Expected Utilities of Lotteries

[ag]

Moderatorial:

Just make sure you eventually attribute the paradox

[ag]

well... since $1 million would certainly make me very happy...

I'd choose A.  There's no chance, however remote, of winning nothing.

That opens up options C and D to a bit more leeway, so I'd choose D.

is this supposed to be an investing trick?

Does it make me a conservative investor with the bulk of my investments with a few risky possibly high-yield ones mixed in?

You should think of it not as two consecutive choices, but as a pair of hypothetical choices, of which you are only going to be making one in reality. If you still choose A and D, your preferences over monetary lotteries are not linear in probabilities and, hence, do not have an expected utility representation (whatever that means). Most likely (though I have no way to tell), you are thinking of possible feelings of disappointment in case the 1% probability event occurs in lottery B.

BTW, this is known as Allais paradox.

[ag]

The point of vNM expected utility theory is that in order for taking expectations of utilities to make sense you need linearity in probabilities. That is, e.g.,  if probability distribution over three outcomes is given, by probability weights p1,p2 and p3, then, in the space of probabilities indifference curves are straight parallel lines and may be represented by a utility function of the form ap1+bp2+cp3 (i.e., linear in proabilities).  If that is not the case, you can't meaningfully take expectations of utility.

Allais paradox demonstrates, why it may happen that preferences are not linear: in this case, if you imagine how unhappy you'd be if that 1% chance of zero payoff would materialize, you might choose not to take the risk. However, unlike the change from 0% to 1%, the change in likelyhood of a bad oucome from 10% to 11% is not going to be truly observable: you can always believe, you wouldn't have gotten anything anyway. Hence, the possible preference reversal.

Prospect theory is one of the possible relaxations of the linearity assumption, of course.  If what I am trying to model involves, say, potential feelings of regret, it could well be among the tools worth using. It's not immune from its own paradoxes, though

[ag]

Which utilities are you talking about? Though, in both case your conjecture would be wrong.

If Bernoulli (the ones defined on money), then the moment you have them and as long as people prefer more money to less (at least weakly) preferences can't go against FOSD by definition of FOSD.

If the utilities on lotteries (on probabilities), then it's not the matter of slant, but of the shape. Violating the independence axiom (which you inevitably must be doing if you are having a utility representation, but not an expected utility representation) means that your indifference curves are not straight lines/planes, but rather bending curves/surfaces. Then, of course, you could disagree w/ FOSD, but it won't be the matter of slant.

I wish I had a blackboard here The graphical illustration of the proofs is very neat and transparent.

[ag]

Ok, that's the key thing: your indifference curves are not parallel straight lines. The moment this happens, you no longer have expected utilities. Once that's the case, there is no reason to respect the stochastic dominance, of course: you may violate it, you may not violate it, but that's already not that important.

I think, we should go step by step. Have you seen the proof of the expected utility theorem?

[ag]

If I have time, I will do a sketch one of these days. It's worth understanding.

[ag]

For all matters of basics go here and you won't be wrong

http://arielrubinstein.tau.ac.il/Rubinstein2007.pdf

[ag]

It is all about the marginal utility of money.

The posited paradox (which is not really a paradox, just that the marginal utility of money is not linear),

No, it's not about marginal utility of money - actually, as far as economists are concerned there is nothing called marginal utility of money, but that is another matter. The paradox is, actually, that many people demonstrate a pattern of behavior (choosing A and D or, more unusually, B and C) that cannot be explained by taking expectations of any utility function over money: linear, concave, convex, increasing, decreasing or whatever. There is just no utility function over money that would work here. Try it ))

In this case, probably, people wood feel extremely upset if they had a guaranteed 1 mln, but lost it (the 1% chance in lottery B), whereas if they choose D the 1% decrease in the probability of winning may be seen as negligible: if you loose you can always say you'd have lost anyway. So, in case you don't get anything having picked lottery B you'd get additional disutility from missing out on an assured outcome had you picked A. That sort of a feeling cannot be modeled no matter what utility function over money you pick - you need something else.

[ag]

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

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.

[ag]

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

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.

[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.

[ag]

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.