Interpersonal Utility Comparisons (New Developments)

Abstract

Recent developments on interpersonal utility comparisons rely on various interpretations of ‘utility’ indicators and combine in various degrees the ‘subjective’ appreciation of the social states by each individual and their ‘objective’ evaluation by the ethical observer. In a formal welfarist approach, interpersonal comparisons are specified by invariance conditions on social welfare functionals or on social welfare orderings. Interpersonal comparisons have also been introduced through scoring methods.

Distributive justice, whether in normative economics or in collective choice theory, can hardly be treated without introducing some interpersonal comparisons. But does this mean considering interpersonal utility comparisons? The term ‘utility’ has received so many different interpretations that the distinguishing mark of the utility approach to the evaluation of social states by an ethical observer is simply that it assigns to each member of the collectivity a unidimensional individual indicator. These indicators combine in various ways the ‘subjective’ appreciation of the social states by each individual and their ‘objective’ evaluation by the observer.

Bentham introduced an interpersonally summable notion of utility based on the objective property of things to procure either pain or pleasure, but it is the subjective reinterpretation given by J.S. Mill, for whom utility means ‘pleasure itself’ and ‘exemption of pain’, which has prevailed (Mongin and d’Aspremont 1998). In Harsanyi’s (1977) reformulation of utilitarianism, the ethical observer is, with equal chance, any one of the individuals and obeys the rationality conditions of decision-making under risk. The criterion is expected utility computed from the individual von Neumann −Morgenstern (NM) utility functions, but ‘corrected’ for factual errors and ‘censored’ for anti-social attitudes.

Pareto has clearly distinguished between the objective notion of utilité (in the Bentham sense) and the subjective notion of ophélimité, as an ordinal measure of actual preference satisfaction. The latter has become the dominant concept in economics (the only concept in the new welfare economics), forbidding interpersonal utility comparisons and, ultimately, reducing preferences to observable individual choices (revealed preference theory). But Samuelson (1947) insisted that welfare economics cannot avoid ethical and interpersonal assumptions, and Arrow (1951) derived an ‘impossibility theorem’. For a set N = {1, 2, ..., n} of individuals and a set X = {x, y, ...} of (more than two) social states, there is no acceptable social welfare function, associating every profile of individual preference orderings with one ‘collective’ preference ordering of X, and satisfying weak Pareto (if all strictly prefer one state to another, so should society) and independence of irrelevant alternatives (if individual preferences are modified except for a subset of several alternatives, then the collective preference should not be modified on this subset). Such a social welfare function can only be dictatorial: one individual imposes his strict preference. Since all Arrow’s assumptions concern ‘preference satisfaction’, modifying them and reinterpreting ‘utility’ involve ethical considerations.

Rawls’s (1971) principles of justice are agreed upon in some original negotiation where all irrelevant personal features (including personal conceptions of the good) are ignored. Also chosen behind this ‘veil of ignorance’ is an ‘index of primary goods’ defined as an objective indicator of the fundamental resources (except for liberties and access to occupations, preliminarily and equally divided) allocated to each person to promote his own conception of the good. If such an indicator can be called ‘utility’, it is not in the sense of ‘happiness’ or ‘preference satisfaction’. Desires (however intense) and tastes (however inexpensive) are not relevant per se. A similar view is represented by Sen’s (1992) notion of capabilities, that is, the set of doings or ‘functionings’ available to a person, leading to an ‘index of functionings’. What is at stake here, as in other theories concerned by opportunities (for example, Roemer (1996)), are the objectively defined conditions allowing individuals to exercise their freedom.

Extended Sympathy, Social Welfare Functionals and Welfarism

To formally examine the role of interpersonal utility comparisons in social choice, it is usual to start within the framework introduced by Sen (1970), in which the basic ingredient is a utility profile given by a real-valued function U defined on elements (x,i) of the Cartesian product X × N. The function U can be seen as a vector of individual indicators U_i(x) ≡ U(x, i), or as the extended utility function of an individual, evaluating from a moral viewpoint what it is to be anyone in any social state (exercising ‘extended sympathy’ or ‘empathy’). Moreover, if for individual i one interprets the name i as designating all the characteristics of i, then one could look at the function U as a fundamental utility function (Harsanyi 1977), itself a representation of ‘human nature’ (Kolm 1972), which would then justify why every individual, when adopting the viewpoint of an ethical observer, should have the same extended utility function (or at least the same fundamental preference). Lack of identity could lead to a dictatorial ethical observer (see Suzumura (1996)).

The fundamental utility approach is also used in econometric estimations to define ‘adult-equivalent scales’ and different forms of exact aggregation (Blackorby and Donaldson 1991; Christensen et al. 1975; Deaton and Muellbauer 1980). Other measurement techniques, such as that which uses the number of ‘just-noticeable-differences’ between two alternatives (discussed in (Arrow 1951)) or the ‘social indicators’ approach using questionnaires about degree of happiness (discussed in (Fleurbaey and Hammond 2004; Hammond 1991)) are differently founded.

The U function is a very flexible informational basis to start with. For every x, we can denote U_x the utility vector (U(x, 1), ..., U(x, n)) in ℝ^N. Taking all functions U in some domain D determines the set of all admissible utility vectors. Sen’s (1970) concept of social welfare functional (SWFL) associates every admissible extended utility function U in D with one (collective) preference ordering R_U. We denote I_U and P_U the corresponding indifference and strict preference relations. Using this notation, Pareto indifference means ‘U_x = U_y implies xI_Uy’ and strong Pareto requires in addition that ‘U_x ≥ U_y and U_x ≠ U_y implies x P_Uy’. Also, Arrow’s independence of irrelevant alternatives can be weakened to binary independence, whereby for any two functions U and V with equal values on two social states x and y we have xR_Vy ⇔ xR_Uy.

An alternative framework is to define directly a social welfare ordering (SWO) denoted R^* on the set of admissible utility vectors. If the set of admissible utility vectors is large enough (for example, equal to ℝ^N ), then, under Pareto indifference and binary independence, the two frameworks coincide: u = U_x and v = U_y implies uR^*v ⇔ xR_Uy. This is called welfarism and is an extreme form of consequentialism. All the information required for social evaluation is contained in the final utility values. Under welfarism, strong Pareto (SP) reduces to the condition that u ≥ v, and u ≠ v implies uP^*v (P^* denoting strict collective preference).

Invariance Axioms

Measurement theory (Krantz et al. 1971; Roberts 1980) associates with different measurement scales the associated meaningful statements. We are interested in meaningful statements about intrapersonal and interpersonal comparisons of utility. For instance, the Arrowian informational basis for SWFLs requires that only intrapersonal level comparisons are meaningful: R_U = R_V whenever, for every i and all x, y,  U(x, i) ≥ U(y, i) if and only if V(x,  i) ≥ V(y,  i). Another example (for this and others see Bossert and Weymark (2004)) is to consider meaningful interpersonal comparisons of utility differences: R_U = R_V whenever, for all w, x, y, z and all i, j, U(w,  i) − U(x,  i) ≥ U(y,  j) − U(z,  j) if and only if V(w,  i) − V(x,  i) ≥ V(y,  j) − V(z, j).

The more standard way (for example, Sen (1977)) to specify the measurability and comparability properties of ‘utility’ is to introduce invariance transformations ϕ = (ϕ₁, ϕ₂, … , ϕ_n), each ϕ_i being a real-valued function on ℝ. In the Arrowian framework, we get the invariance axiom of ordinality and non-comparability (ON): if each ϕ_i is increasing and if for every x, V(x, i) = ϕ_i(U(x, i)), then R_U = R_V. Corresponding to interpersonal comparisons of utility differences, we get cardinality and unit-comparability (CU): if each ϕ_i is a positive affine transformation (ϕ_i(u_i) = a_i + bu_i, with b > 0) and if, for all (x,i), V(x,i) = a_i + bU(x,i), then R_U = R_V.

But there is a third way of specifying such conditions. We have stated these two axioms as restrictions on SWFLs. Under welfarism, they can be translated into axioms on SWOs, as

$$ {\displaystyle \begin{array}{l}\mathrm{ON}:\mathrm{for}\kern0.5em \mathrm{any}\kern0.5em \mathrm{increasing}\kern0.5em {\phi_i}^{'}s,\hfill \\ {}\kern1em {uR}^{\ast }v\iff \left({\phi}_1\left({u}_1\right),\dots, {\phi}_n\left({u}_n\right)\right)\hfill \\ {}\times {R}^{\ast}\left({\phi}_1\left({v}_1\right),\dots, {\phi}_n\left({v}_n\right)\right);\hfill \\ {}\mathrm{CU}\kern0.5em :\kern0.5em \mathrm{for}\ \mathrm{any}\ {{\mathrm{a}}_i}^{'}s,b>0,\hfill \\ {}\kern1em {uR}^{\ast }v\iff \left({a}_1+{bu}_1,\dots, {a}_n+{bu}_n\right)\hfill \\ {}\times {R}^{\ast}\left({a}_1+{bv}_1,\dots, {a}_n+{bv}_n\right).\hfill \end{array}} $$

Invariance axioms determine the informational basis for social evaluation. They should not be considered as purely factual. By specifying the kind of information that a social evaluation can or cannot use, these axioms are taking an ethical stance. But their strong ethical implications are better measured when combined with other axioms. To illustrate, the following axiom allows for (and only for) comparisons of utility differences that are intrapersonal. It is cardinality and non-comparability (CN): for any a_i’s and positive b_i’s, if V(x, i) = a_i + b_iU(x, i), for all (x, i), then R_U = R_V. Such a SWFL version of this axiom does not exclude interpersonal utility comparisons. It does allow us to compare ratios of utility differences of the sort (U(w, i) − U(x, i))/(U(y, i) − U(z, i)) between different individuals, and hence to compare measures of risk aversion in case X is specified as a set of lotteries and each U_i(x) as an NM utility function. But the possibility of such comparisons is erased under welfarism, under which cardinal non-comparability reduces to ordinal non-comparability and, with strong Pareto, implies dictatorship (by Arrow’s theorem). Under welfarism, ON becomes equivalent to CN: for any a_i’s, positive b_i’s,

$$ {\displaystyle \begin{array}{l}{uR}^{\ast }v\iff \left({a}_1+{b}_1{u}_1,\dots, {a}_n+{b}_n{u}_n\right)\hfill \\ {}\times {R}^{\ast}\left({a}_1+{b}_1{v}_1,\dots, {a}_n+{b}_1{v}_n\right).\hfill \end{array}} $$

If CN is replaced by CU and dictatorship is excluded by anonymity (any utility vector is socially indifferent to any of its permutations), then the only possibility is the pure utilitarian SWO: uR^*v if and only if ∑_i∈Nu_i ≥ ∑_{i∈ N}v_i (d’Aspremont and Gevers 1977).

This characterization of utilitarianism is directly related to Harsanyi’s aggregation theorem since, under welfarism (and NM preferences), cardinality and unit-comparability become equivalent to NM-independence of the collective preference ordering (Mongin and d’Aspremont 1998).

By giving priority to liberties and access to occupations, Rawls clearly departs from welfarism, even in a formal sense. However, to allocate other primary goods, a common index V(x_i) is fixed, where x_i is the vector of primary goods to be allocated to individual i. Letting, for an allocation x = (x₁, … , x_n), U(x, i) = V(x_i), we can fall again into the welfarist formal framework. Since the index is common to all, the associated invariance axiom is ordinality and comparability

$$ {\displaystyle \begin{array}{l} OC:\kern1em \mathrm{for}\kern0.5em \mathrm{any}\kern0.5em \mathrm{increasing}\kern0.5em \widehat{\phi},\hfill \\ {}{uR}^{\ast }v\iff \left(\widehat{\phi},\left({u}_1\right),\dots, \widehat{\phi}\left({u}_n\right)\right)\hfill \\ {}\times {R}^{\ast}\left(\widehat{\phi}\left({v}_1\right),\dots, \widehat{\phi}\left({v}_n\right)\right).\hfill \end{array}} $$

Two other axioms are clearly required by Rawls: anonymity and strong Pareto, the latter being the reason why equal distribution of all primary goods is not the agreed-upon solution. To any u in ℝ^N, one can associate a (re)ordered vector u_i(⋅) with same components in {v ∈ ℝ^N : v₁ ≤ v₂ ≤ … ≤ v_n}, the set of ordered utility vectors. Minimal equity requires that one should never give priority to the best-off individual over the worst-off. Then, under separability (in choosing between two utility vectors the indifferent individuals should not be taken into account), the solution is the ‘lexicographic maximin’ (leximin) SWO : for any u, v in ℝ^N, uP^*v if and only if, for some k, 1 ≤ k ≤ n, u_i(k) > v_i(k), and, ∀j < k, 1 ≤ j ≤ n, u_i(j) = v_i(j). Leximin formalizes Rawls’s ‘difference principle’. Other concepts of opportunity equalizations can be so translated into welfarist terms (see Maniquet (2004)).

From a formal viewpoint, in the preceding result only utility levels are both intrapersonally and interpersonally comparable. If we add the same possibility for utility differences we have full comparability, that is

$$ {\displaystyle \begin{array}{l} FC:\kern1em \mathrm{for}\kern0.5em \mathrm{any}\kern0.5em a,\kern0.5em b>0,\hfill \\ {}\kern1em {uR}^{\ast }v\iff \left(a+{bu}_1,\dots, a+{bu}_n\right)\hfill \\ {}\times {R}^{\ast}\left(a+{bv}_1,\dots, a+{bv}_n\right).\hfill \end{array}} $$

With this type of invariance and the same other assumptions (Deschamps and Gevers 1978), the SWO R^* can be either leximin or utilitarianism, but in a weak sense (that is ${\sum}_{i\in N}{u}_i>{\sum}_{i\in N}{v}_i$ implies uP^*v).

Many other invariance axioms can be introduced (for example, (d’Aspremont 1985)). Let us give only two more, ratio-scale measurability, without or with interpersonal comparisons of utility:

$$ {\displaystyle \begin{array}{l}\mathrm{RN}:\kern1em \mathrm{for}\ \mathrm{any}\kern0.5em \mathrm{positive}\kern0.5em {b_i}^{'}\mathrm{s},\kern1em \mathrm{and}\kern0.5em u,\hfill \\ {}\kern0.5em v\in {\mathbb{R}}^{\mathrm{N}},\kern1em {uR}^{\ast }v\iff \left({b}_1{u}_1,\dots, \kern0.5em {b}_n{u}_n\right)\hfill \\ {}\times {R}^{\ast}\left({b}_1{v}_1,\dots, \kern0.5em {b}_n{v}_n\right);\mathrm{RC}:\kern1em \mathrm{for}\ \mathrm{any}\ \mathrm{positive}\ b,\kern1em \mathrm{and}\kern0.5em u,\hfill \\ {}\kern0.5em v\in {\mathbb{R}}^N,\kern1em uR{v}^{\ast}\iff {b}_u{R}^{\ast } bv.\hfill \end{array}} $$

With ratio-scale measurability the origin is fixed, so that, under RN, utility levels are interpersonally comparable if they are of opposite sign (below or above the zero line). Moreover, under RC, all utility levels and differences are comparable. These axioms are most often applied on a positive domain (denoted ${\mathbb{R}}_{++}^N$). Under RN, ratios of utilities or percentage changes in utility are interpersonally comparable. If we add SP, then a continuous SWO on ${\mathbb{R}}_{++}^N$ can only be the Nash bargaining solution with status quo point normalized to zero: for some positive θ_i’s, uR^*v if and only if ${\prod}_{i=1}^n{u}_i^{\lambda_i}\ge {\prod}_{i=1}^n{v}_i^{\lambda_i}$. Under RC and SP the set of continuous and anonymous SWOs is characterized by all homothetic, increasing, continuous and symmetric functions on ${\mathbb{R}}_{++}^N$ (for these and other results, see Bossert and Weymark (2004)).

Beyond Welfarism: Scoring and Fair Allocation Rules

Even when ‘utility’ simply represents actual preference satisfaction, it can be severely adjusted by the ethical observer (or rule designer). This is the case for various voting rules or more generally for scoring rules. These rules violate binary independence in one way or another, so that welfarism is excluded. Voting methods, such as Borda’s, are generally not acceptable for social evaluation, but some related rules are better candidates (Moulin 1988). Another ‘scoring’ method is relative utilitarianism (axiomatized by Dhillon and Mertens (1999)). Each U_i(.) is supposed to have both a maximum ${U}_i^{\mathrm{max}}$ and a minimum ${U}_i^{\mathrm{min}}$ and to represent a NM preference ordering on X and the observer associates to it a NM utility function V_i(.) ordinally equivalent to U_i(.), and then, through individual affine transformations, defines the scoring function ${S}_i(x)=\left({V}_i(x)-{V}_i^{\mathrm{min}}\right)/\left({V}_i^{\mathrm{max}}-{V}_i^{\mathrm{min}}\right).$ The score S_i(x) is the same whatever the arbitrarily chosen NM utility function V_i(.) representing the preference ordering underlying the initial utility function U_i(.), and measures of curvatures are preserved. The SWFL F is taken to be pure utilitarianism applied to the scores. It satisfies ordinality and non-comparability with respect to the utility functions U_i(x) representing the individual preferences. Since the scores are defined as ratios of utility differences, they could be interpersonally compared, but, because their aggregation is utilitarian, it relies only on interpersonal comparisons of differences of scores.

Other concepts have been proposed in the literature on fair allocations, excluding welfarism in terms of the initial utility functions, but ending up applying some SWFL to some recalibrated utility. An example is the Pareto efficient egalitarian-equivalent allocation concept (Pazner and Schmeidler 1978). To illustrate, let $\overline{\omega}$ in ${\mathbb{R}}_{+}^L$ be the vector of total quantities of L goods and X be the set of feasible allocations (x₁, ... x_n): each x_i is in ${\mathbb{R}}_{+}^L$ and ${\sum}_{i=1}^n{x}_i\le \overline{\omega}$. If U(x,i) = U_i(x_i) is increasing in each argument and continuous then one can define an ordinally equivalent function V_i(x_i) such that ${U}_i\left({x}_i\right)={U}_i\left({V}_i\left({x}_i\right)\overline{\omega}\right)$. A Pareto efficient allocation $\widehat{x}$ in X is egalitarian-equivalent if ${U}_i\left({\widehat{x}}_i\right)={U}_i\left({V}_i\left({\widehat{x}}_i\right)\overline{\omega}\right)$ and ${V}_i\left({\widehat{x}}_i\right)={V}_j\left({\widehat{x}}_j\right)$ for all i,j. As observed in Fleurbaey and Hammond (2004), if individual preferences are convex, $\widehat{x}$ can be obtained by applying the leximin SWFL on the V_i(x_i)’s. Again, starting with a concept defined in terms of purely ordinal and non-comparable utilities (the U_i’s), we end up comparing and equalizing utility levels in terms of the V_i’s.

See Also

• Arrow’s Theorem

• Bargaining

• Equality of Opportunity

• Fair Allocation

• Interpersonal Utility Comparisons

• Pareto Principle and Competing Principles

• Revealed Preference Theory

• Risk Aversion

• Social Choice

• Social Choice (New Developments)

• Social Welfare Function

• Utility

• Welfare Economics