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

Question

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.

Thank You in Advance.

Answer 1

Short answer

• 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:

From Wikipedia's article on Arrow's impossibility theorem:

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

2. 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.

• Yes, there exists extensions of Arrow's impossibility theorem when the domain of the social welfare function is the set of vNM utility profiles, see corollary 4.1 of Social Choice with Interpersonal Utility Comparisons : A Diagrammatic Introduction, by Blackorby, Donaldson and Weymark

Long answer

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.

Answer 2

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”