Allais paradox

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.

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:

Several studies involving hypothetical and small monetary payoffs, and recently involving health outcomes, 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 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". Each experiment gives 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, and both 2A and 2B give an outcome of nothing). If this 89% ‘common consequence’ is disregarded, then the gambles will be left offering the same choice.

It may help to re-write the payoffs. After disregarding the 89% chance of winning — the same 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 should also now be seen as the same choice.

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 ${\displaystyle L_{3}}$.

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