## Background

Predictit no longer has probabilities by states, but Smarkets, Betfair, and Polymarket do. The Betfair and Polymarket probabilities sum to 100, but the Smarkets probabilities do not. For Smarkets, the following was done. Let Di be the Smarkets probabiity that the Democrats win state i, and let Ri be the Smarkets probability that the Repubicans win state i. The Democratic probability was then taken to be Di/(Di + Ri).There are 8 states in play for the 2024 presidential election. Table 1 lists in alphabetic order the states assumed to go Democratic, the 8 swing states, and in alphabetic order the states assumed to go Republican. The swing states are ranked by the Smarkets probability for the Democrats. The Betfair probabilities are in parentheses, and the Polymarket probabilities are in brackets. The following is a discussion of the data.

## Data collected from Smarkets, (Betfair), and [Polymarket] at 5pm EST on September 24, 2024

The 8 swing states are:

state
| prob
| votes
| sumvotes
| |

Michigan | 65.0 (66) [68] | 15 | 241 | |

Wisconsin | 62.3 (63) [58] | 10 | 251 | |

Pennsylvania | 54.3 (53) [52] | 19 | 270 | pivot |

Nevada | 51.6 (53) [51] | 6 | ||

Georgia | 41.7 (39) [38] | 16 | ||

North Carolina | 40.9 (43) [44] | 16 | ||

Arizona | 39.4 (37) [37] | 11 | ||

Florida | 12.2 (13) [15] | 30 | ||

52.3, (53.6), [51.0] = Smarkets, (Betfair), [Polymarket] market probability that the Democrats win the presidential election.

"sumvotes" is the sum of the electoral votes of all the states ranked above the state plus the state's vote. (See Table 1.) 270 votes are needed to win. You can see that Pennsylvania is the pivot state. If the Democrats take Pennsylvania and all the states ranked above it, they get 270 votes (assuming 1 vote from Nebraska). Of the states ranked below Pennsylvania, the Democrats could also win by not taking Pennsylvania, but, say, taking Nevada and Georgia. This would, of course, violate the ranking assumption.

According to the ranking assumption, the probability that the Democrats get a majority in the electoral college is the probability that they win the pivot state, Pennsylvania, which is 54.3 for Smarkets, 53 for Betfair, and 52 for Polymarket. The market probability that the Democrats win the presidential election (in the electoral college) is 52.3 for Smarkets, slightly lower than 54.3. For Betfair the two numbers are 53.6 and 53. For Polymarket the two numbers are 51.0 and 52. It is thus roughly the case that the behavior of the market participants is consistent with their using the ranking assumption.

## Data collected from Smarkets and Betfair at 5pm EST on September 11, 2024

The 8 swing states are:

state
| prob
| votes
| sumvotes
| |

Michigan | 60.2 (61) | 15 | 241 | |

Wisconsin | 60.1 (60) | 10 | 251 | |

Nevada | 50.6 (52) | 6 | 257 | |

Pennsylvania | 49.5 (51) | 19 | 276 | pivot |

Georgia | 42.0 (40) | 16 | ||

Arizona | 39.2 (42) | 11 | ||

North Carolina | 35.0 (42) | 16 | ||

Florida | 11.9 (15) | 30 | ||

52.3 (51.4) = Smarkets (Betfair) market probability that the Democrats win the presidential election.

"sumvotes" is the sum of the electoral votes of all the states ranked above the state plus the state's vote. (See Table 1.) 270 votes are needed to win. You can see that Pennsylvania is the pivot state. If the Democrats take Pennsylvania and all the states ranked above it, they get 276 votes. Of the states ranked below Pennsylvania, the Democrats could also win by not taking Pennsylvania, but, say, taking Georgia. This would, of course, violate the ranking assumption.

According to the ranking assumption, the probability that the Democrats get a majority in the electoral college is the probability that they win the pivot state, Pennsylvania, which is 49.5 for Smarkets and 51 for Betfair. The market probability that the Democrats win the presidential election (in the electoral college) is 52.3 for Smarkets, higher than 49.5. For Betfair the two numbers are 51.4 and 51. The Betfair numbers are thus consistent with the market participants using the ranking assumption, but not Smarkets.