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Tradesports Arbitrage

Has Tradesports internally mispriced the chance of the Democrats retaking the Senate? That is, do the chances of the Dems winning the individual races they need "add up" to the same number as the chances of them winning back the chamber as a whole? Mind you, this is an entirely difference question from whether or not Tradesports has accurately priced the contracts. I'm only looking at whether or not Tradesports is internally consistent with itself.

(I'm not the first one to think about this. A Google search turned up this somewhat amateurish attempt, along with these more sophisticated ones.)

Tradesports is a futures market based on real-world events. If you buy a contract, you pay, say $0.56 for a contract stating that the Republicans will hold the House. If they do, the contract expires with a value of $1.00, and you make $0.19. If Speaker Bela Pelosi is sworn in on Jan. 1, then the contract, like promises to control spending and defend the country, expire worthless, and you lose your $0.81. Ideally, the sum of the prices on a given event should equal 1. And the prices of equivalent events should equal each other. If they don't then an arbitrage opportunity exists, which means free money, which means drinks for everyone.

For instance, if the contract giving control to the Republicans is selling for $0.49, and the contract giving control to the Democrats is selling for $0.49, then you only have to pay $0.98 to buy one of each, kind of like what business is doing.

As for the notion of equivalent events, here's a simple example made complex. Suppose I can bet on two flips of a coin. I can bet on each flip by itself, and I can bet on the end result of both flips. In the real world, there's a 50% chance of getting heads, a 50% chance of getting tails, and a 25% chance of getting tails both times. So the prices of the contracts for tails on the first flip = 0.50, tails on the second flip = 0.50, and tails both times, 0.25. Betting the two tails separately is the same event as betting the two tails together.

Now suppose the market thinks that tails is a 60% likelihood, or a 2-3 bet. The chance of two tails should be priced at $0.36. (0.6 x 0.6 = 0.36) If it's not, if it's still priced at $0.25, then there's an arbitrage opportunity based on the idea that the market will discover & correct this discrepancy. Either the 0.25 is right, and tails is overpriced, or the 0.6 is right, and the combination is under-priced, or they're both wrong. But the two numbers are inconsistent with each other.*

So. When I did the math earlier, the contract for the Senate remaining Republican was selling at $0.816. If you take all the possible Senate race outcomes, and multiply them together, and add up the probabilities of those combinations that give the Democrats the Senate, do you get $0.816? If not, there's an arbitrage opportunity, because the market's not pricing the equivalent events the same.

Excel's a wonderful thing.

There are 33 Senate races contested this year. Currently, the Dems hold 18 of the seats, the Republicans 15. To take the Senate, the Dems need to pick up 6 seats. That means after Nov. 7, they need to have 24 of these seats to the Republicans' 9. I found the contract prices for the Democrat and the Republican in each of these races (counting Lieberman as a Democrat). Now, with 33 races, there are 2^33, or 8,589,934,592 possible combinations. In order to simplify things, I took as given any races where one party or another was judged to have a 95% or greater. Of those races, 13 go to Dems (also giving them the Socialist Bernie Sanders), and 6 to Reps. This means that Dems need to win 11 of the remaining 14 races to get to 24.

The contested races (and the Democrat's contract prices) are: Arizona (.09), Maryland (.65), Michigan (.90), Minnesota (.90), Missouri (.48), Montana(.80), Nevada (.09), New Jersey (.43), Ohio (.76), Pennsylvania (.84), Rhode Island (.80), Tennessee (.35), Virginia (.40), and Washington (.88). With 14 races, the number of possible outcomes is only 2^14, or 16384, which Excel can handle. If you write down all the possible outcomes, calculate the probability of each - as determined by Tradesports traders - and add up the likelihoods of those outcomes where the Dems win 11 seats, you get... 7.27%

This means that collectively, Tradesports prices the chances of the Dems taking the Senate at 18.4%, but taken race-by-race, they only get a 7.27% chance. Now, it's clear that the individual elections are not independent events, even though Tradesports lets you bet on them that way. A single event of national significance could swing voters all over the country, affecting every race that's in play, and given the Dems' percentages in the races in play, it would take a bigger event to swing the electorate Republican than to tip, say, Tennessee and Virginia to the Democrats.

If I move the cutoff to 0.80 from 0.95, that tips Arizona & Nevada to the Republicans, and Michigan, Minnesota, Montana, Pennsylvania, Rhode Island, and Washington to the Dems. Even then, the percentage only moves to 10.88%.

But overall, Tradeports seems to be saying either that 1) it can't find the races to put the Dems over the top, or 2) some individual Republicans are getting a benefit of the doubt they don't deserve.

Sooner or later, the market's got to figure this out. Doesn't it? If it does, the beer's on me.

*If the coin actually is fair, and the contracts are consistent, that doesn't mean you hedge perfectly by buying one and selling the other. That's because you're adding the results of the two individual tosses to determine one payout, and multiplying the results to determine the other payout. If anyone's interested, I email me and I'll send you the math.


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