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Published by De Gruyter August 7, 2020

# On the Observational Implications of Knightian Uncertainty

Kevin A. Hassett and Weifeng Zhong

# 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,[14] 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,[15] 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.[16]

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
Business43
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

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Received: 2019-04-27
Accepted: 2020-03-27
Published Online: 2020-08-07

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