“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%
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.
“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.