Arrow’s Impossibility & Validity of Social Welfare and Pareto Improvement Analysis

The utilitarian SWF assumes that individuals' utility functions are quasilinear in money. Thus, they violate Arrow's Unrestricted Domain axiom, but satisfy the other axioms. If you accept quasilinearity of utility as a (at least approximately) valid assumption, then you can work with this SWF.

Let $X$ be the nonempty set of alternatives, $mathcal{P}_X$ the set of preference relations on $X$ and $N={1,ldots,n}$ a finite set of agents. Under the universal domain condition, Arrow's theorem concerns functions from the set $mathcal{P}_X^N=underbrace{mathcal{P}_Xtimes mathcal{P}_Xcdotstimesmathcal{P}_X}_{ntext{ times}}$ to $mathcal{P}_X$. Utilitarianism does not just depend on preferences alone, and can therefore not be formulated in this setting.

Still, one can formulate Arrow's theorem in terms of utility functions. So let $mathcal{U}_X$ be the space of real-valued functions on $X$, interpreted as "utility functions." I'll explain the scare-quotes later. We are now looking at functions from $mathcal{U}_X^N$ to $mathcal{P}_X$. Since utility functions determine preferences, we can do now the same things we could do before and more.

Now there are two ways to formulate Arrow's condition of independence of irrelevant alternatives in the new setting; adapting the other conditions is straightforward.

The first form is:

IIA1: The function $phi:mathcal{U}_X^Ntomathcal{P}_X$ satisfies IIA1 if for any two profiles of utility functions $u=(u_1,ldots,u_n)$ and $u'=(u_1',ldots,u_n')$, and any two alternatives $x,yin X$, such that $u_i(x)geq u_i(y)$ if and only if $u_i'(x)geq u_i'(y)$ for all $iin N$, we have $xphi(u) y$ if and only if $xphi(u') y$.

Note that IIA1 has implicitly two components. The first says that the social ranking of two alternatives depends only on the utility values for these two alternatives. The second says that only the preference ranking matters, but not the "intensity." If we only keep the first, we get

IIA2: The function $phi:mathcal{U}_X^Ntomathcal{P}_X$ satisfies IIA2 if for any two profiles of utility functions $u=(u_1,ldots,u_n)$ and $u'=(u_1',ldots,u_n')$, and any two alternatives $x,yin X$, such that $u_i(x)-u_i(y)=u_i'(x)-u_i'(y)$ for all $iin N$, we have $xphi(u) y$ if and only if $xphi(u') y$.

IIA2 is actually weaker than requiring that the utility for both alternatives are the same in both profiles; only the differences need to be the same.

Now, with the other axioms suitably reformulated, IIA1 implies that all such functions $phi:mathcal{U}_X^Ntomathcal{P}_X$ must be dictatorial, while IIA2 is compatible with utilitarianism.

Now, so far this was a purely mathematical issue. In particular, we treated utility functions as being somehow objective things and not just representations of preferences. They also measure well-being in a way that is comparable across agents. So utilitarianism uses information that is not included in purely positive models of economic behavior, in which only preferences matter. Now, one can discuss how sensible it is to compare the well-being of different agents and what information one needs for certain SWFs. There has been a research program starting in the late 1970s of discussing the exact informational assumptions one uses for such comparisons of well-being. A nice introduction to the topic is given in the 1998 book "Theories of distributive justice" by John E. Roemer. Roemer directly relates all this to Arrow's theorem, so his book will give you a very extensive answer.

Arrow impossibility theorem presupposes:

1. that individual preferences can be expressed ordinally, but not cardinally, and

2. that "Pareto", "relevance" (i.e. “ Pareto Efficiency” and “Independence of Irrelevant Alternatives”) and "no dictatorship" are necessary conditions for democratic elections.

The first point stems from the difficulties that the founders of decision theory had with comparing the cardinally expressed preferences of different people (how do you compare my 5 points with your 4 points?). Arrow, therefore, rejected the cardinal expression of preferences "because it makes no sense". Truly curious, given that von Neumann had shown years before how the cardinal preferences of different individuals could be compared (in Theory of Games and Economic Behavior of 1944). In addition, it is almost trivial to find options for which cardinal preferences make more sense than ordinal ones. Consider the following options:

A. You will receive 100 dollars.

B. You will receive 100 dollars and will be beaten to death.

C. You will be beaten to death.

Which better expresses your preferences, ordinal A > B > C or cardinal

A = 10000000000000000000000000000000000000000000 points,

B = 1 point

C = 0 points?

Thus, electoral systems with cardinal expressions of preferences should not be rejected. But if we take them into account then Arrow's theorem is no longer relevant. Consider an electoral system in which voters are free to assign a certain number of points to each option (e.g. 0 to 9 points) and which finally ranks the options according to the number of points obtained. It is "Pareto", "relevant" and "non-dictatorial". Arrow's theorem does not apply to this system (because Arrow's theorem a priory excluded such systems).

The second point is also questionable. Although "Pareto" and "non-dictatorship" may be thought as unquestionable democratic principles, "relevance" can not. Consider the example, by which Condorcet in the 1780s questioned Borda’s electoral systems. This is an example with the following distribution of votes:

30 A > B > C

1 A > C > B

10 C > A > B

1 C > B > A

10 B > C > A

29 B > A > C

Thus, 30 voters prefer the order A > B > C, 1 voter prefers A> C> B, and so on. Borda’s electoral system which gives 2 points for 1st position, 1 point for 2nd position and 0 points for 3rd position, gives the final order B > A > C. (It can be proved that any decreasing distribution of points from the first to the third position gives the same result.)

Condorcet’s objection is that in a direct duel A wins B (because 41 voters put A ahead of B and 40 voters put B ahead of A) and A wins C (because 60 voters put A ahead of C and 12 voters put C ahead of A). So, A is Condorcet's winner which in direct duels wins all other options, and Borda's system does not declare him the winner. Condorcet concludes that Borda’s system is invalid.

Notice that this is precisely an argument from "relevance". Condorcet’s winner (in his example it is A) is determined by the mutual duels of two options, regardless of how the other options are ranked. This is exactly what "relevance" requires. In other words, Condorcet's objection is that Borda's system does not respect Arrow's (almost two centuries younger) requirement of "relevance."

The question is whether Condorcet is right, i.e. whether "relevance" is relevant. Consider a simple majority choice between two options A and B. Suppose there are 2,100 voters in play. If we detect among them 1000 who prefer A > B and 1000 who prefer B > A, we can ignore those 2000 votes (because they cancel each other). The result of the election is determined by the remaining 100. Analogously, 30 voters in Condorcet’s example:

10 A > B > C

10 B > C > A

10 C > A > B

cancel each other out, so we can ignore them. Namely, each of the options has the same number (in this case 10) of first, second and third places. With the same argument we can ignore 3 voters with preferences:

1 A > C > B

1 C > B > A

1 B > A > C

But when in Condorcet’s example, we ignore those voters whose votes are annulled, the decision is made by the remaining voters:

20 A > B > C

28 B > A > C

And the winner is B.

Condorcet is wrong, as is the Arrow’s “Independence of Irrelevant Alternatives”.