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.
Mathematical Appendix
Proof of Proposition 1
Note that any equilibrium price π has to satisfy (i)
Substitute (2) into (3) and rewrite the aggregate demand as
Hence,
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
for any
Note that
F is non-decreasing since it is a distribution function. It follows that
We prove the results by examining two cases.
Case 1:
Given
where
and
It is easily verified that G is a mean-preserving spread of F, with two new atoms created at points
where the last equality holds because
for any
Case 2:
Given
where
and
It is easily verified that H is a mean-preserving spread of F, with two new atoms created at points
where the last but second equality holds because
Proof of Proposition 3
Decompose
and
Similarly, an increase in ϵ shifts the demand curve inwards (i.e.,
Proof of Proposition 4
Let
where
where the second equality follows from integration by parts. Since
Case 1:
The equilibrium condition is rewritten as
Rearranging terms and dividing both sides by
Case 2:
The equilibrium condition is rewritten as
Rearranging terms yields
Note that the left-hand side of equation (A1) is strictly increasing in π. Thus, the solution
Next, we show
Since
Case 3:
The equilibrium condition is rewritten as
Rearranging terms yields
Similar to Case 2, the solution
Next, we show
where the last inequality holds because
Proof of Corollary 2
Recall that
The symmetry of
to which
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.
Event category | Number of markets |
---|---|
Art | 60 |
Business | 43 |
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
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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