“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.
“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.