Betting Strategies, Market Selection, and the Wisdom of Crowds

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1 Betting Strategies, Market Seection, and the Wisdom of Crowds Wiemien Kets Northwestern University David M. Pennock Microsoft Research New York City Rajiv Sethi Barnard Coege, Coumbia University Santa Fe Institute Abstract We investigate the imiting behavior of trader weath and prices in a simpe prediction market with a finite set of participants having heterogeneous beiefs. Traders bet repeatedy on the outcome of a binary event with fixed Bernoui success probabiity. A cass of strategies, incuding (fractiona) Key betting and constant reative risk aversion (CRRA) are considered. We show that when traders are wiing to risk ony a sma fraction of their weath in any period, beief heterogeneity can persist indefinitey; if bets are arge in proportion to weath then ony the most accurate beief type survives. The market price is more accurate in the ong run when traders with ess accurate beiefs aso survive. That is, the surviva of traders with heterogeneous beiefs, some ess accurate than others, aows the market price to better refect the objective probabiity of the event in the ong run. Introduction Prediction markets have attracted considerabe attention in recent years, both as vehices for the aggregation of diverse opinions, and as independent objects of research (Chen and Pott 22; Cowgi, Wofers, and Zitzewitz 29; Ungar et a. 22). A centra question in the anaysis of such markets concerns the interpretation and accuracy of prices. In the case of a singe event, with prices determined by a static market cearing condition, the answer depends cruciay on the preferences of traders. When traders have og utiity, the equiibrium price equas the budget-weighted average of trader beiefs (Pennock 999). This remains approximatey true for other preference specifications as ong as traders are riskaverse (Gjerstad 24; Wofers and Zitzewitz 26). Under risk-neutraity, however, the equiibrium price corresponds to a quantie of the beief distribution, which can be quite distant from the average beief (Manski 26). In markets with betting on a sequence of events, the interpretation of prices is more chaenging, because the weath of traders change endogenousy as the outcomes of bets are reaized. This is the setting with which the present paper is concerned. Specificay, we consider the imiting distributions of trader weath and market prices when betting occurs over an infinite sequence of periods. In each period traders Copyright c 24, Association for the Advancement of Artificia Inteigence ( A rights reserved. Nisarg Shah Carnegie Meon University bet against each other on the outcome of a binary event with fixed (but unknown) Bernoui probabiity. The sizes of their bets depend on their weath and their beiefs reative to the market price, and the price is determined by market cearing. Trader preferences beong to a cass that incudes constant reative risk aversion (CRRA) utiity, as we as (fractiona) Key betting, both defined beow. We show that when preferences are such that traders bet ony a sma portion of their weath in any given period, the imiting weath distribution is non-degenerate in the sense that traders with beiefs of varying degrees of accuracy have positive expected weath in the imit. In this case, the market price in the ong run refects the objective probabiity of the event on which bets are paced. In contrast, when traders pace arge bets (as a proportion of their weath) then ony the trader with the most accurate beief survives. In this case the market price comes to refect this trader s beief, which may not be cose to the objective probabiity of the event. Hence the surviva of reativey inaccurate traders is necessary for the market to make accurate forecasts. This surviva is ensured if traders are reuctant to pace arge bets reative to their income, for instance because of high eves of risk aversion. That is, high eves of risk aversion not ony aow inaccurate beiefs to survive onger, but aow their surviva in the imit (in cases where they previousy did not survive) and cause the market forecast to become more accurate. Our work is reated to a iterature on market seection that dates back at east to Key (956), with important contributions by De Long et a. (99), Bume and Easey (992; 26), Sandroni (2), and Beygezimer, Langford, and Pennock (22), among many others. Bume and Easey (26) and Sandroni (2) show that when markets are compete and traders can optimay aocate weath across periods, traders who survive have the most accurate beiefs (assuming that they a have a common discount factor). These resuts require that traders make optima consumption and asset aocation pans over an infinite horizon, and do not appy to our setting, in which there is no consumption and asset aocation is myopic. De Long et a. (99) show, in an overapping-generations mode, that traders with incorrect beiefs can survive, since they take on more risk, and this is rewarded with greater expected returns. By contrast, in our setting, the surviva of traders with inaccurate beiefs requires them to take on ess risk.

2 Preiminaries We consider a prediction market with n traders who make a sequence of bets against each other on binary events. Let denote the weath of trader i at the end of period t, with wi t wi > denoting the initia weath eves. In each period t, the outcome of the event X t is an independent Bernoui tria with success probabiity p ; this probabiity is unknown to the traders and never reveaed. Trader i s beief about the success probabiity is denoted p i. These beiefs are exogenousy given and are not updated over time, athough traders with inaccurate beiefs may face oss of weath and eventua disappearance from the market. Traders pace bets in each period based on their beiefs, weath, and the market price. If the market price is p m, a buy order (a bet for the outcome) costs p m, and pays off if the outcome is positive, and otherwise. Simiary, a se order (a bet against the outcome) costs p m, and pays off if the outcome is negative, and otherwise. (In practice, a trade of one unit occurs when a buyer pays p m and a seer pays p m to the exchange; the entire sum is deivered to the buyer if the event occurs and to the seer otherwise.) We assume that orders can be of arbitrary size, incuding fractiona units. At any given price p m, each trader has an optima order size, which can be aggregated across traders to obtain a net demand to buy or se. A competitive equiibrium price is one at which the buy and se orders are in baance, so that the aggregate net demand is zero. At this price a orders can be executed. Let p t m denote the market cearing price in period t. At the end of each period, the outcome is reveaed, and the traders bets pay off accordingy. Weath is redistributed among traders, but remains constant in the aggregate. This process is then repeated indefinitey. We normaize the weath of the traders such that n i= w i =. Since weath is ony redistributed, we have n i= wt i = for a t. We assume that each trader i uses a betting strategy represented by a function b i : [, ] [, ] that denotes the fraction of his weath that the trader wishes to bet as a function of the market price. That is, with weath w and market price p, the trader woud bet wb i (p) of his weath. This specification aows for a variety of common betting strategies, some of which are discussed beow. We assume that each b i is fixed over time, and satisfies the foowing mid conditions.. b i (p m ) for a p m [, ], i.e., the trader cannot bet more than his weath. 2. b i (p m ) = if and ony if p m = p i, i.e., the trader does not bet anything if and ony if the market price matches his beief. 3. b i (p m ) < if p m (, ), i.e., the trader does not bet his entire weath except when betting for or against the event is free. This rues out risk-neutraity. 4. The farther the market price from his beief, the higher fraction of his weath the trader bets. That is, b i (p a ) b i (p b ) for p a p b p i and b i (p a ) b i (p b ) for p a p b p i. Specia cases of popuar betting strategies that satisfy our assumptions incude fractiona Key betting and myopic betting with constant reative risk aversion (CRRA) preferences. If p denotes the beief of the trader, Key betting, the optimizer of og utiity, is given by the foowing formua: { p pm p b(p m ) = m if p m p, p m p p m if p m > p. Fractiona Key betting is simiar, but with reduced exposure to risk. Specificay, a c-key trader bets a fixed fraction c of the share of weath that a Key trader with the same beief woud bet. As shown by Beygezimer, Langford, and Pennock (22), a c-key trader acts ike a Key trader whose beief is a inear combination of his own beief and the market price, given by cp + ( c)p m. The higher the vaue of c, the ower the weight on the market price. Hence, c can be interpreted as the eve of confidence in one s own beief. The η-crra betting strategy paces the optimum myopic bet according to the expected vaue of the power utiity (isoeastic utiity) function of weath. Specificay, if w denotes the end-of-the-period weath, the bettor maximizes the expected vaue of { w η u(w) = η η >, η, og(w) η =. It is easiy verified that this strategy satisfies our assumptions. Ceary, η-crra with η = corresponds to Key betting. Hence, both c-key and η-crra can be seen as generaizations of Key betting, where the risk aversion or confidence of the trader can vary. Without oss of generaity, assume that the traders are sorted by their beiefs, i.e., p i p i+ for a i, and that < p < p n <. That is, no trader is subjectivey certain of the outcome, and not a beiefs are identica. It is easiy seen that the market cearing price p t m in a periods must ie stricty between p and p n as ong as both these traders have positive weath. Therefore, trader n consistenty bets that the event wi occur, and trader consistenty bets against it. Theoretica Resuts In this section, we present a few theoretica resuts that wi provide support for our empirica observations in the next section. Expected Weath and Prices Our first resut shows that the surviva of traders with heterogeneous beiefs is beneficia for the accuracy of the market as a whoe. Theorem. Consider two traders with beiefs p and p 2, and weath w t and w2 t respectivey at the end of period t. If E[w w t t ] = w t and E[w2 w t 2 t ] = w2 t, then one of the foowing conditions hods. See Wofers and Zitzewitz (26) for an expicit expression for bet size given η-crra preferences.

3 . Ony trader survives, i.e. w t = and w2 t =. 2. Ony trader 2 survives, i.e. w2 t = and w t =. 3. Both traders survive with unique weath in (, ), and the market price in the next period is p t m = p. Proof. Note that E[w w t t ] = w t and E[w2 w t 2 t ] = w2 t are equivaent because w t + w2 t = w t + w2 t =. In period t, trader 2 bets w2 t b 2 (p t m) for the outcome (because p p t m p 2 ). Due to the Markov property of his weath and that the success probabiity of the event is p, we have E [ w2 t w t 2 ] = w t 2 = w t 2 [p ( ) b 2 (p t m) + b 2(p t m) p t m + ( p ) ( b 2 (p t m) ) ] [ b 2 (p tm) pt m p p t m ]. () Now, E [ w2 t w 2 t ] = w t 2 hods if w2 t = (thus w t = ). It aso hods if b 2 (p t m) =, which happens if and ony if p t m = p 2 due to our assumptions on the betting strategies. However, the atter must impy w2 t = and w t =, otherwise trader woud bet a positive amount whie trader 2 wi bet nothing, vioating the fact that p t m is the market cearing price. Finay, if neither of the above cases hod (i.e., both traders survive), then we must have p t m = p in Equation (). In this case, the market cearing equation at price p, namey w t b (p ) p = wt 2 b 2 (p ) p, (2) and the weath normaization equation w t + w2 t = yied unique vaues for w t and w2 t. These vaues further beong to (, ), because the numerators on both sides of Equation (2) are non-zero as both traders survive in this case. We remark that Theorem computes the fixed points of the weath and price updates, and not their stationary distributions. Hence, it ony provides weak evidence that surviva of both traders woud ead the expected market price to converge to p (under the condition that expected future weath equas current weath). We now show that with two traders, in the extreme cases of p p and p p 2, ony trader and ony trader 2 survive, respectivey. This reaization of conditions and 2 of Theorem comes with a stronger notion of convergence amost sure convergence. Theorem 2. For two traders with beiefs p and p 2, if p p (resp. p p 2 ), then as t,. amost surey, w t and w t 2 (resp. w t 2 and w t ), and 2. amost surey, p t m p (resp. p t m p 2 ). Proof. Without oss of generaity, we prove the case of p p. First, we show that w2 t amost surey. Consider Equation () from the proof of Theorem. It is easy to check that p t m > p p impies E[w2 w t 2 t ] < w2 t. That is, {w2} t t is a bounded super-martingae. By Doob s super-martingae convergence theorem, w2 t converges amost surey. It remains to show that the imit is. Amost sure convergence of w2 t impies amost sure convergence of w, t thus of p t m. If the imit of w2 t were greater than, the imit of p t m woud be stricty between p and p 2. Hence, the ratio E [ w2 t w 2 t ] /w t 2 woud converge to a constant ess than, which is a contradiction. Thus, amost surey, w2 t converges to, and w t and p t m converge to and p, respectivey. Key traders Beygezimer, Langford, and Pennock (22) anayzed markets where a traders use Key betting. They showed that in such a market, the market cearing price is the weathweighted average of the trader beiefs, and gave an expicit formua for weath updates over time. Proposition. When a traders use Key betting, for every period t, p t m = n i= wt i p i. Proposition 2. When a traders use Key betting, for every period t and every trader i, w t i = w i t = ( ) X pi p m We can now derive the foowing resut. ( ) X pi p. m Theorem 3. If a traders use Key betting and a unique trader j maximizes p og(p j ) + ( p ) og( p j ), then as t,. amost surey, wj t and wt i for a i j, and 2. amost surey, p t m p j. Proof. Consider the most accurate trader j and any other trader i. From Proposition 2, w t j w t i = α t = px j ( p j ) X t = px i ( p i ), X where α = wj /w i is a constant. Next, we take ogarithm on both sides and divide by t. The term og(α)/t goes to as t because we assume non-zero initia weath. Using the strong aw of arge numbers, the ogarithms of the numerator and the denominator converge to p og(p j ) + ( p ) og( p j ) and p og(p i )+( p ) og( p i ) respectivey. Since the former is greater than the atter by our assumption, we have that im t (og(wj t) og(wt i ))/t > amost surey. This can happen ony when im t wi t = amost surey. Since this hods for a i j, we have im t wj t = amost surey. Finay, using Proposition, we have that im t p t m = p j amost surey.

4 Bume and Easey (992) obtained a simiar resut in a somewhat different context, with assets in positive net suppy, traders investing a fixed fraction of their weath in each period, and the tota weath varying based on the bets paced and the state reaized. Fractiona Key Traders As noted above, a fractiona Key trader with beief p acts ike a Key trader whose beief is a inear combination of p and the market price p m. Using the methods in Beygezimer, Langford, and Pennock (22), Propositions and 2 can be generaized to a market where a traders use c-key betting with the same c. Proposition 3. When a traders use c-key betting with the same c, we have that for a t, p t m = n j= w t j (c p j +( c) p t m) p t m = n j= w t j p j. Proposition 4. When a traders use c-key betting with the same c, for every period t and every trader i, w t i = w i t ( c pi + ( c) p t m = p m ) X ( (c pi + ( c) p t m) p m ) X Athough we were not abe to sove for the imiting weath anayticay, one can derive an update rue for the weath distribution using Propositions 3 and 4. Consider the simpe case of two traders with beiefs p and p 2. Let F (t) denote the CDF of the distribution of the weath of trader at the end of period t, i.e., that of w t. Then, F (t) (x) = p F (t ) (f (x)) + ( p )F (t ) (f (x)), where f and f defined beow are the weath update functions in case the outcome is positive and negative respectivey. f (x) = x c p + ( c) p m p m, f (x) = x c ( p ) + ( c) ( p m ) p m, p m = x p + ( x) p 2. In the experiments described beow, we observe that both traders survive for ower vaues of c. In fact, we conjecture (as verified by our simuations) that as c,. E[w t ] (p 2 p )/(p 2 p ), and 2. E[w t 2] (p p )/(p 2 p ). Using Proposition 3 and the inearity of expectation, we then have E[p t m] p, as t.. Simuation Resuts We performed extensive simuations using markets having different combinations of traders with heterogeneous beiefs and strategies, and with different vaues of the Bernoui success probabiity p. We are interested in the imiting (expected) trader weath and market price. In each of the foowing simuations, the market is run for 5 iterations. The average weath of each trader and the average market price over the ast iterations is used as an estimate of the expected vaues from the corresponding stationary distributions. These estimates are further averaged over independent runs, and the 5th and the 95th percenties are used for confidence intervas. In our simuations, we are interested in the stationary distributions of the market price p t m and the trader weath w t i. Uness specified otherwise, we say that a trader survives if his expected imiting weath is positive, and vanishes if it is zero; the reader shoud distinguish this from stronger notions of surviva, e.g., in which the imit infimum of the weath must converge to a positive vaue amost surey. Key and fractiona Key betting First, we anayze markets with Key and fractiona Key traders. We begin with a market that has two c-key traders: trader with beief p =.3 and trader 2 with beief p 2 =.8. We are interested in the imiting behavior of trader weath and market price. We present three experiments: First, we vary c from to, then we anayze the imiting case of c =, foowed by the imiting case of c =.. We choose c =. as a proxy for c = because our assumptions do not aow c =. Varying c. To vary the Key fraction c, we fix the objective probabiity of the event to p =.6. Thus, trader 2 is more accurate than trader. Figures (a) and (b) show the imiting expected weath and market price respectivey in this case. Note that at c = (the right end of the figure), ony the most accurate trader (trader 2) survives and the market price converges to his beief, as predicted by Theorem 3. At c =., however, both traders survive, and the imiting market price coincides with the objective event probabiity p, which matches our conjecture in the previous section. In fact, the surviva of both traders is observed for a arge range of vaues of c. These observations aso generaize to the case of more than two traders. Due to ack of space, we ony provide graphs for the case of 3 traders. Figures (c) and (d) show the imiting expected weath and market price respectivey when an additiona trader with beief p 3 =.5 is added to the market. It can be seen that a three traders survive at c =., but the east accurate trader quicky vanishes as c increases. As c further approaches, ony the most accurate trader 3 survives. The market price aso graduay shifts from p to the beief of the most accurate trader. These experiments are consistent with our hypothesis that when the traders are wiing to bet reativey ow fractions of their weath, even the ess accurate traders survive indefinitey, and in turn hep improve the accuracy of the market

5 Weath vs Key Fraction vs Key Fraction Weath vs Key Fraction vs Key Fraction p=.3 p= Key Fraction (c) (a) n = 2, trader weath Key Fraction (c) (b) n = 2, market price p=.3 p=.5 p= Key Fraction (c) (c) n = 3, trader weath Key Fraction (c) (d) n = 3, market price Figure : Varying the Key fraction c from. to. Weath vs vs Weath vs vs p=.3 p= (a) n = 2, trader weath (b) n = 2, market price p=.2 p=.5 p= (c) n = 3, trader weath (d) n = 3, market price Figure 2: The extreme case of c = (fu Key betting). prediction by bringing the market price coser to the objective event probabiity p. The c = case. Next, we investigate the extreme cases in detai. When both traders use Key betting, Figures 2(a) and 2(b) show the imiting expected weath and market price respectivey as a function of the objective event probabiity p. We see that ony the most accurate trader survives and the market price converges to his beief, as predicted by Theorem 3. In fact, the transition point between trader s dominance and trader 2 s dominance is observed at ˆp.56, which is very cose to the theoretica transition point p predicted by Theorem 3. The atter is obtained by soving p og p + ( p ) og( p ) = p og p 2 + ( p ) og( p 2 ). A the above observations aso hod for markets with more than two Key traders; see, e.g., the imiting expected trader weath and market price in Figures 2(c) and 2(d) respectivey when a trader with beief p 3 =.5 is added. The endpoints of the region of p in which the new trader has the most accurate beief can be checked to coincide with the transition points predicted by Theorem 3. The c case. On the other end of the spectrum, we expore the case of c =. in detai. Figures 3(a) and 3(b) show the imiting expected weath and market price respectivey as a function of the objective event probabiity p. We see that in the extreme cases of p p and p p 2, ony the most accurate trader survives, and the market price converges to his beief, as predicted by Theorem 2. For the remaining case of p (p, p 2 ), we observe that both traders survive, and the market price consistenty coincides with p. This matches our conjecture in the previous section accuratey. 2 Further, this is in contrast to the case of c = where ony the most accurate trader survives, and the market price converges to his beief, away from p. For more than two.-key traders, we observe that the market price sti matches p so far as p is stricty between the owest and the highest of the beiefs; otherwise, Theorem 2 appies. However, this does not aow prediction of unique trader weath from Proposition 3 anymore. See, e.g., the imiting expected trader weath and market price in Figures 3(c) and 3(d) respectivey when a trader with beief p 3 =.5 is added. As p goes from.3 to.8, 3 the weath of the new trader first increases starting from, achieves its peak near p 3 =.5, and then decreases back to. CRRA betting Reca that the parameter η in CRRA betting strategy denotes the risk aversion of the trader. Hence, increasing η owers the sizes of the bets that maximize the traders utiity. This corresponds to decreasing c in the c-key betting strategy. We anayze a market with two CRRA traders, trader with beief p =.3 and trader 2 with beief p 2 =.8. In this case, we are interested in the imiting expected trader weath and imiting expected market price as the risk aversion parameter η varies. 2 If the market price coincides with p, then the trader weath must match our conjecture due to Proposition 3. 3 If p / (.3,.8), then a beiefs ie on one side (either above or beow) the objective probabiity and hence, from Theorem 2, ony the most accurate beief type survives.

6 Weath vs vs Weath vs vs p=.3 p= (a) n = 2, trader weath (b) n = 2, market price p=.3 p=.5 p= (c) n = 3, trader weath (d) n = 3, market price Figure 3: The extreme case of c =. (fractiona Key betting with amost zero fraction) Weath vs Risk Aversion p=.3 p= CRRA Parameter (η) (a) Varying η, trader weath vs Risk Aversion CRRA Parameter (η) (b) Varying η, market price Weath vs p=.3 p= (c) η = 2, trader weath vs (d) η = 2, market price Figure 4: Market with two CRRA traders. Varying η. Figures 4(a) and 4(b) show the imiting expected weath and market price when η varies from to 2 and the objective event probabiity is fixed at p =.6. We see that this market performs very simiary to the market with fractiona Key traders. At η =, we recover the Key behavior where ony the most accurate trader survives and the market price converges to his beief (away from p ). As η increases, we begin to observe surviva of the ess accurate trader as we, and the expected market price quicky converges to p. Once again, we concude that when traders ower the sizes of their bets, the ess accurate traders survive, and the market price more accuratey refects the objective event probabiity. We do not need to anayze the boundary case of η =, because it is identica to Key betting. Next, we next anayze the other boundary case of very high risk aversion. Consistent with the previous experiment, we choose η = 2, which turns out to be high enough to match the behaviour of.- Key betting perfecty. The η = 2 case. Figures 4(c) and 4(d) show the imiting expected weath and market price respectivey as a function of the objective probabiity p. The behavior is remarkaby identica to the market where both traders used.-key betting. In the extreme cases, ony the most accurate trader survives, and the expected market price converges to his beief. As p varies from p to p 2, the weath of the two traders decrease/increase ineary whie the expected market price consistenty coincides with p. Again, both traders survive and the accuracy of the market price is astonishingy high because the traders owered their bet sizes. Discussion Our main conceptua contribution is that when traders decrease their bet sizes, the ess accurate traders aso survive aong with the more accurate ones, and this heps improve the accuracy of the market as a whoe; in the ong run, the market price refects the objective probabiity of the event on which bets are paced. Our findings suggest a number of directions for future research. It woud be usefu to expore, using more fine-grained experiments, the dependence of imiting trader weath and price behavior on factors such as the number of participants in the market, the eve of heterogeneity in their beiefs, the overa accuracy of beiefs, and the composition of betting strategies. Further theoretica work, incuding a proof of our conjecture that high eves of risk-aversion impy the surviva of heterogeneous beief types, might uncover interesting connections between the risk-aversion an beief of a trader and the surviva of the trader. We aso view our mode as a stepping stone to more reaistic modes of prediction markets, e.g., to a mode where in every iteration, the objective event probabiity p is samped from a distribution, and the beiefs of the traders are samped from a distribution concentrated around p. Because the beief of a trader is a fresh noisy estimate of the objective event probabiity in each iteration, ony the betting strategy pays a roe in determining the surviva of the trader in this case. We beieve that our theoretica anaysis coud provide some insight into anayzing such more compex modes. This coud deepen our understanding of the conditions under which prediction markets provide accurate forecasts.

7 Acknowedgments Sethi and Shah were visiting and partiay supported by Microsoft Research New York City during the course of this work. References Beygezimer, A.; Langford, J.; and Pennock, D. M. 22. Learning performance of prediction markets with Key bettors. In Proceedings of the th Internationa Joint Conference on Autonomous Agents and Muti-Agent Systems (AA- MAS), Bume, L., and Easey, D Evoution and market behavior. Journa of Economic Theory 58():9 4. Bume, L., and Easey, D. 26. If you re so smart, why aren t you rich? Beief seection in compete and incompete markets. Econometrica 74: Chen, K. Y., and Pott, C. 22. Information aggregation mechanisms: Concept, design and fied impementation. Socia Science Working Paper no. 3. Pasadena: CaTech. Cowgi, B.; Wofers, J.; and Zitzewitz, E. 29. Using prediction markets to track information fows: Evidence from Googe. Working Paper. DeLong, B.; Scheifer, A.; Summers, L.; and Wadmann, R. 99. Noise trader risk in financia markets. Journa of Poitica Economy 98: Gjerstad, S. 24. Risk aversion, beiefs, and prediction market equiibrium. Working Paper, Economic Science Laboratory, University of Arizona. Key, J A new interpretation of information rate. Be System Technica Journa 35(4): Manski, C. F. 26. Interpreting the predictions of prediction markets. Economics Letters 9(3): Pennock, D. M Aggregating Probabiistic Beiefs: Market Mechanisms and Graphica Representations. Ph.D. Dissertation, The University of Michigan. Sandroni, A. 2. Do markets favor traders abe to make accurate predictions? Econometrica 68: Ungar, L.; Meors, B.; Satopää, V.; Baron, J.; Tetock, P.; Ramos, J.; and Swift, S. 22. The good judgment project: A arge scae test of different methods of combining expert predictions. AAAI Technica Report FS-2-6. Wofers, J., and Zitzewitz, E. 26. Interpreting prediction market prices as probabiities. NBER Working Paper No. 22.

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