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alyssavance comments on Expected utility without the independence axiom - Less Wrong
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The Allais Paradox is not about risk aversion or lack thereof; it's about people's decisions being inconsistent. There are definitely situations in which you would want to choose a 50% chance of $1M over a 10% chance of $10M. However, if you would do so, you should also then choose a 5% chance of $1M over a 1% chance of $10M, because the relative risk is the same. See Eliezer's followup post, Zut Allais.
Turning a person into a money pump also isn't about playing the same gamble a zillion times (as any good investor will tell you, if you play the gamble a zillion times, all the risk disappears and you're left with only expected return, which leaves you with a different problem). The money pump works thusly: I sell you gamble A for $5. You then trade with me gamble A for gamble B. You then sell me back gamble B for $4. I then sell you gamble A for $5... wash, rinse, repeat. Nowhere in the cycle is either gamble actually paid out.
Are you sure you're responding to the right person here?
1) I wasn't claiming that Allais is about risk aversion.
2) I was claiming it doesn't show an inconsistency (and IMO succeeded).
3) I did read Zut Allais, and the other Allais article with the other ridiculous French pun, and it wasn't responsive to the point that Gray Area raised. (You may note that a strapping lad named "Silas" even noted this at the time.)
4) You cannot substantiate the charge that you should do the latter if you did the former, since no negative consequence actually results from violating that "should" in the one-shot case. You know, the one people were actually tested on.
ETA: (I think the second paragraph was just added in tommccabe's post.)
My point never hinged on it being otherwise.
Okay, and where in the Allais experiment did it permit any of those exchanges to happen? Right, nowhere.
Believe it or not, when I say, "I prefer B to A", it doesn't mean "I hereby legally obligate myself to redeem on demand any B for an A", yet your money pump requires that.
The problem is that you're losing money doing it once. You would agree that c(0) > c(-2), yes? If they are willing to trade A for B in a one-shot game, they shouldn't be willing to pay more for A than for B in a one-shot - you don't trade the more valuable item for the less valuable. That their preferences may reverse in the iterated situation has no bearing on the Allais problem.
Edit: The text above following the question mark is incorrect. See my later comment quoting Eliezer for the correct statement.
Again, if suddenly being offered the choice of 1A/1B then 2A/2B as described here, but being "inconsistent", is what you call "losing money", then I don't want to gain money!
But that's not what's happening the paradox. They're (doing something isomorphic to) preferring A to B once and then p*B to p*A once. At no point do they "pay" more for B than A while preferring A to B. At no point does anyone make or offer the money-pumping trades with the subjects, nor have they obligated themselves to do so!
Consider Eliezer's final remarks in The Allais Paradox (I link purely for the convenience of those coming in in the middle):
You're right insofar as Eliezer invokes the Axiom of Independence when he resolves the Allais Paradox using expected value; I do not yet see any way in which Stuart_Armstrong's criteria rule out the preferences (1A > 1B)u(2A < 2B). However, in the scenario Eliezer describes, an agent with those preferences either loses one cent or two cents relative to the agent with (1A > 1B)u(2A > 2B).
Your preferences between A and B might reasonably change if you actually receive the money from either gamble, so that you have more money in your bank account now than you did before. However, that's not what's happening; the experimenter can use you as a money pump without ever actually paying out on either gamble.
Yes, I know that a money pump doesn't involve doing the gamble itself. You don't have to repeat yourself, but apparently, I do have to repeat myself when I say:
The money pump does require that the experimenter make actual futher trades with you, not just imagine hypothetical ones. The subjects didn't make these trades, and if they saw many more lottery tickets potentially coming into play, so as to smooth out returns, they would quickly revert to standard EU maximization, as predicted by Armstrongs's derivation.
"Potentially coming into play, so as to smooth out returns" requires that there be the possibility of the subject actually taking more than one gamble, which never happens. If you mean that people might get suspicious after the tenth time the experimenter takes their money and gives them nothing in return, and thereafter stop doing it, I agree with you; however, all this proves is that making the original trade was stupid, and that people are able to learn to not make stupid decisions given sufficient repetition.
The possibility has to happen, if you're cycling all these tickets through the subject's hands. What, are they fake tickets that can't actually be used now?
There are factors that come into play when you get to do lots of runs, but aren't present with only one run. A subject's choice in a one-shot scenario does not imply that they'll make the money-losing trades you describe. They might, but you would have to actually test it out. They don't become irrational until such a thing actually happens.
"What, are they fake tickets that can't actually be used now?"
No, they're just the same tickets. There's only ever one of each. If I sell you a chocolate bar, trade the chocolate bar for a bag of Skittles, buy the bag of Skittles, and repeat ten thousand times, this does not mean I have ten thousand of each; I'm just re-using the same ones.
"They might, but you would have to actually test it out. They don't become irrational until such a thing actually happens."
We did test it out, and yes, people did act as money pumps. See The Construction of Preference by Sarah Lichtenstein and Paul Slovic.
You can also listen to an interview with one of Sarah Lichtenstein's subjects who refused to make his preferences consistent even after the money-pump aspect was explained:
http://www.decisionresearch.org/publications/books/construction-preference/listen.html
"1) I wasn't claiming that Allais is about risk aversion."
The difference between your preferences over choosing lottery A vs. lottery B when both are performed a million times, and your preferences over choosing A vs. B when both are performed once, is a measurement of your risk aversion; this is what Gray Area was talking about, is it not?
"Believe it or not, when I say, "I prefer B to A", it doesn't mean "I hereby legally obligate myself to redeem on demand any B for an A""
Then you must be using a different (and, I might add, quite unusual) definition of the word "preference". To quote dictionary.com:
pre⋅fer /prɪˈfɜr/ [pri-fur] –verb (used with object), -ferred, -fer⋅ring. 1. to set or hold before or above other persons or things in estimation; like better; choose rather than: to prefer beef to chicken.
What does it mean to say that you prefer B to A, if you wouldn't trade B for A if the trade is offered? Could I say that I prefer torture to candy, even if I always choose candy when the choice is offered to me?
Typo: Did you mean "prefer A to B"?
I prefer B to A does not imply I prefer 10B to 10A, or even I prefer 2B to 2A. Expected utility != expected return.
I agree pretty much completely with Silas. If you want to prove that people are money pumps, you need to actually get a random sample of people and then actually pump money out of them. You can't just take a single-shot hypothetical and extrapolate to other hypotheticals when the whole issue is how people deal with the variability of returns.
Strictly speaking, Eliezer's formulation of the Allais Paradox is not the one that has been experimentally tested. I believe a similar money pump can be implemented for the canonical version, however -- and Zut Allais! shows that people can be turned into money pumps in other situations.
"I prefer B to A does not imply I prefer 10B to 10A, or even I prefer 2B to 2A. Expected utility != expected return."
Of course, but, as I've said (I think?) five times now, you never actually get 2B or 2A at any point during the money-pumping process. You go from A, to B, to nothing, to A, to B... etc.
For examples of Vegas gamblers actually having money pumped out of them, see The Construction of Preference by Sarah Lichtenstein and Paul Slovic.
No, it's not, and the problem asserted by Allais paradox is that the utility function is inconsistent, no matter what the risk preference.
I don't see anything in there that about how many times the choice has to happen, which is the very issue at stake.
If there's any unusualness, it's definitely on your side. When you buy a chocolate bar for a dollar, that "preference of a chocolate bar to a dollar" does not somehow mean that you are willing to trade every dollar you have for a chocolate bar, nor have you legally obligated yourself to redeem chocolate bars for dollars on demand (as a money pump would require), nor does anyone expect that you will trade the rest of your dollars this way.
It's called diminishing marginal utility. In fact, it's called marginal analysis in general.
It means you would trade B for A on the next opportunity to do so, not that you would indefinitely do it forever, as the money pump requires.
"When you buy a chocolate bar for a dollar, that "preference of a chocolate bar to a dollar" does not somehow mean that you are willing to trade every dollar you have for a chocolate bar, nor have you legally obligated yourself to redeem chocolate bars for dollars on demand (as a money pump would require), nor does anyone expect that you will trade the rest of your dollars this way."
Under normal circumstances, this is true, because the situation has changed after I bought the chocolate bar: I now have an additional chocolate bar, or (more likely) an additional bar's worth of chocolate in my stomach. My preferences change, because the situation has changed.
However, after you have bought A, and swapped A for B, and sold B, you have not gained anything (such as a chocolate bar, or a full stomach), and you have not lost anything (such as a dollar); you are in precisely the same position that you were before. Hence, consistency dictates that you should make the same decision as you did before. If, after buying the chocolate bar, it fell down a well, and another dollar was added to my bank account because of the chocolate bar insurance I bought, then yes, I should keep buying chocolate bars forever if I want to be consistent (assuming that there is no cost to my time, which there essentially isn't in this case).