37 Social Welfare Functionals
So far, we have studied methods that use either rankings or grades to make social choices. Now we turn to a more general framework that uses utility functions (Chapter 10) to represent voters’ preferences. We assume that each voter submits a cardinal utility function (Section 10.2) that assigns a real number to each candidate, representing the voter’s subjective evaluation of the candidate.
37.1 The Framework
Let \(X\) be a set of alternatives with at least three elements, and \(V\) be a finite set of voters. For any set \(X\), let \(\mathcal{U}(X)\) be the set of all functions from \(X\) to the real numbers. An utility profile is a function \(\mathbf{U}\) that assigns to each voter \(i\) a utility function \(\mathbf{U}_i \in \mathcal{U}(X)\).
Definition 37.1 (Social Welfare Functional) Suppose that \(X\) is a set of alternatives. A Social Welfare Functional (SWFL) is a function \(f\) that maps profiles of utility functions to a relation on \(X\). For each profile \(\mathbf{U}\), the relation \(f(\mathbf{U})\) represents how society ranks the alternatives given the voters’ utilities in \(\mathbf{U}\).
A natural way to aggregate utilities is to sum them across all voters.
Definition 37.2 (Sum Utilitarian SWFL) The Sum Utilitarian SWFL, denoted \(f_U\), ranks alternatives as follows: for any alternatives \(x\) and \(y\):
- \(x\) is ranked above \(y\) if and only if \(\sum_i \mathbf{U}_i(x) > \sum_i \mathbf{U}_i(y)\).
- \(x\) and \(y\) are tied if and only if \(\sum_i \mathbf{U}_i(x) = \sum_i \mathbf{U}_i(y)\).
To see how this works, consider the following profile with three voters \(v_1\), \(v_2\), and \(v_3\) and three alternatives \(x, y\), and \(z\):
\[\begin{array}{cccc} \mathbf{U} & x & y & z \\\hline v_1 & 3 & 1& 8\\ v_2 & 3 & 2& 1\\ v_3 & 1 & 4& 1\\ \hline Sum & 7 & 7 & 10\\ \end{array}\]The Sum Utilitarian SWFL ranks \(z\) above both \(x\) and \(y\) because the sum of utilities for \(z\) (10) is higher than both \(x\) and \(y\) (7), and ranks \(x\) tied with \(y\) because they have the same sum.
37.2 Arrow’s Axioms
Interestingly, the Sum Utilitarian SWFL satisfies versions of Arrow’s axioms, including non-dictatorship. Let’s verify each axiom for \(f_U\):
- Universal Domain: \(f_U\) produces an ordering for any profile of utility functions
- Rationality: For any profile \(\mathbf{U}\), the social preference relation \(f_U(\mathbf{U})\) is complete and transitive because the ordering is based on comparing real numbers, which is complete and transitive.
- Pareto: If \(\mathbf{U}_i(x) > \mathbf{U}_i(y)\) for all voters \(i\), then the sum of utilities for \(x\) must be greater than the sum for \(y\), so \(x\) is ranked above \(y\).
- Independence of Irrelevant Utilities: If \(\mathbf{U}_i(x) = \mathbf{U}'_i(x)\) and \(\mathbf{U}_i(y) = \mathbf{U}'_i(y)\) for all voters \(i\), then the sums for \(x\) and \(y\) will be the same in both profiles, producing the same ranking between them.
This seems to contradict Arrow’s Impossibility Theorem! However, as we’ll see in the next section, there’s a catch involving the interpersonal comparison of utilities.
37.3 Cardinal-Measurability
While Sum Utilitarianism seems to escape Arrow’s impossibility result, Arrow raised a fundamental objection about comparing utilities between different people. As he wrote:
“The viewpoint will be taken here that interpersonal comparison of utilities has no meaning and, in fact, that there is no meaning relevant to welfare comparisons in the measurability of individual utility…” (Social Choice and Individual Values, p. 9)
The core of Arrow’s argument is that utility functions are only meaningful up to linear transformations (Section 10.2.1). That is, if \(u\) is a utility function representing someone’s preferences, then \(u'(x) = \alpha \times u(x) + \beta\) (where \(\alpha > 0\)) represents exactly the same preferences. For example, if \(u\) assigns utilities \((3,2,0)\) to alternatives \((a,b,c)\), all of the following utilities represent the same preferences:
- \(u'\): \((32, 22, 2)\) [multiply by 10, add 2]
- \(u''\): \((0.75, 0.5, 0)\) [multiply by 0.25]
This creates a serious problem for the Sum Utilitarian SWFL. As Arrow notes:
``Even if… we should admit the measurability of utility for an individual, there is still the question of aggregating the individual utilities. At best, it is contended that, for an individual, his utility function is uniquely determined up to a linear transformation; we must still choose one out of the infinite family of indicators to represent the individual, and the values of the aggregate (say a sum) are dependent on how the choice is made for each individual. In general, there seems to be no method intrinsic to utility measurement which will make the choices compatible…” (Social Choice and Individual Values, pp.~10-11)
Consider again our earlier example profile \(\mathbf{U}\) with three voters and three alternatives:
\[\begin{array}{cccc} \mathbf{U} & x & y & z \\\hline v_1 & 3 & 1& 8\\ v_2 & 3 & 2& 1\\ v_3 & 1 & 4& 1\\ \hline \end{array}\]The following two profiles represent the same preferences as \(\mathbf{U}\), since they differ only by linear transformations of individual utilities:
- Profile \(\mathbf{U}'\) (multiply \(v_2\)’s utilities by 100):
- Profile \(\mathbf{U}''\) (multiply \(v_2\)’s utilities by 100 and \(v_3\)’s utilities by 100):
However, the Sum Utilitarian SWFL produces different rankings for these profiles:
- \(f_U(\mathbf{U})\) ranks \(z\) above \(x\) and \(y\) (which are tied)
- \(f_U(\mathbf{U}')\) ranks \(x\) above \(y\) above \(z\)
- \(f_U(\mathbf{U}'')\) ranks \(y\) above \(x\) above \(z\)
As Arrow notes, there is no principled way to choose which of these profiles is the “correct” one. The problem isn’t just that Sum Utilitarianism gives different rankings - it’s that any method attempting to aggregate utilities across people must somehow justify treating one representation of preferences as more legitimate than another.
This suggests that any reasonable social welfare functional should satisfy what’s called Cardinal Measurability Invariance (CM-invariance): it should produce the same social ranking for any two profiles that represent the same preferences up to linear transformations of individual utilities.
Definition 37.3 (Cardinal Measurability Invariance (CM-invariance)) A SWFL \(f\) satisfies CM-invariance if for any two profiles \(\mathbf{U}\) and \(\mathbf{U}'\) that are equivalent up to linear transformations of individual utilities (that is, there exist \(\alpha_i > 0\) and \(\beta_i\) such that \(\mathbf{U}'_i(x) = \alpha_i \times \mathbf{U}_i(x) + \beta_i\) for each voter \(i\) and alternative \(x\)), then \(f(\mathbf{U}) = f(\mathbf{U}')\).
The Sum Utilitarian SWFL violates CM-invariance, as our example shows. This leads to an extension of Arrow’s Impossibility Theorem, developed by Amartya Sen, showing that any SWFL satisfying basic axioms plus CM-invariance must be dictatorial.
Theorem 37.1 (Sen’s Theorem) Suppose that \(X\) is a set of alternatives with at least three elements, and \(V\) is a finite set of voters. There is no Social Welfare Functional that satisfies Universal Domain, Rationality, Pareto, Independence of Irrelevant Utilities, Cardinal Measurability Invariance, and Non-Dictatorship.