The Allais paradox is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory.

## Statement of the problem

The Allais paradox 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:

 Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 100%$1 million 89% Nothing 89% Nothing 90% Nothing 1% $1 million 11%$5 million 10% $5 million 10% Several studies[1] involving hypothetical and small monetary payoffs, and recently involving health outcomes,[2] have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted that it was reasonable to choose 1A alone or 2B alone. However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is inconsistent with expected utility theory. According to expected utility theory, the person should choose either 1A and 2A or 1B and 2B. The inconsistency stems from the fact that in expected utility theory, equal outcomes (eg.$1 million for all gambles) added to each of the two choices should have no effect on the relative desirability of one gamble over the other; equal outcomes should "cancel out". In each experiment the two gambles give the same outcome 89% of the time (starting from the top row and moving down, both 1A and 1B give an outcome of $1 million with 89% probability, and both 2A and 2B give an outcome of nothing with 89% probability). If this 89% ‘common consequence’ is disregarded, then in each experiment the choice between gambles will be the same – 11% chance of$1 million versus 10% chance of $5 million. After re-writing the payoffs, and disregarding the 89% chance of winning — equalising the outcome — then 1B is left offering a 1% chance of winning nothing and a 10% chance of winning$5 million, while 2B is also left offering a 1% chance of winning nothing and a 10% chance of winning $5 million. Hence, choice 1B and 2B can be seen as the same choice. In the same manner, 1A and 2A can also be seen as the same choice, i.e:  Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance$1 million 89% $1 million 89% Nothing 89% Nothing 89%$1 million 11% Nothing 1% $1 million 11% Nothing 1%$5 million 10% \$5 million 10%

Allais presented his paradox as a counterexample to the independence axiom.

Independence means that if an agent is indifferent between simple lotteries ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$, the agent is also indifferent between ${\displaystyle L_{1}}$ mixed with an arbitrary simple lottery ${\displaystyle L_{3}}$ with probability ${\displaystyle p}$ and ${\displaystyle L_{2}}$ mixed with ${\displaystyle L_{3}}$ with the same probability ${\displaystyle p}$. 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 ${\displaystyle L_{3}}$ increases, ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ 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 ${\displaystyle L_{3}}$.

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), rank-dependent expected utility by John Quiggin, and regret theory. 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 of Daniel Kahneman and Amos Tversky. Identical items will result in different choices if presented to agents differently (e.g. a surgery with a 70% survival rate vs. a 30% chance of death).

The main point Allais wished to make is that the independence axiom of expected utility theory may not be a valid 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 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 conclude that the expected utilities of the preferred is greater than the expected utilities of the second choices, or,

### Experiment 1

${\displaystyle 1U(\1{\text{ M}})>0.89U(\1{\text{ M}})+0.01U(\0{\text{ M}})+0.1U(\5{\text{ M}})}$

### Experiment 2

${\displaystyle 0.89U(\0{\text{ M}})+0.11U(\1{\text{ M}})<0.9U(\0{\text{ M}})+0.1U(\5{\text{ M}})}$

We can rewrite the latter equation (Experiment 2) as

${\displaystyle 0.11U(\1{\text{ M}})<0.01U(\0{\text{ M}})+0.1U(\5{\text{ M}})}$
${\displaystyle 1U(\1{\text{ M}})-0.89U(\1{\text{ M}})<0.01U(\0{\text{ M}})+0.1U(\5{\text{ M}})}$
${\displaystyle 1U(\1{\text{ M}})<0.89U(\1{\text{ M}})+0.01U(\0{\text{ M}})+0.1U(\5{\text{ M}}),}$

which contradicts the first bet (Experiment 1), which shows the player prefers the sure thing over the gamble.