Preference and choice disposition

Here is an initially plausible thesis about deliberation that I've long felt is right:

1. Normally, when choosing between options, you are more likely to choose an option X than an option Y if and only if you prefer X to Y.

Now consider a normal case where you must choose between four options:

1. Cookies and Earl Gray
2. Cookies and chamomile
3. Crackers and Earl Gray
4. Crackers and chamomile.

Suppose that you mildly prefer cookies to crackers and Earl Gray to chamomile, so that you have a 60% chance of choosing cookies over crackers and 60% chance of choosing Earl Gray over chamomile. Further, suppose you have no preferences about the combination as such. Then the chances of your four options are:

• P(A) = 36%

• P(B) = 24%

• P(C) = 24%

• P(D) = 16%.

Clearly, you can choose A. But notice an odd thing. You can organize your deliberation to start as follows: First choose between A and the disjunction B or C or D. (Maybe then if you chose B or C or D, you decide between B and C or D. And finally if you chose C or D, you decide between C and D.) However, here is another plausible principle of rationality:

2. Normally, if you prefer A to every disjunct of a disjunction, you prefer A to the disjunction.

Organizing your deliberation as above is not abnormal. So, (2) should still apply. You prefer A to B, and you prefer A to C, and you prefer A to D. So you should prefer A to the disjunction B or C or D, and hence you prefer A over the disjunction B or C or D. Thus, by (1) you are more likely to choose A than the disjunction B or C or D. In particular, the chance of your choosing A isn’t 36%, but must be more than 50%.

Hence, (1) predicts that the probabilities of our choices greatly depend on how one coarse-grains the options in one’s deliberation. This shouldn’t be. I thus reject (1). Which makes me sad, as it was a very neat thesis.