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Coalition Formation in Nondemocracies?

Daron Acemoglu

MIT

Georgy Egorov

Harvard

Konstantin Sonin

New Economic School

December 2007

Abstract

We study the formation of a ruling coalition in nondemocratic societies where institutions do not

enable political commitments. Each individual is endowed with a level of political power. The ruling

coalition consists of a subset of the individuals in the society and decides the distribution of resources. A

ruling coalition needs to contain enough powerful members to win against any alternative coalition that

may challenge it and it needs to be self-enforcing, in the sense that none of its subcoalitions should be able

to secede and become the new ruling coalition. We present both an axiomatic approach that captures

these notions and determines a (generically) unique ruling coalition and the analysis of a dynamic game

of coalition formation that encompasses these ideas. We establish that the subgame perfect equilibria

of the coalition formation game coincide with the set of ruling coalitions resulting from the axiomatic

approach. A key insight of our analysis is that a coalition is made self-enforcing by the failure of its

winning subcoalitions to be self-enforcing. This is most simply illustrated by the following example: with

“majority rule,” two-person coalitions are generically not self-enforcing and consequently, three-person

coalitions are self-enforcing (unless one player is disproportionately powerful). We also characterize the

structure of ruling coalitions. For example, we determine the conditions under which ruling coalitions

are robust to small changes in the distribution of power and when they are fragile. We also show that

when the distribution of power across individuals is relatively equal and there is majoritarian voting, only

certain sizes of coalitions (e.g., with “majority rule,” coalitions of size 3, 7, 15, 31, etc.) can be the ruling

coalition.

Keywords: coalition formation, political economy, self-enforcing coalitions, stability.

JEL Classi…cation: D71, D74, C71.

?We thank Attila Ambrus, Salvador Barbera, Jon Eguia, Irina Khovanskaya, Eric Maskin, Benny Moldovanu,

Victor Polterovich, Andrea Prat, Debraj Ray, Muhamet Yildiz, three anonymous referees, and seminar par-

ticipants at the Canadian Institute of Advanced Research, MIT, the New Economic School, the Institute for

Advanced Studies, and University of Pennsylvania PIER, NASM 2007, and EEA-ESEM 2007 conferences for

useful comments. Acemoglu gratefully acknowledges …nancial support from the National Science Foundation.

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1Introduction

We study the formation of a ruling coalition in a nondemocratic (“weakly institutionalized”)

environment. A ruling coalition must be powerful enough to impose its wishes on the rest of the

society. A key ingredient of our analysis is that because of the absence of strong, well-functioning

institutions, binding agreements are not possible.1This has two important implications: …rst,

members of the ruling coalition cannot make binding o¤ers on how resources will be distributed;

second, and more importantly, members of a candidate ruling coalition cannot commit to not

eliminating (sidelining) fellow members in the future. Consequently, there is always the danger

that, once a particular coalition has formed and has centralized power in its hands, a subcoalition

will try to remove some of the original members of the coalition in order to increase the share

of resources allocated to itself. Ruling coalitions must therefore not only be powerful enough to

be able to impose their wishes on the rest of the society, but also self-enforcing so that none

of their subcoalitions are powerful enough and wish to split from or eliminate the rest of this

coalition. These considerations imply that the nature of ruling coalitions is determined by a

tradeo¤ between “power” and “self-enforcement”.

More formally, we consider a society consisting of an arbitrary number of individuals with

di¤erent amount of political or military powers (“guns”). Any subset of these individuals can

form a coalition and the power of the coalition is equal to the sum of the powers of its members.

We formalize the interplay between power and self-enforcement as follows: a coalition with

su¢cient power is winning against the rest of the society and can centralize decision-making

powers in its own hands (for example, eliminating the rest of the society from the decision-

making process). How powerful a coalition needs to be in order to be winning is determined

by a parameter ?. When ? = 1=2, this coalition simply needs to be more powerful than the

rest of the society, so this case can be thought of as “majority rule.” When ? > 1=2, the

coalition needs “supermajority” or more than a certain multiple of the power of the remainder

of the society. Once this …rst stage is completed, a subgroup can secede from or sideline the

rest of the initial winning coalition if it has enough power and wishes to do so. This process

continues until a self-enforcing coalition, which does not contain any subcoalitions that wish to

engage in further rounds of eliminations, emerges. Once this coalition, which we refer to as the

ultimate ruling coalition (URC), is formed, the society’s resources are distributed according to

some pre-determined rule (for example, resources may be distributed among the members of this

1Acemoglu and Robinson (2006) provide a more detailed discussion and various examples of commitment

problems in political-decision making. The term weakly-institutionalized polities is introduced in Acemoglu,

Robinson, and Verdier (2004) to describe societies in which institutional rules do not constrain political interactions

among various social groups or factions.

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coalition according to their powers). This simple game formalizes the two key consequences of

weak institutions mentioned above: (1) binding agreements on how resources will be distributed

are not possible; (2) subcoalitions cannot commit to not sidelining their fellow members in a

particular coalition.2

Our main results are as follows. First, we characterize the equilibria of this class of games

under general conditions. We show that a ruling coalition always exists and is “generically”

unique. Moreover, the equilibrium always satis…es some natural axioms that are motivated

by the power and self-enforcement considerations mentioned above. Therefore, our analysis

establishes the equivalence between an axiomatic approach to the formation of ruling coalitions

(which involves the characterization of a mapping that determines the ruling coalition for any

society and satis…es a number of natural axioms) and a noncooperative approach (which involves

characterizing the subgame perfect equilibria of a game of coalition formation). We also show

that the URC can be characterized recursively. Using this characterization, we establish the

following results on the structure of URCs.

1. Despite the simplicity of the environment, the URC can consist of any number of players,

and may include or exclude the most powerful individuals in the society. Consequently, the

equilibrium payo¤ of an individual is not monotonic in his power. The most powerful will

belong to the ruling coalition only if he is powerful enough to win by himself or weak enough to

be a part of a smaller self-enforcing coalition.

2. An increase in ?, that is, an increase in the degree of supermajority needed to eliminate

opponents, does not necessarily lead to larger URCs, because it stabilizes otherwise non-self-

enforcing subcoalitions, and as a result, destroys larger coalitions that would have been self-

enforcing for lower values of ?.

3. Self-enforcing coalitions are generally “fragile.” For example, under majority rule (i.e.,

? = 1=2), adding or subtracting one player from a self-enforcing coalition necessarily makes it

non-self-enforcing.

4. Nevertheless, URCs are (generically) continuous in the distribution of power across indi-

viduals in the sense that a URC remains so when the powers of the players are perturbed.

5. Coalitions of certain sizes are more likely to emerge as the URC. For example, with

majority rule (? = 1=2) and a su¢ciently equal distribution of powers among individuals, the

URC must have size 2k? 1 where k is an integer (i.e., 1, 3, 7, 15,...). A similar formula for the

size of the ruling coalition applies when ? > 1=2.

2The game also introduces the feature that once a particular group of individuals has been sidelined, they

cannot be brought back into the ruling coalition. This feature is adopted for tractability.

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We next illustrate some of the main interactions using a simple example.

Example 1 Consider two agents A and B. Denote their powers ?A> 0 and ?B> 0 and assume

that the decision-making rule requires power-weighted majority, that is, ? = 1=2. This implies

that if ?A> ?B, then starting with the coalition fA;Bg, the agent A will form a majority by

himself. Conversely, if ?A< ?B, then agent B will form a majority. Thus, “generically” (i.e.,

as long as ?A6= ?B), one of the members of the two-person coalition can secede and form a

subcoalition that is powerful enough within the original coalition. Since each agent will receive

a higher share of the scarce resources in a coalition that consists of only himself than in a

two-person coalition, two-person coalitions are generically not self-enforcing.

Now, consider a coalition consisting of three agents, A, B and C with powers ?A, ?Band

?C, and suppose that ?A< ?B< ?C< ?A+ ?B. Clearly, no two-person coalition is self-

enforcing. The lack of self-enforcing subcoalitions of fA;B;Cg implies that fA;B;Cg is itself

self-enforcing. To see this, suppose, for example, that fA;Bg considers seceding from fA;B;Cg.

They can do so since ?A+ ?B> ?C. However, we know from the previous paragraph that the

subcoalition fA;Bg is itself not self-enforcing, since after this coalition is established, agent B

would secede or eliminate A. Anticipating this, agent A would not support the subcoalition

fA;Bg. A similar argument applies for all other subcoalitions. Moreover, since agent C is

not powerful enough to secede from the original coalition by himself, the three-person coalition

fA;B;Cg is self-enforcing and will be the ruling coalition.

Next, consider a society consisting of four individuals, A;B;C and D. Suppose that we

have ?A= 3;?B= 4;?C= 5; and ?D= 10. D’s power is insu¢cient to eliminate the coalition

fA;B;Cg starting from the initial coalition fA;B;Cg. Nevertheless, D is stronger than any

two of A;B;C. This implies that any three-person coalition that includes D would not be self-

enforcing. Anticipating this, any two of fA;B;Cg would decline D’s o¤er to secede. However,

fA;B;Cg is self-enforcing, thus the three agents would be happy to eliminate D. Therefore, in

this example, the ruling coalition again consists of three individuals, but interestingly excludes

the most powerful individual D.

The most powerful individual is not always eliminated. Consider the society with ?A=

2;?B= 4;?C= 7 and ?D= 10. In this case, among the three-person coalitions only fB;C;Dg

is self-enforcing, and it will eliminate the weakest individual, A, and become the ruling coalition.

This example also illustrates why three-person coalitions (22? 1 = 3) may be more likely than

two-person (and also four-person) coalitions.3

3It also shows that in contrast to approaches with unrestricted side-payments (e.g., Riker, 1962), the ruling

coalition will not generally be a minimal winning coalition (the unique minimum winning coalition is fA;Dg,

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Although our model is abstract, it captures a range of economic forces that appear salient

in nondemocratic, weakly-institutionalized polities. The historical example of Stalin’s Soviet

Russia illustrates this in a particularly clear manner. The Communist Party Politburo was the

highest ruling body of the Soviet Union. All top government positions were held by its members.

Though formally its members were elected at Party meetings, for all practical purposes the

Politburo determined the fates of its members, as well as those of ordinary citizens. Soviet

archives contain execution lists signed by Politburo members; sometimes a list would contain

one name, but some lists from the period of 1937-39 contained hundreds or even thousands

names (Conquest, 1968).

Of 40 Politburo members (28 full, 12 non-voting) appointed between 1919 and 1952, only 12

survived through 1952. Of these 12, 11 continued to hold top positions after Stalin’s death in

March 1953. There was a single Politburo member (Petrovsky) in 33 years who left the body

and survived. Of the 28 deaths, there were 17 executions decided by the Politburo, 2 suicides,

1 death in prison immediately after arrest, and 1 assassination.

To interpret the interactions among Politburo members through the lenses of our model,

imagine that the Politburo consists of …ve members, and to illustrate our main points, suppose

that their powers are given by f3;4;5;10;20g. It can be veri…ed that with ? = 1=2, this …ve-

member coalition is self-enforcing. However, if either of the lower power individuals, 3;4;5; or

10; dies or is eliminated, then the ruling coalition consists of the singleton, 20. If, instead, 20

dies, the ultimate ruling coalition becomes f3;4;5g and eliminates the remaining most powerful

individual 10. This is because 10 is unable to form an alliance with less powerful players. While

the reality of Soviet politics in the …rst half of the century is naturally much more complicated,

this simple example sheds light on three critical episodes.

The …rst episode is the suicides of two members of the Politburo, Tomsky and Ordzhonikidze,

during 1937-38. An immediate implication of these suicides was a change in the balance of power,

something akin to the elimination of 5 in the f3;4;5;10;20g example above. In less than a year,

11 current or former members of Politburo were executed. Consistent with the ideas emphasized

in our model, some of those executed in 1939 (e.g., Chubar, Kosior, Postyshev, and Ezhov) had

earlier voted for the execution of Bukharin and Rykov in 1937. The second episode followed

the death of Alexei Zhdanov in 1948 from a heart attack. Until Zhdanov’s death, there was a

period of relative “peace”: no member of this body had been executed in nine years. Monte…ore

(2003) describes how the Zhdanov’s death immediately changed the balance in the Politburo.

The death gave Beria and Malenkov the possibility to have Zhdanov’s supporters and associates

which has the minimum power among all winning coalitions).

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in the government executed.4The third episode followed the death of Stalin himself in March

1953. Since the bloody purge of 1948, powerful Politburo members conspired in resisting any

attempts by Stalin to have any of them condemned and executed. When in the Fall of 1952,

Stalin charged two old Politburo members, Molotov and Mikoian, with being the “enemies of

the people,” the other members stood …rm and blocked a possible trial (see Monte…ore, 2003, or

Gorlizki and Khlevniuk, 2004). After Stalin’s death, Beria became the most powerful politician

in Russia. He was immediately appointed the …rst deputy prime-minister as well as the head

of the ministry of internal a¤airs and of the ministry of state security, the two most powerful

ministries in the USSR. His ally Malenkov was appointed prime-minister, and no one succeeded

Stalin as the Secretary General of the Communist Party. Yet in only 4 months, the all-powerful

Beria fell victim of a military coup by his fellow Politburo members, was tried and executed. In

terms of our simple example with powers f3;4;5;10;20g, Beria would correspond to 10. After

20 (Stalin) is out of the picture, f3;4;5g becomes the ultimate ruling coalition, so 10 must be

eliminated.

Similar issues arise in other dictatorships when top …gures were concerned with others becom-

ing too powerful. These considerations also appear to be particularly important in international

relations, especially when agreements have to be reached under the shadow of the threat of

war (e.g., Powell, 1999). For example, following both World Wars, many important features of

the peace agreements were in‡uenced by the desire that the emerging balance of power among

states should be self-enforcing. In this context, small states were viewed as attractive because

they could combine to contain threats from larger states but they would be unable to become

dominant players. Similar considerations were paramount after Napoleon’s ultimate defeat in

1815. In this case, the victorious nations designed the new political map of Europe at the

Vienna Congress, and special attention was paid to balancing the powers of Britain, Germany

and Russia, to ensure that “... their equilibrium behaviour... maintain the Vienna settlement”

(Slantchev, 2005).5

Our paper is related to models of bargaining over resources, particularly in the context of

political decision-making (e.g., models of legislative bargaining such as Baron and Ferejohn,

1989, Calvert and Dietz, 1996, Jackson and Moselle, 2002). Our approach di¤ers from these

papers, since we do not impose any speci…c bargaining structure and focus on self-enforcing

4In contrast to the two other episodes from the Soviet Politburo we discuss here, the elimination of the

associates of Zhadanov could also be explained by competition between two groups within the Politburo rather

than by competition among all members and lack of commitment, which are the ideas emphasized by our model.

5Other examples of potential applications of our model in political games are provided in Pepinsky (2007),

who uses our model to discuss issues of coalition formation in nondemocratic societies.

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ruling coalitions.6

More closely related to our work are the models of on equilibrium coalition formation, which

combine elements from both cooperative and noncooperative game theory (e.g., Peleg, 1980,

Hart and Kurz, 1983, Greenberg and Weber, 1993, Chwe, 1994, Bloch, 1996, Mariotti, 1997,

Ray, 2007, Ray and Vohra, 1997, 1999, 2001, Seidmann and Winter, 1998, Konishi and Ray, 2001,

Maskin, 2003, Eguia, 2006, Pycia, 2006). The most important di¤erence between our approach

and the previous literature on coalition formation is that, motivated by political settings, we

assume that the majority (or supermajority) of the members of the society can impose their

will on those players who are not a part of the majority. This feature both changes the nature

of the game and also introduces “negative externalities” as opposed to the positive externalities

and free-rider problems upon which the previous literature focuses (Ray and Vohra, 1999, and

Maskin, 2003). A second important di¤erence is that most of these works assume the possibility

of binding commitments (Ray and Vohra, 1997, 1999), while we suppose that players have

no commitment power. Despite these di¤erences, there are important parallels between our

results and the insights of this literature. For example, Ray (1979) and Ray and Vohra (1997,

1999) emphasize that the internal stability of a coalition in‡uences whether it can block the

formation of other coalitions, including the grand coalition. In the related context of risk-sharing

arrangements, Bloch, Genicot, and Ray (2006) show that stability of subgroups threatens the

stability of a larger group.7Another related approach to coalition formation is developed by

Moldovanu and Winter (1995), who study a game in which decisions require appoval by all

members of a coalition and show the relationship of the resulting allocations to the core of a

related cooperative game.8

Finally, Skaperdas (1998) and Tan and Wang (1999) investigate

coalition formation in dynamic contests. Nevertheless, none of these papers study self-enforcing

coalitions in political games without commitment, or derive existence, generic uniqueness and

characterization results similar to those in our paper.

The rest of the paper is organized as follows. Section 2 introduces the formal setup. Section

3 provides our axiomatic treatment. Section 4 characterizes subgame perfect equilibria of the

6See also Perry and Reny (1994), Moldovanu and Jehiel (1999), and Gomes and Jehiel (2005) for models of

bargaining with a coalition structure.

7In this respect, our paper is also related to work on “coalition-proof” Nash equilibrium or rationalizability,

e.g., Bernheim, Peleg, and Whinston (1987), Moldovanu (1992), Ambrus (2006). These papers allow deviations

by coalitions in noncooperative games, but impose that only stable coalitions can form.

considerations are captured in our model by the game of coalition formation and by the axiomatic analysis.

8Our game can also be viewed as a “hedonic game” since the utility of each player is determined by the

composition of the ultimate coalition he belongs to. However, it is not a special case of hedonic games de…ned

and studied in Bogomolnaia and Jackson (2002), Banerjee, Konishi, and Sonmez (2001), and Barbera and Gerber

(2007), because of the dynamic interactions introduced by the self-enforcement considerations. See Le Breton,

Ortuno-Ortin, and Weber (2008) for an application of hedonic games to coalition formation.

In contrast, these

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extensive-form game of coalition formation. It then establishes the equivalence between the

ruling coalition of Section 3 and the equilibria of this extensive-form game. Section 5 contains

our main results on the nature and structure of ruling coalitions in political games. Section 6

concludes. The Appendix contains the proofs of all the results presented in the text.

2 The Political Game

Let I denote the collection of all individuals, which is assumed to be …nite. The non-empty

subsets of I are coalitions and the set of coalitions is denoted by C. In addition, for any X ? I,

CXdenotes the set of coalitions that are subsets of X and jXj is the number of members in X.

In each period there is a designated ruling coalition, which can change over time. The game

starts with ruling coalition N, and eventually the ultimate ruling coalition (URC) forms. We

assume that if the URC is X, then player i obtains baseline utility wi(X) 2 R. We denote

w(?) ? fwi(?)gi2I.

Our focus is on how di¤erences in the powers of individuals map into political decisions. We

de…ne a power mapping to summarize the powers of di¤erent individuals in I:

? : I ! R++;

where R++ = R+n f0g. We refer to ?i? ? (i) as the political power of individual i 2 I.

In addition, we denote the set of all possible power mappings by R and a power mapping ?

restricted to some coalition N ? I by ?jN(or by ? when the reference to N is clear). The power

of a coalition X is ?X?P

Coalition Y ? X is winning within coalition X if and only if ?Y> ??X, where ? 2 [1=2;1)

is a …xed parameter referring to the degree of (weighted) supermajority. Naturally, ? = 1=2

i2X?i.

corresponds to majority rule. Moreover, since I is …nite, there exists a large enough ? (still less

than 1) that corresponds to unanimity rule. We denote the set of coalitions that are winning

within X by WX. Since ? ? 1=2, if Y;Z 2 WX, then Y \ Z 6= ?.

The assumption that payo¤s are given by the mapping w(?) implies that a coalition cannot

commit to a redistribution of resources or payo¤s among its members (for example, a coalition

consisting of two individuals with powers 1 and 10 cannot commit to share the resource equally

if it becomes the URC). We assume that the baseline payo¤ functions, wi(X) : I ? C ! R for

any i 2 N, satisfy the following properties.

Assumption 1 Let i 2 I and X;Y 2 C. Then:

(1) If i 2 X and i = 2 Y , then wi(X) > wi(Y ) [i.e., each player prefers to be part of the

URC].

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(2) For i 2 X and i 2 Y , wi(X) > wi(Y ) () ?i=?X> ?i=?Y( () ?X< ?Y) [i.e., for

any two URCs that he is part of, each player prefers the one where his relative power is greater].

(3) If i = 2 X and i = 2 Y , then wi(X) = wi(Y ) ? w?

URCs he is not part of].

i[i.e., a player is indi¤erent between

This assumption is natural and captures the idea that each player’s payo¤ depends positively

on his relative strength in the URC. A speci…c example of function w(?) that satis…es these

requirements is sharing of a pie between members of the ultimate ruling coalition proportional

to their power:

wi(X) =

?X\fig

?X

=

?

?i=?X

0

if i 2 X

if i = 2 X

.(1)

The reader may want to assume (1) throughout the text for interpretation purposes, though this

speci…c functional form is not used in any of our results or proofs.

We next de…ne the extensive-form complete information game ? = (N;?jN;w(?);?), where

N 2 C is the initial coalition, ? is the power mapping, w(?) is a payo¤ mapping that satis…es

Assumption 1, and ? 2 [1=2;1) is the degree of supermajority; denote the collection of such

games by G. Also, let " > 0 be su¢ciently small such that for any i 2 N and any X;Y 2 C, we

have

wi(X) > wi(Y ) =) wi(X) > wi(Y ) + 2" (2)

(this holds for su¢ciently small " > 0 since I is a …nite set). This immediately implies that for

any X 2 C with i 2 X, we have

wi(X) ? w?

i> ".(3)

The extensive form of the game ? = (N;?jN;w(?);?) is as follows. Each stage j of the game

starts with some ruling coalition Nj (at the beginning of the game N0= N). Then the stage

game proceeds with the following steps:

1. Nature randomly picks agenda setter aj;q2 Njfor q = 1.

2. [Agenda-setting step] Agenda setter aj;qmakes proposal Pj;q2 CNj, which is a subcoalition

of Njsuch that aj;q2 Pj;q(for simplicity, we assume that a player cannot propose to eliminate

himself).

3. [Voting step] Players in Pj;qvote sequentially over the proposal (we assume that players in

Njn Pj;qautomatically vote against this proposal). More speci…cally, Nature randomly chooses

the …rst voter, vj;q;1, who then casts his vote vote ~ v (vj;q;1) 2 f~ y; ~ ng (Yes or No), then Nature

chooses the second voter vj;q;26= vj;q;1, etc. After all jPj;qj players have voted, the game proceeds

to step 4 if players who supported the proposal form a winning coalition within Nj (i.e., if

fi 2 Pj;q: ~ v (i) = ~ yg 2 WNj), and otherwise it proceeds to step 5.

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4. If Pj;q = Nj, then the game proceeds to step 6. Otherwise, players from Njn Pj;q are

eliminated and the game proceeds to step 1 with Nj+1= Pj;q (and j increases by 1 as a new

transition has taken place).

5. If q < jNjj, then next agenda setter aj;q+12 Nj is randomly picked by Nature among

members of Njwho have not yet proposed at this stage (so aj;q+16= aj;rfor 1 ? r ? q), and the

game proceeds to step 2 (with q increased by 1). If q = jNjj, the game proceeds to step 6.

6. Njbecomes the ultimate ruling coalition. Each player i 2 N receives total payo¤

X

where If?gis the indicator function taking the value of 0 or 1.

The payo¤ function (4) captures the idea that an individual’s overall utility is the di¤erence

Ui= wi(Nj) ? "

1?k?jIfi2Nkg,(4)

between the baseline wi(?) and disutility from the number of transitions (rounds of elimination)

this individual is involved in. The arbitrarily small cost " can be interpreted as a cost of

eliminating some of the players from the coalition or as an organizational cost that individuals

have to pay each time a new coalition is formed. Alternatively, " may be viewed as a means

to re…ne out equilibria where order of moves matters for the outcome. Note that ? is a …nite

game: the total number of moves, including those of Nature, does not exceed 4jNj3. Notice

also that this game form introduces sequential voting in order to avoid issues of individuals

playing weakly-dominated strategies. Our analysis below will establish that the main results

hold regardless of the speci…c order of votes chosen by Nature.9

3Axiomatic Analysis

Before characterizing the equilibria of the dynamic game ?, we take a brief detour and introduce

four axioms motivated by the structure of the game ?. Although these axioms are motivated

by game ?, they can also be viewed as natural axioms to capture the salient economic forces

discussed in the introduction. The analysis in this section identi…es an outcome mapping ? : G ?

C that satis…es these axioms and determines the set of (admissible) URCs corresponding to each

game ?. This analysis will be useful for two reasons. First, it will reveal certain attractive

features of the game presented in the previous section. Second, we will show in the next section

that equilibrium URCs of this game coincide with the outcomes picked by the mapping ?.

More formally, consider the set of games ? = (N;?jN;w(?);?) 2 G. Holding ?;w and ?

…xed, consider the correspondence ? : C ? C de…ned by ?(N) = ?(N;?jN;w;?) for any N 2 C.

9See Acemoglu, Egorov, and Sonin (2006) both for the analysis of a game with simultaneous voting and a

stronger equilibrium notion, and for an example showing how, in the absence of the cost " > 0, the order of moves

may matter.

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We adopt the following axioms on ? (or alternatively on ?).

Axiom 1 (Inclusion) For any X 2 C, ?(X) 6= ? and if Y 2 ?(X), then Y ? X.

Axiom 2 (Power) For any X 2 C, Y 2 ?(X) only if Y 2 WX.

Axiom 3 (Self-Enforcement) For any X 2 C, Y 2 ?(X) only if Y 2 ?(Y ).

Axiom 4 (Rationality) For any X 2 C, for any Y 2 ?(X) and for any Z ? X such that

Z 2 WXand Z 2 ?(Z), we have that Z = 2 ?(X) () ?Y< ?Z.

Motivated by Axiom 3, we de…ne the notion of a self-enforcing coalition as a coalition that

“selects itself”. This notion will be used repeatedly in the rest of the paper.

De…nition 1 Coalition X 2 P (I) is self-enforcing if X 2 ?(X).

Axiom 1, inclusion, implies that ? maps into subcoalitions of the coalition in question (and

that it is de…ned, i.e., ?(X) 6= ?). It therefore captures the feature introduced in ? that players

that have been eliminated (sidelined) cannot rejoin the ruling coalition. Axiom 2, the power

axiom, requires a ruling coalition be a winning coalition. Axiom 3, the self-enforcement axiom,

captures the key interactions in our model. It requires that any coalition Y 2 ?(X) should be

self-enforcing according to De…nition 1. This property corresponds to the notion that in terms

of game ?, if coalition Y is reached along the equilibrium path, then there should not be any

deviations from it. Finally, Axiom 4 requires that if two coalitions Y;Z ? X are both winning

and self-enforcing and all players in Y \Z strictly prefer Y to Z, then Z = 2 ?(X) (i.e., Z cannot

be the selected coalition). Intuitively, all members of winning coalition Y (both those in Y \ Z

by assumption and those in Y nZ because they prefer to be in the URC) strictly prefer Y to Z;

hence, Z should not be chosen in favor of Y . This interpretation allows us to call Axiom 4 the

Rationality Axiom. In terms of game ?, this axiom captures the notion that, when he has the

choice, a player will propose a coalition in which his payo¤ is greater.

At the …rst glance, Axioms 1–4 may appear relatively mild. Nevertheless, they are strong

enough to pin down a unique mapping ?. Moreover, under the following assumption, these

axioms also imply that this unique mapping ? is single valued.

Assumption 2 The power mapping ? is generic in the sense that if for any X;Y 2 C, ?X= ?Y

implies X = Y . We also say that coalition N is generic or that numbers f?igi2Nare generic if

mapping ?jN is generic.

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Intuitively, this assumption rules out distributions of powers among individuals such that two

di¤erent coalitions have exactly the same total power. Notice that mathematically, genericity

assumption is without much loss of generality since the set of vectors f?igi2I2 RjIj

not generic has Lebesgue measure 0 (in fact, it is a union of a …nite number of hyperplanes in

RjIj

++that are

++).

Theorem 1 Fix a collection of players I, a power mapping ?, a payo¤ function w(?) such that

Assumption 1 holds, and ? 2 [1=2;1). Then:

1. There exists a unique mapping ? that satis…es Axioms 1–4. Moreover, when ? is generic

(i.e. under Assumption 2), ? is single-valued.

2. This mapping ? may be obtained by the following inductive procedure. For any k 2 N, let

Ck= fX 2 C : jXj = kg. Clearly, C = [k2NCk. If X 2 C1, then let ?(X) = fXg. If ?(Z) has

been de…ned for all Z 2 Cnfor all n < k, then de…ne ?(X) for X 2 Ckas

?(X) =argmin

A2M(X)[fXg

?A,(5)

where

M(X) = fZ 2 CXn fXg : Z 2 WXand Z 2 ?(Z)g.(6)

Proceeding inductively ?(X) is de…ned for all X 2 C.

The intuition for the inductive procedure is as follows. For each X, (6) de…nes M(X) as

the set of proper subcoalitions which are both winning and self-enforcing. Equation (5) then

picks the coalitions in M(X) that have the least power. When there are no proper winning and

self-enforcing subcoalitions, M(X) is empty and X becomes the URC), which is captured by

(5). The proof of this theorem, like all other proofs, is in the Appendix.

Theorem 1 establishes not only that ? is uniquely de…ned, but also that when Assumption

2 holds, it is single-valued. In this case, with a slight abuse of notation, we write ?(X) = Y

instead of ?(X) = fY g.

Corollary 1 Take any collection of players I, power mapping ?, payo¤ function w(?), and

? 2 [1=2;1). Let ? be the unique mapping satisfying Axioms 1–4. Then for any X;Y;Z 2 C,

Y;Z 2 ?(X) implies ?Y= ?Z. Coalition N is self-enforcing, that is, N 2 ?(N), if and only if

there exists no coalition X ? N, X 6= N, that is winning within N and self-enforcing. Moreover,

if N is self-enforcing, then ?(N) = fNg.

Corollary 1, which immediately follows from (5) and (6), summarizes the basic results on self-

enforcing coalitions. In particular, Corollary 1 says that a coalition that includes a winning and

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self-enforcing subcoalition cannot be self-enforcing. This captures the notion that the stability

of smaller coalitions undermines the stability of larger ones.

As an illustration to Theorem 1, consider again three players A, B and C and suppose that

? = 1=2. For any ?A< ?B< ?C< ?A+ ?B, Assumption 2 is satis…ed and it is easy to see

that fAg, fBg, fCg, and fA;B;Cg are self-enforcing coalitions, whereas ?(fA;Bg) = fBg,

?(fA;Cg) = ?(fB;Cg) = fCg. In this case, ?(X) is a singleton for any X. On the other hand,

if ?A= ?B= ?C, all coalitions except fA;B;Cg would be self-enforcing, while ?(fA;B;Cg) =

ffA;Bg;fB;Cg;fA;Cgg in this case.

4Equilibrium Characterization

We now characterize the Subgame Perfect Equilibria (SPE) of game ? de…ned in Section 2 and

show that they correspond to the ruling coalitions identi…ed by the axiomatic analysis in the

previous section. The next subsection provides the main results. We then provide a sketch of

the proofs. The formal proofs are contained in the Appendix.

4.1 Main Results

The following two theorems characterize the Subgame Perfect Equilibrium (SPE) of game ? =

(N;?jN;w;?) with initial coalition N. As usual, a strategy pro…le ? in ? is a SPE if ? induces

continuation strategies that are best responses to each other starting in any subgame of ?,

denoted ?h, where h denotes the history of the game, consisting of actions in past periods

(stages and steps).

Theorem 2 Suppose that ?(N) satis…es Axioms 1-4 (cfr. (5) in Theorem 1). Then, for any

K 2 ?(N), there exists a pure strategy pro…le ?K that is an SPE and leads to URC K in at

most one transition. In this equilibrium player i 2 N receives payo¤

Ui= wi(K) ? "Ifi2KgIfN6=Kg: (7)

This equilibrium payo¤ does not depend on the random moves by Nature.

Theorem 2 establishes that there exists a pure strategy equilibrium leading to any coalition

that is in the set ?(N) de…ned in the axiomatic analysis of Theorem 1.10This is intuitive in view

of the analysis in the previous section: when each player anticipates members of a self-enforcing

ruling coalition to play a strategy pro…le such that they will turn down any o¤ers other than K

10It can also be veri…ed that Theorem 2 holds even when " = 0. The assumption that " > 0 is used in Theorem

3.

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and they will accept K, it is in the interest of all the players in K to play such a strategy for any

history. This follows because the de…nition of the set ?(N) implies that only deviations that

lead to ruling coalitions that are not self-enforcing or not winning could be pro…table. But the

…rst option is ruled out by induction while a deviation to a non-winning URC will be blocked

by su¢ciently many players. The payo¤ in (7) is also intuitive. Each player receives his baseline

payo¤ wi(K) resulting from URC K and then incurs the cost " if he is part of K and if the

initial coalition N is not equal to K (because in this latter case, there will be one transition).

Notice that Theorem 2 is stated without Assumption 2 and does not establish uniqueness. The

next theorem strengthens these results under Assumption 2.

Theorem 3 Suppose Assumption 2 holds and suppose ?(N) = K. Then any (pure or mixed

strategy) SPE results in K as the URC. The payo¤ player i 2 N receives in this equilibrium is

given by (7).

Since Assumption 2 holds, the mapping ? is single-valued (with ?(N) = K). Theorem 3

then shows that even though the SPE may not be unique in this case, any SPE will lead to

K as the URC. This is intuitive in view of our discussion above. Because any SPE is obtained

by backward induction, multiplicity of equilibria results only when some player is indi¤erent

between multiple actions at a certain nod. However, as we show, this may only happen when a

player has no e¤ect on equilibrium play and his choice between di¤erent actions has no e¤ect on

URC (in particular, since ? is single-valued in this case, a player cannot be indi¤erent between

actions that will lead to di¤erent URCs).

It is also worth noting that the SPE in Theorems 2 and 3 is “coalition-proof”. Since the

game ? incorporates both dynamic and coalitional e¤ects and is …nite, the relevant concept of

coalition-proofness is Bernheim, Peleg, and Whinston’s (1987) Perfectly Coalition-Proof Nash

Equilibrium (PCPNE). This equilibrium re…nement requires that the candidate equilibrium

should be robust to deviations by coalitions in all subgames when the players take into account

the possibility of further deviations. Since ? introduces more general coalitional deviations

explicitly, it is natural to expect the SPE in ? to be PCPNE. Indeed, if Assumption 2 holds, it

is straightforward to prove that the set of PCPNE coincides with the set of SPE.11

4.2 Sketch of the Proofs

We now provide an outline of the argument leading to the proofs of the main results presented

in the previous subsection and we present two key lemmas that are central for these theorems.

11A formal proof of this result follows from Lemma 2 below and is available from the authors upon request.

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Consider the game ? and let ? be as de…ned in (5). Take any coalition K 2 ?(N). We will

outline the construction of the pure strategy pro…le ?Kwhich will be a SPE and lead to K as

the URC.

Let us …rst rank all coalitions so as to “break ties” (which are possible, since we have not yet

imposed Assumption 2). In particular, n : C !?1;:::;2jIj? 1?be a one-to-one mapping such

that for any X;Y 2 C, ?X> ?Y) n(X) > n(Y ), and if for some X 6= K we have ?X= ?K,

then n(X) > n(K) (how the ties among other coalitions are broken is not important). With

this mapping, we have thus ranked (enumerated) all coalitions such that stronger coalitions are

given higher numbers, and coalition K receives the smallest number among all coalitions with

the same power. Now de…ne the mapping ? : C ! C as

?(X) = argmin

Y 2?(X)

n(Y ). (8)

Intuitively, this mapping picks an element of ?(X) for any X and satis…es ?(N) = K. Also,

note that ? is a projection in the sense that ?(?(X)) = ?(X). This follows immediately since

Axiom 3 implies ?(X) 2 ?(?(X)) and Corollary 1 implies that ?(?(X)) is a singleton.

The key to constructing a SPE is to consider o¤-equilibrium path behavior. To do this,

consider a subgame in which we have reached a coalition X (i.e., j transitions have occurred

and Nj = X) and let us try to determine what the URC would be if proposal Y is accepted

starting in this subgame. If Y = X, then the game will end, and thus X will be the URC. If,

on the other hand, Y 6= X, then the URC must be some subset of Y . Let us de…ne the strategy

pro…le ?Ksuch that the URC will be ?(Y ). We denote this (potentially o¤-equilibrium path)

URC following the acceptance of proposal Y by X(Y ), so that

X(Y ) =

?

?(Y )

X

if Y 6= X;

otherwise.

(9)

By Axiom 1 and equations (8) and (9), we have that

X = Y () X(Y ) = X: (10)

We will introduce one …nal concept before de…ning pro…le ?K. Let FX(i) denote the “fa-

vorite” coalition of player i if the current ruling coalition is X. Naturally, this will be the weakest

coalition among coalitions that are winning within X, that are self-enforcing and that include

player i. If there are several such coalitions, the de…nition of FX(i) picks the one with the

smallest n, and if there are none, it picks X itself. Therefore,

FX(i) = argmin

Y 2fZ:Z?X;Z2WX;?(Z)=Z;Z3ig[fXg

n(Y ).(11)

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