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

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.

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## closed as off-topic by Benoît Kloeckner, Carlo Beenakker, Stefan Kohl, Karl Schwede, Ricardo AndradeDec 11 '13 at 21:00

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What do you mean, "How does it apply?" It doesn't apply. The objects are different. So, the assumptions aren't satisfied. –  Douglas Zare Sep 11 '13 at 20:32
@DouglasZare That's why I was asking about generalizations. –  Keshav Srinivasan Sep 11 '13 at 22:12
I do not think this question is suitable here, notably because it is ill-defined mathematically (as pointed out below). You do not define marginal utility of money, which is in fact something you must add to the model. My guess: each voter has a number $a$ that represents how much he or she is willing to pay to increase its own utility by a unit. Then, this $a$ is only defined up to a multiplicative constant. The point is then that we want to identify $(u,a)$ with $(bu, a/b)$ for all $b>0$: the two multiplicative constants are related. –  Benoît Kloeckner Dec 11 '13 at 8:53
That said, as long as you do not explain how to go from preferences to utilities and marginal utility of money (which you should probably call utility of money in this context), you do not have a question. –  Benoît Kloeckner Dec 11 '13 at 8:54
This question appears to be off-topic because it is about economics, and is ill-defined mathematically. –  Benoît Kloeckner Dec 11 '13 at 8:55

There exists adaptions of Arrow's theorem to von Neumann-Morgenstern preferences. see for example Theorem 4.3 here.

the weighted utilitarianism you propose violates the independence axiom, and one can multiply each of the utility functions by some positive number. This changes the SWF, but not the preferences over lotterie represented.

There is an extensive literature on informational requirements and utility comparisons in social choice developed by Gevers, Sen, and others. A fairly comprehensive survey is Social welfare functionals and interpersonal comparability by d'Aspremont and Gevers. For a mor leisurely overview, see chapter 1 of Theories of Distributive Justice by Roemer.

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Concerning scaling the utility function by a constant, wouldn't the marginal utility be scaled by the same constant, so that the utility divided by the marginal utility would be invariant under such transformations? See here: math.stackexchange.com/questions/477723/how-can-a-social-welfare-function-be-a-l‌​inear-combination-of-von-neumann-morgens/477924#477924 –  Keshav Srinivasan Oct 1 '13 at 14:22
@KeshavSrinivasan Hm, what exactly is the marginal utility of money? Usually, it should be the derivative, but that is a function and not a constant. –  Michael Greinecker Oct 1 '13 at 21:42
I got the notion of the weights being invarsely proportional to the marginal utility of money in order to achieve Kaldor-Hicks efficiency from Brad Delong. See e.g. his short paper here: docs.google.com/gview?url=http://holtz.org/Library/… –  Keshav Srinivasan Oct 2 '13 at 1:18
@KeshavSrinivasan What BradDelong is doing is choosing the weights so as to make a certain allocation the maximum. He doesn't really explain why that is possible, but he seems to be using something like what is in section 7 here. So you have to fix some allocation first, before talking about the marginal utility of money. –  Michael Greinecker Oct 2 '13 at 7:31
Why do we need the coefficients in front of the utility functions to be constants in the first place? What's wrong with letting them be functions? –  Keshav Srinivasan Nov 19 '13 at 0:18

First, Arrow's Theorem says that no map $$L(A)^n\rightarrow L(A)$$ can simultaneously satisfy a certain list of properties.

You are trying, more or less, to construct a counterexample where $A$ is replaced by $G(A)$.

There are (at least) two problems with your idea.

1) You are not constructing a map $$L(G(A))^n\rightarrow L(G(A))$$ Instead you are constructing a map $$V^n\rightarrow L(G(A))$$ where $V\subset L(G(A))$ consists of those linear orders that satisfy the vonNeumann-Morgenstern axioms. Therefore there's no reason Arrow's Theorem should apply.

2) Your proposal is ill-defined because it gives me no clue how to normalize the functions $u_i$. (I have no idea what "marginal utility of money" means in the general context of lotteries over states of the world.) You can set the weights $a_i$ arbitrarily, but you still need a map from preferences to utility functions (which are, a priori, well-defined only up to affine transformations). What makes you think you can construct that map in such a way that your proposed mechanism will satisfy all of the Arrow conditions? In particular, what makes you think that any such map will satisfy the independence of irrelevant alternatives?

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Steve, re 1), you're right that I should have said maps from $V^N$, not the full space $L(G(A))^N$. But can't we similarly restrict the codomain to V? Are you familiar with Harsanyi (1955)? darp.lse.ac.uk/papersdb/Harsanyi_(JPolE_55).pdf It deals with a social planner who has a vnM utility function that's a linear combination of the vnM utility functions of each individual. Now Harsanyi's theorem was a single-profile result, i.e. it involved a fixed set of individual preferences, but there have been various multi-profile generalizations of it, like Mongin: tinyurl.com/q5emxwl –  Keshav Srinivasan Sep 11 '13 at 22:32
Re 2), I've heard that Arrow's theorem doesn't apply when the number of voters is infinite, but I wasn't aware that it requires the set of states to be finite. Do you have a source that explains why infinitely many states aren't allowed? And what do you mean by "non-standard analysis is possible"? You mean analysis not in accord with Arrow's reasoning? –  Keshav Srinivasan Sep 11 '13 at 22:40
Re 3), we don't need to define "marginal utility of money" to be definable in the context of lotteries, just in the context of states. I thought that one of the reasons why it makes sense for the $a_i$'s to be inversely proportional to the marginal utilities of money was so that $a_i u_i$ is invariant under scalar multiples of $u_i$, since whenever $u_i$ is scaled by k, a_i is multiplied by 1/k. Am I wrong? –  Keshav Srinivasan Sep 11 '13 at 22:46
When the number of voters is infinite, one arrives at the Gibbard-Satterthwaite theorem (en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem), that the preference relation is obeying the preference relation provided by an ultrafilter on the voters. This implies Arrow's theorem, since every ultrafilter on a finite set is principal, but in the infinite context, there can be nonprincipal ultrafilters. –  Joel David Hamkins Sep 11 '13 at 22:52
Keshav: This is not a blog or a discussion site, and it's the wrong place to go fumbling around asking other people to figure out whether there's any sense to be made out of your vague ideas. You haven't defined $M_u$ and you haven't given a well-defined map from vN/M preferences to utility functions. It would really be quite inappropriate to carry this further. –  Steven Landsburg Sep 12 '13 at 2:20