# 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 > 2 ϵ and (ii) π < 1 2 ϵ . If (i) does not hold, then π * q i + 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

X F ( π ) = ϵ π ϵ q + ϵ π π ( 1 π ) w d F ( q ) + π + ϵ 1 ϵ q ϵ π π ( 1 π ) w d F ( q ) .

Hence, X F ( π ) = 0 if and only if

ϵ π ϵ ( q + ϵ π ) d F ( q ) + π + ϵ 1 ϵ ( q ϵ π ) d F ( q ) = 0

ϵ π ϵ ( q π ) d F ( q ) + π + ϵ 1 ϵ ( q π ) d F ( q ) + ϵ π ϵ ϵ d F ( q ) π + ϵ 1 ϵ ϵ d F ( q ) = 0

E F ( q ) π π ϵ π + ϵ ( q π ) d F ( q ) + ϵ [ F ( π ϵ ) + F ( π + ϵ ) 1 ] = 0

E F ( q ) π + π ϵ π + ϵ F ( q ) d q [ ( q π ) F ( q ) ] π ϵ π + ϵ + ϵ [ F ( π ϵ ) + 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 , 1 ϵ ] , endowed with the Lévy metric , where

( G 1 , G 2 )

inf { ε > 0 | G 1 ( q ε ) ε G 2 G 1 ( q + ε ) + ε for all q [ ϵ , 1 ϵ ] }

for any G 1 , G 2 G . Let F be the subset of G that satisfies π F * = E F ( q ) for any F F . 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 F F and any δ > 0 , there exists some G G \ F such that ( F , G ) < δ .

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

lim q [ E F ( q ) + ϵ ] F ( q ) F ( E F ( q ) ϵ ) .

We prove the results by examining two cases.

Case 1: lim q [ E F ( q ) + ϵ ] F ( q ) > F ( E F ( q ) ϵ ) .

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

G ( q ) { F ( q ) if q [ ϵ , E F ( q ) ϵ δ 1 ) , F ( E F ( q ) ϵ ) if q [ E F ( q ) ϵ δ 1 , E F ( q ) + ϵ + δ 2 ) , F ( q ) if q [ E F ( q ) + ϵ + δ 2 , 1 ϵ ] ,

where δ 1 , δ 2 > 0 are such that function g G F satisfies conditions

E F ( q ) ε δ 1 E F ( q ) + ε + δ 2 g ( q ) d q = 0

and

max { g ( E F ( q ) ϵ δ 1 ) , g ( E F ( q ) + ϵ + δ 2 ) } = δ 2 .

It is easily verified that G is a mean-preserving spread of F, with two new atoms created at points E F ( q ) ϵ δ 1 and E F ( q ) + ϵ + δ 2 . By construction, this implies that

E G ( q ) ϵ E G ( q ) + ϵ G ( q ) d q = E F ( q ) ϵ E F ( q ) + ϵ G ( q ) d q = E F ( q ) ϵ E F ( q ) + ϵ [ F ( q ) + g ( q ) ] d q = ϵ + E F ( q ) ϵ E F ( q ) + ϵ g ( q ) d q < ϵ ,

where the last equality holds because F F , and the inequality holds because g ( E F ( q ) + ϵ ) < 0 , which implies E F ( q ) ϵ E F ( q ) + ϵ g ( q ) d q < 0 . Since E G ( q ) ϵ E G ( q ) + ϵ G ( q ) d q < ϵ , G G \ F . Finally, let ρ be the uniform metric, that is,

ρ ( G 1 , G 2 ) sup { | G 1 ( q ) G 2 ( q ) q [ ϵ , 1 ϵ ] }

for any G 1 , G 2 G . By construction, ρ ( F , G ) = δ 2 . Since the Lévy metric is bounded by the uniform metric from above, that is, ( G 1 , G 2 ) ρ ( G 1 , G 2 ) for any G 1 , G 2 G , we have ( F , G ) δ 2 < δ .

Case 2: lim q [ E F ( q ) + ϵ ] F ( q ) = F ( E F ( q ) ϵ ) .

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

H ( q ) { F ( q ) if q [ ϵ , E F ( q ) ϵ ) , F ( E F ( q ) ϵ ) + δ 3 if q [ E F ( q ) ϵ , E F ( q ) + ϵ + δ 4 ) F ( q ) if q [ E F ( q ) + ϵ + δ 4 , 1 ϵ ] ,

where δ 3 , δ 4 > 0 are such that function h H F satisfies conditions

E F ( q ) ϵ E F ( q ) + ϵ + δ 4 h ( q ) d q = 0

and

max { δ 3 , h ( E F ( q ) + ϵ + δ 4 ) } = δ 2 .

It is easily verified that H is a mean-preserving spread of F, with two new atoms created at points E F ( q ) ϵ and E F ( q ) + ϵ + δ 4 . By construction, this implies that

E G ( q ) ε E G ( q ) + ε H ( q ) d q = E F ( q ) ε E F ( q ) + ε H ( q ) d q = E F ( q ) ε E F ( q ) + ε [ F ( q ) + h ( q ) ] d q = ε + E F ( q ) ε E F ( q ) + ε h ( q ) d q = ε + 2 ε δ 3 > ε ,

where the last but second equality holds because F F , and the last equality follows from the construction of H. Since E G ( q ) ϵ E G ( q ) + ϵ H ( q ) d q > ϵ , H G \ F . Finally, similar to Case 1, we have ρ ( F , H ) = δ 2 and, hence, ( F , H ) < δ . Q.E.D.

### Proof of Proposition 3

Decompose X F ( π ) into the aggregate supply (shorts) S F ( π ) and the aggregate demand (longs) D F ( π ) , where

S F ( π ) = ε π ε q + ε π π ( 1 π ) w d F ( q ) , D F ( π ) = π + ε 1 ε q ε π π ( 1 π ) w d F ( q ) ,

and S F ( π F * ) = D F ( π F * ) in equilibrium. We show that an increase in ϵ shifts the supply curve inwards. That is,

d S F ( π ) d ϵ = 0 + ϵ + ϵ π π ( 1 π ) w d F ( ϵ ) ϵ π ϵ ϵ q + ϵ π π ( 1 π ) w d F ( q ) < 0.

Similarly, an increase in ϵ shifts the demand curve inwards (i.e., d D F ( π ) d ϵ < 0 ). It follows that the equilibrium quantity of trade— S F ( π F * ) , or D F ( π F * ) —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 ) ε q F ( q ) d q . It follows from the definition of F that

Φ ( q ) = ε q F ( q ) d q = { ( 1 m ) Φ ( q ) i f q [ ε , p ) ( 1 m ) Φ ( q ) + m ( q p ) i f q [ p , 1 ε ] ,

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

π = E F ( q ) + Φ ( π + ϵ ) Φ ( π ϵ ) ϵ = 1 2 ϵ Φ ( 1 ϵ ) + Φ ( π + ϵ ) Φ ( π ϵ ) ,

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 π ϵ .[16]

Case 1: π ϵ p π + ϵ .

The equilibrium condition is rewritten as

π = 1 2 ϵ ( 1 m ) Φ ( 1 ϵ ) m ( 1 ϵ p ) + ( 1 m ) Φ ( π + ϵ ) + m ( π + ϵ p ) ( 1 m ) Φ ( π ϵ ) .

Rearranging terms and dividing both sides by 1 m yields

π Φ ( π + ε ) + Φ ( π ε ) = 1 2 ε Φ ( 1 ε ) .

Case 2: p > π + ϵ .

The equilibrium condition is rewritten as

π = 1 2 ε ( 1 m ) Φ ( 1 ε ) m ( 1 ε p ) + ( 1 m ) Φ ( π + ε ) ( 1 m ) Φ ( π ε ) .

Rearranging terms yields

(A1) π 1 m Φ ( π + ε ) + Φ ( π ε ) = 1 2 ε Φ ( 1 ε ) + ( p ε ) m 1 m .

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 p π ^ + ϵ , where π ^ is the equilibrium price in Case 1, the right-hand side of equation (A1) converges to 1 2 ϵ Φ ( 1 ϵ ) + π ^ m 1 m , 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 * ( 1 ϵ ) < 1 2 ϵ , which implies π * ( 1 ϵ ) < 1 ϵ in part 1 of the proposition. Let L H S ( π ) and R H S ( p ) denote the left- and right-hand sides of equation (A1), as functions of π and p, respectively. Note that

L H S ( 1 2 ε ) R H S ( 1 ε ) = [ 1 2 ε 1 m Φ ( 1 ε ) + Φ ( 1 3 ε ) ] [ 1 2 ε Φ ( 1 ε ) + ( 1 2 ε ) m 1 m ] = Φ ( 1 3 ε ) > 0.

Since L H S is strictly increasing in π, the solution to the equation when p = 1 ϵ  must be smaller than 1 2 ϵ .

Case 3: p < π ϵ .

The equilibrium condition is rewritten as

π = 1 2 ε ( 1 m ) Φ ( 1 ε ) m ( 1 ε p ) + ( 1 m ) Φ ( π + ε ) + m ( π + ε p ) ( 1 m ) Φ ( π ε ) m ( π ε p ) .

Rearranging terms yields

(A2) π 1 m Φ ( π + ε ) + Φ ( π ε ) = 1 2 ε Φ ( 1 ε ) + ( p + ε ) m 1 m .

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

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

L H S ( 2 ε ) R H S ( ε ) = [ 2 ε 1 m Φ ( 3 ε ) + Φ ( ε ) ] [ 1 2 ε Φ ( 1 ε ) + 2 ε m 1 m ] = [ Φ ( 1 ε ) Φ ( 3 ε ) ] [ ( 1 ε ) 3 ε ] < 0 ,

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

### Proof of Corollary 2

Recall that π ^ is identified by equation

π ^ Φ ( π ^ + ε ) + Φ ( π ^ ε ) = 1 2 ε Φ ( 1 ε ) .

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

π ^ [ π ^ + Φ ( 1 π ^ ϵ ) ] + Φ ( π ^ ϵ ) = 1 2 ϵ [ 1 2 ϵ + Φ ( ϵ ) ]

Φ ( 1 π ^ 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 category Number of markets
Art 60
Chess 52
Climate & Weather 861
Construction & Engineering 9
Current Events 1540
Education 1
Entertainment 8715
Fine Wine 5
Foreign Affairs 87
Legal 310
Media 10
Politics 5460
Real Estate 2
Science 20
Social & Civil 30
Technologies 65
Transportation 11

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 category Total observations Observations per bin
(50 bins) (30 bins)
Politics 897 18 30
Entertainment 1157 23 39
Full sample 2509 50 84

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