Fair Betting, and Glitches in the Matrix

Last time, I talked about why betting is a good idea. I didn’t mention a key point — if you want your betting to help you the most, you should pick either setting the odds however you want, or which side of the bet to pick — not both.

Why?

Let’s say I was interested in understanding how people use technology a bit better. To test my intuitions, I said that I think Pokemon Go will fizzle out, and by December 1st, predicted it won’t be in the top 10 App Downloads, nor in the top 5 Apps by total minutes used. How certain am I? Not absolutely certain — that would be crazy. Instead, I am 75% sure — and I’m willing to bet on that basis. Lets say I’d be willing to bet $100 at those 3:1 odds that this will happen.

If I actually believe my estimate of 3:1, I should be approximately indifferent to which side of the bet I take — I should be happy to gain, say, $25 and keep my $75 stake if Pokemon Go fizzles as I thought, or gain $75 with a $25 stake if it doesn’t. Otherwise, I don’t actually believe the 75% number I gave, I actually believe some higher number.

Is this really true? Let’s try a though experiment; I might not be willing to take the other side of that 3:1 bet. Still, I would take the other side of the bet at 99:1 odds. In that case, I’d get paid $99 with a $1 stake if I’m wrong— which seems like a much better deal. In fact, unless I’m nearly certain that Pokemon Go will fizzle, I should take this bet. This means I actually think the probability is between 75% and 99%. But fuzzy probabilities aren’t coherent — and if I get to pick which side of each bet I take, this would let me continue being incoherent about what I expect to occur.

Now there are people who say, with varying degrees of philosophical rigor, that this is fine. But we wanted to make bets to train ourselves to understand the world better. They claim it’s OK not to notice a glitch in the matrix, and they can stay happy ignoring discrepancies in their own models of the world. Perhaps they wish to, as Heinlein put it, avoid the “uncertainty of reason” to “bask at the warm fire of [irrationality]” — but why are you doing so?

As a side point, if you have incoherent probabilities, but still let other people pick the side of the bet, or bet based on incoherent probabilities, you’re susceptible to what is called a dutch book — a set of bets that loses you money no matter what. That’s usually a bad thing, and so incoherent probabilities don’t get along well with willingness to bet. That’s a large part of why the people who care about coherent probabilities also tend to be OK with betting, and those that don’t think coherent probabilities matter avoid them.

So there are two options for coherent betting; you can either pick the odds and state how much you’d bet, and let anyone who wants to do so pick the side they prefer, or you can look at the odds someone else is offering, and pick a side. Neither of these allows incoherence, but only the second allows you to expect a profit.

Slight algebraic sidebar — You can’t make money offering odds based on the probability you expect, since you get the stake×odds if you win, and lose the stake if you lose. If the odds are, as in our earlier example, 3:1, you bet $25 to win $75, which is 25×(3/1). If you took the other side of the bet, your stake is $75, and the odds are 1:3, so you get $25 if you win, which is 75×(1/3). Your expected payout from any wager is the sum of the probability of each outcome times the payoff — p×o×s -(1-p)× s. If you pick the probability, it reflects your best guess, and so the expected payout is 0 — since the odds are, by definition, (1-p):p, you end up with (1-p)×s -(1-p)×s = 0.

On the other hand, if someone else picks the odds, you choose the side you prefer, and can make an expected profit, at least if your estimate of the probabilities are better than theirs. But that means they got to pick the odds, and/or the amounts — which has some interesting implications I’ll get to in the next post.