Shady Gambles

Roger White has proven, to my satisfaction, that perfectly rational agents could not have imprecise credences. In this post, I want to explain away a potential source of contrary intuitions. It seems perfectly reasonable -- and perhaps even rationally required -- to refrain from accepting bets on issues you are completely ignorant about. But if I were required to always have precise credence, then there are many (intuitively dubious) bets that I would have to accept. For example, suppose I have no evidence for or against a proposition P. If I must give it a precise credence of 0.5, then I must accept any better-than-even bet (e.g. costing $10 if P is false, and paying $15 if P is true).

The worry is that this leaves me very vulnerable to exploitation from more knowledgeable dealers. They might offer me deals that seem tempting in my ignorance, but which they have carefully set up so that the option I'm expected to choose is in fact a loser. The offering of a bet is itself a piece of new evidence: since the bookie is out to rip me off, an offer that looks "too good to be true" probably is. So it's a good practical rule of thumb to be disposed not to accept bets that others are keen to offer you.

Of course, we want to bracket these practical considerations for philosophy's sake. To avoid any fear of shady dealings or manipulation, we may suppose the details of the bet were determined by some completely random process. In that case, my intuitions sharpen up considerably. It no longer seems permissible to reject a better-than-even bet. Let me now offer an argument to back this up.

We saw in the previous post that a better-than-even Bet A can be combined with its converse Bet B (i.e. offering the same, better-than-even payoffs for the opposite result) to yield a sure win ($5 in the case of $15 vs. -$10 payoffs). So, even if you don't know anything else, you at least know that it's more desirable to take both bets than neither. The expected payoff is (say) $5 rather than zero.

But now suppose you are offered the following: we flip a coin, and if it lands heads you're committed to Bet A only, and if tails you get Bet B only. Is this a game you should accept? It seems so. If you play it over and over again, you can expect to make ever increasing winnings ($5 for every two games, on average). And it's perfectly random, so your one-shot expected utility must be positive too.

So, if you're offered Bet A alone (assuming the background conditions are such that the details of this bet were selected via some random, non-shady process), you should take it. A sharp credence of 0.5 won't leave you vulnerable in these properly sanitized conditions.