Thomas_B.

The Allais Paradox

I think I missed something on the algebraic inconsistency part...

If there is some rational independent utility to certainty, the algebraic claims should be more like this:

- U($24,000) + U(Certainty) > 33/34 U($27,000) + 1/34 U($0)
- 0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0)

This seems consistent so long as U(Certainty) > 1/34 U($27,000).

I'm not committed to the notion there is a rational independent value to certainty, I'm just not seeing how it can be dismissed with quick algebra. Maybe that wasn't your goal. Forgive me if this is my oversight.

"A number of commenters, yesterday, claimed that the preference pattern wasn't irrational because of "the utility of certainty", or something like that. One commenter even wrote U(Certainty) into an expected utility equation."

It was not my intent to claim "the preference pattern wasn't irrational," merely that your algebraic modeling failed to capture what many could initially claim was a salient detail of the original problem. I hope a reread of my original comment will find it pleading, apologetic, limited to the algebraic construction, and sincere.

I should have mentioned that I thought the algebraic modeling was a very elegant way to show that the diminishing marginal utility of money was not at play. If that was its only purpose, then the rest of this is unnecessary, but I think you can use that construction to do more, with a little work.

Here's one possible response to this apparent weakness in the algebraic modeling:

If you can simply assert that Allais's point holds experimentally for arbitrarily increasing values in place of $24k and $27k (which I'm sure you can), then we find this proposed "utility of certainty" (or whatever more appropriate formulation you prefer*) increasing with no upper bound. The notion that we value certainty seems to hold intuitive appeal, and I see nothing wrong with that on its face. But the notion that we value certainty above all else is more starkly implausible (and I would suspect demonstrably untrue: would you really give your life just to become certain of the outcome of a coinflip?).

I was trying to make the argument stronger, not weaker, but I get the impression I've somehow pissed all over it. My apologies.

*I've read your post on Terminal Values three times and haven't yet grokked why I can't feed things like knowledge or certainty into a Utility function. Certainty seems like a "fixed, particular state of the world," it seems like an "outcome," not a "action," and most definitely unlike "1." If the worry is that certainty is an instrumental value, not a terminal value, why couldn't one make the same objection of the $24,000? Money has no inherent value, it is valuable only because it can be spent on things like chocolate pizza. You've since replaced the money with lives, but was the original use of money an error? I suspect not... but then what is the precise problem with U(Certainty)?

I should clarify that, once again, I bring up these objections not to show where you've gone wrong, but to show where I'm having difficulties in understanding. I hope you'll consider these comments a useful guide as to where you might go more slowly in your arguments for the benefit of your readers (like myself) who are a bit dull, and I hope you do not read these comments as combative, or deserving of some kind of excoriating reply.

I'll keep going over the Terminal Values post to see if I can get it to click.