Institutions of Proportionality andthe Evolution of Groups∗

Macartan HumphreysAugust 2007

Abstract

I examine the evolution of group structures under discretetime Markov processes for settings in which institutions are de-signed to benefit groups in proportion to their size. I study threeforms of proportionality: proportionality in the division of pri-vate goods, proportionality in decision making power over pub-lic policies, and proportionate jurisdictional power over policiesthat affect clubs. These institutions are studied for contextsvarying along three dimensions: first as a function of within groupdecision making processes; second as a function of limitations onthe acceptable number of groups; and third as a function of therule governing the entry into groups. For each case I examine theset of stable states; since these are often multiple I use stochasticstability as a refinement. Outcomes, I find, are very sensitive toboth institutions and context; broadly however, private and clubproportionality lead to more fragmented structures with morehomogenous membership while public proportionality leads togreater assimilation and greater within group heterogeneity.

DRAFT: NOT FOR CITATION

∗Prepared for APSA 2007. My thanks to Kanchan Chandra and Eric Dickson foruseful discussions on an earlier draft of this paper.

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1 Introduction

I examine the evolution of group demographies under different institu-tional conditions. The institutional arrangements I examine–institutionsof proportionality–are ones that are introduced in deeply divided so-cieties as a response to those divisions. Whereas in such contexts pastscholarship has focussed on the implications of these institutions forinter-group behavior conditional on a group structure (Lijphart 1977,Horowitz 1991), I consider here the implications of these institutionsfor the group structure itself. Understanding such processes is relevantto addressing questions of the form: ‘are demographies likely to be-come more or less ethnically fragmented as a result of consociationalinstitutions?’ or ‘are groups likely to become more or less internallyhomogenous?’ or ‘when are federal systems more likely to consolidateor dissolve?’ The answer depends entirely on the manner in which pro-portionality is introduced and it depends on the context into which itis introduced. Indeed it is possible to show that the set of equilibriumstructures are disjoint under different institutions.I focus on three forms of proportionality: proportionality in the di-

vision of private goods, proportionality in decision making power overpolicies affecting everyone in society and proportionate jurisdictionalpower over policies that may affect different subgroups.For each of these I consider tendencies for changes in membership

of groups. There are many ways to think about the dynamics of groupformation. Models can emphasize leader effects or “mass” effects; theycan focus on historical features, informational effects or the psychologicaleffects that may obtain when individuals, as members of officially identi-fied groups, are treated in a similar manner. Here I focus on one simpleaspect of institutions of proportionality: the manner in which they tiethe inputs that an individual can have over a political process and thebenefits that may accrue to an individual from such processes to charac-teristics of the group in which she claims membership. In focussing onthe relation between individuals and groups I also focus on individualincentives as a vehicle for demographic change, examining conditionswhen individuals have incentives to change their group membership andthe incentives that groups have to accept such a change. While thisapproach comes with analytic advantages, it prevents us from studyingstrategic attempts at social engineering by leaders or by groups.In the analysis I examine different ways in which the context of deci-

sion making varies: first I distinguish between societies in which decisionsthat are beyond the reach of political institutions are made in a hier-archical manner and those in which these decisions are made in a moreegalitarian manner.

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Next I distinguish between two ways in which demographies mayevolve, conducting separate analysis for “fluid” situations in which iden-tity change is relatively easy, requiring only the desire to shift betweengroups on the part of a given individual, and situations in which iden-tities are more “sticky” in the sense that for an individual to shift fromone group to another requires not simply a desire to do so on the partof the individual but a willingness on the part of the receiving group torecognize the individual as one of their own (see also Greenberg (1978,1979)).1

The dynamic processes studied are similar to those examined byLaver and Benoit (2003) who consider the evolution of party systemsas individuals leave parties to join others in order to increase their shareof a legislative pie. Like Laver and Benoit the types of changes I lookfor in the evolution of groups all entail single movements by individualsfrom one group or another, and I consider a demography “stable” whenno individual is willing and able to change group membership.The institutional settings I examine are however very different to

those studied by Laver and Benoit. Whereas Laver and Benoit studymajoritarian settings in which the benefits of office are allocated on thebasis of majority status (in fact according to Shapley values) I examineconsociational settings in which institutions are designed to minimizebetween group competition. And whereas Laver and Benoit assumesymmetry across all actors, much of the important variation in modelsconsidered here is driven by variation in attributes between actors, eithertheir strength of their policy preferences. In this sense while the evo-lutionary dynamics studied in Laver and Benoit are driven by betweengroup differences, the dynamics here are driven more by within groupdifferences. In addition the focus of this study is on variation acrossinstitutional settings and thus examination is conducted separately fordifferent institutional rules as well as for different types of transitiondynamics.Although the range of possible institutions and contexts I examine

is narrow, it is rich enough to generate a surprising degree of variationin the ways that group demographic structures may evolve. It turns outthat it is not possible to make simple claims of the form: proportionalsystems are likely to lead to greater (or less) fragmentation. However,

1Other conditions for group change exist and remain to be examined–in sometreatments for example trasnsitions may occur or be prevented contrary to the de-sires of the individual and the receiving group, either because an individual can beejected from a group (Dréze and Greenberg (1980)), abducted into a group, or forciblyretained in a given group; or, from the receiving groups perspetive, if a group is pre-vented from or required to take on members.

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the variation that obtains is highly structured and gives rich insight intothe effects of different types of systems in different contexts. The results,summarized in the concluding section, suggest that in systems with pro-portionality over private goods, all demographies are stable in egalitariancontexts, in hierarchical contexts however, demographies tends towardsgreater fragmentation when identities are sticky and yield no equilibriumstates when identities are fluid. In contrast, under “public proportional-ity,” egalitarian systems tend towards a reduction in fragmentation withindividuals attempting to join groups that are both large and internallydiverse. This tendency towards consolidation is reduced in more hierar-chical contexts where demographies may consist of a large core and a setof smaller groups. Under club proportionality, the role of hierarchy isless pronounced; in all cases, when identities are fluid there is a tendencytowards greater fragmentation with groups containing like types; withless fluidity all demographies are stable.

2 Preliminaries

2.1 Dynamic Processes and EquilibriumOur interest is in examining the evolution of demographic structuresover time. For each point in time I assume that each individual inN = {1, 2, ...n} is a member of some group, Gj. The collection of groups,Γ, is a partitioning of N in the sense that everybody is in some group(∪j=1,2,...mGj = N) and no person is in two groups ( Gj ∩Gk = ∅ for j6= k). The size of group j, nj = |Gj| is given by the number of players inthe group and the relative size of group j is denoted λj, where λj = nj/n.States. A state is a partitioning of N defined uniquely up to the

reordering of cells and of individuals within cells (thus {{1, 2}, {3, 4}} isconsidered the same state as {{4, 3}, {1, 2}}). A state, Γ, is consideredadmissible if its cardinality, |Γ|, does not exceed some exogenous insti-tutional limit on the number of possible group, m.2 The universe of allstates G for a given process is the set of all admissible states.For a population of size n and a limit ofm ≤ n admissible groups, the

number of admissible states is given by |G|=Pm

k=1 S(n, k), where S(n, k)is the Stirling number of the second kind for n, k. For m = n, |G| isthe nth Bell number (Rota 1964), a number which grows very rapidly inn (indeed Bell referred to them as “exponential numbers”). The set ofpartitions for n = 4 is 15. For n = 10 the Bell number is 115, 975.

2Other plausible criteria for admissability could use information on the numberof individuals in each group.

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Labels are useful for some particular states or types of state. Forany player i and partition Γ, we let Γ(i) denote the partition in Γ thatcontains i. We say that a state is atomistic if there is only one agentin each partition. We say that a state is the “grand coalition” if allagents are in the same partition. We say that a pair of groups, A, B,is an ordered pair with respect to α and write A >α (<α)B if for alla ∈ A and b ∈ B, αa ≥ (≤)αb; partitioning Γ is a complete orderedpartitioning if all pairs are ordered.Markov processes. To the finite state space G we associate a

discrete time Markov process, summarized by a matrix of time homoge-nous transition probabilities, P . An element of P gives the probabilityof transitioning from one element of G to another. Conditional on beingin state Γk in period t, PΓkΓh denotes the probability of being in state hin period t + 1, (P 2)ΓkΓh denotes the probability of being in state h inperiod t + 2 and so on. Given this structure we review some standardterminology (see Young 1998). We say that state Γ0 is accessible fromΓ if for some t, (P t)ΓΓ0 > 0; states Γ and Γ0 communicate if each isaccessible from the other. A set of states is a communication classif each pair communicates and is a recurrent class if it is a communi-cation class from which no state outside the class is accessible. A stateis absorbing if it is a singleton recurrent class. Hence, a state is ab-sorbing if from that state it is only possible to move to that state. Aprocess is irreducible if there is only one recurrent class, consisting ofthe entire state space. A distribution, µ, is a stationary distributionif µP = µ.The asymptotic frequency distribution of a process be-ginning at µ0 is µ0P

∞. Finally, P is ergodic if µ0P∞ does not dependon µ0; that is the starting state is not relevant for determining the longrun outcome.Equilibrium states. We say that a state Γ is an equilibrium of

the process P if it is absorbing, that is if PΓΓ = 1.

Example 1 Two examples of Markov processes are given in (1).

P =

⎡⎣1 0 0120 12

0 1212

⎤⎦ , P 2=⎡⎣1 0 0121414

141412

⎤⎦ , P∞=⎡⎣1 0 01 0 01 0 0

⎤⎦ , Q =⎡⎣1 0 00 1

434

0 3414

⎤⎦ , Q∞=⎡⎣1 0 00 .25 .250 .25 .25

⎤⎦(1)

Matrix P , describes a process in which from state 1 it is only possibleto move to state 1. State 1 is an absorbing state; it is stable in thesense that once a process reaches such a state it does not leave it. Fromstate 2 it is possible to move to state 1 directly but it is also possible tomove to state 3. From state 3 it is not possible to move to state 1 directly

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although it is possible to move indirectly through state 2. PP = P 2 showsthe probability of being in each state after two periods, confirming thatit is possible to move in two steps from 3 to 1 but also from 2 back to 2.The asymptotic distribution of states for this process is given by P∞; inthis case every row is identical, indicating that the process is ergodic: theultimate distribution does not depend on where you start. For this processwherever you start you will always reach state 1 eventually and stay therefrom there on out. Matrix Q is a more difficult matrix. Again state 1 isa stable state, but in this case states 2 and 3 do not communicate with1; they form a recurrent class; once inside this class the process cyclesamong states in the class. As can be seen from Q∞ the process is notergodic.

Stochastically stable states. In some instances it is possible fora process to produce many equilibria. In such cases we seek refinementsthat allow us to select among equilibria on the basis of some notion oftheir likelihood. The notion we employ here is that of stochastic stability;the stochastic stability criterion requires that we expect states to obtainin the long run in the presence of small errors by players. Formally,given a process P let P ε denote a regular perturbed Markov processif (i) P ε is irreducible for small ε, (ii) lim

ε→0P ε = P and (iii) P ε

ΓΓ0 > 0 for

ε > 0 → 0 < limε→0

P εΓΓ0

εr(Γ,Γ0) < ∞ for some scalar r(Γ,Γ0) which we term

the resistance associated with the transition Γ,Γ0. Let µε denote the(unique) stationary distribution associated with P ε. We say that a stateΓ is stochastically stable if lim

ε→0µεΓ > 0 (Young 1998).

Stage games. The transition matrices studied in this paper aregenerated through a simple game played in each period. In each periodone individual and one group (possibly the individual’s own group, andpossibly an empty group) is randomly selected. The individual thendecides whether she wants to join the selected group or not (we confineattention to pure strategies). In fluid environments a decision to joinis sufficient to generate a transition. In sticky environments the movehas to be ratified by any non-empty receiving group using a sequentialup-down vote given some exogenous ordering of members. A transitionis attractive if the first condition is satisfied, acceptable if the secondcondition is satisfied, and potential if both are satisfied. An equilibriumcan therefore also be defined as a state for which no transitions arepotential. For acceptability we restrict attention here to the case whereunanimity is required for accepting a member although for two of thethree cases considered the main results are consistent with any rule thatreflects the preferences of any subset of members.

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Once a decision is made to change groups (or not), payoffs are allo-cated according to the demographic structure and institutions in ques-tion (described more fully below). Finally we assume that in calculat-ing their decisions individuals take account of their stage payoffs only.This requires minimal sophistication: they need to calculate for examplewhat the profile of policies will be if they leave a given group and joinanother group and not simply consider the payoffs they would achievefrom membership of a given group without taking into account how thedemographic change will alter those payoffs. It does not require howeverthat agents calculate payoffs that will arise in future periods, possibly asa result of future demographic changes. Clearly we can also view eachstage of the process as a sequence of simple games in which players placeno value on the future and let the transitions correspond to the Nashequilibria of these games.

Perturbed stage games. There are many ways to generate per-turbed transition matrices. For most of the analyses we generate per-turbed matrices, P ε, by assuming that individuals make transition-facilitating mistakes with probability ε. Thus for example if {{1, 2}, {3}}is preferred by only one of players 2 and 3 to {{1}, {2, 3}} then (in asticky environment) a single error is required for this transition; this oc-curs with probability ε, conditional on this encounter taking place. If itis preferred by neither then two errors are required, which occurs withprobability ε2 (conditional on the encounter). If it is preferred by boththen the transition occurs with probability 1−ε−ε2 (conditional on theencounter). Other types of perturbations could be considered by treat-ing errors of joining and errors of accepting differently or by relating theprobability of errors to the utility gains or benefits that correspond tothe error.

2.1.1 Within Group Decision Making.

I simplify within-group decision making processes considerably and as-sume that the outcome of group decision making depends in a very sim-ple manner on a collection of player characteristics, given by the vectorα ∈ Rn+. For a given individual, αi describes the agent’s relative influenceand can be thought of as an index of the individual’s wealth, patience,leverage, aggression or some other measure of personal power. I assumethat αi is strictly positive, although it may be arbitrarily small.I assume that the outcomes of group processes is given by the weighted

average of the ideals of the members of a group, where the weights aregiven by the relative size of the αi terms. Such outcomes can be sus-tained by either cooperative or non-cooperative solution concepts. For

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distributive games such an outcome corresponds to what would obtain inthe limity for example if each player had preferences over income givenby ui(y) = yαi for αi ∈ (0, 1) and groups engaged in alternating offersbargaining under a unanimity rule (implementing the Nash bargainingsolution) (see Binmore 1987); for voting games such an outcome cor-responds to the results predicted by the mean voter theorem (Caplinand Nalebuff 1991), in games with probabilistic voting, in which voterweights are given by (αi)i∈Gj , or in games with efficient bargaining andtransfers (Deirmeier and Merlo 1990).

2.2 Institutional ArrangementsI embed the three forms of proportionality in the context of a singlemodel. Each agent i ∈ N has preferences over both public goods and pri-vate goods given by a simple quasilinear utility function with quadraticloss of the form:

ui(yi, π) = yi −Xn

j=1(πj − πi)

2 (2)

where π ∈ Rn, is a vector of realizations of some public policy, πi ∈ R1is players i’s preferred policy and yi is an income allocation to i. Notethat in effect we consider only one dimension of policy but we allowthe possibility of observing multiple and possibly diverse applications ofpolicy in that area. For example, we could imagine that there is a singledimension of left-right policy but that different left-right policies mightnonetheless be implemented with respect to education, health and socialwelfare. For simplicity we assume that there are exactly n equally andadditively valued realizations of the policy outcome.Now consider three approaches to institutionalizing proportionality.

The first is the proportionate allocation of individual benefits to groups:the private benefits of public policy are allocated proportionately. Thesecond provides each group with a monopoly over decision making ina share λj of issues of public concern: policy jurisdictions are allocatedproportionately. The final approach provides each group with monopolypower over all issue areas but the decisions they make in their areasapply only to their own members; in other words subject jurisdictionsare allocated proportionately. Abstractly, the three approaches can beseen in terms of the allocation of public goods, of private goods andof club goods and in what follows we refer to the three approaches asprivate proportionality, public proportionality and club proportionality.The most obvious case of club proportionality for ethnic groups is inan ethnofederalist system; but jurisdictions can be divided up in a non-geographic manner also; for example decisions with respect to aspects

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of law, can be determined differently for different groups, as is the casefor family law for example for different religious groups in Israel.We represent these three approaches formally as follows:Under private proportionality, the choice of public goods is indepen-

dent of the demography, and hence we ignore it in the representation ofplayer utility; however, each group j is provided with a share λj =

|Gj |Nof

some stock of private goods (normalized to unity). The within-group al-location of private goods to its members is determined internally bythe group. The utility to player i, a member of group A, from a givendemography under private proportionality is then given by:

ui(Γ) =nAn

αiPA αk

(3)

Under public proportionality, the allocations of private goods, yi, isindependent of the group demography and normalized to 0 for all players;however each group A is given monopoly control over decision makingin a share λA of policy areas. The utility to player i from a given de-mography under public proportionality is then given by:

ui(Γ) = −mXk=1

nkn

ÃXh∈Gk

αhπhPh∈Gk αh

− πi

!2(4)

Under club proportionality, the choice of private goods is again in-dependent of the demography outside a player’s group, but public deci-sions (over all n realizations) made by a group affect only the group’sown members. The utility to player i, a member of group A, from agiven demography under private proportionality is then given by:

ui(Γ) = −µX

A

αkπkPA αh

− πi

¶2(5)

2.2.1 Variation within and between groups

For some of the results that follow I draw on the notion of within grouphomogeneity and between group heterogeneity or vice versa. Given ouruse of quadratic loss functions we can describe these notions in a partic-ularly simple way in which within group homogeneity implies betweengroup heterogeneity and vice versa.Let a(x) denote the mean of a vector x and let a(x)α denote the

weighted average of x given some weighting vector α. For compactnesswe also write x(A) for the mean of the subvector of x corresponding togroup A, x(A) = a((xi)i∈A).

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Let the “heterogeneity” of a group Gj (with respect to x) begiven by ρ(A) = a([(xi)i∈A − x(A)]2); analogously define the “weightedheterogeneity” of A, given some weighting vector α as: ρ(A)α ≡a¡[(xi)i∈A − a((xi)i∈A)α]2

¢α.

The (weighted) between group heterogeneity of a coalition structure,ρ(Γ)λ, is given by the heterogeneity of the (possibly weighted) means ofeach of the groups in Γ, with a weighting vector given by the relativesize (or sum of weights) of the groups.The average within group heterogeneity a(ρ(Γ))λ is then proportion-

ate to the sum of squared distance of all players from the means of theirown groups.Usefully, for a given demographic structureG, the total heterogeneity

ρ(N) is equal to the sum of the between group heterogeneity and thewithin group heterogeneity. In particular:

ρ(N)=1

n

mXj=1

Xi∈Gj

[x∗i − x]2

= a(ρ(Γ))λ + ρ(Γ)λ

A consequence of this is that conditional upon a population N , a risein within group heterogeneity resulting from a changing demographicstructure necessarily corresponds to a decline in between group hetero-geneity and vice versa.

2.2.2 Genericity.

Throughout I assume that player strength and preference parameters aregeneric in the sense that given any two non-equivalent sets of players,the averages of the parameters over these sets parameters are distinct.For example if 2 players have weights given by 1,2, then the weight ofa third person cannot be exactly one of the rational numbers in the setX = {1, 1

2, 2, 3} although an infinity of other positive real number are

admissible.

3 Results I: Private Proportionality

Under private proportionality the private income of player i ∈ N is givenby yi = nA

nαiPA αk

. More compactly this can be written: yi = αiαA

1n. Thus

an individuals’s reward is her egalitarian share, 1n, multiplied by a factor

αiαA, that reflects whether a player’s weight is above or below the average

for her group.

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Note that the payoffs take a very simple form in this game. In par-ticular, an individual’s payoff is a function only of the numbers in herown group and of the strengths of the members of her own group. Itdoes not depend on other aspects of the partitioning, and in particularon the group membership or powers of influence of individuals not in theplayer’s own group.

3.1 Egalitarian Private SystemsAs a benchmark consider a situation in which all bargaining weights areuniform. In this case, it is evident that yi = 1

nfor all players. But

in this case, payoffs are independent of the demographic structure Γ.Hence, although, formally, goods are distributed via group membership,the proportionality and equidivision rules make the form of Γ irrelevant:all demographies are stable. This is the case whether demographies arefluid or sticky. This, though simple, is the first result. Under privateproportionality the interesting dynamics take place within inegalitarianstructures. We examine these next.

3.2 Inegalitarian Private SystemsUnder private proportionality any interesting distinction between de-mographies depends on some notion of possible inequalities within groups.Making use of the strength indices, α, described above, we refer to indi-vidual i whose weight is below (above) the average weight of group A asa “weak” (“strong” ) relative to A. Clearly payoffs of players who areweak in their groups are below 1

nwhile payoffs of strong players exceed

1n.To characterize equilibrium in this game more generally, we consider

two cases, the case with no binding constraint on the number or size ofgroups (m ≥ n) and the case with such a constraint (m < n).

Proposition 2 For any stable state Γ, |Γ| = min(m,n).

The proposition follows immediately from the observation that fromour genericity condition there is at least one loser in every group withmore than one member. If new groups are permissible such losers havean incentive to join them. For the m ≥ n case the proposition impliesthe following corollary.

Corollary 3 If m ≥ n then a state is an equilibrium of a sticky processif and only if it is the atomistic structure.

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The corollary implies both that stable demographies exist in m ≥ ncases for sticky structures and that they are highly egalitarian–since inthe atomistic structure all players receive the same payoff.The properties of stable demographies are more complex in situations

where there is a limit on either the number or the minimum size of groupsm < n. If there are less permissible groups than there are players andso singletons can not always be formed, then the properties of stablestructures will be different. Weak members of groups will not always beable to separate from groups in order to claim their share of the pie unlessthey can find another group willing to accept them. In the absence ofsuch groups we may observe in-group heterogeneity and multiple stabledemographies that are not payoff equivalent.Individuals prefer to join groups that are weak relative to them.

Wanting to find themselves in groups in which their bargaining poweris strong, they find a change desirable only if it improves their relativestatus. More formally:

Lemma 4 (Attractiveness) [Inegalitarian, Private]A shift from groupA to group B is desirable for player i if and only if the average strengthof group B members, augmented by i’s membership, is lower than theaverage strength of group A, including i’s membership

In fluid structures transitions depend only on the preferences ofwould-be joiners. From Lemma 4 we can then work out whether ornot equilibria exist under fluid processes. As shown next, they do not.

Proposition 5 No states are stable under fluid hierarchical processes.

For sticky structures, which we will focus on from here on out, achange is acceptable if and only if the incoming member would becomea weak member of the new group. It turns out that the acceptabilityof a member is uniform across members of the receiving group and sothe internal decision-making rule for admission is unimportant: if onemember of a group supports a new entrant, all members do. We stateand prove these claims next.

Lemma 6 (Acceptability) [Inegalitarian, Private] Individual i ∈ Awill be acceptable to an arbitrary member of group B 6= A if and only ifshe is weak relative to B.

The intuition for this is the following: the distributive rule employedallocates a share of the pie to a group as a function only of the group’ssize but not of the characteristics of its members; hence whenever weak

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members join, their take from the group is smaller than the value theybring to the group, leaving a surplus from which everyone else in thegroup can benefit.3 All members receive a share of this surplus (even ifthey are weak), in proportion to their strength.These conditions for attractiveness and acceptability provide the ba-

sis for the necessary and sufficient conditions for stability. These aregiven next.

Proposition 7 [Inegalitarian, Private] A demographic structure, Γ, isstable under a sticky process if and only if |Γ| = min(m,n) and for eachweak player there is no group in which she would be less weak (but stillweak).

Under a sticky structure, every weak player is in the group in whichshe is relatively least weak. Intuitively this implies a homogenizing ef-fect, suggesting that individuals should cluster in groups with individ-uals of similar strength. This intuition is partly correct. The followingproposition provides conditions under which such clustering by size in-duces stability, providing conditions for when movements are impossiblebetween two groups that are ordered with respect to α.

Proposition 8 No movement is possible between groups A and B ifA <α B , and nA − 1 ≤ nB.

One immediate corollary is of this proposition is the following:

Corollary 9 Equilibria exist; in particular any partitioning that is com-pletely ordered on α is stable if for each j,Gj >α Gj−1 implies nj ≥nj−1 − 1.

Example 10 Given a process with n = 4 and m = 2, and labelling play-ers such that α1 < α2 < α3 < α4 Corollary 9 implies that {{1}, {2, 3, 4}}and {{1, 2}, {3, 4}} are both stable states irrespective of the cardinal valueof the player weights. The ordered partition {{1, 2, 3}, {4}} is not how-ever guaranteed to be stable.

The condition in Proposition 8, applied to all pairs of groups, pro-vides a sufficient condition for a state to be stable. For n > 3 howeverthe condition is not necessary as the next example indicates.

3This result clearly stands in contrast to situtations in which inter-group relationsaffect shares; for example to one in which the group’s share is a function of the numberof “high quality” members.

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j

h

1

α

6

Figure 1: An example of an equilibrium with high within group hetero-geneity.

Example 11 Consider a process with two groups and four players ofsize α1 = 1, α2 = 5 − ε,α3 = 5 + ε and α4 = 6, for small ε. In thisgame the most inegalitarian partitioning G = {{1, 4}, {2, 3}} is stable.This equilibrium is illustrated in Figure 1. Weak as he is in negotiatingwith the strongest player (receiving 2

7), the smallest player is even weaker

negotiating with two large players (receiving 311).

The example shows that equilibria are not always characterized byordered partitions But to a minimal extent however, all stable parti-tionings are at least partially rankable. In particular we can show thefollowing:

Proposition 12 In every stable demography, the mean strength of thegroup containing the jth weakest player is no larger than the mean strengthof the jth weakest group.

Attractiveness of Equilibrium. We have already established inCorollary 9 that equilibrium always obtains for sticky Markov processes.In addition however we can show that such processes will always lead tosome equilibrium state. We establish this property next.Lemmas 4 and 6 describe the conditions for when individual move-

ments will be potential. They also give insights into the paths thatdemographies may take as they evolve towards a stable state. For anyfeasible transition we can show that only weak members of groups willshift and only then to groups in which they will remain as weak mem-bers. Once they move, the group to which they move will be on averageweaker than the group from which they moved; the group they leavebehind will, as a result be stronger on average. These regularities sug-gest that in the dynamic process we should observe a general movement

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of weak players from stronger to weaker groups. However these gen-eral trends notwithstanding, the dynamic process can hold a number ofimportant non-monotonicities.It is possible to find demographies from which movements may take

place from relatively strong to relatively weak groups or from weak tostrong groups. It is also possible that movements are potential both fromgroup A to B and from group B to A. These features raise the prospectthat the dynamic process may not eventually result in the attainment ofa stable demography: they raise the concern that stable demographies,while retentive, may not be attractive. In fact, this is not the case.Since the state space is finite, all paths either lead to cycles or to steadystates; we now demonstrate that they invariably lead to a steady stateby showing that they cannot produce cycles.

Proposition 13 All recurrent classes contain only one state. Further-more the average weight of the weakest group declines monotonically dur-ing this process.

3.3 Complete characterization of equilibrium for theprivate goods case, with n = 4,m = 2.

We now characterize equilibria and stochastic dynamics in the n = 4,m = 2 case assuming a sticky structure and inegalitarian within grouppolitics characterized by generic weighting vectors normalized such thatP

αi = 1.The universe of states, G, contains just 8 elements which we label as

follows.ΓI = { {1, 2, 3, 4}, ∅ }ΓII = { {1}, {2, 3, 4} }ΓIII = { {2}, {1, 3, 4} }ΓIV = { {3}, {1, 2, 4} }ΓV = { {4}, {1, 2, 3} }ΓV I = { {1, 2}, {3, 4} }ΓV II = { {1, 4}, {2, 3} }ΓV III = { {1, 3}, {2, 4} }

The question is which of these states are stable and which of theseare “likely.” Using ordinal information alone we have from Proposition8 and its corollary that ΓII and ΓV I are stable states, from Proposition2 that ΓI is unstable (since the transition ΓI → ΓII is feasible) and fromProposition 7 that ΓIII is unstable since a switch by 1 to transition toΓV I is potential. Our propositions do not tell us however whether or notΓIV , ΓV ,ΓV II or ΓV III are equilibria.

15

Indeed a full examination of all possible structures confirms that nopropositions could could establish further which states are necessarilystable or not using ordinal information alone; each of the remainingstates may or may not be stable depending on the exact value of α.

Proposition 14 For games with n = 4 and m = 2, and α normalizedsuch that

Pα = 1 the equilibrium states are:

A. ΓII, ΓIV , ΓV , and ΓV I if α1 > 2−5α33

B. ΓII, ΓV , and ΓV I if α1 < 2−5α33

and α1 > 2−5α43

and α1 > 2α2−α3C. ΓII, ΓV , ΓV I and ΓV III if α1 < 2−5α3

3and α1 >

2−5α43

and α1 <2α2 − α3D. ΓII, ΓV I, ΓV II and ΓV III if α1 < 2−5α3

3and α1 <

2−5α43.

  α1<[2‐5α3]/3 

α1<[2‐5α4]/3 

α1<2α2 ‐ α3 

N

N 

N  Y 

Y 

Y

A 

Equilibrium States: 

II, IV, V, VI 

 

Example:  .15, .16, .33, .36 

B

Equilibrium States: 

II, V, VI 

 

Example: .02, .03, .40, .45 

C

Equilibrium States: 

II, V, VI, VIII 

 

Example: .05, .15, .20, .60 

D

Equilibrium States: 

II, VI, VII, VIII 

 

Example:  .01, .32, .33, .34 

Figure 2: Taxonomy of equilibrium states for n = 4, m = 2.

Figure 2 illustrates the set of equilibria identified by Proposition 14.We see that each of the four states ΓIV , ΓV ,ΓV II and ΓV III may besustained as equilibria for some values of α but not others.The proposition can be used to characterize the degree of within

group heterogeneity that can be sustained in m = 2, n = 4 cases. Thegreatest value that the sum of squared errors can take (SSE), is for theα vector (close to) (0, 0, 0, 1), with SSE = 3

4. For such cases within group

heterogeneity is minimized at state ΓV which is indeed an equilibriumin this case. Indeed it can be shown that ΓV is an equilibrium wheneverSSE exceeds .11. We see here that a necessary and sufficient conditionfor the high within group heterogeneity equilibrium that we examinedin Example 11 is that α1 < 2−5α4

3, a condition that can only be satisfied

for α4 < 25. From the Proposition however we can see that although this

state has the highest within group heterogeneity among possible states,

16

it occurs only when the total amount of heterogeneity is low. The totalsum of squares and the within cluster sum of squares are maximized forgroup D cases for the α vector (0, 1

5, 25, 25), which admits a total sum of

squared errors of only 520

2+ 120

2+ 320

2+ 320

2= 11

100and a total within group

heterogeneity of 10100. ΓIV is also a heterogenous case, but this too only

arises when overall heterogeneity is low. The most heterogenous casefor which ΓIV can be sustained is for α vector close to (0, 0, 25 ,

35), which

yields a within group heterogeneity (for ΓIV ) of 24100. Thus whenever the

most heterogenous within-group states are equilibrium overall within-group heterogeneity is low.Finally, we consider stochastic stability. The question of interest is

whether for a given process, the equilibria with the greatest within groupheterogeneity can be sustained even as stochastically stable equilibria.The answer is yes and we illustrate with the case already discussed inExample 11.For the case given in Example 11, the perturbed transition matrix

for this case, P ε is given by:

P ε =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

7−3ε8

18

ε8

ε8

ε8

0 0 0ε8

8−4ε8

0 0 0 ε8

ε8

ε8

ε3

80 7−2ε2−ε3

80 0 1

8ε2

8ε2

8ε3

80 0 7−ε−ε2−ε3

80 ε2

8ε8

18

ε3

80 0 0 7−2ε−ε3

8ε8

18

ε8

0 ε8

ε8

ε2

8ε2

88−2ε−2ε2

80 0

0 ε3

8ε2

8ε2

8ε8

0 8−ε−2ε2−ε38

0

0 ε8

ε2

8ε8

ε8

0 0 8−3ε−ε28

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦This perturbation arises when, with probability ε an individual ac-

cepts a change to the status quo that is against her interest. Some typesof moves require a single error for— for example P ε

13 requires only thatplayer 2 mistakenly decides to leave the grand coalition. Others, such asP ε37 require the compounding of two errors, in this case, player 3 mistak-enly choosing to leave {1, 3, 4} to join player 2, and player 2 mistakenlyaccepting him. Other moves require 3 errors, for example the move P ε

72,which involves 4 mistakenly choosing to leave and 2 and 3 mistakenlyaccepting him. Thus the three element path from the grand coalition,to one in which 2 is on her own, is then joined by 3 and finally by 4 re-quires a substantial compounding of errors and occurs with probabilityε8ε2

8ε3

8= ε6

8.

Note that this is a regular perturbed matrix of P for which 0 <limε→0

P εΓΓ0

εr(Γ,Γ0) <∞ for precisely the resistances given in the matrix r.

17

r =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 1 1 1 0 0 01 0 0 0 0 1 1 13 0 0 0 0 0 2 23 0 0 0 0 2 1 03 0 0 0 0 1 0 10 1 1 2 2 0 0 00 3 2 2 1 0 0 00 1 2 1 1 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦For any P ε we can find the corresponding stationary distribution, µε

by solving µεP ε = µε (computationally this can be done by locating thenormalized eigenvector corresponding to the transpose of P ε given aneigenvalue of 1). A graph showing the evolution of µε is given in Figure3. The figure demonstrates that State ΓV II is the unique stochasticallystable equilibrium. Notably however with moderate errors, state ΓV Ialso occurs with high frequency.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.050.10.150.20.250.30.350.40.450.5

I

II

III

IV

V

VI

VII

VIII

Figure 3: The horizontal axis gives the probability that a given playerwill make an error, ε ∈ [.5, 0). On the vertical we show the proportion oftime spent at each state for any given ε. It can be seen that as ε vanishesthe share of time spent in state V II (ΓV II) tends to 1.

We can also demonstrate analytically that state ΓV II is the uniqueSSE in this case by calculating the stochastic potential associated witheach state (Young 1998, Theorem 3.1). These are given by 3 for ΓV II and4 for all other equilibria. The lower stochastic potential for ΓV II arisesin fact because given this equilibrium, leaving state ΓV II is difficult sincethe pair {1, 4} do not want new entrants and no other group wants towelcome 4; any transitions other than ΓV II → ΓV , thus requires more

18

than one error; in addition any subsequent movement from ΓV to anystate other than ΓV II requires further errors.We have then that this outcome can be sustained as a unique sto-

chastically stable state and so stochastic stability can not be relied uponto select internally homogenous coalition structures. A full characteri-zation of stochastic stability is still required for these games

4 Results II: Public Proportionality

4.1 Egalitarian Public SystemsIn the case of public proportionality we have that utility depends onthe preferences of all players in all groups. Not only does a player careabout which other player is in her group, she also cares about whichother players are in other groups, since each of these affects the aggregatepolicy outcome.Whereas in the case of private goods we identified incentives to frag-

ment, in this case we find incentives to consolidate.Our first proposition shows that complete integration is an equilib-

rium in this game in egalitarian settings. Furthermore, all other equi-librium approximate this equilibrium; while it is not the case that fullintegration is the unique equilibrium, any equilibrium other than thegrand coalition is similar to the grand coalition in the sense that eachgroup contains an internally diverse preference structure similar to thatof the grand coalition, and payoffs at every equilibrium approximatepayoffs at the grand coalition. These features are summarized in thenext proposition.

Proposition 15 In an egalitarian polity (whether fluid or sticky): (i)the grand coalition is stable (and efficient); (ii) shifts are potential if andonly if they reduce between group heterogeneity; and (iii) all demographicchanges shift payoffs towards those payoffs that are provided by the grandcoalition

Corollary 16 (No Cycling) All recurrent classes contain only one state.

Proof. Follows from part (ii) of Proposition 15

The proof of the Proposition is given in the appendix but some ofthe intuition of the proof is worth noting. That the grand coalitionis an equilibrium follows from the concavity of player utility functions.With concave preferences all players prefer the policy outcome that cor-responds to the average of the ideals of all players to the average of the

19

utility arising from their own ideal and the utility arising from the aver-age of all other ideals. The same logic can be used to show not simplythat the grand coalition is an equilibrium, but also that no equilibriumcan contain singletons.From this we can calculate the maximum number of groups in any

stable structure:

Proposition 17 For any stable structure Γ, |Γ|≤Pm

k=2 S(n, k), whereS(n, k) is the Stirling number of the second kind for n, k.

More generally, the condition for player a to be willing to switch fromgroup A to group B in this setting is (for nA > 1):−nA (πA − π∗i )

2 − nB (πB − π∗i )2

< −(nA − 1)³nAπA−π∗inA−1 − π∗i

´2− (nB + 1)

³nB πB+π

∗i

nB+1− π∗i

´2Or, more compactly:

(πA − π∗i )2

(πB − π∗i )2 <

nBnB + 1

nA − 1nA

(6)

This condition is more likely to hold if i is relatively centrist withrespect to his own group and relatively extreme with respect to anothergroup. It is also most likely to hold when groups are large, in which casethe right hand side rises to unity and all that is required for a switchis that some player prefers the mean position of the other group lessthan the mean position of her own group. This results in a narrowingof the distance between the groups and hence the relationship betweenindividual motivations to switch and the global narrowing of betweengroup heterogeneity. Finally we find that in this game the incentivesof players are strongly aligned: in the egalitarian case the condition fora switch for i from group j to group h to be acceptable to some thirdplayer m is in fact independent of m’s own ideal, depending only on thesmoothing effects of j’s shift. An implication of this last feature is thatevery shift is Pareto improving.Although it is possible to have equilibria other than the grand coali-

tion, the proposition suggests that any such equilibria are similar interms of payoffs to all players to what would obtain under the grandcoalition. More precisely, any equilibrium must be characterized by lowbetween-group and high within-group heterogeneity. In particular, asshown in the next proposition, all equilibrium coalitional structures are“cross cutting” and the ordered partitions that we saw to be equilibriain the Private goods case are never equilibria in the Public goods case.

Proposition 18 No complete ordered partitioning is stable.

20

4.2 Complete characterization of equilibrium for thepublic goods case, with n = 4,m = 2.

We know from Proposition 15 that in the four person case no partitionsinvolving singletons are equilibria and from Proposition 18 that no or-dered partitions are equilibria. These two propositions then rule outtwelve of fifteen possible structures leaving only three:{{1, 2, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}.The first of these, the grand coalition, we know to be stable. The

only remaining question from Proposition 15 is whether the other two,non-ordered pairs, can be equilibrium structures. The answer is thatthey can, but in the spirit of the proposition, only if they produce lowbetween group homogeneity.To study these two cases consider a state containing two pairs, L and

H with πL = 0 and πH = 1. An individual in group H will switch if:(1−π∗i )

2

(π∗i )2 < NL

NL+1NH−1NH

and will stay put if³

π∗i1−π∗i

´2≤ NL+1

NL

NHNH−1 . Consider

some player h in H with π∗i ∈ max{(πk)k∈H}. For this individual π∗h > 1and so the condition not to switch is π∗h

π∗h−1≤q

NL+1NL

NHNH−1 or :

π∗h ≥

qNL+1NL

NHNH−1q

NL+1NL

NHNH−1 − 1

(7)

The right hand side of (7) tends to infinity as the number in each groupincreases. For NL = NH = 2 it takes a minimum of approximately 2.4;stability stable then requires that the highest player of group H has anideal of at least 2.4 and the lowest of less than −0.4.Consider next some player l in L with π∗l < πL = 0. By the same

reasoning, in this case for a movement not to be desirable, we need(π∗l )

2

(1−π∗l )2 ≥ NH

NH+1NL−1NL

or:

π∗l ≤1

1−q

NLNL−1

NH+1NH

(8)

The RHS is negative but takes on its highest value in situations withfewer players in each group; in the case with two two- player groups theRHS takes the value 1

1−√3' −1.4.

Hence a stable four person game in which there is not complete inte-gration requires heterogeneity at least as large as would obtain if playersin one group have ideals −1.4, 1.4, and players in the other group haveideals −.4, 2.4. Graphically the overlap implied by these ideals is shown

21

 

0 1

H

L

Figure 4: This figure shows the minimal degree of cross cuttingness oftwo group in equilibium under public proprtionality.

in Figure 4. Note the within group heterogeneity is so large relativeto the between group heterogeneity that some members in each actuallyprefer the policies set by the other group to those set by their own group;they remain in their own group to balance the policy preferences of theirother group members.This implies that for stability we require that the share of to-

tal heterogeneity accounted for by within group heterogeneity is about2

4−√3> 88%. Hence even in the two group two player case–which

admits the maximum rigidity–any equilibrium in which some betweengroup heterogeneity persists, this within group heterogeneity can notaccount for more than 12% of total heterogeneity.4

Finally we consider the question of stochastic stability. In this processthe structures with trios and singletons act as bridges: a change fromthese to any of the equilibrium state is achievable with zero resistance.If resistance is only incurred by those leaving equilibrium structures (influid systems), then each of the three states has a stochastic potentialof 2. If however resistance also arises from the requirement of accep-tance (in sticky systems), then paired structures are stochastically sta-ble states. A move away from a paired situation is resisted both by the

4Note that in this 2 group case, minimal total heterogeneity is given by:

ρ(N) =( 11−√3−.5)2+( −1

1−√3−.5)2+(

√3√

3−1−.5)2+(

√3−2√3−1−.5)

2

4 . This can be decomposed into

within group heterogeneity, given by: ρw =( 11−√3)2+( −1

1−√3)2+(

√3√

3−1−1)2+(

√3−2√3−1−1)

2

4 andbetween group heterogeneity, given by:ρb =

12

¡12

¢2+ 1

2

¡12

¢2.

To see that that 88% figure is the minimum share note that our normalization ofπh = 0, πj = 1, implies that any ε increase in the position of the player with thehighest value of π∗i , say π

∗4 corresponds to a fall in the value of the other member of

group j, say π∗2 leaving πj and π unchanged. Since π∗4 > π and π∗2 < π this produces

a rise in ρ with no change in ρb. Hence we have a decline in ρb/ρ and a correspondingrise in ρw/ρ.

22

mover and by the accepting pair, for a total resistance of 3. In contrasta move from the grand coalition requires resistance only from the mover.In a case with two pair equilibria, the stochastic potential of each pairis 4, while the stochastic potential of the grand coalition is 6.The fact that the grand coalition is not selected as the unique sto-

chastically stable equilibrium (or in some cases not selected at all) issurprising given the fact that it is the unique Pareto optimal structure.It arises in part because of the lack of sophistication of the agents. Anintuition from sub-game perfection would suggest that if one player in apair left to form a triple the other member would soon follow to every-one’s benefit. Such farsightedness is not admitted here however.

4.3 Inegalitarian Public SystemsIn the case in which players are likely to vary in their ability to affectthe outcome of decisions made within their groups, the characterizationof the equilibrium is more involved.

The condition for i to want to shift from A to B is:

(nB + 1)

µPB αkπk + αiπiPB αk + αi

− π∗i

¶2+ (nA − 1)

µPA αkπk − αiπiPA αk − αi

− π∗i

¶2<

nB

µPB αkπkPB αk

− πi

¶2+ nA

µPA αkπkPA αk

− πi

¶2With some manipulation this simplifies to:

(πh − π∗i )2

(πj − π∗i )2 <

ΨA

ΨB

for ΨB > 0

(πh − π∗i )2

(πj − π∗i )2 >

ΨA

ΨBfor ΨB < 0 (9)

whereΨA = nA−(nA−1)³ P

A αkPA αk−αi

´2andΨB = (nB+1)

³ PB αkP

B αk+αi

´2−

nB.The trade-off then depends on the distance between a player’s ideal

and the policy of each of the two groups. The comparative statics dependon the sign of ΨB–a measure of a player’s weakness in group B. Thisrenders interpretation difficult. Nonetheless a number of regularities canbe inferred.

23

• A bandwagonning effect obtains: a switch is more likely if B islarge and A is small. To see this note that both ΨA and ΨB aredecreasing in nA.

• Avoiding weakness. If a player is strong in the receiving groupB such that αi >

³pnB + n2B − nB

´αB and hence ΨB < 0, but

weak in his own group A, αi <³nA −

pn2A − nA

´αA, such that

ΨA > 0, then a switch is always desirable, independent of thepolicy positions of the two groups. Conversely if a player is weakin the receiving group, but strong in his own group, then a switchis never desirable.

• A Polarization effect can obtain if a player is especially weak inboth groups; in the sense thatΨA > 0, andΨB > 0. For reasonablylarge n these conditions obtain when αi <

αA2, αB2. In this case a

player will prefer to be in the group that is like him–in shiftinggroups he has little direct impact on policy in any area, one wayor the other, but by teaming up with like types he can at leastmake sure that his interests get represented indirectly through theassignment of power to his group. This effect is especially strong ifthe player is strong in the like group relative to the unlike group.

• A Homogenizing effect obtains for individuals that are strong inboth groups, ΨA < 0, ΨB < 0. In this case the condition is morelikely to hold when (πB − πi)

2 is large relative to (πB − πi)2–the

more extreme a player is in a group (relative to his position in hisown group) the more likely he is to join that group.

Hence a player always weighs two features—the gain from joining a likegroup by increasing the number of issue areas controlled by like types,and the gain from joining an unlike group that arises from exerting amoderating influence on the group. Relatively influential players aredriven more by the latter condition, weaker players by the former. Allplayers however are attracted by large groups.Note that the condition for strength here is always satisfied for players

with average strength or above. The strength of the condition dependson the size of the groups. ΨB < 0 is always satisfied for players withαi >

αB2and does not depend greatly on nB. Condition ΨA < 0 is

more demanding for low nA but tends to the condition αi >αA2as nA

increases. Importantly, it follows that there are always strong players forwhom nA > αA

2. Furthermore, letting group A denote the weaker group,

there are always players for whom switching is optimal if and only ifbetween group inequality can be sufficiently reduced by the switch.

24

Such strong players might not switch if they are already having amoderating effect on their own group.These results suggest that in inegalitarian polities, an equilibrium

is likely to be characterized by groups that contain multiple influentialplayers with diverse preferences, alongside weak players with centristpreferences, and satellite groups of like minded weak players with ex-treme preferences. These equilibrium partitions have however not yetbeen studied.

5 Results III: Club Proportionality

5.1 EquilibriumIn the third case we examine a player’s utility depends on the policypreferences of the members of her own group but not the preferences ofmembers of other groups.In this setting, the condition for player i to be willing to switch from

group A to group B is:

−µP

B αkπk + αiπiPB αk + αi

− πi

¶2> − (πA − πi)

2

or, equivalently:

(πB − πi)2

(πA − πi)2 <

µ1 +

αiPB αk

¶2(10)

Clearly a player always has an incentive to switch to an empty groupwhenever this is feasible, giving rise to analogues results to those fromour study of private goods.

Proposition 19 For any stable state Γ, |Γ| = min(m,n).

Corollary 20 If m ≥ n then a state is an equilibrium if and only if itis the atomistic structure.

What of acceptability under club proportionality? It is immediatefrom out genericity condition that any new entrant into a group willresult in a change in policy for a group which will in turn produce atleast one loser. Thus, under the unanimity rule considered here thereis always opposition to new entrants and so we expect no transitions insticky polities with club proportionality.

Proposition 21 All states are stable under sticky processes.

25

In consequence, we focus here on out on fluid polities for which atransition requires only that there exists an individual willing to changegroups. For such contexts a necessary and sufficient condition for sta-bility is that for every pair of groups A and B for all i in group A:

(πB − πi)2

(πA − πi)2 ≥

µ1 +

αinBαB

¶2> 1 (11)

This is worth comparing to the condition derived for the public goodscase:

(πA − πi)2

(πB − πi)2 ≥

nBnB + 1

nA − 1nA

> 1 (12)

Strikingly, this comparison tells us that there is no overlap betweenpublic goods equilibria (with equal weights) and club goods equilibria.In the public goods case fully ordered partitions were equilibria whereasin the club good case condition 11 tells us that only ordered groups arestable: each player should be strictly closer to the center of his own groupthan to the center of all other groups. In particular a player cannot beprecisely on the border of two groups because in such a situation hewould have an incentive to change groups (with the immediate resultthat the policies of that group would be closer to him next period thatthe policies of his present group).

5.2 Complete characterization of equilibrium for theclub goods (equal weights) case, n = 4,m = 2 .

Consider a game with four equally weighted players. Normalize thepositions such that π1 = 0, and π4 = 1 and π1 < π2 < π3 < π4.From Condition 11 and Proposition 19 we have that of 8 states there areonly 3 candidate equilibria: the 3 two-group ordered partitions, Γ1 ={{1}, {2, 3, 4}}, Γ2 = {{1, 2}, {3, 4}} and Γ3 = {{1, 2, 3}, {4}}. Notethat player 2 is decisive over Γ1 and Γ2. He will prefer Γ2 if and only ifπ2 − π2

2< π2+π3+1

3− π2, or equivalently, if and only if:

π3 <7

2π2 − 1 (13)

Furthermore among the set of ordered partitions, Γ2 is the only possi-ble deviation from Γ1. Hence Γ1 is an equilibrium if and only if Condition13 holds. Similarly 3 is decisive over Γ3 and Γ2 and prefers 3 if and onlyif π3 − π2+π3

3< π3+1

2− π3, or equivalently:

26

π3 <2

7π2 +

3

7(14)

Since among the set of ordered partitions, Γ2 is the only possibledeviation from Γ3, Γ3 is an equilibrium if and only if Condition 14 holds.If neither condition holds then since Γ1 and Γ3 are the only ordereddeviations from Γ2, Γ2 is an equilibrium. Finally we note that it ispossible that two equilibria exist if and only if π3 < 7

2π2 − 1 and π3 <

27π2+

37. Given the restriction that π2 < π3 there is only a narrow range

in which both of these conditions hold simultaneously. For these casesthe two centrist groups are sufficiently similar that they prefer to clubtogether with a third player rather than divide and make alliances withthe extremes. These equilibria are illustrated in Figure 5.Stochastic stability is required only for the case where two equilibria

obtain. In this case both states are stochastically stable as a result ofthe symmetry of the situation; the stochastic potential of each state is2.

Figure 5: The set of equilibrium states for 0 = π1 < π2 < π3 < π4 = 1combinations.

27

5.3 Equilibrium and Centroidal Voronoi Tessela-tions

The necessary condition for equilibrium given in Equation 11 has a usefulgeometric interpretation. Given a set of m distinct generating points,x = {x}mi=1 a Voronoi tesselation on X ⊂ R1 is a partitioning of X intom regions {V }mi=1 such that each point y ∈ X is in region Vi if andonly if y is closer to xi than to any other point in x. Given a measurefunction f , over X, a centroidal Voronoi tesselation is a tesselation withthe property that the mean of each partition (under f) is its generatingpoint.It can be seen from equation 11 that every equilibrium partition must

be a Centroidal Voronoi tesselation onR1 with generating points given bythe policies of each group and the player ideals of each group containedwithin the corresponding Voronoi cell of the tesselation. However, asthe following example demonstrates, not every centroidal tesselation isan equilibrium.

Example 22 Consider a game with n = 4, m = 2, equal weights andpreference parameters given by π1 = −2, π2 = 2, π3 = 3, π4 = 4,π5 = 5.Restricting attention only to ordered non-empty groups we have 4 possiblepartitions, (Γ1)4i=1,such that j ∈ G1(Γi) if and only if i ≤ j. ThusΓ1 = {{1}, {2, 3, 4, 5}} generates a centroidal tesselation with generatingpoints −2 and 3.5. Γ2 = {{1, 2}, {3, 4, 5}} also generates a centroidalVoronoi tesselation with generators 0 and 4. Player 2 is decisive overthese two states, but she is closer to 3.5 than to 0, hence although it isa centroidal Voronoi tesselation, Γ2 is not an equilibrium state.

The reason why one of the centroidal Voronoi tesselations in Example22 is not an equilibrium is because in state Γ2 player 2 compares not thepresent policy she receives in Group 1 (0) to what is produced in theGroup 2 (4), but rather what would be achieved in Group 2 were she tojoin. In effect, she anticipates her influence.There are two circumstances in which we can ignore this effect; first

if the size of all groups are large such that individual influence is small;or second if players are not sufficiently strategic to take their future in-fluence into account. Let us call such a condition a “diminished expectedmarginal effect condition” (demec).Under demec we have from a modified form of Equation 11 for

which equilibrium states correspond exactly to the centroidal Voronoitesselations of Γ (or, for discrete distributions, k-means clusters). Muchis known about clusters of these forms and so it is worth considering

28

properties of equilibrium under these assumptions.5

In older work on the subject Fisher (1958) demonstrated the ex-istence of k-means clusters and showed that they are always orderedpartitionings It is also well known that for any number m the partition-ing of a space into m sets that minimizes within group variance is givenby a centroidal Voronoi tesselations (for a review see Du et al 1999), assuch we have that the state that minimizes within group heterogeneity isan equilibrium of our system. This establishes existence but it also hasstriking normative properties, standing in stark contrast to the result onpublic goods, given in Proposition 15; there we found that demographicevolution minimized rather than maximized between group heterogene-ity. Finally with demec we can draw on a large body of work thatdescribes the evolution of partitions towards equilibrium.MacQueen (1964, 1967) shows that a simple algorithm can be used

to identify centroidal Voronoi tesselations. We can provide an easy in-terpretation of MacQueen’s algorithm as follows. Start with a set ofk leaders that dictate policy for their group. Then randomly select anadherent and let the adherent join the group whose policies he prefers.Then redefine the policy for this group based on the updated member-ship. Select another individual and repeat. As the size of the populationincreases the sum of within group variance that can arise from allocationof cells to the ensuing set of Voronoi points converges. A stronger resultdue to Pollard establishes that the Voronoi points themselves convergeto the minimizing set as the population increases.Lloyd (1982) proposes a faster algorithm that is also consistent with

models of adaptive play. Begin with any demography and identify theset of political outcomes associated with this demography (under ourgenericity condition we take this set to be composed of k distinct out-comes). Then let all individuals simultaneously select the groups whosepolicies they prefer, producing a new demography. Identify the new setof political outcomes. Then repeat the process, letting all individuals re-

5This problem is closely related to questions examined in two other areas. One isthe study of Tiebout public goods. Tiebout equilibria are used in settings in which“the consumer-voter may be viewed as picking that community which best satisfieshis preference pattern for public goods.” If it is the case that the choices of a givencommunity in turn depends on the membership of a community, then the problemis that studied by Caplin and Nalebuff (1992). Voronoi diagrams have a number ofother applications in politics. Most obviously they can be used to illustrate the set ofvoters that support a given candidate in multicandidate elections, assuming sincerevoting. They can also be used to study districts and the location of public goods(Simeone et al). For the study of ethnic politics they can also be used to link types to“stereotypes”: given a collection of stereotypes (Voronoi points) a Voronoi diagramclassifies individuals according to which stereotype they most closely resemble (for arecent study see Fryer and Jackson 2002).

29

select their preferred group again. The process converges to a centroidaltesselation. Du et al (2006) show that if the measure of types over thespace is positive and smooth then the process is globally convergent inthe one dimensional case.6

6 Conclusion

Our inquiry reveals tendencies for how demographic profiles may evolveunder different institutions of proportionality and under different as-sumptions of the conditions under which group change is feasible. Thecontext we examine abstracts greatly from many important features ofdemographic change. The analysis very deliberately abstracts from be-tween group competition; the between group allocation is treated asgoverned entirely by a set of rules; in practice however these rules arethemselves a subject of contestation. Beyond this, other absent featuresinclude the possibility of elite manipulations, the existence of affectiveties to groups, and limitations on group formation that might derivefrom physiological or other sticky attributes (Posner 2005, Chandra andBoulet 2003 ).Despite the simplicity of the model however it has generated a rich

array of propositions regarding how group structures might respond toinstitutional rules. These results are summarized in Figure 6.

The overall message is that very distinct patterns emerge from eachof the three types of proportionality considered, with outcomes rangingfrom a wide multiplicity of equilibria, to the absence of equilibria, fromstrong tendencies to fragmentation, to stagnation, to strong pressures forassimilation. Most strikingly the different systems also have dramaticimplications for not just he number and size of groups but their internalcomposition with in some cases a strong tendency towards coalitions oflike types and others a tendency to join with unlike types in a bid tobalance.Much of the dynamics examined here warrant further and deeper

exploration; equilibrium has not been fully characterized in the case ofpublic goods and hierarchical structures, the dynamics of the club casehave not been explored for cases in which the demec condition does nothold. Furthermore, in many cases it seems plausible that results could befound with weaker assumptions on preferences and institutional struc-tures. Finally this theoretical work has been developed entirely without

6In addition Du et al provide approximations for the size of each group at con-vergence.

30

SOCIAL CONTEXT INSTITUTIONAL RULES

Transition Rules

Limits on admissible states?

Within group bargaining Private Public Club

Sticky

(Desire + Acceptance)

m=n

Egalitarian All demographies are stable

Tendency to consolidation; Few

groups. Minimal between group heterogeneity

Complete Fragmentation

Hierarchical Complete fragmentation. Tendency to consolidation

although polarized structures possible

Complete Fragmentation

m<n

Egalitarian All demographies are stable

Tendency to consolidation; Few

groups. Minimal between group heterogeneity

All demographies with |Γ|=min(m,n) are stable

Hierarchical

Tendency to fragmentation;

minimization of within group heterogeneity

Tendency to consolidation although polarized structures possible

All demographies with |Γ|=min(m,n) are stable

Fluid

(Desire only)

m=n

Egalitarian

All demographies are

stable

Tendency to consolidation; Few

groups. Minimal between group heterogeneity

Complete Fragmentation

Hierarchical No Equilibrium Tendency to consolidation

although polarized structures possible

Complete Fragmentation

m<n

Egalitarian All demographies are

stable

Tendency to consolidation; Few

groups. Minimal between group heterogeneity

Tendency to fragmentation

minimization of within group heterogeneity

Hierarchical No Equilibrium Tendency to consolidation

although polarized structures possible

Tendency to fragmentation

minimization of within group heterogeneity

Figure 6: Summary of analysis: Demographic Evolution as a function ofinstitutions and social context

reference to actual cases. Yet there is a wide field of application, andgreat scope for generating empirical tests of the prepositions developedhere, either for the study of the evolution of ethnic demographies, forchanges in party systems, or for population movements in federal sys-tems.

31

7 Appendix (Proofs)

Lemma 4Proof. GroupB is attractive to i ∈A iff: |A|

nαiPk∈A αk

< |B|+1n

αiPk∈B αk+αi

⇔Pk∈B αk+αi|B|+1 <

Pk∈A αk|A| ⇔ α(B ∪ {i}) < α(A).

Proposition 5.Proof. With m > 1 any equilibrium must contain at least two non-

empty groups which we label A and B. From our genericity assumption,either α(A) < α(B) or α(A) > α(B). Assume without loss of generalitythat α(A) > α(B). There exists some player i ∈ A with αi ≤ α(A). Butthen αi ≤ αA and α(B) < α(A) which implies α(B∪{i}) < α(A). To seethis assume to the contrary that α(B ∪ {i}) ≥ α(A) then α(B ∪ {i}) ≥α(A) > α(B) so α(B ∪ {i}) > α(B) and so αi > α(B). However, wewould then also have α(B∪{i}) ≥ α(A) ≥ αi and so α(B∪{i}) ≥ αi andso αi ≤ α(B), a contradiction. With α(B∪{i}) < α(A) Lemma 4 impliesthat a move is desirable and hence, under a fluid process, potential.

Lemma 6Proof. The condition for a member h of group B to be willing to

accept a new member i into her group is given by:

|B|N

αhPk∈B αk

<|B|+ 1N

αhPk∈B αk + αi

⇔αi <

Pk∈B αk|B|

⇔αi < α(B)

This condition depends only on the weight of player i and a popu-lation statistic of group B, and hence the same condition applies for allincumbent members of group B.

Proposition 7.Proof. The conditions for stability follow directly from lemmas 6

and 4: αi < α(B) and α(B ∪ {i}) < α(A). The latter condition impliesin particular that αi

α(B∪{i}) >αi

α(A).

Proposition 9

32

Proof. Assume G2 >α G1. Clearly since all members of G2 arestronger than all members in G1 no movement from G2 to G1 is possible.We focus therefore on a movement of some player i from group G1 to G2.Normalize weights such that

PG1∪G2 αk = 1 and define γ =

PG1\{i} αk.

Note that the average weight of group 1 is γ+αin1, the average weight of

group 2 is 1−γ−αin2

. Since the movement is acceptable it must be thatαi <

1−γ−αin2

.Since player i wants to shift we have:

αiγ + αi

n1 <αi1− γ

(n2 + 1)

Which yields Condition 15:

n1n2 + 1

(1− γ)− γ < αi (15)

(a). In addition we have αi <γ

n1−1 , which follows from the fact that,i is weak in his own group. Together with Condition 15 this impliesn1n2+1

(1− γ)− γ < γn1−1 or equivalently: γ >

n1−1n1+n2

(b). We also have from the fact that the partition is ordered thatγ

n1−1 <1−γ−αin2

, that is, the average of the other members in G1 areof lower weight than the average of G2. Reorganizing this conditiongives: αi < 1− γ − γn2

n1−1 . Then combining this with Condition 15 givesn1n2+1

(1 − γ) − γ < 1 − γ − γn2n1−1 . This condition can never be satisfied

n2 = n1−1. We then focus on the cases n2 ≥ n1; for these the conditionis equivalent to γ < n1−1

n1+n2. But this condition is incompatible with the

analogous conditions from (b). Hence the condition can not be satisfiedfor n2 ≥ n1 − 1.

Proposition 12Proof. Label the players such that αi < αi+1 and the groups such

that α(j) < α(j+1) for all i < n and j < |Γ|. Assume first that Γ(1) > 1,that is, that the weakest player is not in the weakest group. Since 1 isacceptable to all groups, she will move if there is any group such thatα(j) < α(Γ(1)). But there is such a group, in particular, α(1) < α(Γ(1)).Hence in a stable demography the weakest player must be in the weakestgroup. We now show that in any equilibrium, for any s if player s is inone of the smallest s groups, then player s+ 1 is a member of on of thesmallest s + 1 groups. Given that players 1, 2, ..., s are in the smallests groups, player s + 1 is acceptable at least to all groups larger than s.Furthermore, if Γ(s+ 1) > s+ 1 then group s+ 1 is desirable to players+1 since α(s+1) < α(Γ(s+1)). The proposition follows by induction.

33

Proposition 13:Proof. We establish the first part of the Proposition by showing

that no cycles are possible. Assume that over the course of some cyclefrom time 1 to time T > 1 exactly k groups are involved in changesin membership. Let αt ∈ Rk denote the vector of average strengths ofthese k groups in time t. Without loss of generality, let the groups besuch that k is the weakest group: α1k ∈ min(α1). In any cycle we haveα1 = αT and so in particular, min(αT ) = α1k.Consider a move by i ∈ Gh to Gj. From Lemma 4 we have that

αt(Gj ∪ {i}) < αth, or equivalently αt+1j < αth. To be acceptable to Gjwe must have that i is weak relative to j which gives αt+1j < αtj Henceαt+1j is strictly weaker than the weakest of αtj and αth. This has twoimplications:(i) If in any period t a player i ∈ Gh moves to Gj then αt+1j <

min(αtj, αth).

(ii) In any two consecutive periods min(αt) ≤ min(αt+1).Let s denote the first period in which group k is involved in a de-

mographic change and let j denote the group with which it is involved.From (i) we have min(αt+1) ≤ min(αt+1j , αt+1k ) < min(αtj, α

tk) ≤ α1k.

Then from (ii) we have min(αT ) < min(α1) contradicting the fact thatαT = α1.

Proposition 14Proof. We make use of the following nested structure of the condi-

tions.A Claim. (i) α1 <

2−5α43

implies α1 <2−5α33

(ii) α1 <2−5α43→

α2 >α1+α32

(iii) α1 > 2−5α33→ α1 >

2−5α43

(i) The first part is immediate: α1 < 2−5α43

implies α1 < 2−5α33

since α3 <α4. For (ii) we need to rule out both α1 <

2−5α43

and α2 <α1+α32. Note

that with α1 <2−5α43, and α3 < α4, we have α2 < α1+α3

2<

2−5α43

+α42

=1−α43, hence α1 + α2 + α3 + α4 <

2−5α43+ 1−α4

3+ 2α4 = 1. (iii) is implied

by (i), (ii) implies that α2 < α1+α32→ α1 >

2−5α33

B. We then proceed to establish conditions for equilibrium for eachstate. Given the nested nature of these conditions we have the followingclaims. Thus, we have ΓV is an equilibrium if and only if α1 > 2−5α4

3.

To check this claim note the admissible transitions from ΓV involve PV,I ,PV,V I , PV,V II , PV,V III . Now: PV,I = 0, since it is opposed by receivers1, 2 and 3; PV,V I = 0 since it is opposed by mover 3, PV,V III = 0since it is opposed by mover 2; everything then depends on PV,V II ;this transition is always acceptable to receiver 4 but is attractive to 1if and only if: α({1, 4}) < α({1, 2, 3}), or equivalently if and only if:α1+α42

< 1−α43↔ α1 <

2−5α43. Thus ΓV is an equilibrium if and only if

34

α1 >2−5α43. Since, from (i) α1 < 2−5α4

3implies α1 < 2−5α3

3, we have that

ΓV is an equilibrium for cases A, B, and C only. The proof follows thesame method for each of the other three states ΓV I ,ΓV II and ΓV III .

Proposition 15.Proof. (i) To check that complete integration is an equilibrium, we

show that for an arbitrary player, i, contributing to the decisions of thecommittee of the whole, N , on every realization of the policy choice ispreferred to monopoly power over one realization of policy choice andno input over all others.Setting

PN αh = 1, we need to show:

−ÃXh∈N

αhπh − π∗i

!2> 0− (n− 1)

⎛⎝ Xh∈N\{i}

αhπh1− αi

− π∗i

⎞⎠2

However this inequality we can see is equivalent to:

−ÃXh∈N

αhπh − π∗i

!2> − n− 1

(1− αi)2

ÃXh∈N

αhπh − π∗i

!2Which reduces to the simple quadratic: αi2 − 2αi + 1

n< 0. Since

we have αi ≤ 1, we can determine that this inequality holds for any

αi > 1−q

n−1n∼= 1

2n.

In other words, in a minimally egalitarian setting in which all indi-viduals have a bargaining strength of at least half of what the averagebargaining strength is, then the completely consolidated demography isan equilibrium.(ii) The previous logic shows not simply that the grand coalition is an

equilibrium, but also that no equilibrium can contain singletons. Henceto examine the more general case of a switch we consider the conditionin which for two groups A, B with nA, nB > 1, some player i ∈ A iswilling to switch to group B:

−(nB + 1)µP

B πk + π∗inB + 1

− π∗i

¶2− (nA − 1)

µPA πk − π∗i(nA − 1)

− π∗i

¶2>

−nBµP

B πknB

− π∗i

¶2− nA

µPA πknA

− π∗i

¶2For πA 6= π∗i and nA, nB > 1, this condition reduces to the following

inequality:(πB − πi)

2

(πA − πi)2 >

nAnA − 1

nB + 1

nB(16)

35

Between group heterogeneity is given by:

ρ(G|λ) = λB(

PB πknB

− π)2 + λA(

PA πknA

− π)2 +X

Gz∈Γ\(A∪B)

λz(πz − π)2

This between group heterogeneity declines due to a shift by i fromgroup A to B iff:

nB + 1

n

µPk∈B πk + πinB + 1

− π

¶2− nBn(

Pk∈B πknB

− π)2

<nAn(

PA πknA

− π)2 − nA − 1n

(

Pk∈A πk − πinA − 1

− π)2

This condition reduces to:

nBnB + 1

(πB2 − π2i

nB− 2πBπi) >

nAnA − 1

(πA2 − 2πAπi +

π2inA)

which in turn reduces to:

(πB − πi)2

(πA − πi)2>nB+ 1

nB

nAnA − 1

But this is exactly Equation 16, the condition for player i to wantto move to group B. Hence: a player will be willing to move to fromgroup A to group B if and only if such a move will reduce between groupheterogeneity.The final feature we need to check is whether a player that is willing

to move will be accepted to the new group. If so then all moves thatreduce between group heterogeneity are potential.In the egalitarian case the condition for a switch for i from group A

to group B to be acceptable to some third player s is:

−(nB + 1)µP

N πk + πinB + 1

− πs

¶2− nA

µPA πknA

− πs

¶2>

−nBµP

B πknB

− πs

¶2− (nA + 1)

µPA πk + πinA + 1

− πs

¶2

Multiplying out we find that all terms with πs cancel. In other wordsthe condition for a switch to be preferable is the same for all players. If

36

i prefers to change group, so will all members of the group she leavesbehind and all members of the group she seeks access to. This establishes(iii)

Proposition 18Proof. Consider two groups, A and B, part of a stable complete

ordered partitioning. Assume without loss of generality that πA <πB and set πA = −1, πB = 1. Consider some player i in A withπ∗i ∈ min{(πk)k∈A} < −1. This player will not wish to change onlyif: (

1+π∗i )2

(1−π∗i )2 ≥ nB

nB+1nA−1nA. The LHS is decreasing in π∗i and takes on its

highest values for low π∗i . Now, given any nA with πA = −1, let π∗LBdenote the minimum value taken by players in Group B, given our or-dering, π∗j < π∗LB for all j in A and so

(nA−1)π∗LB+π∗inA

> −1. This gives alower bound on π∗i : π∗i > −(nA − 1)π∗LB. π∗i > −nA − (nA − 1)π∗LB.Since π∗LB ≤ 1, the lowest bound is given by π∗i > 1−2nA. Hence for anynA the greatest possible value for

(1+π∗i )2

(1−π∗i )2 is

(2−2nA)2(2nA)

2 = (1−nA)2(nA)

2 and the

condition is: nA−1nA≥ nB

nB+1or nA ≥ 1+nB. By a similar logic for a player

j from B with π∗j ∈ max{(πk)k∈B} > 1 we have π∗j < 2nB − 1. Given

the condition for this player not to want to change, (1−π∗j)

2

(1+π∗j)2 <

nAnA+1

nB−1nB,

we have (nB−1)2(nB)

2 < nAnA+1

nB−1nB

and so nB−1nB

< nAnA+1

or nB < 1 + nA.

37

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