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Department of Economics
Working Paper Series
Coalition Formation in
Non-Democracies
Daron Acemoglu
Georgy Egorov
Konstantin Sonin
Working Paper 06-33
November 30, 2006
(Originally titled Coalition Formation
in Political
*REVISED:
Games)
Dec. 31, 2007
& Sept.
14,
2009
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at
Coalition Formation in Nondemocracies*
Daron Acemoglu
Georgy Egorov
Harvard
MIT
Konstantin Sonin
School
New Economic
December 2007
Abstract
We
study the formation of a ruling coalition
in
nondemocratic societies where institutions do not
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
enable political commitments.
of the coalition formation
A
Each individual
game
is
coincide with the set of ruling coalitions resulting from the axiomatic
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
approach.
key insight of our analysis
coalition.
Keywords:
JEL
coalition formation, political economy, self-enforcing coalitions, stability.
Classification: D71, D74, C71.
*We thank
Attila
Ambrus, Salvador Barbera, Jon Eguia,
Victor Polterovich, Andrea Prat, Debraj Ray,
Muhamet
Irina
Khovanskaya, Eric Maskin, Benny Moldovanu,
anonymous referees, and seminar par-
Yildiz, three
Canadian Institute of Advanced Research, MIT, the New Economic School, the Institute
Advanced Studies, and University of Pennsylvania PIER, NASM 2007, and EEA-ESEM 2007 conferences
useful comments. Acemoglu gratefully acknowledges financial support from the National Science Foundation.
ticipants at the
Electronic
copy available
at:
http://ssrn.com/abstract=948585
for
for
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/coalitionformatiOOacem
Introduction
1
We
study the formation of a ruling coalition in a nondemocratic ("weakly institutionalized")
environment.
society.
A
ruling coalition
A key ingredient of our
institutions, binding
members
must be powerful enough to impose
analysis
members
that, once a particular coalition has
of a candidate ruling coalition cannot
in the future. Consequently, there
is
tradeoff between "power"
we
formally,
amount
rest of the society,
enough and wish
powers in
making
its
in order to increase the share
but also self-enforcing so that none
to split
from or eliminate the
number
consider a society consisting of an arbitrary
of political or military powers
is
own hands
How
process).
a.
(for
determined by a
is
(
"guns" )
.
Any
equal to the
of individuals with
subset of these individuals can
sum
of the powers of
When a =
Once
continues until a
members.
a coalition with
powerful a coalition needs to be in order to be winning
1/2, this coalition simply needs to be
can be thought of as "majority
coalition needs "supermajority" or
rest of the initial
its
example, eliminating the rest of the society from the decision-
rest of the society, so this case
of the society.
rest of this
winning against the rest of the society and can centralize decision-making
is
by a parameter
to not
always the danger
formalize the interplay between power and self-enforcement as follows:
power
commit
and "self-enforcement"
form a coalition and the power of the coalition
sufficient
be distributed;
in its hands, a subcoalition
These considerations imply that the nature of ruling coalitions
coalition.
will
first,
Ruling coalitions must therefore not only be powerful enough to
itself.
of their subcoalitions are powerful
We
on how resources
formed and has centralized power
be able to impose their wishes on the
different
offers
remove some of the original members of the coalition
of resources allocated to
More
that because of the absence of strong, well-functioning
make binding
more importantly, members
eliminating (sidelining) fellow
will try to
wishes on the rest of the
agreements are not possible. 1 This has two important implications:
of the ruling coalition cannot
second, and
is
its
more than a
this first stage is completed, a
winning coalition
if it
self- enforcing coalition,
some pre-determined
is
When a >
1/2, the
power of the remainder
subgroup can secede from or sideline the
has enough power and wishes to do
so.
This process
which does not contain any subcoalitions that wish to
engage in further rounds of eliminations, emerges. Once
ultimate ruling coalition (URC),
determined
more powerful than the
rule."
certain multiple of the
is
this coalition,
which we
refer to as the
formed, the society's resources are distributed according to
rule (for example, resources
may be
distributed
among
the
members
of this
Acemoglu 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
1
among
various social groups or factions.
Electronic copy available
at:
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game
coalition according to their powers). This simple
weak
mentioned above:
institutions
formalizes the two key consequences of
binding agreements on
(1)
are not possible; (2) subcoalitions cannot
commit
how
resources will be distributed
to not sidelining their fellow
members
in a
particular coalition. 2
Our main
results are as follows. First,
We
under general conditions.
we
show that a
Moreover, the equilibrium always
unique.
characterize the equilibria of this class of
ruling coalition always exists
and
is
games
"generically"
some natural axioms that are motivated
satisfies
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
satisfies
characterizing the
that the
URC
a
number
subgame
of natural axioms)
perfect equilibria of a
can be characterized recursively.
following results on the structure of
1.
and a noncooperative approach (which involves
game
of coalition formation).
if
be a part of a smaller self-enforcing
2.
An
increase in a, that
show
URCs.
URC
can consist of any number of players,
and may include or exclude the most powerful individuals
belong to the ruling coalition only
also
Using this characterization, we establish the
Despite the simplicity of the environment, the
equilibrium payoff of an individual
We
in the society.
not monotonic in his power.
is
he
is
Consequently, the
The most powerful
will
powerful enough to win by himself or weak enough to
coalition.
an increase in the degree of supermajority needed to eliminate
is,
'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 a.
3.
a
=
Self-enforcing coalitions are generally "fragile." For example, under majority rule
1/2), adding or subtracting
(i.e.,
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
5.
remains so when the powers of the players are perturbed.
Coalitions of certain sizes are
majority rule (a
URC
URC
must have
=
1/2)
size 2
fc
and a
—
1
more
likely to
emerge as the URC. For example, with
sufficiently equal distribution of
where k
size of the ruling coalition applies
is
an integer
when a >
(i.e.,
1, 3, 7,
powers among individuals, the
15,...).
A
similar formula for the
1/2.
"The 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.
Electronic copy available
at:
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We
next illustrate some of the main interactions using a simple example.
Example
1 Consider
A
two agents
and B. Denote
powers
their
>
7,4
that the decision-making rule requires power-weighted majority, that
that
if
> 7B
7^4
then starting with the coalition {A, B}, the agent
,
Conversely,
himself.
as long as 7^4
if
jA <
7 S ), one
7^
subcoalition that
-y
B
B then agent
,
members
of the
will
and 75 >
is,
A
a =
will
and assume
1/2. This implies
form a majority by
form a majority. Thus, "generically"
of the two-person coalition can secede
(i.e.,
and form a
powerful enough within the original coalition. Since each agent will receive
is
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
< jB < jc <
Jq, and suppose that j A
The
enforcing.
self-enforcing.
They can do
7,4
To
so since 7,4
B}
+ 7# >
is itself
7c- However, we
similar
B}
C
and
with powers 74, 7 B and
no two-person coalition
Clearly,
7b-
see this, suppose, for example, that {A,
C}
implies that {A, B,
argument applies
this,
know from
agent
A
C}
is self-
itself
is
considers seceding from {A, B, C}.
the previous paragraph that the
not self-enforcing, since after this coalition
would secede or eliminate A. Anticipating
A
+
lack of self-enforcing subcoalitions of {A, B,
subcoalition {A,
{A, B}.
B
coalition consisting of three agents, A,
is
established, agent
B
would not support the subcoalition
Moreover, since agent
for all other subcoalitions.
C
is
not powerful enough to secede from the original coalition by himself, the three-person coalition
{A, B,
C}
is
self-enforcing
and
will
be the ruling coalition.
Next, consider a society consisting of four individuals, A, B,
=
have j A
B,C}
{A,
3,7#
=
4,7^
=
5,
and
"f
D =
10.
D's power
starting from the initial coalition {A,
is
C
and D.
insufficient to eliminate the coalition
B,C}. Nevertheless,
D
two of A, B, C. This implies that any three-person coalition that includes
enforcing. Anticipating this, any two of {A,
{A, B,
C}
is
B,C} would
self-enforcing, thus the three agents
Suppose that we
is
stronger than any
D
would not be
self-
decline D's offer to secede. However,
would be happy to eliminate D. Therefore, in
example, the ruling coalition again consists of three individuals, but interestingly excludes
this
the most powerful individual D.
The most powerful
2,
is
B =
4,
C =
self-enforcing,
7
individual
and 7 D
and
it
=
is
not always eliminated.
will eliminate the
This example also illustrates
why
It
also
shows that
7^4
the three-person coalitions only {B, C,
=
D}
weakest individual, A, and become the ruling coalition.
three-person coalitions (2
two-person (and also four-person) coalitions.
2
—
1
=
3)
may be more
likely
than
3
approaches with unrestricted side-payments (e.g., Riker, 1962), the ruling
minimal winning coalition (the unique minimum winning coalition is {A, D},
in contrast to
coalition will not generally be a
among
10. In this case,
Consider the society with
Although our model
abstract,
is
captures a range of economic forces that appear salient
it
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
Though formally
of the Soviet Union. All top government positions were held by
members were
its
Politburo determined the fates of
archives contain execution
one name, but some
names (Conquest,
practical purposes the
as those of ordinary citizens.
list
would contain
1968).
Of
There was a
(28
full,
12 non-voting) appointed between 1919
and 1952, only 12
these 12, 11 continued to hold top positions after Stalin's death in
single Politburo
member
(Petrovsky) in 33 years
who
left
and survived. Of the 28 deaths, there were 17 executions decided by the Politburo,
I
death in prison immediately after
To
interpret the interactions
arrest,
and
1
five
members, and to
illustrate
It
member
either of the lower
self-enforcing.
ultimate ruling coalition becomes {3,
individual 10. This
is
because 10
is
the reality of Soviet politics in the
this simple
The
example sheds
first
episode
during 1937-38.
is
light
4,
first
the suicides of two
If,
instead, 20
5} and eliminates the remaining most powerful
half of the century
on three
1/2, this five-
is
less
naturally
powerful players. While
much more
complicated,
critical episodes.
members
of the Politburo,
Tomsky and Ordzhonikidze,
An immediate implication of these suicides was a change in the balance of power,
II current or former
in our model,
=
power individuals, 3,4,5, or
unable to form an alliance with
something akin to the elimination of 5
earlier
if
2 suicides,
our main points, suppose
then the ruling coalition consists of the singleton, 20.
10, dies or is eliminated,
dies, the
However,
body
lenses of our model,
can be verified that with a
that their powers are given by {3,4,5, 10,20}.
is
the
assassination.
among Politburo members through the
imagine that the Politburo consists of
coalition
Soviet
from the period of 1937-39 contained hundreds or even thousands
lists
survived through 1952.
1953.
members, as well
its
all
signed by Politburo members; sometimes a
lists
Of 40 Politburo members
March
elected at Party meetings, for
members.
its
some
members
in the {3, 4, 5, 10, 20}
example above. In
less
of Politburo were executed. Consistent with the ideas
of those executed in 1939
(e.g.,
than a year,
emphasized
Chubar, Kosior, Postyshev, and Ezhov) had
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"
(2003) describes
how
:
no member of
body had been executed
in nine years. Montefiore
the Zhdanov's death immediately changed the balance in the Politburo.
The death gave Beria and Malenkov
vhich has the
this
minimum power among
all
the possibility to have Zhdanov's supporters and associates
winning
coalitions).
in the
government executed. 4 The third episode followed the death of Stalin himself
1953.
Since the bloody purge of 1948, powerful Politburo
members conspired
attempts by Stalin to have any of them condemned and executed.
Stalin charged
members stood
and Khlevniuk, 2004). After
of the ministry of internal affairs
ministries in the
Stalin's death, Beria
USSR. His
ally
fell
the
first
any
in the Fall of 1952,
became the most powerful
politician
deputy prime-minister as well as the head
and of the ministry of
state security, the
two most powerful
Malenkov was appointed prime- minister, and no one succeeded
Communist
Stalin as the Secretary General of the
Beria
in resisting
firm and blocked a possible trial (see Montefiore, 2003, or
He was immediately appointed
in Russia.
March
two old Politburo members, Molotov and Mikoian, with being the "enemies of
the people," the other
Gorlizki
When
in
Party. Yet in only 4 months, the all-powerful
victim of a military coup by his fellow Politburo members, was tried and executed. In
terms of our simple example with powers
20 (Stalin)
is
out of the picture, {3,
4,
{3, 4, 5, 10, 20},
Beria would correspond to
10.
After
5} becomes the ultimate ruling coalition, so 10 must be
eliminated.
Similar issues arise in other dictatorships
when top
were concerned with others becom-
figures
ing too powerful. These considerations also appear to be particularly important in international
relations, especially
war
(e.g.,
when agreements have
to be reached under the
Powell, 1999). For example, following both
shadow
of the threat of
World Wars, many important features
the peace agreements were influenced by the desire that the emerging balance of power
of
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
1815.
Similar considerations were paramount after Napoleon's ultimate defeat in
players.
In this case, the victorious nations designed the
Vienna Congress, and
special attention
and Russia, to ensure that
(Slantchev, 2005).
Our paper
is
"...
new
political
map
of
Europe
at the
was paid to balancing the powers of Britain, Germany
maintain the Vienna settlement"
their equilibrium behaviour...
5
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
papers, since
we do not impose any
specific bargaining structure
In contrast to the two other episodes from the Soviet Politburo
Zhadanov could
we
differs
and focus on
from these
self-enforcing
discuss here, the elimination of the
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.
5
Other 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.
associates of
also
ruling coalitions.
More
6
closely related to our
work are the models of on equilibrium
coalition formation,
combine elements from both cooperative and noncooperative game theory
(e.g.,
which
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 difference between our approach
and the previous
on coalition formation
literature
assume that the majority
that, motivated
is
members
(or supermajority) of the
by
political settings,
we
of the society can impose their
on those players who are not a part of the majority. This feature both changes the nature
will
of the
and
game and
free-rider
A
second important difference
is
that most of these works assume the possibility
commitments (Ray and Vohra, 1997,
no commitment power.
and the
results
opposed to the positive externalities
problems upon which the previous literature focuses (Ray and Vohra, 1999, and
Maskin, 2003).
of binding
also introduces "negative externalities" as
1999), while
we suppose
that players have
Despite these differences, there are important parallels between our
insights of this literature. For example,
Ray
(1979) and
Ray and Vohra
1999) emphasize that the internal stability of a coalition influences whether
it
(1997,
can block the
formation of other poalitions, 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.
Moldovanu and Winter
members
of a coalition
7
Another related approach to coalition formation
(1995),
who study
a
game
Finally,
Skaperdas (1998) and Tan and
coalition formation in
dynamic
coalitions in political
games without commitment, or derive
contests. Nevertheless,
developed by
which decisions require appoval by
in
and show the relationship of the resulting allocations
related cooperative game. 8
is
Wang
all
to the core of a
(1999) investigate
none of these papers study self-enforcing
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.
6
Section 4 characterizes
See also Perry and Reny (1994), Moldovanu and Jehiel (1999), and
subgame
Gomes and
perfect equilibria of the
Jehiel (2005) for
models of
bargaining with a coalition structure.
is also related to work on "coalition-proof Nash equilibrium or rationalizability,
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. In contrast, these
considerations are captured in our model by the game of coalition formation and by the axiomatic analysis.
Our 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 defined
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.
'
e.g.,
In this respect, our paper
game
extensive-form
of coalition formation. It then establishes the equivalence between the
and the
ruling coalition of Section 3
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
The Appendix
concludes.
The
2
X
Let
I
are coalitions
all
and the
individuals,
which
=
w(-)
{wi{-)} ieI
Our
define a
focus
URC
the
if
is
X, then player
is
on how
R++ = R+
We
\ {0}.
restricted to
some
Coalition
the
is
number
coalition
is
=
refer to 7,
C T by j\n
(or
within
YCX
is
WX
-
Since
The assumption
commit
members
C
in
We
X,
X.
game
We
forms.
denote
into political decisions.
We
different individuals in I:
7(2) as the political power of individual
i
6
I,
power mappings by 1Z and a power mapping 7
by 7 when the reference to
winning within coalition
that corresponds to unanimity rule.
X by
X
N
is
clear).
The power
7^ = Yltex 7r
corresponds to majority rule. Moreover, since
1)
any
H-+!
set of all possible
N
map
X
if
and only
if
jy > ajxi where a £
a fixed parameter referring to the degree of (weighted) supermajority.
than
of
(URC)
ruling coalition
powers of individuals
differences in the
we denote the
X
C. In addition, for
obtains baseline utility Wi (X) £ R.
i
power mapping to summarize the powers of
of a coalition
and \X\
The non-empty
to be finite.
.
In addition,
is
X
N, and eventually the ultimate
7 :J
where
assumed
a designated ruling coalition, which can change over time. The
is
starts with ruling coalition
assume that
is
denoted by
set of coalitions is
denotes the set of coalitions that are subsets of
In each period there
the results presented in the text.
all
Game
Political
denote the collection of
subsets of
Cx
contains the proofs of
a >
1/2,
if
Y,
Z
£
J
is finite,
We
WX
,
to a redistribution of resources or payoffs
consisting of two individuals with powers
if it
becomes the URC).
any
i
We
1
Naturally,
a = 1/2
a
(still less
there exists a large enough
denote the set of coalitions that are winning
then
that payoffs are given by the
[1/2, 1)
YDZ^
mapping w
among
its
0.
(•)
implies that a coalition cannot
members
(for
example, a coalition
and 10 cannot commit to share the resource equally
assume that the baseline payoff functions, W{ (X)
:
1
x C
— R
>
for
£ N, satisfy the following properties.
Assumption
(1) If
URC].
i
£
1 Let
X
i
and
£
i
1 and X,Y
<£
6 C. Then:
Y, then W{ (X)
>
W{ (Y)
/i.e.,
each player prefers to be part of the
ForieX andieY
(2)
any two URCs that he
(3) If
URCs
he
X
£
i
and
not part
is
This assumption
on
is
Wi{X) >
,
part
is
=
W{ (Y)
= w~
a player
[i.e.,
is
is
[i.e.,
for
greater].
indifferent between
natural and captures the idea that each player's payoff depends positively
URC. A
sharing of a pie between
is
lx < 7y )
<=^>
of].
his relative strength in the
requirements
(
each player prefers the one where his relative power
of,
£ Y, then wt (X)
i
«=> 7^/7* > 7^7^
(Y)
iu;
example of function
specific
members
w ()
that satisfies these
of the ultimate ruling coalition proportional
to their power:
(x)=
The reader may want
specific functional
We
form
assume
Jihx Hi
(
=
throughout the text
(1)
the
is
Assumption
games by
and a G
Q. Also, let e
though this
not used in any of our results or proofs.
is
7
initial coalition,
1,
ex
for interpretation purposes,
game T = (N,j\n,w(-)
next define the extensive-form complete information
C
TV G
to
fxn{i}
is
[1/2, 1) is the
>
be
w ()
the power mapping,
a),
mapping that
a payoff
is
,
where
satisfies
degree of supermajority; denote the collection of such
sufficiently small
such that for any
i
G
N
and any X,
Y
G C, we
have
Wi (X)
(this holds for sufficiently small e
X
any
G C with
i
>
=»
Wi (Y)
>
I
since
extensive form of the
starts with
some
2.
of
game V
ruling coalition
game proceeds with the
1.
is
a
>
Wi (Y)
+ 2e
(2)
This immediately implies that for
finite set).
G X, we have
wt (X)
The
Wi (X)
=
Nj
- wT >
{N,~f\x,w
(•)
,a)
e.
(3)
is
Each stage j of the game
as follows.
game
(at the beginning of the
=
iVo
N). Then the stage
following steps:
Nature randomly picks agenda setter aj >q G Nj
[Agenda-setting step]
Nj such that aJj9 G Pj iq
a^
Agenda setter
=
1.
makes proposal Pj tQ G
we assume
(for simplicity,
for q
Cjv,-
,
which
is
a subcoalition
that a player cannot propose to eliminate
himself)
3.
Nj
[Voting step] Players in Pj tQ vote sequentially over the proposal (we assume that players in
\ Pj iQ automatically vote against this proposal).
the
first
voter, Ujo.i,
who then
chooses the second voter
to step 4
{i
G
Pj,
q
:
if
v
players
(i)
=
Uj,
9) 2
More
casts his vote vote v
7^ ^',9,11
who supported
e^ c
-
After
(fj/,
specifically,
g ,i)
all \Pj q\
t
Nature randomly chooses
G {y,n} (Yes or No), then Nature
players have voted, the
game proceeds
the proposal form a winning coalition within Nj
y} G Wjv,)i an d otherwise
it
proceeds to step
8
5.
(i.e.,
if
4.
If
=
Pj tQ
game proceeds
Nj, then the
eliminated and the
game proceeds
to step
to step
Otherwise, players from Nj \ Pj tq are
6.
=
with Nj+i
1
Pj A (and j increases by
as a
1
new
transition has taken place).
5.
If
<
q
members
\Nj\,
Nj who have not yet proposed
of
game proceeds
6.
then next agenda setter
where
is
Ii-.i
The
by
1).
ruling coalition.
Ui
£ Nj
9 +i
is
randomly picked by Nature among
^
at this stage (so aj, 9 +i
to step 2 (with q increased
Nj becomes the ultimate
aj,
If
q
=
€
i
<
and
this individual is involved in.
eliminating
some of the
(4)
or
1.
disutility
The
the difference
is
from the number of transitions (rounds of elimination)
can be interpreted as a cost of
arbitrarily small cost e
players from the coalition or as an organizational cost that individuals
is
may be viewed
formed. Alternatively, e
where order of moves matters
for the
game form
means
as a
outcome. Note that V
is
game: the total Dumber of moves, including those of Nature, does not exceed 4|iV|
also that this
6.
i
have to pay each time a new coalition
to refine out equilibria
to step
N receives total payoff
payoff function (4) captures the idea that an individual's overall utility
(•)
and the
q),
= w (Nj)-eJ2 1 < k < I {ieNk },
j
the indicator function taking the value of
between the baseline W{
<
r
game proceeds
\Nj\, the
Each player
aj jT for 1
a finite
Notice
.
introduces sequential voting in order to avoid issues of individuals
Our
playing weakly-dominated strategies.
analysis below will establish that the
hold regardless of the specific order of votes chosen by Nature.
main
results
9
Axiomatic Analysis
3
Before characterizing the equilibria of the dynamic
four axioms motivated by the structure of the
by game
T, they can also
game
satisfies
T.
we take a
The
analysis in this section identifies an
these axioms and determines the set of (admissible)
game presented
that equilibrium
URCs
of this
game
correspondence
<fr
salient
:
=
(TV,
C =1 C defined by
economic forces
outcome mapping $
URCs
:
Q
=4
corresponding to each
we
will
show
in the next section
coincide with the outcomes picked by the
games F
and introduce
First, it will reveal certain attractive
in the previous section. Second,
formally, consider the set of
fixed, consider the
brief detour
Although these axioms are motivated
T.
This analysis will be useful for two reasons.
features of the
More
game
T,
be viewed as natural axioms to capture the
discussed in the introduction.
C that
game
w (•) a 6 Q- Holding
(TV) = $ (TV, 7|yv, w, a) for
^\n,
<f>
mapping
,
)
$.
w
and a
any
N E C.
7,
See 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 e > 0, the order of moves
may
matter.
We
adopt the following axioms on'
X
Axiom
1
Axiom
2 (Power) For any
Axiom
3 (Self-Enforcement) For any
Axiom
4 (Rationality) For any
(Inclusion) For any
X
on $).
(or alternatively
<f>
EC,
Y
E C,
(X)
<f>
E
^
and
[X) only
4>
X
if
Y
E
E C, for any
Y
E C,
if
Y
YE
<p
E
YC
(X), then
<p
X.
%.
{X) only
YE
if
<j>
(Y).
[
Z
Wx and Z E
E
we have
(Z),
<j>
Motivated by Axiom
"selects itself
that
it is
1,
we
Z
that
<=> i Y < 7 Z
4>{X)
<£
E 4>{X) and
X E P (X)
inclusion, implies that
defined,
i.e.,
(X)
<f>
^=
0).
self-enforcing
is
maps
<j>
It
if
XE
<fi
(X).
into subcoalitions of the coalition in question (and
self-enforcing according to Definition
T,
if
deviations from
and
Y
coalition
it.
self-enforcing
Finally,
and
Axiom
by assumption and those
Z
.
Axiom
if
two coalitions Y,Z
strictly prefer
members
Y
to Z, then
Z
of winning coalition
because they prefer to be in the
Y
E
C
£
Y
URC)
X
cfi
At the
first
glance,
game
Assumption
7^
is
We
which
Axioms 1-4 may appear
this
(X) should be
is
his payoff
is
(i.e.,
(both those
Z
mY(~\Z
Y
to Z;
Axiom
4 the
strictly prefer
that,
call
cannot
when he has
the
greater.
relatively mild.
cp is
Nevertheless, they are strong
single valued.
generic in the sense that
also say that coalition
are both winning
Moreover, under the following assumption, these
<f>.
unique mapping
2 The power mapping 7
X = Y.
axiom captures the notion
T, this
enough to pin down a unique mapping
axioms also imply that
4>
(X)
should not be chosen in favor of Y. This interpretation allows us to
choice, a player will propose a coalition in
mapping
the self-enforcement axiom,
requires that any coalition
4 requires that
in7nZ
Y\Z
Rationality Axiom. In terms of
implies
3,
the power
This property corresponds to the notion that in terms
1.
Intuitively, all
in
It
Axiom
2,
reached along the equilibrium path, then there should not be any
players
all
be the selected coalition)
hence,
is
T that players
therefore captures the feature introduced in
captures the key interactions in our model.
game
such that
define the notion of a self-enforcing coalition as a coalition that
axiom, requires a ruling coalition be a winning coalition.
of
X
.
that have been eliminated (sidelined) cannot rejoin the ruling coalition.
,
Z C
any
for
This notion will be used repeatedly in the rest of the paper.
.
Definition 1 Coalition
Axiom
3,
X
N
is
generic.
10
if for
any
X,Y
E C,
7^ = 7y
generic or that numbers {7;}j e /v are 9 eneric
if
Intuitively, this
assumption rules out distributions of powers among individuals such that two
different coalitions
have exactly the same total power. Notice that mathematically, genericity
assumption
much
without
is
loss of generality since the set of vectors {ji}i j
€
not generic has Lebesgue measure
(in fact,
it is
number
a union of a finite
^++
€
that are
of hyperplanes in
R&).
Theorem
1 Fix a collection of players 1, a
Assumption
1.
(i.e.
This mapping
= {X
£ C
been defined for
all
is
6 C
n
Axioms
that satisfies
cp
1^4. Moreover,
be obtained by the following inductive procedure.
for
such that
when 7
is
generic
single-valued.
C
Clearly,
fc}.
Z
<fi
may
<f>
=
\X\
:
2),
w (•)
7, a payoff function
Then:
[1/2, 1).
There exists a unique mapping
under Assumption
2.
Ck
and a £
1 holds,
power mapping
all
= U k€N C k
n <
If
.
X
e C
then define
k,
4>(X)=
4>
1
then
,
(X) for
argmin
=
let <f>(X)
X
6 C
k
For any k E N,
{X}.
let
has
If 4>{Z)
as
7^,
(5)
AeM(X)u{X}
where
M (X) = {Z e Cx
Proceeding inductively
The
4>
(X)
is
:Z£Wx
{X}
\
defined for all
intuition for the inductive procedure
andZe4> (Z)}
(6)
.
X G C.
is
For each X,
as follows.
(6) defines
the set of proper subcoalitions which are both winning and self-enforcing.
picks the coalitions in
M. (X)
self-enforcing subcoalitions,
The proof
(5).
Theorem
2 holds,
it is
1
[1/2, 1).
Y,Z £
(p
single- valued.
=
N
Let
<f>
1,
<\>
is
becomes the URC), which
is
in the
(5)
there are no proper winning
is
then
and
captured by
Appendix.
uniquely defined, but also that
In this case, with a slight abuse of notation,
be the
=
X
power mapping
when Assumption
we write
7, payoff function
unique mapping satisfying Axioms 1^4.
7^. Coalition
C N,
X
enforcing, then (p(N)
Corollary
X
other proofs,
collection of players 1,
(X) implies 7y
is self-
all
empty and
When
Equation
as
<f>
(X)
= Y
{Y}.
there exists no coalition
if
is
establishes not only that
Corollary 1 Take any
a 6
M (X)
of this theorem, like
(X)
instead of
that have the least power.
M. (X)
j^
=
N,
N
is self- enforcing,
and
Then for any X,Y,Z G
that
is,
N
and
that is winning within
w (),
N€
<f>
(N),
if
self- enforcing.
and only
C,
if
Moreover,
{N}.
which immediately follows from
enforcing coalitions. In particular, Corollary
1
(5)
and
(6),
summarizes the basic
results
on
self-
says that a coalition that includes a winning and
11
be
self-enforcing subcoalition cannot
self-enforcing.
This captures the notion that the stability
of smaller coalitions undermines the stability of larger ones.
As an
a =
illustration to
that {A}, {B}, {C}, and {A,
=
1,
({B, C})
=
B,C}
7,4
+
7b> Assumption 2
(X)
({A, C})
if
jA
= 7 B = 7C
{{A,
B} {B, C}
4
Equilibrium Characterization
,
We now
,
,
all
{A,
{C}. In this case,
coalitions except {A,
C}}
characterize the
the proofs.
Subgame
The next
and
satisfied
it is
cfi
is
a singleton for any
easy to see
B})
=
{B},
X. On the other hand,
B, C} would be self-enforcing, while
({A, B,
<f>
C})
=
Perfect Equilibria (SPE) of
game T
defined in Section 2
and
to the ruling coalitions identified by the axiomatic analysis in the
We
subsection provides the main results.
then provide a sketch of
proofs are contained in the Appendix.
Main Results
4.1
The
(-W)
The formal
and suppose that
in this case.
show that they correspond
previous section.
is
C
and
are self-enforcing coalitions, whereas 4>{{A,
4>
(p
B
consider again three players A,
7a < 7b < 7c <
For any
1/2.
Theorem
following two' theorems characterize the
i\n,
w
,
Oi)
with
initial coalition
N. As
Subgame Perfect Equilibrium (SPE) of game T
usual, a strategy profile
o
in
T
SPE
a
is
if
=
a induces
continuation strategies that are best responses to each other starting in any subgame of T,
denoted Th, where h denotes the history of the game, consisting of actions in past periods
(stages
and
Theorem
K
£
<j)
steps).
2 Suppose that
<j>
(N)
satisfies
Axioms
1-4 (cfr.
(N), there exists a pure strategy profile ctk that
most one
transition. In this equilibrium player
=
i
£
N
is
(5) in
an
eI{ ie x}I{A^K}-
This equilibrium payoff does not depend on the
random moves
Theorem
that
is
Wi (K)
SPE and
URC K
leads to
in the set
<p
(TV)
denned
in the axiomatic analysis of
when each
by Nature.
Theorem
player anticipates
ruling coalition to play a strategy profile such that they will turn
can also be verified that Theorem 2 holds even when e
3.
12
=
in at
(7)
2 establishes that there exists a pure strategy equilibrium leading to
of the analysis in the previous section:
It
Then, for any
1).
receives payoff
-
Ui
Theorem
0.
1.
This
members
down any
The assumption that
e
is
any coalition
intuitive in
view
of a self-enforcing
K
offers other
than
>
Theorem
is
used
in
and they
will accept
K,
This follows because the definition of the set
history.
K to play such a strategy for any
in the interest of all the players in
it is
<j>
(N) implies that only deviations that
lead to ruling coalitions that are not self-enforcing or not winning could be profitable. But the
first
by
option
is
sufficiently
many
The
players.
initial coalition
N
is
not
Notice that Theorem 2
payoff in (7)
is
URC
if
given by
results in
K
URC. The payoff player
as the
if
The
= K.
i
N
G
Then any (pure or mixed
receives in this equilibrium is
(7).
Since Assumption 2 holds, the mapping
then shows that even though the
K
is
2.
3 Suppose Assumption 2 holds and suppose (f>(N)
SPE
receives his baseline
stated without Assumption 2 and does not establish uniqueness.
next theorem strengthens these results under Assumption
strategy)
Each player
also intuitive.
is
be blocked
will
K and then incurs the cost e he part of K and the
equal to K (because in this latter case, there will be one transition).
payoff Wi (K) resulting from
Theorem
URC
ruled out by induction while a deviation to a non-winning
as the
URC. This
is
intuitive in
is
<f>
SPE may
=
single- valued (with <fi(N)
not be unique in this case, any
view of our discussion above. Because any
by backward induction, multiplicity of equilibria
when some
results only
between multiple actions at a certain nod. However,
we show,
as
this
may
K). Theorem 3
SPE
will lead to
SPE
player
is
is
obtained
indifferent
only happen
when
a
player has no effect on equilibrium play and his choice between different actions has no effect on
URC
(in particular, since
(f>
is
single-valued in this case, a player cannot be indifferent between
actions that will lead to different
It is also
game T
URCs).
worth noting that the
SPE
in
Theorems
2
and 3
incorporates both dynamic and coalitional effects and
coalition-proofness
is
is
Since the
"coalition-proof.
is finite,
the relevant concept of
Bernheim, Peleg, and Whinston's (1987) Perfectly Coalition-Proof Nash
Equilibrium (PCPNE). This equilibrium refinement requires that the candidate equilibrium
should be robust to deviations by coalitions in
the possibility of further deviations.
explicitly,
is
it is
natural to expect the
straightforward to prove that the set of
T
to be
PCPNE
PCPNE.
Indeed,
if
coalitional deviations
Assumption
coincides with the set of
SPE.
2 holds,
it
11
Sketch of the Proofs
4.2
We now
provide an outline of the argument leading to the proofs of the main results presented
in the previous subsection
11
in
subgames when the players take into account
T introduces more general
Since
SPE
all
A
and we present two key lemmas that are central
formal proof of this result follows from
Lemma
2
below and
13
is
for these
available from the authors
theorems.
upon
request.
Consider the game
T and
let
<fi
be as defined
Take any coalition
in (5).
ax
outline the construction of the pure strategy profile
which
6
<j>
SPE and
be a
will
K
{N).
We
lead to
will
K
as
URC.
the
Let us
first
rank
imposed Assumption
that for any X,
€
In particular,
2).
n C
:
^ n PO
> 1y
C, 'Jx
n(X) > n(K) (how
then
this
Y
among
the ties
—
<
{!,..'.,
> n 00
given higher numbers, and coalition
Now
define the
K
an<^
>
2^ —
l} be a one-to-one
^
some
f° r
X
other coalitions are broken
mapping, we have thus ranked (enumerated)
the same power.
we have not yet
coalitions so as to "break ties" (which are possible, since
all
'
—
C
>
is
K
we have 7^ =
not important).
7^-,
With
coalitions such that stronger coalitions are
all
number among
receives the smallest
mapping x C
^
mapping such
coalitions
all
with
as
X (X) = argminn(y).
(8)
Y&4>(X)
mapping
Intuitively, this
note that
Axiom
x
1S
a projection in the sense that
3 implies
The key
x (X) 6
ef>
tf>
x
(X)
SPE
is
1
any
for
(x (-^0)
(x (X)) and Corollary
to constructing a
subgame
consider a
picks an element of
and
= X (X).
implies that
satisfies
x (N)
X
=
K- Also,
This follows immediately since
4>
{x (X))
is
a singleton.
to consider off-equilibrium path behavior.
which we have reached a coalition
in
X
(i.e.,
To do
this,
j transitions have occurred
»
= X)
and Nj
and
us try to determine what the
let
starting in this subgame. If
on the other hand,
profile
URC
ok
Y
7^
Y=
X, then the game
X, then the
such that the
URC
will
URC
Vx[n
By Axiom
1
and equations
(8)
and
(9),
We
will introduce
one
vorite" coalition of player
coalition
player
i.
among
If
smallest n,
final
i if
and thus
will end,
Y
X
we have
Fx
(i)=
picks
be the
will
accepted
URC.
<^=»
1
If,
so that
[
otherwise.
X
">
that
ip
x (Y)=X.
the current ruling coalition
it
is
x 00
tp
(10)
concept before denning profile ax- Let
there are none,
X
Y
by
coalitions that are winning within
if
proposal
denote this (potentially off-equilibrium path)
is
X.
itself.
X, that are
{i)
self-enforcing
Fx
(i)
denote the
"fa-
and that include
picks the one with the
Therefore,
n (Y)
argmin
Ye{Z:ZcX,Z€Wx,x(Z)=Z,Z3i}U{X}
14
Fx
Naturally, this will be the weakest
there are several such coalitions, the definition of
and
if
We
\
X=Y
would be
must be some subset of Y. Let us define the strategy
be x(F).
following the acceptance of proposal
URC
(11)
we
Similarly,
stage. This
define the "favorite" coalition of players
again the weakest coalition
is
among
Y C X
members
those favored by
{argmin n (Z)
{Z-3ieY:Fx (i)=Z}
X
X
starting with
at the current
of Y, thus
iiY^0;
(12)
otherwise.
Equation (12) immediately implies that
For
The
is
first
part
is
X
all
€ C
:
Fx (0) = X
and
The second
part follows, since for
true by definition.
and
feasible in the minimization (11),
Fx (X) =x(X).
has the lowest number
it
(13)
all
n among
winning
self-
Z
that
all
enforcing coalitions by (8) and (5) (otherwise there would exist a self-enforcing coalition
is
winning within
Therefore,
it is
Now we
=
v (h)
and
satisfies
7 Z < 7 X (A')> which would imply that
the favorite coalition of
are ready to define profile
supposed to move
v
X
we
if
agenda-setter
a
after this history
all i
x (X) and thus Fx (X) = x
£
proposal P).
Axiom
violates
4).
(X).
a^. Take any history h and denote the player who
=
a(h)
if
are at a voting step deciding on
who made
<fi
x(X)
6 x{X),
i
after h,
we
are at an agenda-setting step,
some proposal
P
(and in this case,
is
and
a be the
let
Also denote the set of potential agenda setters at this
stage of the
game by A.
Then ok
the following simple strategy profile where each agenda setter proposes his favorite
is
Finally, recall that
game
coalition in the continuation
proposal
P
prefer to
come
n denotes a vote
(given current coalition
P
excludes
{agenda-setter a proposes
P = Fx
the
if
URC
following
him
of "No"
X) and
and y
is
a vote of "Yes".
each voter votes "No" against
or he expects another proposal that he will
shortly.
(
voter v votes
if
_
Fx
(A)
,
P f Fx
(A U {a})
,
and 1fx (A) <
1->I>
X {F)\
otherwise.
y
\
;
either v £ ipx (P) or
v £
<^
(a)
(14)
In particular, notice that v £
Fx
(A) and
different coalition proposal that will
the
game
if
the current voting
weakly higher payoff under
fails,
P ^ Fx (AU
be made by some future agenda
and ^PxiA) —
With the
is
lib
this alternative proposal.
similar to a "truth-telling" strategy; each individual
(constrained by what
feasible)
strategy profile
&k
and votes
defined,
constitute the bulk of the proof of
{a}) imply that voter v
(P)
is
part of a
setter at this stage of
implies that this voter will receive
This expression makes
it
makes proposals according
clear that <jk
is
to his preferences
truthfully.
we can
Theorem
2.
15
state the
main lemma, which
will essentially
For this lemma, also denote the set of voters
who
already voted "Yes" at history h by
V=P\
Then,
Lemma
V~ U
the set of voters
,
{v}) denotes the set of voters
who
who
already voted "No" by
V
.
will vote after player v.
Consider the subgame Th of game T after history h in which there were exactly jh
1
and
transitions
is
(V + U
V+
the current coalition be
let
X. Suppose
that strategy profile
ax
defined in (14)
played in Th- Then:
(a) If
V+
R = FX
h
is
{i
€ {v}
U
and
at agenda-setting step,
UV
:
votes y in
i
R = Fx (A U {a}),' if h is
Wx, then the URC is R = tpx (P)>
URC
the
ax} 6
is
at voting step
an<^ otherwise
(A).
(b) If
h
the voting step
is at
Ui
Otherwise
(if
proposal
=
Wi (R)
P
P
and proposal
-
e (j h
+ 1{p^x)
will be rejected
or
if
player
will be accepted,
h
(!{ieP}
at
is
i
X
e
receives payoff
+ ^R^pfyeR}))
(!5)
agenda- setting step), then player
G
i
X
receives payoff
Ui
The
=
-
Wi (R)
lemma
intuition for the results in this
of the strategy profile
ok- In particular, part
immediately from 'o~k- For example,
if
we
favorite coalition of the set of remaining
+ I{RjLX}I{i€R})
£ (j h
is
(16)
straightforward in view of the construction
(a) defines
what the
agenda
setters, given
by
A U {a}.
and voters
will vote
n
(
them
for
in the
This
URC will be the
is
an immediate
each player will propose his
"No" ) against current proposals
more preferred outcome
o~k will induce a
,
This follows
will be.
are at an agenda-setting step, then the
implication of the fact that according to the strategy profile ctk
favorite coalition
URC
if
the strategy profile
remainder of this stage.
Part (b)
simply defines the payoff to each player as the difference between the baseline payoff, Wi (R), as
URC R
a function of the
Given
URC
is
will
Lemma
be
clear that
1,
K and
defined in part
Theorem
will
it
2 then follows
no player can profitably deviate
Lemma
for any
by
if
strategy profile
will lead to
1
the same
in
by
(in
ax
ax
for the case in
a
SPE
subgame
URC
T^, will lead to
(because in this case
ax
defined in (14),
it
which Assumption 2 holds by estab-
and payoffs
=
as those in
{K}. Let
pure or mixed strategies) and for any history
in the
is
transitions.
any subgame.
2 Suppose Assumption 2 holds and 4>(N)
SPE a
a and
SPE
any
and the costs associated with
be reached with at most one transition). With
The next lemma strengthens Lemma
lishing that
(a),
the
same
URC
ax
Lemma
1.
h, the equilibrium plays
and
Then
be defined in (14)-
induced
to identical payoffs for
each
player.
Since
<p
(N)
=
{K}, Theorem 3 follows as an immediate corollary of this lemma (with h
16
=
0).
The Structure
5
we present
In this section,
result
of Ruling Coalitions
(Theorems
2
and
on the structure
several results
we
3),
make
will
Assumption
and a G
1
Given the equivalence
use of the axiomatic characterization in
Theorem
game T = (N, j, w () a) with w
satisfying
Throughout, unless stated otherwise, we
1.
URCs.
of
a
fix
,
we assume
[1/2, 1). In addition, to simplify the analysis in this section,
throughout that Assumption 2 holds and we also impose:
Assumption 3
For no X,
Y
X CY the equality 7y
6 C such that
Assumption 3 guarantees that a small perturbation
make
it
Similar to Assumption
winning.
measure
(in fact, it coincides
ocy
x
is
satisfied.
of a non-winning coalition
assumption
this
2,
=
=
does not
only in a set of Lebesgue
fails
with Assumption 2 when a
Y
All proofs are again
1/2).
provided in the Appendix.
Robustness
5.1
We
start with the result that the set of self-enforcing coalitions
ogy); this
T
=
is
A {N),
,
a), a
C
vector {7i}j € 7v
by
power mapping 7
IR++-
D en °te
and the subset of A
3
1.
The
for which coalition
which
N
stable)
games are
"close,"
N
is self- enforcing,
corollary of
—
7i|.
Proposition
1.
$
1
A
will
1Z,
6
>
(TV, 7',
standard topol-
Note that
for
game
given by a \N\ -dimensional
Lemma
3
set
that
is
will
(N, 7,
w (•)
such that
,
if 7'
URC. To
:
N —
>
w, a).
17
(i.e.,
the subset of power
= A(N) \ S
(N)
3,
is
also dense in
within either
(N).
A (N),
N
its
is
subsets
not
self-
R++.
S (N) orAf(N).
the distributions of powers in two different
if
H
N
subsets for which coalition
its
A(N)
a) with
:
let J\f
have the same
{7' e
is
)
>
Assumptions 2 and
lies entirely
not change the
=
is
w a =
1
S (N) and
with the sup-metric, so that
Consider T
=$
(TV, {7j}j jv
e
by
K+i- The
^-neighborhood of 7
There exists
(N, 7, w,a)
<E>
S (TV), and
then these two games
the set of mappings 7,
|7j
explicitly j\n)
set of power allocations that satisfy
weak players
sufficiently
sup i€ j
is
Each connected component ofA(N)
An immediate
more
(or
(TV) for
enforcing, A/" (AT), are open sets in
2.
(in the
the subset of vectors {7i} i6 ^ that satisfy Assumptions 2 and 3
distributions for which coalition
Lemma
open
not only interesting per se but facilitates further proofs.
w (•)
(N, 7,
is
p
(TZ, p) is
<
also that the inclusion of
lies
this proposition,
endow
a metric space with ^(7,7')
=
6}.
[1/2, 1).
K++
and
and prove
state
(7, 7')
a 6
URC
Then:
within 5 -neighborhood of 7, then
There exists
2.
$
>
5'
such that
£ [1/2,
if a'
1) satisfies \a!
<
a\
then
6',
<E>
(N, 7, w,a)
=
(iV,7, w, a').
3.
Let
N = N\ U N2
<
with j N2
8,
<f>
=
(TVi)
This proposition
N\ and N2
with
U
(JVi
<j>
rules of the
N2
disjoint.
Then, there exists 5
>
such that for
all
N2
).
intuitive in
is
some degree of continuity and
will
view of the results in
Lemma
3.
It
implies that
URCs have
not change as a result of small changes in power or in the
game.
Fragility of Self-Enforcing Coalitions
5.2
Although the structure of ruling coalitions
within the society,
may
it
be
robust to small changes in the distribution of power
is
more
fragile to
out, to
be such a sizable shock when a
Proposition 2 Suppose a
1.
not
2.
X
If coalitions
and
=
Y
=
member
proposition shows that
of the self-enforcing coalition turns
1/2.
1/2 and fix a power mapping 7
such that
The next
sizeable shocks.
in fact the addition or the elimination of a single
is
—
XnY=
1 — R++.
Then:
*
:
are both self- enforcing, then coalition
X UY
self- enforcing.
If
X
a
is
self- enforcing coalition,
X U {i}
then
for
i
£
X
and
X \ {i}
for
i
£
X
are not
self- enforcing.
The most important
implication
is
that,
=
under majority rule a
1/2, the addition or the
elimination of a single agent from a self-enforcing coalition makes this coalition no longer
enforcing. This result motivates our interpretation in the Introduction of the
Soviet Russia following
random deaths
of Politburo
self-
power struggles
in
members.
Size of Ruling Coalitions
5.3
Proposition 3 Consider T
1.
Suppose a
players
for any
N,
\N\
m^
=
=
n,
N,
\N\
w (•)
,
a).
n and
m such that 1
<
m
and a generic mapping of powers 7 such
<
m ^ 2,
n,
that \cp(N)\
2 there exists a self- enforcing ruling coalition of size
Suppose that a
players
(TV, 7,
1/2, then for any
self- enforcing coalition
2.
=
=
n,
of size
>
m.
there exists a set of
= m.
In particular,
However, there
is
no
2.
1/2, then for any
n and
m
such that
and a generic mapping of powers 7 such
18
1
<
m
<
that \<j>(N)\
n, there exists a set of
=
m.
These
results
show that one can say
about the
relatively little
size
URCs
when a = 1/2,
and composition of
without specifying the power distribution within the society further (except that
coalitions of size 2 are not self-enforcing). However, this
is
largely
due to the
can
fact that there
be very unequal distributions of power. For the potentially more interesting case in which the
distribution of power within the society
some
for
more power and there
integer k
(i.e.,
said about the
In particular, the following proposition shows that, as long as larger
size of ruling coalitions.
coalitions have
much more can be
relatively equal,
is
majority rule (a
is
== 1/2),
be the
coalitions of size 3, 7, 15, etc.) can
—
only coalitions of size 2 fc
URC
(Part
1
Part 2 of the
1).
proposition provides a sufficient condition for this premise (larger coalitions are more powerful)
to hold.
The
rest of the proposition generalizes these results to societies
Proposition 4 Consider T
Let
1.
Ix > 7y
=
a = 1/2 and suppose
For
the condition
some A >
exists
a £
a) with
\/X,Y € C
1/2.
X,Y
>
£ C such that \X\
<f>(N)
=
N
if
\Y\
and only
if
we have
\N\
= km
under these conditions, any ruling coalition must have
\X\
:
a >
[1/2, 1).
Then
have greater power).
where k m = 2 m - 1, m 6 Z. Moreover,
m — 1 for some m £ Z.
size k m = 2
2.
,
any two coalitions
that for
e -, larger coalitions
(i-
w (•)
(N, 7,
with values of
>
\Y\ =>
7^ > 7y
to hold, it is sufficient that there
such that
\N\
EItSuppose a £
3.
such that \X\
N
4>(N)
=
where
[z\
\Y\
> a\Y\
(\X\
and only
if
,
resp.)
enforcing
and suppose
<
\N\
a\Y\,
=
fc
if
>
whenever
and only
if
that
resp.J
m>Q where
denotes the integer part of
There exists 6
4-
a
if
[1/2, 1)
7
1
is
^
we have
k\^ a
x >
\X\
(
X
=
1
x > a lY (ix < a 7y>
and TO>Q = \km -i^ a ja\ +
-y
/c
C Y C
for
N
Then
resp.).
1
17 )
m
>
1,
(]X\
<
z.
ccyy (ix
= km a
-
such that for any two coalitions
such that maxijgjv {li/lj}
-y
1
< a lYi
for some
^
This proposition shows that although
it
m
is
<
resP-)-
1
+
<5
implies that \X\
> a\Y\
In particular, coalition
(where k ma
is
impossible to
X
£ C
is self-
defined in Part 3).
make any
general claims about
the size of coalitions without restricting the distribution of power within the society, a tight
characterization of the structure of the
in
URC
is
possible
when
individuals are relatively similar
terms of their power.
5.4
Power and the Structure of Ruling Coalitions
One might expect
size of the
that an increase in
URC. One might
a
—the supermajority requirement—cannot decrease the
also expect that
if
an individual increases
19
his
power
(either exoge-
nously or endogenously) this should also increase his payoff. However, both of these are generally
,
not true. Consider the following simple example:
is
self-enforcing
is
now
when a =
1/2, but
w (•)
not self-enforcing
be given by
21,
player E,
Then
(1).
when 4/7 < a <
a self-enforcing and winning subcoalition. Next, consider
coalition
(3, 4,
5)
7/12, because (3,4)
game T with a — 1/2 and
five
E with powers A = A = 2, 7 S = iB = 10, 7 C = c = 15, 7 D = iD = 20,
iE = 40. Then $ (N, 7, w, a) = {A, D, E}, while $ (N, 7', w, a) = {B, C, D}, so
players A, B, C, D,
7s =
is
let
and
who
-y'
-y'
-y
$
the most powerful player in both cases, belongs to
is
(N, 7, w, a) but not to
$(iV,V,iu, a).
We
summarize these
An
Proposition 5
1.
exists a society
N,
<fr
(JV, 7,
increase in
a may
W, a) and X' e
$ (N, 7, w, a/),
a/Z i
^ j,
applies even
7,-
<
when
7'-
j
Intuitively, higher
sucft.
is
the
ftoi
\X\
a, a' 6 [1/2, 1), such that a'
>
[1/2, 1),
je$ (AT, 7,
\X'\
two mappings 7, 7'
u;,
a), 6ui j ^
$
less likely to
The
stable.
first
:
N
(TV, 7',
— M++
»
but for
there
all
X
6
—
7^
satisfying
r
yi
w, a). Moreover, this result
cases, i.e. j[
=
7^
< j3 <
7'-
for
all
certain coalitions that were otherwise non-self-enforcing into self-
enforcing coalitions. But this impUes that larger coalitions are
and
> a
is,
and jx > j x ,.
most powerful player in both
a turns
That
reduce the size of the ruling coalition.
power mapping 7 and
There exists a society N, a £
2.
/or
a
results in the following proposition (proof omitted).
emerge as the ruling
coalition.
now
less likely to
makes
This, in turn,
be self-enforcing
larger coalitions
more
part of the proposition therefore establishes that greater power or "agreement"
requirements in the form of supermajority rules do not necessarily lead to larger ruling coalitions.
The second
part implies that being
powerful player. This
with
less
is
more powerful may be a disadvantage, even
for the intuitive reason that other players
may
powerful players so as to receive higher payoffs.
ruling coalition. This question
Proposition 6 Consider
•
•
1
most
prefer to be in a coalition
This latter result raises the question of when the most powerful player
7ii
for the
1\N\
is
the
is
be part of the
addressed in the next proposition.
game V (N, 7, w
an increasing sequence.
then either coalition -N
will
is self- enforcing
If
7^
(•)
,
a)
with
_1
£ (a Ej=2
a €
lj/ i 1
[1/2,1),
~ a ) a E^'i"
>
or the most powerful individual, \N\,
URC.
20
and suppose
is
1
Ijl
(
l
that
~ a )j
>
not a part of the
Concluding Remarks
6
We
The ab-
presented an analysis of political coalition formation in nondemocratic societies.
sence of strong institutions regulating political decision-making in such societies implies that
individuals competing for power cannot
able to
to
commit
commit
make binding promises
(for
and they
to a certain distribution of resources in the future)
to abide
by the coalitions they have formed. This
example, they
be un-
will
will also
be unable
latter feature implies that
once
a particular ruling coalition has formed, a subcoalition can try to sideline some of the original
members. These considerations imply that ruling
be
self- enforcing, in
sideline
some
of the
coalitions in
nondemocratic societies should
the sense that there should not exist a self-enforcing subcoalition that can
members
of this ruling coalition. This implies that coalition formation in
such political games must be forward-looking; at each point, individuals have to anticipate
future coalitions will behave. Despite this forward-looking element,
we showed that
ruling coalitions can be determined in a relatively straightforward manner.
presented both an axiomatic analysis and a noncooperative
established that both approaches lead to the
same
game
self-enforcing
In particular,
of coalition formation,
set of self-enforcing ruling coalitions.
can be characterized recursively (by induction),
over, because such coalitions
how
it is
we
and
More-
possible to
characterize the key properties of self-enforcing ruling coalitions in general societies.
Our main
results
show that such ruling
coalitions always exist
and that they are generically
unique. Moreover, a coalition will be a self-enforcing ruling coalition
possess any subcoalition that
sufficiently
is
if
powerful and self-enforcing.
and only
We
also
if it
does not
demonstrated
that, although equilibrium ruling coalitions are robust to small perturbations, the elimination
of a
member
of a self-enforcing coalition corresponds to a "large" shock
and may change the
nature of the ruling coalition dramatically. This result provides us with a possible interpretation
for
the large purges that took place in Stalin's Politburo following deaths of significant figures.
Finally,
we showed
that although ruling coalitions can, in general, be of any
restrict attention to societies
where power
is
relatively
coalition
must be of the
size 2
—
1,
where k
is
an
a =
and
is
depends on some of our assumptions. In particular, the assumption that there
is
crucial
both
1/2, the ruling
integer).
Naturally, the result that the ultimate ruling coalition always exists
to future divisions of resources
once we
equaUy distributed, we can make strong
predictions on the size of ruling coalitions (for example, with majority rule,
fc
size,
for the
genetically unique
is
no commitment
uniqueness and the characterization results.
Other assumptions can be generalized, however, without changing the major results in the paper.
For instance, the payoff functions can be generalized so that individuals
21
may sometimes
wish to
be part of larger
coalitions,
without affecting our main results.
Other interesting areas of study
Section
5.
some
For example, we saw that individuals with greater power
suggests that individuals
guns) or they
Some
in this context relate to
may
may
of the results presented in
may end up worse
off.
This
voluntarily want to relinquish their power (for example, their
wish to engage in some type of power exchange before the game
is
played.
of these issues were discussed in the working paper version of our paper, Acemoglu,
Egorov
and Sonin
(2006),
and developing these themes
to be a fruitful area for future research.
are to extend these ideas to
games that
in the context of
more concrete problems appears
The two most important
are played repeatedly
challenges in future research
and are subject to shocks, and to
relax the assumption that individuals that are sidelined have no say in future decision-making.
Relaxing the latter assumption
political decision-making in
is
particularly important to be able to apply similar ideas to
democratic
societies.
22
Appendix
Proof of Theorem
We
(j>{X) in (5) (Step 1).
the unique
2 holds,
4> is
Step
Z
mapping
Axioms 1-4 (Step
<
A
\X\
is
and thus
2)
establish that
M.
and that
this is
when Assumption
(A') is well-defined
The argmin
has already been defined for Z.
<$>
selects the
it
minimum
a subset of I, which
X. This implies that
set includes
we
3). Finally,
at each step of the induction procedure,
also well defined, because
smaller than 2^1;
Axioms 1-4 (Step
satisfies
and the mapping
in (6)
These four steps together prove both parts of the theorem.
single valued (Step 4).
in (6) satisfies \Z\
M (A)
the properties of the set
first
then prove that <p(X)
satisfying
Note that
1:
Consider
1:
this
of a finite
number
because
set in (5) is
number
of elements (this
is
Non-emptiness follows, since the choice
is finite).
procedure uniquely defines some mapping
<p
(which
is
uniquely defined, but not necessarily single- valued).
Step
given by
cases
4>
To
the
Axiom
verify that
>
Axiom
6 C.
Y
Axiom
take
1,
E
Y
<f>
(X)
3
=
is satisfied,
<f>
because either 4>{X)
1 is satisfied,
(X) contains only winning (within A)
first case,
\X\
X
X
so <f>{X) contains only subsets of
(5),
Finally,
4>
Take any
2:
such that 4>{X)
Y
6
(Y), while in the latter,
when
4 holds trivially
6 4>{X) and
Z C A,
=
|A|
<p
(Y) by
Z
Wx
£
is
and
1)
or
satisfied.
is
or
Y
6 M. (A). In
(6).
only one winning coalition.
Z
is
G
By
<j){Z).
If
construction of
we have that
(A),
Y
argmin
6
/
AeM(X)U{X}
Note
G
since there
1,
such that
Y
2
Y=X
Either
<f>(X).
=
\X\
0. Furthermore, in both
=£
and thus Axiom
coalitions,
take any
= A" (if
also that
Z
£
M (A) U {A} from
Z
(6).
Then,
$
argmin
yA
.
if
jA
,
AeM(X)u{X}
we must have ^ z > jy, and
Step
exist
<f>
<p
(A)
3:
and
=
4>'
We
next prove that Axioms 1-4 define a unique mapping
^ <p that satisfy these
(A) = A; this is because
<fi'
Therefore, there must exist k
and there
Y
e
(p
exists
(X) and
hypothesis).
We
first
fact that
We
Axiom
vice versa, which completes the proof that
X
Y
£ C, \X\
(£
will
$ (A).
now
=
axioms. Then, Axioms
(p
^
(A)
>
1
k,
such that
and
<j>
such that for any
Take any
Z
(p
&
(A)
4>'
^
(A)
A
(jJ
1
and
C Cx and
with
\A\
(A) (such
Z ^ Y and Z C
< jy H Y = A,
A
(by
Axiom
1).
<
then ^ z
imply that
if
[A|
=
the same applies to
we have
k,
Z
exists
< jy
Y
by Axiom
$.
(f>(A)
=
1,
then
</>'
(A).
4>
(^)>
1
and
Z f Y
by
(p(X).
follows immediately from the
Now, consider the case
23
Suppose that there
(A). Without loss of generality, suppose
derive a contradiction by showing that
prove that ^ z
2
<f>.
4 holds.
Y ^
X, which
implies
<
\Y\
k (since
YC
<
however, since \Y\
=
4>{A)
Y
1
£
Note
Axiom
£
</>'
Z
£
<f>'
z < 7y,
Z
4>
$l
£ 4>{X).
Since
Z
Wx
=
(Y) (by the hypothesis that for any
<f>'
£
(X) implies that
<j>
Z
Next, since
(Y).
£
Y
(X),
<p'
£
Wx
£
Y
and
A
Y
,
£
with
£
</>'
(Y);
<f>
<
\A\
k,
(Y) and
7z < 7y.
X
Z ^
Y
3,
(X) implies (from Axioms 2 and
£ 4>(Z) by hypothesis. Since
that
Y
~j
Y
Y
and
2
(Y)
<f>
4 implies that
also that
Moreover, since
Z
we have
k,
and thus
4>'{A))
(X),
(p
By Axioms
X).
Y
and therefore
£ 4>{X)
Y
£ <j>(X),
Z
,
£
<
\Z\
<j>{Z)
£ <p(Y),Y £
3) that
k (since
Z
Z
£
Z C
Wx
X).
Z
and
£
(Z).
<j>'
This again yields
Wx, 7z < 7y> Axiom 4 implies
Wx, lz < 7y> Axiom 4 implies that
,
£
{X), yielding a contradiction. This completes the proof that Axioms 1-4 define at most
one mapping.
Step
follows.
4:
>
\X\
If
Y,Z £ C
Suppose Assumption 2 holds.
then <fi{X)
1,
=
such that yy
given by
is
=
\X\
If
(5);
1,
= {X}
then 4>{X)
since under
and the conclusion
Assumption 2 there does not
exist
7z>
argmin
7^
A&M(X)u{X}
must be a
completes the proof of Step
Proof of Lerrlma
of
T
(the
number
Base.
If T/i
X. In
1:
is
lemma
is
F
(X)
is
a singleton and
subgame
proved by induction on the
(p is
single- valued.
maximum length
of histories
implies that the
URC
means that the current
is
coalition
is
some
not eliminated before receives Wi (R)
Step. Suppose that the result
Case
1:
The
V
current step
:
i
votes y in
is
—
must be
R = X = ip x
{X)
proven
ok) £
for all
Wx
and
= Fx
(0) and
(15)
and
i
it
does
who was
(16).
proper subgames of F^. Consider two cases.
Then, proposal
voting.
is
which coincides with both
ejh,
X
voting over the last agenda setter's proposal
not depend on v's vote. Moreover, there are no more eliminations, hence each player
V + L){i £ {v} U
This
I\).
voting by the last voter v
this case,
<j)
4.
This
of steps in
\X\,
includes one last step only, this
the current step
P=
any
singleton. Consequently, for
P
(recall the definitions of
as the set of future voters, the set of those that have voted
be accepted
will
V,
h and the
V~
if
and only
if
and V + from the text
set of those that
have voted
URC will be X if P = X, while if P ^ X, the game will have
a transition to P, after which the URC will be Fp (P) = x (P) by induction (recall that after
transition the game proceeds to an agenda-setting step). In both cases, the URC R = ipx (P)If P = X, player i £ X gets Wi (X) — ejh, which equals (15). If P ^= X, player
£ P receives
y respectively).
If
P is accepted,
the
i
Wi (R)
—e
w~ —
ejh,
(jh
+
1
+ i{R^p\l{i£R\)
,
which
which again equals (15) because
rejected, then the
i
£
X\P obtains
the other hand,
if
proposal
in this case equals (15), while player
i
^ y (P)
game ends when the voting ends
24
if
C
P.
A=
On
(then
P
is
R = X = Fx (0) = Fx (A)
and each player
with some
6
b
X
Note
if
X, then
=
ipx (P)
receives
current step
P
that such
first
P±
The
X
£
i
and the remaining
= x {P),
R = Fx
(J3
U
{6})
=
(P)
being
B =A\
{A) will be the
but for
is
{b}.
URC, and
P = Fx
(a)
^
X
Note that
V-
we must have X (P)
ip
P = Fx (A\J
x
(P)
1S
P = Fx (a).
if P = X, while
= P by (11), so
to propose
P- Indeed, this automatically holds
6 ipx (P) W1U vote
i
= Fx
game continues
0, the
set of agenda-setters
P- Consider two subpossibilities. First, suppose
suggests, each player
A ^
if
agenda-setting; suppose player a
satisfies ipx
(P)
x
tp
that
(16)) or,
given by (16).
is
is
which equals
ejh,
we know by induction
the payoff player
2:
—
gets Wi (R)
as agenda-setter
In the latter case,
Case
X
6
i
{a}).
Then, as (14)
necessarily winning within
P = X it follows from X = tpx (P) s VVx, while if P ^ X, then, as we just showed,
ipx (P) = P = Fx (a) which is winning by (11). This means that if proposal P = Fx {A U {a}),
it is accepted, and R = P = Fx (A U {a}) both in the case P = X and P ^ X (in the latter case,
R = ipx (P) by induction, and ipx (P) = P)- Payoffs in this case are given by (16) because there
X:
if
are no
P
more
transitions
— e.
get the additional
>
for 'yp iA)
x
Fx
(a)
=
7j/>
as (14) suggests,
= P
all
We know
rejected.
and
.
=
by
N = X, K = R).
To do
Lemma
Fx
(A) 6
X, and only
players in
Wx
(16)
ok
we deduce
0,
P, and thus
will vote against proposal
R = Fx {A)
and the payoffs are given by
P
will
(16).
be
This
1.
2: Profile
The theorem
is
is
Lemma 1 to the
= x (-W) = K, and
involves only pure strategies. Applying
URC
that the
which equals
we show that there
this
P^
{A U {a}) which leads to a contradiction. But then,
by induction that then
stage where h
if
P ^ Fx (A U {a}). Then 7fx M) — Ttyv(.p)'
minimum in (12) for Y = A U {a} ^
is reached at
P = Fx
and thus
Proof of Theorem
payoffs are given
and exactly one more transition
Second, consider the case
players in
completes the proof of
first
P=X
would imply that
(p)
ipx (P)
if
(7)
under
=
because jh
therefore proved
if
gk
is -F/v
(N)
(there were
we
no eliminations before
establish that profile
no profitable one-shot deviation, which
ax
is
a SPE.
sufficient since
is
T
is
finite.
Suppose, to obtain a contradiction, that there
h; since
only one player moves at each history, this
start with the former case,
Subcase
better off
if
1:
which
deviating to
voter v
is
is
he voted
n.
h can
a one-shot profitable deviation after history
is
either voter v or agenda-setter a. Let us
then subdivided into two subcases.
suppose voter v votes y
in
a^
(this
means
v 6 i^x (P)
In profile ok, the votes of players
V) do not depend on the vote
if
is
of player v.
Hence,
if
who
proposal
P
we
restrict attention to this case.
25
P), but would be
vote after voter v (those in
is
accepted in equilibrium,
result in rejection, but not vice versa. This deviation
pivotal, so
c
may
From Lemma
1,
only be profitable
the
URC
will
be
ip
x
proposal
(-P) if
and
between
since deviation
Second,
o~k)-
Assumption
P=X
In the
for
some
£ X; the
<
(A))
Fx
since
1fx {A) =
(A)
± X
(P)> an<^
7v>
if
so
would Fx
the same. Hence, deviation to
w v (Fx
would get
—e
(A))
case
Y = AU
FX (A)
if ip
(A)
x
j£
^
X=
ip
X
x
1
and
is
(A) and
w~ —
1i/j
URC
URC
is
is
x
I{FX (A)?X}
X. In
(A), ^iFx(A)
—
lib
(P) and he receives
wv (Fx
> 1 {P^X}
(A))
+
is
P
less
(1
+
1
{i>
(P)).
Then
1fx {A) =
X
±
number
ip
IP)-
lij>
(P)
x
=
minimization problem, so
w v (Fx
ip
(A))
= w v (ip x
(P)),
(P) would contradict
x
of additional transitions
an<^
(P)>
wv
is
wv
n
P
will
if
Px
in profile
Again,
If
Fx
(A)
x
(P),
to P,
and
from
v ^
X
ip
W1 ^ proceed in one step, so v will
then result in at most one additional
(A))
—
e (jh
+
1)-
This implies that the
from deviating to
y,
thus
v 6 ipx (P)> then, as implied by equation
By Lemma
(A\J {a}).
(Fx (A)) —
-
URC
part of transition from
is
¥"
w v (Fx
¥"
(A U {a}).
1,
if
v votes h, the
+ I{FX (A)^X})> if ne votes y, the
+ I{p/A'} (l + hi'xiP)^}))- But nfFx (A) <
e (jh
x
(P))
x
(P)), so the deviation could only be profitable for v
(ip
(tp
P
P ^ Fx
off voting y.
can only change the
it
at least as large as his payoff
x (P)¥=P})-
(2) implies
considered in
y.
to ipx (P)
than
s (j h
This can onl y be true
this case, however, strict inequality
w v (ip x
'=
(Fx (A)
G P, so v
(v
1)
the other hand,
> wv
is
Consider two possible cases.
pivotal.
is
(A) and he receives payoff
imphes
lip x (P)
On
not profitable.
Fx
Fx
ij)
v
no
h
payoff of player v from voting
v e
or
IP)
lij>
(P)) due to
this case either
while the second
<
x
x ip) implies that voter v must vote h
transition). Voting
transition, so v obtains a payoff
(14),
if
e (jh
more
m
(f° r
This contradiction proves that
T^M) <
implies that transition from
is
= P
(ip
not profitable because with or without this deviation player v
is
only profitable
participate in exactly one
deviation
< wv
(A))
{a} cannot be achieved at
(P))} tne
change other voters' votes,
voting y gives v exactly
Lemma
(P)
In the second case,
(P)
(ip
suppose that voter v votes h in gk, but would be better
2:
this deviation does not
to ipx (P)
an d
feasible in this
is
ax, which contradicts the assumption that he votes
Subcase
x
is trivial
or
either
x (P)) = w v {Fx (A))
P ^ Fx (Au{a}). First,
wv
if
w v (Fx
ip
and
be
(P) simply because v votes y in
x
would imply
first
+ I{fx (A)^lx})-
(jh
Fx
This, together with v £
n
ip
of transitions will
in the first case
(P)>
7i/>
{A U {a}), then
a contradiction.
and only
P ^ X)
if
7px (A) —
in (12) for
n(ip x (P)) and
is
P
number
rejected; the
is
showed that either 1fx {A)
just
minimum
case, the
n(Fx
But we
to
(p); this
P = Fx
1).
P
(A) (recall that v €
7i/>
i
X
{A),
instead
{A U {a}) which
Fx
Fx
if
Third,
Lemma
first
>
instead ryp i J\)
x
if
of transitions matters only
Fx
€
profitable, v
P = Fx (i)
or
because
t^
is
if
1.
the proof of
and
number
(A)
from
2 (including transition
Let us prove that v €
(see (2)).
Fx
accepted and
is
the second case, so the
1 in
P
P
P
7fx m) <
that deviation for v
26
7,/,
is
if
FA-
ip\ holds,
(A)
+
X and
and therefore
ip
x
wv
(P)
if
=P=
(Fx (A)) >
not profitable. This completes the proof
may be
that no one-shot deviation by a voter
The remaining
case
is
accepted
is
as follows:
Fx
(A)
if
Q
if
this inequality
minimization problem
so
strict,
w a (Fx (A)) = w a (P),
then
both cases a participates
in
Q cannot be rejected under
URC would be Fx (A). If
were rejected, then the
is
Consider two subcases. Subcase
set.
proposed; in that case,
feasible in
is
Q ^
where agenda-setter a has a best response
does not belong to the best response
P
profitable.
(11),
P, and
ok- The reason
Fx
6
is
(A), then coalition
w a {Fx
which means that
suppose coalition
1:
profile
i
P = Fx (a)
(A))
< wa
(P).
If
wa (Fx (A)) < w a (P), then deviation to Q is not profitable; if
the either Fx (A) = P = X or neither Fx (A) nor P equals X, but
in the
same number
(0 or
1,
respectively) of extra transitions, so
P and Q is the same and the deviation is not profitable either. however,
^ Fx (A), then proposing Q will give a payoff w~ — ejh while proposing P will give at least
w a (P)—e (jh + 1) (again, ip x (P) = P for P = Fx (a)), so deviation is again not profitable. This
proves that Q must be accepted, which immediately implies that ip x (Q) £ Wx (only members of
ipx (Q) vote f° r Q m a K, see (14)), and then Q 6 Wx because ipx (Q) c Q- We next prove that
this immediately implies
^x (Q) = Q- Suppose, to obtain a contradiction, that ipx (Q) ¥"
from proposing
utility
If,
i
Q'->
Q
7^
X
and thus
ip
x
(Q)
Moreover, the fact that
X (Q)
=
V'x (Q)-
I
n
= X (Q)%isa
If
a proposed
addition, any player
because ipx (Q)
'fix
(Q)
=
Q)> an(I fr°
n
also vote
if
Q
m
¥"
Fx (A U
(14)
x
(Q) instead of Q,
projection implies that
i
who
therefore would participate in voting for ipx
case, too,
tp
ip
votes y
(Q)'>
{a}) implies
x
(V'x (Q))
if
Q
= ^x (x (Q)) =
proposed
is
Q ^ Fx (A U
if ip
ipx (Q) would result in only one transition while proposing
Q is a
Q must
x
(Q)
Q
would
if
x
hold.
=
having
a
is
have proved that coalition
Q ^ P implies n (P) < n (Q). But in that
P instead of Q as the URC, even if it means an
a,
and
indifferent,
because the number of transitions
Q = ^x (Q))- These arguments
P will be accepted under axSubcase
2:
is
Q
P
is
case either
rejected
if
P = ip x (P)
1S
winning within X, but
is
=
satisfy
proposed would
Q- Finally, for a to
7p <
-Jq
extra transition) or
the same because both
proposed.
Q
is
Clearly,
otherwise the payoffs under the two proposals are identical and
Since
would
in that
feasible in minimization (11)
is
together imply that deviation to
suppose coalition
and
better than proposing Q, which contradicts the
1S
propose Q, a G
i
1S
(Q)
Q
ax
=
result in two. Agenda-setter
best response for a and establishes that ipx (Q)
for
(x (Q))
proposed, but proposing
assumption that
We
X
part of ipx (Q),
{a}) (otherwise
were proposed. Therefore, ipx (Q) would be accepted
ip
is
moreover, he would vote y under
one can see that anyone who votes n
a must be in ipx {Q)i so proposing
would be accepted, too.
it
Q
is
-y
P =
-fq (then
P = tp x
(P) an d
not profitable for a
Q
when
must be accepted,
for
not a profitable deviation.
not accepted, then, from (14),
27
(then a prefers
7fx {A) —
^-ip
x (p
)
and
P
Fx (A U
7^
< n(P)
be rejected. In both cases, n(Q)
=
Fx
of winning coalition
Q
we can show that
in the- previous case
>
accepted, (14) implies that either "(Fx{.A)
is
members
otherwise
for
Q
Since
a e Q.
As
{a})-
libxiQ) ox
Q
(A) would vote against
(in the first case
because
=
€ Wx, i>x (Q)
*jq
Q = ^X
in
=
Q> an d
(AL) {a}),
ax and Q would
< 1fx (A) ^
(Q)
7,/,
P = Fx {a) is feasible in minimization (12) for
Y = Au{a}, and P ^ Q). This means, however, that P cannot be the outcome in minimization
= a because Q is also feasible (Q e Wx, 6 Q, and x (Q) = Q because ip x (Q) = Q
(11) for
(P)
lib
iPi ano
-
i
n the second case because
i
and
a,
Q ^ X
P = Fx
where the
by construction of
(a)
profile
<
from n (Q)
latter follows
ax
in (14). Therefore, there
the agenda setting step either. This completes the proof of
Lemma
Proof of
Base:
If
2:
This, however, contradicts that
n(P)).
no profitable deviation at
is
Theorem
2.
This proof also uses induction on the number of steps in
only one step remains, then the current ruling coalition
must be voting by the
P = X made
v over proposal
last voter
IV
some X, and
is
by the
this step
last agenda-setter.
X will be the URC, and each
each players in N \ X will receive the same payoff
Regardless of the vote (and therefore in either profile), coalition
player
S
i
X
will receive payoff
under both
profiles,
player faced
is
W{ (X)
—
ejh',
because the intermediate coalitions and the number of transitions each
the same because histories up to the last step are identical.
i
Step. Take any history h and denote the
payoff to player
'
i
profile
a and ax
profile
a
is
when
by Ui
b plays action £
played thereafter.
is
It
when ax
is
Then both £ K and
is
must
£
if
will result in the
consider the case where £
i
same
^ £K
together with
(2),
Rx
if
are
URCs
£
if
N
£
play of profile ax, will result in
after the first elimination,
is
optimal for player b
if
=
Ub
and
played,
is
same payoff
for b
vice versa.
profile
URC
and
0,
profile
payoffs). This
Both action
.
£K
because both are
is
£,x are played
=
result in the
- Wb
by
b,
(Rx)\
<
,
w^ (Rx)
7^
28
wb
>
ax
URC
and
same
when
£
=
£x-
Now
accompanied by equilibrium
Lemma
1:
most one more elimination. This,
2e and Wb (Rq)
respectively.
profile
followed by equilibrium play
clearly true
and action £
any, equilibrium play will have at
First, consider the case u>b (Rq)
ax
or 2 additional eliminations, as follows from
1,
a with a
(£o)-
(then by induction, action £
implies that \wb (Rq)
an d
if
show that both action £ followed by equilibrium play of
the same payoff for ah players
a
ax
profile
yield the
and action £ K followed by equilibrium play of the same
of profile
some action
and the
b
same both
the
is
be an action played in
£
played thereafter. Thus Ub (£#•)
therefore suffices to
if
true
ax and
subgame Fh by
induction this value
Consequently,
Let £ K be the action played by b in profile
optimal
By
(£).
played in subgames of Th, the same
non-zero probability.
player to act in
first
=
There are two
Wb (Rx), where Rq and
possibilities.
then b € Rq, b G Rx, hence j Rq
= Jr k
,
and by Assumption
transitions
—
{Ro)
i
If
£ (jh
Rq
+
= Rk
then each player
transitions,
X \ Ro
6
i
and £K
Lemma
agenda-setting, for in that case
w~ —
gets
equilibrium play involves only one more transition.
and
proposal P; moreover, both action £
again by
Lemma
X
£
i
X
receives the
P
to
same payoff
Second, consider the case Wb(Ro)
K
Fx
is
(b)
that b
and from
not be part of (this happens
P
if
the case under consideration b £
the proposal of some coalition
an extra transition to
and
Ro
Q
but
= Fx
(A)
gets Wi (Rq)
P, then
b
—
= Rk,
<x^-).
votes y, he receives
But
Lemma
this
1
he receives
means that Ub
(y) j^
is
ip
e
x
step of
first
i
(P)
Rk = Fx
is
first
Now
is
and £K
is
and
b
so Ub (£o)
£
and
£K
Fx
Q
If
is
must be
that b
is
K
.
agenda-setter;
Lemma
{A) which b
=
may
— ejh-
w^
the
1,
may
or
Action £
= wb ~
URC
rejected, the
—
£ (jh
+ 1)
while
b is voting over
if fi is
i
£ tih
+
is
1)
must be
G
X \ Rq
some proposal
played (or vice versa, but this
member
Rq and Rr. Therefore,
of
,
¥"
is
Ub
On
the other hand,
is
b
(£a')>
which implies that
b
votes n, then by
if b
only one transition in which
b is
eliminated.
But
cannot be indifferent
thus yielding a contradiction.
is
indifferent.
But any player
i
because in this case the payoffs are entirely determined by history
of the step of induction,
or £
accepted, but b £ Rq, then b has
have therefore proved that after either of the two actions £
X
two
in that case the
y and the other h. For the action to matter,
^ Rq, b $ R^).
because there
£
some
(he does not participate in the second transition, which will
(fi),
i
is
the set of would-be agenda-setters. In
suppose that
played and rejected
1)
P
Th
step of
voting over
T/j is
Then, as implied by
{A) and Ub (£#)
Q
first
P- This establishes that each
¥"
£ Rq will receive Wi (Rq)
Ub
the same and each player
the
But
transition.
accepted) or
A
6 Rq gets
i
acceptance of this proposal, for a
Suppose
.
recall that voter b is not a
w^ — ejh,
between the two actions
We
b
is
Therefore,
^fc(Co)-
w~ — e(jh+
happen under ok because
if
eventually eliminated, so he receives Ub (£o)
=
y
P
such that b & Q.
£ P. Then one of actions £
impossible under
the
P = Fx (b).
if
ejh in the case of either action.
if
to
= Wb(Rx) = w^
Rk, hence
and each player
proposal must be accepted
P
rejected); here
Q^P
is
this contradicts Ub (£.k)
is
to Rq, then player
regardless of whether b plays action £
part of (this happens
is
Then
will result in
then action £K corresponds to making proposal
URC
X obtains utility W{ (Rq) — ejh
action £ K played under profile <jk, the
would lead to only one extra
1,
additional transitions are from
player
£
between
indifferent
is
of
Consider the case where there are
ejh-
if
The number
cases.
and
This cannot be the case
.
implies that
1
£
i
X
exactly one transition from
is
exactly two transitions both after £
rejection,-
both
in
more
an d player
1)
same
the
is
b participates in all transitions
there
If
URC
so that the
,
same (because
there are no
both actions.
for
w
also the
is
and £ K ).
£
2,
and therefore of
Lemma
29
2.
and £ K
6
h.
is
N\X
played, the
is
URC
indifferent, too,
This completes the proof
Proof of Theorem
the entire
in
game
SPE must
any
given by
X ^Y
is
The proof
follows immediately
starting with history h
=
The lemma then
0.
Lemma
A{N) may
set
of hyperplanes given by equations
and by equations
= af x
-yy
implies that the
K, and
i-e.
2 to
URC
must be
payoffs
2.
(Part 1) The
3:
Lemma
from the application of
coincide with that under the strategy profile uk,
Lemma
number
finite
which
implied by
(7), as
Proof of
a
T,
3:
au
f° r
X,Y
£
=
^x
R++ by
be obtained from
P (N)
f° r
^Y
X,Y
au
such that
P (AT)
e
X cY.
subtracting
such that
These hyperplanes
are closed sets (in the standard topology of R++), hence, a small perturbation of powers of a
generic point preserves this property (genericity). This ensures that
A(N)
S
dense because hyperplanes have dimension lower than \N\. The proofs for
S (N) = R++ and
by induction. The base follows immediately since
Now
suppose that we have proved this result
N
{fi}ieN>
self-enforcing
is
Now
within N.
perturbation
and only
if
if
and
(N) are
(AT)
M (N) =
if
and only
if
J\f
are open sets.
For any distribution of powers
\N\.
small, then the set of winning coalitions
is
set; it is
take some small (in the sup-metric) perturbation of powers
self-enforcing
is
<
an open
there are no proper winning self-enforcing coalitions
same
set of proper self-enforcing coalitions is the
{7^}
for all k
is
{"f'i}
ieN
If this
-
the same, and, by induction, the
is
Therefore, the perturbed coalition
as well.
the initial coalition with powers {7^}
self-enforcing;
is
which
completes the induction step.
(Part 2) Take any connected component
open
in
A
in the
topology induced by
topology.
Also, (S
given that
A
(N) n A) n {Af (N) n A)
either entirely within
S (N)
Proof of Proposition
a,
$
N
with \N\
(N, 7, w, a)
described in
is
=
n.
=
{N}.
We
or
N (N).
1:
The
Now
Both S (N) n
(and, in turn, by
=
A
K++) by
and
N
S (N) D A
or J\f [N)
nA
n
=
A
are
A, which,
is
empty. Hence,
If
N=
A
lies
This completes the proof.
first
two parts follow by induction.
suppose that this
is
true for
all
N with
\N\
$
<
1, for
n; take
any 7 and
any society
(N, 7, w, a), which
In particular, Assumptions 2 and 3 imply that the set
1.
(AT)
definition of induced
and {S [N)T\ A) U (A/ (AT) n A)
then use the inductive procedure for determining
Theorem
identical for
A (N)
connected, implies that either
is
A C A (N).
r (N, 7,10, a), F (N,j',w,a), and T (N,j,w,a'), provided
that 5
M. (N)
is
is
in (6)
sufficiently
small (the result self-enforcing coalitions remain self-enforcing after perturbation follows from
Lemma
3).
implies that
1
and
Moreover,
$
if
5
(N, 7, w, a)
is
small, then
7^ > 7y
= $ (N, 7', w, a) = $
is
(AT, 7,
equivalent to
w,
a').
7^ >
j'
Y
.
Therefore, (5)
This completes the proof of parts
2.
The proof
of part 3
is
also
by induction.
Let |Ai|
straightforwardly. Suppose next that the result
is
30
=
true for n.
For n
n.
If
5
is
—
1
the result follows
small enough, then
<f>
(N\)
is
N = N\ U N2; we also know that
winning within
that there exists no
in
X
C N\U N2
N\ U N2 and has 7^ <
(i.e.,
it is
=
such that (j>(X)
X
Consider
7xnNi where the
that
is
X € P (N\ U N2)
X n N\.
JVi,
By
definition,
to verify
winning
self-enforcing,
not the case
is
does not coincide with
7^ >
Imni) > Ix
^
by hypothesis. This string of inequalities implies that
a proper subset of N\, thus must have fewer elements than n. Then, by induction, for
is
small enough
and thus
within
lies
strict inequality follows
1
X n N\
part that
its
X
(i.e.,
Jmjji))- Suppose, to obtain a contradiction, that this
that the minimal winning self-enforcing coalition
<p(Ni)).
Thus we only need
self-enforcing.
X
U
within N\
S,
{X n N\) —
<f>
C
winning within
is
implies that the hypothesis
is
Proof of Proposition
2:
of the two
is
(since
X
7^/jVi)
But
TVi).
true for
n
+
1.
and winning within N\
^ 7x
(since 4>{N\)
X
stronger than
is
is
foUows directly from Part
is
always self-enforcing. For the case of elimination: suppose that
is
self-enforcing.
This
coalition.
Then, by Part
is
is
adding this person back
a self-enforcing coalition
M of
powers to form coalition iV would yield
the set of players.
If
m=
=
sequence recursively: 7j
2,
(j>(N)
the statement
1,
7
> Xw=i
fc
m
size
=
7,/,(/v1
Fix
=
2, 3,
lj for
all
fc
m
.
3.
The
stronger
XUY
is
X U Y.
in
wrong, and the coalition
it is
2.
n —
= M).
is trivial.
> Ix an d
)
will result in a non-self-enforcing
(then adding
cj>{M)
Ni,
winning
Therefore,
URC
(Part 1) Given Part 3 of Proposition
3:
C
since coalition of one player
1,
a contradiction which completes the proof of Part
Proof of Proposition
that, there
1,
X UY.
not the
it is
iV*i)
it is
Y or vice versa.
not equal to
not the minimal winning self-enforcing coalition, and so
(X n
the minimal self-enforcing
is
This completes the proof of part
a winning self-enforcing coalition that
it
<j>
(since
this contradicts the inequality
(Part 1) Either
(Part 2) For the case of adding,
However,
self-enforcing).
is
self-enforcing
is
This implies that
N2).
coalition that
X
N\. Therefore,
X
=
(X)
<j)
.
.
>
,
m players
Let
2
to
1, it is sufficient
i
M
G
show
with negligible
=
{l,...,m} be
and construct the following
m—
1,
=
7m
~! ^
Y2T=i 7j
*s
straightforward to check that numbers {7i}; e j^ are generic. Let us check that no proper winning
coalition within
M
to check that \X\
player
m
is
>
self-enforcing.
no
for
2,
Take any proper winning coalition X;
single player forms a
(with power 7 m ), then
it
winning coalition.
excludes some player k with k
it is
straightforward
If coalition
< m;
his
X
power 7
includes
fc
>
2
by
construction. Hence,
m—
=
7m
Yl
which means that player
If
X
does not include
r
y
m
m
,
is
7j
-
1
m—
> Yl
Tj
-
7fc
> 7x\{m},
stronger than the rest, and thus coalition
M
then take the strongest player in X; suppose
However, by construction he
is
stronger than
all
31
is
it
non-self-enforcing.
is k,
other players in X, and thus
X
k
<
is
m—
not
1.
self-
M
enforcing. This proves that
is-
then one of the players, say player
self-enforcing coalition.
(Part 2) The proof
=
follows: 7j
7
2,
Proof of Proposition
\X\
fc
fc
lj for all
^ |y\JT| => i x ^ 1y\x
m 's in Part 1
m = 2m — 1 =
and
|_2fc
=
fc
1.
X
1,
The
2, 3,
,
.
"*
is
satisfied
m-
1,
=
7m
a YJj^i
==**
X
lx ^ alY
—
Q =
f° r
2
—
1
N
X \ (X n Y)
implies
>
such that \X\
and Y'
\Y\
1
=
-
lj
= Y \ (X D Y),
which do not
Zjex> 7,7> < ZjeY' 7i/A, and thus
1-
,
(
7 ,/A -
l)
if
m -i =
2
fc
false,
is
Then the same
-
\X\
3.
^ a
To
see
-*=*>
\Y\
m_1 —
1
then
by induction.
intersect, so that
£ieX
C Y C N,
and
1,
result follows
we have j x < jY
a winning
Moreover, the sequences of
1/2-
(Part 2) Suppose, to obtain a contradiction, that the claim
X,Y C
is
cannot be self-enforcing.
since for any
,
and thus the desired
1
and Assumption 2 holds,
2
recursive sequence should be constructed as
...
in Part 3 are equal since k\
m _iJ +
=
\X\
(Part 1) This part follows as a special case of Part
4:
note that the condition in Part 4
this,
if
stronger than the other one, and thus {i}
is
identical to Part
> a £^lj
fc
i,
But then, by Corollary
is
However,
self-enforcing.
X'
inequalities hold for
< T,jev
< £^ yi ( 7j./A -
£jex'
+ |X'|
some
that for
i.e.,
Jj
lj-
=
This
l)+|y'|.
Rearranging, we have
isM-M<Eff-i).-E(?-^ £
However, X' and y' do not intersect, and therefore this violates
(17).
It-
1
!-
This contradiction com-
pletes the proof of Part 2.
(Part 3) The proof
enforcing,
it
for q
=
and \N\
=
\N\.
\N\
size at least
a
If
(|_A;
The base
by induction.
is
is
a one-player coalition
trivial:
TV
I
I
^ km
for
= 1. Now assume the claim has been proved for all q < \N\, let us prove
= km for some m, then any winning (within N) coalition X must have
m _i/aJ +
1)
> k m -\
any m, then take
(if it
has smaller size then
m
such that
the strongest k m -\ individuals. This coalition
(this follows since
would have
k m -\
at least
self-
k\
7^ <
By
a~f N ).
such coalitions are not self-enforcing, and this means that the grand coalition
If
is
> a
a share
[k m -i/a\
of
power
is
fc
m _i <
<
self-enforcing
= a(k m —
if all
\N\
1)
>
individuals
a|7V|,
km-
Now
is
induction,
all
self-enforcing.
take the coalition of
by induction.
It is
also
winning
which means that this coalition
had equal power, but
since this
is
the
strongest k m -\ individuals, the inequality will be strict). Therefore, there exists a self-enforcing
winning
coalition, different
from the grand
coalition.
This implies that the grand coalition
is
non-self-enforcing, completing the proof.
(Part 4) This follows from Part 3 and Proposition
Proof of Proposition
that includes
|7V|,
6: Inequality
3.
7^1 > a Yl]=2 Ijl
but excludes even the weakest player
32
will
U—
a)
implies that any coalition
not be self-enforcing.
The
inequality
7|jvi
< a Y^j=2
Ijl
(1 ~~
Therefore, either TV
is
a)
implies that player \N\ does not form a winning coalition by himself.
self-enforcing or
(j>
(TV)
does not include the strongest player.
33
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