# Does vNM rationality depend on the good chosen?

The von Neumann-Morgenstern theorem states that, assuming a person's preferences under risk satisfy certain rationality axioms, then there exists a utility function u, the von Neumann utility function, such that the person will tend to maximize the expected value of u. For this reason, the hypothesis that people satisfy the von Neumann-Morgenstern rationality axioms is known as expected utility theory. Now in my question here, I asked about a confusion I had concerning the Ellsberg paradox, one of the major challenges to expected utility theory. But as I thought more about it, it seems to me that the fundamental issue was more general than the Ellsberg paradox.

Consider some good X. Let $L_1$ be the lottery that gives you a guaranteed 2 units of X, and let $L_2$ be the lottery that gives you a 50% chance of 1 unit of X, and a 50% chance of 3 units of X. Then my question is, is it always consistent with the von Neumann-Morgenstern axioms to prefer lottery $L_1$ to lottery $L_2$? Or are there some goods X such that it's inconsistent with the vNM axioms to prefer $L_1$ to $L_2$?

Well, suppose that good X is dollars. Then if u is the vNM utility function, then the expected utility of $L_1$ is equal to $u(2)$, and the expected utility of $L_2$ is equal to $.5u(1) + .5u(3)$. And it's certainly not inconsistent with the vNM axioms for $u(2)$ to be greater than $.5u(1) + .5u(3)$; that just means the person has diminishing marginal utility/risk aversion.

But now let's choose a different good X. Consider a raffle where four raffle tickets are placed in a hat, a single ticket is pulled out, and whoever the ticket belongs to wins a 100 dollars. Let good X be raffle tickets in this raffle. Then the probability of winning a 100 dollars is equal to .25 times the number of tickets you own. So under this scenario, the expected value of $L_1$ is $.5u(100)$, and the expected value of $L_2$ is $(.5)(.25)u(100)+(.5)(.75)u(100) = .5u(100)$. Thus the expected value of the two lotteries are equal, so it is irrational to prefer $L_1$ to $L_2$.

So what is going on here? How is preferring 2 units of X to a 50% chance of 1 unit of X and a 50% chance of 3 units of X vNM-rational for one good X but not vNM-rational for another? Just to add to the absurdity, what if someone were making a decision between lottery $L_1$ and $L_2$ where good X is dollars, and then was planning to use the money he made to buy raffle tickets? Then wouldn't the choice between $L_1$ and $L_2$ where X is dollars reduce to the choice between $L_1$ and $L_2$ where X is raffle tickets?

Any help would be greatly appreciated.

• The Ellsberg paradox has to do with subjective expected utility, where people can have any beliefs about the likelihood of some event, consistent with basic probability theory. Von Neumann and Morgenstern only treated the case of objective probabilities. Jun 13 '15 at 11:46

In the Von Neumann-Morgenstern theory, arguments of the utility function are final consumption goods. The utility function is extended over lotteries by linearity but it does not really make sense to assume that the individual "consumes" the lotteries L1 and L2 in your scenario with the raffle tickets: what he ultimately consumes is the 100 dollars monetary prize. Therefore there is no reason to expect something like diminishing marginal utility over lottery tickets - but only over final monetary prizes.

This property comes from the independence axiom: intuitively, the decision-maker views randomizations over lotteries and randomizations over final consumption goods differently. This behavioral pattern (that you find paradoxical) is usually called Reduction of Compound Lotteries. In your scenario, it says that the agent views the lottery L2 and the lottery that is computed by compounding the second-order probabilities as equivalent objects. There are some reasons to question the relevance of this axiom if you want to dig deeper (for instance, it has been found that Ellsberg-like behavior is associated with an aversion to compound lotteries).

I hope it helps.

• "In the Von Neumann-Morgenstern theory, arguments of the utility function are final consumption goods." No. There is a reason the theory was introduced in their book on game theory. Jun 13 '15 at 11:46
• Can you prove that the independence axiom implies the reduction of compound lotteries (from first principles, rather than using the vNM utility function)? Because I think I believe in the independence axioms, but I have a hard time swallowing the reduction of compound lotteries. In fact when analyzing by behavior in regard to the Ellsburg paradox, it was my attitude towards compound lotteries rather than my attitude toward Knightian uncertainty that led me to deviate from vNM-rational behavior. Jun 13 '15 at 14:06
• @Keshav Srinivasan I suggest you read the following article: bc.edu/content/dam/files/schools/cas_sites/economics/pdf/… which will probably answer your questions much better than I would do ;-)
– Oliv
Jun 13 '15 at 15:19
• OK, so it looks like the result I'm looking for is stated in part c of theorem 2, which states "Mixture independence, compound independence, and time neutrality imply the reduction axiom". This result is proven on page 370, in the paragraph which starts with "Assume next that..." But the proof is rather hard to follow, because the steps aren't explained. Could you explain the steps of the proof and how they're justified, or present the proof more simply? Jun 14 '15 at 4:45
• By the way, I'm now in the position where I think I believe in mixture independence, compound independence and time neutrality (which I think is a misnomer), but I'm still having trouble swallowing the reduction of compound lotteries! Jun 14 '15 at 4:48