Monday, April 15, 2019

When being right doesn't pay

Like me, you might have naively speculated that the more truth you know, the better you’ll do in gambling scenarios. But this is mistaken, at least when taken in the strong sense that there is a guarantee of doing better (or even just as well).

For instance, suppose that Alice and Bob are betting on two coin flips. Alice has credence 1/2 for heads for each coin. Bob has credence 1 for heads for the first coin (maybe because he peeked) and credence 1/2 for the second coin. The house happens to offer Alice and Bob this bet:

  • you get $10 if the first coin is heads and you pay $16 if the second coin is heads.

And as it happens both coins are heads.

Alice calculates the expected payoff at (1/2)⋅$10 − (1/2)⋅$16 = −$3 and declines. Bob calculates the expected payoff at $10 − (1/2)⋅$16 = $2 and accepts. But of course the actual payoff on double heads is −$6, so Bob is worse off than Alice for having been right about the first coin.

Can we at least say that in the long run Bob will be better off (financially, maybe not morally) for peeking than Alice? Yes, if the house offers the same bet each time and the coins are fair and Bob accepts the bet whenever he sees the first coin to be heads. But if the house also peeks at the coins and varies the offering based on the outcome, and Bob doesn’t notice this variation, then the house can fleece Bob (e.g., the house can offer the above bet whenever both coins are heads, and in all other cases offer some tiny bet worth a penny).

So in what sense can truth be guaranteed to help? Well, if you are betting on a single proposition, you will do better (or at least no worse) the closer your credence is to the actual truth value (where 0 is falsehood and 1 is truth).

1 comment:

IanS said...

That’s why sensible gamblers take care to specify their bets in advance. :-)

The problem is not what Bob knows, but what he doesn’t know (viz. that the house is peeking and setting its bets accordingly).

Strictly, Bayesian decision theory applies to scenarios fully defined in advance. In principle, a scenario should specify all the information you will be given and the circumstances in which you will be given it, and all your possible choices and the circumstances in which they will be offered.

In your example, the fact that the house offers large bets if both coins land heads and small ones otherwise is properly part of the scenario. If Bob knows it, he will reject the large bets, accept the small bets if he sees that the first coin is heads, and reject the small bets otherwise. Then he will sometimes win and never lose. If Alice too knows what the house is up to, she will reject all the bets. (I’m assuming both coins are fair and that Bob knows the first outcome with certainty.) So Bob’s positive expectation will be greater than Alice’s zero – his peeking helps him.

Outside textbooks and standard gambling setups, we rarely have well-defined scenarios. We know what actually happened, but we are forced to make assumptions about what might have happened. (e.g. that the same bet would have been offered regardless of the random outcomes). If these assumptions are wrong, we may make bad decisions. This is one of the many problems with trying to apply Bayesian decision theory in practice..