Note: I wrote this on Saturday morning before the events of Sunday hilariously vindicated the conclusions below. My point was to illustrate how bets are not probabilities, though they do use them. I examined the simplest market, PredictIt, because it is easiest to explain. Though because of the events, I wish I had used something like Polymarket instead, because they don’t limit bets. As a compromise, I add an update at the end with a Polymarket example. I leave the rest of the text as it stood Saturday.
Prediction markets, so called, are hot again, mostly because of the Presidential election. I was asked on Twitter to comment on what these things mean, and what Nate Silver’s 538 stuff might be in contrast. I pointed my questioner to a recent piece by Scott Locklin with the apt title “Prediction markets are vile midwittery“, and which I think explains prediction markets well.
I don’t think I can improve on his explanation, except to flesh out some material which Locklin assumed, material that is unfamiliar to those who asked me about the markets and who didn’t follow Locklin’s explanation.
PredictIt is one of the big markets in the Presidential Election. As of this writing (last Saturday morning), Yes contracts for Trump winning the election were 64 cents. No contracts were 37 cents. Biden’s Yes contracts were only 16 cents. The Cackler ‘s Yes contracts were going for almost double at 26 cents.
The question is: what do these numbers mean?
Locklin says they’re not predictions. He’s right. They’re bets. It’s gambling. But the numbers can be taken as predictions, as we’ll see, but only because anything can be taken as predictions.
It works like this. Somebody, not PredictIt, but a real person with a PredictIt account, is selling Trump Yes contracts for 64 cents. Others will offer different prices. You can buy them. You can later try to sell them, or you can keep them, waiting for the Event, which is of course the Election. All winning contracts garner 1 dollar. So if you bought a Trump Yes for 64 cents, and Trump loses, you lose your 64 cents. If Trump wins, you win 100 – 64 = 36 cents.
Less the vig, or the vigorish, i.e. the transaction cost. Which is huge and rivals horse betting (which is usually just under 15% in parimutuel betting). PredictIt takes 10% of your profits, then adds a 5% fee on all withdraws. Ouch. If all you had in was that 64 cents, after paying the bookie you only take out only 30 cents (I’m assuming they round down).
PredictIt caps contracts in an event at \$850. You can deposit, say, \$1,000 with them. You can buy 1,328 Trump Yes contracts, which costs \$850, and you can, for instance, let the other \$150 lay fallow. Win or lose on the contracts, you will still pay 5% of that \$150, or \$7.50, to get your unspent money back. That’s a sure loss. Which you’d know, and therefore be you might be tempted to spend it buying contracts, which increases your risk.
You might think it is an idiotic move to let money lay fallow in the account. But you might be caught with contracts you want to sell, or just did, or you are hoping to buy more at a later date. And this might get you: “You cannot withdraw funds until 30 days after your initial deposit. ” Dude. And in any case that 5% withdraw vig looms. (Does even Amex have the cojones to squeeze that much in their credit card transactions?)
Here is a picture of the change in this market. (Bizarrely, they invert our usual red/blue divide; Biden is red.)
Suppose you bought some Trump Yes shares back on 22 April, which then cost about 43 cents. Today those contracts are going for 64 cents. You need money for some project. Or you might be worried those susceptible to propaganda will soon flood the market. Just look at that trading volume (bluish bars) soar! So you might sell your Yes shares now, and get something close to 64 – 43 = 21 cents profit (17 cents after the vig). But if you had a lot of 43 cent shares, selling them all at once will lower the current Yes price. If enough early buyers do this, the price will drop noticeably. Supply and demand, baby.
Does that therefore mean Trump has a lower chance of winning?
PredictIt says so: “The price of a share, between 1 and 99 cents, corresponds to the market’s estimate of the probability of an event taking place. ” And “rationalists” like Scott Alexander insist on it. Locklin links to Alexander’s Prediction Market FAQ, which we’ll critique below. But, of course, the lowered price might not have anything to do with Trump winning or losing, and instead reflects the desire of some sellers to cash out. I shouldn’t have to say, but the “rationalists” make me say, that people need and gamble money for all kinds of reasons. Not all can bet the same amounts, risk tolerance varies, not everybody knows about or cares for prediction markets, and on and on.
The point should be clear. The price is not a direct measure of the chance of a Yes or No, though we grant it is part of it. (Technically, if all bettors used the same decision rule and it was a one-to-one function of probability, we could back out probability; but rules vary by individual bet, so we cannot.) The price is instead a mixture of the profit motives and finances of the gamblers, and on what they think other people will think about Yes and No, not just today but in the future. Most gamblers are not solely thinking of the evidence for or against a Yes (or No). Though, we agree, some of them will be thinking in those terms. Are these the smartest ones? The ones with the best political savvy? How do we separate, from the price alone, how much is caused by intelligence of event and how much from all the other monetary concerns? I have no idea.
RATIONALISTS
Scott Alexander, who belongs to the Curtis “Use More Words: No, More, More, And Still, More” Yarvin School of Writing, has had this quip on his sites for years (go to the bottom): “P(A|B) = [P(A)*P(B|A)]/P(B), all the rest is commentary.” Long time readers and those taking the Class (free! no sign up!) know this is a stunted form of Bayes’s theorem. I don’t know how seriously he takes it. I do know nobody should give it any weight at all.
For two reasons: (1) There is no such thing as P(A), and (2) There is no such thing as P(B). Because nothing has a probability, and probability is not a subjective “feeling”. You only get probabilities with respect to evidence assumed, e.g. Pr(B|E) or Pr(A|C) and it’s the “E”s and “C”s which are important.
The point is relevant here, because Alexander insists those prices are probabilities for the events (and so can be manipulated by his formula).
As said, Trump Yes contracts are going for 64 cents. Harris for 26 cents. Biden for 15 cents. That’s 105 cents. Then there 4 cents for Gruesome Newsom and 6 cents for six other horrible possibilities (like Buttgheigh). That’s 115 cents. If the bets were probabilities, they’d have to sum to 100 cents. So they are not probabilities. There’s the vig again, which has to be accounted for. Tim Scott has a 1 cent Yes contract. Does anybody believe he has a 1% chance of winning? Or even 1/115 chance?
It’s easy enough to adjust the money to fit between 0 and 1: divide the individual’s Yes contract price by the total Yes prices. So Trump is 64/115 ~ 56%. What a difference! (The same is done in horse betting.)
Now, you can take this as a prediction, but as I said, this is because you can take anything as a prediction. You just assume it is. The right hand side of the probability calculations has been supplied. The math works. I can consult my crystal ball (yes, I have one; well, glass, anyway). That’s a prediction, too.
Since we can take these as predictions, it makes sense to ask how good they are. Alexander defines them as “canonical”, i.e, prediction markets “will always be at least as trustworthy as any individual expert, and it cannot be biased.” We have already seen they can be biased, for instance toward people cashing out for any number of reasons, or cynically cashing in, riding news waves, just like in a stock market. It is is also obviously false, or it should be obvious, that markets are as trustworthy as any individual expert. Because the best (small e) expert will be the one who guesses correctly, and does not follow the market. Alexander assumes that such people will automatically seek out prediction markets to lord it over lesser beings. Which is why he trusts the markets.
Yet it’s far from true that everybody will enjoy betting on a prediction market, especially the more intelligent. There’s the huge vig, which drives smart money to better investments. And people’s risk tolerance and attitude to gambling is not uniform. Bets are decisions, they are not probabilities. Bets do use probabilities, but they are not the same as probabilities.
Alexander also says prediction markets “will be at least as accurate as any other source of information” in the long run. This brings us to the Voting Fallacy, which I wrote about at length here (blog, Substack). The more people use prediction markets, the less useful they will be, because the ignorant will change prices in unpredictable and less informational ways.
One example should convince you. How much should people bet on slot machines? How much do they bet? Do those bets represent the probabilities of wins?
If you still doubt, explain to us how are people doing with actual voting. I’ve said it more times than you care to hear, but we have people who do not even know how many states there are voting for President. Alexander, in a way, gets this, and so he insists over and over that we know predictive markets are “canonical” because smart money would move in and fleece these sheep.
But they can’t. At least with PredictIt. As said, a person’s limit for a particular contract, like the Presidential election, is capped at \$850. And, worse, no more than 5,000 people at a time are allowed in the market (search for “5000” here). Given the trading volume in this picture, we must be seeing people jumping in and out of this market in a hurry.
Supposing a fellow with $850 knows this coming election will be fortified, as in previous elections. In other words, he knows the fix is in. He can buy Biden Yes contracts for 15 cents, and when all the absentee and mail-in votes are “counted”, he’ll win, in total, \$4,093 after the vig. This \$850 bet would likely move the Biden Yes contract price up a cent, maybe two cents.
This is the best case scenario, where somebody has perfect knowledge—and who is all in with prediction markets. Or suppose there are 5,000 really smart guys out there, which is not an unreasonable number out of 7+ billion. These smart guys would of course all bet the same way. No fleecing could occur.
The usual case scenario with more intelligent platers, I think, is one in which smarter money enters at the beginning of the market and takes advantage of the early big jumps, betting what other less intelligent people will do, not what will happen to the Event. Locklin makes references to this in “predicting volatility”. But, as we have seen above, this moves the price toward money making, and not toward the Event. Towards the end of the market, smarter money will have moved out, or it will stick with early positions that now seem like sure things.
You can go on like that, imagining all kinds of scenarios, but the gist is the same. The bets are not probabilities. The big error Alexander makes is assuming bets are probabilities, and vicey versey. It just isn’t so.
Alexander (and Locklin quotes him) says “you should trust a prediction market above your own opinion.” Which is circularly self-refuting, as Locklin points out. Because if everybody trusted the prediction market over their own opinion, then the market could never get started! Or it would be captured by the first lunatic to make a bet, which everybody thereafter would be bound to follow because that’s what the “unbiased” market says. Or, as Locklin says, “Yeah, OK then, why should anyone bet against what the prediction market says, charlie?”
A revelation is how these markets officially validate the bets-as-predictions. They evaluate their markets at 8 am on election day, using the market’s value at that point. In their favor, this is when most of the information for the gamble will be in, and the markets’ bets will most resemble probabilities (and most likely will be near 0 and 1). But it’s also the point where there usually is less new money in the market. In other words, it’s not a fair assessment of their accuracy.
Take a point before election day. Look at the curve above again. Let’s assume it’s November, and the Regime did not have enough oomph to fortify the election sufficiently, and Trump has won. Evaluate the predictive performance of the market on 21 April. Biden is 54 cents to Trump’s 45 cents. The market stank it up.
This leads to another point Locklin made, but which many don’t grasp. The Cackler’s stock went up. But after the debate. (Update: Biden’s went down, dramatically!, after he dropped out.) Prediction markets are therefore, as Locklin says, lagging indicators. At best, and ignoring all the non-ignorable caveats about betting, they are a summary of current thoughts.
Lest I go on to Alexander-like lengths, that’s enough. You have the idea. I can do 538 another time, if there’s interest. But I warn you, it won’t be that interesting.
UPDATE
It seems Polymarket’s vig is 4.3%, which I got from summing the prices of all current Yes bets on the election (Monday morning), which come to \$104.30. I don’t believe they charge a withdraw fee. I welcome corrections. Polymarket makes you use crypto to bet (which is not Bitcoin, but another, which you have to convert to). After Biden dropped out, Polymarket put out a tweet stating user AnonBidenBull lost \$2 million on his Biden bet.
Incidentally, these markets have ton of cheesy “proposition bets”, and these seem to be the of bets; and prop bets are ideal for luring in suckers.
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