You have two choices: A) A lottery that guarantees $ 1 M vs B) where you have a 10% chance of winning $ 5M, 89% chance for 1 M and 1% chance of nothing. Which one will you choose? If I write them in a different format:
| A | $ 1M (1) |
| B | $ 5M (0.1); $ 1M (0.89); $ 0 (0.01) |
Having chosen one of the above two, you have another one to choose from. C) A lottery with an 11% chance of $ 1 M and 89% chance of nothing vs D) a 10% chance of winning $ 5M, 90% chance of nothing.
| C | $ 1M (0.11); $ 0M (0.89) |
| D | $ 5M (0.1); $ 0 (0.9) |
Allais (1953) argued that most people preferred A and D. What is wrong with that?
Expected Value
If the person had followed the expected value theory, she could have chosen B and D:
A) $ 1M x 1 = $ 1M
B) $ 5M x 0.1 + $ 1M x 0.89 + $ 0 x 0.01 = $ 1.39 M
C) $ 1M x 0.11 + $ 0M x 0.89 = $ 0.11 M
D) $ 5M x 0.1 + $ 0 x 0.9 = $ 0.5 M
Expected Utility
Since the person chose A over B, clearly, it was not the expected value but an expected utility that governed her. Mathematically,
U($ 1 M) > U($ 5 M) x 0.1 + U($ 1 M) x 0.89 + U($ 0M) x 0.01
Now, collect the U($ 1 M) on one side, add U($ 0M) x 0.89 on both sides, and simplify.
U($ 1 M) – U($ 1 M) x 0.89 > U($ 5 M) x 0.1 + U($ 0M) x 0.01
U($ 1 M) x 0.11 > U($ 5 M) x 0.1 + U($ 0M) x 0.01
U($ 1 M) x 0.11 + U($ 0M) x 0.89 > U($ 5 M) x 0.1 + U($ 0M) x 0.01 + U($ 0M) x 0.89
U($ 1 M) x 0.11 + U($ 0M) x 0.89 > U($ 5 M) x 0.1 + U($ 0M) x 0.9
Pay attention to the last equation. What are you seeing here? The term on the left side is the expected utility equation corresponding to option C, and the one on the right side is option D. In other words, if A > B, then C > D. But that was violated in the present case.
