“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.
There is a certain U(certainty) in a game, although there might be better ways to express it mathematically. How do you know the person hosting the game isn’t lying to you an really operating under the algorithm: 1A. Give him $24,000 because I have no choice. 1B. Tell him he had a chance to win but lost and give nothing.
In the second situation(2A 2B) both options are probabilities and so the player has no choice but to trust the game host.
Also, I am still fuzzy on the whole “money pump” concept. “The naive preference pattern on the Allais Paradox is 1A > 1B and 2B > 2A. Then you will pay me to throw a switch from A to B because you’d rather have a 33% chance of winning $27,000 than a 34% chance of winning $24,000.”
Ok, I pay you one penny. You might be tricking me out of one penny(in case you already decided to give me nothing) but I’m willing to take that risk.
“Then a die roll eliminates a chunk of the probability mass. In both cases you had at least a 66% chance of winning nothing. This die roll eliminates that 66%. So now option B is a 33⁄34 chance of winning $27,000, but option A is a certainty of winning $24,000. Oh, glorious certainty! So you pay me to throw the switch back from B to A.”
Yes yes yes, I pay you 1 penny. You now owe me $24,000. What? You want to somehow go back to a 2A 2B situation again? No thanx. I would like to get my money now. Once you promised me money with certainty you cannot inject uncertainty back into the game without breaking the rules.
I’m afraid there might still be some inferential distance to cover Eliezer.