# 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) *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

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 ^{[14]} the proposition is equivalent to the claim that

Note that ^{[15]} it remains to be shown

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

We prove the results by examining two cases.

**Case 1: **

Given *G* from *F* as

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 *ρ* be the uniform metric, that is,

for any

**Case 2: **

Given *H* from *F* as

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 *H*. Since

### Proof of Proposition 3

Decompose

and *ϵ* shifts the supply curve inwards. That is,

Similarly, an increase in *ϵ* shifts the demand curve inwards (i.e.,

### Proof of Proposition 4

Let *F*, i.e., *F* that

where

where the second equality follows from integration by parts. Since *p*, the equilibrium price depends on the position of *p* relative to ^{[16]}

**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 *p*, respectively. Note that

Since

**Case 3: **

The equilibrium condition is rewritten as

Rearranging terms yields

Similar to Case 2, the solution *p*, and it converges to

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.

### Table 4:

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

### Table 5:

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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**Received:**2019-04-27

**Accepted:**2020-03-27

**Published Online:**2020-08-07

© 2020 Walter de Gruyter GmbH, Berlin/Boston