Economics Education and Research Consortium
Working Paper Series
ISSN 1561-2422
No 04/13
Achieving stability in heterogeneous societies
Multi-jurisdictional structures, and redistribution policies
Alexey Savvateev
This project (03-055) was supported
by the Economics Education and Research Consortium
All opinions expressed here are those of the author
and not those of the Economics Education and Research Consortium
Research dissemination by the EERC may include views on policy,
but the EERC itself takes no institutional policy positions
Research area: Public Economics
JEL Classification: D70, H20, D73
SAVVATEEV A.V. Achieving stability in heterogeneous societies: multi-jurisdictional structures, and
redistribution policies. — Moscow: EERC, 2005.
Consider a “linear world” populated by several agents. These agents’ locations are identified with optimal variety of a
horizontally differentiated local public good. Agents are to be partitioned into several communities (hereafter, groups),
and each group chooses a variety of public good to be produced and consumed by members of that group via the
majority voting procedure. It is shown that a stable partition may fail to exist, where stability means that no potential
group would like to secede and form a new community. At the same time, compensation schemes are proposed which
guarantee the existence of a stable partition. Small societies are studied in detail, as well as certain special types of
distributions of agents’ locations.
Keywords. Russia, stability, partitions, redistribution, core of a cooperative game.
Acknowledgements. I wish to thank Shlomo Weber, Serguei Kokovin, Victor Polterovich, Valerii Marakulin, Michel
Le Breton, Anna Bogomolnaya and all the participants of many seminars, including EERC workshops, Seminar on
Mathematical Economics at Central Economics and Mathematics Institute and in the Institute for Mathematics in the
Siberian Branch of Russian Academy of Sciences. Special thanks are to my wife, Ekaterina, who supported me during
all this project.
Substantial part of the project was accomplished while being a research fellow at the Institute for Mathematics in the
Siberian Branch of Russian Academy of Sciences, Novosibirsk.
Alexey Savvateev
Central Economics and Mathematics Institute and New Economic School
and Institute for Theoretical and Experimental Physics, Moscow
117418, Moscow, Nakhimovskii Prospekt, 47, room 816
Tel.: +7 (095) 332 46 06
Fax: +7 (095) 129 37 22
E-mail: [email protected]
 Alexey Savvateev, 2005
Contents
Introduction
3
Model specification
4
Formalization of a “linear world” benchmark . . . . . . . . . . . . . . . . . . . . .
4
Specification of the median rule correspondence . . . . . . . . . . . . . . . . . . . .
5
Coalitionary stability and the core . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Alternative stability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Stability of one country
8
Hedonic case: an overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Stability conditions: hedonic case . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Stability conditions: unconstrained median correspondence. The case of an oddnumbered world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Stability conditions: unconstrained median correspondence. The case of an evennumbered world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Stability of an equidistantial country . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Multi-jurisdictional stability
16
First observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Small societies (|N | = 2, 3)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Geometric representation of the problem . . . . . . . . . . . . . . . . . . . . . . . .
17
The case of |N | = 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Stability of a uniform world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Strong stability: a general counter-example
19
Negative result for |N | = 5 and unconstrained median choice . . . . . . . . . . . .
19
A strict instability result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
A universal counter-example in numbers . . . . . . . . . . . . . . . . . . . . . . . .
21
Finite replica societies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
0.1
Finite types atomless distributions . . . . . . . . . . . . . . . . . . . . . . . .
22
An unstable bi-polar world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
A construction of a finite counter-example . . . . . . . . . . . . . . . . . . . . . . .
23
Redistribution policies
25
A model with partial compensation . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Full compensation: A Rawlsian case (i.e. “Socialism”) . . . . . . . . . . . . . . . .
25
Socialist games have a nonempty P-Core . . . . . . . . . . . . . . . . . . . . . . . .
26
Bibliography
28
Draft policy brief
29
2
Introduction
Consider a “linear world” comprised by several heterogenous agents differing in their attitudes
towards horisontally differentiated local public good. The space of varieties of this good
available for production is assumed to be unidimentional — a real line. We will also use a
term “location”, instead of “variety”. Unlike the standard Tiebout 1956 model, agents are
immobile. An agent’s geographical location is by default identified with the location of his
optimal, or ideal, variety of the public good.
Agents are being split into several groups, each group financing its own variety of the
public good for its members to consume. The group structure is such that each agent belongs
to exactly one of these groups. It is assumed that the location of the public good to be
chosen is being formed endogenuously within each group, via the majority voting. Agents’
preferences are assumed to be single-peaked on the space of alternative locations, which
results in a median voter’s choice selection.
Within this framework, a group formation process is being studied. Members of a group
face a trade-off between economies of scale in sharing fixed costs of production of a chosen
variety of the public good, and personalized distance costs. The latter ones stem from the fact
that the variety chosen by the group is not, generally, identical to that mostly preferred by
a given member of the group. Questions posed in this environment is whether there exists a
stable partition of the world into several groups, where stability requirements will be specified
below.
There are many applications of this setting. To mention a few: theory of clubs and
the club formation problem (Buchanan 1965; Littlechild 1973; Sandler and Tschirhard 1980;
Makarov 2003); facility location model (Topkis 1998; Goemans, Skutella 2002); group formation of Le Breton, Weber 2003; public project (Mas-Colell 1980); generally, there are many
dimensions of a local public good’s differentiation in these applications, but, as the first step,
one is to study the unidimensional environment.
Various modifications and applications were studied extensively in literature, and each
of these applications has its own specific features. Those close to our setting are Guesnerie
and Oddou 1979, 1981, 1988, Greenberg and Weber 1986, 1993, Demange 1994, Demange
and Guesnerie 1997, Konishi, Le Breton and Weber 1997a-e, Haimanko, Le Breton and
Weber 2003. Among them, there exist no attempts to analyze stable group structures in a
general linear world.
A formal model introduced below is being historically designed for the study of the so
called country formation problem (see Alesina, Spolaore 1997 and Bolton, Roland 1997).
Within this approach, one can study political issues such as threats of a secession, stability of
a country or of a federation of several countries, and so on. Instead of “agents”, we then talk
about “regions” (communities, ethnic or cultural groups, nations, etc.), and a given variety
of the local public good is being interpreted as the unique, aggregated policy parameter (say,
the “location of the government”).
3
What distinguishes our approach from the existing literature is that we operate the nontransferable utility (NTU) framework. Papers that study the opposite case of transferable
utilities establish a number of positive results on existence and efficiency of stable partitions.
(The very term “efficiency” makes sense only in the transferable utility (TU) framework; in
the model considered, it virtually has no sense.) Among those papers devoted to TU models
of country formation are Wooders 1978; Haimanko, Le Breton and Weber 2002, 2003; Le Breton, Weber 2001; Alesina, Spolaore 1997; Bolton, Roland 1997. Close topics are discussed,
within the framework of location models, in Wei 1991, and Goemans, Skutella 2002 (in the
latter one, authors use linear programming techniques in the analysis of stability). The only
paper where NTU-game is considered at least partly is Cechlarova, Dahm, Lacko 2001, where
the case of a uniform distribution of the regions in the world is studied, but even there general
existence result was not obtained. Unlike them, we consider general distributions regions.
The paper is organized as follows. In Section 1, the model is introduced. Section 2 describes questions to be analyzed, within the model considered. Section 3 characterizes conditions of stability of the world as the unique country. Section 4 introduces multi-jurisdictional
federations, and establishes stability of such federations in special cases, as well as presents a
counter-example (“an unstable world”). Section 5 introduces the atomless world model; with
the help of this model, a universally unstable world is constructed where there is no even way
to specify a subset of a median correspondence which would retain stability. Section 6 shows
that if some sort of compensation of unsatisfied agents is available within groups, namely,
when costs are divided equally among the members of each group, a stable multi-jurisdictional
federation always exists. This partly explains the stability of the former socialist federations,
such as Soviet Union or Ugoslavia, as well as suggests a somewhat arguable way to keep
Russian federation from the break-up.
Model specification
Formalization of a “linear world” benchmark
We define here a discrete (finite) version of a basic model. An infinite (atomless) formalization
will be introduced later on in the paper.
Let N = {1, 2, . . . , n} be the finite set of agents, and I = [−T, T ] be the set of possible
varieties of local public goods (hereafter, locations). Each agent t ∈ N has symmetric singlepeaked preferences over I and, therefore, could be identified with his ideal location pt ∈ I. The
cumulative distribution of agents (or their ideal points) is denoted by F , and it is determined
by the set of ideal points {pt }t∈N . F generates the measure µ: µ(S) = |S| for any S ⊂ N .
Without loss of generality, we assume that p1 ≤ p2 ≤ · · · ≤ pn .
Each coalition S ⊂ N , if it forms, has to choose its location in I, and bear the fixed
cost g > 0 that it shares equally among its members. In addition to the monetary burden,
each agent has a disutility of distance between his ideal point, and the choice made by the
coalition to which he belongs. We assume that, if coalition S forms and chooses location p
4
in I, the utility of each i in S is given by
vt (p, S) = −|pt − p| −
g
.
|S|
(1)
For every coalition S, denote by M (S) the set of its medians.
The question here is the existence of a stable partition
P = {S1 , . . . , SK }
(2)
where N = S1 t · · · t SK in this non-transferable utility (NTU) framework, when the location
choices of S are limited by M (S). The term “stability” will be discussed later on. There are
in fact several questions arising in this environment, also stated and discussed below. We
complete this section with the following useful notions.
Definition 1 A coalition S is called consecutive if for every three agents i < j < k , i, k ∈ S
implies j ∈ S. A partition P is called consecutive if every coalition S ∈ P is consecutive.
Definition 2 For any coalition S ⊂ N , we denote by supp[S] ⊂ R the smallest segment on
the real line which contains all pt for t ∈ S. We call supp[S] a support of a coalition S.
Notice that a coalition is consecutive if and only if its support does not contain agents
from other coalitions.
Specification of the median rule correspondence
In what follows, we will use a term group as a synonymous to the term coalition, and,
sometimes, a term center of a group as a synonymous to the term location chosen by a
coalition.
One could notice that M (S) is either a singleton, or a segment, depending on whether
|S| is odd or even. In the latter case, still, M (S) could degenerate into a singleton if the two
median agents in |S| are characterized by one and the same location, p̂ ∈ I. However, if this
is not the case, there is a luft in choosing a precise location in M (S), and the rules governing
this choice are to be defined. In other words, one is to specify a correspondence
m : 2N ⇒ R,
(3)
such that ∀S ∈ 2N we have m(S) ⊂ M (S). This correspondence prescribes which medians
are available for group S as its locational parameters, if it forms. Four examples of such a
correspondence arise naturally; here they are.
C1 m(S) ≡ M (S); this is the case of unrestricted locational choice. This case is the
principal one, both because it looks more realistic than others, and due to the fact that its
analysis sheds light on other cases, as will be demonstrated below.
C2 “Hedonic case”. This is the case when, once a group forms, its location is being
uniquely determined. When talking about hedonic case, we will always mean that m(S) is a
middlepoint of a segment M (S).
5
Two more cases are slightly less important, but still are of a certain interest.
C3 m(S) contains only (two) endpoints of M (S);
C4 m(S) contains two endpoints and the middlepoint of M (S).
Coalitionary stability and the core
Each specification of m(S)-correspondence completes the description of the environment under study. After specifying this correspondence, one should explain what is meant by a stable
partition. Again, there are several (at least, three) alternative notions of stability. In addition, consult Alesina, Spolaore 1997 and Cechlarova, Dahm and Lacko 2001 for a number of
other interpretations of stability.
S1: Coalitionary stability. This is a strong requirement that no coalition would find
it profitable to organize a new group, breaking away the partition structure. Formally:
Definition 3 A pair (P, T ), where P = {S1 , . . . , SK } is a partition of I and T = {p1 , . . . , pK }
with pk ∈ m(Sk ) for all k = 1, . . . , K, is called coalitionary stable if there is no coalition S,
together with a location p ∈ m(S), such that
vt (p, S) > vt (pk , Sk )
(4)
for all t ∈ S, where t ∈ Sk ∈ P . (Such coalition we be called a blocking coalition). We say
that a partition (2) is stable, if there exists a bundle T from this definition, such that a pair
(P, T ) is stable.
We could state this requirement in terms of coalitionary game theory. Given any society
N , we construct a cooperative game without sidepayments. Namely, this is the game of N
players in which the set V (S) of feasible payoff allocations for a coalition S ⊂ N consists of
vectors {v = (vt )t∈S ∈ R|S| }, such that there exists p ∈ m(S) with the property
∀t ∈ S
vt < vt (p, S).
(5)
Such a game satisfies all standard assumptions for games without sidepayments (Danilov 2002;
Greenberg, Weber 1986). Therefore, one can apply usual techniques in the analysis of this
game. Let me remind the definition of a core.
Definition 4 A core of a cooperative game is the set of all non-dominated allocations v ∈
V (N ), i.e. of such allocations that ∀S ⊂ N the induced allocation vS does not lie inside the
set V (S). Here, the induced allocation is by definition the vector (vi )i∈S ∈ R|S| .
Cooperative games associated with societies under consideration do not always satisfy
the superadditivity property. This means that it is not necessary optimal to form the grand
coalition (we will return to this question earlier). Therefore, we modify the definition of a
core:
6
Definition 5 A core of a coalition partition form (say, a P-Core) of a cooperative game is
a set of non-dominated allocations v = {vt }t∈N for which there exists a pair (P, T ) from the
definition 3 such that
∀Sk ∈ P
∀t ∈ Sk
vt ≤ vt (pk , Sk ).
(6)
Elements from P-Core which correspond to the trivial partition N = N are exactly those lying
in the core of the game, treated in the standard sense.
Now, we can conclude that a pair (P, T ) is coalitionary stable if and only if the corresponding payoff vector lies in the core of a coalition partition form of the game that was
introduced above.
Notice that the game under consideration is the one associated with the provision of the
horizontally differentiated public good, when the variety (or the location), not the amount of
the good, is to be chosen. In the case of vertical differentiation, the corresponding game is
always supermodular, hence, its core is nonempty. The case of horisontal differentiation is
much more complicated.
To conclude this subsection, I present a theorem that was proved in Greenberg, Weber 1986. It will be useful for the analysis below. In order to formulate this theorem, let me
introduce the concept of a quasistable partition.
Definition 6 A partition P is called quasistable if there exists a bundle of locations, T such
that the payoff vector of a corresponding pair (P, T ) could not be improved by any consecutive
coalition (compare to the Definition 3).
Theorem 1 (Greenberg, Weber 1986) For an arbitrary linear world characterized by a bundle
(p1 ≤ · · · ≤ pn ), there exists a quasistable consecutive partition.
Now, suppose that we identify a certain class of partitions, for which the existence of a
blocking coalition implies the existence of a consecutive blocking coalition. Then, to prove that
a stable partition exists for a given linear world, it is suffice to establish that the quasistable
consecutive partition (which exists according to theorem 1) lies within this class. We make
use of this observation in what follows.
Alternative stability concepts
Although our analysis will mostly be concerned with the concept of coalitionary stability,
there are other questions of interest that may arise in this environment naturally. We will
introduce two stability concepts, in addition to that of coalitionary stability.
S2: Individual stability. This is the requirement that no agent would like to switch to
another group (or to organize his “individual” group), provided other members of the new
group would not object to accept him. Formally:
Definition 7 A pair (P, T ), where P = {S1 , . . . , SK } is a partition of I and T = {p1 , . . . , pK }
with pk ∈ m(Sk ) for all k = 1, . . . , K, is called individually stable if there exists no agent t,
7
together with a certain group S = Sr ∈ P or the empty group S = ∅, and with p ∈ m(S t t),
such that
vt (p, S t t) > vt (pk , Sk );
vi (p, S t t) ≥ vi (pr , S) ∀i ∈ S,
(7)
where t ∈ Sk ∈ P .
This type of stability is, generally, implied by the strong stability, since a coalition S t t is
characterized as the one in which one member increases his payoff, while others nondecrease
it. Normally, they would then also strictly increse their payoffs, thus forming with the agent
t the very coalition that would block the existing partition.
Third type of stability, however, is a different story. It requires that no agent would like
to change his group irrespective of the will of the accepting group’s members. One should
then ask: if we are not in a hedonic case, how then we define the location of a new group?
Generally, there is no way to give a universally correct answer; usually, however, the
requirement is that the migrating agent will affect the location parameter of the new group
minimally. In our linear world, this requirement specifies new location uniquely. At least, for
the case of the unconstrained median rule correspondence, this means the following.
S3: Nash stability.
Definition 8 A pair (P, T ), where P = {S1 , . . . , SK } is a partition of I and T = {p1 , . . . , pK }
with pk ∈ M (Sk ) for all k = 1, . . . , K, is called a Nash equilibrium, or Nash stable if there
exists no agent t, together with a certain group S = Sr ∈ P or the empty group S = ∅ and
with p ∈ M (S t t) ∩ M (S), such that
vt (p, S t t) > vt (pk , Sk ),
(8)
where t ∈ Sk ∈ P ; for S = ∅, we put M (S) = [−T, T ]. One can easily notice that M (S t t) ∩
M (S) is always single-valued.
In what follows, however, we mainly are concerned with issues of strong stability. Next
section deals with stability of the whole federation regarded as the trivial partition N = N .
Stability of one country
We begin our analysis with the following question: Under which conditions, a trivial partition
N = N of the world is stable? That is, when the whole world could comprise the only one
country, experiencing no threats of a secession by its regions? Formally, when the cooperative
game associated with this world has a nonempty core, treated in the orthodox sense?
Obviously, at the same time we are studying a question of inner stability of a single group
Si ∈ P entering an arbitrary partition P . In what follows, we say that a group (or a coalition)
S ⊂ N is prone to a secession, if this group blocks the trivial partition N = N .
8
Hedonic case: an overview
Let us first of all consider the hedonic case, that is, when the median correspondence m(S)
is single-valued, namely, for any coalition S ⊂ N we require that m(S) should coincide
with the middlepoint of the segment M (S), once the group S is formed. Our principal case
m(S) ≡ M (S) is being postponed until next section; it is a little bit more tedious.
Consider a group N . Without loss of generality, we assume that Its center is located
at 0 (otherwise, contemplate a horisontal shift). If the group is odd-numbered, denote its
members’ locations as
p−n , . . . , p−1 , p0 = 0, p1 , . . . , pn ;
(9)
if it is even-numbered, it is characterized by a bundle
p−n , . . . , p−1 , p1 , . . . , pn .
(10)
Moreover, from now on we put g = 1 (otherwise, one can contemplate a homotetic transformation of the real line with the coefficient equal to g1 ).
We begin our analysis with the following observation:
Lemma 0.1 If a subgroup S ⊂ N is prone to a secession, then 0 ∈
/ supp[S].
Proof: Wherever a center p of S be, those agents from S for whom 0 is closer than p
would unambiguously oppose a secession, due to the formula (1): for them, both common
and personalized costs would increase.
Proof is complete.
Actually, we have proved that if S is prone to secession and, say, its support lies inside
the positive ray, supp[S] ⊂ (0, +∞), then necessarily p/2 ∈
/ supp[S].
Notice, in addition, that due to this lemma, the analysis of stability splits into two disjoint
studies: the analysis of left-hand stability (i.e. stability of the negative tail of the group
N ), and that of right-hand stability, which means stability of the positive tail. The two
counterparts are symmetric; we take the positive tail of the group, thus analyzing secession
threats of various subgroups S ⊂ N with positive supports: supp[S] ⊂ (0, +∞).
Lemma 0.2 If a group S is prone to secession, than there is a consecutive group S 0 with the
same center and the same size, which is also prone to secession.
Proof: Indeed, wherever a center p of S be, we take, instead of members of S, consecutive
[ |S|+1
2 ]
agents both to the left and to the right of p, and the very agent in p, if the group S
is an odd-numbered one.
Call this new coalition by S 0 , and choose its center in p again (it is possible, by construction). Then, S 0 is prone to secession too. This fact, in turn, follows directly from
9
Lemma 0.3 Consider a group S ⊂ N , denote its center by p > 0. Then, S is prone to a
secession if and only if the following conditions hold:
supp[S] ⊂ (0, +∞);
2q > p +
1
|S|
−
(11)
1
|N | ,
where q is the left border of supp[S], that is, the location of the leftmost agent in S.
Proof of Lemma 0.3: Necessity of the first condition is demonstrated above. Next, the
group is prone to a secession if and only if every group’s member is prone to a secession. We
will show that, actually, it is equivalent to claiming that the leftmost member of a group is
prone to a secession. Indeed, take any member t ∈ S located at pt . He is prone to a secession
if
pt +
1
1
> |p − pt | +
.
|N |
|S|
(12)
Consider two cases separately: pt > p and pt < p. If the former case holds, then the
inequality (12) reduces to
pt +
1
|N |
> pt − p +
p>0+
1
|S|
2p > p +
1
|S|
⇔
1
|N | ⇔
1
1
|S| − |N | ;
−
(13)
In particular, we can see that if a center of S is prone to a secession, than all the members
of S located to the right of the center are prone to a secession too, and vise versa. In fact,
all the statements that a given member of a group S, which is located not to the left of the
center, is prone to a secession, are equivalent.
In the latter case of pt < p, the inequality (12) reduces to
2pt > p +
1
|S|
−
1
|N | .
(14)
The smalle is pt , the more rigid this condition is. Therefore, the most rigid condition
implies all the other ones. And this is the condition that the leftmost agent is prone to a
secession, i.e. the condition (11).
Proof of Lemma 0.3 is complete.
Now, to prove Lemma 0.2, one just notices that, for a new, consecutive group S 0 , the
condition (11) could only be relaxed, relative to the initial group S: nothing changes but the
location of the leftmost agent, which could only increase (at least, stay the same).
Proof of Lemma 0.2 is complete.
Stability conditions: hedonic case
Now we can state that the list of necessary and sufficient conditions for stability of a group
N with members’ locations specified by (9) or by (10) looks as follows: for every pt to the
right of the center, we claim that pt could not be the leftmost member of a seceding subgroup
S of a size 1, 2, . . . , n − t + 1.
10
Precisely, it gives us the following list of inequalities (for convenience, we have introduced
the following notation in the inequalities below: ph+1/2 = (ph + ph+1 )/2), which are necessary
and sufficient conditions for right-hand stability:
∀(t, l) : t ∈ {1, . . . , n};
l ∈ {1, . . . , n − t + 1}
2pt ≤ p(2t+l−1)/2 +
1
l
−
1
|N | ,
(15)
where |N | = 2n, if N is an even-numbered group, and |N | = 2n + 1, otherwise. And there
are similar conditions for left-hand stability.
Examples. Begin with |N | = 2, with its members’ locations at p−1 and p1 . This group
is stable iff p−1 ≥ −1/2 and p1 ≤ 1/2. Obviously this simply mean that p1 − p−1 ≤ 1, i.e.,
the distance between the two agents should not be greater than 1.
The case |N | = 3 is quite simple as well. We have the world specified as (p−1 , p0 = 0, p1 ),
and there are two conditions necessary and sufficient for stability: p−1 ≥ −2/3 and p1 ≤ 2/3.
In both cases, which is quite expected, these conditions, generally, speak about not too
diversified country.
However, one could not state that, if some of locations decrease, stable group is bound to
remain stable. The problem is that, while group becomes less diversified on average, some
coalitions became so homogenous that they become prone to a secession.
And here is an example. It arises for |N | = 4; so let me first list necessary and sufficient
conditions in this case (group members’ locations are (p−2 ≤ p−1 (≤ 0 ≤)p1 ≤ p2 )). I omit
left-hand stability conditions, for they are just the same, up to a symmetry. Some simple
calculations are skipped.
p2 ≤ 3/4,
p1 ≤ min p2 , (p2 )/3 + 1/6.
(16)
Now, consider a group N with members’ locations given by (−0.5, −0.3, 0.3, 0.5). Conditions for stability hold, hence, the group N is stable. Let then two farmost agents to join the
other two, such that the new group N 0 is specified by (−0.3, −0.3, 0.3, 0.3). A new group is
less heterogenuous, still it is unstable! Namely, a coalition S = {3, 4} is prone to a secession,
and the second inequality in (16) fails to hold.
Stability conditions: unconstrained median correspondence. The case of an
odd-numbered world
Returning back to our principle case of unconstrained median rule correspondence, which
changes occur in our picture?
If the whole world N is odd-numbered, its center is still uniquely specified; therefore,
again, a stability requirement splits into left-hand and right-hand stability requirements.
However, secession threats become more tuff.
To demonstrate this, take, for instance, right-hand stability. First of all, all the existing
secession threats associated with the hedonic case are still valid; therefore, odd-numbered
11
groups N that are stable in the unconstrained median choice environment will certainly be
stable in the hedonic environment too.
Moreover, odd-numbered coalitions generate equal secession threats in both environments,
for their center is all in all uniquely determined. The same is true for those even-numbered
coalitions whose two median members’ locations coincide. In what follows in the unconstrained median choice environment, we will refer to such groups or coalitions as hedonic;
in the hedonic environment, all coalitions are hedonic, so to speak, and we will not use this
term there.
More generally, a hedonic coalition is a coalition for which, no matter how to specify a
median choice rule, the location of its center is uniquely determined: M (S) is a singleton.
Now, consider an even-numbered coalition S whose support is a non-degenerated segment
lying inside the positive ray. As we have known from the previous section, such a group’s
secession possibilities are bounded by the leftmost agent’s preferences: the coalition S is
prone to a secession iff this agent is prone to a secession, which in turn is equivalent to the
second inequality of (11).
But now, as we are in the unconstrained median choice environment, we know that p not
necessarily coincides with the middlepoint of a segment between the two median members of
S; there is a possibility of choosing a center p of our group S coinciding with the location
of the leftmost of the two median agents. But this somewhat relaxes the second inequality
of (11)! Therefore, one can conclude that, under a presumption of unconstrained median
choice, stability of the trivial partition N = N of the whole odd-numbered world requires
more than stability in the hedonic case.
We illustrate this by means of the following example. Consider a world N comprised of
5 agents1 , which locations are given by (p−2 = −0.7, p−1 = −0.4, p0 = 0, p1 = 0.4, p2 = 0.7).
This world is symmetric, hence, it is suffice to check only right-hand stability conditions.
They indeed hold, for we have
0.7 < 1 − 1/5;
2 · 0.4 < (0.4 + 0.7)/2 + 1/2 − 1/5 = 0.85.
(17)
Hence, N is stable in the hedonic case. At the same time, in the case of unconstrained median
correspondence, it looses stability, since a coalition {4, 5} with the center p located in the
point 0.4 is prone to a secession: instead of 0.6 = 0.4 + 0.2, the 4th agent pays just 0.5.
One can write down a list of stability conditions in the unconstrained median choice
environment; they repeate (15), with a new agreement on what is meant by ph+1/2 . Now, we
simply put ph+1/2 = ph . Except for this modification, the whole system of inequalities (15)
does not change2 .
1
Stability of the world N with |N | = 3 does not depend on the precise specification of a median choice
rule.
2
One could have noticed that in the new system, there are abundant inequalities, which actually follow
from other ones. Those are inequalities written for odd-numbered coalitions in which the farmost member of
N does not participate. Also, only one inequality for |S| = 1 is bounding, namely, that for this farmost agent
(the latter is true also in the hedonic case).
12
Stability conditions: unconstrained median correspondence. The case of an
even-numbered world
Things become more complicated if a group is even-numbered: now, by assuming that its
center is being located at 0, we are not perfectly determining that group’s identity.
To be more precise, recall that a positive simultaneous shift of all the locations of members
of a group does not influence any of stability issues. In the hedonic environment, we specify
the bundle of locations of the group N by fixing the location of its center, say, in 0. If the
group is hedonic itself, again, its members’ locations are being uniquely determined by the
choice of its center’s location, even in the unconstrained median choice environment.
Now, consider a world N for which M (S) is a non-degenerate segment, i.e. the world
is not hedonic, when considered as a unique group. Then, by defining a group members’
locations via the bundle
p−n , . . . , p−1 , p1 , . . . , pn ,
(18)
and assuming that its center is being located at 0, we implicitly kill the freedom in choosing
a location of the center, p elsewhere inside the segment [p−1 , p1 ]. In order to make use of
this freedom while analyzing stability issues, one should examine all the points from the
segment M (S) as possible locations of the center of the group N . Now, the two counterparts
of stability, that is, right-hand and left-hand stability are not more independent, and the
closer the center of the group to the right median agent’s location, the less tight right-hand
stability requirement is, and the more tight the left-hand stability. And the world N is stable
(or, equivalently, secession-proof) if there exists at least one such location p ∈ [p−1 , p1 ], for
which both requirements are satisfied.
Let us now take a formal and precise account of these considerations. For the purposes
of convenience, it is easier to fix the location of the center at 0, thus varying locations of the
group’s members, within the feasible limits. Take right-hand stability conditions
∀(t, l) : t ∈ {1, . . . , n};
2pt
≤
l ∈ {1, . . . , n − t + 1}
p(2t+l−1)/2
+
1
l
−
(19)
1
|N | ,
written down under the assumption that the center of the world N is being located at 0,
where ph+1/2 = ph .
If the center shifts by ∆ to the right, this is equivalent to having the center staying at 0,
while all the locations pt shifting by ∆ to the left. Than, as one can learn from (19), every
inequality is being relaxed exactly by ∆.
In order to make use of this observation, as well as to make this observation clear, we
define the following secession indicator (SI) ρ:
ρ[p1 , . . . , pn ] =
2pt − p(2t+l−1)/2 −
max
(t,l): t∈{1,...,n};
l∈{1,...,n−t+1}
1
1
+
.
l
|N |
(20)
Obviously, a country (10) with its center located at 0 is right-hand stable iff ρ[p1 , . . . , pn ] ≤ 0.
And it is left-hand stable iff ρ[−p−1 , . . . , −p−n ] ≤ 0.
13
Now, we can see that simultaneous shift of locations p1 , . . . , pn to the left by ∆ decreases
secession indicator exactly by ∆, for every expression inside maximization in (20) decreases
by ∆. At the same time, left-hand part of a country should logically be shifted to the left by
∆ too, which reflects the very fact that this shift is solely due to the center of N approaching right-hand part of the world, hence, distancing from the left-hand part. Henceforth,
ρ[−p−1 , . . . , −p−n ] increases exactly by ∆, in line with this shift.
Now, the final step of our analysis is to construct the following ultimate secession indicator
(USI) λ just as the sum of the two values of ρ:
λ[N ] = λ[p−n , . . . , p−1 , p1 , . . . , pn ] = ρ[p1 , . . . , pn ] + ρ[−p−1 , . . . , −p−n ].
(21)
USI does not depend on the specific choice of the center’s location inside M (S), thus
being a correctly defined functional over the space of all groups N with |N | = 2n. However,
one should be aware of the fact that, in order to calculate λ, he should specify the list of
locations (10) of members of that group in such a way that p−1 < 0 < p1 ; otherwise the
indicator would not work correctly.
Let us summarize the knowledge obtained above.
Theorem 2 The world N , specified by (10) with p−1 < 0 < p1 , is stable, or secession-proof,
if and only if the following conditions hold:
ρ[0, p2 − p1 , . . . , pn − p1 ] ≤ 0;
ρ[0, p−1 − p−2 , . . . , p−1 − p−n ] ≤ 0;
(22)
λ[N ] ≤ 0.
Proof: Actually, almost everything was prepared earlier for this result. The violation
of any of the two first inequalities (22) automatically means that the world is unstable,
because even placing the center to the nearmost median agent’s location could not keep the
corresponding part of the world from a secession by some of its coalitions (“regions”). And
even if both hold, while the third one being violated, one would not find a center’s location
in such a way that will satisfy both left-hand and right-hand parts of the world. The proof
of the opposite direction is just a triviality.
Proof is complete.
An interesting corollary of the principles of stability of one group is the following property:
Lemma 0.4 If pn − p1 ≤ 1/n then the partition N = N is stable.
It means that if a country is not much diverse in agents’ preferences, then it is easily kept
together, without any secession threats.
Now, it’s time to realize what we get. As we already know, for odd-numbered groups,
switching from the hedonic case to that of unconstrained median choice leaves less chances
for stability. What about even-numbered group?
Generally, the answer is ambiguous: although secession threats become more tough, a
freedom in choosing a group’s center’s location somehow smooth these threats. This could
14
help if a group is not symmetric: there exist worlds with |N | = 6 for which the stability in
the case of unconstrained median rule correspondence is assured, while they are unstable in
the hedonic environment. At the same time, if a group is symmetric, it never could be the
case.
Now, we write the stability conditions for the world of 4 agents. Denote by lt the distance
between neighbors: lt = pt − pt−1 , where t = 2, 3, 4. The bundle l2 , l3 , l4 then perfectly
characterize our group; one can see that the stability conditions of this group are
l1 + l2 ≤ 1,
l2 + l3 ≤ 1,
l2 ≤ 1/2,
(23)
l1 + l2 + l3 ≤ 3/2.
From now on, we will frequently use a bundle l2 , . . . , ln of distances to characterize the
group, instead of the bundle p1 , . . . , pn . Moreover, we can put g = 1, once and for all, due to
the scale invariance of the problem under investigation.
Stability of an equidistantial country
For an equidistantial society (l, l, . . . , l) there is an additional assertion:
Assertion 0.1 If a subgroup S 0 is prone to secession, than the farmost subgroup of the same
size is prone to secession too.
This is obvious, and gives a compact set of stability conditions for the stable equidistantial
group. Namely, it is claimed that far-distant 1-agent group, 2-agent group, 3-agent group etc.
are secession-proof. Omitting tedious formulas, let us note that all these conditions are upperbounds on l, and thus there exists one condition guaranteeing the other ones. Therefore, for
every n there exists k such that, provided far-distant k-agent subgroup is secession-proof, the
whole group will be stable.
The answer for big n’s:
k≈
n
√ .
3+ 6
(24)
Necessary and sufficient asymptotic conditions for stability:
l≤
where κ =
κ
,
n2
(25)
√
2(2+ 6)
√ .
√
6(3+ 6)
Compare this to Alesina, Spolaore 1997, and Cechlarova, Dahm, Lacko 2001. An interesting result is that stability requires extremely low mutual distances, so the stable group of
the size n should be very densitive (be of a size less than 1/n as a whole). When recalling the
political interpretation, look at the Japan, please! It seems quite stable, and has the minimal
size. Opposite case is our Russia (unless one is to claim that geographical distance is not
vorrelated with preference distance in our country).
15
In addition, I present the following table, where for every n = 1, . . . , 10 the crucial k
and the cutoff value of l are given for which the equidistantial group becomes unstable by
n k
l
n k
l
secession of a k-agent far-distant subgroup:
1
-
-
6
2
2/9
2
1
1
7
2
5/28
3
1
2/3
8
2
3/20
4
1,2
1/2
9
2
7/54
5
2
3/10
10
3
7/75
Multi-jurisdictional stability
Once a country as a group is typically not stable (there exist regions which are prone to
secession), one may ask whether a country can be kept stable as a federation, i.e. there exists
a stable partition of the world into countries (jurisdictions, groups). We now address these
issues of multi-jurisdictional stability. Let us start with preliminary work.
First observations
The set Υ of all partitions of a set N is a partially ordered set. Namely, we say that a partition
P is contained in a partition P 0 if P could be obtained from P 0 by coarsering (i.e further
decomposition). This is an ordering, and the set Υ forms a complete lattice. This lattice
contains the set of all consecutive coalitions as a sublattice. We can assert the following.
Assertion 0.2 There are 2n−1 consecutive partitions. The set Ξ of consecutive partitions is
isomorphic (as a partially ordered set) to the power set of the (n − 1)-elemented set.
Indeed, consider a set of neighbor pairs {[12], [23], . . . }. Every consecutive partition is
identified with a subset of this set of pairs — the subset that contains all the pairs belonging
to one and the same group. It is easy to check that this is a one-to-one correspondence.
In the following assertion, recall that by quasistable partition we mean a partition P for
which there exists a bundle of locations, T such that the payoff vector of a corresponding
pair (P, T ) could not be improved upon by any consecutive coalition.
Assertion 0.3 If lt+1 = pt+1 − pt > 1 then every quasistable (hence, every stable) partition
contains in the partition N = {1, . . . , t} t {t + 1, . . . , n}.
Proof by contradiction: let S be any group that contains a pair (j, h) with j ≤ t and
h ≥ t + 1. Then, the three cases are different:
(i) when the two parts of a group, S ∩{1, . . . , t} and S ∩{t+1, . . . , n} are equally-numbered
and multivalued (both contain more than 1 agent);
(ii) when they both are single-valued (hence, |S| = 2), and
(iii) when the two parts contain different numbers of agents.
16
In the cases (i) and (ii), the center p of S is located somewhere between pt and pt+1 . If
the first case takes place, then consider that part of S whose agents all are far from p by
more than 0.5. Take two of them, and let them secede with the center coinciding with the
agent who is closer to p. This would increase the payoff of both.
In the second case, it is obvious that the agent who is far from the center more than by
0.5 will secede, because he gives up more than gains (his gains are exactly 0.5).
And if the third case takes place, then the center is located within the bigger part of S,
hence, outside the segment (pt , pt+1 ) of no less than unit length. So, agents from the second,
less part of S are far from p by more than 1, which is absurdic (everyone from the latter part
would be better off by seceding unilaterally). Proof is complete.
The important corollary of this theorem is the fact that our analysis is confound to
the cube {(l2 , . . . , ln )} | lt ≤ 1}. Outside this cube, the problem could be divided into several
n−1
problems of lower dimension. Being a compact in R+
, this cube is a good object to analyse.
Additionally, this cube is a complete lattice with respect to the standard ordering.
Small societies (|N | = 2, 3)
One easily notes that when there are 2 agents (i.e. only one parameter l of distance between
them), a stable partition contains one group {1, 2} with the center, for instance, located at
the point
p1 +p2
2
if l ≤ 1 and the two distinct countries {1}, {2} if l ≥ 1. In this case, the
whole space R+ of possible societies is being divided into two disjoint parts each of them
being characterized by the unique stable partition.
Things become more complicated when there are 3 regions characterized by the bundle
(l1 , l2 ). If l1 , l2 ≤ 2/3 then the partition N = {1, 2, 3} is stable (where the center by assumption coincides with p2 ); if l1 ≥ max{l2 , 2/3}, l2 ≤ 1 then a partition N = {1} t {2, 3} with
centers in p1 and in min{p2 , p3 − 1/2} is stable; if both are greater than 1 then a partition
N = {1} t {2} t {3} is stable. One can draw all the cases on the following diagram (see
picture 3).
Below, after introducing of a powerful geometric technique, we will proof that the 4-agent
societies are multi-jurisdictionally stable too.
Geometric representation of the problem
Let us introduce a 2-dimensional space (i.e. a plane) with a horizontal axis representing
coordinates of agents, and a vertical axis representing payoffs of agents. Now suppose that
a group S is formed, and the location p = p(S) ∈ m(S) is chosen. Imagine a “mountain”
assotiated with the group S whose peak is located in p, and whose height is equal to −1/|S|.
(One can think of an “undersea” rocky mountains, because all our peaks have negative
height.) The mountainsides are straight lines with the slope of exactly 1 (or, equivalently, of
45 degrees).
Each agent t ∈ S is located at the mountainside corresponding to the horizontal coordinate
pt .
His “height” is equal exactly to his payoff in the group S.
17
Now consider a partition P . Then, every agent and, generally, every point could be found
on all the mountainsides corresponding to groups from this partition. We say that agent t
from the group Sj is visible under the partition P if his height measured with respect to his
group Sj is not lower than his height measured with respect to any other group’s mountain.
Definition 9 A partition P is called regular if all the agents are visible. If this is not a case,
than there exists a pair t ∈
/ S such that the height of t measured with respect to the group S
is higher than that measured with respect to its own coalition, Sj . The group S will then be
called a covering group for t.
A picture 1 presents two possible geometric representations of the group formation problem. First case corresponds to a regular partition, and the second is not.
The case of |N | = 4
We start the analysis of stability, with the help of our enreached geometric intuition. The
next assertion is a triviality.
Assertion 0.4 A trivial partition N = N is regular.
Clearly, if the agent t would change his group to one of his covering groups, S, he would
be unambiguously better off, regardless of the new location of the S’s center. The point is
that other agents could not approve such a migration (due to a possible center’s location
change). But, if the covering group S is an odd-numbered group, the center could remain
the same! And then, obviously, all the agents in a new group S t i will be better off.
This observation needs to be stated explicitly.
Theorem 3 A covering group, S from a stable consecutive partition must be even-numbered.
An irregular stable partitions must contain even-numbered groups.
The second assertion concerns 2-agent groups.
Assertion 0.5 A 2-agent group could not be a covering group even in a quasistable consecutive partition (it means stable with respect to consecutive coalitions).
Proof. By contradiction: let t ∈
/ S and |S| = 2 is a covering group for t in a quasistable
consecutive partition, with the center p = p(S). Consider a group Sj 3 t. Without a loss of
generality, let p(S) < pt . It is clear from a geometric representation that every r ∈ Sj which
lies in the segment (p(S), pt ) is also covered by S (wherever the center of Sj be located). So,
consider r0 ∈ Sj which is most close to p(S) in this segment. Redenote it by t again.
Then, either this segment contains one of S-members, h ∈ S (and then the secession-prone
consecutive coalition would be {h, t} with the center located in ph ), or p(S) = ph for h ∈ S
(then, the secession-prone consecutive coalition would be {S t t}). Proof is complete.
Corollary. The irregular stable partition must contain a group of 4 or more agents.
We now are ready to state the following result (proved to be useful in the analysis).
18
Theorem 4 If there is a consecutive regular partition, than the existence of a secession-prone
coalition implies the existence of a secession-prone consecutive coalition.
Corollary. A regular quasistable partition is overall stable.
Proof is more or less obvious from the picture 1. Indeed, consider the two boundary
agents of a seceding coalition, S. Wherever a center of S be located, it is obvious that under
this center, all the agents between the two considered are better off. Hence, one can replace
S with the one of the same size, but consecutive. So, if one establishes that a quasistable
partition is regular, it means at the same time that this partition is stable.
Now, we are ready to prove the existence result for the case of |N | ≤ 4, exploring theorem 4. Indeed, consider a quasistable partition. If it is N = N , it is regular, hence, stable; if
not, and if it is irregular, it must contain a group of 4 agents. But then, the grand coalition
is secession-prone, which is a consecutive coalition — a contradiction.
Proof is complete.
Stability of a uniform world
Things get worse for |N | ≥ 5. But, before presenting a negative result, let me state the
existence result for the uniform societies (l, . . . , l). First steps towards this result see in
Cechlarova, Dahm and Lacko 2001.
Theorem 5 For an arbitrary uniform society represented by a bundle (l, . . . , l) of distances,
there exists a stable partition into more or less equal-sized countries of the assymptotic population of
1
√
2 l
when l → 0.
The intuition of this result is the following: imagine that the society is very big. Then,
consider a country of the size k such that the borders of this country minimize costs. Easily,
k=
1
√
.
2 l
A partition of the society into such countries is obviously stable, since any potential
group would not convince its border agents to secede.
However, for general types of distributions, this positive result does not remain to hold.
Strong stability: a general counter-example
Negative result for |N | = 5 and unconstrained median choice
First of all, let us present the society without a stable partition when m(S) = M (S). This
society contains 5 agents. We already know that this is the minimal counter-example.
Assertion 0.6 A stable partition may fail to exist.
Proof: Consider a society with five agents, N = {1, 2, 3, 4, 5}, whose locations are given
by p1 = p2 = 0, p3 = p4 = p5 =
19
60 .
We show that there exist no stable partitions. Let us
make few observations:
19
(i) The grand coalition is not stable. Indeed, in this case, agents 1 and 2 pay
1
5
+ 19
60 =
31
60 ,
while in a two-person coalition their contribution would be smaller, namely, 12 .
(ii) No two agents stay separately alone as they would be better off by staying together.
Indeed, in the coalition say {1, 5} each agent pays
1
2
+
19
120
< 1.
(iii) No partition into three coalitions is stable. Indeed, by (ii), a partition into one triple
and two singletons is not stable. However, given a partition into one singleton and two pairs,
agents 3, 4, 5 would be better off by staying together.
(iv) No partition that contains a singleton is stable. Indeed, since 1 and 2 contribute
31
60
in the grand coalition, none of them would stay alone and contribute 1. Consider a partition
{1, 2, 3, 4}, {5}, where the first group chooses the location l, 0 ≤ l ≤
19
60 .
Let l0 =
19
60
− l.
1
0
4 + l , and since 3, 4, 5 can go separately and
1
. This means that the contribution
contribute 31 , we have the inequality 14 + l0 ≤ 13 , or l0 ≤ 12
1
29
of agents 1 and 2 is at least 14 + 19
60 − 12 = 60 . But then agents {1, 2, 5} would obviously be
The contribution of agents 3 and 4 is
better off by being together.
(i)-(iii) imply that a stable partition must consist of two coalitions.
Thus, by (iv),
there are only three candidates to constitute a stable partition: P1 = {3, 4, 5}, {1, 2},
P2 = {2, 4, 5}, {1, 3}, P3 = {1, 2, 5}, {3, 4}.
Consider P1 . Note that a group {1, 2, 4, 5} could make the first two members better off
(relatively to P1 ) by choosing l0 less than
inequality should be satisfied:
1
4
+l <
1
12 . In
1
0
2 or l
order to make 1 and 2 better off the following
>
1
15 .
Thus, by jointly choosing location l0
1 1
within the interval ( 12
, 15 ), agents 2, 4, 5 and 1 would be better off than in P1 .
To complete our examination, note that agent 1 would be better off by joining 2, 4, 5,
rather than staying with 3, whereas agent 5 would prefer join 3 and 4 rather than stay with
1 and 2. Thus, neither P2 nor P3 is stable.
The proof is complete.
The essence of this counter-example becomes clear from picture 2. It is the same as in
the celebrated “dividing a dollar” cooperative game. We have three parties of agents here:
{1, 2}, {3, 4} and {5} which “buy” each other in a cyclical manner. Indeed, suppose, first,
that the last two cooperate and form a group {3, 4, 5}. Then, the party {1, 2} offers {3, 4}
to cooperate and locate the center in the point, say,
29
120 ,
where both parties will be better
off (one can observe from the picture 2 that the corresponding mountain of the height −1/4
covers all the involved agent’s previous locations).
But, having such a group {1, 2, 3, 4}, a party {5} offers a cooperation to the first party,
{1, 2}. This again increases the payoff of all three agents from {1, 2, 5} (it is observed from
picture 2: a mountain of the height −1/3 with the peak located at the 0-point covers all the
three agents’ previous locations on the diagram).
Finally, a party {3, 4} offers {5} to join them again: a group {3, 4, 5} guarantees all its
members the higher payoff than before the cooperation. And this cycle repeats itself, etc.
etc. At the same time, the union of all groups, {1, 2, 3, 4, 5} is unstable: a coalition {1, 2}
wishes to secede.
20
To complete the discussion of this counter-example, let me stress that, in accordance
with (Greenberg, Weber 1986) theorem, there exists a partition which is quasistable. It is
the partition N = {1, 2, 3, 4} t {5}, with the center of the group {1, 2, 3, 4} located in, say,
the point
29
120 .
No consecutive secession-prone coalition exists here.
A strict instability result
Our next goal is to generalize this example. Essentially, stated below is the central result of
the analysis.
Theorem 6 Whatever correspondence m(S) ⊂ M (S) is given, there is a society for which
the (coalitionary) stable partition fails to exist.
The plan of the proof is the following one. First, we define new objects, namely, replica
societies and a atomless society lying in the base of all replicas of a given society . Then,
we introduce a stability notion for atomless societies, and call unstable atomless societies
universally unstable. We explain the meaning of this term by proving that if a atomless society
is universally unstable, then its replicas are unstable in the usual sence, for sufficiently big
values of M . All these constractions correspond to the case of unconstrained median choice,
m(S) = M (S).
Next step will be to prove that the atomless society lying in the base of the counterexample above is universally unstable, and more than this, that every pair (P, T ) is blockaded
by a coalition from very concise family of coalitions.
Further on, we introduce a fictitious agent and place him to the point of the atomless
society corresponding to 29/120 in the initial counter-example. By this, we will accomplish
a very importent task: now, every possible pair (P, T ) is blockaded by a hedonic coalition
(i.e. the coalition which location is unambiguously defined for every correspondence m;
hedonic coalitions are either odd-numbered, or even-numbered but have the median segment
generating to one point).
But if every partition is blockaded by a hedonic coalition, than such a society is clearly a
universal counter-example for all correspondences m! What is rest is to construct a replica
version of this universally unstable division society, and this replica version will present a
counter-example in the usual, discrete sence.
A universal counter-example in numbers
Let 400 agents live in the point 0; 1 agent live in the point 29/24000; and 600 agents live in
the point 19/12000. No matter how to specify a correspondence m(S) ⊂ M (S), this society
does not admit for a stable partition. For instance, it means that stable partition fails to
exist in this society for hedonic case; for the case when locations could be placed only in the
endpoints of the median segment; only the endpoints or a middlepoint etc. — one could not
retain stability using instrument m(S) ⊂ M (S). What is left is to try redistribution policies,
which is extensively discussed below.
21
Finite replica societies
Definition: Consider an arbitrary society N which is represented by a bundle p1 ≤ p2 ≤
· · · ≤ pn . Denote distances between points by li = pi+1 − pi , i = 1, . . . , n − 1. By M-replica
society corresponding to the initial one we mean the society of M N agents with distances
{rj }j=1,...,M N −1 defined as
∀j : M - j rj = 0;
rM i =
∀i = 1, . . . , n − 1
li
M.
(26)
In other words, replica society is the initial society multiplied by M and concised by M times,
simultaneously. One can recall replica economies, but in our case we should not only clone
every agent, but to shrink the distances between them.
There is an intimate connection between a society and its replicas. Indeed, if one looks at
the geometric representation of a replica society, he finds that the picture does not change,
except for a scale transformation. From the coalition formation point of view, however, one
observes much greater opportunities in replicas: there are 2M N coalitions now, instead of 2N ,
as was initially. That is why one cannot state that if a society admits a stable partition, then
all its replicas admit it too; neither one can state the opposite. These questions are open, by
now.
One can thought of replica societies as of the initial society in which we allow for agents
to divide (i vyshlo u nego v otvete dva zemlekopa i dve treti). For instance, one can consider
a coalition {(1/3)1; (2/5)4} consisting of one third of the agent 1 and of two fifth of the
agent 4, etc. What matters is the possibility to define payoffs in such coalitions, which is
straightforward. Then, an initially stable partition may become unstable since, say, such a
coalition which was not allowed to form initially could improve the payoffs to one third of
the agent 1 and two fifth of the agent 4. And in the opposite direction, a stable partition
may apper and consist of such fractionalized coalitions.
0.1
Finite types atomless distributions
However, certain conclusions could be made for stability of higher replicas. The point is that,
for such replicas, the space of coalitions approach the space of all the possible non-integer
coalitions in the initial society. Taking this limit, and normalizing this society to consist of
mass 1 of agents allocated in several points, we get the following basic definition.
Definition: An atomless normalized society (AMS) corresponding to a society (l2 , . . . , ln )
is the allocation of masses (shares) 1/n in the points 0 = r1 , r2 , . . . , rn which are defined by
rt − rt−1 = nlt ,
t = 2, . . . , n.
(27)
This object is designed in order to apply powerful techniques of calculus in our essentially
discrete framework. We mean by a quasi-coalition the allocation of masses xt ≤ 1/n in
corresponding points. Sometimes we will skip the adjanctive quasi, when it could not come
to mass.
22
Now, if not all the points are pairwise different, we can collect masses located in coinsiding
points. AMS-s obtained from normal societies will then be characterized be rational masses;
irrational AMS-s could be approached by big-numbered societies. What I like to stress is that
the space of abstract (rational or irrational) AMS-s is the crucial object to study. Reformulate
the definition of an abstract AMS.
Definition: An abstract AMS is just a finite probability distribution over the real line.
It is characterized by a pair (U, X), where U = {p1 , . . . , pf } ⊂ R is a finite subset of points
on the real line, and X = {x1 , . . . , xf } is a probability distribution over the set U . (Quasi)coalition is defined as the collection of masses Y = {y1 , . . . , yf } allocated in the points of U
yi ≤ xi . A partition, hence, is a family of coalitions {Y 1 , . . . , Y d }
d
P
with the requirement that ∀i = 1, . . . , f
yij = xi .
such that ∀i = 1, . . . , f
j=1
One can now generalize all the definitions of stability for this case. We say that a given
AMS is universally unstable if there exists no partition into coalitions which is coalitionproof. For abstract AMS-s, the same questions as before could be asked; the only point is
that the notion of Nash stability is not clear; we omit the discussion of Nash stability here,
concentrating on strong stability issues. The following hypothesis, I beleive, is true.
Conjecture. If a given AMS is isuniversally unstable, then there exists a standard society
N approaching this AMS which does not allow for a stable partition.
An unstable bi-polar world
Instead of proving this general assertion, we concentrate on the AMS assotiated with our
counter-example; namely, this is the following AMS:
U = {0, 19/12}; X = {0.4, 0.6}.
(28)
I will prove the following, strengthened version of the main conjecture.
Theorem 7 The AMS define above is universally unstable. Moreover, every partition could
be blockaded by a coalition belonging to one of the following three types: those with the location
at 0; those with the location at 19/12; those with the location at 29/24. And there exists the
unique constant > 0 such that every partition is blockaded by a coalition whose members all
improve their payoff by no less than .
We will postpone the proof of this theorem; first of all, I would like to demonstrate how
to construct our wanted counter-example in the standard sence, having this theorem proved.
A construction of a finite counter-example
Let us put a mass less than /10 to the point 29/24. Then, for every partition in new situation
we construct the corresponding partition with the same locations by omitting this new mass.
It is always possible, at least up to /10-changes for coalitions centered in 29/24. Then, we
see that this partition is blockaded by say a coalition S. If the center of S lies in 29/24, we
23
take this mass and add it to S (this mass will opt for this, because S improves the payoff of
both sides, hence, of 29/24 as well). On this way, we constructed an (abstract) AMS that is
universally unstable and is blockaded only by hedonic coalitions.
We now left with the storage of /2 for approximating our AMS with rational ones, closer
that /2. And the rational one is replaced with the corresponding very big society.
Proof is complete.
The proof of theorem 7.
We have three different types of coalitions here: {x ≥ y} with the median located at
0; {x ≤ z} with the location at 19/12; and {x = x} with the location inside the interval
(0, 19/12). Now, consider an arbitrary partition {Y 1 , . . . }, and suppose it is stable.
Lemma 0.5 There is at most one coalition with the location in 0. The same with the point
19/12.
Indeed, the union of two such coalitions unambigously increases the payoffs to all the
participants. This leads to the following assertion.
Lemma 0.6 The coalition with the location in 19/12 has the mass not less than 0.2.
Indeed, the union of the other coalitions has at least the same mass in the point 0, as in
the point 19/12. Hence, the difference between these masses should belong to the coalition
centered in 19/12.
Lemma 0.7 There is at most one coalition of the sort x = x. (Hence, any stable partition
should contain at most 3 coalitions.)
Indeed, if there are two or more symmetric coalitions, then in one of such coalitions y = y
we have y ≤ 0.2. Then, the right y would joint the coalition centered in 19/12, with obvious
increase of everyone involved in the union.
Now, we are ready to prove that no symmetric coalition enters any stable partition.
Indeed, consider the location of its center, r. Denote our symmetric coalition by x = x.
Lemma 0.8 x ∈ [12/31, 0.4].
The right endpoint is obvious; in order to establish the left endpoint inequality, write
down two conditions which mean that, respectively, left half and right half of this symmetric
coalition would not want to organize the “endpoint” coalition:
1/0.4 ≥ 1/(2x) + r;
1/o.6 ≥ 1/(2x) + 19/12 − r.
(29)
Summing them up we get x ≥ 12/31, after little rearrangements.
Now, use the second inequality of (29), and estimate x by 0.4 from above. We get r ≥ 7/6.
24
Lemma 0.9 The coalition centered in 0, and comprised of all the mass located in 0 and of
0.2 of the mass located in 19/12 and not involved in a symmetric coalition, improves upon
the partition under consideration.
To demonstrate this, first notice that, in view of x ≥ 12/31, maximum mass of the
coalition centered in 19/12 is 1 − 24/31 = 7/31; hence, they pay at least 31/7 > 4. Those in
the symmetric coalition that live in the 0-point pay minimum 1/0.8 + 7/6 > 2. At the same
time, in the proposed coalition those in the point 19/12 pay 1/0.6 + 19/12 < 4, while those
in the point 0 pay just 1/0.6 < 2. Q.E.D.
What we get now is that the only possible stable partition is that of two coalitions, 0.4
in the point 0 and 0.6 in the point 19/12. But such a prtition is blockaded by the symmetric
0.4 = 0.4 coalition with the center located in 29/24.
The proof is perfectly completed.
Now, let us return back to the positive (optimistic) results.
Redistribution policies
A model with partial compensation
Once a stable partition fails to exist, one may suggest that certain redistribution policies
could retain stability. We will be interested in only one kind of policies, where far-distant
agents are partly compensated, and payoffs are
vt (α, p, S) = −α|pt − p| − c̄,
(30)
where
|S|c̄ + α
X
|p − pt | = 1 +
t
X
|p − pt |.
(31)
t
Here α ∈ [0, 1] is a redistribution parameter: α = 1 means that compensation does not take
place at all, and α = 0 means full compensation (everyone pays the same amount). The
question is: for which α a stable partition exists?
Positive (and negative) results are obtained for the full-compensation case, when α = 0.
Full compensation: A Rawlsian case (i.e. “Socialism”)
Let us now turn to the case where compensation is available, and state a positive result in
the case of full compensation (α = 0).
Theorem 8 If group’s formation is followed by equally dividing the overall (common and
personalized) costs, then every society admits a stable partition. Moreover, generally this
stable partition is unique.
At the same time, one could not guarantee the existence of a consecutive partition in this
case. Here is the example of a society for which the only stable partition is not consecutive.
25
An example. Let (l1 , . . . , l6 ) = (1/3, 0, 0, 0, 0, 1/3). Then, a partition N = {1, 7} t
{2, 3, 4, 5, 6} is the only stable partition.
Now turn to the proof of theorem (8). It follows from the basic property of the class
of so-called “socialist” games, which is dual to the class of T-U-games (T-U-games means
transferable utility games).
We say that a game is a socialist game if every coalition S guarantees equal payoffs to its
members. More severe, we require that V (S) is defined by the formula
|S|
V (S) = (v(S), . . . , v(S)) − R+ .
(32)
Socialist games have a nonempty P-Core
Theorem 9 Every socialist game has a nonempty core in a coalition partition form. Moreover, there exists an element in the core which is individually stable. If all the 2n − 1 payoff
numbers, {v(S)}S⊂N,S6=∅ are different, than the core is singlevalued.
Proof: Consider any S1 ∈ ArgmaxS {v(S)} that is not contained in any other group with
this property. It will be the first group in a partition. Then, turn to a “factor-game” with
players’ set N \ S1 , and notice that it is a socialist game too. Then, use inductive argument.
We get a partition N = S1 t S2 t . . . .
I claim that this partition is stable and individually stable. Indeed, consider any S 0 prone
to secession. It could not cross S1 since v(S1 ) > v(S 0 ) by construction. Then, again use the
inductive argument. For individual stability, use the maximum requirement.
If all the v(S) are different, then any stable partition must include S1 , otherwise S1 would
be prone to secession. And so on — again by the inductive argument.
Proof is complete.
26
A geometric representation of the problem
the surface of the ocean
•
@
@•
@
@
•
x1
•
@
x2 x3
y2
6
@•
@
@
@
x
x5
4
@ x
@ •6
@@
Regular mounain system
{••}{•}
{•}{•}{•}
the surface of the ocean
•
•
(1, 1)
•
@
2/3
@
@
x1 x2 x3
{• • •}
@
•
@
@
x4 @
x5
•@
{•}{••}
2/3
@@
Irregular mounain system
A case N = 3
the surface of the ocean
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
x1 , x2
•
x3 , x4 , x5
•
A counter-example
27
y-1
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Policy brief
In many social, political and economic situations individuals form groups rather than operate
on their own. For example, communities are formed in order to share the costs of production
of local public goods among the residents, or workers join a labor union in order to attain a
better working contract. In these situations individuals utilize the increasing returns to scale
provided by large groups.
On the other hand, given the heterogeneity of agents’ characteristics and tastes, the
decision-making process of a large group may lead to outcomes quite undesirable for some of
its members. This observation supports the claim that benefits of size are not unlimited and,
on some occasions, a decentralized organization is superior to a large social structure. Thus,
instead of a grand coalition containing the entire population, we may observe the emergence
of group structures which consist of groups smaller than the grand coalition.
When we consider the political economy of country formation and stability, there are
arguments towards keeping a given country (i.e. the union of regions) together. Then the
major question is the nature of requrements that guarantee the stability of the entire country.
For example, is such a diversed country as Russia is stable? And how the diversity of its
citizens has been transformed in heterogeneity and disparity of its regions?
Second observation is that the reason for the existence of groups that contain more than
one agent but less than the entire society lies in the conflict between increasing returns to
29
scale, on one hand and heterogeneity of agents’ preferences, on the other. A natural and
important question is whether societies that are not efficient in operating as only one group,
could be partitioned into several groups under certain federal structure, in such a way that this
group structure would be stable, with respect to possible threats of individuals constituting
a society.
It is quite natural to expect that the existence of stable structures is crucially dependent
on the dimensionality of the policy space. In general, the severity of preferences’ divergence
raises when the number of policy dimensions increases, in which case the search for a stable
group structure becomes more challenging. We will have focus here on unidimensional policy
spaces (say, a geographical distance from the Center in our leading political interpretation),
which nevertheless is sufficient to demonstrate that there could be situations, and societies,
which does not admit for a stable group structure, needless to say about their stability as a
union.
It is important to examine whether there is the set of policy instruments available to the
government in order to mitigate these adverse effects of population heterogeneity, which will
at least retain the existence of a stable group structure. Since changing population political
preferences is not an easy option (and certainly neither is changing geographical facts), what
remains consists of fiscal measures.
It is shown that the central government may try to target a dissasatisfied or disadvantaged
group by transferring to it (directly or indirectly) a part of the tax revenue (e.g., Navarra
and Basque country in Spain, South of of Italy, Atalantic provinces in Canada, the Western
provinces in China). We characterized the particular form these stabilizing transfers could
take, establishing that in societies where costs whithin each group are divided equally among
members, a stable group structure always exists.
It would be interesting then to compare our theoretical conclusions with the actual transfer
schemes implemented in various countries and, especially, to contrast the transfer formulas
in Russia and other CIS countries with the redistribution mechanisms employed in Europe,
Asia, Australia and North and South America.
30