The Allais Paradox has been a puzzle for standard economics, and rationality theory in general for fifty-odd years. Roughly, economics believes that people have stable preferences (for things like risk). If you don't like La Wik, here's Less Wrong on the same topic, explaining that it's not a simple solution.
The paradox suggests that risk preference is not stable.
Which do you prefer for choice 1:
Choice 1A: 100% chance of winning $1M
Choice 1B: 89% chance of winning $1M, 1% chance of winning $0, and 10% chance of winning $5M
Which do you prefer for choice 2:
Choice 2A: 11% chance of winning $1M, 89% chance of winning $0.
Choice 2B: 10% chance of winning $5M, 90% chance of winning $0.
Most people choose 1A, and 2B. Mathematical analysis of people preferences for risk fails rather robustly against this truth...assuming the assumption that people's preferences for risk are stable.
Eric Falkenstein uses his killer insight: Envy explains economics/investing/etc. BETTER than greed to solve the paradox. Go peek, if that's your kind of thing.