = \u03c4 > 0 \u2200j \u2265 1, then the aggregation weights satisfy\nj = 1 + \u03c4 /b > 1 \u2200j = 1,\u00b7\u00b7\u00b7 , n\u2212 1, i.e. LMSR assigns progessively higher\nweights to traders arriving later in the market\u2019s lifetime when they all exhibit identical constant risk\naversion. This seems to be a reasonable aggregation principle in most scenarios wherein the amount\n0 = \u03c4 /b which\nindicates that the weight of the market\u2019s baseline belief in the aggregate may be higher than those of\nsome of the trading agents if the market maker has a comparatively high loss parameter. This strong\neffect of the trading sequence on the weights of agents\u2019 beliefs is a signi\ufb01cant difference between the\none-shot trader setting and the market equilibrium setting where each agent\u2019s weight is independent\nof the utility function parameters of her peers.\nConvergence: If agents\u2019 beliefs are themselves independent samples from the same distribution P\nover [0, 1], i.e. \u03c0i \u223ci.i.d. P \u2200i, then by the sum laws of expectation and variance,\n\nthe inequalities(cid:101)\u03b1n\nof information in the world improves over time. Moreover, in this situation, (cid:101)\u03b1n\n1 /(cid:101)\u03b1n\n\n0 )E\u03c0\u223cP [l(\u03c0)] ; Var [l(pn)] = Var\u03c0\u223cP [l(\u03c0)](cid:80)n\n\nthe(cid:101)\u03b1n\n\ni )2.\nHence, using an appropriate concentration inequality (Boucheron et al. [2004]) and the properties of\ni \u2019s, we can show that, as n increases, the market log-odds ratio l(pn) converges to E\u03c0\u223cP [l(\u03c0)]\n\nwith a high probability; this convergence guarantee does not require the agents to be Bayesian.\n\n0 l(p0) + (1 \u2212(cid:101)\u03b1n\n\nE [l(pn)] =(cid:101)\u03b1n\n\ni=1((cid:101)\u03b1n\n\n3.2 LMSR as LinOP for an atypical utility with decreasing absolute risk aversion\nTheorem 3. If myopic agent i, having a subjective belief \u03c0i \u2208 (0, 1) and a risk-averse utility func-\ntion satisfying our criteria, trades with a LMSR market with parameter b and current instantaneous\nprice pi\u22121, then the market\u2019s updated price pi is identical to a linear opinion pool between the\ncurrent price and the agent\u2019s subjective belief, i.e.\n\npi = \u03b2i\u03c0i + (1 \u2212 \u03b2i)pi\u22121,\n\nfor some constant \u03b2i \u2208 (0, 1),\n\nif and only if agent i\u2019s utility function is of the form\n\nui(c) = ln(exp((c + Bi)/b) \u2212 1),\n\nc \u2265 \u2212Bi,\n\n(8)\n\n(9)\n\ni (c)/u(cid:48)\n\nwhere Bi > 0 represents agent i\u2019s budget, the aggregation weight being \u03b2i = 1 \u2212 exp(\u2212Bi/b).\nThe proof is in Section 2.2 of the Supplementary Material. The above atypical utility function has\nits domain bounded below, and possesses a positive but strictly decreasing Arrow-Pratt absolute\nrisk aversion measure (Mas-Colell et al. [1995]) Ai(c) = \u2212u(cid:48)(cid:48)\nb(exp((c+Bi)/b)\u22121) for\nany b, Bi > 0.\nIt shares these characteristics with the well-known logarithmic utility function.\nMoreover, although this function is approximately linear for large (positive) values of the wealth c,\nit is approximately logarithmic when (c + Bi) (cid:28) b.\nTheorem 3 is somewhat surprising since it is logarithmic utility that has traditionally been found to\neffect a LinOP in a market equilibrium (Pennock [1999], Beygelzimer et al. [2012], Storkey et al.\n[2015], etc.). Of course in this paper, we are not in an equilibrium / convergence setting, but in light\nof the above similarities between utility function (9) and logarithmic utility, it is perhaps not unrea-\nsonable to ask whether the logarithmic utility-LinOP connection is still maintained approximately\nfor LMSR price evolution under some conditions. We have extensively explored this idea, both an-\nalytically and by simulations, and have found that a small agent budget compared to the LMSR loss\nparameter b seems to produce the desired result (see Section 3 of the Supplementary Material).\n\ni(c) =\n\n1\n\n7\n\n\fi = (1 \u2212 \u03b2i)pi\u22121, pmax\n\ni\n\ni + (1 \u2212 \u03c0i)pmin\n\ni\n\nNote that, unlike in Theorem 2, the equivalence here requires the agent utility function to depend on\nthe market maker\u2019s loss parameter b (the scaling factor in the exponential). Since the microstructure\nis assumed to be common knowledge, as in traditional MSR settings, the consideration of an agent\nutility that takes into account the market\u2019s pricing function is not unreasonable.\nSince the domain of utility function (9) is bounded below, we can derive \u03c0i-independent bounds on\n= \u03b2i + (1 \u2212 \u03b2i)pi\u22121. Hence,\npossible values of pi from Lemma 2: pmin\n, i.e. the revised price is a linear interpolation\nequation (8) becomes pi = \u03c0ipmax\nbetween the agent\u2019s price bounds, her subjective probability itself acting as the interpolation factor.\nAggregation weight and budget constraint: Evidently, the aggregation weight of agent i\u2019s be-\nlief, \u03b2i = (1 \u2212 exp(\u2212Bi/b)), is an increasing function of her budget normalized with respect to\nthe market\u2019s loss parameter; it is, in a way, a measure of her relative risk tolerance. Thus, broad\ncharacteristics analogous to the ones in Section 3.1 apply to these aggregation weights as well, with\nthe log-odds ratio replaced by the actual market price.\nBayesian interpretation: Under the mild technical assumption that agent i\u2019s belief \u03c0i \u2208 (0, 1) is\nrational, and her budget Bi > 0 is such that \u03b2i \u2208 (0, 1) is also rational, it is possible to obtain positive\nintegers ri, Ni and a positive rational number mi\u22121 such that \u03c0i = ri/Ni and \u03b2i = Ni/(mi\u22121+Ni).\nThen, we can rewrite the LinOP equation (8) as pi = ri+pi\u22121mi\u22121\n, which is equivalent to the poste-\nmi\u22121+Ni\nrior expectation of a beta-binomial Bayesian inference procedure described as follows: The forecast\nevent X is modeled as the (future) \ufb01nal \ufb02ip of a biased coin with an unknown probability of heads.\nIn episode i, the principal (or aggregator) has a prior distribution BETA(\u00b5i\u22121, \u03bdi\u22121) over this prob-\nability, with \u00b5i\u22121 = pi\u22121mi\u22121, \u03bdi\u22121 = (1 \u2212 pi\u22121)mi\u22121. Thus, pi\u22121 is the prior mean and mi\u22121\nthe corresponding \u201cpseudo-sample size\u201d parameter. Agent i is non-Bayesian, and her subjective\nprobability \u03c0i, accessible to the aggregator, is her maximum likelihood estimate associated with the\n(binomial) likelihood of observing ri heads out of a private sample of Ni independent \ufb02ips of the\nabove coin (Ni is common knowledge). Note that mi\u22121, Ni are measures of certainty of the aggre-\ngator and the trading agent respectively, and the latter\u2019s normalized budget Bi/b = ln(1+Ni/mi\u22121)\nbecomes a measure of her certainty relative to the aggregator\u2019s current state in this interpretation.\nSequence of one-shot traders and convergence:\ni=1(1 \u2212 \u03b1i), (cid:101)\u03b2n\nIf all agents have utility (9) with potentially\ndifferent budgets, and trade with LMSR myopically once each, then the \ufb01nal aggregate market\nj =\nj from Section 3.1\nj . Moreover, if \u03c0i \u223ci.i.d. P \u2200i, then we can proceed exactly as in Section 3.1 to show\n\n0 p0 +(cid:80)n\nprice is given by pn = (cid:101)\u03b2n\ni=1(cid:101)\u03b2n\n(cid:81)n\ni=j+1 (1 \u2212 \u03b2i) \u2200j = 1, . . . , n \u2212 1, (cid:101)\u03b2n\ncarry over to(cid:101)\u03b2n\n\n0 =(cid:81)n\ni \u03c0i, which is a LinOP where (cid:101)\u03b2n\nn = \u03b2n. Again, all intuitions about(cid:101)\u03b1n\n\n\u03b2j\n\nthat, as n increases, pn converges to E\u03c0\u223cP [\u03c0] with a high probability.\n\n4 Discussion and future work\n\nWe have established the correspondence of a well-known securities market microstructure to a class\nof traditional belief aggregation methods and, by extension, Bayesian inference procedures in two\nimportant cases. An obvious next step is the identi\ufb01cation of general conditions under which a MSR\nand agent utility combination is equivalent to a given pooling operation. 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Perspectives, 18(2):107\u2013126, 2004.\n\n9\n\n\f", "award": [], "sourceid": 1389, "authors": [{"given_name": "Mithun", "family_name": "Chakraborty", "institution": "Washington Univ. in St. Louis"}, {"given_name": "Sanmay", "family_name": "Das", "institution": "Washington University in St. Louis"}]}