Allais paradox

The Allais paradox, more neutrally described as the Allais problem, is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory. The problem arises when comparing participants' choices in two different experiments, each of which consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows:

Presented with the choice between 1A and 1B, most people choose 1A. Presented with the choice between 2A and 2B, most people choose 2B. This is inconsistent with expected utility. The point is that both gambles give the same outcome 89% of the time (the top row; $1 million for Gamble 1, and zero for Gamble 2), so, in expected utility, these equal outcomes should have no effect on the desirability of the gamble. If the 89% ‘common consequence’ is disregarded, both gambles offer the same choice; a 10% chance of getting $5 million and 1% chance of getting nothing as against an 11% chance of getting $1 million. (It may help to re-write the payoffs. 1A offers an 89% chance of winning 1 million and a 11% chance of winning 1 million, where the 89% chance is irrelevant. 2B offers an 89% chance of winning nothing, a 1% chance of winning nothing, and a 10% chance of winning 5 million, with the 89% chance of nothing disregarded. Hence, choice 1A and 2A should now clearly be seen as the same choice, and 1B and 2B as the same choice).

Allais presented his paradox as a counterexample to the independence axiom (also known as the "sure thing principle" of expected utility theory. Independence means that if an agent is indifferent between simple lotteries $$L_1$$ and $$L_2$$, the agent is also indifferent between $$L_1$$ mixed with an arbitrary simple lottery $$L_3$$ with probability $$p$$ and $$L_2$$ mixed with $$L_3$$ with the same probability $$p$$. Violating this principle is known as the "common consequence" problem (or "common consequence" effect).  The idea of the common consequence problem is that as the prize offered by $$L_3$$ increases, $$L_1$$ and $$L_2$$ become consolation prizes, and the agent will modify preferences between the two lotteries so as to minimize risk and disappointment in case they do not win the higher prize offered by $$L_3$$.

Difficulties such as this gave rise to a number of alternatives to, and generalizations of, the theory, notably including prospect theory, developed by Daniel Kahneman and Amos Tversky, weighted utility (Chew) and rank-dependent expected utility by John Quiggin. The point of these models was to allow a wider range of behavior than was consistent with expected utility theory.

Also relevant here is the framing theory by Daniel Kahneman and Amos Tversky. Identical items will result in different choices if presented to agents differently (i.e. a surgery with a 70% survival rate vs. a 30% chance of death) However, the main point Allais wishes to make, is that the independence axiom of expected utility theory may not be a necessary axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the analysis of the gamble as a whole. However, this overlooks the notion of complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B, there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense of disappointment if you were to pick that gamble and lose, knowing you could have won with 100% certainty, if you had chosen 1A. This feeling of disappointment however, is contingent on the outcome in the other portion of the gamble (i.e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or choices independently of the other choices presented, as the independence axiom requires, and thus is a poor judge of our rational action(1B cannot be valued independently of 1A as the independence or sure thing principle requires of us). We don't act irrationally when choosing 1A and 2B, rather expected utility theory is not robust enough to capture such "bounded rationality" choices that in this case arise because of complementarities.

Mathematical Proof of Inconsistency
Using the values above and a utility function of u(W), where W is wealth, we can demonstrate exactly how the paradox manifests.

Because the typical individual prefers 1A to 1B and 2B to 2A, we can write conclude that the expected utilities of the preferred is greater than the expected utilities of the second choices, or,

$$1.00U(1m) > 0.89U(1m) + 0.01U(0) + 0.1U(5m)$$

$$0.89U(0) + 0.11U(1m) < 0.9U(0) + 0.1U(5m)$$

We can rewrite the latter equation as,

$$0.11U(1m) < 0.01U(0) + 0.1U(5m)$$

$$1U(1m) - 0.89U(1m) < 0.01U(0) + 0.1U(5m)$$

$$1U(1m) < 0.01U(0) + 0.1U(5m) + 0.89U(1m)$$

Which contradicts the first bet which shows the player prefers the sure thing over the gamble.