In this paper, we discuss the question of approximate stability in group partition (Alesina-Spolaore-like) models. Consider a world with a finite number of agents living along a line. The coordinate may represent a location in some geographical or virtual space. A subgroup of agents may build a facility at some point and enjoy benefits from it. Each facility costs some amount , and every agent pays for her transportation. These costs may be somehow redistributed. We seek for a stable division of the whole society into subgroups. Two main stability concepts are considered. Migrational stability means that no individual may decrease his cost by changing the group. Coalitional stability means that no new group may emerge and decrease the costs of all its members. We relax these concepts: an individual must substantively decrease her cost in order to break the structure, otherwise the change is not worth it. It was shown by Bogomolnaia et al that for some rules there are no stable configurations. We try to maximize the approximation rate for which the counterexamples still exist. Specifically, we study three cases: coalitional stability without redistribution, migrational stability with the central median rule and migrational stability with Rawlsian redistribution. We found the worlds with instability 6.2%, 2.5% and 9.6% respectively. In the first case we prove that this is the maximal rate for all bipolar worlds.
the groups must be disjoint. For instance, in the USA a citizen may be a registered supporter of only one political party. In many countries a person must have a unique address of residence, thus the municipalities or regions may be treated as non-intersecting communities. If a group of tourists choose where to go on a particular day, then everyone may choose only one excursion.
club good deteriorates when the number of users rises. This is why there could be no grand coalition, but some coalitional structure.
In this paper we study the question of horizontal diferentiation . The club good has some features and all agents have preferences over them. The cost of organizing a club and providing the club good is constant and does not depend on the number of users. Thus, there are two opposite forces. On the one hand, large groups are better since they provide economy of scale. Each member pays less share of the total cost. On the other hand, small (and homogeneous) groups are better since they can tune the features of the good precisely under members’ preferences. The main question is whether these forces can always equilibrate each other. And how do details of the model influence the answer?
vidual wishes to change her community unilaterally in order to reduce her costs. Coalitional stability suggests that no group of agents would like to secede from the club system, organize a new club and thus reduce the costs of all members. The distribution of agents may be described in diferent ways.
may be of three types. It may be discrete, when a finite set of agents live at some specific points.
atomic, when a continuum of agents live at a finite number of points. The costs may be redistributed between the agents in some way: usually, less satisfied agents may be subsidized or pay a less share of the fixed cost. The combinations of these specifications lead to a family of models. In some of them an existence theorem may be proven. In some others an example without a stable partition may be constructed.
This study considers the frameworks without an equlibrium. We relax the notions of stability: suppose that a threatening individual or group must not only get better after the change, but decrease their costs by a considerable amount. Of course, if this amount is large enough, the stable configuration will always exist. Our research question is to find the maximal size of this amount when the configuration is still unstable. We consider three uni-dimensional frameworks: • Coalitional stability without cost redistribution; • Migrational stability without cost redistribution; • Migrational stability with egalitarian redistribution.
group partition.
The discussion about local public goods started in the 1950s. Samuelson [1] presented a model
that demonstrated ineficiency of decentralized public good provision. The main idea was that the
freerider problem is inevitable. If an agent buys a public good by himself, then he cares only about his
own benefits, not about the others. This is why it will be always underfinanced. Tiebout [
Tiebout’s intuition. In particular, the researchers tried to justify the existence and eficiency of the suggested equilibrium. The early endeavours included the papers by Westhof [3] and Bewley [4]. A thorough analysis was done by Greenberg and Weber [5] who introduced the notions of migrational and coailitional stability and justified both of them by quotes from the Tiebout’s paper. All these eforts were done in the framework of vertical diferentiation: the good was homogeneous, but the agents difered in willingness to pay for it and were self-selecting in jurisdictions with diferent levels of public good.
and Spolaore [6]. They introduced a one-dimensional parameter that describes both the local public good and the agents’ preferences. They interpreted the model in the spirit of political science: the parameter describes a geographical location, the coalitions are nations and the facilities are their capitals. Each citizen pays taxes to finance the capital and needs to visit it sometimes. The paper also considered the concepts of migrational and coalitional stability.
ous papers. In particular, the issues of nonuniform density or discrete configuration, various cost redistribution rules and multiple dimensions were studied. In some frameworks it was shown that an equilibrium does always exist, in other cases very involved counterexamples were constructed.
does always exist. Weber et al [8] present an example where no migrational equilibrium may exist under the egalitarian redistribution rule. Bogomolnaia et al [9], [10] and Savvateev [11] studied discrete models with a finite number of agents and presented several examples in diferent frameworks where no equilibrium exists. On the other hand, Musatov et al [12], Savvateev et al [13] and Marakulin [14] presented rather general existence theorems in the continuous framework.
sented, including the definitions of approximate stability. In Sect. 3 we extensively analyze the case of coalitional stability of a bipolar world. In Sect. 4 and Sect. 5 we present a preliminary analysis of migrational stability in two settings: egalitarian redistribution and the central median rule. Finally,
This model is similar to the models in [9] and [8]. A continuous model may be found, for instance, in [12]. In our model we have an agent set = {1, … , }. Every agents “lives” at location , where 1 ≤ ⋯ ≤ . A community is a nonempty subset of . It may build a facility at some point . We consider two possible rules of determining : • The median rule. The point is a median of , i.e., the sets { ∣ ≤ } and { ∣ ≥ } must have equal sizes. If contains an even number of agents and there is a segment of medians, then any median will fit.
middle point of this segment is chosen.
• The central median rule. The same as before, but if there is a segment of medians, than the
median location minimizes the total cost. Secondly, it wins over any other option by majority voting. the set of suitable medians under the specified rule. It will be either a segment or a singleton. The central median rule, moreover, always returns a unique point. In the sequel we denote by med( ) two rules of distributing the total cost:
the cost of maintenance does not depend neither on the number of users, nor on the location of the facility. Denote this cost by . Apart of this cost, all members should be transported to the location. The transportation cost of an agent at point to the facility at point equals ⋅ | − |. We consider • The private cost rule. Every agent pays for himself, so the total cost of agent within coalition with facility at equals where | | is the cardinality of . • The Rawlsian (egalitarian) rule. The total cost is redistributed such that every agent pays an equal share. Here we have the following cost function: 1 | | | | (, , ) =
+ ⋅ | − |, (, , ) = ( + ∑ | − |).
∈ notation. 1, … ,
. We analyze the following notions of stability: Note that under the central median rule the facility is uniquely located, so we can omit in the Suppose that the whole society is divided into communities 1, … , with facilities at medians • Migrational (or Nash) stability. No individual wishes to change his community. Formally, for every agent belonging to community
and any other community it holds that (, ) ≤ (, ∪ { }). defined, or can be chosen arbitrarily among possible variants.
Note that agent anticipates that she would change the community she joins and the distribution of costs within it. The existing members of may become better of or worse of, it does not matter. The median of is predetermined, and the median of ∪ { } is either uniquely communities
and all agents ∈ ∩ it holds that • Coalitional (or core) stability. No group of agents wish to establish a new community such that all members become better of. Formally, there is no set with a median such that for all (, ,
) < (, , ), where is the median of .
Note that migrational and coalitional notions of stability are incomparable. Namely, if an individual changes the jurisdiction, then the old members of the new jurisdiction may become worse of due to the change of the median. Thus, they would not like to admit the new member. Thus, this change would be a threat for migrational stability but not for coalitional stability. Vice versa, it may occur that no individual move may decrease the cost, but a collective change may decrease the costs of all participants.
Suppose that changing the jurisdiction or establishing a new one is costly: for instance, it takes some efort for negotiation. In this case a structure would be unstable only if some change may substantively decrease the cost of all involved agents. We consider two measures of instability: • Absolute, or additive, instability. A threat is credible only if all agents performing the move order to compare diferent unstable societies with each other.
decrease their costs by some absolute value . Of course, the scale should be fixed somehow in • Relative, or multiplicative, instability. A threat is credible only if all agents performing the move decrease their costs by some fraction . This definition does not depend on the scale.
(⋅)). Now define two measures of absolute instability of a group structure: Definition 1. 1, … , , respectively. Denote by ( ) the community that contains agent and by ( ) the respective median. Denote by (, ) the cost of in , i.e. (, ( ), ( )). (Where (⋅) denotes either (⋅) or Consider a society structure , i.e. a partition = 1 ⊔ ⋯ ⊔ with facilities at points • The absolute migrational instability of the structure is the value • The absolute coalitional instability of the structure is the value Δ ( ) = max( (, ) − ∈ min =1,…, ∈med( ∪{ })
(, ∪ { }, )); Δ ( ) = ⊂, ≠∅, ∈med( ) m∈in( (, ) − (, , max )).
Definition 2. relative instability:
Keep all the notation from the previous definition. Define the following measures of • The relative migrational instability of the structure is the value • The relative coalitional instability of the structure is the value Δ ( ) = max ∈ (, ) − min
(, ∪ { }, ) =1,…, ∈med( ∪{ }) (,
) Δ ( ) = ⊂, ≠∅, ∈med( ) ∈ max min (, ) − (, , (, ) ) We call the structure -stable in some sense, if its instability in this sense is not greater then .
Note that instead of taking the maximin of (, )− (, , ) we may take the minimax of (,(,, ) ) . Later we refer to this value as to the new-to-old ratio. (, )
Problem 1. Find the least upper bounds on Δ( 1, … , ) in diferent variations. Which worlds are the least stable under which rules? And which rules produce the least stable worlds?
Partial solutions to this problem include computing the values of Δ( 1, … , ) for particular worlds, ifnding parameters with large instability, establishing some upper bounds and exact solution for some families of worlds (for instance, parametric).
This question was previously studied by the last two authors in [16]. That paper was focused on bipolar worlds, where the agents live in just two points. It was found out that the least absolutely stable configuration is the one with 3 agents at one point and 4 agents at another point. For cost parameters = = 1 the distance is 254 ≈ 0.208. The analysis of relative instability was done numerically. It was shown that the maximal possible relative instability is somewhere between 0.061 and 0.063. In this paper we provide an exact solution and prove that the maximal instability in a bipolar world is (asymptotically) √65 − 8 ≈ 0.0622. Our conjecture is that this bound holds for any world.
Consider a bipolar world with agents at point 0 and agents at point . W.l.o.g. we postulate that = = 1 and ≤ . It is also instrumental to think that the weight of a single agent is 1 , so the total weight of the left agents is 1 and the total weight of the right agents is = ≥ 1. We co nsider several specific structures.
Definition 3.
• Union is the structure with one grand coalition and the facility at point . • Federation is the structure with two coalitions: the left coalition with all agents at 0 and the right coalition with all agents at . • Mixed structure is a structure where one coalition includes equal number of agents from both poles and has a facility somewhere in between (we call such a coalition ambiguous). Among those structures we consider the MaxAmbigs, where one coalition includes a unit weight of agents from each pole and the other contains agents of weight − 1 from the right. The former coalition will be also referred to as the MaxAmbig. These atructures are parameterized by the facility position ∈ [0, ]. • Pseudofederation is a structure where one coalition includes all agents from one pole and some agents from the other, and the second coalition is formed by the remaining agents of the other pole. We distinguish right pseudofederations where the first coalition includes all agents from the right and left pseudofederations where it includes all agents from the left. All pseudofederations may be parameterized by a single parameter ∈ (−1, ). If < 0, then the dispersed coalition consists of | | agents from the left and agents from the right. If > 0, then it consists of mass-1 agents from the left and agents from the right. In case = 0 the structure turns into the Federation. In case ∈ {−1, } the structure becomes the Union.
tions. We claim that the most stable configuration may be of one of three types: Union, MaxAmbig or Pseudofederation( ) where ∈ [0, − 1]. In the sequel we prove this assertion and find the world where the relative instability of these structures is maximal. In fact, we proceed in the opposite way.
It was shown by Savvateev [17] that in an unstable structure it must hold that > 0.5, < 1.62, < + 1 and < 1 (the exact bounds are tighter, but these are enough for us). In the sequel we take these bounds as granted. Now consider all credible threats for each structure and calculate the new-to-old ratios for them.
Lemma 1. If the Union is unstable, then the set of all left agents wins the most from the secession.
agents may want to secede. But the cost is the least possible in the coalition of all left agents. So this set has the maximal willingness to secede.
Note that the new-to-old ratio is 1/ ( 1+ + ).
1 Lemma 2. Let Pseudofederation( ), where ∈ [0, − 1], be unstable. If the Union is also unstable, then the group that wins the most from the secession is either the MaxAmbig group or the set of all right agents. Proof. Recall that > 0 means that there is a group with all agents from the left and some agents from the right. Suppose that some group with the center on the left wins from the secession. Since the left agents are better of, the group must be larger then the initial left group. If all left agents join the group, they are even better of. Some agents from the right must also be in that group and be better of. Thus, the right agents will join the seceding group until the center is still at 0. So, the limit case is the MaxAmbig group.
group. Thus it must include agents from the right group and the MaxAmbig group is a greater threat.
expanded to the whole right pole. If there were some agents from the left, then the grand coalition will be a greater threat again. But we assume that the Union is unstable, so the set of all left agents reduces the costs after seceding from the Union. By transitivity, it means that the left pole becomes better of when seceding from the Pseudofederation, which is a contradiction.
possible threats and all agents participating in a threat. Consider two possibilities: • MaxAmbig threat with median . Here the agents from the left compare the new cost 12 + to the tohldatcwositth1+1o u.rTphaeraamgeentetsrsfrtohmeltahtteerriigshgtrceoamteprathrean21 +th e−f ortmoeeri,tshoerw e−1shooruld1 maximizes the ratios 1+ + . It can be shown Plugging this back to the ratios, we get the value (1+ )(1+ )( − ) . 1+ 211++1 and 21 +−1 − . Since the first ratio grows with and the second one decreases, the maximum of them is the largest when they are equal. Solving the equation, we get = 21 + − (1+ )(1+ ) . 1+ • The right pole threat. Here the agents from the undivided coalition compare 1 to 1 and the new-to-old ratio is max 1 − , (1+ + )
1+ { }
.
agents from the right part of the dispersed coalition compare 1 to 1
1+ + . Th us t h−e maximal Lemma 3. The minimax of the new-to-old ratio in a Pseudofederation( ) is achieved when The minimum is reached simultaneously for the right pole threat and for the MaxAmbig threat = (1+ ) 1− . with = 21 + − (1+ )(1+ ) . The corresponding new-to-old ratio is 1 − 21(−1+ ) 1+ .
tion. It can be shown that MaxAmbig is the most credible threat for close to 0 and for close to − 1 and the right pole is the most credible for the intermediate values of . Thus, the value of the most credible threat has two local peaks. It can be checked that the left one is higher and corresponds to the claimed values of and .
Lemma 4. Let MaxAmbig( ) be unstable. If the Union and the Federation are also unstable, then the group that wins the most from secession is either the right pole, or the left pole united with the part of the right pole not in the ambiguous coalition.
Proof. It is clear that an ambiguous coalition cannot be a credible threat. Indeed, it cannot be larger than MaxAmbig, and the facility becomes further from one half of the agents. The Union cannot be credible if it is unstable, analogously to lemma 2. If a credible threat has the facility on the right, then it must contain agents from both the ambiguous coalition and the remaining right coalition. Thus, the whole right pole must be more credible. Similarly, if it has the facility on the left, it must contain some agents from the right. Otherwise, the Federation would be stable. Thus, the coalition containing the left part of the ambiguous coalition and the whole right coalition must be the most credible.
Lemma 5. The maximal new-to-old ratio is minimal in a MaxAmbig( ) when = 2 . Proof. If the right pole is a credible threat, then the agents from the right part of the ambiguous coalition compare 1 to 12 + − and the agents from the right coalition compare 1 to 1 . Similarly, if the left pole plus the right coalition is a credible threat, then the agents from the left c−o1mpare 1 to 21 + , and the agents from the right compare 1 + to 1 . Note that −1 1 2 + − <
+ < 1 2 1
+ 2 1 <
1 − 1 where the second inequality arises from < 1 and the last one from < 2. Thus, in the first case the right half of the ambiguous coalition is relatively less satisfied than the right coalition. To determine the instability, we should now compare three ratios: 1 / ( 21 + − ), 1 / ( 21 + ) and ( 1 + ) / −11 . It is fr/o(m21 (+1 2+) >)( ( −1 +1) <) 1/,−1w1 .hIinchdefeodll,oiwtissferqoumiv a<lent to 2 > (1 + )(1 + )( − 1). Since < 1
1 and < + 1. clear that the maximum of the first two values is minimal when = 2 . Then it can be shown that 1, it follows
Theorem 1. The maximal new-to-old ratio is minimal among Union, Pseudofederation( ) and MaxAmbig( ) for = √685−3 ≈ 0.6328 and = 1+√213/5 ≈ 1.3062. The relative instability is = √65 − 8 ≈ 0.06226 and the parameter of Pseudofederation is = 1(1−+ ) ≈ 0.0813.
Proof sketch. We should minimize the maximum of (, ) = 1/ ( 1+1 + ), (, ) = 1 − 21(−1+ ) and . It can be shown that is decreasing by both variables, is increasing by both (, ) = 1 / ( 21 + 2 ) and is decreasing by and increasing by . Thus, if the maximum is reached by only one function, then it can be decreased by changing some variable. If it is reached by and simultaneously, then it can be decreased by rising . If it is reached by and , then it can be decreased by lowering . And if it is reached by and , then it equals 2( +1) and thus can be decreased by lowering . Thus, the maximum is minimal only if all three maxima3 n+d1s are equal. Solving the equations, one can obtain the claimed values.
Theorem 2. In any bipolar world, the most unstable configuration consists of at most three jurisdictions, of which: • At most one locates the facility at 0, at most one locates it at and at most one locates it somewhere else; • Among coalitions with the facilities at 0 and at most one contains agents from the other pole.
in any other configuration some coalition wishes to break out with willingness greater than √65 − 8.
Theorem 3. In any bipolar world, the most unstable configuration is either the Union, or a Pseudofederation( ) for ∈ [0, − 1], or a MaxAmbig( ).
biguous but not maximal; a Pseudofederation( ) with ∉ [0, −1], or a configuration with 3 coalitions.
• If one coalition is ambiguous but not maximal, then some threat is more credible than the most credible one in the MaxAmbig with the same median. Thus, MaxAmbig is more stable. • The relative instability of Pseudofederation( ) increases when > − 1. If ∈ [0, 1], then the relative instability of Pseudofederation(− ) is higher than the one of Pseudofederation( ).
Thus, the minimal instability of a Pseudofederation is reached for ∈ [0, − 1]. • In the case of three coalitions several subcases are considered. If the ambiguous coalition is rather small, it will be much better of when joining some other coalition. If it is large, then it can be shown that MaxAmbig is more stable. For intermediate sizes it can be directly shown that the instability is larger than √65 − 8. 4. Approximate migrational stability under egalitarian redistribution
The example is the following: there are 6 agents living at 3 points. Agents 1 lives at point 0, agents 2 and 3 live at point and agents 4, 5 and 6 live at point + . In [8] the parameters are the following: = 1.9, = 2.2. The idea behind the result is the following. The grand coalition is not stable because agent 1 wishes to secede and not to subsidize the others’ transportation. Then agent 2 would also like to secede for the same reason (the configuration is ({1}, {2}, {3, 4, 5, 6})). Then agent 3 would like to join agent 2: ({1}, {2, 3}, {4, 5, 6}). Then agent 1 would also like to join: ({1, 2, 3}, {4, 5, 6}) (Note that agents 2 and 3 may become worse of, but in the migrational setting it does not matter.) But now agent 3 would like not to subsidize agent 1’s transportation and would better join the right coalition: ({1, 2}, {3, 4, 5, 6}). Now agent 1 would like to be alone, and we return to ({1}, {2}, {3, 4, 5, 6}).
presents the initial configuration, the final configuration, the agent who wishes to move, the values she compare (the initial cost reduced by and the final cost) and the resulted bound on .
Old config. New config. Agent Ineq. on util. Ineq. on {1, 2, 3, 4, 5, 6} {1}, {2, 3, 4, 5, 6} 1 {1}, {2, 3, 4, 5, 6} {1}, {2}, {3, 4, 5, 6} 2 {1}, {2}, {3, 4, 5, 6} {1}, {2, 3}, {4, 5, 6} 3 +411 −(1 −> ) +>1 21 {1}, {2, 3}, {4, 5, 6} {1, 2, 3}, {4, 5, 6} 1 3
{1, 2, 3}, {4, 5, 6} {1, 2}, {3, 4, 5, 6} 3 +31 (1 − ) > 1+4 The analysis of the inequalities lead to the following re s+2u1l(t1: − ) > 1 {1, 2}, {3, 4, 5, 6} {1}, {2}, {3, 4, 5, 6} 1 +36 +1 (1 − ) > 1 2 5+1 (1 − ) > 1 < ++33 +−15 < 22 −+41 < −+11 < 2−3 < 4 4+1+−43 < −+11 Theorem 4. In every world of the described type, the relative instability is at most ≈ 0.0963 (the exact value is the real root of the equation 8 3 − 24 2 + 23 − 2 = 0). The corresponding values of the distances are = 2 − 3 ≈ 1.711 and = 24−+2 ≈ 2.2665. Every value of instability below this threshold is achievable. Proof idea. We should find the maximal for which all inequalities are satisfiable. It may be shown that the binding inequalities are < 22 −+41 , < 2− and < 4 +1−3 . Solving the system of corresponding equalities yields the claimed values. 3 4 +4 5. Approximate migrational stability under the central median rule
bility under no-redistribution and central median rules. Recall that the latter means that if a group has an even number of members, then the facility is located at the middle of the segment of medians.
The considered example is the following: agent 1 lives at point 0, agents 2 and 3 live at point , agent 4 lives at point + and agent 5 lives at point + + . In [10] the values are = 3203 , = 51 and = 85 . The idea behind the instability result is the following: agents 2, 3 and 4 are always in the same coalition. But agent 5 prefers to be in the main coalition if agent 1 does not belong to it, and agent 1 prefers to be in the main coalition if agent 5 belongs to it. Thus we have the following cycle of configurations:
({1}, {2, 3, 4}, {5}) → ({1}, {2, 3, 4, 5}) → ({1, 2, 3, 4, 5}) → ({1, 2, 3, 4}, {5}) → ({1}, {2, 3, 4}, {5}).
{1}, {2, 3, 4}, {5} {1}, {2, 3, 4, 5} {1, 2, 3, 4, 5} {1, 2, 3, 4}, {5}
{1}, {2, 3, 4, 5} {1, 2, 3, 4, 5} {1, 2, 3, 4}, {5} {1}, {2, 3, 4}, {5}
5 1 5 1
= 14 and = 8407 would fit. Every value of instability below this threshold is achievable. Theorem 5. In every world of the described type, the relative instability is at most = 41−√1601 ≈ 0.0247. 40 √ The corresponding values of the distances are = 1601−9 ≈ 0.775 and a range of and : for instance, 40 Proof idea. It turns out that only the conditions on are binding. For the claimed value of the values are equal. It can be checked that for the specified values of and both other inequalities are problem hard for some computational complexity class?
works. We believe that the considered examples are either the most unstable worlds or close to them.
other possible approach is the algorithmic one. Suppose that an algorithm gets the total description of a world and a number . Can we eficiently compute whether this world is -stable? Or is this
how typical are our examples? It seems that a bipolar world may occur in diferent situations, like twin cities or cat and dog lovers. And the situation that one pole is about 30% larger than the other is also realistic. For instance, it is how much Minneapolis is greater by population than Saint Paul. Secondly, what are the real values of ? It seems that in long-existing structures it can be much larger than several percents, so the participants may want to make a new coalition but then see the cost and “all of the sudden we’re scared to change a thing”. A study of such real cases would be an interesting research topic.
[1] P. A. Samuelson, The pure theory of public expenditure, The Review of Economics and Statistics [3] F. Westhof, Existence of equilibria in economies with a local public good, Journal of Economic [4] T. F. Bewley, A critique of Tiebout’s theory of local public expenditures, Econometrica: Journal of the Econometric Society (1981) 713–740. [5] J. Greenberg, S. Weber, Strong Tiebout equilibrium under restricted preferences domain, Journal of Economic Theory 38 (1986) 101–117. [6] A. Alesina, E. Spolaore, On the number and size of nations, The Quarterly Journal of Economics 112 (1997) 1027–1056. [7] O. Haimanko, M. Le Breton, S. Weber, Voluntary formation of communities for the provision of public projects, Journal of Economic Theory 115 (2004) 1–34. [8] S. Weber, A. Savvateev, M. Le Breton, A. Bogomolnaia, The Egalitarian sharing rule in provision of public goods, Economics Bulletin 8 (2005) 1–5. URL: https://ideas.repec.org/a/ebl/ecbull/ eb-05h40002.html. [9] A. Bogomolnaia, M. Le Breton, A. Savvateev, S. Weber, Stability under unanimous consent, free mobility and core, International Journal of Game Theory 35 (2007) 185–204. [10] A. Bogomolnaia, M. Le Breton, A. Savvateev, S. Weber, Stability of jurisdiction structures under the equal share and median rules, Economic Theory 34 (2008) 525–543. [11] A. Savvateev, Uni-dimensional models of coalition formation: non-existence of stable partitions,
Moscow Journal of Combinatorics and Number Theory 2 (2012) 49–62. [12] D. V. Musatov, A. V. Savvateev, S. Weber, Gale–Nikaido–Debreu and Milgrom–Shannon: Communal interactions with endogenous community structures, Journal of Economic Theory 166 (2016) 282–303. [13] A. Savvateev, C. Sorokin, S. Weber, Multidimensional free-mobility equilibrium: Tiebout revisited, 2018. URL: https://arxiv.org/abs/1805.11871. [14] V. Marakulin, Spatial equilibrium in a multidimensional space: an immigration-consistent division into countries centered at barycenter, in: International Conference on Mathematical Optimization Theory and Operations Research, Springer, 2019, pp. 651–672. [15] J. Drèze, M. Le Breton, A. Savvateev, S. Weber, “Almost" subsidy-free spatial pricing in a multidimensional setting, Journal of Economic Theory 143 (2008) 275–291. [16] A. Golman, D. Musatov, Approximate coalitional equilibria in the bipolar world, in: Y. Evtushenko, M. Jaćimović, M. Khachay, Y. Kochetov, V. Malkova, M. Posypkin (Eds.), Optimization and Applications. OPTIMA 2018, volume 974, Springer, 2019, pp. 512–526. [17] A. Savvateev, An analysis of coalitional stability in a bipolar world, Journal of New Economic Association 17 (2013) 10–44. In Russian.