Abstract

We develop a model of a prediction market with ambiguity and derive testable implications of the presence of Knightian uncertainty. Our model can also explain two commonly observed empirical regularities in betting markets: the tendency for longshots to win less often than odds would indicate and the tendency for favorites to win more often. Using historical data from Intrade, we further present empirical evidence that is consistent with the predicted presence of Knightian uncertainty. Our evidence also suggests that, even with information acquisition, the Knightian uncertainty of the world may be not “learnable” to the traders in prediction markets.

JEL Classification: D81; G13; L83

Corresponding author: Weifeng Zhong, Mercatus Center, George Mason University, Arlington, 22201, VA, USA, E-mail:

Article note: We appreciate helpful comments from Julian Chan, Jon Hartley, and Joe Sullivan, and we thank Cody Kallen for excellent research assistance. All errors are our own.

Mathematical Appendix

Proof of Proposition 1

Note that any equilibrium price π has to satisfy (i) π>2ϵ and (ii) π<12ϵ. If (i) does not hold, then π*qi+ϵ for all i, which means any trader will have either a long position or a zero position—not an equilibrium. Similarly, if (ii) does not hold, no trader will have a long position, which cannot be an equilibrium either.

Substitute (2) into (3) and rewrite the aggregate demand as

XF(π)=ϵπϵq+ϵππ(1π)wdF(q)+π+ϵ1ϵqϵππ(1π)wdF(q).

Hence, XF(π)=0 if and only if

ϵπϵ(q+ϵπ)dF(q)+π+ϵ1ϵ(qϵπ)dF(q)=0
ϵπϵ(qπ)dF(q)+π+ϵ1ϵ(qπ)dF(q)+ϵπϵϵdF(q)π+ϵ1ϵϵdF(q)=0
EF(q)ππϵπ+ϵ(qπ)dF(q)+ϵ[F(πϵ)+F(π+ϵ)1]=0
EF(q)π+πϵπ+ϵF(q)dq[(qπ)F(q)]πϵπ+ϵ+ϵ[F(πϵ)+F(π+ϵ)1]=0,

where the last step follows from integration by parts. Simplifying and rearranging terms yields the stated expression in the proposition. Q.E.D.

Proof of Proposition 2

Let G be the space of distribution functions over [ϵ,1ϵ], endowed with the Lévy metric , where

(G1,G2)
inf{ε>0|G1(qε)εG2G1(q+ε)+εforallq[ϵ,1ϵ]}

for any G1,G2G. Let F be the subset of G that satisfies πF*=EF(q) for any FF. Since the Lévy metric metrizes the weak topology, the proposition is equivalent to the claim that F is nowhere dense in (G,).

Note that F is closed. Since a set is nowhere dense if and only if the complement of its closure is dense, it remains to be shown G\F is dense, that is, for any point in G, there is a sequence from G\F converging to that point. It is thus enough to show, for any FF and any δ>0, there exists some GG\F such that (F,G)<δ.

F is non-decreasing since it is a distribution function. It follows that

limq[EF(q)+ϵ]F(q)F(EF(q)ϵ).

We prove the results by examining two cases.

Case 1: limq[EF(q)+ϵ]F(q)>F(EF(q)ϵ).

Given δ>0, we construct a distribution function G from F as

G(q){F(q)ifq[ϵ,EF(q)ϵδ1),F(EF(q)ϵ)ifq[EF(q)ϵδ1,EF(q)+ϵ+δ2),F(q)ifq[EF(q)+ϵ+δ2,1ϵ],

where δ1,δ2>0 are such that function gGF satisfies conditions

EF(q)εδ1EF(q)+ε+δ2g(q)dq=0

and

max{g(EF(q)ϵδ1),g(EF(q)+ϵ+δ2)}=δ2.

It is easily verified that G is a mean-preserving spread of F, with two new atoms created at points EF(q)ϵδ1 and EF(q)+ϵ+δ2. By construction, this implies that

EG(q)ϵEG(q)+ϵG(q)dq=EF(q)ϵEF(q)+ϵG(q)dq=EF(q)ϵEF(q)+ϵ[F(q)+g(q)]dq=ϵ+EF(q)ϵEF(q)+ϵg(q)dq<ϵ,

where the last equality holds because FF, and the inequality holds because g(EF(q)+ϵ)<0, which implies EF(q)ϵEF(q)+ϵg(q)dq<0. Since EG(q)ϵEG(q)+ϵG(q)dq<ϵ, GG\F. Finally, let ρ be the uniform metric, that is,

ρ(G1,G2)sup{|G1(q)G2(q)q[ϵ,1ϵ]}

for any G1,G2G. By construction, ρ(F,G)=δ2. Since the Lévy metric is bounded by the uniform metric from above, that is, (G1,G2)ρ(G1,G2) for any G1,G2G, we have (F,G)δ2<δ.

Case 2: limq[EF(q)+ϵ]F(q)=F(EF(q)ϵ).

Given δ>0, we construct a distribution function H from F as

H(q){F(q)ifq[ϵ,EF(q)ϵ),F(EF(q)ϵ)+δ3ifq[EF(q)ϵ,EF(q)+ϵ+δ4)F(q)ifq[EF(q)+ϵ+δ4,1ϵ],

where δ3,δ4>0 are such that function hHF satisfies conditions

EF(q)ϵEF(q)+ϵ+δ4h(q)dq=0

and

max{δ3,h(EF(q)+ϵ+δ4)}=δ2.

It is easily verified that H is a mean-preserving spread of F, with two new atoms created at points EF(q)ϵ and EF(q)+ϵ+δ4. By construction, this implies that

EG(q)εEG(q)+εH(q)dq=EF(q)εEF(q)+εH(q)dq=EF(q)εEF(q)+ε[F(q)+h(q)]dq=ε+EF(q)εEF(q)+εh(q)dq=ε+2εδ3>ε,

where the last but second equality holds because FF, and the last equality follows from the construction of H. Since EG(q)ϵEG(q)+ϵH(q)dq>ϵ, HG\F. Finally, similar to Case 1, we have ρ(F,H)=δ2 and, hence, (F,H)<δ. Q.E.D.

Proof of Proposition 3

Decompose XF(π) into the aggregate supply (shorts) SF(π) and the aggregate demand (longs) DF(π), where

SF(π)=επεq+εππ(1π)wdF(q),DF(π)=π+ε1εqεππ(1π)wdF(q),

and SF(πF*)=DF(πF*) in equilibrium. We show that an increase in ϵ shifts the supply curve inwards. That is,

dSF(π)dϵ=0+ϵ+ϵππ(1π)wdF(ϵ)ϵπϵϵq+ϵππ(1π)wdF(q)<0.

Similarly, an increase in ϵ shifts the demand curve inwards (i.e., dDF(π)dϵ<0). It follows that the equilibrium quantity of trade—SF(πF*), or DF(πF*)—has to be smaller as the degree of ambiguity increases. Q.E.D.

Proof of Proposition 4

Let Φ denote the integral of F, i.e., Φ(q)εqF(q)dq. It follows from the definition of F that

Φ(q)=εqF(q)dq={(1m)Φ(q)ifq[ε,p)(1m)Φ(q)+m(qp)ifq[p,1ε],

where Φ is the integral of F. The equilibrium condition becomes

π=EF(q)+Φ(π+ϵ)Φ(πϵ)ϵ=12ϵΦ(1ϵ)+Φ(π+ϵ)Φ(πϵ),

where the second equality follows from integration by parts. Since Φ(q) has a kink at point p, the equilibrium price depends on the position of p relative to π+ϵ and πϵ.

Case 1: πϵpπ+ϵ.

The equilibrium condition is rewritten as

π=12ϵ(1m)Φ(1ϵ)m(1ϵp)+(1m)Φ(π+ϵ)+m(π+ϵp)(1m)Φ(πϵ).

Rearranging terms and dividing both sides by 1m yields

πΦ(π+ε)+Φ(πε)=12εΦ(1ε).

Case 2: p>π+ϵ.

The equilibrium condition is rewritten as

π=12ε(1m)Φ(1ε)m(1εp)+(1m)Φ(π+ε)(1m)Φ(πε).

Rearranging terms yields

(A1)π1mΦ(π+ε)+Φ(πε)=12εΦ(1ε)+(pε)m1m.

Note that the left-hand side of equation (A1) is strictly increasing in π. Thus, the solution π* to the equation is a continuous and strictly increasing function of p. Furthermore, as pπ^+ϵ, where π^ is the equilibrium price in Case 1, the right-hand side of equation (A1) converges to 12ϵΦ(1ϵ)+π^m1m, and the solution to the equation converges to π^. In other words, the equilibrium price is continuous at point p=π^+ϵ.

Next, we show π*(1ϵ)<12ϵ, which implies π*(1ϵ)<1ϵ in part 1 of the proposition. Let LHS(π) and RHS(p) denote the left- and right-hand sides of equation (A1), as functions of π and p, respectively. Note that

LHS(12ε)RHS(1ε)=[12ε1mΦ(1ε)+Φ(13ε)][12εΦ(1ε)+(12ε)m1m]=Φ(13ε)>0.

Since LHS is strictly increasing in π, the solution to the equation when p=1ϵ must be smaller than 12ϵ.

Case 3: p<πϵ.

The equilibrium condition is rewritten as

π=12ε(1m)Φ(1ε)m(1εp)+(1m)Φ(π+ε)+m(π+εp)(1m)Φ(πε)m(πεp).

Rearranging terms yields

(A2)π1mΦ(π+ε)+Φ(πε)=12εΦ(1ε)+(p+ε)m1m.

Similar to Case 2, the solution π* to equation (A2) is continuous and strictly increasing in p, and it converges to π^ as pπ^ϵ. Hence, the equilibrium price is continuous at point p=π^ϵ as well.

Next, we show π*(ϵ)>2ϵ, which implies π*(ϵ)>ϵ in part 1 of the proposition. Again, let LHS(π) and RHS(p) denote the left- and right-hand sides of equation (A2). Note that

LHS(2ε)RHS(ε)=[2ε1mΦ(3ε)+Φ(ε)][12εΦ(1ε)+2εm1m]=[Φ(1ε)Φ(3ε)][(1ε)3ε]<0,

where the last inequality holds because Φ is the integral of distribution function F over [ϵ,1ϵ]. Since LHS is strictly increasing in π, the solution to the equation when p=ϵ must be larger than 2ϵ. Q.E.D.

Proof of Corollary 2

Recall that π^ is identified by equation

π^Φ(π^+ε)+Φ(π^ε)=12εΦ(1ε).

The symmetry of F implies Φ(1x)=(xϵ)Φ(x) for any x[ϵ,1ϵ]. Thus, the equilibrium condition becomes

π^[π^+Φ(1π^ϵ)]+Φ(π^ϵ)=12ϵ[12ϵ+Φ(ϵ)]
Φ(1π^ϵ)Φ(π^ϵ)=Φ(ϵ)=0,

to which π^=0.5 is the only solution. Q.E.D.

Data Appendix

The historical data of Intrade was archived by Ipeirotis (2013) and is available on GitHub. Table 4 lists all the categories of events and the number of markets within each category. We complete the dataset by creating an outcome variable and recording how each random event had turned out. The outcome equals 1 if an event occurs, and it equals 0 if its complement event occurs.

Table 4:

Intrade data: event categories and number of markets.

Event categoryNumber of markets
Art60
Chess52
Climate & Weather861
Construction & Engineering9
Current Events1540
Education1
Entertainment8715
Fine Wine5
Foreign Affairs87
Legal310
Media10
Politics5460
Real Estate2
Science20
Social & Civil30
Technologies65
Transportation11

Some markets have correlated outcomes, because they are about the same, uncertain circumstances. For example, concerning the 2012 U.S. Republican Party presidential nominee, there are 53 separate markets corresponding to 53 possible winners, including Mitt Romney, Rick Santorum, Ron Paul, Newt Gingrich, and “any other individual” not specified by the prediction platform. To avoid such correlation in the observations, for each group of these correlated markets, we randomly select one market into the aggregate sample and disregard the rest.

The total number of selected markets included in the final analysis also shown in Table 5. The table lists the number of observations—the total as well as the number of observations per percentile bin—for political events, entertainment events, and the full sample. The dataset is skewed towards political and entertainment events, as the two categories together accounts for 82% of the full sample.

Table 5:

Intrade data: number of observations in final analysis.

Event categoryTotal observationsObservations per bin
(50 bins)(30 bins)
Politics8971830
Entertainment11572339
Full sample25095084

References

Ali, M. M. 1977. “Probability and Utility Estimates for Racetrack Bettors.” Journal of Political Economy 85 (4): 803–15.https://doi.org/10.1086/260600.Search in Google Scholar

Andrews, D. W. K. 2003. “Tests for Parameter Instability and Structural Change with Unknown Change Point: A Corrigendum.” Econometrica 71 (1): 395–97.https://doi.org/10.1111/1468-0262.00405.Search in Google Scholar

Andrews, D. W. K., and P. Werner. 1994. “Optimal Tests when a Nuisance Parameter is Present Only Under the Alternative.” Econometrica 62 (6): 1383–414.https://doi.org/10.2307/2951753.Search in Google Scholar

Asch, P., B. G. Malkiel, and R. E. Quandt. 1982. “Racetrack Betting and Informed Behavior.” Journal of Financial Economics 10 (2): 187–94.https://doi.org/10.1016/0304-405x(82)90012-5.Search in Google Scholar

Berg, J. E., F. D. NelsonT. A. Rietz2008. “Prediction Market Accuracy in the Long Run.” International Journal of Forecasting 24 (2): 285–300.https://doi.org/10.1016/j.ijforecast.2008.03.007.Search in Google Scholar

Berg, J., R. ForsytheF. NelsonT. Rietz. 2008. “Chapter 80 Results from a Dozen Years of Election Futures Markets Research.” In Handbook of Experimental Economics Results, Vol. 1, edited by C. R. Plott, and V. L. Smith, 742–51. Elsevier.Search in Google Scholar

Boole, G. 1854. An Investigation of the Laws of Thought: On which are Founded the Mathematical Theories of Logic and Probabilities. New York: Dover Publications.Search in Google Scholar

Caballero, R. J., and A. Simsek. 2013. “Fire Sales in a Model of Complexity.” The Journal of Finance 68 (6): 2549–87.https://doi.org/10.1111/jofi.12087.Search in Google Scholar

Caballero, R. J., and A. Krishnamurthy. 2008. “Collective Risk Management in a Flight to Quality Episode.” The Journal of Finance 63 (5): 2195–230.https://doi.org/10.1111/j.1540-6261.2008.01394.x.Search in Google Scholar

Cain, M., D. Law, and D. Peel. 2000. “The Favourite-Longshot Bias and Market Efficiency in UK Football betting.” Scottish Journal of Political Economy 47 (1): 25–36.https://doi.org/10.1111/1467-9485.00151.Search in Google Scholar

Cao, H. H., T Wang, and H. H. Zhang. 2005. “Model Uncertainty, Limited Market Participation, and Asset Prices.” Review of Financial Studies 18 (4): 1219–51.https://doi.org/10.1093/rfs/hhi034.Search in Google Scholar

Chow, G. C. 1960. “Tests of Equality Between Sets of Coefficients in Two Linear Regressions.” Econometrica 28 (3): 591–605.https://doi.org/10.2307/1910133.Search in Google Scholar

Chu, C. S. J., K. Hornik, and C. M. Kuan. 1995a. “The Moving-Estimates Test for Parameter Stability.” Econometric Theory 11 (4): 699–720.https://doi.org/10.1017/s0266466600009695.Search in Google Scholar

Chu, C. S. J., K. Hornik, and C. M. Kuan. 1995b. “MOSUM Tests for Parameter Constancy.” Biometrika 82 (3): 603–17.https://doi.org/10.1093/biomet/82.3.603.Search in Google Scholar

Dow, J., and S. R. da Costa Werlang. 1992. “Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio.” Econometrica 60 (1): 197–204.https://doi.org/10.2307/2951685.Search in Google Scholar

Ellsberg, D. 1961. “Risk, Ambiguity, and the Savage Axioms.” The Quarterly Journal of Economics 75 (4): 643–69.https://doi.org/10.2307/1884324.Search in Google Scholar

Epstein, L. G., and M. Schneider. 2007. “Learning under Ambiguity.” The Review of Economic Studies 74 (4): 1275–303.https://doi.org/10.1111/j.1467-937x.2007.00464.x.Search in Google Scholar

Epstein, L. G., and M. Schneider. 2010. “Ambiguity and Asset Markets.” Annual Review of Financial Economics 2 (1): 315–46.https://doi.org/10.1146/annurev-financial-120209-133940.Search in Google Scholar

Fountain, J., and G. W. Harrison. 2011. “What Do Prediction Markets Predict?” Applied Economics Letters 18 (3): 267–72.https://doi.org/10.1080/13504850903559575.Search in Google Scholar

Gandhi, A., and R. Serrano-Padial. 2015. “Does Belief Heterogeneity Explain Asset Prices: The Case of the Longshot Bias.” The Review of Economic Studies 82 (1): 156–86.https://doi.org/10.1093/restud/rdu017.Search in Google Scholar

Gilboa, I., and D. Schmeidler. 1989. “Maxmin Expected Utility with Non-Unique Prior.” Journal of Mathematical Economics 18 (2): 141–53.https://doi.org/10.1016/0304-4068(89)90018-9.Search in Google Scholar

Gjerstad, S. 2005. Risk Aversion, Beliefs, and Prediction Market Equilibrium. Economic Science Laboratory Working Paper.Search in Google Scholar

Guidolin, M., and F. Rinaldi. 2010. “A Simple Model of Trading and Pricing Risky Assets under Ambiguity: any lessons for policy-makers?” Applied Financial Economics 20 (1-2): 105–35.https://doi.org/10.1080/09603100903262939.Search in Google Scholar

Huber, P. J., and E. M. Ronchetti. 2009. Robust Statistics, 3rd ed. Hoboken, NJ: Wiley.Search in Google Scholar

Kahneman, D., and A. Tversky. 1979. “Prospect Theory: An Analysis of Decision under Risk.” Econometrica 47 (2): 263–91.https://doi.org/10.2307/1914185.Search in Google Scholar

Keynes, J. M. 1921. A Treatise on Probability. London: Macmillan.Search in Google Scholar

Klibanoff, P., M. Marinacci, and S. Mukerji. 2005. “A Smooth Model of Decision Making under Ambiguity.” Econometrica 73 (6): 1849–92.https://doi.org/10.1111/j.1468-0262.2005.00640.x.Search in Google Scholar

Knight, F. H. 1921. Risk, Uncertainty and Profit. Boston, MA: Houghton Mifflin Co.Search in Google Scholar

Manski, C. F. 2006. “Interpreting the Predictions of Prediction Markets.” Economics Letters 91 (3): 425–29.https://doi.org/10.1016/j.econlet.2006.01.004.Search in Google Scholar

Marinacci, M. 2002. “Learning from Ambiguous Urns.” Statistical Papers 43 (1): 143–51.https://doi.org/10.1007/s00362-001-0092-5.Search in Google Scholar

Milgrom, P., and N. Stokey. 1982. “Information, Trade and Common Knowledge.” Journal of Economic Theory 26 (1): 17–27.https://doi.org/10.1016/0022-0531(82)90046-1.Search in Google Scholar

Quandt, R. E. 1986. “Betting and Equilibrium.” The Quarterly Journal of Economics 101 (1): 201–07.https://doi.org/10.2307/1884650.Search in Google Scholar

Routledge, B. R., and S. E. Zin. 2009. “Model Uncertainty and Liquidity.” Review of Economic Dynamics 12 (4): 543–66.https://doi.org/10.1016/j.red.2008.10.002.Search in Google Scholar

Shin, H. S. 1992. “Prices of State Contingent Claims with Insider Traders, and the Favourite-Longshot Bias.” The Economic Journal 102 (411): 426–35.https://doi.org/10.2307/2234526.Search in Google Scholar

Snowberg, E., and J. Wolfers. 2010. “Explaining the Favorite–Long Shot Bias: Is it Risk-Love or Misperceptions?” Journal of Political Economy 118 (4): 723–46.https://doi.org/10.1086/655844.Search in Google Scholar

Sutherland, W. A. 1975. Introduction to Metric and Topological Spaces. Oxford: Clarendon Press.Search in Google Scholar

Thaler, R. H., and W. T. Ziemba. 1988. “Anomalies: Parimutuel Betting Markets: Racetracks and Lotteries.” Journal of Economic Perspectives 2 (2): 161–74.https://doi.org/10.1257/jep.2.2.161.Search in Google Scholar

Ulrich, M. 2013. “Inflation Ambiguity and the Term Structure of U.S. Government Bonds.” Journal of Monetary Economics 60 (2): 295–309.https://doi.org/10.1016/j.jmoneco.2012.10.015.Search in Google Scholar

Weitzman, M. 1965. “Utility Analysis and Group Behavior: An Empirical Study.” Journal of Political Economy 73 (1): 18–26.https://doi.org/10.1086/258989.Search in Google Scholar

Wolfers, J., and E. Zitzewitz. 2006. Interpreting Prediction Market Prices as Probabilities. National Bureau of Economic Research Working Paper 12200.Search in Google Scholar