38 The Problem of Interpersonal Utility Comparisons

Faced with the impossibility theorems of Arrow (Theorem 34.1) and Sen (Theorem 37.1), social choice theorists have considered weakening the Cardinal Measurability (Definition 37.3) axiom by allowing interpersonal comparisons of utility. This means assuming we can meaningfully compare and aggregate utilities across different individuals. However, this raises a deep philosophical question: Can we actually make such comparisons? As Arrow argues:

“It requires a definite value judgment not derivable from individual sensations to make the utilities of different individuals dimensionally compatible and still a further value judgment to aggregate them according to any particular mathematical formula… It seems to make no sense to add the utility of one individual, a psychic magnitude in his mind, with the utility of another individual.” (Social Choice and Individual Values, p. 11)

The following example from Resnik (1987) illustrates the issues that arise. Suppose that Mary and Sam are deciding on a vacation destination. They have three options: seashore, museums, or camping. Their rankings are:

Mary: \(seashore \mathrel{P} museums \mathrel{P} camping\)
- The seashore is the only option Mary finds bearable.
- She strongly dislikes camping.
- Museums are somewhere in between.
Sam: \(camping \mathrel{P} museums \mathrel{P} seashore\)
- All options are acceptable to him.
- He has a preference for camping.
- The differences between options are small.

Now suppose they try to use the Sum Utilitarian SWFL to make their decision. The first step is to assign utilities to each option:

\[\begin{array}{cccc} & seashore & museums & camping \\\hline Mary & 20 & 10 & 9\\ Sam & 86 & 93& 100\\ \hline Sum & 106 & 103 & 109\\ \end{array}\]

The Sum Utilitarian SWFL would rank camping above seashore above museums based on the sum of the utilities. However, this analysis raises several concerns:

Intensity of Preferences: Mary argues her preferences are more intense than Sam’s, suggesting her utilities should use a different scale - perhaps 0 to 1000 instead of 0 to 100. If we rescaled her utilities this way, seashore would become the winner.
Cultural Expression: Sam counters that he feels things just as deeply but expresses them differently due to his cultural background. His seemingly moderate utilities might represent equally strong feelings.

These disagreements point to fundamental questions about interpersonal utility comparisons:

Is Mary’s preference for the seashore really stronger than Sam’s for camping, or is she just more expressive?
Even if some people do have stronger preferences, how could we know this?
Does comparing Mary’s and Sam’s utilities make any more sense than comparing a dog’s preference for bones with a horse’s preference for oats?
Even if we could make such comparisons, should they influence social decisions?

These questions reveal that interpersonal utility comparisons are problematic on two levels: epistemologically (how can we know which utility representations are “correct”?) and morally (should preference intensity influence group decisions?). This helps explain why Arrow insisted on avoiding interpersonal comparisons in his framework for social choice.

38.1 Relative Utilitarianism

One way to address some of the concerns raised above is to focus on relative rather than absolute utilities. The idea is simple: before summing utilities, normalize each voter’s utilities to a common scale. More precisely, assuming a finite set of alternatives, for each voter \(i\):

Find their maximum utility \(M_i\) and minimum utility \(m_i\)
Transform their utilities using the formula: \[\mathcal{K}(\mathbf{U})_i(y) = \frac{\mathbf{U}_i(y) - m_i}{M_i - m_i}\]

This normalization (called Kaplan normalization) maps each voter’s utilities to the \([0,1]\) interval, with their most-preferred alternative(s) getting utility 1 and their least-preferred alternative(s) getting utility 0.

However, as Arrow noted, this approach is “extremely unsatisfactory” because it violates Independence of Irrelevant Utilities. Here’s an example showing why: Consider three voters (Alice, Bob, and Cora) evaluating alternatives \(x\), \(y\), and \(z\):

\[\begin{array}{c|ccc} \mathbf{U} & x & y & z \\ \hline Alice & {1} & {.9} & 0 \\ Bob & {1} & {.9} & 0 \\ Cora & {.5} & {1} & 0 \\ \end{array}\]

\[\begin{array}{c|ccc} \mathbf{U}' & x & y & z \\ \hline Alice & {1} & {.9} & 1 \\ Bob & {1} & {.9} & 1 \\ Cora & {.5} & {1} & .5 \\ \end{array}\]

The Independence of Irrelevant Utilities principle requires that since the utilities of \(x\) and \(y\) are the same in both profiles, then their rankings should also remain the same. Now, consider the Kaplan-normalized utilities:

\[\begin{array}{c|ccc} \mathcal{K}(\mathbf{U}) & x & y & z \\ \hline Alice & {1} & {.9} & 0 \\ Bob & {1} & {.9} & 0 \\ Cora & {.5} & {1} & 0 \\\hline Sum & 2.5 & 2.8 & 0 \\ \end{array}\]

\[\begin{array}{c|ccc} \mathcal{K}(\mathbf{U}') & x & y & z \\ \hline Alice & {1} & {0} & 1 \\ Bob & {1} & {0} & 1 \\ Cora & {0} & {1} & 0 \\\hline Sum & 2 & 1 & 2 \\ \end{array}\]

In profile \(\mathbf{U}\), relative utilitarianism ranks \(x\) above \(y\) because \(x\) has a total utility of \(2.5\) in the normalized profile, while \(y\) has \(2.8\) in the normalized profile. However, in profile \(\mathbf{U}'\), the ranking reverses: \(y\) is ranked below \(x\). This reversal occurs even though the utilities of \(x\) and \(y\) are unchanged across the two profiles, which directly violates Independence of Irrelevant Utilities.