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.

[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.

[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?)

[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.

[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.

[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.

[phk]

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.