Interdependent Utilities, Preference Indeterminacy,
and Social Networks
Yann Bramoullé¤
December 2001
Abstract
Interdependent utilities constitute the only model of interdependent preferences in
which agents truly respect the others’ preferences. I identify and analyze two consequences of this respect: preference indeterminacy and social network e¤ects. Suppose
that the utility of Juliet depends on the utility of Romeo. If the utility of Romeo depends on the utility of Juliet as well, utilities become recursively de…ned. I show how
multiple utilities and misery traps can emerge from such feedback loops. If Juliet likes
her father, but her father does not like Romeo, this creates a negative indirect tie from
Juliet to Romeo. Juliet lives an internal con‡ict between her primary love to Romeo and
the disliking to him induced by her love to her father. I investigate how the social links
between agents are modi…ed when they consider how the others are themselves connected to others. Multiple utilities and indirect links correspond to important aspects
of real situations in which agents’ interests are entangled and yet, have been overlooked
by the literature on interdependent preferences.
THEMA, Université of Paris X-Nanterre and Department of Agricultural and Resource
Economics, 2200 Symons Hall, University of Maryland, College Park, MD 20742.
Email: bramoulle@arec.umd.edu
¤
I would like to thank Antoni Calvo-Armengol and Marc Nerlove for their early encouragement and com-
ments, Robert Chambers and Rachel Kranton for numerous discussions, and Gary Becker for his electronic,
yet invaluable support. I gratefully acknowledge …nancial support from the French Ministry of Agriculture
and the University of Maryland.
1
“Traditional economics has been based on methodological individualism. Until quite
recently, with some rare exceptions, it has not been appreciated that this method can be, or
perhaps I should say, should be, extended in describing social decisions to include dependence
of individuals’ utility on the utility or actions of others.” Akerlof (1997, p.1005).
1
Introduction
In many economic and social situations, agents do not interact anonymously but have
speci…c social links with each other. Gary Becker early advocated an integration of social
interactions within the microeconomic framework (see Becker (1974)). There is now a large
body of research on social interactions, which successfully expanded the scope of rational
choice theory and greatly improved our understanding of real markets (see e.g. Becker
(1981), Manski (2000), Rauch and Cassela (2001)). Within consumer theory, the most
direct way to represent a speci…c link that agent A has towards agent B is to assume that
the utility function of A depends on certain characteristics of B. For example, if agent A is
altruistic towards agent B, the utility function of A could depend positively on B’s income
or on B’s utility level. Such interdependent preferences would constitute a special case of
externalities, if it were not for interdependent utilities. Utilities are interdependent when
agents’ utility functions depend on the utility levels of other agents. Interdependent utilities
have notably been used to model altruism within the family1 .
Interdependent utilities are di¤erent from other models of interdependent preferences.
They constitute the only model in which agents truly respect the preferences of other agents.
This respect has the mathematical implication of de…ning the utility levels implicitly. To
understand how the utility levels depend on the allocations of resources, one …rst has to solve
a system of equations. Thus, interdependent utilities have an additional layer of complexity
1
Works using interdependent utilities to model altruism include Becker (1974, 1981, ch.8), Bergstrom
(1989a, 1989b, s.6, 1997, 1999), Bruce and Waldman (1991), Dutta and Michel (1998), Hori (1992, 1997),
Hori and Kanaya (1989), Horowitz and Wang (2001), Kimball (1987), Kolm (1969), Loury (1981), Ray
(1987), and Stark (1995).
2
with respect to standard models. Before any question of behavior and choice, the de…nition
of preferences has to be clari…ed.
Confronted with this di¢culty, researchers have generally reduced it to traditional models. In this paper, I take an alternative approach and focus on the speci…city of interdependent utilities. I show that interdependent utilities model two e¤ects that cannot be captured
by other microeconomic models: indeterminacy in the preferences and social network effects. These two e¤ects are logical consequences of the model and correspond to important
aspects of real situations in which people’s happinesses or interests are entangled. These
two e¤ects have been overlooked by the economic literature on interdependent preferences.
First, interdependence engenders indeterminacy. If the utility of Romeo depends on the
utility of Juliet, which depends on the utility of Romeo, which depends on the utility of
Juliet,... ; both utilities recursively depend on each other. This social recursivity arises
because agents recognize that the other’s utility may depend on their own. I show that
in certain cases, interdependent utilities lead to multiple or even in…nite utilities. When
utilities are multiple, it is not su¢cient to know the bundles of goods allocated to each
individual and their preferences to determine their utility levels. The reduced form utilities
are intrinsically indeterminate.
Multiple utilities seemingly contradict the principle of well-behaved preferences. In
fact, almost all papers based on interdependent utilities assume that the utility functions
are contracting2 . Contraction insures that the system of equations de…ning the utility levels
possesses one and only one solution. However, contraction is a strong assumption, di¤erent
in nature from the traditional assumptions made on utility functions like monotonicity and
quasi-concavity. I interpret contraction and argue that its scope may not be as broad as
its use. I then seek to understand what happens when the assumption of contraction is
dropped.
My …rst result states that the number and stability of the admissible utility levels
are invariant to increasing transforms of the utility functions. This provides an ordinal
2
Or that they rely on a natural generalization of contraction, called p-contraction, which implies existence
and uniqueness as well, see section 3.1.
3
legitimacy to multiple utilities. I then develop a simple analytical example of altruistic
interdependent utilities leading to a misery trap. A misery trap is a situation in which
interdependent utilities possess two stable solutions and one Pareto dominates the other.
I use this example to illustrate various aspects of multiples utilities. How do the utilities
depend on the consumption levels? How does the prospect of multiplicity vary with the
magnitude of altruism? Finally, I analyze concave altruism between agents. I show that for
two and three agents, concave altruism can lead to multiple utilities, but not to multiple
stable utilities. This result partly extends to n agents.
Multiple utilities are reminiscent of multiple equilibria that emerge in general from
models of social interactions (see e.g. Brocke and Durlauf (2001)). Yet, multiple utilities
are multiple equilibria of a unique kind, because they are not behavioral. Before any decision
of the agents, the process through which they de…ne their satisfaction and their preferences
might lead to di¤erent alternatives.
Second, interdependent utilities model social network e¤ects in the de…nition of agents’
preferences. ‘Social networks’ refer to the idea that agents share non uniform pairwise links
with each other, and that these links have potentially important economic consequences3 .
In general, interdependent preferences are related to social networks, since they specify
for each pair of agent how one’s utility function depends on the other’s characteristics.
Interdependent utilities especially model indirect ties between agents. If the utility of Aaron
depends on the utility of Barnaby and the utility of Barnaby depends on the utility of Chris,
Aaron has an indirect interest in Chris through Barnaby. Indirect links arise when agents
recognize that the other’s utility may depend on others’ utilities.
Because interdependent utilities de…ne the utility levels as solutions of a system of
equations, interdependent utilities do not model one, but two types of social networks. I
call these two types the primary social network and the induced social network. The primary
3
There is a recent and growing literature on the economics of social networks. Certain articles investigate
the strategic formation of social links, e.g., Bala and Goyal (2000), Jackson and Wolinsky (1996), Kranton
and Minehart (2001). Other articles analyze the e¤ect of a …xed social structure framing the choices of the
agents, e.g., Chwe (2000), Morris (2000), Young (1998, ch.6).
4
network describes the direct links between agents, i.e., how one’s utility function depends
on the other’s utility level. Solving the interdependence system induces modi…cations in the
way agents are linked with each other. Agents alter their links to take into account how the
others are themselves connected to others. I call the network emerging from this process
the induced network. The induced network is the actual network upon which agents base
their decisions and choices. I investigate how the primary social network is transformed
into the induced social network.
When utilities are linear, the induced network is deduced from the primary network
through a well-known matrix formula, familiar from the theory of Leontief input-output
matrixes. I use elementary concepts of social network theory to understand the structural
properties of this transformation. I …rst study the transmission of dependence, abstracting
from the sign and amplitude of the links. The induced network always possesses more links
than the primary network. I show that this expansion in general is highly structured. The
induced network can be decomposed in completely connected subgroups, linked to each
other through an acyclic graph. In a purely altruistic world, altruism spills over through
the indirect links. Sometimes this expansion is dramatic, and I give examples of primary
networks with few connections that lead to completely connected induced networks.
More ambiguous phenomenons arise when agents have both positive and negative links,
e.g., have both allies and enemies. In such situations, the induced link between two agents
may well be in opposition with the primary link. I characterize the class of structures on
which the primary and the induced networks are always in accordance. In doing so, I link
interdependent utilities with a fundamental notion developed by sociologists working on
social networks: structural balance.
The remainder of the paper is organized as follows. In section 2, I de…ne interdependent
utilities and discuss how they di¤er from other models of interdependent preferences. In
section 3, I focus on the social recursivity founding interdependent utilities and on the
prospect of multiple solutions. In section 4, I study the social network e¤ects and analyze
how the primary social network is transformed into the induced social network. I discuss
some empirical aspects of multiple utilities and indirect links in section 5 and I conclude in
5
section 6.
2
De…nition of interdependent utilities
I …rst give a general de…nition of interdependent utilities (see also Bergstrom (1999)). I
then brie‡y present the two other ways through which economists model interdependent
preferences: consumption externalities and Bergson utilities. Interdependent utilities di¤er
from these other ways on a crucial issue. They constitute the only model in which agents
truly take the others’ preferences into account.
Consider an economy composed of n agents and k goods. Denote by ci = (ci1 ; :::; cik ) 2
Rk+ the allocation of goods to agent i; i.e., cij denotes the amount of good j allocated to
agent i. Denote by c = (c1; :::; cn) 2 Rnk
+ the collection of individual allocations. Preferences
are represented by utility levels, denoted by ui , and utility functions, denoted by vi . Denote
by u = (u1; :::; un ) 2 Rn the vector of utility levels and by v = (v1 ; :::; vn) 2 Rn the vector
of utility functions4 . The utility function vi of agent i depends on his allocation ci. When
preferences are interdependent, vi depends on other parameters as well.
De…nition 2.1
Utilities are said to be interdependent if
1. the utility functions vi depend on the other agents’ utility levels u1 ; :::; ui¡1; ui+1 ; :::; un ,
and
2. the utility levels satisfy the following set of equations
u1 = v1 (c1; u2; :::; un )
:::
un = vn (cn; u1 ; :::; un¡1)
This set of equations is called the interdependence system.
The …rst condition expresses that concern for others takes the shape of a concern for
their utility levels. The second condition says that the utilities are consistently de…ned.
4
As in Bergstrom (1999), I assume that the range of the utility functions is the entire real line.
6
Interdependent utilities model altruism as well as envy. Under interdependent utilities,
agent i is said to be altruistic, or benevolent, towards agent j when vi is an increasing
function of uj. Agent i is envious, or malevolent, towards agent j, when, on the contrary, vi
is a decreasing function of uj . De…nition 2.1 generalizes in a straightforward manner to an
in…nite number of agents. When v is di¤erentiable in u, I denote by ruv(c; u) the jacobian
of v with respect to u, i.e., the n by n matrix whose (i; j) entry is
@vi
@uj (c; u):
Besides interdependent utilities, economists have used two general models to represent
interdependent preferences within consumer theory: consumption externalities and so-called
Bergson utilities5 . Consumption externalities mean that the utility functions vi depend on
the allocations of the other agents in a comprehensive fashion. Formally, interdependent
preferences are represented by consumption externalities if
u1 = v1(c1 ; c2; :::; cn )
:::
un = vn (c1; c2; :::; cn)
Bergson utilities are based on the assumption that agents possess private utility functions
u
^i that only depend on their private bundle of goods ci . Bergson utility functions vi then
only depend on these private utilities. Formally,
u1 = v1 (^
u1(c1); u
^2(c2 ); :::; u
^ n(cn ))
:::
un = vn (^
u1(c1 ); u
^2(c2); :::; ^un(cn ))
In fact, Bergson utilities constitute the speci…c case of consumption externalities where
v1; :::vn are weakly separable in (c1; :::; cn ) and have the same aggregators.
Interdependent utilities, Bergson utilities, and consumption externalities are based on
di¤erent assumptions regarding the type of social interaction. In the literature on altruism,
5
Game theory is of course built on interdependent preferences as expressed by the externality formulation:
agent’s payo¤s depend on others’ actions.
7
authors have especially distinguished pure altruism and paternalistic altruism6 . Pure altruists are supposed to respect the preferences of the other, whereas paternalistic altruists
know what is good for the other better than he or she does. Formally, paternalistic altruism
has been represented as an externality and pure altruism with Bergson or interdependent
utilities. For example, models are based on the externality representation in Hochman and
Rodgers (1969), Kohlberg (1976), Leininger (1986), Pollak (1988), and Schall (1972). Bergson utilities appear in Bergstrom (1989, s.6), Parks (1991), and Rader (1980). Utilities
are assumed to be interdependent in Becker (1974, 1981, ch.8), Bergstrom (1989a, 1989b,
s.6, 1997, 1999), Bruce and Waldman (1991), Dutta and Michel (1998), Hori (1992, 1997),
Hori and Kanaya (1989), Horowitz and Wang (2001), Kimball (1987), Kolm (1969), Loury
(1981), Ray (1987), and Stark (1995).
I would argue that Bergson utilities do not really represent pure altruism. With Bergson
utilities, agents respect the others’ private preferences, but do not take into account their
social preferences. Thus, pure altruism should itself be divided in pure private altruism,
when agents respect the others’ private preferences, and pure social altruism, when agents
respect the others’ social or complete preferences. To model pure social altruism, concern for
others has to be a concern for their actual utility levels. Therefore, interdependent utilities
constitute the only way to model situations where agents respect the complete preferences
of the others. In the remainder of this paper, I seek to explore the consequences of this
property.
3
Social recursivity and multiple utilities
In this section, I focus on the …rst consequence of the respect for the others’ preferences
that characterize interdependent utilities. When agents recognize that other’s utility might
6
The literature on envy and interdependent preferences is less developed than the literature on altruism.
Important exceptions include the works of Frank (1984, 1985) on status and p ositional externalities and VaiLam (1995). In these papers, envy is represented as a consumption externality. An increase in the income
or status of the other decreases one’s own utility. Alternatively, envy can be modelled with interdependent
utilities: an increase in the utility of the other decreases one’s own utility.
8
depend on their own, utilities become recursively de…ned and the possibility of multiple
utilities arises. Mathematically, the interdependence system is a multidimensional …xed
point system. I discuss the relation between interdependent utilities and individual recursive utilities. Contraction and stability are two basic notions in the theory of …xed-point
systems that I introduce and interpret. Then I show that despite their cardinal formulation,
interdependent utilities have an ordinal foundation. Finally, I focus on non-contracting interdependent utilities. I study an example leading to a misery trap and I analyze the case
of concave altruism.
De…nition 2.1 has two noteworthy consequences. On the one hand, utility functions
are well-de…ned at the individual level. Individuals have unambiguous preferences de…ned
over states of the world composed of their allocations and the others’ utility levels. On the
other hand, this unambiguity breaks down at the society’s level. The vector of utility levels
u is not explicitly, but recursively de…ned. Using vectorial notations, the interdependence
system is written as
u = v(c; u)
which means that the interdependence system is a multidimensional …xed point problem.
Denote by v(c; :) the function from Rn to Rn ; u ! v(c; u). For any consumption vector c,
the vector of utility levels u is a …xed point of v(c; :). Mathematics of …xed point systems
are familiar to economists, notably because they underlie John Nash’s proof of existence of
a Nash equilibrium and Arrow and Debreu’s proof of existence of a competitive equilibrium
in a Walrasian economy. However, interdependent utilities are di¤erent from best-responses
and demand curves. I do not intend here to expand the theoretical understanding of …xed
point systems, but only to make sense of this understanding in the context of interdependent
utilities.
In the literature, ‘recursive utilities’ usually refer to intertemporal recursive utilities7 .
Intertemporal recursive utilities and interdependent utilities both de…ne the utility levels
7
Koopmans (1960) introduced intertemporal recursive utilities as a compact way to represent a certain
class of intertemporal preferences. Subsequent authors have reversed this approach, see e.g., Boyd (1990),
Lucas and Stokey (1984), Streufert (1990). They consider the aggregator as the primary representation of
9
in a recursive manner. This similarity has been emphasized by several papers that study
intergenerational altruism and assume interdependent utilities between a denumerable number of agents8 . These papers notably share two particular assumptions. First, every agent
has one and only one descendant. And second, all agents have the same utility function.
These assumptions induce a strong formal similarity between intergenerational interdependent utilities and intertemporal recursive utilities that would not hold under more general
assumptions, e.g., if agents could marry and have more (or less) than one child.
In addition, there is an essential di¤erence between both models. Intertemporal recursive
utilities rely on an individual and internal recursivity - “my utility today depends on my
utility tomorrow” - , whereas interdependent utilities rely on a social and external recursivity
- “my utility depends on your utility, which depends on mine”. This di¤erence might play
an important role when making sense of multiple utilities. In the context of intertemporal
preferences, multiple utilities would imply an internal indeterminacy. On the contrary, with
interdependent utilities individuals are internally determinate. The indeterminacy arises
because of the interactions between the agents.
My analysis is framed in a static and …nite setting. I abstract from the complexities associated with the intertemporal framework (dynamic consistency, in…nite number of agents).
This allows me to study the core properties of interdependent utilities under very general
assumptions.
3.1
Contraction
Contraction is a crucial property insuring existence, uniqueness, and stability of the solutions to a …xed point system. A natural generalization of contraction, called p-contraction,
has the same e¤ect. Almost all papers based on interdependent utilities rely on the assumption that the utility functions are contracting or p-contracting.
Let f be a function from R n into Rn. If t is an integer, denote by f (t) the function from
the preferences and look for conditions on this aggregator that insure well-behaved reduced form preferences.
These conditions generally include an assumption of contraction.
8
These papers include Dutta and Michel (1998), Hori (1992, 1997), Hori and Kanaya (1989), Kimball
(1987), Loury (1981), Ray (1987).
10
Rn into Rn de…ned recursively as f (1) = f and f (t+1) = f ± f (t) : Let S be a subset of Rn .
P
Consider the usual Euclidian norm on Rn , i.e., jjxjj2 = ni=1 x2i .
De…nition 2.1
f is contracting on S if and only if f(S) ½ S and 9· < 1 :
8x; y 2 S; jjf (x) ¡ f (y)jj · ·jjx ¡ yjj:
f is p-contracting on S if and only 9t > 0: f (t) is contracting on S.
Notably, if f is contracting, all its elements fi must be contracting as well. Therefore
if for some i, fi is not contracting, neither is f . By the Contraction Mapping Theorem,
when f is contracting or p-contracting on S, the …xed point system x = f (x) has a unique
solution on S; see Stokey and Lucas (1989, p. 50-53). The following Lemma characterizes
p-contraction for linear functions.
Lemma 2.1
A linear function is p-contracting on Rn if and only if its eigenvalues are all strictly less
than 1 in modulus.
Lemma 2.1 is important because many existing models of interdependent utilities are
based on linear utility functions. Research based on interdependent utilities can be classi…ed as follows. Becker (1974), Bergstrom (1989b), Horowitz and Wang (2001), and Stark
(1995) study linear interdependent utilities, under assumptions equivalent to p-contraction.
Becker (1981) and Bruce and Waldman (1991) consider non-recursive interdependent utilities, which are always p-contracting (see section 4.2). Dutta and Michel (1998), Hori (1997),
Hori and Kanaya (1989), Loury (1981) study linear and contracting interdependent utilities
between a denumerable number of agents. Hori (1992) study non-linear and contracting
interdependent utilities between a denumerable number of agents. Kolm (1969) analyze
marginal interdependent utilities ; the assumption of p-contraction is implicit in his reasoning p.154. Finally, Ray (1987) examines interdependent utilities between a denumerable
number of agents without assuming contraction or p-contraction. He obtains existence, but
not uniqueness of the solutions to the interdependence system.
11
Contraction is a convenient mathematical assumption that has been little discussed and
interpreted. For linear altruistic utilities, p-contraction is related to diagonal-dominance
and implies that certain agents place more weight on their welfare than on the welfare of
others (see section 4.1). Examples of altruistic self-sacri…ce show that, on the contrary,
people often place more weight on the other’s welfare than on their own. For instance,
women have a widespread tendency to sacri…ce themselves for their children or husbands,
see Rubenstein (1997). Moreover, this interpretation is not valid for non-linear functions.
In general, contraction implies that, globally, changes in the utilities of others translate
into proportionally smaller changes in one’s own utility. Therefore, a utility function is not
contracting if locally, a change of the others’ utilities induce a proportionally higher change
in one’s own utility. This possibility does not appear to be particularly extreme, especially
in the familial situations to which interdependent utilities are usually applied.
The example presented in section 3.4 illustrates how non-contracting interdependent
utilities can lead to multiple solutions, even when the absolute magnitude of the interdependence is low. When the assumption of contraction is dropped, possibility of multiple
solutions arises and raises the traditional question of solutions’ selection. In this respect,
stability is one fundamental notion that allows one to discard certain solutions.
3.2
Stability
The concept of stability has been somewhat neglected in the literature on interdependent
utilities9 . Stability is a basic, yet fundamental notion of …xed point theory. Especially,
unstable solutions to the interdependence system should not be considered as valid solutions.
This simple remark allows one to solve an apparent paradox pointed out by Theodore
Bergstrom: when altruism is strong, altruistic interdependent utilities lead to negative
marginal utilities of consumption.
9
Except in Bergstrom (1989a, 1997). When the utility functions are contracting, stability always holds,
hence needs not be stated explicitly. Since I analyze utility functions that are not contracting, stability
becomes an indispensable concept.
12
Denote by B(^
u; ") the ball of center u
^ 2 Rn and radius " > 0; i.e.
B(^
u; ") = fu 2 Rn : ku ¡ u
^k · "g
The following de…nition is standard, see e.g. Stokey and Lucas (1989, p.140).
De…nition 3.1
A solution ^
u of the interdependence system is stable if there exists " > 0 such that for
every u 2 B(^
u; "); the sequence (u(t) )t2N satisfying u(0) = u and 8t 2 N; u(t+1) = v(c; u(t) )
converges to u
^.
The following classic properties apply. When v(c; :) is continuously di¤erentiable, a
…xed point u of v(c; :) is stable if and only if ruv(c; u) is p-contracting, see e.g. Stokey and
Lucas (1989, th. 6.5). When v(c; :) is contracting or p-contracting on S, its unique …xed
point on S is stable.
Consider now the example presented in Bergstrom (1989a, 1999)10 . Two agents have
benevolent and additively separable interdependent utilities
v1 = u
^ 1 + ®u2
v2 = u
^ 2 + ¯u1
where u
^1; u
^2 are private utilities and ®; ¯ > 011 : When ®¯ 6= 1, the interdependence
system has the following unique solution
1
(^
u1 + ®^
u2 )
1 ¡ ®¯
1
=
(^
u2 + ¯u
^1)
1 ¡ ®¯
u1 =
u2
When ®¯ > 1, an awkward consequence is that utilities are decreasing in consumptions.
Following the imaginative formulation of Bergstrom (1989a), should one conclude that ‘true
lovers hate spaggethis?’
10
See also Bernheim and Stark (1988), Stark (1995). Becker (1974) analyze Cobb-Douglas interdependent
utilities between two agents, which are equivalent to linear utilities under a logarithmic transform. These
authors discard the case ®¯ > 1 on a priori grounds.
11
In the remainder of the paper, greek letters systematically denote strictly positive parameters.
13
I would argue that stability provides an appropriate concept to solve this paradox12 .
E¤ectively, the previous solution is stable if ®¯ < 1 and unstable if ®¯ > 1 (see Proposition
4.1). When behavioral models lead to multiple equilibria, economists usually discard the
unstable solutions. One reason is that dynamic processes of equilibrium formation in general
do not converge to unstable equilibria. In addition, even if the system happens to be at
the unstable equilibrium, small perturbations will make the system diverge from it. The
same arguments should apply to interdependent utilities13 . Thus in the example, when ®¯
is greater than 1, the solution is unstable and should not be considered as valid any more.
Clearly, this only defers the problem. If the only candidate solution is not valid, what
should one conclude?
One possible answer is that valid solutions exist, but they are in…nite. Speci…cally,
when ®¯ > 1, the utility levels in simple processes of utility formation tend to (¡1; ¡1)
or (+1; +1). Do in…nite utilities make sense? In a cardinal interpretation of utilities, +1
actually represents the maximum level of satisfaction and ¡1 the minimum. If utilities are
interpreted as actual psychological and cognitive levels of pleasure or su¤ering, +1 refers
to extreme happiness and ¡1 to extreme unhappiness. Perhaps the conclusion should be
that ‘true lovers are very happy (or very unhappy)’14 . In any case, in…nite utilities are
caused by the linear shape of the utility functions and would not arise if decreasing returns
in the other’s utility were su¢ciently strong (see section 3.5).
12
Bergstrom (1989a) mentions the unstability of the solution, without concluding that it solves the para-
dox.
13
In real situations where utilities are likely to be interdependent, solving the interdependence system
probably corresponds to a psychological and social process. On the one hand, people communicate their
levels of satisfaction and dissatisfaction to other people with whom they interact. On the other hand, they
adjust and respond to the perceived levels of satisfaction of the others in accordance with their preferences.
I do not develop here a formal theory of utility formation in the context of interdependent utilities. Notice,
however, that De…nition 3.1 is based on such a process, see also Bergstrom (1989a, 1997).
14
The readers worried about which alternative prevails are referred to the general argument developed
in Krugman (1991). Which solution is selected depends on how history and expectations interplay in the
underlying dynamic process of utility formation.
14
3.3
Ordinal foundation
The former discussion on in…nite utilities calls attention to what one might think to be an
uncomfortable feature of interdependent utilities. Interdependent utilities are apparently
based on a cardinal representation of the preferences.15 I show here that view is partly
incorrect. Speci…cally, the number and stability of the solutions to the interdependence
system are invariant through increasing transforms of the utility levels.
When utilities are interdependent, de…ning transforms of the utility levels requires some
care. As Bergstrom (1989a, p.171 and 1999, p.4) points out, if the utility level ui is an
argument of the utility function vj , vj has to be modi…ed when considering a transform of
ui . Formally, consider interdependent utility functions vi . Let ' be a continuous increasing
transform from R to R. Without loss of generality16 , assume that the transform ' is applied
to u1.
De…nition 3.2
The transforms of the interdependent utilities vi through ' are the interdependent utilities
vi0 such that
v10 (c 1; u2; :::; un) = '[v1 (c1 ; u2; :::; un )]
v20 (c 2; u1; u3; :::; un) = v2[c2 ; '¡1 (u1); u3 ; ::; un ]
:::
v0n(cn ; u1 ; u2 ; :::; un¡1) = vn[cn ; '¡1(u1); u2; :::; un¡1]
The transform of agent 1’s utility function has to be compensated for in all the other
utility functions that depend on agent 1’s utility level.
15
Most economists would think that consumer theory under certainty is …rmly rooted in ordinality. Yet,
some eminent economists disagree. Allais (1988) develops a cardinal and psychological interpretation of
utility. Harsanyi (1988) acknowledges that people make interpersonal comparisons of utility.
16
Multidimensional transforms are obtained by successive applications of unidimensional transforms.
Therefore, Prop osition 3.1 applies to multidimensional transforms as well.
15
Proposition 3.1
The number and the stability of the solutions to the interdependence system are invariant
to continuous increasing transforms.
Proposition 3.1 means that interdependent utilities possess an ordinal foundation. This
indicates that there ought to exist a rigourous construction of socially recursive preferences
only based on ordering relations. Such a construction is probably not straightforward and
I do not undertake it here.
The clari…cation of the notions of contraction and stability were necessary preliminaries
of the analysis of multiple utilities, to which Proposition 3.1 provides an ordinal legitimacy.
The following two sections study examples of non-contracting interdependent utilities.
3.4
An example of misery trap
I now develop a simple analytical example of non-concave altruism between two individuals17 . The interdependence system possesses a low utility, or misery, trap for low allocations,
which vanishes at high allocations.
With two agents, interdependent utilities can be understood graphically: For given allocations c1 and c2 the solutions to the interdependence system are the points of intersection
between the two curves u1 = v1(c1; u2 ) and u2 = v2(c2; u1) in the plane (u1; u2). An inter@v1 @v2
section point represent a stable solution i¤ j @u
j < 1, which means that at this point v2
2 @u1
intersects v1 from above.
Consider the following system of interdependent utility functions:
v1(c1; u2) = u
^1 (c1) + ®u2
(1)
1
3
v2(c2; u1) = u
^2 (c2) + ¯u1
where u
^ 1 and u
^ 2 are two private utility functions such that u
^1 ; u
^ 2 ¸ 0. Both agents are
altruistic towards each other. Agent 1 has a linear utility function. Agent 2 has a S-
shaped utility function, i.e., convex at low levels of the other’s utility level and concave
17
Examples of non-contracting interdep endent utilities leading to misery traps can easily be constructed
for more than 2 agents. Models with 2 agents capture the essence of social recursivity in a minimal manner.
16
at high levels. The solution to this interdependence system is described in the following
proposition. Introduce the parameter U = 2(®¯=3)3=2 .
Proposition 3.2
If u
^ 1(c1) + ®^
u2(c2) < U, the interdependence system 1 has three Pareto ordered solutions.
The lowest and the highest ones are stable, the intermediate one is unstable.
If u
^ 1(c1) + ®^
u2(c2) > U , the interdependence system 1 has one stable solution.
This proposition is illustrated graphically in …gure 1. For low levels of consumptions,
the interdependence system has two stable solutions. By analogy with the idea of poverty
trap, I call the lowest solution a misery trap. Both agents are benevolent towards each
other and yet, they might end up at utility levels lower than their private utility levels and
lower than their utility levels in the other solution. This provides another example of a
process through which mutual altruism might be socially harmful, di¤erent in nature from
the processes analyzed in Bernheim and Stark (1988).
The misery trap exists for all values of the parameters ® and ¯. Thus, even apparently low levels of altruism can lead to multiple utilities. To understand the origin of this
multiplicity, let us examine the jacobian of v:
2
ruv(c; u) = 4
0
®
¡ 23
1
¯u
1
3
0
3
5
The eigenvalues of ru v are less than 1 in modulus if and only if
3
1
1
¡2
®¯ju1 3 j < 1 , ju1j > ( ®¯) 2
3
3
There is a range of values around 0, independent on the allocations, on which both eigenvalues of r uv are greater than 1 in modulus. Two related phenomenon take place on this
range. First, changes in the utility of agent 1 induce proportionally higher changes in the
utility of agent 2. Second, the unstable solution, when it exists, belongs to it (see Appendix). Non-contraction stems from the fact that the marginal utility of agent 2 with respect
to the utility level of agent 1
@v2
@u1
is in…nite at 0. However,
17
@v2
@u1
does not need to be in…nite
for the range of non-contraction to exist. It is su¢cient that
that
@v2
j @u
j
1
>
@v2
@u1
is high enough, speci…cally
@v1 ¡1
j @u
j:
2
In summary, the marginal utility of agent 2 with respect to the utility level of agent 1
is in…nite at 0: This induces a range of utility levels on which the utility functions are not
contracting, which, in turn, is related to the existence of an unstable solution. However,
the unstable solution and multiple solutions do not always exist. Figure 2 depicts how
the solutions vary with the allocations. If the allocations are su¢ciently high and despite
the presence of the range of non-contraction, the misery trap disappears. Even when the
utility functions are not contracting, the interdependence system can lead to a unique stable
solution. In this example, when people are privately su¢ciently happy, they can not get
caught in depressing feedback loops.
Finally, how does the solution vary when the level of altruism varies? It can be checked
that if ® or ¯ (or both) increase, the range of consumption levels leading to multiple solutions
expands (see Appendix). In this example, when altruism increases, the prospect of multiple
solutions is ampli…ed as well.
This simple example illustrates the possibility of multiple utilities. Multiple utilities …nd
their origin in the shapes of the utility functions, i.e., linear and S-shaped 18 . Notably, the
utility function of agent 2 is neither concave nor convex. Although S-shaped utility functions might have interesting interpretations19 , one might want to know how interdependent
utilities behave under more traditional assumptions.
18
Prop osition 3.2 can easily b e generalized as follows. Consider utility functions satisfying
v1(c 1; u2)
=
up1 (c1 ) + ®u2
v2(c 2; u1)
=
up2 (c2 ) + v2 (u1)
with v2(0) = 0; v2 increasing, convex on ] ¡ 1; 0] and concave on [0; +1[. Additionnally, suppose that
@v2
1
(0) >
@u1
®
Then the interdep endence system has three Pareto-ordered solutions, the two extremes being stable, for
low allocations and a unique stable solution for high allocations.
19
S-shaped utility functions arise when there are decreasing returns in absolute value for both high and
low utility levels.
18
3.5
Concave altruism
In the last part of this section, I analyze concave altruism between agents. I show that for
two and three agents, the interdependence system can have multiple solutions, but cannot
have multiple stable solutions. This property extends to n agents when the solutions to the
interdependence system are Pareto-ordered.
De…nition 3.3
Interdependent utilities vi are said to represent concave altruism if 8i; j, the utility function vi is twice di¤erentiable, increasing and concave in the utility level ui, i.e.,
and
@ 2v i
@u2j
· 0. Concave altruism is strict if 8i; j;
@vi
@uj
> 0 and
@ 2 vi
@u2j
@vi
@uj
¸0
< 0:
Under concave altruism, agents are benevolent towards each other and their altruism
exhibits decreasing returns to scale. When n = 2, the general solution of an interdependence
system representing concave altruism can easily be determined graphically (see Figure 3 and
Appendix for a formal proof).
Proposition 3.3
When n = 2, an interdependence system under strict concave altruism can have one stable
solution, one unstable solution, or two Pareto ordered solutions, the …rst one being unstable
and the second one stable.
Which case occurs depends on how utility functions behave when the other’s utility
level tends to in…nity. For example, when the utility functions are bounded from above,
the interdependence system always has a stable solution. Proposition 3.3. says that for
two agents, decreasing returns in the others’ utilities allows one to avoid multiple stable
solutions. I now study if this property generalizes to more than two agents.
Theorem 3.1
For any n; if u is a stable solution of an interdependence system under concave altruism
there is no other solution Pareto greater than u.
19
Theorem 3.1 relies on the fact that when a solution is stable, the jacobian of the utility
functions at this solution must be p-contracting. Because marginal utilities are decreasing,
one can show that if the jacobian is p-contracting at u, then it is p-contracting for every
u0 ¸ u. Through a global unicity argument , it implies that there is no other solution on
fu0 ¸ ug. Theorem 3.1 notably means that if the solutions of the interdependence system
are Pareto ordered, there can be at most one stable solution.
Lemma 3.1
When n = 3, the solutions of an interdependence system in which all agents are strictly
altruistic are Pareto-ordered.
As a direct consequence of Lemma 3.1 and Theorem 3.1, we obtain the following result.
Corollary 3.1
With n = 2 or 3, an interdependence system under strict concave altruism cannot have
multiple stable solutions.
Lemma 3.1 apparently does not generalize to n ¸ 4. Therefore, it is possible that for
n ¸ 4, certain interdependent utilities representing concave altruism lead to multiple stable
solutions. There might exist natural additional properties on the utility functions that
would insure the existence of a unique stable solution. Their existence and exact form are
open research questions.
When utilities are interdependent, the utility levels are socially recursively de…ned. Under the assumptions that the utility functions are contracting or p-contracting, the interdependence system has a unique and stable solution. Contraction makes social recursivity
innocuous. When the assumption of contraction is dropped, the possibility of multiple utilities appears. Multiple utilities mean that the preferences of the agents are intrinsically
indeterminate. One could argue that preference indeterminacy is undesirable and should be
rejected on a priori grounds. Yet, multiple utilities are as plausible as multiple equilibria
emerging from behavioral models. Poverty traps and misery traps both arise from feedback
loops in certain underlying parameters.
20
When the interdependence system has multiple solutions, one can still describe their
number, stability, and how they vary with the consumption levels. I illustrated this method
on two examples of altruistic and non-contracting interdependent utilities.
4
Social network e¤ects
In this section, I study the second consequence of the respect for the others’ preferences that
characterize interdependent utilities. When agents recognize that the others are themselves
connected to others via their utility functions, social network e¤ects emerge in the de…nition
of individuals’ preferences. In general, a social network is a collection of pairwise links
between agents. The concept of social network becomes indispensable when the traditional
economic assumption of anonymous and uniform interactions between agents does not apply.
Interdependent utilities are related to social networks in two ways. On the one hand, in
real situations that can be plausibly modelled by interdependent utilities, the dependence of
one agent’s utility function on another agent’s utility level is the formal expression that the
…rst agent has a link of psychological, emotional, economic, or social nature to the second
agent. Interdependent utilities are thus used to represent formally a real social network.
On the other hand, any model of interdependent utilities implicitly de…nes two types of
social networks: the primary social network and the induced social network. The primary
social network is composed of the direct links between agents as they appear originally in
the utility functions. The induced social nework correspond to the links between agents
emerging from the reduced form utilites, i.e., after having resolved the interdependence
system. These informal de…nitions will be given precise formal meanings in what follows.
For clarity, I abstract in most of this section from the possibility of multiple utilities
outlined in the previous section, although both analysis could be combined20 . The results
presented are valid for any number of agents. Yet, they may be most insightful for societies composed of many agents with sparse connections between them. That is, when most
agents’ utility functions only depend on a few others’ utility levels. Even if interdepen20
See the remark on acyclic primary graphs in section 4.2. More generally, Theorem 4.1 could b e used to
gain a structural understanding of multiplicity in the utilities.
21
dent utilities have been mainly used to model social interactions within the family, they
may suitably represent social interactions within much larger groups. Potential examples
include extended families, business links between companies, international relations among
countries.
I …rst present two simple examples to clarify the notions of the primary and the induced
social networks.
Example 4.1: the friends of our friends
This example illustrates the case of pure altruism. Suppose that there are 3 agents, with
utility functions de…ned as
v1 = u
^ 1 + ®u2
v2 = u
^ 2 + ¯u3
v3 = u
^3
Agent 1 likes agent 2, who likes agent 3, who is sel…sh. Thus, the primary network
is composed of a positive link from 1 to 2 and a positive link from 2 to 3. Solving the
interdependence system leads to
u1 = u
^1 + ®^
u2 + ®¯ u
^3
u2 = u
^2 + ¯ u
^3
u3 = u
^3
Because of his direct link to agent 2, agent 1 now has a positive indirect link to agent 3. The
induced network possesses an additional link with respect to the primary network. I will
show that, in presence of altruism in general, the induced social network is an expansion of
the primary social network.
Example 4.2: social disloyalty
This example illustrates the possibility of antagonism within the primary network. Sup-
22
pose that there are 3 agents, with utility functions de…ned as
v1 = u
^1 + ®u2 + ¯u3
v2 = u
^2 ¡ °u3
v3 = u
^3
Agent 1 likes agent 2 and agent 3, but agent 2 dislikes agent 3. The solution of the interdependence system is
u1 = u
^1 + ®^
u2 + (¯ ¡ ®°)^
u3
u2 = u
^2 ¡ ° u
^3
u3 = u
^3
Two cases now appear. Depending on the sign of ¯ ¡ ®°, the induced link between 1 and
3 can be positive or negative. Especially if ®° > ¯, the indirect negative link from 1 to 3
through 2 overcomes the direct positive link from 1 to 3. In this case, for agent 1 being
loyal to 2 implies being disloyal to 3. When both negative and positive ties are present,
the induced network is generally in opposition with the primary network on certain links.
I will characterize the particular class of structures on which the induced and the primary
networks are never in opposition.
I now investigate how the primary network is transformed into the induced network. I
…rst present the solution for linear utilities and how this solution generalizes to continuously
di¤erentiable utility functions through the implicit function theorem. These results are wellknown to economists, notably because benevolent linear interdependent utilities are similar
to Leontief input-output matrices. Then, I use concepts of graph theory to obtain insights
on the structure of this transformation. This allows me to link interdependent utilities with
fundamental notions developed by sociologists working on social networks.
23
4.1
Linear utilities
Interdependent utilities are said to be linear if there exist private utility functions ^ui and,
for every i 6= j; real parameters aij such that
vi (ci ; u¡i ) = u
^i (ci) +
n
X
aijuj
j=1
j6=i
I denote by A the n by n matrix whose (i; j) entry is equal to aij and (i; i) entry is 0.
The matrix A in fact directly represents the primary social network. Especially, aij > 0
means that agent i is altruistic towards agent j, whereas aij < 0 means that agent i is
envious towards agent j. In matrix notations, the interdependence system associated to
linear interdependent utilities becomes
u=^
u + Au
The solution to this system is well-known and described in the following proposition (see
e.g. Bergstrom (1999) and Stokey and Lucas (1989, th. 6.3) on stability).
Proposition 4.1
The linear interdependence system has a unique solution if and only if det(I ¡ A) 6= 0. In
this case,
u = (I ¡ A)¡1^
u
This solution is stable if and only if A is p-contracting. In this case,
(I ¡ A)¡1 = I + A + A2 + A3 + :::
Proposition 4.1 means that the induced utilities are linear combinations of the private
utilities. The weight of the private utility of agent j in the induced utility of agent i
is the (i; j) entry of (I ¡ A)1. Therefore, the induced social network corresponds to the
matrix (I ¡ A)¡1. When there are only two agents, A is p-contracting if and only if
ja12 a21j < 1. This is the condition stated by Becker (1974), Bernheim and Stark (1988),
and Stark (1995). When all the primary links are positive, i.e., agents are only benevolent,
24
this model is formally similar to Leontief input-output matrices. Notably, if all the primary
links are positive and A is p-contracting, then all the induced links are positive as well21 .
This linear model has several other applications in social sciences, see e.g. Friedkin (1998)
on the sociological theory of social in‡uence.
The solution for linear utility functions generalizes to non-linear utility functions under
the assumptions of the implicit function theorem, see also Kolm (1969) for the case of one
good. Speci…cally, consider a system of interdependent utility functions vi.
Proposition 4.2 (Implicit Function Theorem)
Assume that 8i, vi is continuously di¤erentiable in (ci; u¡i). Let (¹
c; ¹
u) be a solution of the
interdependence system, i.e., ¹
u = v(¹
c; u
¹) and denote A = r uv(¹
c; ¹
u). If det(I ¡ A) 6= 0,
the interdependence system has a unique solution u(c) around ¹
c such that u
¹ = u(¹
c): This
solution is stable if and only if A is p-contracting. u is continuously di¤erentiable and the
marginal variations of u with respect to the consumption levels at ¹
c are given by 8cij 2 R;
rcij u(¹
c) = [I ¡ A]¡1rcij v(¹
c; u
¹)
By assumption, vi is the only element of v that depends on cij : Therefore, Proposition
4.2 means that 8i; j; k;
@uk
@v
= bki i
@c ij
@c ij
where b ki denotes the (k; i) entry of [I ¡ A] ¡1 . The marginal induced utility of agent k with
respect to the consumption of the good j by agent i is proportional to the marginal primary
utility of agent i with respect to his consumption of good j. The constant of proportionality
is the amplitude of the induced link between k and i.
The linear formula appearing in Propositions 4.1 and 4.2 has an intuitive interpretation
(see also Kolm (1969) and Friedkin (1998)). Suppose that A is p-contracting, so that
21
When all the primary links are positive, economists have made explicit various properties equivalent to
p-contraction. One is that A is a ‘productive’ matrix. Another one is that I ¡ A has a dominant diagonal,
see Bergstrom (1997, 1999) and e.g. Murata (1977) on Leontief input-output matrices.
25
(I ¡ A)¡1 = I + A + A2 + A3 + ::: Entries of the successive powers of A can be interpreted
as follows.
Aij = aij is the direct, primary link from agent i to agent j.
P
A2ij = nk=1 aik akj corresponds to the resulting indirect link from agent i to agent j
through common acquaintances. Agent i has a primary link to agent k, of amplitude aik ,
and agent k has a primary link to agent j, of amplitude aik: Connections between i and k,
and k and j induce an indirect link of order 1 from i to j, whose amplitude is the product
aikakj . A2ij is the sum of all these indirect links of order 1 from i towards j.
P
Similarly, (At+1)ij = ni1;i2 ;:::;it =1 aii1 ai1i2 :::ait¡1 it ait j is the sum of all the indirect links
from i to j along the social chains of t people connecting i and j. In other words, (At+1 )ij
measures the resulting indirect link of order t from i to j.
In summary, the induced link between agent i and agent j is the sum of their primary
link and of all the indirect links that connect them through all possible social chains.
Propositions 4.1 and 4.2 describe in a compact way how the primary network is transformed into the induced network. However, networks are complex mathematical objects
whose properties often cannot be grasped directly from their matrix representation. I now
seek to bring structural insights to this transformation. How does the shape of the primary
network relate to the shape of the induced network? Sociologists have theoretically developed and empirically implemented various concepts in order to better understand the shape
and properties of real social networks, see e.g. Wasserman and Faust (1994). One general
principle is to look for natural divisions of the agents into certain subgroups induced by the
network.
The two results which follow, Theorem 4.1 and 4.2, are based on two fundamental
notions developed in social network theory: ‘cliques’ and ‘balance’. Cliques are maximal
subgroups of agents within which everyone is connected to everyone else. In Theorem 4.1,
I show that the induced network can be decomposed into cliques linked together through a
hierarchical structure. In contrast to the idea of cliques, the notion of balance is concerned
with the signs of the links. A network is balanced if its structure is similar to the structure
of international relations during the Cold War: agents naturally split in two groups of allies,
26
opposed to each other. In Theorem 4.2, I show that primary and indirect links always agree
in sign when the primary network is balanced. If the primary social network is not balanced,
some tension will emerge between the primary links. The statement of both results relies
on elementary notions of graph theory that are introduced in due course.
4.2
Existence of links
I …rst focus on the existence of an induced link between two agents, abstracting from its
magnitude and sign 22 . A graph on a …nite set S is a set of directed links between elements
of S. Equivalently, a graph on [1; n] can be viewed as a n by n matrix whose (i; j) entry is
1 if there is a link from i to j and 0 otherwise. Two di¤erent graphs on the set of agents
naturally underlie any system of interdependent utilities.
De…nition 4.1.
The primary graph associated with a system of interdependent utilties is such that i is
linked to j when vi depends on uj , i.e., j’s utility level is an argument in i’s utility function.
The induced graph associated with the same system is such that i is linked to j when ui
depends on cj , i.e., after having resolved the interdependence system, j’s allocation a¤ects
i’s utility level.
The primary and the induced graphs are reduced forms of the primary and the induced
social networks. They contain information on the existence of the links between agents,
but not on their sign or magnitude. The induced graph can be deduced from the primary
graph by a natural process that has important implications for its shape. A path from i
to j in a graph g is a sequence of agents i1; :::; il such that gii1 = gi1 i2 = ::: = gil j = 1 and
i; i1 ; :::; il ; j are all distinct. By assumption, there is a path from every agent i to herself
(i.e., l = 0 in the former statement). Agent j is reachable from agent i in the graph g if
there is a path from i to j in g. The following lemma describes the fundamental property
22
If we assume that interdependent utilities are altruistic and linear, represented by the p-contracting
matrix A, the central question of this section can be speci…ed as follows. Knowing the pairs (i; j) for which
a ij > 0, how to deduce the pairs (i; j) for which (I ¡ A)¡1
ij > 0?
27
relating the primary graph to the induced graph.23
Lemma 4.1
There is a link from i to j in the induced graph if and only if there is a path from i to j in
the primary graph.
This lemma allows us to rely on the well-known properties of the relation of reachability
to understand the structure of the induced graph. Some additional de…nitions are …rst
needed. Agent i and agent j are mutually reachable if i is reachable from j and j is
reachable from i. A graph is connected if every agent is reachable from every other
agent in the graph. A cycle in a graph g is a sequence of agents i; i1; :::; il ; i such that
gii1 = gi1i2 = ::: = gil i = 1 and i; i1; :::; il are all distincts. An acyclic graph is a graph
without cycles. A complete subgraph of g is a subset S 0 of S such that 8i 6= j 2 S0 ; gij = 1:
A clique of g is a maximal complete subgraph of g, i.e., a complete subgraph S 0 such that
there is no complete subgraph S00 ! S 0 .
Classical results on reachability can be summarized as follows (see Harary et al. (1965)).
Let g be a graph on S. Let g¹ be the graph on S such that ¹gij = 1 if and only if j is reachable
from i in g (if g is the primary graph associated to an interdependence system, ¹g is its
induced graph).First, reachability is a transitive relation. Since reachability is re‡exive,
it means that reachability de…nes a pre-ordering on S and that mutual reachability is an
equivalence relation on S. The classes of equivalence with respect to mutual reachability
are called components of the graph g. Components de…ne a partition of the set of agents.
Second, the components of g are the cliques of g¹. The graph g induces a graph on the set of
components, called the condensation of g; as follows. There is a link from one component
to another if and only if there is a link in g from one agent in the …rst component to
another agent in the second component. Third, there is a path from i to j in g if and only if
23
In the presence of negative links, it might happen that a primary link between two agents has a non-zero
value while its corresp onding induced link has zero value. For instance, in example 4.2, when ° = ®¯, the
indirect negative link from 1 to 3 exactly compensate for the direct positive link, and the utility of agent 1
ends up b eing independent of the allocation of agent 3. This possibility is non generic and is assumed away
in the establishment of Lemma 4.1.
28
there is a path from i0s component to j0 s component in the condensation of g. And fourth,
condensation is acyclic. All these concepts are illustrated in …gure 4.
Theorem 4.1
The cliques of the induced graph partitions the set of agents into the components of the
primary graph. The primary graph induces an acyclic graph on the components, called the
condensation. There is an induced link between two agents if and only if there is a path
between their components in the condensation.
Theorem 4.1 delineates the general structure of a society based on interdependent utilities. A component is a set of agents such that the utility of every agent in this set depends on
the allocations of all the other agents of the set. Components are complete interdependent
groups of the society.
Components relate to each other through an acyclic relation. Acyclicity is a crucial
property. It means that the induced network is based on a hierarchy and that components
are ordered (although two components might not be comparable). For example, it is possible
to assign numerical levels ni to the components Ci in such a way that: if Ci is linked to Cj in
the condensation then ni > nj (see Harary et al. (1965, th.10.1)). Another consequence is
that there always exist a ‘self-interested’ component, i.e., a component within which nobody
is related to the exterior of the component (see Harary et al. (1965, th.3.8)).
One special case is when the primary graph itself is acyclic. That is, there is no loop in
the interdependence system. When the primary graph is acyclic, every agent constitutes her
own component and the induced graph is acyclic as well. In fact, in this case interdependence
is not recursive, and the interdependence system has a unique and stable solution24 . A
simple example of this is provided by models of interdependent utilities between two agents,
among which one is sel…sh, see Becker (1981), Bruce and Waldman (1991).
A more important special case is when the primary graph is connected. If the primary
graph is connected, the induced graph is complete. Every agent ends up actually linked to
24
If the primary graph is acyclic, one can show that there is an integer t such that v(c; :)(t) does not
depend on u, hence the vector of utility functions is obviously p-contracting.
29
every other agent. A noteworthy phenomenon is that primary graphs can be connected even
when they possess few links. Two examples of this are the ‘ring’ and the ‘star’. Suppose
…rst that agents are located around a circle and suppose that the utility function of every
agent depends on her two direct neighbors’ utility levels. The primary graph of this interdependence system is called a ring. The induced graph is complete: the utility level of every
agent ends up depending on the consumption levels of all the other agents. Second, suppose
that agents are divided in two groups: one agent, called the center, and all the other agents,
called periphery agent. Every periphery agent’s utility function only depends on the center’s
utility level. The center’s utility function depends on all the periphery agents’ utility levels.
The primary graph of this interdependence system is called a star. Again, the induced
network is complete. Both types of graphs have played an important role in the theoretical literature and constitute key stylized types of real social networks. They illustrate two
di¤erent processes through which sparse primary networks can lead to densely connected
induced networks. The ring is an example of how interdependence spreads through local
and geographic interactions. The star shows the prominent role played by agents who have
many connections.
Solving the interdependence expands the connections between people, in certain cases to
a dramatic extent. The induced network results from the transmission of dependence along
the primary links. For example, when society is purely altruistic, the induced network is an
expansion of the primary network. I showed that this expansion leads to a highly structured
network.
When society is not purely altruistic and displays envy as well as altruism, con‡icts
might arise between direct and indirect links. Theorem 4.1 still gives a valid picture of the
unsigned structure underlying the induced network. Yet, the most interesting questions
might concern the signs of the induced links and the possible tensions between the primary
links. As in Example 4.2., people can end up being disloyal to their friends. I now focus on
these questions.
30
4.3
Signed relations
Suppose that the utilities are linearly interdependent and represented by the p-contracting
matrix A. The primary social network is represented by the matrix A and the induced social
network by the matrix (I ¡ A)¡1 = I + A + A2 + :::. An indirect link between two agents i
and j is a product aii1 ai1i2 :::ait¡1it aitj , where (i1 ; :::; it) are t agents and t ¸ 1. Assume that
the primary link from i to herself is equal to 1. We thus know that for every i and j, the
induced link between i and j is the sum of their primary link and all their possible indirect
links. Therefore, an induced link has a di¤erent sign from the primary link if a su¢cient
number of indirect links have a di¤erent sign from it.
De…nition 4.2
A primary network is free of tension if for every pair of agents, their primary link and
all their indirect links have the same sign.
Notably, when a primary network is free of tension, for every pair of agents, their primary
link and their induced link have the same sign. The main result provides a characterization
of the absence of tension using the notion of balance.
De…nition 4.3
A network is balanced if there is a partition of the set of agents in two groups (one of
which may be empty) such that all the positive links relate two agents belonging to the same
group and all the negative link relate two agents belonging to di¤erent groups.
Balance is a central notion introduced and developed by sociologists working on signed
social networks. Balance expresses the absence of frustration within a network25 . It is
therefore naturally related to interdependent utilities.
Theorem 4.2
If the primary network is balanced, then it is free of tension.
25
See e.g. Harary et al. (1965, ch. 13) and Wasserman and Faust (1994, ch. 6). Theoretically, sociologists
have made explicit several equivalent formulations of balance. Empirically, real social networks exhibit a
general tendency to be balanced.
31
If the primary network is connected and free of tension, then it is balanced.
Theorem 4.2 says that balance is closely related to the alignement between the primary
and the indirect links. Only in a dual (or primal) world do the indirect links agree with
the primary links. If the primary network is not connected, characterization of the absence
of tension is not straightforward. It is possible to construct examples of networks that are
free of tension but not balanced26 .
Corollary 4.1
If the primary network is balanced, then the induced network is balanced. Moreover, their
partititions in two groups are identical.
The property of being balanced is preserved when solving the interdependence system.
However, the reverse property is not true. If the induced network is balanced, it does not
imply that the primary network is balanced. Example 4.2 provides a simple illustration
of this. When ®° > ¯, the induced network is composed of a positive link from 1 to 2, a
negative link from 2 to 3, and a negative link from 1 to 3, hence is balanced (the two groups
are f1; 2g and f3g). However, the primary network is not balanced.
Theorem 4.2 expresses a natural relation between the primary and the induced network
in terms of a structural property of the primary network.
In summary, the induced link between two agents is the sum of their primary link and
of all their indirect links. This has sharp implications for the shape of the graph underlying
the induced network. In general, the induced graph is composed of completely connected
subgroups linked together through a hierarchy. It also implies that when agents have positive
as well as negative links, some tension will generally emerge between the primary links. That
is, except in the special case where the primary network is balanced.
26
The following example was p ointed out to me by Eugene Johnsen. Consider 4 agents linked in the
following way: 1 has a positive link to 3 and a negative link to 4. 2 has a positive link to 3 and to 4. 3 and
4 have no links.
32
5
A discussion of some empirical aspects
I now discuss some empirical aspects of multiple utilities and indirect links. First, I would
argue that multiple utilities are related to important …ndings of interactional psychology.
Psychologists from the Mental Research Institute in Palo-Alto have done a considerable
work to document and analyze psychological patterns emerging from repeated interactions
among small groups of people, see e.g. Watzlawick et al. (1962). One central conclusion
is the prevalence of harmful situations. Many relations between husbands and wives or
parents and children get trapped in self-enforcing misery. Most of the time, this happens
despite the good will of everyone. More importantly, they showed that these relational
vicious circles could often be solved by apparently small changes in the way people interact
and communicate27 . This led to the successful development of the so-called ‘brief therapy’,
see e.g. Fish et al. (1982).
This means that these relational vicious circles are real misery traps, i.e., situations in
which people could be happy but are not. Real misery traps and misery traps emerging from
interdependent utilities are similar in many respects. They are both caused by altruistic
interactions between people. In both cases, people’s welfares can greatly improve without
corresponding changes in external conditions. Of course, solving a real misery trap must
involve certain behavioral changes, however small or unexpected. This is not uncompatible
with interdependent utilities, though. Any process of utility formation must be based on
behavioral assumptions on how agents communicate their utility levels and perceive the
utility levels of others. Hence, a fully developped theory of interdependent utilities would
have to include strategic aspects. In any case, real misery traps are common in familial
relations although they cannot be modelled by traditional models of altruism within the
family.
27
These changes might appear small from an economic point of view, because they do not involve any
change in external conditions. From a psychological point of view, these changes are critical. For example, a
general and surprisingly e¤ective therapeutic principle is what Watzlawick et al. (p.236-240) call ‘prescribing
the symptom’. To be told to do something that one has always done spontaneously can induce dramatic
changes in behavior.
33
Although in di¤erent settings, indirect ties are crucial components of real situations
as well. They might explain why the level of altruism observed from people’s behavior is
sometimes more important than what might be expected a priori. In traditional societies,
one of the main social role of marriage is to bring closer the two families (or tribes, clans, see
e.g. Fox (1967)). The strong bond created between the two spouses is intended to extend
to the two groups to which they belong, in a way reminiscent of the expansion of altruism
caused by interdependent utilities.
More generally, indirect links appear everytime an agent A acts on the situation of
another agent C at the demand of an agent B. Societies, or industries, in which personal
networks play an important role provide numerous examples of such ‘transmission’ of social
capital.
International relations between countries should provide an appropriate …eld of study
of the social network e¤ects of interdependent utilities. When a country A gives support to
another country B, it might be because they have a long history of alliance and cooperation
in common. Or it might be because country A is a traditional ally of country C, which
itself is closely linked to country B. Country A would then help country B for the interest
of country C. On the contrary, for cultural, social, or religious reasons, country A might
be in …erce opposition with country D, itself in con‡ict with country B. Support from A
to B would then be aimed at weakening country D. When deciding which side of a con‡ict
to join, countries certainly weigh the direct and the indirect interets involved.
6
Conclusion
Interdependent utilities constitute the only model of interdependent preferences in which the
agents truly respect the others’ preferences. This respect has two consequences. Multiple
and in…nite utilities arise from the recognition that the other’s utility may depends on one’s
own and the feedback loop that this re‡ection engenders. Indirect links arise from the
recognition that the other’s utility may depend on others’ utilities.
Results presented in this paper could be extended in a number of directions. Interdependent utilities can be given a rigorous ordinal foundation by explicitly constructing
34
socially recursive preferences based on ordering relations (see section 3.3). The analysis
of non-contracting concave altruism can be further developed for n ¸ 4 (see section 3.5).
What kind of additional properties would insure that there cannot exist multiple stable
solutions? In general, the study of non-contracting utility functions should be a fertile …eld
for theoretical analysis. The result on balance (Theorem 4.2) only constitutes a …rst step
of the analysis of signed relations. To study primary networks that are not free of tension,
one could de…ne a degree of tension and a degree of unbalance and one could understand
how they relate. A reasonable conjecture might be that the more unbalanced the primary
network, the higher the antagonism between the primary and the induced networks.
In most of the research on altruism and envy, economists need to have operational models
that explain how people’s concerns for others translate into choices and decisions. In short,
economists need to work with reduced form utilities. Despite this need, the assumption
of interdependent utilities has been fairly popular. This paper attempted to clarify the
speci…city of interdependent utilities.
35
7
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40
8
Appendix
Proof of Lemma 2.1
Let A be a linear function from Rn into Rn. First, suppose that At is contracting. Let
¸ be an eigenvalue of A and x 2 R n an associated eigenvector. We have
jjAt x ¡ At 0jj < jjx ¡ 0jj ) j¸jt jjxjj < jjxjj ) j¸j < 1
Reciprocally, suppose that all the eigenvalues of A are strictly less than 1 in modulus.
To prove the result, I use the Jordan representation of A, see e.g. Stokey and Lucas (1989,
p. 144). Let (¸1; :::; ¸n) 2 Cn be the eigenvalues of A: Let t be an integer greater than or
equal to n and x = (x1; :::; xn) be a vector of C n. De…ne jjxjj1 = supi jxij and ¸ = supi j¸i j.
jj jj1 is a norm on Cn . The properties of the Jordan representation lead to
jjAtxjj · P (t)¸t¡n jjxjj1
where P (t) is a certain polynomial of t independent of the ¸i ’s and the xi’s. Since all
norms are equivalent in Cn , there is a constant M such that 8y 2 Cn ; jjyjj1 · Mjjyjj: This
leads to
jjAtxjj · MP(t)¸t¡n jjxjj
If t is high enough, MP (t)¸t¡n becomes lower than 1, which shows that At is contracting.
Proof of Proposition 3.1
De…ne the application © from Rn into R n such that ©(u1; u2; :::; un) = ('(u1 ); u2; :::; un).
First, note that
u = v(c; u) , ©(u) = v0 (c; ©(u))
Since © is a bijection from Rn to R n, this means that © establishes a bijection from
the set of solutions of the interdependence system u = v(c; u) to the set of solutions of the
transformed interdependence system u = v0 (c; u): Therefore, both interdependence systems
have the same number of solutions (possibly in…nite).
To show the second part of the Proposition, consider u
^ a stable solution of u = v(c; u).
©(^
u) is the corresponding solution of u = v0(c; u). Let " > 0 be such that all sequences
41
originating in the ball of center u
^ and radius " and satisfying u(t+1) = v(c; u(t) ) tend to ^
u:
Since © is continuous, there exists "0 > 0 such that
u 2 B(©(^
u); "0) ) ©¡1(u) 2 B(^
u; ")
Take u 2 B(©(^
u); "0 ) and de…ne the sequence u(t) such that u(0) = u and u(t+1) =
v0 (c; u(t) ): Thus, the sequence ©¡1(u(t) ) satis…es ©¡1 (u(0) ) 2 B(^
u; ") and ©¡1(u(t+1) ) =
v(c; ©¡1(u(t) )). By stability of u
^, it means that ©¡1(u(t) ) tends to ^
u as t tends to in…nity.
By continuity of ©, it implies that u(t) tends to ©(^
u). Hence ©(^
u) is a stable solution of the
transformed interdependence system. The reverse sense can be obtained by applying the
same reasoning to ©¡1. Finally, note that when ' and v are di¤erentiable, it can be shown
that ru v0 and ru v have the same characteristic polynomial, hence the same eigenvalues.
Graphically, an increasing transorm of a utility level corresponds to an axis’ rescaling in
Rn .
Proof of Proposition 3.2
The interdependence system is equivalent to u1 = u
^ 1 + ®u2 and '(u1) = 0; with '(u) =
u
^1 + ®^
u2 + ®¯u1=3 ¡ u: Derivatives of ' are as follows
d'
(u) = 1=3®¯u¡2=3 ¡ 1
du
d2'
(u) = ¡2=9®¯u¡5=3
du2
Therefore,
d2 '
du2
2
> 0 if u < 0; and ddu'2 < 0 if u > 0. This means that
d'
du
is increasing on ] ¡
d'
1; 0[ and decreasing on ]0; +1[: Since limu!¡1 d'
du = limu!+1 du = ¡1 and lim u!0
+1;
d'
du
d'
du
=
is negative on ] ¡ 1; ¡A[, positive on ] ¡ A; +A[, and negative on ] + A; +1[, with
A = (®¯=3)3=2: Thus, ' is decreasing, increasing, and decreasing on the same segments.
Since limu!¡1 ' = +1 and limu!+1 ' = ¡1 and ' is continuous, checking the signs of
'(¡A) and '(A) are su¢cient to know the signs of '.
'(¡A) = u
^1 + ®^
u2 ¡ ®¯A1=3 + A = u
^1 + ®^
u2 ¡ U
'(A) = u
^1 + ®^
u2 + ®¯A1=3 ¡ A = u
^1 + ®^
u2 + U > 0
42
If '(¡A) < 0, the interdependence system has one solution on ] ¡ 1; ¡A[, one solution
on ] ¡ A; 0[ (because '(0) > 0), and one solution on ] + A; +1[: If '(¡A) = 0; the inter-
dependence has two solutions: ¡A, and another one located on ] + A; +1[: If '(¡A) > 0,
the interdependence system only has one solution, located on ] + A; +1[:
For continuously di¤erentiable utilities and two agents, a solution is stable if and only
1 @v2
if j @v
@u2 @u1 j < 1. Here,
@v1 @v2
@u2 @u1
3=2 ,
1 @v2
= 1=3®¯u¡2=3
, and j @v
1
@u2 @u1 j < 1 () ju1j > (®¯=3)
ju1 j > A.
Denote by D(®; ¯) = f(x1; x2) 2 R2+ : x1 + ®x2 · 2(®¯=3)3=2g the domain of private
utilities that leads to multiple solutions. It is a linear domain, whose intersection points
with the axis are (0; 2®1=2(¯=3)3=2 ) and (2(®¯=3)3=2; 0). Both points increase when (®; ¯)
increase, hence it is true for D as well.
Proof of Proposition 3.3
The interdependence system is u1 = v1(c1; u2) and u2 = v2(c2 ; u1 ): By substitution, this
is equivalent to '(u1) = 0 and u2 = v2(c2; u1), where ' is the function from R into R
u ! '(u) = v1(c1; v2(c2; u)) ¡ u
' is twice di¤erentiable and
d'
@v2
@v
(u) =
(c2; u) 1 (c1; v2) ¡ 1
du
@u1
@u2
2
2
2
d '
@ v2
@v1
@v2
2 @ v1
(u)
=
(c
;
u)
(c
;
v
)
+
[
(c
;
u)]
(c 1; v2 )
2
1
2
2
2
du2
@u1
@u2
@u1
@u22
Thus,
d2 '
du2
< 0 and ' is strictly concave. A strictly concave function can be zero in at
most two points. Therefore, the interdependent system has at most two solutions.
@v1 @v2
A solution is stable if and only if j @u
j<1,
2 @u1
d'
du
< 0: A strictly concave function that
is zero at two numbers must have its derivative positive at the …rst number and negative
at the second one.
Proof of Theorem 3.1
The proof unfolds in two steps. First, we show that if ruv(u) is p-contracting, then for
every u0 ¸ u, ru v(u0 ) is p-contracting as well. A matrix A is p-contracting if and only if
43
At ! 0 when t ! 1. Consider now two matrixes A and B such that 0 · B · A. Since
both matrixes are positive, we have
0 · (B t+1)ij =
n
X
i1;i2 ;:::;it =1
bii1 bi1 i2 :::bit¡1 it bitj ·
n
X
aii1 ai1 i2 :::ait¡1 it ait j = (At+1 )ij
i1 ;i2;:::;it=1
which means that 8t; 0 · B t · At : Hence if A is p-contracting, At ! 0 when t ! 1,
hence B t ! 0 when t ! 1 and B is p-contracting. By concavity, for every u0 ¸ u;
0 · ruv(u0 ) · ruv(u). Therefore, r uv(u) is p-contracting implies that for every u0 ¸ u,
ruv(u0 ) is p-contracting as well.
Second, consider two Pareto-ordered solutions u1 · u2 such that u1 is stable. This
means that ruv(u1) is p-contracting, hence ruv is p-contracting for every u such that
u1 · u · u2 . This means that the jacobian of u ¡ v(c; u) is a P-matrix on fu1 · u · u2 g:
Since u1 ¡ v(c; u1) = 0 = u2 ¡ v(c; u2 ), by the Gale-Nikaido theorem, we have u1 = u2 (see
e.g. Murata (1977)).
Proof of Lemma 3.1
Consider two distinct solutions u and u0 and without loss of generality suppose that
u1 < u01 : This means that
v1(c1; u2 ; u3 ) < v1(c1; u02; u03 )
therefore, since v1 is increasing in (u2; u3); we cannot have u2 > u02 and u3 > u03 : Suppose,
for example, that u2 · u02. It implies that
u03 = v3(c3; u01 ; u02 ) ¸ v3(c3; u1; u2 ) = u3
and u · u0 .
Proof of Lemma 4.1
Suppose that there is a path from i to j in the primary graph. It means that there is
a sequence of agents i1 ; :::; il such that vi depends on ui1 ; vit depends on uit+1 , vil depends
on uj. Appropriate successive substitutions in the interdependence system along this path
show that ui depends on uj, hence on c j. Reciprocally, suppose that there is a link from
i to j in the induced graph. Through the interdependence system, ui depends on ci and
44
on certain ut , t 2 T ½ [1; n]=fig. Because ui depends on cj , at least one of these ut has
to depend on c j as well. Either j 2 T and there is a path between i and j in the primary
graph, or j 2
= T and we have to investigate one step further. By repeating this operation,
we will eventually reach uj through a path of dependence in the utility functions.
Proof of Theorem 4.2
On the one hand, suppose that the primary network A is balanced and consider the two
groups of its decomposition. The order of an indirect link is the number of agents present in
the social chain that de…nes it. I show the result by recurrence on the order of the indirect
links. The hypothesis of the recurrence is as follows.
Ht : for every pair of agents (i; j), aij and all the indirect links between i and j of order
less than or equal to t have the same sign.
Assume that t = 1. Take (i; j) a pair of agent and k a third agent. If i and j belong to
the same group, then aij ¸ 0. If k belongs to this group as well, aik ¸ 0 and akj ¸ 0, hence
aikakj ¸ 0. If k belongs to the other group, aik · 0 and akj · 0, hence aik akj ¸ 0. If i and
j belong to di¤erent groups, then aij · 0. If k belongs to i’s group, aik ¸ 0 and akj · 0,
hence aikakj · 0: If k belongs to j’s group, aik · 0 and akj ¸ 0, hence aik akj · 0: In all
the cases, aij and aikakj have the same sign. Hence H1 is true.
Assume that t ¸ 2 and that Ht¡1 is true. Consider an indirect link of order t between
i and j aii1 ai1 i2 :::ait¡1 it aitj . Since aii1 ai1i2 :::ait¡1 it is an indirect link of order t ¡ 1 between
i and it , by Ht¡1 , it has the same sign as aiit : Therefore, aii1 ai1 i2 :::ait¡1it aitj has the same
sign as aiit aitj , which, by H1; has the same sign as aij . Therefore Ht is true.
By recurrence, Ht is true for every t:
On the other hand, suppose that for every pair of agents, their primary link and all their
indirect links have the same sign. Using the concepts presented in Harary et al. (1965, ch.
13), it means that the primary network is both cycle-balanced (all cycles are positive)
and path-balanced (all path joining two points have the same sign). We can then rely on
Theorem 13.11 (p. 356), which states that cycle-balance implies that all components are
balanced. Since the primary network is connected, it is equal to its only component, hence
is balanced.
45
Figure 1
u2
Two stable solutions and
one unstable solution
v1 (c1 ,u2 )
v2 (c2 ,u1 )
u1
u2
One stable solution
v1 (c1 ,u2 )
v2 (c2 ,u1 )
u1
Figure 2
Solutions of the interdependence system as functions of the private utility of agent 1, under
the assumption that û2 =0. The dashed line represents the unstable solution.
u1
U
û1
u2
U
û1
Figure 3
u2
v1 (c1 ,u2 )
One stable solution
v2 (c2 ,u1 )
u1
u2
v2 (c2 ,u1 )
One unstable solution
v1 (c1 ,u2 )
u1
u2
One unstable and
one stable solution
v1 (c1 ,u2 )
v2 (c2 ,u1 )
u1
Figure 4
The primary graph: an arrow from agent 1 to agent 2 means that the
utility function of agent 1 depends on the utility level of agent 2.
The components : sets of agents whose utility levels are affected by the consumption levels of all
the other agents of the set.
The condensation: an acyclic graph related the components induced by the primary graph
There is a link from agent 1 to agent 2 in the effective graph if and only there is a path from
agent 1’s component to agent 2’s component in the condensation.