Coalition Formation in NonDemocracies
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 selfenforcing, 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 subgameperfect 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 selfenforcing by the failure of its winning subcoalitions to be selfenforcing. This is most simply illustrated by the following example: with "majority rule", twoperson coalitions are generically not selfenforcing and consequently, threeperson coalitions are selfenforcing (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 1, 3, 7, 15, etc.) can be the ruling coalition. Copyright © 2008 The Review of Economic Studies Limited.

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Page 1
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 selfenforcing, 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 selfenforcing by the failure of its
winning subcoalitions to be selfenforcing. This is most simply illustrated by the following example: with
“majority rule,” twoperson coalitions are generically not selfenforcing and consequently, threeperson
coalitions are selfenforcing (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, selfenforcing 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 EEAESEM 2007 conferences for
useful comments. Acemoglu gratefully acknowledges …nancial support from the National Science Foundation.
Page 2
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, wellfunctioning
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 selfenforcing 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 “selfenforcement”.
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 selfenforcement as follows: a coalition with
su¢cient power is winning against the rest of the society and can centralize decisionmaking
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 selfenforcing 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 predetermined 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 politicaldecision making. The term weaklyinstitutionalized 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.
1
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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 selfenforcement 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 selfenforcing 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 nonself
enforcing subcoalitions, and as a result, destroys larger coalitions that would have been self
enforcing for lower values of ?.
3. Selfenforcing coalitions are generally “fragile.” For example, under majority rule (i.e.,
? = 1=2), adding or subtracting one player from a selfenforcing coalition necessarily makes it
nonselfenforcing.
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.
2
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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 decisionmaking rule requires powerweighted 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 twoperson 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
twoperson coalition, twoperson coalitions are generically not selfenforcing.
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 twoperson coalition is self
enforcing. The lack of selfenforcing subcoalitions of fA;B;Cg implies that fA;B;Cg is itself
selfenforcing. 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 selfenforcing, 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 threeperson coalition
fA;B;Cg is selfenforcing 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 threeperson 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 selfenforcing, 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 threeperson coalitions only fB;C;Dg
is selfenforcing, and it will eliminate the weakest individual, A, and become the ruling coalition.
This example also illustrates why threeperson coalitions (22? 1 = 3) may be more likely than
twoperson (and also fourperson) coalitions.3
3It also shows that in contrast to approaches with unrestricted sidepayments (e.g., Riker, 1962), the ruling
coalition will not generally be a minimal winning coalition (the unique minimum winning coalition is fA;Dg,
3
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Although our model is abstract, it captures a range of economic forces that appear salient
in nondemocratic, weaklyinstitutionalized 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 193739 contained hundreds or even thousands
names (Conquest, 1968).
Of 40 Politburo members (28 full, 12 nonvoting) 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 selfenforcing. 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 193738. 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).
4
Page 6
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 primeminister 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 primeminister, and no one succeeded
Stalin as the Secretary General of the Communist Party. Yet in only 4 months, the allpowerful
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 selfenforcing. 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 decisionmaking (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 selfenforcing
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.
5
Page 7
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 freerider 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 risksharing
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 selfenforcing
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 “coalitionproof” 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 selfenforcement considerations. See Le Breton,
OrtunoOrtin, and Weber (2008) for an application of hedonic games to coalition formation.
In contrast, these
6
Page 8
extensiveform game of coalition formation. It then establishes the equivalence between the
ruling coalition of Section 3 and the equilibria of this extensiveform 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 nonempty
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].
7
Page 9
(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 extensiveform 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. [Agendasetting 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.
8
Page 10
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 weaklydominated 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 (SelfEnforcement) 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 selfenforcing 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 selfenforcing 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 selfenforcement axiom,
captures the key interactions in our model. It requires that any coalition Y 2 ?(X) should be
selfenforcing 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 selfenforcing 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 singlevalued.
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 selfenforcing. Equation (5) then
picks the coalitions in M(X) that have the least power. When there are no proper winning and
selfenforcing 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 singlevalued. 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 selfenforcing, that is, N 2 ?(N), if and only if
there exists no coalition X ? N, X 6= N, that is winning within N and selfenforcing. Moreover,
if N is selfenforcing, 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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selfenforcing subcoalition cannot be selfenforcing. 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 selfenforcing 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 selfenforcing, 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 14 (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 selfenforcing
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 selfenforcing or not winning could be pro…table. But the
…rst option is ruled out by induction while a deviation to a nonwinning 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 singlevalued (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 singlevalued 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 “coalitionproof”. Since the
game ? incorporates both dynamic and coalitional e¤ects and is …nite, the relevant concept of
coalitionproofness is Bernheim, Peleg, and Whinston’s (1987) Perfectly CoalitionProof 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 onetoone 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 selfenforcing 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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