Jan 21, 2008

37 comments

**Continuation of**: The Allais Paradox, Zut Allais!

Judging by the comments on Zut Allais, I failed to emphasize the points that needed emphasis.

**The problem with the Allais Paradox is the incoherent pattern 1A > 1B, 2B > 2A. **If you need $24,000 for a lifesaving operation and an extra $3,000 won't help that much, then you choose 1A > 1B

**Expected utility is not expected dollars.** In the case above, the utility-distance from $24,000 to $27,000 is a tiny fraction of the distance from $21,000 to $24,000. So, as stated, you should choose 1A > 1B and 2A > 2B, a quite coherent combination. **The Allais Paradox has nothing to do with believing that every added dollar is equally useful.** That idea has been rejected since the dawn of decision theory.

**If satisfying your intuitions is more important to you than money, do whatever the heck you want.** Drop the money over Niagara falls. Blow it all on expensive champagne. Set fire to your hair. Whatever. **If the largest utility you care about is the utility of
feeling good about your decision, then any decision that feels good is
the right one.** If you say that different trajectories to the same outcome "matter emotionally", then you're attaching an inherent utility to conforming to the brain's native method of optimization, whether or not it actually optimizes. Heck, **running around in circles from preference reversals** could feel really good too. **But if you care enough about the stakes that winning is more important than your brain's good feelings about an intuition-conforming strategy, then use decision theory.**

**If you suppose the problem is different from the one presented**
- that the gambles are untrustworthy and that, after this mistrust is
taken into account, the payoff probabilities are not as described -
then, obviously, **you can make the answer anything you want.**

Let's say you're dying of thirst, you only have $1.00, and you have
to choose between a vending machine that dispenses a drink with
certainty for $0.90, versus spending $0.75 on a vending machine that
dispenses a drink with 99% probability. Here, the 1% chance of dying
is worth more to you than $0.15, so you would pay the extra fifteen
cents. You would also pay the extra fifteen cents if the two vending
machines dispensed drinks with 75% probability and 74% probability
respectively. **The 1% probability is worth the same amount whether or
not it's the last increment towards certainty.** This pattern of decisions is perfectly coherent. **Don't confuse being rational with being shortsighted or greedy.**

*Added:* A 50% probability of $30K and a 50% probability of $20K, is not the same as a 50% probability of $26K and a 50% probability of $24K. If your utility is logarithmic in money (the standard assumption) then you will definitely prefer the latter to the former: 0.5 log(30) + 0.5 log(20) < 0.5 log(26) + 0.5 log(24). **You take the expectation of the utility of the money, not the utility of the expectation of the money.**