What does Arrow's theorem say about Kaldor-Hicks social welfare functions with von Neumann-Morgenstern utility?

Let $A$ be the set of all possible states of the world, let $G(A)$ be the set of all "lotteries" or "gambles", i.e. the set of all probability distributions over $A$. Now consider an individual with a preference ordering of the various lotteries in $G(A)$. Then the von Neumann-Morgenstern theorem states that, assuming the individual's preferences obeys certain rationality conbditions, there exists a function $u: A \rightarrow \mathbb{R}$, such that the individual's preference ordering maximizes the expected value of $u$. Moreover, the function $u$ is unique up to linear transformations, i.e. maximizing the expected value of $u$ and maximizing the expected value of $a + bu$ yield equivalent results.

Now consider a society with N individuals, where each individual's preferences obey the von Neumann Morgenstern axioms. Then we can define a social welfare function $W = a_1u_1 + a_2u_2 + ... + a_Nu_N$, where $u_i$ is the von Neumann-Morgenstern utility function for the $i^{\textrm{th}}$ individual, and $a_i$ is the reciprocal of the marginal utility of money for the $i^{\textrm{th}}$ individual. As shown in this thread, $W$ is well-defined, because it's invariant under linear transformations of the $u_i$'s. More importantly for our purposes, it is my understanding that maximizing $W$ will achieve a Kaldor-Hicks optimal result. (Can someone back me up on this, and preferably tell me where I can find a proof?)

My question is, how does Arrow's impossibility theorem apply to a social preference ordering based on Kaldor-Hicks efficiency? Specifically, given two outcomes in $A$, what would happen if we let the social ordering prefer the outcome that has a greater value of W? Arrow's theorem, as usually stated, is about rules that are maps from $L(A)^N$ to $L(A)$, i.e. rules that take each individual's preference ordering on A, and then spit out a social preference ordering on A. ($L(A)$ is the set of linear orders on the set $A$.)

But the rule I'm describing is not just based on each individual's preference ordering on $A$ (their preferences for certain outcomes), but on their von Neumannn-Morgenstern utility function $u$, i.e. on their preference ordering on $G(A)$ as well (their preferences under uncertainty). So are there generalizations of Arrow's theorem that deal with maps from $L(G(A))^N$ to either $L(G(A))$ or failing that, maps from $L(G(A))^N$ to $L(A)$, as is the case with the rule I'm describing? If an extension of Arrow's theorem does apply, what does it say about this rule? What conditions does the rule obey or not obey?

Any help would be greatly appreciated.

• The utility doesn't obey the Arrow prerequisites. All we get knowing optimality is that there is at least one outcome of a trial/election/choice that has maximum utility (for some utility function). But that's pretty obvious. It doesn't mean a rule exists with the Arrow conditions to always choose that maximum. – ex0du5 Aug 31 '13 at 23:39
• @exodu5 I'm afraid I don't understand your comment. When you say "the utility", what utility are you talking about? You say a rule doesn't exist, but I clearly specified one: rank outcome x higher than outcome y if W(x)>W(y). And W is entirely determined by each individual's preference ordering on G(A). What conditions does that rule not satisfy? – Keshav Srinivasan Sep 1 '13 at 0:31
• I was referring to the same function you are. I did not say your function did not exist. I said just because the optimal choice (or choices) exist according to that utility does not mean it obeys the Arrow conditions (like Pareto efficiency, to be specific). – ex0du5 Sep 1 '13 at 1:23
• @exodu5 What are you talking about? The rule I specified definitely obeys Pareto efficiency. Suppose each individual prefers outcome x to outcome y. Then $u_i(x) > u_i(y)$ for all i, so W(x) > W(y), and thus the social ordering prefers outcome x to outcome y. – Keshav Srinivasan Sep 1 '13 at 1:44
• That's what I was referring to, but I see my last comment was worded bad. I mean there are multiple conditions to Arrow's impossibility. You have chosen one to be satisfied (Pareto efficiency), but the others then cannot all be covered in the same fashion. Typically, you face independence of irrelevant assumptions on the generic points, but there are edge cases with dictators, etc. To be clear: defining a preference function is an independent process here. Your description is not really a generalisation. – ex0du5 Sep 1 '13 at 2:17

• Arrow's theorem holds for social welfare function from the set of orderings profiles into the set of orderings, no matter whether the set of alternatives is a set of lotteries, or any other set of alternatives (see edit at the end of the post).
• No, Arrow's impossibility theorem does not apply to social welfare function which take sets of utility profiles as domains, as Arrow acknowledged himself:

Voting systems that use cardinal utility (which conveys more information than rank orders; see the subsection discussing the cardinal utility approach to overcoming the negative conclusion) are not covered by the theorem.2

1. Interview with Dr. Kenneth Arrow: "CES: Now, you mention that your theorem applies to preferential systems or ranking systems. Dr. Arrow: Yes CES: But the system that you're just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems. Dr. Arrow: And as I said, that in effect implies more information. Archived January 14, 2013, at the Wayback Machine.

The fact that -- as you mention -- the domain of your social welfare $W$ function is the space of vNM utilities implies that $W$ does not satisfy Arrow's "Independence of Irrelevant Alternative" axiom. Relaxing this axiom allows for the existence of social welfare functions satisfying the other Arrovian axioms.

To be more precise, as shown by Fleurbaey and Mongin's "The news of the death of welfare economics is greatly exaggerated" and others, Independence of Irrelevant Alternatives is equivalent to the combination of two weaker axioms:

• Ordinal Noncomparability : the ranking of alternatives depends only on the individual orderings of the allocation. In utility terms, the social ranking is invariant to any increasing transformation of the individual utility levels.
• Binary Independence : the ranking of two alternatives depends only on people's utilities at these two alternatives (and not on the relative ranking of these two alternatives with respect to some third, fourth, ... alternative)

Your social welfare function $W$ satisfies Binary Independence but violates Ordinal Noncomparability which is the reason why it does not satisfy the conditions of the Arrow's impossibility theorem. Obviously, as $W$ does not satisfy the conditions of the theorem, the theorem does not apply to it (hopefully this answers your first question?).

Now the whole question is "what do you replace Ordinal Noncomparability with ?".

(If your want to read more about what forgoing Ordinal Non-comparability implies -- and why it is a bad idea, you may want to read "Inequality, income, and well-being" by Koen Decancq, Marc Fleurbaey and Erik Schokkaert.)

If you just discard the axiom without replacing it by a weaker constraint on the way your social welfare function should react to transformations of the utility profiles, then you allow for the existence of a plethora of social welfare functions satisfying efficiency, binary independence and non-dictatorship on a universal domains of preferences.

However, if you do add weaker restrictions, you might run into impossibility results again. Because you speak of vNM utility functions, it is interesting to consider the case of affine transformations. An alternative to Ordinal Noncomparability which is relevant with these kinds of preferences is

• Cardinal Noncomparability : the ranking of alternatives depends only on the individual orderings of lotteries. In utility terms, the social ranking is invariant to any affine transformation of the individual utility levels.

Then you recover an impossibility result if you slightly strengthen Arrow's efficiency condition, as shown in Corrolary 4.1 of Social Choice with Interpersonal Utility Comparisons : A Diagrammatic Introduction, by Blackorby, Donaldson and Weymark

Corrolary 4.1 (roughly): If a social-welfare function satisfies Unrestricted domain, Binary Independence, Strong Pareto (i.e. if some are made better of and no-one is made worse-off, we have a social improvement, even if some individuals are not strictly better-off) and Cardinal Noncomparability, it must be a dictatorship

EDIT following OP question

The theorems I mentioned are valid for any set of alternatives. Whether the alternatives are lotteries or non-stochastic outcomes (or really whatever else) does not alter the validity of the theorems.

I think limiting the domain of vNM functions without weakening IIA would not be enough to prevent us from running into impossibilities. Here is a rather informal argument. vNM only restricts the ranking of lotteries, not the ranking of the "degenerated" lotteries through the function $u(.)$. So if $A = \{a,b,c,\dots\}$ is the set of degenerated lotteries, $u : A \rightarrow \mathbb{R}$ is not constrained.

So assume that Arrow's theorem does not hold on a vNM domain. This means that there exists a social welfare function $F$ satisfying the axioms of the theorem (except for unrestricted domain) on $G(A)$. Thus there exists a subrelation of $F$ over the set of degenerated lotteries $A$, say $\tilde{F}$, which also satisfies the axioms on $\tilde{F}$. But this implies that $\tilde{F}$ satisfies the axioms of Arrow's theorem on a set of alternatives $A$ for an unrestricted domain of preferences over $A$, a contradiction.

Hope this helps.

• I agree that you can't have an Arrow-like result for a map from $L(G(A)^N$ to $L(A)$, i.e. a rule that takes individual vNM utility functions and merely constructs a ranking. But what about a map from $L(G(A)^N$ to $L(G(A))$, i.e. a rule that takes individual vNM utility functions and constructs a vNM social welfare function? Or equivalently, a rule that takes individual rankings over lotteries and constructs a social ranking over lotteries. – Keshav Srinivasan Feb 9 '14 at 16:19
• @ Keshav : Sorry I had made a major edit to my question. I refreshed my memory and remembered some important results which might be viewed as Arrow-like result for social welfare function with domain in the space of utilities. – Martin Van der Linden Feb 9 '14 at 16:31
• What does unrestricted domain mean? Does that mean that we consider all possible utility functions, not just von Neumann-Morgenstern utility functions? What if we restricted ourselves to vNM utility functions? Would that stop us from proving a similar theorem? What would happen if, as I suggested in my last comment, we have a rule that constructs not merely a ranking over $A$, but a ranking over $G(A)$ as well. – Keshav Srinivasan Feb 9 '14 at 16:38
• @ Keshav : I do not think that the fact that the set of alternatives be made of lotteries makes any difference. The theorems I mention are valid for any set of alternatives. Whether the alternatives are lotteries or non-stochastic outcomes (or really whatever else) does not alter the validity of the theorems. Now what could be of importance is your assumption that agents have vNM utilities because that restricts the domain of preferences and the theorems rely heavily on the unrestricted domain assumption. On that I have no direct answer... – Martin Van der Linden Feb 9 '14 at 16:38
• Just to make sure there is no ambiguity : Arrow's theorem holds when the social ranking is over the set of alternatives over which agents have preferences. So if agents have preferences over G(A) the social ranking is over G(A). It requires that a set of axioms be satisfied AND that the social ranking apply to any possible ordering profile over the set of alternatives (unrestricted domain). There are examples of restricted domains on which Arrow's theorem does not apply (e.g. so-called single-peaked domains). I'll try to think about the vNM case in an edit to my answer. – Martin Van der Linden Feb 9 '14 at 16:52

Using the arrows imposibility theorem show that the “idea of social welfare function to determine a unique Point of maximum social welfare is not only utopian, but in principle impossible”

• Didn't you see this part of my question? "Arrow's theorem, as usually stated, is about rules that are maps from $L(A)^N$ to $L(A)$, i.e. rules that take each individual's preference ordering on A, and then spit out a social preference ordering on A. ($L(A)$ is the set of linear orders on the set $A$.) But the rule I'm describing is not just based on each individual's preference ordering on $A$ (their preferences for certain outcomes), but on their von Neumannn-Morgenstern utility function $u$, i.e. on their preference ordering on $G(A)$ as well (their preferences under uncertainty)." – Keshav Srinivasan Jan 17 '14 at 15:11