Ask The Wizard #408
It seems I can arbitrage the winner of the 2024 presidential election by betting on Harris at Pinnacle and Trump on PredictIt. Am I right? If I want to lock in the same profit regardless of who wins, how should I divide my bets?
You're absolutely right! Here are the odds if we convert them to the European decimal format, which shows how much you get back (including your original wager) for one unit bet.
| Side | Pinnacle | PredictIt |
|---|---|---|
| Harris | 2.490000 | 2.127660 |
| Trump | 1.574713 | 1.886792 |
You can tell there is an arbitrage situation is if the sum of the inverses of the best you can get on both sides is less than 1. In this case, (1/2.49) + (1/1.886792) =~ 0.931606.
The inverse of this sum is the expected return you can expect, including return of winning wagers, if you bet in such a way to lock in the same profit regardless of who wins. In this case 1/0.931606 = 1.073415. That means for every dollar you bet, you'll get back $1.07, or a 7.34% advantage.
To balance your bets in this situation, for every dollar you bet on Harris at Pinnacle, bet $1.32 on Trump at PredictIt. In other words, bet 43.1% on Harris and 56.9% on Trump.
In 2020, there were seven states that were decided by less than 3%. Assuming each of these could go to either Harris or Trump with a 50/50 chance each, what is the probability each candidate wins and an electoral college tie?
Here are the 2016 electoral college votes, counting only states with more than a 3% winning margin. The rest I count as a "battleground."
- Trump: 217
- Biden: 227
- Battleground: 94
Of the 94 battleground votes, Trump got 15 votes (North Carolina) and Biden got the rest, making the final totals:
- Trump: 232
- Biden: 306
The following table shows the percent of the vote each candidate received in 2020, the difference (Biden minus Trump), and the electoral votes for the state.
| State | Biden | Trump | Difference | Votes |
|---|---|---|---|---|
| North Carolina | 48.59% | 49.93% | -1.34% | 15 |
| Arizona | 49.36% | 49.06% | 0.30% | 11 |
| Wisconsin | 49.45% | 48.82% | 0.63% | 10 |
| Georgia | 49.47% | 49.24% | 0.23% | 16 |
| Pennsylvania | 49.85% | 48.69% | 1.16% | 20 |
| Nevada | 50.06% | 47.67% | 2.39% | 6 |
| Michigan | 50.62% | 47.84% | 2.78% | 16 |
If we assume each of these seven states were decided by a coin flip, then here is the probability of each outcome.
- Harris wins = 60.16%
- Trump wins = 36.72%
- Tie = 3.12%
In case you're wondering, here are the scenarios resulting in an electoral college tie, listed as those states Trump winning:
- Wisconsin, Georgia, Pennsylvania, Nevada
- North Carolina, Nevada, Michigan
- North Carolina, Arizona, Pennsylvania, Nevada
In case you're wondering, in the event of an electoral college tie, the House of Representatives would have a special election to decide the president (and the Senate deciding the vice president), with each state getting one vote. I do not know how each state delegation decides who gets this vote. The fair thing to do would be to give it to the winner of that state. If that happened in 2024, Trump would likely get the majority of the states. I personally am curious what would happen if that vote was a 25-25 tie.
Imagine there are two busses, as follows:
- Bus A arrives at the bus stop exactly once an hour.
- Bus B arrives at a random time, uniformly distributed, every hour, with the range starting and ending on the hour (for example 3:00 PM to 4:00 PM).
Assuming no passengers own a watch and simply choose a bus and then wait for the next one to arrive. Your questions are as follows:
- What is the average time between arrivals for bus A?
- What is the average time between arrivals for bus B?
- What is the waiting time for bus A?
- What is the waiting time for bus B?