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I've been reading up on the von-Neumann and Savage proofs for the existence of an expected utility representation. I've also been reading critiques of the expected utility hypothesis, especially the Allais paradox.

Its obvious that the Allais paradox "disproves" von-Neumann: this can be shown numerically using commonly used utility functions such as log-utility. However, I'm less sure why Allais and Savage are inconsistent with each other. Its easy to find "subjective" probabilities that resolve the Allais paradox while still retaining the expected utility form. To see this, consider the outcomes of the Allais paradox:

$$M^T=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 5 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 5 & 1 & 0 \\ 1 & 0 & 0 & 5 \\ 1 & 1 & 0 & 5 \\ 1 & 5 & 0 & 5 \\ 1 & 0 & 1 & 5 \\ 1 & 1 & 1 & 5 \\ 1 & 5 & 1 & 5 \end{bmatrix}$$

Where each column represents one of the four gambles.

These gambles have the following corresponding "physical" probabilities:

$$ p=\begin{bmatrix} 0.008 \\ 0.713 \\ 0.080 \\ 0.001 \\ 0.088 \\ 0.001 \\ 0.001 \\ 0.079 \\ 0.009 \\ 0.000 \\ 0.010 \\ 0.001 \end{bmatrix} $$

Using a log utility with initial wealth of $100,000, I get 13.911 "utils" for the sure million; 14.040 utils for the 89% chance for 1 million, 1 percent for 0, and 10% chance for 5 million; 11.777 utils for the 89% chance of 0 and 11% chance for 1 million; and 11.906 utils for the 90% chance of 1 million and 1% chance of 5 million. Clearly this violates Allais' paradox.

However, consider the following subjective probabilities:

$$q=\begin{bmatrix} 0.04 \\ 0.16 \\ 0.02 \\ 0.04 \\ 0.07 \\ 0.18 \\ 0.01 \\ 0.13 \\ 0.07 \\ 0.18 \\ 0.03 \\ 0.07 \end{bmatrix}$$

Using these probabilities, I get 13.911 "utils" for the sure million; 13.785 utils for the 89% chance for 1 million, 1 percent for 0, and 10% chance for 5 million; 12.880 utils for the 89% chance of 0 and 11% chance for 1 million; and 13.440 utils for the 90% chance of 1 million and 1% chance of 5 million. This provides probabilities that are consistent with Allais' paradox.

How or why is Savage's proof inconsistent with Allais' paradox?

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  • $\begingroup$ Why do you need lotteries with so many outcomes to make your point? Allais paradox is about inconsistencies when lotteries are combined en.wikipedia.org/wiki/Allais_paradox Regarding Savage, it shows a violation of his "sure-thing principle". $\endgroup$ – Bayesian Jul 15 at 8:44
  • $\begingroup$ I have just the 4 lotteries. I've enhanced the state space. This allows the lotteries to be combined for easier analysis and comparison. I'm trying to understand why the sure thing principle is necessary in this case. $\endgroup$ – user9403 Jul 15 at 23:56

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