Expected Utilities of Lotteries
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  Economics (Moderator: Torie)
  Expected Utilities of Lotteries
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Poll
Question: Choose what best describes your preferences
#1
Gamble A and Gamble C
 
#2
Gamble A and Gamble D
 
#3
Gamble B and Gamble C
 
#4
Gamble B and Gamble D
 
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Total Voters: 22

Author Topic: Expected Utilities of Lotteries  (Read 8297 times)
Torie
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« Reply #25 on: December 20, 2009, 06:38:38 PM »
« edited: December 20, 2009, 06:40:41 PM by Torie »

What do you call the thingy then, when it is far more painful to your when you net worth drops from 1 million to zero, than from 2 million to 1 million, if not something that involves the word "utility."  The balance of your post I did not understand very well, and I think my insurance analogy is apt. One pays something to avoid a catastrophic loss, and giving up a bunch of money if one gets lucky to secure avoiding a  catastrophic loss, is what drives the need for insurance.

Sometimes, intuitive common sense cuts to the chase better than complex formulas, in my experience. I am reasonably confident I have this one right, but perhaps not.
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ag
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« Reply #26 on: December 20, 2009, 06:59:55 PM »

There are no complex formulas, but what you call "common sense" does not describe the paradox here. Your intuition is simply wrong. It may be zillion times "more painful" (whatever that means) for you net worth to drop from 1mln to zero then for it to drop from 100 mln to 2 mln, but still, whoever is choosing A over B should be choosing C over D. Just try it Smiley

The problem here is not with risk-aversion (that's what you really are trying to get to): no matter how high the risk aversion, the paradox will not be explained. The problem is that with preferences like this you don't just care about outcomes and their probabilities, but also about "what could have happened".  You need either to model that explicitly, or you have to do smtgh "funny" with probabilities: i.e., you may assume that people overweigh the small probability of getting nothing. Unless you either incorporate the disutility of having let the bird in you hands out, or do something strange with probabilities, nothing, literally nothing you do to utility of wealth would get you this behavior.


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Torie
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« Reply #27 on: December 20, 2009, 07:04:43 PM »

I will ponder that comment. I still think I am right and you are wrong, but I need to put my lawyer hat on to try to tackle it all, and well, I don't "work" on Sundays. And if you are right, I will have learned something, so it is a win win for me either way, since this is an issue that really interests me, the risk and reward thing. It is what drives my important choices in life. Thanks Ag.
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ag
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« Reply #28 on: December 20, 2009, 08:03:40 PM »

I will ponder that comment. I still think I am right and you are wrong, but I need to put my lawyer hat on to try to tackle it all, and well, I don't "work" on Sundays. And if you are right, I will have learned something, so it is a win win for me either way, since this is an issue that really interests me, the risk and reward thing. It is what drives my important choices in life. Thanks Ag.

Well, I am an econ prof. And my hat is on Smiley

Let me do it very simply. If you choose A over B it means that

u(1 mln) > 0.89*u(1 mln) + 0.1*u(5 mln)

Now, subtract 0.89*u(1 mln) from both sides of the inequality and you'll get

0.11*u(1mln) > 0.1*u(5 mln)

Which, of course, implies that you should choose C over D.

So, unless you do smthg funny w/ probabilities or let the utility depend not only on the ammount of money, but also on something else there is no way A and D could be consistent choices, irrespective of what your u is.

Google Allais paradox.
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Torie
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« Reply #29 on: December 20, 2009, 08:58:27 PM »

I understand the expected return thing, but my point is that risking a sure thing, is different than risking something that is a long shot anyway. That is entirely sensible. That indeed is the way most of us run our lives really, who are rational when it comes to money matters.
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ag
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« Reply #30 on: December 20, 2009, 10:00:29 PM »
« Edited: December 20, 2009, 10:02:47 PM by ag »

I understand the expected return thing, but my point is that risking a sure thing, is different than risking something that is a long shot anyway. That is entirely sensible. That indeed is the way most of us run our lives really, who are rational when it comes to money matters.

Well, that one IS the right intuition. But it has exactly nothing to do w/ decreasing marginal utility of money. It might be an "increasing (more properly, varying) marginal utility" of probabilities, but you are then playing w/ probabilities, not w/ monetary ammounts.

Of course, there is nothing irrational about having preferences like this. You'd be violating something else - what economists call independence - but rationality would be intact. There will be a utility function representation of such preferences - just not expected utility representation. You'd loose a tool, but there is no reason why such preferences couldn't exist. In fact, the point of the ostensible paradox, they do.
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phk
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« Reply #31 on: December 20, 2009, 11:11:48 PM »
« Edited: December 21, 2009, 10:34:46 PM by phknrocket1k »



I think the issue is that people don't weight changes in probability equally across all ranges of the (P, PI(P)) function.

The dotted line prescribes EU weights on the Prospect Theory weighting function (assuming the independence axiom is not violated).  I think the Allais Paradox preferences can be described by a convex utility function (correct me if I'm wrong here).

The whole issue is that instead of a 1-to-1 correspondence (as in a slope of 1) between changes in probability and the weight given to that change in probability.

Shifting from hypothetical Gamble W: ($1 million, 1) to Gamble X: ($1 million, .99; $0, .01) is more painful than shifting from Gamble Y: ($1 million, .30; $0, .70) to Gamble Z: ($1 million, .29; $0, .71).

Intuition being that shifting from 100% to 99% seems more painful than shifting from 30% to 29%. Even if it was just a 1% shift either way.
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ag
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« Reply #32 on: December 21, 2009, 11:56:33 AM »

That's one way of modeling it. Or else, you could expand the state space to include things that could have (but didn't) happen. One could think of an experiment to figure that out.

My own view is that very small probabilities are indeed treated differently. How - that's an empirical question. Any model you might propose will, eventually, stumble into a paradox, however, and, for the most part, the EU model is doing fine.
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