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PK comments on Zut Allais! - Less Wrong

23 Post author: Eliezer_Yudkowsky 20 January 2008 03:18AM

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Comment author: PK 21 January 2008 05:06:10PM 0 points [-]

"There are various other games you can also play with certainty effects. For example, if you offer someone a certainty of $400, or an 80% probability of $500 and a 20% probability of $300, they'll usually take the $400. But if you ask people to imagine themselves $500 richer, and ask if they would prefer a certain loss of $100 or a 20% chance of losing $200, they'll usually take the chance of losing $200. Same probability distribution over outcomes, different descriptions, different choices."

Ok lets represent this more clearly. a1 - 100% chance to win $400 a2 - 80% chance to win $500 and 20% chance to win $300

b1 - 100% chance to win $500 and 100% chance to lose $100 b2 - 100% chance to win $500 and 20% chance to lose 200%

Lets write it out using utility functions.

a1 - 100%*U[$400] a2 - 80%*U[$500] + 20%*U[$300]

b1 - 100%*U[$500] + 100%*U[-$100]? b2 - 100%*U[$500] + 20%*U[-200%}?

Wait a minute. The probabilities don't add up to one. Maybe I haven't phrased the description correctly. Lets try that again.

b1 - 100% chance to both win $500 and lose $100 b2 - 20% chance both win $500 and to lose $200, leaving an 80% chance to win $500 and lose $0

b1 - 100%*U[$500 - $100] = 100%*U[$400] b2 - 20%*U[$500-$200] + 80%*[$500-$0] = 80%*U[$500] + 20%*U[$300]

This is exactly the same thing as a1 and a2. More importantly however is that the $500 is just a value used to calculate what to plug into the utility function. The $500 by itself has no probability coefficient and therefore it's 'certainty' is irrelevant to the problem at hand. It's a trick using clever wordplay to make one believe there is a 'certainty' when none is there. It's not the same as the Allais paradox.

As for the Allais paradox, I'll have to take another look at it later today.