# Allais Paradox

## Definition

### Health Economics

This is a famous paradox of expected utility theory that has caused some to question the validity of the theory and therefore those bits of health economics that use it. Suppose a subject has the following choices under uncertainty:

Gamble A: A 100 per cent chance of receiving $1 million.

Gamble B: A 10 per cent chance of receiving $5 million, an 89 per cent chance of receiving $1 million, and a 1 per cent chance of receiving nothing.It is a matter of fact that most people choose A over B, even though the expected pecuniary value of B is $1.39 million. Presumably, people have a marked preference for certainty over uncertainty. In terms of expected utility they are revealing (where > indicates preference and U utility) that:

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

and, subtracting 0.89U($1m) from each side of the inequality, we get:

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

Now present the same subject with a further two gambles:

Gamble C: An 11 per cent chance of receiving $1 million, and an 89 per cent chance of receiving nothing.

Gamble D: A 10 per cent chance of receiving $5 million, and a 90 per cent chance of receiving nothing.Most people choose D over C. In terms of expected utility, they are revealing that:

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

Now, as expected utility theory permits, subtract 0.89U($0) from each side to get:

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

which is the opposite of what was chosen in the first choice situation. Expected utility theory does not permit of this possibility because preferring A to B implies preferring C over D.