Like Allais’ Paradox, Machina’s Paradox is a thought experiment which seems to lead people to violate the independence axiom of expected utility theory.

Suppose there were two gambles, and you could choose to take part in one of them. In gamble A you have a 99% chance of winning a trip to Venice and a 1% chance of winning tickets to a really great movie about Venice. (No, really, it’s a totally kick ass movie about Venice, maybe with Sophia Loren in it.) In gamble B, you have a 99% chance of winning a trip to Venice and a 1% chance of just staying at home.

Now, suppose you prefer the trip over the movie and the movie over staying home.

Well, the independence axiom would dictate that:

u(Movie) > u(Home) implies (0.99)u(Trip) + (0.01)u(Movie) > (0.99)u(Trip) + (0.01)u(Home)

So, the independence axiom insists you must choose gamble A. And maybe you personally would. On the other hand, maybe you’d be so depressed by losing the almost-sure win of the trip, that the movie reminding you of Venice would just bring you down. Knowing your temperament is such, you could decide to take gamble B so you’ll never be confronted with the movie outcome.

That’s the story, anyway. What it tries to capture is that the independence axiom doesn’t allow for regret, which could well affect how people make risky choices.

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