## 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.