A Simple Money Pump for Some Limited Version of Independence, as Well as Another Argument for It

Why it's irrational to prefer A to b but a .5 probability of A and .5 probability of B to a certainty of A.

Suppose that you prefer apples to bananas. However, you prefer a 50% probability of apples and a 50% probability of bananas to a 100% probability of apples. Note, in this case it wouldn’t be anything about the selection procedure that makes you prefer it—it would instead have to do with the possible prospects. It is of course not irrational to be intrinsically excited by uncertainty, and thus prefer uncertainty. Here, I will argue that this is irrational—though Gustafsson has a very compelling money pump for this in his book.

Suppose you start with an apple. Someone offers you one penny to trade it for a die roll, wherein if the die rolls 1-3, it will give you a banana and 4-6 will give you an apple. You take that deal. Now there are two possibilities—either you roll 1-3 or 4-6. Suppose you roll 1-3—you’re guaranteed to get a banana. Someone offers you a trade where you trade the banana for an apple at the cost of one penny. You take that deal because bananas are worse than apples. Thus, regardless of whether you get 1-3 or 4-6, you’ll eventually, through a sequence of beneficial trades, end up with an apple. But an apple is that same as a 100% chance of getting an apple. So you trade that for one penny and restarting the die roll. Thus, you’re now down a few cents and back where you started—this can repeat. Thus, this results in people spending all of their money going in circles—something that’s clearly irrational.

Here’s another argument for this. The following principle seems plausible

Knowledge expansion: if you know that either A or B will be true but you don’t know which one, and conditional on A some choice X is better than another choice X, and conditional on B they’re both equal, you should prefer X to Y.

This is very plausible—if something can be better than another thing but can’t be worse, then it is better overall.

But this entails independence. Suppose that you prefer apples to bananas. Your offer is a .5 probability of apples and a .5 probability of pears vs a .5 probability of bananas and a .5 probability of pears (assume that you’ll only get the pairs conditional on not getting the other option). Assume that this will be determined by a die roll—if it’s 1-3 you get an apple or banana, if it’s 4-6 you get a pear.

If you get 1-3 you regret the trade—after all, you get something you want less rather than more. If you get 4-6 you’re indifferent. Thus, by knowledge expansion you should prefer .5 probability of apples and .5 probability of pears to .5 probability of bananas and .5 probability of pears.