Damn Lies, Dismal Science and the Electoral College

Bruce D. Kothmann
10 August 2004

I recently made a wonderful discovery: the political markets at TradeSports.com. Here is how it works: Tradesports defines "contracts" which offer a fixed payment of $10 based on the outcome of a future event. For example, there is a "Bush Wins" contract that will pay out if President Bush is reelected in November. Tradesports does not buy or sell these contracts, but only acts as a broker to allow their users to do so: users may post offers to buy or sell "Bush Wins" contracts; Tradesports merely matches buyers with sellers. If someone is offering to buy "Bush Wins" for $6 and you offer to sell a contract for $6, Tradesports will take $6 from the buyer's account, and $4 from your account, placing the total of $10 in a sort of escrow. (Note that selling "Bush Wins" is exactly like buying "Bush Doesn't Win"; you don't have to own a share of "Bush Wins" to sell it!) When the final outcome is known, the winner gets the $10 from the escrow. (Tradesports also takes a small transaction fee from buyer and seller; this is how they make money.) For convenience, the posted prices of contracts are multiplied by 10, so that a contract bought for $5.53 is posted on the trading board as a price of 55.3 units.

An {argument} can be made that the price of a contract on "Bush Wins" will on average be the same as the price for a contract on a random event, if the probability of the random event (in percent) were equal to the price of the contract (in units). In this sense, the "probability" of Bush winning the election is equal to the price of a contract on "Bush Wins". (In the formal mathematical sense, speaking of the "probability" that Bush is reelected is meaningless: presidential elections are not random events that can be repeated over and over again until sufficient data are gathered to arrive at an accurate estimate of the liklihood of a given outcome. Nevertheless, it must be true in some lay sense that the "probability" of Bush being sworn in next January is higher than the "probability" of Ralph Nader taking the oath, and the market price of a contract of "Bush Wins" provides a very useful quantitative notion of the "odds" of Bush winning.)

Okay, so you can have fun going to Tradesports and betting on who will win in November. But you can have a lot more fun than that. It turns out that Tradesports offers contracts not only on Bush winning the entire election, but also on Bush winning each state! These data can be used to generate a map of the united states, with each state colored according to the probability that Bush wins that state:

(See the archive for a complete history of such maps.) The solid bar on the right shows the number of electoral votes that Bush would get if he won every state where he is selling at or above 50.0. The dashed bar on the right shows the "expected value" of electoral votes for Bush, which is the sum of the probability of Bush winning each state multiplied by the number of electoral votes in that state. The expected value has the advantage that it doesn't show jump changes when a state fluctuates around 50.0 in the market, but it has the disadvantage that Bush's roughly 10-percent chance of winning California's 55 electoral votes counts for 5.5 "expected" electoral votes--more than the actual 3 he will get from winning Wyoming. On balance, I prefer the "expected" total, because it moves with changes in the market prices of Ohio and Florida, even if these don't cross 50.

It is interesting to compare the above continuously varying map coloration to more conventional election maps, which typically give only a rough indication of how well the predicted outcome in a given state is known.

There is more fun to be had with the state-by-state data than merely drawing maps. Treating the outcome in each individual state as a statistically independent random event with a probability equal to the price, you can apply elementary probability theory to compute the odds of Bush receiving any given number of electoral votes. (Anyone interested in the details, please let me know.):

(Aside: That this curve looks like the prototypical "bell curve" (Gaussian) is a demonstration of the central value theorem, which states that any event that is the result of combining many other random events with arbitrary density functions (in this case, the very non-Gaussian all-or-nothing densities of the 50 states plus DC) will approach a Gaussian!) Note that there is a slightly higher than 1-percent chance that the election will result in a tie (269-269), in which case the outcome would be decided by a vote in the House of Representatives (which the Tradesports market shows has a roughly 90-percent chance of remaining Republican). A tie is not at all far-fetched: if Kerry wins all the states that Gore won plus New Hampshire and West Virginia, the split at 269 is a reality. (Try it Yourself!)

Finally, the probability density function can be integrated to obtain a probability distribution function, which tells the odds that Bush will receive at least a certain number of electors:

Also shown on this graph are the prices of the Tradesports contracts for Bush receiving at least a certain number of electors (e.g., the contract on "Bush Gets at Least 200 Electors" was selling for 84). The price of the "Bush Wins" contract has been plotted at 269, because Bush will win if he gets at least 269 electors (assuming a successful tie-breaker in the House). You will notice that the overall election outcome prices do not preciesly match the integrated state-by-state prices (although the trend is certainly right). At first, it appears that this reveals an inconsistency in the market, and implies that you should buy at the low end and sell at the high end. But the stronger argument is that the market knows that the outcomes in each state are not independent random events, as postulated for this calculation. In other words, if North Carolina is a 75-percent Bush state, and South Carolina is an 80-percent Bush state, the chances of Bush winning both of them is not 60 percent (i.e. 75-percent of 80-percent), but probably closer to 75 percent, because if some rare event were to cause North Carolina to vote against Bush, it is quite likely that South Carolina would go against him, too. Therefore, the market prices show a larger "spread" in the distribution, due to this "bundling" of likely outcomes. The market also has a higher probability of Bush winning the overall contest outright, perhaps because he holds slim margins in Ohio and Florida, so that Kerry's statistically expected electoral votes may not actually materialize in the single realization of the election outcome.

Some other interesting notes:

• The Tradesports price of "Bush Wins the Popular Vote" is between 5 and 10 percent lower than the price of "Bush Wins". The market is predicting what is, historically speaking, the quite unusual outcome of a popular vote loss with an electoral college win!
• There is no "Kerry Wins" contract (Ralph Nader aside, you effectively buy this by selling "Bush Wins"). But for some reason, there is a "Kerry Wins the Popular Vote" contract, as well as a "Someone Else Wins the Popular Vote" contract. Suppose that each of these 3 contracts were selling for 50. You could sell one of each, and, since at most one can pay out, have an automatic profit of 50. On the other hand, if the 3 contracts were selling for 25, you could sell and have an automatic profit of 25. It is a testament to the market mechanisms that the prices always adjust themselves so that the sum of the prices of these 3 contracts always comes out very close to 100!

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Links

President Elect

This is a fantastic site for presidential election information, including colored maps of all previous elections and interesting commentary and discussion.

Unfutz Blog

This has a wonderful list of links to a great variety of election predictions, most of which show a colored map of some kind.

Sam Wang's State Poll Meta Analysis

This site does with polling data everything I do with the TradeSports data, and more! His calculations require the extra step of converting state polling results into a probability of a candidate winning. He also has a groovy red-blue map with the state sizes distorted to reflect the number of electors they control. You can see the nearly even split very clearly.

Mystery Pollster

This is a great blog with information about polls.

Rasmussen Reports

This has excellent daily nationwide polling and lots of state-by-state polling, as well as a frequently updated electoral college prediction.

Electoral College Calculator

The Opinion Journal site has a very elegant electoral college calculator that even lets you save your results to update later. Very nice. Another Calculator includes the split-elector possibilities in Maine and Nebraska.

Iowa Electronic Markets

This is an alternative to TradeSports with no commission, but it isn't as good. Their "winner takes all" market is based only on the popular vote, and it doesn't have state-by-state contracts. The market does support academic research, though, for example to compare the relative accuracy of markets and polls.

Federal Register Electoral College Site

This site has exhaustive historical data and other information, including an electoral college FAQ.

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{argument}: Suppose instead of presidential elections, you were betting on the result of a coin flips. If you could buy a $10 payout for "heads" at a cost of only $3, you would probably do it. Why? Because, if you took the bet over and over again, on average, every two times you play (at a cost of $6), you get back one payout of $10, for a total net profit of $4. Your average ("expected") profit per play is $2. As long as the cost of the "contract on heads" is below $5, your "expected" return on investment will be positive.

The problem with this "game", of course, is that no one will sell you a "contract on heads" for less than $5 if they know the coin is honest (i.e., that it has an equal chance of coming up heads or tails). To make the game more interesting, suppose the coin is dishonest, so that it is somehow more likely to come up heads, but you don't know how much more likely. You are sitting at a table with another player, and both of you get to examine the coin, in an effort to discern the likely outcome. Your assessment is that it has been weighted so that heads has a 20-percent chance of hitting. In this case, you might offer to "buy" from the other player a contract on heads for $1, thinking that, on average, for every 10 tosses (and $10 you pay him), heads will hit twice (giving you a payout of $20), and you will make a profit (of $10, or $1 for every toss). But suppose the other player thinks the coin is more heavily weighted, and the odds of heads are only 10-percent. He won't sell for $1, because he thinks that for every 10 tosses (for which he gets $10), heads will hit once (requiring him to pay you $10), and he will only break even. So, instead, he offers to sell for more than $1. As long as his offer is less than $2, you might buy, because by your estimation, you will make a profit. If the negotiated price comes out to $1.50, each player would anticipate an equal average ("expected") profit of $0.50 per toss. If we include a large number of players in the game, there will be a great variety of offers to buy and sell, and the current price will {more or less} reflect the average of these players estimations of the probability of a contract payout (that is, of the coin coming up heads).

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{more or less}: The more I thought about this, the more complicated it seemed. Although setting the price as the average of the two players' estimates of the underlying probability will yield an equal profit in absolute terms, the return on investment will be quite different. Recalling the way Tradesports works, for a contract price of 15, the buyer puts in $1.50, but the seller must contribute $8.50. Therefore, the expected profit of $0.50 represents 33-percent for the buyer but only about 6-percent for the seller. (If both parties look for an equal rate of return on investment, the price would have to be about $1.80.) But the difference in return is consistent with the so-called "risk premium" associated with more volatile investments. Consider what happens if there is a mere 5-unit swing in the market price. If the price goes from 15 to 10, the buyer can sell, but he will lose 33-percent of his investment. On the other hand, if the price goes from 15 to 20, the seller can buy (effectively leaving this market), and he will lose only 6-percent. Because of the potential for much larger losses, the buyer will probably demand a higher average (expected) rate of return. So, perhaps the equal absolute return price is reasonable.

I am also a bit unsure what effect the Tradesports commission has on the price. One question I ended up asking myself was, "Why do people play roulette?" A typical US roulette wheel has 36 slots for the numbers 1 through 36, half red and half black, plus green "0" and "00" slots. If you bet on red or black, the odds of winning are 18 out of 38 (47.3-percent), and the payout is twice your bet. So, your average ("expected") return is about 95 cents for every dollar you bet (a negative 5-percent rate of return). I suppose the "pleasure premium" (the enjoyment of sitting at the roulette wheel and the free drinks, plus the reasonable possibility for modest winnings) offsets the negative rate of return. (By comparison, that people regularly play state-sponsored "daily number" drawings, in which the expected return is usually around negative 50 percent, is just a depressing reflection of the bleak outlook of the poor in America.)

Finally, one might argue that transactions in these markets doesn't really reflect your best guess of what the probability of the election outcome is, but rather your best guess at what other people will think the probability of the election outcome will be. In other words, even if you think Bush is likely to lose, you might buy a contract on "Bush Wins" at 50 if you think that someone else will be willing to buy it from you tomorrow for 60. Ultimately, though, the price must be driven to the actual expected outcome, because that is the basis of the payout. I think the risk of price distortion due to "day trading" in these markets is much lower than it is in real markets--for example, in the pricing of the google IPO. (Speaking of the google IPO, if you want to get in on the action without buying google stock, Tradesports offers contracts on "google closes higher in the first day after IPO"!)

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