In the early sixties, Nobel-prize winning economist Paul Samuelson sat down in the cafeteria at MIT and had a short conversation that soon became legendary in economic circles. He asked his lunch partner if he would accept the gamble heads you win $100, tails you lose $50. Any economist would call this a great deal - its expected value is $25. But his lunch partner turned it down. So would most people. Humans are generally quite risk-averse. But then his lunch partner (sadly, no one knows who it was) countered that he would take the gamble if Samuelson would let him repeat it 100 times in a row.
That was strange. Samuelson felt the same urge himself, but it sounded deeply wrong. He went back to his office and quickly proved that this pair of preferences is irrational. Irrational doesn't mean the same as risk-aversion. Irrational means having totally inconsistent preferences. And Samuelson elegantly proved that if you are risk-averse enough to reject the single gamble, you must also reject the bundle of 100 gambles.
[I won't give the proof here. But think about it this way: imagine you have already taken 99 gambles, now the last one is equivalent to the one-gamble offer, so to be consistent you have to reject it. Well, if you reject that one, then the 99th one is equivalent to the one-shot gamble, so you have to reject it too. Then the 98th one. You keep up this logic and pretty soon you have to reject all the gambles.]
In the 60's, it was enough to prove that human preferences can be paradoxical, chuckle at human nature, and leave it at that. Nowadays we like to go further. We ask why does this provably irrational behavior is so seductive. Most people, even if you carefully explain the math, would still reject the one-shot but accept the hundred-shot. I know I would. And that needs some explanation.
Here's my theory. In the long run, following a strategy of accepting gambles like Samuelson's gamble pays off an average of $25. But in the short run, averages are meaningless. People tend to focus on the short term, so we choose based on the probability of actually winning or losing money. Samuelson's one-shot gives you a 50% chance of losing money. The hundred repetition offers you less than 1% chance of losing money. I simulated the hundred gambles 10,000 times on my computer and lost money only 7 times. Those are pretty good odds. So the paradox goes away if you consider this bias towards the short term.
OK so why do people preferentially consider short term outcomes? That's a fascinating question that no one can answer. I think it has to do with impulsivity and self-control, and that's a big part of what I study in my lab these days. I'll keep you updated.
Risk and uncertainty: A fallacy of large numbers (1963)
Paul Samuelson, Scientia, 98, 108-113.
The mean, the median, and the St. Petersburg Paradox (2009)
Benjamin Y. Hayden and Michael L. Platt, Judgment and Decision Making vol. 4 (4), p. 256-273.