JEL Classifications

Interdependent preferences arise in economic theory in the study of both individual decisions and group decisions. We imagine that a decision is required among alternatives in a set X and that the decision will depend on preferences between the elements in X. If the preferences represent different points of view about the relative desirability of the alternatives, of if they are based on multiple criteria that impinge on the decision, then we encounter the possibility of interdependent preferences.

There are two predominant approaches to interdependent preferences, the synthetic and the analytic. The synthetic approach begins with a set of preference relations on X and attempts to aggregate them into a holistic representative preference relation on X. This is done in social choice theory, where each original relation refers to the preferences of an individual in a social group. The aggregate relation is then referred to as a social preference relation. The synthetic approach also appears in studies of individual preferences, as when an individual rank-orders the alternatives for each of a number of criteria and then seeks a holistic ranking that combines the criteria rankings in a reasonable way.

In contrast, the analytic approach begins with a holistic preference relation on X and seeks to analyse its internal structure. This may involve a decomposition into components of preference, or it may concern trade-offs between factors that describe interactive contributions to overall preferences.

The synthetic approach often considers a list (≫1, ≫2, … , ≫n) of preference relations on X, where x ≫ iy could mean that person i prefers x to y, or that an individual prefers x to y on the basis of criterion i. The problem may then be to specify a holistic relation ≫ = f(≫1, ≫2, … , ≫n) for each possible n-tuple of individual relations.

The analytic approach often begins with X as a subset of the product X1 × X2 × … × Xn of n other sets. It considers a holistic is preferred to relation ≫ on X and asks how ≫ depends on the Xi considered separately or in combination. Under suitably strong independence assumptions it may be possible to define ≫i for each i in a natural way from ≫ on X, and perhaps to establish a functional dependence of ≫ on the ≫i. However, interdependencies among the factors will often preclude such a simple resolution.

Historical Remarks

During the rise of marginal utility analysis in the latter part of the 19th century (Stigler 1950), the utility of each commodity bundle in a set X = X1 × X2 × … × Xn was thought of as an intuitively measurable quantity. Founders such as Jevons, Menger and Walras regarded x as preferred to y precisely when u(x), the utility of x, is greater than u(y). Their analytic approach ignored interdependencies since they used the independent additive utility form u(x) = u1(x1) + … + un(xn).

Later writers such as Edgeworth, Fisher, Pareto and Slutsky discarded the additive decomposition for the general interdependent form u(x1, x2, … , xn). Their ordinalist view of utilities as a mere reflection of a preference ordering remains dominant, and they considered interactive effects among goods, such as complementarities and substitutabilities. A fine example of interdependent analysis appears in Fisher (1892).

Fisher was also one of the first people to mention explicitly the interpersonal effect on individual utility (Stigler 1950, p. 324). This occurs when one’s utility and consequent demand depend on other people’s consumption and could generally be expressed by ui(x1, … , xn) as consumer i’s utility when xj denotes the commodity bundle of consumer j. Pigou (1903) considered the interpersonal effect in modest detail, and Duesenberry (1949) explored it in greater depth, but it has never been a prominent concern in economic theory.

Early examples of the synthetic approach in social choice theory come from Borda and Condorcet in the late 1700s. They asked: Given a list of voter preference rankings on a set X of m ≥ 3 nominees, what is the best way of selecting a winner? Borda’s answer was to assign m, m – 1,…,1 points to each first, second,…, last place nominee in the rankings and to elect the nominee with the largest point total.

Condorcet advocated the election of a nominee who is preferred by a simple majority of voters to each other nominee in pairwise comparisons. Black (1958) contains an excellent review of their work and the proposals of later writers. The debate over good election methods continues today (Brams and Fishburn 1983).

The turning point for social choice theory was Arrow’s (1951) discovery that a few appealing conditions for aggregating individual preference orders on three or more candidates into social preference orders were jointly incompatible. The avalanche of research set off by Arrow’s discovery is represented in part by Sen (1970, 1977), Fishburn (1973), Pattanaik (1971), and Kelly (1978).

In the area of risky decision theory, we envision a risky alternative as a probability distribution x on potential outcomes in a set C and observe that such decisions involve multiple factors since they entail both chances and outcomes. Bernoulli (1738) argued that a reasonable person will choose a risky alternative from a set X of distributions that maximizes his expected utility ∑x(c)u(c). He proposed that u be assessed without reference to chance since he held an intuitive measurability view of utility. Consequently, his approach is wholly synthetic.

Little changed in the foundations of risky decisions during the next two centuries. Then, in a complete turnabout, von Neumann and Morgenstern (1944) introduced the analytic approach by beginning with a preference relation ≫ on X. Axioms for ≫ on X were shown to imply the existence of a real valued function u on C such that, for all x and y in X, x ≫ y precisely when x has greater expected utility than y, and u is to be assessed on the basis of comparisons between distributions. With a few exceptions, most notably Allais (1953), subsequent research has adopted the von Neumann-Morgenstern approach.

In the rest of this essay we comment further on multiattribute preferences under ‘certainty’, interdependent preferences in risky decisions, and social choice theory.

Multiattribute Preferences

We assume throughout this section that ≫ is a strict preference relation on X = X1 × X2 × … × Xn. A given Xi could represent amounts of commodity i, consumption bundles available to person i, levels of income and/or consumption in period i, or values that elements in X might have for criterion i. Also let u on X and ui denote real valued functions.

A non-empty proper subset N of {1, 2, …, n} is defined to be ≫-independent if, for all xN and yN in the product of the Xi over N and for all z(N) in the product of the Xi over i not in N,

$$ {\displaystyle \begin{array}{l}\left( xN,z(N)\right)\gg \left( yN,z(N)\right)\gg \left( xN,w(N)\right)\hfill \\ {}\times \gg \left( yN,w(N)\right).\hfill \end{array}} $$

Most research for ≫ on X involves ≫-independence for some N, but this need not exclude elementary notions of preference interdependencies. Two models that presume all N to be ≫-independent are the additive model (see Krantz et al. 1971)

$$ x\gg y\iff {u}_1\; \; \left({x}_1\right)+\dots +{u}_n\; \; \left({x}_n\right)>{u}_1\left({y}_1\right)+\dots +{u}_n\; \; \left({y}_n\right), $$

and the lexicographic model (Fishburn 1974a) that places a value hierarchy on the factors.

Relationships between factors in the additive model and the more general model x ≫ y ⇔ u(x) > u(y) with u continuous, are often characterized by indifference maps or iso-utility contours. Interdependence arises in the lexicographic model from the fact that a small change in one factor overwhelms all changes in factors that are lower in the hierarchy.

Situations in which only some of the N are ≫-independent are reviewed by Keeney and Raiffa (1976, ch. 3) and Krantz et al. (1971, ch. 7). Among other things, these models allow complete reversals in preferences over one factor at different fixed levels of the other factors. This, of course, is a very strong form of interdependence under which all N may fail to be ≫-independent.

Other general models for interdependent preferences are discussed by Fishburn (1972) for finite sets, and by Dyer and Sarin (1979) when u is viewed in the intuitive measurability way.

Models that explicitly incorporate the interpersonal effect in economic analysis have been investigated by Pollak (1976) and Wind (1976), among others. Pollak explores the influence of several versions of interdependence among individuals on short-run and long-run consumption within a group. Using models of demand that are locally linear in others’ past consumption, he concludes that the distribution of income need not be a determinant of long-run per capita consumption patterns. Wind’s work is representative of empirical approaches to the influence of others on an individual’s choice behaviour.

Risky Decisions

Interdependent preferences in risky decisions fall into two categories. The first concerns special forms for u(c) = (c1, c2, … , cn) in the context of von Neumann- Morgenstern expected utility theory when the outcome set C is a subset of a product set C1 × C2 × … × Cn. The second focuses on changes in the basic model that occur when the independence axiom that gives rise to the expected utility form ∑x(c)u(c) is relaxed or dropped.

Decompositions of u(c1, c2, … , cn) in the expected utility model have been axiomatized by various people. Reviews and extensions of much of this work appear in Keeney and Raiffa (1976) and Farquhar (1978). The simplest independent decompositions are the additive form and a multiplicative form. The first of these requires x and y to be indifferent whenever the marginal distributions of x and y on Xi are the same for every i. The multiplicative form arises when, for each non-empty proper subset N of {1, …, n}, the preference order over marginal distributions on the product of the Ci for i in N, conditioned on fixed values of the other factors, does not depend on those fixed values.

An example of a more involved interdependent decomposition is the two-factor model (Fishburn and Farquhar 1982) u(c1, c2) = f1(c1)g1(c2) + … + fm(c1)gm(c2) + h(c1), which clearly allows a variety of interactive effects.

In the basic formulation for expected utility, assume that X is closed under convex combinations λx + (1 – λ)y with 0 < λ < 1 and x and y in X. The independence axiom for expected utility asserts that, for all x, y and z in X and all 0 < λ < 1.

$$ x\gg y\Rightarrow \uplambda x+\left(1{-} \uplambda \right)z\gg \uplambda y+\left(1{-} \uplambda \right)z. $$

Systematic violations of this axiom uncovered in experiments by Allais (1953), Kahneman and Tversky (1979), and MacCrimmon and Larsson (1979) among others, have led to new theories of risky decisions (Kahneman and Tversky 1979; Machina 1982; Chew 1983; Fishburn 1982) that do not assume independence. Machina (1982) proposes a model that approximates expected utility locally but not globally. Fishburn (1982) weakens the usual transitivity and independence assumptions to obtain a non-separable model x ≫ y ⇔ φ(x, y) > 0 that allows preference cycles.

Related interdependent generalizations of Savage’s subjective expected utility model for decisions under uncertainty are developed by Loomes and Sugden (1982) and Schmeidler (1984).

Social Choice

Many problems in social choice theory are related to Condorcet’s phenomenon of cyclical majorities. This phenomenon occurs when voters have transitive preferences yet every nominee is defeated by another nominee under simple majority comparisons. The simplest example has three nominees and three voters with x ≫ 1y ≫ 1z , z ≫ 2y and y ≫ 3z ≫ 3x ; x beats y, y beats z, and z beats x. Borda’s point- summation procedure can fail to satisfy Condorcet’s majority-choice principle, and it is notoriously sensitive to strategic voting. Moreover, all summation procedures based on decreasing weights for positions in voters’ rankings are sensitive to nominees who have absolutely no chance of winning, but whose presence can affect the outcome.

Various problems and paradoxes for multicandidate elections that arise from combinatorial aspects of synthetic methods are discussed by Fishburn (1974b), Niemi and Riker (1976), Saari (1982) and Fishburn and Brams (1983). Analyses of strategic voting, which suggest that no sensible election method is immune from manipulation by falsification of preferences, are reviewed in Kelly (1978) and Pattanaik (1978).

Arrow’s (1951) theorem offers a striking generalization of Condorcet’s cyclical majorities phenomenon. Suppose X contains three or more nominees, each of n voters can have any preference ranking on X, and an aggregate ranking ≫ = f(≫1, ≫2, … , ≫n) is desired for each list (≫1, ≫2, … , ≫n) of individual rankings. The question addressed by Arrow is whether there is any way of doing this that satisfies the following three conditions for all x and y in X:

  1. 1.

    Pareto optimality: if x≫iy for all i, then x > y;

  2. 2.

    Binary independence: the aggregate preference between x and y depends solely on the voters’ preferences between x and y;

  3. 3.

    Non-dictatorship: there is no i such that x ≫ y whenever x ≫ iy. Arrow’s theorem says that it is impossible to satisfy all three conditions.

Several dozen related impossibility theorems have subsequently been developed by others. Many of these are noted in Kelly (1978) and Pattanaik (1978). As well as multi-profile theorems, like Arrow’s, that use different lists of preference rankings to demonstrate impossibility, there are single-profile theorems (Roberts 1980) that use only one list with sufficient variety in the rankings to establish impossibility.

Impossibility theorems, voting paradoxes, and results on strategic manipulation highlight the difficulty of designing good election procedures. Recent research to alleviate such problems (Dasgupta et al. 1979; Laffont and Moulin 1982) focuses on the design of preference-revelation mechanisms (generalized ballots) and aggregation procedures that encourage people to vote in such a way that the outcome will agree with some theoretically best decision based on the true but unknown preferences of the voters. Other work, such as that on approval voting (Brams and Fishburn 1983), continues to search for simple synthetic methods that minimize the problems that beset these methods.

See Also