You have to use the right distance measure for the right purpose. The coherence proofs on Bayes's Theorem show that if you want the distance between probabilities to equal the amount of evidence required to shift between them, you have no choice but to use the log odds.

What the coherence proofs for the expected utility equation show, is more subtle. Roughly, the "distance" between probabilities corresponds to the amount of one outcome-shift that you need to compensate for another outcome-shift. If one unit of probability goes from an outcome of "current wealth + $24,000" to an outcome of "current wealth", how many units of probability shifting from "current + $24K" to "current + $27K" do you need to make up for that? What the coherence proofs for expected utility show, and the point of the Allais paradox, is that the invariant measure of distance between probabilities for this purpose is the usual measure between 0 and 1. That is, the distance between ~0 and 0.01, or 0.33 and 0.34, or 0.99 and ~1, are all the same distance.

You've got to use the right probability metric to preserve the right invariance relative to the right transformation.

Otherwise, shifting you in time (by giving you more information, for example about the roll of a die) will shift your perceived distances, and your preferences will switch, turning you into a money pump.

Neat, huh?

Okay, just one more question, Eliezer: when are you going to sit down and condense your work at Overcoming Bias into a reasonably compact New York Times bestseller?

The key word is compact. It's a funny thing, but I have to write all these extra details on the blog before I can leave them out of the book. Otherwise, they'll burst out into the text and get in the way.

So the answer is "not yet" - there are still too many things I would be tempted to say in the book, if I didn't say them here.

"What the coherence proofs for expected utility show, and the point of the Allais paradox, is that the invariant measure of distance between probabilities for this purpose is the usual measure between 0 and 1. That is, the distance between ~0 and 0.01, or 0.33 and 0.34, or 0.99 and ~1, are all the same distance."

In this example. If it had been the difference between .99 and 1, rather than 33/34 and 1, then under normal utility of money functions, it would be reasonable to prefer A in the one case and B in the other. But that difference can't be duplicated by the money pump you choose. The ratios of probability are what matter for this. 33/34 to 1 is the same ratio as .33 to .34.

So it turns out that log odds is the right answer here also. If the difference in the log odds is the same, then the bet is essentially the same.

## Comments (43)

OldHmmm.... I thought the point of your article at http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/ was that the difference between 1 and .99 was indeed much larger than, say, .48 and .49.Heh, I wondered if someone would bring that up.

You have to use the right distance measure for the right purpose. The coherence proofs on Bayes's Theorem show that if you want the distance between probabilities to equal the amount of evidence required to shift between them, you have no choice but to use the log odds.

What the coherence proofs for the expected utility equation show, is more subtle. Roughly, the "distance" between probabilities corresponds to the amount of one outcome-shift that you need to compensate for another outcome-shift. If one unit of probability goes from an outcome of "current wealth + $24,000" to an outcome of "current wealth", how many units of probability shifting from "current + $24K" to "current + $27K" do you need to make up for that? What the coherence proofs for expected utility show, and the point of the Allais paradox, is that the invariant measure of distance between probabilities for

thispurpose is the usual measure between 0 and 1. That is, the distance between ~0 and 0.01, or 0.33 and 0.34, or 0.99 and ~1, are all the same distance.You've got to use the right probability metric to preserve the right invariance relative to the right transformation.

Otherwise, shifting you in time (by giving you more information, for example about the roll of a die) will shift your perceived distances, and your preferences will switch, turning you into a money pump.

Neat, huh?

Okay, just one more question, Eliezer: when are you going to sit down and condense your work at Overcoming Bias into a reasonably compact New York Times bestseller?The key word is

compact. It's a funny thing, but I have to write all these extra details on the blogbefore I can leave them out of the book. Otherwise, they'll burst out into the text and get in the way.So the answer is "not yet" - there are still too many things I would be tempted to say in the book, if I didn't say them here.

"What the coherence proofs for expected utility show, and the point of the Allais paradox, is that the invariant measure of distance between probabilities for this purpose is the usual measure between 0 and 1. That is, the distance between ~0 and 0.01, or 0.33 and 0.34, or 0.99 and ~1, are all the same distance."

In this example. If it had been the difference between .99 and 1, rather than 33/34 and 1, then under normal utility of money functions, it would be reasonable to prefer A in the one case and B in the other. But that difference can't be duplicated by the money pump you choose. The ratios of probability are what matter for this. 33/34 to 1 is the same ratio as .33 to .34.

So it turns out that log odds is the right answer here also. If the difference in the log odds is the same, then the bet is essentially the same.