Dutch Books and Bayesian Betting
The last two posts spent a fair amount of time trying to explain why you should be willing to bet, and why you need to offer fair odds. This one explains why, without a few critical caveats, that’s actually terrible advice.
A guy was interviewing at some investment bank on Wall Street, and he’s asked if he’d bet at even odds on a fair coin. He says yes — he’s not risk averse, he’ll fit right in on Wall St. He asked if he’d still bet at even odds after the coin came up tails 10 times — and being knowledgeable about probability theory, he says yes, the gambler’s fallacy is a fallacy, and the coin is as fair now as it was before. He’s then told that there’s a guy on the trading floor who trained himself to flip fair coins on whichever side he wanted, and taking the other side of a bet someone else offers is a stupid move. Of course, the moral of the story is that you’d better be a Bayesian, and make sure your model includes a non-0 prior on something being wrong.
But in fact, this is effectively the argument against the dutch book argument — offering fair odds based on limited knowledge to others is a great way to be taken advantage of. The fact that you’re consistent doesn’t imply that you’re right, and others are much more interested in betting when they know something you don’t — or think they do. This shouldn’t be a problem, given a bit more sophisticated of an approach than I’ve been advocating so far, but that requires just a bit of explanation.
Let’s say I have a prior belief that a coin is almost certainly fair. That means that, being slightly risk-averse, I’ll take almost any bet that gives me an edge over even odds. But if someone comes up to me and says they want to bet $50,000 and they’ll give me 1:1 odds that the coin will land heads 5 times in a row, even if I know they are good for the money, I’m not interested. Why? Because, as before, my model contains a non-0 probability that something is wrong, and that the coin will not be fair — I was only nearly certain before. Unlike the interview, however, I haven’t seen the coin flip, so a naive approach might assume there is nothing to update on.
In fact, however, I have a more complete model of the world than just my model of the coin. I have a fairly strong prior that people don’t offer large-money bets at unusually favorable odds unless there is some reason to do so. Usually it’s because they are planning on making money somehow. This significantly changes my estimate of “something is wrong” in my model of the coin flips. Of course, the reason may be that they are hosting a game show, and so I might glance around for surreptitious video cameras before declining the ostensibly favorable bet.
Does this violate my earlier claim that I should be willing to state odds, and bet on them? Or pick a single probability, and bet at any odds that give me a favorable edge? I think not. Instead, it simply allows me to update that probability on the basis of information I receive in the form of bets that are offered. If a mathematician offers to bet a lot of money at even odds that, using a coin of my choosing, flipped by me, the sequence “Heads Tails Tails” will appear before “Heads Heads Tails”, I’m going to look into it a bit before assuming that it’s actually a fair bet. My subjective prior might have been that these seem to be equally likely, but even without building a markov chain model, I should update away from that assessment on the basis of the bet offered.
Based on this, there are a couple of things I’d like to add as caveats to the general rule that it’s good to bet. First, most bets should be small — large bet offers should make you suspicious, and you certainly shouldn’t bet more than you can afford to lose. (In fact, never bet more than a fraction of what you can afford to lose proportionate to your estimated edge.) Second, your willingness to bet should be focused on areas where you want to develop a better understanding of the world — and short term, deterministic but unknown, easily resolved bets are better training for most tasks than long term, complex, or mostly random subjects.
