Expected Utilities of Lotteries

[phk]

Pick a lottery from each set and mark the poll.

First set of lotteries

Gamble A
100% chance of winning $1 million.

Gamble B
89% chance of winning $1 million.
1% chance of winning nothing.
10% chance of winning $5 million.

Second set of lotteries

Gamble C
89% of winning nothing.
11% chance of winning $1 million.

Gamble D
90% chance of winning nothing.
10% chance of winning $5 million.

[Хahar 🤔]

B and D

[ag]

Moderatorial:

Just make sure you eventually attribute the paradox

[Sewer]

B and D

[Ban my account ffs!]

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?

[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.

[Јas]

B and D

^^

Surely one would have to be very risk-averse to choose either option A or C?

[phk]

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.

Basically for the first set of gambles

If A>B than

U(1 million) > .89U(1 million) + .01U(0) + .10U(5 million)
.11U(1 million) > .10U(5 million)

....and for the second

If D > C

.10U(5 million) + .90U(0) >.89U(0) + .11U(1 million)
.11U(1 million) < .10U(5 million)

I fixed the U(0) to zero.

Ag as for being an vNM-EU maximizer.... doesn't it simply mean that you weight movements in probability equally? Like going from 100% to 99% should have the same weight as going from 45% to 44%.

Though the whole point of Prospect Theory is that people don't weight probabilities in this way (which was designed to resolve this).

[phk]

B and D

^^

Surely one would have to be very risk-averse to choose either option A or C?

Yeah. I would see as A and C being a universally risk averse person. B and D being a universally risk loving person.

Wheras any other combination is inconsistent.

[I spent the winter writing songs about getting better]

A and D

[Filuwaúrdjan]

What if you have a little voice in your head (alright, not the best way of putting that) that informs you that gambling is SINFUL!!!!!?

[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

[phk]

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

Are you talking of having the indifference curves slanted the wrong way (where you'd be indifferent between two lotteries, even though one first order stochastic dominates the other?)

[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.

[Beet]

So only 3 people (including myself) voted with the Allais hypothesis in this poll?

[phk]

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.

Are you familiar with the Machina Triangle?

Here's what I'm talking about


Not exactly the original paradox. But an Allais style paradox definitely.




Technically speaking D' FOSD D (just by how the triangle diagram works). Movements up and to the left are strictly increased preference.

Yet D' and D lie on the same indifference curve.

Though I think Kahneman and Tversky got around this (rather sloppily) by saying any lottery that is "better" (maybe FOSD) will dominate and the rest will get thrown out.

[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?

[phk]

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?

I haven't seen the proof of the formal theorem, no.

[ag]

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

[k-onmmunist]

I would take A and D

[dead0man]

b and d

[phk]

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

Is there a link to the proof? I'v been searching on Google to no avail.

[ag]

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

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

[Torie]

It is all about the marginal utility of money. Most people would ascribe a much higher marginal utility to the first one million, then to subsequent millions, but the odds here are so extreme, and the expected cost of risk reduction so high, ie the expected return from taking more risk so high, that even given our marginal utility curve, most would still prefer to take more risk for more money here.

The posited paradox (which is not really a paradox, just that the marginal utility of money is not linear), is about folks willing to take more risk if the chances of landing a substantial amount of money are small anyway, as opposed to being in the bag, so in that sense if you lose, the odds are with the second bet that you would have ended up with nothing anyway whichever bet you picked, while in the first the odds of getting a substantial amount of money taking the safer bet were 100%.  Most of us would fall for the paradox if the numbers were considerably less extreme, because if the sum of money is large enough, it would take a very high rate of return indeed for us being willing to risk losing it all. That is why we buy insurance, which has a negative expected rate of return (administrative costs have to be paid for), but we still consider it worth it.

[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.