## Sunday, March 09, 2008

### Combining Bets

Suppose you have no idea whether P is true. Is it rationally permissible for you to reject the following bet?

(A) You win \$15 if P is true, and lose \$10 if P is false.

If so, the following bet is presumably also permissible to turn down:

(B) You win \$15 if P is false, and lose \$10 if P is true.

But if someone offers you both bets at once, it would be crazy to turn them down: you can net \$5 no matter the outcome, it's a guaranteed win! We may take this to show that the rational status of bets is not closed under conjunction. It can be rationally permissible to reject A, and permissible to reject B, but impermissible to reject (A and B).

What if you do not know that both bets will be offered? Suppose you are offered A, and permissibly reject it. Then, to your surprise, you are offered B. Are you now rationally required to accept bet B, based on the principle that it would be irrational to reject both? That would be bizarre. Instead, I'd suggest that the rational principle in play is the following:

(Sure Win) It is irrational to knowingly turn down a sure win (unless it comes with opportunity costs, etc.).

It is irrational to reject A and B together, for together they offer a sure win. But the person who rejects A, and is only later offered B, was never offered a sure win. Bet A by itself (with no guarantee that B will be offered too) is not a sure win. So it may be permissibly rejected. And once you've rejected A, bet B by itself is not a sure win either. So it too may now be rejected without violating the Sure Win principle.

Why am I going on about this, you ask? Adam Elga uses this combination of bets to argue that a rational agent must be disposed to accept at least one of them (and so have precise credences). But it seems like a bad argument to me, since the defender of imprecise credence can appeal to the Sure Win principle - as I did above - to explain why the irrationality of the combined bet rejection does not imply the irrationality of rejecting either bet alone. Right?

1. A sure win hypothesis is a mighty risk adverse strategy.

2. Yeah, that's why I added the proviso about opportunity costs: it could be reasonable to turn down a (small) sure win if by taking just one of the two bets you had (sufficient) chance of much larger winnings. But it at least seems clearly irrational to turn down a sure win unless you stand to gain something else by doing so, right?

3. Is it just an assumption that the first bet was permissible to turn down? I'd think the best response to this tension is to deny that you can turn it down, whether or not B is going to be offered.

4. Yeah, I agree with that in my latest post. But note that this only holds if we (are required to) have no less than 40% credence in the winning outcome. If we can have imprecise credence, say spread over the interval [0.2, 0.8], then it is at least permissible - and many advocates of imprecise credence would even say obligatory - to reject any bet that is rejectable according to some value in the credence interval.

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