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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Dynamics and
Stability of Constitutions,
Coalitions,
and Clubs
Daron Acemoglu
Georgy Egorov
Konstantin Sonin
Working Paper 08-1
August 4, 2008
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
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http://www.archive.org/details/dynamicsstabilitOOacem
Dynamics and
Stability of Constitutions, Coalition.s,
and Clubs*
Daron Acemoglu
Georgy Egorov
Harvard
MIT
Konstantin Sonin
New Economic School
August 2008
Abstract
A
dynamic collective decision-making is that the rules that govern the procedures
decision-making and the distribution of political power across players are determined by current
central feature of
for future
decisions.
For example, current constitutional change must take into account
how
the
new constitution
and regulations. We develop a general framework for the
analysis of this class of dynamic problems. Under relatively natural acyclicity assumptions, we provide
a complete characterization of dynamically stable states as functions of the initial state and determine
conditions for their uniqueness. We show how this framework can be applied in political economy,
coalition formation, and the analysis of the dynamics of clubs. The explicit characterization we provide
highlights two intuitive features of dynamic collective decision-making: (1) a social arrangement is made
stable by the instability of alternative arrangements that are preferred by sufficiently many members of
the society; (2) efficiency-enhancing changes are often resisted because of further social changes that they
may pave
will
the
way
for further
changes
in laws
engender.
Keywords: commitment,
constitutions,
dynamic
coalition formation, political economy, stability,
voting.
JEL
Classification: D71, D74, C71.
*Daron Acemoglu gratefully acknowledges financial support from the National Science Foundation and the
Canadian Institute for Advanced Research. We thank John Duggan, Matthew Jackson, Robert Powell, and
conference and seminar participants at the ASSA 2008 (New Orleans), the 2008 Game Theory World Congress,
Berkeley, California Institute of Technology, Harvard and Princeton for useful comments.
Introduction
1
Consider the problem of a society choosing
from adopting a
Naturally, the current rewards
constitution.
its
But, as long as the
specific constitution will influence this decision.
of the society are forward-looking
and
members
may
patient, the future implications of the constitution
be even more important. For example, a constitution that encourages economic activity and
benefits the majority of the population
minority to seize political control.
for a
may
nonetheless create future instability or leave
the society
If so,
—or the majority of
away from adopting such a constitution. Many problems
rationally shy
its
members
in political
room
—may
economy,
club theory, coalition formation, organizational economics, and industrial organization have a
structure resembling this example of constitutional choice.
We
develop a general framework
constitutions, coalitions,
and
for
the analysis of dynamic group-decision-making over
clubs. Formally,
The
of infinitely-lived individuals.
we
consider a society consisting of a finite
society starts in a particular state,
as the constitution of the society, regulating
how economic and
determines stage payoffs and also how the society can determine
for
which can be thought of
political decisions are
its
dynamic game when individuals are
made.
It
future states (constitutions),
example, which subsets of individuals can change the constitution. Our focus
perfect) equilibria of this
number
on (Markov
is
sufficiently forward-looking.
natural acyclicity assumptions, which rule out Condorcet-type cycles,
we prove the
Under
existence
and
characterize the structure of [dynanucally) stable states, which are defined as states that arise
and
An equihbrium
persist.
stable state
</>(so)
stable states
is
represented by a
mapping
as a function of the initial state sq.
(f),
We
which designates the dynamically
show that the
largely independent of the details of agenda-setting
Although our main focus
paragraph, we
is
first start
is
the analysis of the noncooperative
set of
dynamically
and voting protocols.
game
outlined in the previous
with an axiomatic characterization of stable states.
This character-
ization relies on the observation that sufficiently forward-looking individuals will not wish to
support change towards a state (constitution) that might ultimately lead to another,
ferred state. This observation
is
encapsulated in a simple stability axiom.
We
less pre-
also introduce
two
other natural axioms ensuring that individuals do not support changes that will give them lower
utihty.
We
characterize the set of mappings,
<I>,
that are consistent with these three axioms
provide conditions under which there exists a unique
</>
€ $ (Theorem
a singleton, the sets of stable states according to any two
Our main
results are given in
Theorem
2.
voting protocol the equilibria of the dynamic
and that
for
c^ij,
02 €
$
1).
Even when
<& is
and
not
are identical.
This theorem shows that for any agenda-setting and
game we
outline can be represented by
any ^ G $, there exists a protocol such that the equilibrium
will
some
<^
g $
be represented
by
This means that starting with
4>.^
stable state
(sq) for
cp
some
initial state sq,
when such
£ $. Naturally,
(^
any equihbrium leads to a dynamically
unique,
is
all
equilibria result in the
uniquely-defined dynamically stable 0(so).
An
attractive feature of this analysis
that the set of dynamically stable states can be
is
characterized recursively. This characterization
computed using
states can be
ticular state
stable
and
is
dynamically stable only
is
of our approach
—those
commit to
it
not only simple (the set of dynamically stable
also
emphasizes a fundamental insight: a par-
there does not exist another state that
if
that gain additional decision-making power as a result of a reform cannot
refraining from further choices that
would hurt the
characterization result provided in these theorems.
A
dynamically
the natural lack of commitment in dynamic decision-making
is
initial set of
This lack of commitment, together with forward-looking behavior,
1.
is
preferred by a set of players that form a winning coalition within the current state.
At the center
problems
induction), but
is
It also
is
decision-makers.
at the root of the general
leads to two simple intuitive results:
particular social arrangement (constitution, coalition, or club)
is
made
stable not by
the absence of a powerful set of players that prefer another alternative, but because of the
absence of an alternative stable arrangement that
To understand why
constituency.
study the
2.
instabilities that
nated by the payoffs in another
tions in
final general result,
Theorems
1
and
Theorem
2 to hold
subset of M). In particular,
inefficient
—
We
also
Pareto domi-
•
state.
3,
'
'
.
-
-.
provides sufficient conditions for the acyclicity assump-
when
of winning coalitions are satisfied).
may be
-
states belong to an ordered set (e.g.,
single-crossing property or are single peaked (and
variety of environments.
in the sense that they
shows that these results apply when
it
we must thus
certain social arrangements are stable,
changes away from these arrangements would unleash.^
Dynamically stable states can be
Our
preferred by a sufficiently powerful
is
when they
are a
(static) preferences satisfy
a
some minimal assumptions on the structure
This theorem makes our main results easy to apply in a
show that Theorems
which states do not belong to an ordered
The next two examples provide simple
1
and
2 apply in a range of situations in
set.
illustrations of the
dynamic
trade-offs
emphasized by
our approach.
An
additional assumption in the analysis of this
game
is
that there
is
a transaction cost incurred by
all
This assumption is used to prove the existence of a
pure-strategy equihbrium and to rule out cycles without imposing stronger assumptions on preferences (see also
individuals every time there
Examples
3
and
4 in
is
Appendix
a change in the state.
C).
'This result also shows that, in contrast to Riker's (1962) emphasis in the context of political coalition formation
games, the equilibrium will typically not involve a "minimum winning coalition," because the state corresponding
to this coalition
is
generally unstable.
Example
E
and
represents the elite and initially holds power,
M;
invest; (3)
monarchy
(1) absolutist
are three states:
rights for
(2) constitutional
democracy
d,
where
Suppose that stage payoffs
clearly prefers
Both
absolutist monarchy.
in
which
M has greater security and
c,
satisfy
than under absolutist monarchy,
M
E
and
,
E
that
with no checks and no political
and the
influential
privileges of
w^.j (a)
wm (c)
<
democracy
willing to
is
E
disappear.
w^ {d)
<
has higher payoff under constitutional monarchy
M increase
example, because greater investments by
for
There
^
-
(c)
rules,
M.
and
class.
which
M becomes more
we (a) < we (c) means
In particular,
M corresponds to the middle
a, in
monarchy
We {d) < We (a) < We
revenues.
E
(Inefficient Inertia) Consider a society consisting of two individuals,
1
monarchy and
to constitutional
under
least well-off
is
parties discount the stage payoffs with discount factor
/3
G
tax
As
(0, 1).
described above, "states" not only determine payoffs but also specify decision rules. In absolutist
monarchy,
E
that starting in both regimes c and
d,
introduced above, this implies that d
is
=
d.
Therefore,
is
in the following period,
and thus give
E
if
E
=
decides to stay in a forever,
sufficiently small,
/3 is
when
then
Ue
case, the
(d)
=
d.
In contrast, c
there will be a transition to d
chooses a reform towards
is
/?
u;£
(c)
c,
and thus
this will lead to d
+ /3^^^.
payoff
its
< Ue
(no reform)
players are forward looking and
E
c,
cp
a discounted payoff of
t/E (reform)
In contrast,
In terms of the notation
a dynamically stable state, and
starting in regime a,
if,
simplify the discussion, suppose
M decides next period's regime.
not a dynamically stable state, since starting from
4>{c)
To
decides which regime will prevail tomorrow.
Ue
is
and reform
(reform),
then
large,
unique dynamically stable state starting with a
is
a
— that
(a) / (1
will take place.
(no reform)
(/f;
= we
(no reform)
is,
>
4>
—
=
If
However,
In this
C/g (reform).
(a)
/3).
a.
This example, when players are sufficiently forward-looking, illustrates both of the intuitive
results
is
mentioned above.
preferred by those
in c
than
who
First, state a is
made
are powerful in
a.
in a, so the stable state starting
general insight: the set of stable states
small, only d
A
similar
is
stable,
whereas when
game can be used
to
by the
E
Second, both
from a
is
j3 is
stable
is
larger
large,
Pareto
when
act "conservatively"
M would be
both a and d are
that
another
(when
(3 is
stable).
of concessions in wars. For example, a
may
not take place because
could also be used to illustrate
and resist efficiency-enhancing
c,
strictly better off
inefficient. It also illustrates
concession that increases the payoffs to both warring parties
It
and
players are forward-looking
model the implications
change the future balance of power.
instability of another state,
how
it
will
organizations might
restructuring. For instance, the appointment
CEO who
of a
would increase the value of the firm may not be favored by the board of directors
they forecast that the
if
Example
2
(Voting
A
individuals.
of
A''
Wi
(st),
to
some voting
which
CEO
in
would then become powerful and reduce
Clubs) Consider the problem of voting
club
a subset of the society.
is
as a function of the current club
is
tomorrow's club
rule)
st+i.
studies a special case of this environment,
club
St
must take the form Xk
=
their privileges.'^
in clubs.
Each individual
i
The
society consisting
receives a stage payoff
and current club members decide (according
st,
The seminal unpublished paper by Roberts (1999)
where individuals are ordered,
{!,... ,/c} for
some k
=
i
=
1,2,..., A^,
l,2,..,N, and decisions are
any
made by
majoritarian voting. Under a range of additional assumptions Roberts estabhshes the existence
of mixed-strategy (Markovian) equilibria
nests a
more general version
and characterizes some
of this environment
Our model
of their properties.
and enables us
to establish the existence of
a unique dynamically stable club (and a pure-strategy equilibrium) under weaker conditions.
In addition, our approach allows a complete characterization of dynamically stable states
clarifies
the reasons for potential Pareto inefficiency.
These examples
rich set of
illustrate
some of the possible applications
as its
major advantages. Both our
be applied to a range of problems
On
of our approach.
-
We
view the
environments that are covered by our model and the relative simplicity of the resulting
dynamic stable states
and other
•
and
areas.
Some
in political
of these additional
specific results
and the general ideas can
economy, organizational economics, club theory,
examples are discussed
in Section 6.
the theoretical side, Roberts (1999) and Barbera, Maschler, and Shalev (2001) can be
viewed as the most important precursors to our paper. Barbera, Maschler, and Shalev (2001)
study a dynamic game of club formation in which any
a
new
agent.
The
of political reform
precm-sor.
of the club can unilaterally
admit
recent ambitious paper by Lagunoff (2006), which constructs a general
model
and
is
relates reform to the time-inconsistency of induced social rules,
Acemoglu and Robinson's
franchise extension
related
themes as
member
(2000, 2006a)
and
Lizzeri
and Barbera and Jackson's (2004) model
well.
discussed in Section
How
is
another
and Persico's (2004) analyses of
of constitutional stability are
on
these papers can be viewed as applications of our general framework
'
6.
The two papers most
closely related to our
work are Chwe (1994) and Gomes and
Jehiel
Ideas related to this example have been discussed in a number of different contexts. Robinson (1997) and
Bourguignon and Verdier (2000) discuss how a dictator or an oligarchy may refrain from providing productive
public goods or from educational investments, because they may be afraid of losing power. Rajan and Zingales
(2000) also emphasize similar ideas and apply them in the context of organizations. Acemoglu and Robinson
(2006a) construct a dynamic model in which the elite may block technological improvements or institutional
reforms, because they will destabilize the existing regime. Fearon (1996, 2004) and Powell (1998) discuss similar
ideas in the context of civil wars and international wars, respectively.
'
Chwe
(2005).
model where payoffs are determined by states and there are exogenous
studies a
rules governing transitions
Chwe demonstrates
from one state to another.
the relationship
between two distinct notions from cooperative game theory, the consistent and stable
However,
in
Chwe's setup, neither a noncooperative analysis nor characterization results are
possible, while such results are at the heart of our paper.
sets
and our dynamically stable
states
is
related environment with side payments.
payoff to improve
his'
may be
that equilibrium
The
link
make
between Chwe's consistent
Gomes and
discussed further below.
They show that a player may
bargaining position for the future, which
of winning coalitions to
sacrifice his
when the discount
factor
is
More
and are more Hkely when discount factors are large
we
generally,
not present in
instantaneous
They
also
show
small. In contrast, in our
game
transitions to non-stable states in our paper.
inefficient
may emerge
we suspect
Jehiel (and in fact, with side payments,
as
unique
by Example
(as illustrated
also provide a full set of characterization (and uniqueness) results,
Gomes and
study a
Jehiel
related to the unwillingness
is
Pareto dominated outcomes are not only possible in general, but they
equilibria
sets.
1).
which are
that such restilts
are not possible). Finally, in our paper a dynamically stable state depends on the initial state,
while in
Gomes and
Jehiel, as the discount factor
that the ultimate distribution of states does not
Finally, our
work
1956, Ellickson et
al.,
is
1999, Scotchmer, 2001).
important papers
in this
there
is
"ergodicity" in the sense
initial state.
on club theory
While early work
dynamics
and Barbera, Maschler, and Shalev
1,
depend on the
also related to the literature
of recent papers have investigated the
(1999)
tends to
(see, for
in this area
of club formation.
(2001),
admission to a jurisdiction
than an unrestricted right to migrate.
Alesina, Angeloni,
version of Roberts's model to the enlargement of the
who
rest of the
paper
is
a
number
develop an incomplete
EU
may
who show
that the
not be more restrictive
and Etro (2005) apply a
simplified
and Bordignon and Brusco (2003)
study the role of "enhanced cooperation agreements" in the dynamics of
The
static,
In addition to Roberts
contracts theory of club enlargement, and Jehiel and Scotchmer (2001),
for
was
which were discussed above, some of the
area include Burkart and Wallner (2000),
requirement of a majority consent
example, Buchanan,
EU
enlargement.
organized as follows. Section 2 introduces the general environment.
Section 3 motivates and presents our axiomatic analysis, which also acts as a preparation for
our noncooperative analysis. In Section
perfect equilibrium of the
lish
we prove
for
the existence of a (pure-strategy)
Markov
any agenda setting and voting protocol and estab-
the equivalence between these equilibria and the axiomatic characterization in Section
Section 5 shows
set.
dynamic game
4,
how
the results of Sections 3-4 can be applied
when
3.
states belong to an ordered
Section 6 discusses a range of applications of our framework, including the two examples
presented above. Section 7 concludes. Appendix
A
presents the
main proofs omitted from the
Appendix B and C, which are not
text.
some omitted
additional results, examples, and
number
of generalizations,
proofs.
Environment
2
There
is
for publication, contain a
a
is
Time
finite set of players X.
a finite set of states which
X,
of elements of set
so
we denote by
and
\J\
is
and
discrete
infinite,
indexed by
i
(i
>
Tliere
1).
Throughout the paper, |X| denotes the number
S.
denote the number of individuals and states, respectively.
|<5|
States represent both different institutions affecting players' payoffs and procedures for decision-
making
the identity of the ruling coalition in power, the degree of supermajority, or the
(e.g.,
game
weights or powers of different agents). Although our
limited
amount
same procedure
initial state is
distribution. For any
called coalition,
t
>
G S,Wi
(s) is
that Wj (s)
1,
game
the state
s
E
S
decision-making but different payoffs across players.
This state can be thought of as being determined
iS.
&
S
is
A nonempty
determined endogenously.
set of coalitions
by C (that
characterized by a pair
is
(s)}j£j
({lo,
C
is,
,
>
is
the set of
is
Ws) Here,
a (strictly) positive stage payoff assigned to each individual
i
pohtical institutions in state
s.
set of
winning coalitions
in state
s.
for
We
W^
use
is
if in
is
i
is
nonempty
each state
Ws
to
model
This allows us to summarize different political procedures, such
required for decision-making,
state s individual
a dictator,
Ws
Ws
includes
includes
all
all
subsets of
I
if
in state
that form a majority;
coalitions that include i^
Since
function of the state, the procedure for decision-making can vary across states.
Ws
(a) If
(b)
1
X,Y
(Winning Coalitions) For any
e C,
IfX,Ye
X
c
y, and
Ws, then
X eWs
X r\Y
Part (a) simply states that
if
=^
Notice that
game.
We
Ws
some
or the pohtical rules
Y
G S,
•
.
Ws C C
satisfies:
e Ws.
,
:
•
0.
coalition
of the coalition will not reverse this. This
how voting
then
state s
a
is
;
Throughout the paper, we maintain the following assumption.
Assumption
cI
a (possibly
as weighted majority or supermajority rules, in an economical fashion. For example,
a majority
X
set
G I. The restriction
a normalization, making zero payoff the worst outcome.
empty) subset of C representing the
s
a
utility,
or as chosen by Nature according to a given probability
st
and we denote the
subsets of I). Each state
s
for
denoted by sq G
as part of the description of the
is
one of non-transferable
of transfers can also be incorporated by allowing multiple (but a finite set of)
states that have the
The
is
is
X
is
winning
in state s,
then increasing the
size
a natural assmuption for almost any decision rule.
do not specify certain institutional
details,
such as
who makes
proposals,
takes place and so on. These are specified by the agenda-setting and voting protocols of our dynamic
will
show that these only have
a representation of "political rules"
a limited effect on equilibrium outcomes, justifying our focus
on W,, as
,
Part (b) rules out the possibility that two disjoint coalitions are winning in the same state, thus
imposing a form of (possibly weighted) majority or supermajority
is
rule. If
Ws =
0, then state
s
exogenously stable. None of our existence or characterization results depend on whether tliere
exist exogenously stable states.
The
following binary relations on the set of states
S
will
be useful
for
the rest of our analysis.
For x,y E S, we write
X
In this case
we
our purposes
^y
I
itij
:
=
{x)
w, {y)
(1)
.
x and y payoff- equivalent, or simply equivalent. More important for
call states
the binary relation
is
<==^ Vi e
>^2.
For any ^ € 5, ^^
is
defined by
yhz^r <=> {iel .w^{y) >Wi{x)} eWzIntuitively, y >Zz
each of
its
x means that there exists a coalition of players that
members weakly
preferring y to x.
i
is
winning
(in z) ^vith
Note three important features about ^z-
> w^
only contains information about stage payoffs. In particular, Wi (y)
individual
(2)
than
will prefer a switch to state y rather
x.
Whether
(x) does not
or not
First,
it
mean that
he does so depends
on the continuation payoffs following such a switch. Second, the relation ^, does not presume
any type of coordination or collective decision-making among the members of the coalition
question.
on
It
z since
set of
whether the coalition of players weakly preferring y to x
winning
neither
which
coalitions,
we say that y
holds,
in the
simply records the existence of such a coalition. Third, the relation ^,
means that
is
all
state dependent.
weakly preferred to x in
With a
z.
it
Relation
~
say that y
is
.X
^^
we have
The
some
x,
{i
el
strictly preferred to
and whenever
>-,
Wj
x in
2
defined similarly by
'
(3)
~
y and y
~
z,
is
nonempty, we also have x ^^
y,
Finally,
similarly y
some basic properties
on payoff functions and winning
(Payoffs) Payoffs {wt
and
x.
^
then x
need not even be transitive. Nevertheless,
following assumption introduces
Assumption
if (2)
z.
if a;
that for any x,y, z G S, y y^ x implies x ^,
joint restrictions
)^; is
.w,[y)> w^[x)] i^Wz-
clearly defines equivalence classes;
the binary relations ^~ and
have
abuse of terminology,
simply designates that there exists a winning coalition of players, each
y>zX ^=>
we
winning depends on the
is
In light of the preceding comments, this
obtaining a greater stage payoffs in y than in x. Relation
If (3) holds,
slight
conditioned
individuals prefer y to x nor that there will be a change from state x to y
—
dynamic game
is
is
in
hz
for
z.
any x,z E S, we
from Assumption
-f^
implies x
of payoff functions
coalitions.
(s)} ^i-j ^^^ satisfy the following properties:
7
In contrast,
1
)f. y.
and places
(a)
For any sequence of
states si, S2,
(b)
For any sequence of
for
>-Sj Sj
Sj+i
states s,si,
Sj+i ^s Sj for
Assumption
2 plays a
major
z
—that
a cycle of states
is,
all I
.
.
.
Sk in S,
<
j
<
,Sk in
S
<
<
all 1
j
role in our analysis
weaker than "transitivity"). Part
X y,
,
(a) of
Assumption
(x, y, z)
k
-
with
k
-
=>
1
^^^
si
Sfc.
Sj ^-s s
for each 1
=4>
^^
I
si
and ensures
<
j
<
Sk-
"acyclicity" (but
2 rules out cycles of the
X ^s
1
and
Assumption
2 are natural given our focus.
3 (Comparability) For x,y,s G
Assumption 3 means that
z are ;^s-comparable.
assumption
Our main
is
It
if
such that x
(stronger) requirement.
y yg
>-s s,
s,
and x
00 y^
either
'
\
y ^s X or xy-sV-
S
>-s x, z >^ ^ U,
Throughout the paper we suppose
we sometimes impose the following
that they hold. In addition,
considerably
such that in each, a winning coalition of players strictly
Part (b) of Assumption 2 rules out cycles of the form y
Assumptions
is
yyxX,z>-yy,
form
prefer the next state.
z.°
k,
.
'
two states x and y are weakly preferred to
s (in s),
then y and
turns out to be sufficient to guarantee uniqueness of equilibria.^ This
not necessary for the majority of our results, including the general characterization.
results are stated
without this assumption and are then strengthened by imposing
it.
In each period, each individual maximizes his discounted expected utility:
U,[t)
where
/3
e
function Wi
(s)
common
a
(0, 1) is
= {\-l3)Yr ,^^^dr),
discount factor and
introduced in Assumption 2
consider situations in which
/3
is
we can think
(see, in particular,
greater than
(MPE)
of this
of dynamically stable states.
stable
all
t
if
>T
there exists a
— that
is,
MPE
[t)
equation
€
/Sg
will
as given by the payoff
(9) in
(0, 1)
Section
(this
4).
"Neither part of Assumption 2
when
is
time
finite
"necessary" in the sense that
in
is
then characterize the Markov
T
in Definition 2, a state s is
dynamically
such that this equilibrium involves
implied by the other. Examples 5 and 6
either 2(a) or 2(b)
with multiple equilibria. Example 7
We
threshold
Sf
=
a dynamically stable state persists in equilibrium.
of cycles that can arise
It is also
We
(4)
.
dynamic game and investigate the existence and structure
As defined more formally
and a
.
':-
•.,,
of u,
some threshold
derived as an explicit function of payoffs in Appendix A)
perfect equilibrium
,,
if
in
s for
'
'
Appendix C
illustrate the
types
fails.
this
assumption
Appendix C
is
dispensed with,
illustrates this.
it is
easy to construct examples
Axiomatic Characterization
3
^
Before specifying the details of agenda-setting and voting protocols,
we
provide a more abstract
(axiomatic) characterization of stable states. This axiomatic analysis has two purposes. First,
illustrates that the
making aie
it
key economic forces that arise in the context of dynamic collective decision-
largely independent of the details of the agenda-setting
and voting
protocols. Second,
the results in this section are a preparation for the characterization of the equilibrium of the
dynamic game introduced
will
make
main
in the previous section; in particular, our
result,
Theorem
2,
use of this axiomatic characterization.
The key economic
insight enabling an axiomatic characterization
that with sufficiently
is
forward-looking behavior, an individual should not wish to transit to a state that will ultimately
lead to another state that gives her lower utility.
Theorem
ization of {axiornatically) stable states.
This basic insight enables a tight character2 in the next section
between the notions of axiornatically and dynamically stable
More
any
We
(f>
states.
formally, our axiomatic characterization determines a set of
G ^,
(j)
:
S
^S
assigns an axiomatically stable state s°° t
impose the following three axioms on
Axiom
1
Axiom
2 (Stability) If x,y £
Axiom
3 (Rationality) If x,y,z £
(Desirability) If x,y
(E
S
S
S
mappings $ such that
to each initial state sq
g
for
S.
(j).
=
are such that y
are such that y
S
shows the equivalence
=
<p
=
(p{x), then- either y
[x), then y
are such that 2
>-2.
x,
=
;
x or y ^x
x.
4){y).
=
(ind z
'/'(-),
^x
y,
then
y^dix).
All three axioms are natural in hght of
that the society should not (permanently)
is
move from
a winning coahtion that supports this transition.
discussed above;
for
if
what we have discussed above. Axiom
any
if
some
initial state,
mapping
4>
state
is
because there
will eventually
Aiciom 3 imposes the rea.sonable requirement that
both X and y by winning coalitions
All three
axioms
vidual preferences.
in state x,
refer to properties of 0,
Because
indirectly apply to the
it
2 encapsulates the stability notion
cannot be the (ultimate) stable state
be a transition away from
picks state y starting from state x, then
then
if
4>
requires
state x to another state y unless there
Axiom
not dynamically stable,
1
it
this state (and thus
should also pick y starting from
y).
there exists a stable state z preferred to
should not pick y in
x.
but they are closely related to underlying indi-
collective decision-making aggregates individual preferences, they
mapping
4'
that summarizes these collective preferences (for example.
one might think that
supermajority
We
rule,
aggregates individual preferences according to majority rule or weighted
(f)
and so
$
next define the set
and the mapping
states
on).
formally and state the relationship between axiomatically stable
(f).
Definition 1 (Axiomatically Stable States) Let
A
£ S
state s
(axiomatically) stable
is
(fixed points) for
states is
V=
{s
mapping
£ S:
(p
The next theorem
(s)
^
E
(p
=
{s
some
for
s
V^ =
is
€ S:
hold.
The
2.
Any
&
(p
is
^
=
Mi-
'-'
\
non-empty. That
I
<
j
For each k
Mk =
':..
(If there exist
corresponds
If,
<
I
~
<
there exists a
is,
\S\,
fii
2, ...,\S\,
^^
satisfies
Axioms 1-3}.
set of stable states
G $} and the set of
(^
all
to
seMk--
\
more than one
multiple
(p
s
stable
and provides a recursive charwhich shows the equivalence
2,
e
and the mappings
1
$.
and
'•
mapping
4>
satisfying
Axioms
Order the states as
(this is feasible
//,
<
1-3.
/Xj, ..., (Ui^j
>
given Assumption 2(a)). Let
'"
define
•
,
{s e {iJ,i,...,iJ,k-i}
s>~f,^
.
fif,
and(p{s)
=
s}
(5)
.
.
heMk
ifMk^fZ
withzy^^s
€ Mk- $z £ -Mk
'^'ith.
z >-^^
^'
'
s,
we pick any of
^
coincide.
these; this
functions).
set of stable states of
any two mappings
in addition, Assumption 3 holds, then any
for any two mappings 0j and
Proof.
(p
e ^. The
cp
;'-;,;i,,
/
^^'^'''
4.
for
iS;
in the previous section
'.'
Then
The
s
can be recursively constructed as follows.
such that for any
3.
=
4){s)
paves the way for Theorem
It also
,
$
set
0(Mi)
for some
>
(Axiomatic Characterization of Stable States) Suppose Assumptions
1
Then:
1.
5 —
:
G $}.
between the equilibrium of the dynamic game
2
s
{0
establishes the existence of stable states
acterization of such states.
Theorem
=
if (f>{s)
=
4>
4>2
m ^,
(Part 1) To prove existence, we
<P\
first
(s)
and 02
(pj
(p
~
£
02
(^ is
^^
That
is,
V^ =
"payoff-unique" in the sense that
{^) /''^
"^^^
s
£
S
construct the sequence of states
<
/i^, ..., /ii^i
>
such that
if 1
<j-
</ <
|>Si,
10
tlienA^i/,.^.
Mr
•'
{^)
The construction
Q C
states
we obtain
same
the
is
by induction. Assumption 2(a) implies that foFany nonempty collection of
Q
S, there exists z e
such that
Now, suppose we have defined
/ij.
S
result to the collection of states
any x e Q, x
for
< k—
for all j
/i,
\ {/Uj,
.
,
.
where
I,
<
/c
to S,
this results
Then, applying
|.S|.
we conclude that there
y^^-.j },
.
Applying
z.
)/-,
exists
/j,f.
satisfying (7) for each k.
The second
For
=
/c
step
1, let
=
(/ij.)
define
which
satisfy
=
(p (/Li^,)
Suppose we have defined
/i^.
Mk
the collection of states
defined,
=
(j){s)
Mk
^/.. If
and are preferred
s
A;
<
|iSj,
we obtain
To complete the
we need
;:
such that z
=
and
1
implies that
(7),
z
/j.^.,
G
=
z
{/xj,
.
(Part 2) This statement
1-3.
Axiom
1,
because
well-defined, so either
(pifj-k) 7^
Mk =
(7).
and
fJ-k-
But the
violates
Axiom
Therefore,
Mk =
>
fc,
by
i^ij
some
Mk
s
G
7^
0,
Mk
empty, then, we
is
Mk)- Proceeding inductively
2(b) to
in (6) satisfies
cp
3,
Mk-
Axioms
suppose that
Then
z
y^^
for
(pij-i-i)
=
>
I
or
Mk
7^
Mk =
1
and
2
0-
imply
M/ for
<p
Axiom
If
/c,
but in
violates either
1
is
violated.
Mk
7^
we have
(pil^k)
£ -^k-
such that
s )^^^
(p{Pk)-
11
there exists
fif.
now
contradicts
(8).
< /j.j, ..., /ji^i >
(6) at
=
fc
Then by
1.
In this case,
I.
(p
does not
Next, again to obtain a
time at k
>
1.
(p{(p{ni^))
this case (pil^k)
Axiom
If
and
Mk
1
Mk
Then
Finahy,
=
in (5)
and (pihk)
7^
(pil-^-k)
in this case
z
"^iH
(p
y^^
^^.
contradicts
f^k
ij.^
Since
(j){^ii.).
"^
or 2 (or both).
and observe that
But
ii^.
0, the contradiction hypothesis implies
4>{l^k) ^A^t Mfc
>
/
>~^^
does not satisfy Axioms
(p
yielding a contradiction.
(7),
cp (fi/;)
given sequence
if,
for
that
(in
/^/c
is
satisfied.
is
(6) for the first
=
(fik),
(p
not given by
/t;
state
This
combined with condition
fif.,
last condition, z y^^^
is
some
for
1-3.
=
In the latter case,
not given by
(pij^k)
then
Mii so
3 (to see this, take any z £
when
there exists
/
has already been
cp
A4k
Define
\S\.
(8).
and therefore Axiom 3
^
(pil^ij)
Then, Axioms
^i for
where k <
such that
not given by (6) for some k, then
This contradiction implies that
=
1
If
/i;..
Mk
G
z
equivalent to the following:
(p {jj./,) is
0, we must have that
(p{fJ.f-)
=
mapping
z ;^^^ (Pil^-k)-
exist,
^^^
/.t;
contradiction, suppose that
—
/c
Mk
&
Suppose, to obtain a contradiction, that (p{^k)
satisfy
that
is
(7), (p{fJ.k) is
the contradiction hypothesis
is
s
within
/if.
an element of
(p (/i/.) is
^J-k-i]-
,
This means that such z does not
with the property
<
again by induction.
is
In particular, by construction, either <p{Hk)
2.
and
(p(z),
.
.
any
z for
To check Axiom
<p{^k) by (5).
J-^,^
to
(p {fij^)
to verify that
case these axioms trivially hold), or
(p{(p{iij.))
—^ S. This
as in (6).
straightforward for Axioms
and
S
:
for all j
ii^
we can apply Assumption
(p
proof,
i^^
cp
the subset of states where
is
nonempty, then take
is
(such state z exists because
<
This
as in (5).
s
all 2
mapping
to construct a candidate
is
If
—
and
Mk ^
l^k^
then
<p{z)
not be given by
violates
Axiom
=
(p
z).
(6) if
3 (since
5
y^f.
(Mfc), s )^^^
(f>
and
fii^,
(p
=
(s)
s).
This shows that any mapping
one of Axioms 1-3. and completes the proof of part
violates
(Part 3) Suppose, to obtain a contradiction, that V^^
that
6
/Jfc
e P^j <^
fj.j
that 02
and 02
and 01
"=
(Mfc)
<
=
<
I
k.
<
'^i
^,
^^* either
assume that
(Part 4) Suppose Assumption
/U;
3 for
Ti
if
02 (s)
01 (s)
=
(s).
(s) >-s s,
02
01 (s) such that 0i (s)
(s) is
y-
Axioms
suppose that the former
^s
V,
<Pi
(s)
^.s
a 0i-stable state by
s,
1-3, that
=
Vij,^)
<
1
:
<
|iS|
^
T^<j>^
A;
Mfc
G P^^
fij
satisfies
some
there exists
(s) >~s
Then
=
Axiom
(s).
02
fij.
G P^,
=
(/.ifc)
p.f^),
and
02
=
for y
state s such
s if
and only
then imphes
1
or 02 (s)
(s)
02
)-5
0i
(s).
02 (s) there exists
0i (s) (the latter equahty holds
and by part 3 of
2,
we can apply Axioni
sufficient condition for its uniqueness.
=
and
^f^
jj.^
<=> /u
(because 0i
such
implies
(6)
we obtain
02,
implies that 0j (s)
(s) 7^ s 7^
(0i (s))
(6)
establishes the desired result.
is,
the case.
is
and 02
Axiom
shows that a mapping that
1
^;.
>-^^ fi^, m, )-^^ 0i (/x^)
3 to 02
This contradiction completes the proof.
Theorem
a
mapping
we must have that 0^
02-stable state). This implies that
4'2 i^) 7^
2 to
and Assumption 3 implies that either 0i
loss of generality
because 0i
and
and
3/c
^ P^^- Then
and
0j
by
not given
^ P^^ or
^;,
P<pj
Part 3 of this Theorem (that P^^
since 0j {s) oo 02 (s),
5;
>"5 s,
Without
02
Then
P^^.^.
G
mapping
is
3 holds. Suppose, to obtain a contradiction, that 0j
are two non-equivalent mappings that satisfy
that 0j (s)
^
G P^^. Since, by hypothesis,
jii
G P^,. Therefore,
Axiom
1
that
2.
G P^^ and
fij.
^/.
Applying Axioms
but this violates
i-Li,
•?
latter implies that
we have
k,
(^;)
some
The
M(-
^°^ ^^1
loss of generality,
for
("(
=
(M/)
aU j
for
Without
P,/,^-
e P02
fj.j
(f)
Theorem,
this
it
is
also a
and derive the conclusion that
'
Axioms 1-3
and provides
necessarily exists
Even when the uniqueness
condition.
Assumption
3,
does not hold, we know that axiomatically stable states coincide for any two mappings 0i and
02 that satisfy Axioms 1-3.
Theorem
satisfy
also provides a simple recursive characterization of the set of
1
Axioms
1-3.
Intuitively,
Assumption 2(a) ensures that there
such that there does not exist another
as
k
<
=
fii, ..., /2|5|
2,
is
a state
fij.
s y^_^
/Ltj.
Taking
fi^
is
//;,
s >-^^ /i^).
(in
fij.)
When
the set
Mk
as base,
G S,
fi-^
we order the
states
set of states A4fc
(that
fj.^.
state
is,
C
S,
states s such
empty, then there exists no stable state that
is
by members of a winning
we have
coalition. In this case,
nonempty, there exists such a stable state and thus
its
some
exists
and then recursively construct the
and
s
In addition to
it
satisfied
with
=
Alfc
because
is
iS
that includes stable states that are preferred to state
preferred to
When
so that (7)
G
\S\,
...,
that (p{s)
>
s
mappings $ that
(/x^.)
—
s for
(/i;.)
=
fij..
some such
recursive (and thus easy-to-construct) nature, this characterization
is
s.
useful
highlights the fundamental property of stable states emphasized in the Introduction:
is
made
stable precisely by the absence of winning coalitions in
to another stable state
(i.e.,
by the fact that Ai^
12
=
0)-
We
/x^.
favoring a transition
will see that this insight plays
an
important
role in the applications in Section 6.
Part 3 of Theorem
shows that the
1
chosen from $. For different 0's
same
to the
may
initial state
set of stable states T>
$ (when $
in
V
are uniquely determined by preferences
of winning coalitions.^ Finally, part 4 shows that
resulting from an initial state
then
and
S]
might
S2
the same payoff to
We
all
when Assumption
must be equivalent. In other words,
differ in
(j)
not a singleton), the stable state corresponding
but the ranges of these mappings are the same.
differ,
ranges and the set of stable states
is
does not depend on the specific
Tliese
and the structure
3 holds, any stable states
=
if 5]
(/ij
(sq)
and
=
S2
<p
(so),
terms of the structure of winning coalitions, but they must give
individuals.
have motivated the analysis leading up to Theorem
with the argument that,
1
when
agents are sufficiently forward-looking, only axiomatically stable states should be observed (at
least in the "long run",
i.e.,
for
t
> T
for
some
finite T).
The
analysis of the
dynamic
gan:ie in
the next section substantiates this interpretation.
Noncooperative Foundations of Dynamically Stable States
4
We now
Section 2 and characterize the
the
MPE
of this
This game
state;
game capturing dynamic
describe the extensive-form
and
(2)
MPE
of this game.
game and the axiomatic
characterization in
(1) a protocol for
specifies:
The main
interactions in the environment of
result
Theorem
is
agenda-setting using a sequence of mappings,
each
s
1.
sequential
We
(the exact sequence in which votes are cast will not matter).
for
the equivalence between
a sequence of agenda-setters and proposals in each
a protocol for voting over proposals. Voting
Ks be a natural number
is
{'iTslsg^,
£ 5. Then,
tt^ is
and
refer to
defined as a
and
is
described below
represent the protocol for
it
simply as a protocol. Let
mapping
7r,:{l,...,K,}^IU5
each state
for
s
G 5. Thus each
tt^
specifies a finite
sequence of elements from TUiS, where Ks
is
the length of sequence for state s and determines the sequence of agenda-setters and proposals.
In particular,
if tTj (/c)
e I, then
the set of states S. Alternatively,
specified proposal over
it
denotes an agenda-setter
if tTj (fc)
which individuals
£ S, then
vote.
it
who
will
make
directly corresponds to
Therefore, the extensive-form
a proposal from
an exogenously-
game
is
general
relate the set T) to two concepts from cooperative game theory, von Neumannand Chwe's largest consistent set. Even though these concepts generally differ, we
show that under Assumptions 1 and 2, both sets coincide with V. This is an intriguing result, though it does
'
In
Appendix C, we
Morgenstern's stable
set
not obviate the need for our axiomatic characterization in this section or the noncooperative analysis in the next
section, since our
main focus
is
on the mappings (p, which determine the axiomatically or dynamically stable
Von Neumann-Morgenstern's stable set and Chwe's largest consistent set
states as a function of the initial state.
are silent
on
this relationship.
13
enough to include both proposals
exogenous proposals.
Assumption
We make
for
a change to a new state initiated by agenda-setters or
the following assumption on (TTslgg^:
(Protocols) For every state
4
G S, one (or both) of the following two conditions
s
is satisfied:
(a)
TTs
For any
=
(k)
=
[k)
S
\ {s}
£
I
there
,
is
an element k
i
there is
an element k
the "status quo"
proposal.
s) as
proposals or
ensures that either
It
it
of sequence Wg such that
<
\
:
< Kg
k
of sequence
=
state sq ^
0,
S
\s
allows
tTj
contains
all
possible agenda-setters to eventually
all
is
accepted
all
1.
Period
2.
For k
Pk,t
t
=
=
t
>
1),
1,
.
,
.
.
If Pk^t 7^ st-i,
might be determined as part
it
the timing of events
is
some probability
as follows:
Kst-i
i
the
If ""st-i (k)
then there
proposal P^^t
/cth
is
is
£ I, then player
determined as follows.
iTs^_^ (k)
chooses
P^^t
If /T5,_j (/c)
here).
it
set of players
a sequential voting between Pi;t and st~\ (we will show that
is, if
Ykt ^ VVs,_j),
convention that
4.
If P/;
t
is
Each player votes yes
who voted
we do not
if
yes. If Y^t e
it is
(for
or
Pfc,(.)
Wsi-n then
rejected. If
=
P^t
no
(for S(_i).
alternative P^^t
Sf_i, there
is
is
Let
Yk^t
denote the
accepted, otherwise
no voting and we adopt the
in this case Pk^t is rejected.
accepted, then
rejected or
G S, then
G S.
the sequence in which voting takes place has no effect on the equilibrium and
(that
a
begins with state St_i inherited from the previous period.
"'st-i (^)-
specify
make
earlier).
taken as given (as noted above,
distribution). Subsequently (for
than
players will have a
of the description of the environment or determined by Nature according to
3.
such that
Tig
possible states (other
alternatives will be considered or
all
chance to propose (unless a proposal
t
< Kg
k
i.
This assumption implies that either sequence
At
<
1
:
z.
For any player
(b)
TTs
state z E
there
is
we
transition to state st
no voting because
to step 2 with k increased by
1; if
fc
=
P^^t
=
Ks^-_^,
=
st-i
Pfc,t,
and the period ends.
If Pi^^t is
and k < Ks^_^, then the game moves
the next state
is St
=
st-i,
and the period
ends.
5.
In the end of the period, each player receives instantaneous utility u^
Payoffs in this dynamic
game
(t).
'
are given by (4), with
,
^ =
u,(i)
^
|^'i^*)
1
14
!^^'
=
"
^*-^
liSty^St-l
(9)
.,
..,
^
.
'
each
for
G J. In other words,
i
zero payoff.
In
period in which a transition occurs, each individual receives
other periods, each individual
all
the current state
in the
The period
Sj.
of zero payoff can be interpreted as representing a "transaction
cost" associated with the change in the state
piue-strategy
MPE.
game
Since the
one-period "transaction cost" has
a dynamically stable state
Examples
3
and
4 in
A MPE
is
and
introduced to guarantee the existence of a
is
little effect
on discounted
Appendix C demonstrate that
may
fail
if
may
to exist or
the transaction cost
S
G
s
be
to
j3
large,
once (and
is
removed from
in
Appendix C. In what
made within
We
a formal definition. For completeness, such a definition
follows,
we
will
use the terms
an
if there exist
T<
G S, a
initial state s
is
provided
and equilibrium interchangeably.
repeated thereafter.
in the
dynamically stable state
differently, s°° is a
Our
objective
is
We
=
St
'^
dynamically stable state
MPE
strategy profile a,
and
> T.
t
reached by some
is
if it
a
finite
time
T and
them
as a function of the initial
between dynamically and axiomatically stable
also estabUsh the equivalence
We
introduce a slightly stronger version of
first
2(b).
Assumption 2(b)* For any
and
is
5°° for all
to characterize
states characterized in the previous section.
Assumption
S
(£
to determine whether dynamically stable states exist
dynamic game described above and
state So E S.
s°°
set of protocols {'^s}seS'
oo such that along the equilibrium path we have
Put
Sj o^ si for I
<
j
<
sequence of states s,si,
I
<
Moreover,
if
for x,y, s e
.
.
.
,Sk in
S
with
=>
si
^s
Sj
yg
s
for
I
<
j
<
k
k,
Sj+i '^s Sj for
it
MPE
is
MPE
next define dynamically stable states.
Definition 2 (Dynamically Stable States) State
is
(9),
Here payoff-relevant states are different
described above, since the order in which proposals have been
we do not provide
if)
(SPE) where
perfect equihbrium
a given period are also payoff relevant for the continuation game. Since the notion of
familiar,
this
induce cycles along the equilibrium path.
strategies are only functions of "payoff-relevant states."
from the states
will take
payoffs. In particular,
subgame
defined in the standard fashion as a
is
and we
infinitely-repeated
reached, individuals will receive w^ (s) at each date thereafter.^
s is
a (pure-strategy) equilibrium
receives the payoff w^ (st) as a function of
i
S we
all 1
<
-
<
j
s
and y ^^
have x yg
k
1
s,
then y
Sfc.
)/-s
^
Various different alternative game forms also lead to the same results. We chose to present this one because
appears to be the simplest one to describe and encompasses the most natural protocols for agenda setting and
voting.
In particular,
have a move), which
is
it
allows votes to take place over
all
a desirable feature, since otherwise
15
possible proposals (or
some
transitions
all
possible agenda-setters to
would be ruled out by the game form.
Assumption
In addition to cycles of the form y^-^a;, 2>-s2/,a:>-sZ (which are ruled out by
assumption rules out cycles of the form y hs
2(b)), this
y,
and
y ^5
s,
z are payoff-equivalent.
then y
The main
Theorem
1,
)/-s
It
also
result of the
paper
is
hs
^ hs
y,
~,
unless the states
imposes the technical requirement that when x x^
Both requirements of
x.
x, ^
Then
2(a,b) and 4 hold.
summarized
there exists
j3q
G
and
s
assumption are relatively mild.
this
in the following theorem.
(Characterization of Dynamically Stable States) Suppose
2
a;,
(0, 1)
such
Assumptions
we have
the following
all
j3
>
"^^"^
'^
pure-strategy
for
if
that
(3q,
results.
1.
For any ^ £ $ there
game such
and
that
=
St
exists a set of protocols {'n's}ssS
0(so) for any
stays in this state thereafter.
t
>
1; that is, the
=
Therefore, s
game
MPE
a of the
reaches 0(so) after one period
a dynamically stable state.
(j){so) is
Moreover, suppose that Assumption 2(h)* holds. Then:
2.
For any
set of protocols {tTs}^^^ there exists a pure-strategy
the property that for
4>
E
such that s°°
(^
stable.
3.
If,
.•:
';;.
'
.
,,
any
=
..;
;c'---''-,
Assumption 3
in addition,
=
t
>
has
Moreover, there exists
1.
dynamically stable states are axiomatically
all
',';:''
.-.,',„.
s°° for all
MPE a
such
•-
v
.^
-
'
\:'\'^'-^.
.
MPE is essentially unique in the sense that
pure-strategy MPE a induces St '^
(sq) for all
> 1,
holds, then the
for any set of protocols {t^sJ^^S'
where
iS, Sj
Therefore,
4>{so).
,
G
initial state sq
MPE. Any
'^'"2^
t
4>
G $.
•
,
Proof. See Appendix A.
Parts
1
and 2 of Theorem 2 state that the
set of
dynamically stable states and the set of
P defined by axiomatic characterization in Theorem 1 coincide; any mapping £ $
Axioms 1-3) is the outcome of a pure-strategy MPE and any such MPE implements
stable states
(satisfying
(/>
the outcome of some
£ $. This theorem therefore establishes the equivalence of axiomatic
and dynamic characterizations. An important imphcation
is
that the recursive characterization
of axiomatically stable states in (6) applies exactly to dynamically stable states.
The equivalence
of the results of
Theorems
1
and
2
is
intuitive.
Had
the players been short-
sighted (impatient), they would care mostly about the payoffs in the next state or the next few
states that
would
arise along the
equihbrium path
duced next). However, when players are
care
more about payoffs
(as in the concept of
myopic
sufficiently patient, in particular,
in the ultimate state
stability intro-
when
/?
than the payoffs along the transitional
sequently, winning coalitions are not wilhng to
move
16
to a state that
is
>
[Hq,
states.
they
Con-
not (axiomatically) stable
Theorem
according to
1
between the concepts of axiomatically and
this leads to the equivalence
;
dynamically stable states.
To highhght some
between dynamically stable states and states that may
arise
we next introduce a number
and
of corollaries of
Definition 3 (Myopic Stability)
s
G
iS
with
Myopic
in the
s^ € S
state
1
is
stability
would apply
if
made
individuals
is
not true. This
We start with
myopically stable
a simple definition.
there does not exist
if
(axiomatically and dynamically) stable,
is
stable not by the absence of a powerful group preferring change, but by the absence of
preferred by a powerful group. This corollary
is
implication of Theorems
2, in
Corollary
1
1.
s
The
set of
S
State s°° G
G S with
any
and
1
and axiomatically)
a (dynamically
and any
s )-5~ s°°,
myopicaUy
is
particular, of equation (5). Its proof
satisfying
4>
stable states is a subset
Axioms
ofD
s°° is not necessarily
final part of
stable (recall
still
myopicaUy
be stable because
a winning coahtion in 5°°
may
In particular, there
1).
s
^
(if
Another direct implication of
(^{s)
we had
^
s™
an immediate
omitted.
stable state only if for
^(s).
and dynamically
a stable state, but a stable state
is
stable.
the corollary imphes that s°°
Example
1-3, s
is
is
(the set of axiomatically
stable states). In particular, a myopically stable state
may
individuals are shortsiglited,
stated in the next corollary, which emphasizes that a state
is
an alternative stable state that
The
2.
emphasize the difference
choices only considering their implications
next period. Clearly, a myopicaUy stable state
made
2.
A
Theorems
when
to
)^5m s™.
s
but the converse
is
and
of the implications of our analysis so far
may be
stable even
exist a state
such that
.s
and leads to some other state
s'
=
(p
this corollary
^s"^ s°°
(s)
,
if it
s',
which
is
not myopically
s >-soo s°°;
is
but
s°°
not preferred by
then s°^ would not be a stable state).
that forward-looking behavior enlarges the set of
is
stable states.
For the next corollary, we
first
introduce an additional definition.
Definition 4 (Inefficiency) State
there exists a state
State s €
that
S
is
s'
G
S
s
such that w,
(strictly)
€
S
(s')
(strictly)
is
>
Wi
(s)
for
all
Pareto inefficient
i
if
Ws ^
and
G T.
winning coalition inefficient
if
there exists state
s'
G S such
s' '^s s.
Clearly,
if
a state
s is
Pareto
inefficient,
it is
winning coalition
17
inefficient,
but not vice versa.
Corollary 2
A
1.
£ S can
stable state s°°
winning coalition inefficient and Pareto inef-
be
ficient.
Whenever
2.
The
Proof.
s°° is not
first
part again follows from
follows from the fact that
s
myopically stable,
winning coalition
it is
Example
1
The second part
in the Introduction.
s°° is not myopically stable,
if
inefficient.
then there must exist
G
5
such that
«S
^5~ s~.
Ordered States
5
Theorems
1
and 2 provide a complete characterization
states as a function of the initial state sq S
While the former
more
is
5
of axiomatically
and dynamically stable
provided that Assumptions
1
and 2 are
a very natural assumption and easy to check, Assumption 2
difficult to verify. In this section,
we show that when
the set of states
satisfied.
may be somewhat
S admits
a (linear)
order according to which individual (stage) paj^offs satisfy single-crossing or single-peakedness
properties (and the set of winning coalitions
Assumption 2
tions).
main theorems
in a
number
In a
wide variety of circumstances.
we can
which
is
take, without loss of
also without loss of
of individuals,
restrictions
either x
we introduce
and
is
Wj
or X
>
We
z
>
if
(y)
if,
> Wj
for any
y implies Wi {y)
.
,
•
.
When
such an order
to be a subset of R. Similarly, let
set of states
I c
and the
M,
set
certain well-known restrictions on preferences.^ All of the following
T C
<
Wi
i
E
and are thus easy to
M,
for any i,j £
{x)
and Wj
Definition 6 (Single-Peakedness) GivenI
single-peaked (SP)
.
has a natural order, so that any two states
S
any generality,
definitions refer to stage payoffs
(x) implies
.,
any generality. Given these orders on the
crossing condition (SC) holds
> Wi
some natural additional condi-
"greater than" or "less than" y).
Definition 5 (Single-Crossing) Given
Wi {y)
•
S
of applications, the set of states
(e.g.,
satisfies
This result enables more straightforward appUcation of our
is satisfied.
X and y can be ranked
exists,
{W^j^g^
T
{y)
C
5 C
R,
<S
x,y E
S
such that
[x) implies Wi {y)
C
.^
,
R, and {^i {s)} ^^j ^^^
I and
< Wj
verify.
<
i
<
the
,
j
.
single-
and x
<
y,
Wi {x).
R, and {wi (s)}jgjs£5, preferences are
there exists state x such that for any
y,zES,y<z<x
[z).
next introduce a generalization of the notion of the "median voter" to more general
political institutions (e.g., those involving supermajority rules within the society or a club).
See Barbera and Moreno (2008) for the connection between these notions.
18
Definition 7 (Quasi-Median Voter) Given
a
quasi-median voter
some
a, 6
e
R
we have
Denote the
set of
i
X
Ws
G
C
vS
IR,
and {Ws}s^S'
such that
X=
x < y there
last definition
exist
is
i
^'^^
monotonic median voter property
e Mx, j S
Assumption
we
(Weak Genericity)
"^^^
5
My
such that
<
i
it is
nonenapty
if
M, the sets
for each x,y ^
S
j.
members
satisfied
if
Preferences
no player
{wiz
(s)},gi sg^
and
The main
not rule out such indifferences).
winning
the set of
^
coali-
y.
between any two states (though
indifferent
is
or those that are
impose the following weak genericity assumption.
also
such that for any x,y,z G S, x y^ y implies x y^ y or x
is
j
G
general enough to encompass majority and supermajority voting as well
part of a limited franchise). Finally,
tions {yVsjgcs
<
a
:
Median Voter Property) Given T C K and S C
as these voting rules that apply for a subset of players (such as club
5
&T
[j
I is
<by for
i
1 is satisfied).
of winning coalitions {yVs}seS
Assumption
P^o.yer
quasi-median voters in state s by M^. Theorem 3 shows that
Definition 8 (Monotonic
The
K,
G X.
(provided that Assumption
satisfying
any
(in state s) if for
I C
does
it
results of this section are presented in the following
theorem.
Theorem
3
(Characterization with Ordered States) For any I c K, 5 C R, preferences
{wi (s)}jgj5g5, and winning coalition {WaJsPS satisfying Assumption
1.
If single-crossing condition
2(a, b) is satisfied
2.
3.
and monotonic median voter property
and thus Theorem
If preferences are single-peaked
X (lY j^
and thus Theorem 2
1
is
Assumption 5
we have
that:
hold, then
Assumption
applies.
and for any x,y € S and any
0, then Assumption 2(a,b)
If in either part 1 or 2,
1,
satisfied
X
£ Wx,
and thus Theorem
also holds, then
1
Y
G
Wy
we have
applies.
Assumption 2(b)*
is also satisfi.ed
applies.
Proof. See Appendix A.
The
conditions of
Theorem
3 are relatively straightforward to verify
and can be applied
in
a
wide variety of applications and examples. Notice that part 2 of Theorem 3 requires a stronger
condition than the monotonic median voter property.
imphes the monotonic median voter property, so part
under the hypothesis of part
2.
However, the converse
In particular, consider the following environment:
It
1
is
can be verified that this condition
of the
not
theorem continues to be true
true.^*^
there are two states, x
19
<
y,
and two
voters,
i
<
j.
G
A|)|)lic;il,i<)iis
VVc iKiw
ill
we
Nevi'l'l lu'lcss,
Hli|Milalci| in
t,li()S(!
iiiuy
Voiinn
id
ii;.
III
n
1,1)
K.oIhtLs
(I!)!)!)),
Holiei'Ls
(
l!)l)!))
I
mil
I'lxa.inplc
suppose
iinpiise:;
(A:
1
I
/'2 if
)
is
A;
Inlluvvini',
lie
/
•
society eonsisl.s of
liol.li
I'iiiuililii
odd
or
Kl;al.(!K,
Wj
i,
Uie ronn
ol'
i,s
respecl
s( i'iU||,litroi'wa,rd
and
Uial,
A:/'2
if
I
|
A'
iiK'i'easinji diU'crenci's
/
'I'lici
'riii'oicni.s
in
>rciii
(.ji.-ui
and
I
^-
A:
and
Me
jiropeity to
>
lie
I,
same
,
N
]
k] for
\
-
m,
.
iMillowinc;
<
k
<
N.
(lOj
(,s/,)
l.lie
,s;,.
one needs more
a,grccinoill of indivii
'i'liese l,wo votiii)',
liiaj
rules lead
I.
and Me(liaii
VoUii|', I'liiiiililirinni
>
v/i,
\'ic(i
of
sel,
and
his
two
let
,
>
-=>•((;,, (.s^.)
weaken Uoberts's
lis lirst
lei
; idj (s/)
(.s'^.)
are spec.iaJ cases
votini', rules
in pa.i ticiila.i
us
vij
>
assume
(.s^.)
w,
,
l.lmt
a.iul
(11)
.
(.s;)
Uoberts's two
In additinii,
versa.).
sta.lile chilis.
In particular,
"'lis,,) >-w.j{si)
(hut iml
.
,
Markov
siiii;,li' crossini:,,
ni; (si)
i,
,,
('learly, (Id) implies (II)
,
.
esl al ilislies llie exisl.eiiee nl' inixeil sl.iale(',y cciuilil nia, wil.ll
Ihey Imlh lead In
/
,
eoiidil.ion:
vWj (.s/)
even are needed),
is
rules allowed in oin riainework.
•strid
voliii)'.
I
|l,
.sj;
si/e) or ineilian vol,er rule (wliere
to verily that UnliertsV; nindel
R<''i''i'''il
>
(Oj (.na;)
(.s()
{
innjorit.y vdMiii'; williiii a clnl) (wliere in cliili
i\'ely.
of the
for all
apply,
iloi'S iiul,
diHeicnccs
inerea,sin|',
sl.iif'l,
>
a,nd j
A;
rnli's:
A'/'J
and shows
mil, ions
It
inin,
.'(
Ici
arc iiKirc rcsUictivc
.'i
indivi duals, so 7
A^^
corri'sponiliii)', e(|iiililii iiini iml inns, wliieli Uolierl.s calls
V'olei
'i1ii-(iicin
in
wlirn 'riicDicin
'I'liii;;,
llicic a,ic A^
Ilial
I,
a,ll
The
'-.^
vol.<\s for a cliaii|',e in cinli
l\/'2,
inn:,
f:()risi(l('ii'<l
simply api^'al
ciui
a,))| )lir(l
dirccljy.
considers Lwo volanc,
ii,iid
2.
n jim lil
Ijir
we
cnviii inniciil.s
he
ca.ii
ciivironincnUs
nl pnlilicaj ('coiioiiiy
l.lic;;!'
2
a,ii(|
I
'liil>s
(
loi
I,
I
and
I
rlii'oiciii.s
rcsiill.s |)ic)vi<li'il in
iniinlici
a
in inaiiy nl'
l-lial,
'I'lii'dii'ini;
ill
ion
i/,ji,l,
will aJHo src llial
Ix' applii'(
si, ill
(i.l
l-i'l.
show
Wi'
arid
mm Im Imc
.sil.iiaiiniiH,
lilci aiiiic.
.'J.
linw iJic dial
all'
I
variety of
a,
llii'
2
illiisl
.
votiiiji,
rules
,
can he
i'ei)resenled hy the Inllowini', sets of wiimiii|!, coa.litioiis:
WUr
W
I'rcriM'ciu'HM
l)rol'(>rtMU'(\s
i
and
,/
urc
liowovoi'i
iirii
siicli
wiUi two
Ilinl.
|.V
f
\
ic, (.r)
.sl.al.os).
|.\'
< uu
'J(a) iw violal.i'd lor
f-
(.
t
C'
c
:
bill,
(;/),
SiiiipiiHo, in
(iiia.si-ini'diaii voU>i-k in .st.ato.s
AssuinpUon
{'^
-
;
(': |.Vl'l.^A.|
(A-
|-;i)/2 t
{k/2,k/2-\- 1]
Wj {x) >
uij (y).
addition, that W,„
:i:
and
;(/,
---
I'ospoctivoly,
{x,y} (wo liavo
20
•;/
y.^,
>
and
A:/2),
.VI
',l'li<;so
{{»'}
it A-
X]
t.
,
if
A:
is
odd;
is (>veii.
prel'orniic.os
{i,j}],
and
Wy
and tluia nionolonic
x and x >-,; y).
uro .siuglo-poiiUed (so are any
-
{{j}
,
{»',,?'}}.
In Uiat, cnsd,
iiicdiun voter pri)|>cily holds.
Clearly, hoUi
VV,'""'
<
and
>
vol:or properl.y in Di'linilion
can he
by
gnai'anl.ecHJ
he
aJscj
characterizes the stincl. ore of
ii),
/
(.s)
siilliiieiil).
Uohinls's model
l.o
i—i WftliHly A.ssnin|)l imi
Then,
For example, we
a
eoilld allow
'I'lieorein
elnb
in
be
.h/,.
and any
'rh<H)reni
In
I,
G
,s'
.v,
.'!
pnre
a,
his
c.a.sc,
S
(lliou;
linn
Uiis
a,
'Ji
previous
Ironi llie
MI'M and
sl.raie)',y
,
,
more
applies witli fionsidtirably
.'i
degret' of silpermajoiil.y rule in
dillereiil-
lie iiKiiiorniiic iiie<
lioMs.
T)
exisl;enee of
l.lii^
I,
"
set of wiiininp, coalitions nests various majority
degree of supermajority
J.
iJial,
;
can also be verified Ihai
It
(
i
is e,le.a,i'
it
(wiahlishes
a.n<l
for a.riy
(,s')
7/);
aJile cliilis.
si
as well as
I
also aSHiiiiir iJial Assiinipl.idn
lis
IjcI,
iS.
}
a.ssuniinj', l.liai
wnakt^r eondil.ion would
seclion applies
{y^l'l
and
siipeiiiiajoi ity
<
where k/2
/;,;
<
//c
A:
ea.eli eliili.
lollowiijc;
'i'hi'
each
for
iiil<:s:
ajid
votin/^ nikjs.
('ciieral
define the
s(!t
let
/;;,
thr;
wiiiiiini',
a,
coalitions as:
{XeC:\Xns,\>n
Wil =
Then, a
relatively straiglitforwa,rd a,|)pliea,tion of 'riieorein
completeness,
(for
Proposition
or Wl'l,
Wl'l"'',
(i)
'Jlic.
(it)
(ill)
//(
(ui.d
I'>r "I I
>•
mc.dum voters
only odd-Hizc.d
voter rules Ass'iimptuyii
.'I
olso
as a fimctiov. of
(i:ii.d,
oj MI'I',
a,
sharp
stable clubs can be obtajned easily by
—
n, ...,/:,.,,,/,;
the approach
in
|-
n}
Assu'iiipiioii
tlir
'I'lirii
Fi.
vn,
Assv.rn'pldons fl(a,h)
'Ilic.oTcms
I
(i/ii.d
foi'
a fixed
H.oljerts's pafx^r is
also establishes the existence of
(did))
The Structure
In this subsection,
we
c^f
'is
and
(vp'jilicH.
iii.i:(hii.'ii.
v:n,i(/'aj'ly
a|))}lyin(r
Thc'orem
rif
'.'>
to Itoberts's oii)MiiaJ
sta.l,ed in
ehibs
n (and dideient
consideraJdy more
mix(!(l-stj'ate|.',y
of dynajiiics of clubs and the
MPIO
<:a,n
I'ropositioii
I,
also be taj'.en to
value's of k).
It is
i;;
\ti:
tcj
se(,
of
model or
to
aJlow for a
ol the Irjiin of
also noteworthy tha,t
Koberts
dillieiilt a,nd restiictive (th()U(di
for a,ny
//).
Therefore, this application
illustrates the usefulness of the jfeneral characterizati<jn results fMcsented in this
G.2
!'i
w. Lhc cose of iii.ajoTtly ot
dy'ii.d/nncidly st(d}le state
Another' generalization, not
Dl
I'iUi.cr VV,™'''',
in sdJ/isJicd.
HldicH
l.li.cv.
clia,rac,teri/,a,t,ioii
richer set of clubs, l-br example, the fea,sible set
{k
8
Dcjin/i.l/urn.
a/n.d :il.(iblc
tli/ns
by
f/iv<"ii.
.follomviuj ivsulls liold.
avf (dloinrd,
(i/ii.d
cixdil.iov.s
imtttd stide (ebdi).
This proposition shows thai
various geiierali/atioiis.
in.
proposition
followiiij',
(J).
wrinmu/
iinUi
'/"'
I-,
(llj
ciiiJis
li.olds
th.e
Appendi;-;
'i>roj)("rl,y
Ui.r cli,(i:r(i,c.L("riz(iJ.io'ii
tj
in
chiha model,
i/ii.
ji'i-c.fn-nif.cs sal/isfy
iJi.aJ.
MoTV.n'UCT,
d,eA.e:nu.iv,eA
provided
k <
k/2 <
vilicrv.
iJi.v.s
voLmfj
//le
rnoTU)l,o'm.i-
Suppose
hold
2b''
1
th(! ]))oof is
the
estaJilish(!s
.'i
pa.piii'.
Elite CIuIjs
briefly discuss anothei examjile of dyna,mic club lorma,tion,
a simple explicit characterization.
Sujjjjose there ai-e A' individuals
21
1,2, ...,A/
which alKjws
and
A^ slates
Si, S2,
•
1
SN, where
Sfc
=
{1, 2,
Wk
.
.
/c}.
,
.
= Wk
(Sno)
(s„i)
< Wk
(s„3)
< Wk (SnJ
These preferences imply that each player k wants to be part of the
in the club, he prefers to
between two clubs he
of the club
is
be
outcome mapping.
player
2,
so
(f>{s2)
1
=
who
S2-
a player
"elite") one. In addition,
on being
indifferent
is
(13)
and
1
2 is
and thus by Axiom
is
needed
is
for a change.
and only
if
j
=
But
=
S2.
1,
who
(l>(si)
and
is
sj.
preferred
Proceeding inductively, we
2" for n G Z+, and the unique mapping
,....''.,
=
is
S2 is the best state for
different: state S2 is stable
the only such state), so 4>is3)
is
and the unique
we must have
1
and
2(a,b), 2b*,
1,
This club only includes player
stable.
is
In state S3, the situation
Sj is stable if
Axioms 1-3
';
•
i
v
'
that
(f)
•
'
than or equal to
'
/•
.
'
/
'-
.
'
'
Assumptions
2.
If,
instead of (12), for uq
'>JJk
(sns)) then single- crossing condition is satisfied
1,
2(a,b), 2b*,
<
,,
,
and
(13), the following results hold.
1.
and 3
<
ni
hold.
k
-'
•
'
< n2 <
n^,
we have Wk
(Sno)
<
''-,•
(sm) <
''^k
<
i^k (Sns)
(and monotonic median voter property
always satisfied in this example).
Club Sk
"This
.'
;•
the elite club example considered above with preferences given by (12)
winning coalitions given by
is
e R.
i-
following proposition summarizes the above discussion.
Proposition 2 In
3.
club, but conditional
to characterize the set of stable states
[xj denotes the greatest integer less
The
set of
and 2
likes this state best,
and 2 (and
can show that club
where
1
a consensus of players
to S3 by both
satisfies
n^,
(12)
= {XeC:\Xnsk\>k/2}.
First, notice that state si
thus the dictator, and
S2,
<
n2
•
straightforward to verify that this environment satisfies Assumptions
In state
<
j
members, so that winning coalitions are given by
Hence, we can use Theorems
3.^^
<
ni
Suppose that decisions are made by a simple majority rule
of.
Ws,
It is
(more
in a smaller
not part
=
Preferences are such that for any no
is
is
stable if
and only
if
k
=
2^ for n ^Jj'^
'•
.
.
'
•
;
formally siiown in Appendix C. Alternatively, one could consider a slight variation where a player
who does not belong
to either of
any two clubs prefers the larger of the two. In
applied. In particular, with this variation, the single-crossing condition
is
this case,
satisfied
(if to,
and j > i, then i ^ x and thus, j ^ x, and Wj {sy) > Wj (si); conversely, Wj {sy) < Wj
i € Sy, and therefore lu, (sy) < tUi (sx)).
The monotonic median voter condition holds
quasi-median voter
in state Sj to
be
[(j
-I-
1)
/2| £ Ms^; this sequence
22
is
Theorem
3 can also
{sy)
> wt
{s^)
means
(si) for y
be
> x
j e Sy, thus
as well (one can choose
weakly increasing
in j).
The unique mapping
4-
Axioms 1-3
that satisfies
ip
Proof. See Appendix C.
given by (14).
is
^
'
.
Stable Voting Rules cind Constitutions
6.3
Another interesting model that can be analyzed using Theorem 3
(2004) model of self-stable constitutions.
In addition, our analysis shows
Motivated by Barbera and Jackson's model,
The
society takes the form of
J=
a "constitution" represented by a pair
utility
from being
{1,
.
(a, b),
.
,
where a and
=
Wr{{a,b)]
'
a,
6 are integers
(so b
may
= {X
w, (a).
G C
:
|X|
directly corresponds to
between
so that each player
1
and
The
A^.
receives utility
i
(15)
.
In contrast, the set of winning coalitions needed to change the state
W(„.6)
farsighted
somewhat more general frame-
and each state now
A''}
determined by
in state (a, b) is fully
.
us introduce a
let
.
how more
Barbera and Jackson's model.
decision- makers can be easily incorporated into
work.
Barbera and Jackson's
is
>
is
determined by
6
G Z^:
(16)
6}
be interpreted as the degree of supermajority).
In Barbera and Jackson's model, individuals chffer according to the probability with which
they will support a proposal for a specific reform away from the status
determines the (super)majority necessary
the other hand,
is
implementing the reform.
The parameter a
The parameter
Vj\ [{a,b)].
Expected
a.
Ranking individuals according to the probability with which they
satisfy (15)
as follows:
Wi
write
(a),
will
(strict) single-
and are (weakly) single-peaked.
For our analysis here,
we
on
utility is calculated before these preferences are realized
support the reform, Barbera and Jackson show that individual preferences satisfy
crossing
6,
the (super)majority necessary (before individual preferences are realized) for
changing the voting rule
and defines
for
cjuo.
and
(16).
if (a, 6)
(a, 6)
<
and
(a', 6');
and thus w^
us to apply
It
let
us consider any situation in which preferences and winning coalitions
turns out to be convenient to reorder
(a', 6')
satisfy a
a',
then
(a, 6) is
the ordering of states with the
[(a, 6)], satisfies
Theorem
<
all
pairs (a, 6)
located on the
same a
is
on the
left
real line
of {a',b'),
and
unimportant. Suppose that
the single-crossing condition in Definition
5.
This enables
3 to any problem that can be cast in these terms, including the original
Barbera and Jackson model.
Let us next follow Barbera and Jackson in distinguishing between two cases. In the case of
constitutions, any combination (a, 6)
is
allowed, while in the case of voting rules, only the subset
23
of states where a
to
assume
there
is
b
=
6
considered (then a
is
> N/2. Barbera and
Jackson
no alternative voting rule
that {a',b')
is
{a',b')
=
a voting rule or a constitution
call
with
a'
myopic
it
1,
may be
is
natural
(a', 6'))
is
such
equivalent
when we allow
not surprising that
enlarged.'^
following proposition shows summarizes this discussion.
Proposition 3 Consider
1.
Assumptions
2.
There
exist
1,
2(a,b)
mappings
Assumption 5
and 2b* are
(f)^
holds.
Then:
satisfied.
=
for the case of voting rules (a
Axioms
tutions that satisfy
environment and assume that preferences satisfy
the above- described
single-crossing condition and
1-3.
.
b)
..'•
and
for the case of consti-
(p^
'
•
-.
,
In the case of voting rules, the set of self-stable voting rules (in the sense of Barbera and
Jackson) coincides with the set of myopically stable
(a, 6),
where a
=
b,
satisfies (j)y\{a,b)]
=
states.
The
(a, 6).
subset of set of dynamically stable staies. -
4.
constitution
(or, respectively,
b'
Given Corollary
stability.
players to be farsighted, the set of stable states
3.
=
it is
(a, b) self-stable if
preferred to (a, 5) by at least b players. Clearly, this stability notion
to our notion of
The
the voting rule); in both cases
6 is
In particular, any such state
set of self-stable voting rules is a
•'
'
.
.
In the case of constitutions, the set of self-stable constitutions (in the sense of Barbera
and Jackson),
and
the set of myopically stable states
the set of dynamically stable states
coincide.
Proof. See Appendix C.
.
'!::'
-
,!..
\
'.
:,.:,.;: i^,:^
:i
Cocdition Formation in Nondemocracies
6.4
As mentioned above. Theorems
1
and
states does not admit a (linear) order.
game
of the
'"Ill
•;
of
dynamic
2 can
be directly applied in situations where the set of
We now
coalition formation in
particular, the set of stable states
is
enlarged
illustrate
one such example using a modification
Acemoglu, Egorov, and Sonin (2008).
in 'the
case of voting rules, but remains the
case of constitutions. This can be seen as follows: suppose constitution {a'.b')
without loss of generality, we may assume that (a, 6)
which Pareto dominates (a',fc') and thus is preferred to (a,
players;
is
is
preferred to
Pareto-efficient (otherwise
(a, b)
same
by at
in
the
least b
we could pick {a",b")
by these b players). Then these b players also
prefer constitution [a N) to (a, 6), since the payoffs are the same. But constitution (a N) is stable by A.xiom
1. Moreover, it is impossible that all A^ players prefer some other constitution (a", 6") because (a',b'), and thus
{a',N), are Pareto efficient. Hence, if state (a, 6) is not myopically stable, it is also not dynamically stable, for
the players may move to constitution {a',N), which is dynamically stable and preferred to (a,b) by at least
,
6)
,
N
players.
24
Suppose that each state determines the ruhng coahtion
S
states
Members
coincides with the set of coahtions C.
composition of the ruhng coahtion in the next period.
aDowed, which highhghts that the
Each agent
t
I
e
is
assigned a positive
is
of the ruhng coahtion determine the
A
C
transition to any coahtion in
is
does not admit a complete order (one could
set of states
define a partial order over states, though this
and thus the set of
in a society
not particular useful
number
7,,
for the analysis here).-'''
which we interpret as
"political influence"
or "political power." For any coalition A' G C, let
jex
Suppose that payoffs are given by
for
any
G
z
X and
a€
take any
X
any
[1/2, 1) as
E C
=
S}'^
The
restriction to (17) here
is
just for simplicity.
Also,
a measure of the extent of supermajority requirement. Define the set of
winning coahtions as
Clearly, this corresponds to weighted ck-majority voting
tion
X
(with a
=
among members
of the incimibent coali-
1/2 corresponding to simple majority). In addition, suppose that the following
simple genericity assumption holds:
7a-
The
following proposition can
Proposition 4 Consider
1.
Assumptions
1.
= 7r
now be
only
if
X
=
y.
established.
the environment in Acemoghi, Egorov,
2(a,b),
(19)
•
3 are satisfied, so that Theorem
1
and Sonin (2008). Then:
applies
and characterizes
the
axiomatically stable states.
''In Acemoglu, Egorov and Sonin (2008), not all transitions were allowed. In particular, there we focused
on a game of "eliminations" from ruling coalitions in nondemocracies, so that once a particular individual was
eliminated, he could no longer be part of future ruling coalitions (either because he is "killed," permanently exiled,
or is permanently excluded from politics by other means). Moreover, we assumed that payoffs were realized at
the end of the game. Appendix B discusses how the current framework can be generalized so that there are limits
on the feasible set of transitions.
'''This
a special case of the payoff structure in Acemoglu, Egorov and Sonin (2008), where we allowed for any
lu, (X) > u), {Y)\ (2) if i S
is
payoff function satisfying the following three properties: (1) if i e A' and i f Y, then
and i e y then ui, (AT) > Wi {¥) if and only if 7,/7x > 1,/1y'i ^'^'^ (3) i ^
and
X
,
The form
in (17)
is
adopted to simplify the discussion
here.
25
X
i
^ Y, then w, (X)
=
w, (Y).
2.
Moreover, there
an
exists
2(b)* also holds. In
such that Assumption
arbitrarily small perturbation of payoffs
this case,
Theorem 2
also applies
and characterizes the dynamically
stable states.
Proof. See Appendix C.
In
Appendix B we introduce the
how Proposition
possibility of restrictions
on
and sliow
feasible transitions
4 can be generalized to cover the case of political eliminations considered in
We
Acemoglu, Egorov, and Sonin (2008).
also illustrate
how not
allowing previously-eliminated
players to be part of the ruling coalition affects the results and the structure of stable coalitions.
Coalition Formation in
6.5
We
Democracy
next briefly discuss how similar issues arise in the context of coalition formation in democ-
racies, for
example, in coalition formation in
Suppose that there are three parties
sufficient to
form a government.
in turn has
more
ble states.
{{1, 2}
,
in legislative bargaining.
in the parliament,
Suppose that party
seats than party
The
3.
1
and any two of them would be
1, 2, 3,
has more seats than party
initial state is
0, and
,
{2, 3}
,
{1, 2,
3}} for
and a party with a greater share
all 5. First,
suppose that
all
< w^
({1, 2))
other payoffs are defined similarly. In this case,
<
government would be too strong. The equilibrium
minimum winning
coalition.
However, as emphasized
that governments that have a greater
number
<
t(;2
higher payoff even though
is sufficiently
powerful.
coalition, that
continuation
is,
game
''See, for example.
4>{0)
({1,2,3})
it
1,
influential in the coali-
is
({1,3})
0(0) =
< w^
in this
({2, 3});
{2,3}: indeed,
because party
I's
influence
example then coincides
in the Introduction (recall footnote 2), the
minimum winning
u;2({l,3})
W0 = W^ =
.
formation does not necessarily lead to
<
which
coalitions are possi-
we have
< ^3
verified that
neither party 2 nor party 3 wishes to form a coalition with party
in the coalition
more
1^3 ({1, 2, 3})
can be
it
2,
governments are equally strong
of seats in the parliament will be
tion government. Consequently, UI3 (0)
u;2(0)
all
Since any two parties are sufficient to form a government,
{1, 3}
with the
^^
<
coalitions.
To
dynamics of coalition
illustrate this,
suppose
of seats in the parliament are stronger, so that
u;2({2,3})
<
u;2({l,2}).
That
is,
party 2 receives a
a junior partner in the coahtion {1,2}, because this coaUtion
We might
= {1,2}.
then expect that {1,2}
may
indeed arise as the equilibrium
Nevertheless, whether this will be the case depends on the
after coalition {1,2} is formed.
Suppose,
for
example, that after the coalition
Baron and Ferejohn (1986), Austen-Smith and Banks (1988), Baron (1991), Jackson and
Moselle (2002), and Norman (2002) for models of legislative bargaining. The recent paper by Diermeier and Fong
(2008) that studies legislative bargaining as a dynamic game without commitment also raises a range of issues
related to our general framework here.
26
{1,2} forms, party
by
we have
C
:
2 e
starts ruling
=
^{2}
{C'
shows that
number
greater
by
then party
1
itself.
to denote such a feasible transition,
{2,3}, that
is,
i-^
its
the coalition {2,3} emerges as the
This example therefore reiterates,
Theorem
1
in
the fact that {2}
is
=
formally, YVa
= {X & C E
and
:
players except
S
X] Suppose
e
E
.
that there exists y €
such that
S
ip
{x) ==
such that Wi
state y than in absolutist monarchy,
could be arbitrarily large.
d
{y)
a.
It is
state.
not dynamically
democ-
s.
A'^
that for
Example
all
a;
G
5\
{a},
E
holds power.
of
for all
x G
5
\ {a}.
will eventually
> w,
(a),
More
we have that I\ {£} e W^,
In other words, starting with any
end up with democracy. Suppose also
meaning that
In fact, the gap
interactions in the
game described
dynamic
in Section 4.
all
individuals are better off in
between the payoffs
in state y
and those
most straightforward manner, consider the
It is
a, since
collective decision-making
the heart of these inefficiencies.
y
^
d,
1
We
individuals and a set of finite states S.
then clear that for P sufficiently large,
these will lead to state d and thus
(a)
=
If
and emphasizes that commitment problems are
the society could collectively
then these inefficiencies could be partially avoided.
possible, since once state y
is
reached,
E
And
commit
to stay in
yet such a
commitment
can no longer block the transition to
27
some
d.
E
a.
This example illustrates the potential (and potentially large) inefficiencies that can arise
games
is,
then straightforward to verify that Assumptions 1-3 are
not accept any reforms away from
will
2
game.
To understand economic
extensive- form
party
together form a winning coalition. Moreover, there exists a state,
regime other that absolutist monarchy, we
satisfied in this
=
in Corollary 1) that the instability of states
a corresponding to absolutist monarchy, where individual
"democracy," d ^
in a
is
the context of coalition formation in
Consider a society consisting of
start with sq
all
€ K]).
paper immediately
provide a more detailed example capturing the main trade-offs emphasized in
in the Introduction.
is,
1
and Lack of Reform
Inefficient Inertia
that
:
dynamicaUy stable
that can be reached from a state s contributes to the stability of state
We now
G C
greater seat share, that
{1}. In this case, the analysis in this
{2,3} dynamically stable in this case
racies, the insight (discussed after
6.6
= [^
also reasonable to suppose that once
can regain power by virtue of
1
e Cj and thus {2}
=
(f){0)
it is
>
itself,
;
and rule
from the coalition {2,3}, party 2 can also do the same, so that W{2,3}
{2,3} i— {2}. However,
e C
of seats, can sideline party 2
"i—"
(which naturally presumes that W{i_2}
{1}
X] and
What makes
stable
i—»
{1,2}
Similarly, starting
{X e
its
Let us introduced the shorthand symbol
itself.
so that
by virtue of
1,
in
at
state
is
not
6.7
Middle Class and Democratization
Let us next consider a variation of the environment discussed in the previous subsection. Suppose
again that the
there
is
=
initial state is sq
a,
where
Wa = {X
E C
:
E
£ X}. To start with, suppose that
with hmited redistribution, and d2, democracy with extensive redistribution.
GC:PeX}.
Wdi = yVd2= {X
WE
so that
P
is
(a)
< WE
and
{dl)
established, the poor can
(a)
< wp
Given the
< wp
(dl)
{d2)
Wdi =
fact that
,
W(i2
=
{P}^ once
implement extensive redistribution. Anticipating
consider an additional social group,
the middle class
sufficiently
is
this,
numerous
M,
representing the middle class, and suppose that
so that
Wdi^Wd2 = {{M,P},{E,M,P}].
--•
The middle
wp
democratization.
will resist
Now
< WE
(d2)
Suppose that
Suppose
prefers "extensive" redistribution.
democracy
E
democracy
only one other agent, P, representing the poor, and two other states, dl,
class
is
opposed to extensive redistribution, so
also
'"'
WM{a) <WM{d2)
''
'
'
<WM {di)
'
,,
'
'
,,
,
,"'..
'
'
.
/
'
This implies that once state dl emerges, there no longer exists a winning coalition to force
extensive redistribution.
Now
anticipating this,
the franchise). Thus, this example illustrates
such as the middle
class,
E
how
can have a moderating
will
be happy to establish democracy (extend
the presence of an additional powerful player,
effect
on
political conflict
and enable institutional
reform that might otherwise be impossible (see Acemoglu and Robinson, 2006a,
which the middle
6.8
class
may have
Concessions in Civil
wars
also
in a civil
initiate
be used to illustrate how similar issues arise
war with a rebel group, R. The
peace and transition to state
shorthand
"i—>"
civil
introduced in subsection
£ C
:
R£
X}, and naturahy,
6.5,
Wc = {C
£ C
i--»
:
£
r,
where
wr (r) > wr (p).
If
wq
C}
.
engaged
However, using the
r denotes a state in
domestic
(r)
is
The government can
c.
G
sufficiently influential in
wars. Tliis
context of international
in the
denoted by
is
we now have p
28
civil
Suppose that a government, G,
war state
so that
p,
which the rebel group becomes strong and
Wp — {X
in
War
Fearon, 1996, 2004, Powell, 1998).
(see, e.g.,
examples
played such a role in the process of democratization).
Let us briefly consider an application of the ideas in this paper to the analysis of
example can
for
< wq
politics.
(c),
Moreover,
there will be no
peace and
why
(f>{c)
=
c despite the fact that
war may continue
civil
As an
in this case
we may
is
also have
wq
(p)
> wg
i->
R can first disarm partially,
where d denotes the state of partial disarmament.
d,
for
similar to that for inefficient inertia discussed above.
interesting modification, suppose next that the rebel group
in particular, c
The reasoning
(c)-
Moreover, d
i—>
d-p,
where the state dp involves peace with the rebels that have partially disarmed. Suppose that
=
y^dp
{{C, i?}}, meaning that once they have partially disarmed, the rebels can no longer
become dominant
(p
=
(c)
domestic
in
politics.
W)
make a concession changes
Therefore, the ability of the rebel group to
dp.
> wg
In this case, provided that vug (dp)
we have
the set of
dynamically stable states. This example therefore shows how the role of concessions can also be
introduced into this framework in a natural way.
Taxation and Public
6.9
many apphcations
In
Good
Provision
preferences are defined over economic allocations, which are themselves
determined endogenously as a function of political
Here we
in such environments.
good
results can also
be applied
by providing an example of taxation and public
illustrate this
provision.
-
Suppose there are
{1,2,.
.
We
k}.
,
individuals
who
N
individuals
1,2, ...,A''
states
A''
si, S2,
.
.
,
.
where s^
s;v,
{XeC:\Xnsk\>k/2}.
i
is
given by
m,(5j)=E[(l-r,J^+G,J,
where Ai
is
individual
individuals are
i's
productivity (we assume A^
more productive),
E
vote, the tax rate
is
is
Sj.
When
determined by the median.
median voters
>
(20)
for
A-j
i
<
j,
so that lower-ranked
denotes the expectations operator, and Tj
determined when the voting franchises
of two
=
are enfranchised, so that winning coalitions take the form
assume that the payoff of individual
also
and
assume that decisions on transitions are made by an absolute majority rule of
Ws, =
We
Our main
rules.
is
the tax rate
an odd number of individuals are allowed to
When
there
is
an even number of voters, each
gets to set the tax rate with equal probability.
The
expectations in (20)
included because of the uncertainty of the identity of the median voter in this case. Finally,
^sj
=
h
{
5Z;^2 Tsj Ai
j
is
the public good provided through taxation, where h
is
an increasing
concave function.
For the single-crossing property, we require that
Wj
(si+i)
> Wj
{si)
=> Wi (s;+i)
>
Wi
(S;)
and Wi
29
for
any
(s; + i)
i
<
< Wi
j
€
2 and
{si) =4>
Wj
for
any
{Sl + i)
<
s/,
sj+i £
lUj (s/)
<S,
Denoting the equilibrium taxes
sufficient (but not necessary) to
E
(1
-
r,,^
J
Aj
and
in states s;
ensure
- E (1 -
T,,)
E
Aj >
<jt
single-crossing.
by
and
Ts^,^^
t^,
,
the following condition
is
this:
and
since the equihbrium levels of public goods, Gg,
is sufficient for
s;+i
>
«i+i
-
(1
t,,^
G^,^,
J
^-E
(1
-
r,,) ^i,
cancel out from both sides. Therefore,
,
Er,,
(21)
Notes that individual
when determining the tax
i,
rate in
si,
would maximize
(l-r)A, + ^(t
This implies that individual
i
would choose
Ai
~
h
I
Ti
\
the concavity of h
Si+i.
Then, with probability 1/2, the tax
the same in s/+i as in
tax rate
is
s;),
follows that for
i
Ar,
such that
y
^-^m=l
From
it
Tj
V'
<
Am
/
/
^-^m = l
j, Ti
is set
>
s;).
productive individual (then the
More
[taxes] are
have voting rights (club members).
is
we can apply Theorem 3
interestingly, these results
made
to
can
available differentially to
.
.
Conclusion
7
A
who
less
Therefore, (21) holds and
extended to situations where public goods
[imposed on] those
consider a switch from s; to
by the same individual (then the tax rate
characterize the dynamically stable states in this society.
also be
Now
Tj.
and with probability 1/2, by a
greater in Sj+i than in
Am-
central feature of collective decision-making in
many
social situations, such as societies choos-
ing their constitutions or political institutions, or political coalitions, international unions, or
private clubs choosing their membership,
is
that the rules that govern the regulations and proce-
dures procedures for future decision-making, and inclusion and exclusion of members are
made
by the current members and under the current regulations. This feature implies that dynamic
collective decisions
must recognize the implications of current decisions on future
example, current constitutional change must recognize how the new constitution
way
for further
choices.
will
changes in laws and regulations and how these further changes might
For
open the
affect the
long-run payoffs of different players.
We
developed a general framework
for
a systematic study of this class of problems.
We
provided both an axiomatic and a noncooperative characterization of stable states and showed
30
that the set of (dynamically) stable states can be
acterization highhghts that a particular state
state that
makes a winning
s is
coalition (in s) better
stable states need not be Pareto efficient; there
to all individuals, but
We
also
is
itself
affect future
stable
off.
may
if
An
This recursive cliar-
recursively.
there does not exist another stable
implication of this reasoning
is
that
exist a state that provides higher payoffs
not stable.
showed that our framework
have been used
computed
is
general enough to nest various different models that
analyze specific problems in which current collective decisions
in the literature to
These include models of
decision-making procedures.
inefficient inertia (lacl: of
reform) because of fear of changes in the future balance of political power, models of institutional
change and enfranchisement (such
as
Acemoglu and Robinson, 2001, 2006a,
Lizzeri
and Persico,
2004, and Jack and Lagunoff, 2006), models of voting in clubs (such as Roberts, 1999,
and
Barbera, Maschler, and Shalev, 2001), models of the stabihty of constitutions (such as Barbera
in
Jackson, 2004), and models of coalition formation in democracies and nondemocracies.
these cases and in a
number
of others,
literature are special cases of our
we
illustrated
how models
previously studied in the
framework and how our approach highlights the main economic
-
insights in these diverse environments
Although our framework
assumptions.
amoimt
Some
of acyclicity
is
fairly general,
essential).
Others are adopted
often at the cost of further complication.
interesting
our analysis
still relies
on a number of important
of those are necessary for our general approach (for example, a
is
In
Among
for
minimum
convenience and can be relaxed, though
possible extensions,
would be to introduce stochastic elements, so that the
we
believe that
most
set of feasible transitions or
the distribution of powers stochastically vary over time, and to include capital-like state variables
so that
some subcomponents
of the state have
autonomous dynamics. Such extensions would
allow us to incorporate an even larger set of dynamic political games within this framework.
view the analysis of such dynamics as an interesting area for future research.
31
We
A
Appendix
Proof of Theorem 2
7.1
We start
with a
an important
dynamic
lemma about
Theorem
role in the proof of
economy models.
political
MPE
the structure of
and SPE
and
2
in voting
lemma plays
games. This
of independent interest in the contex:t of
is
establishes that there always exists a pure-strategy
It
MPE
(and SPE) in which each individual votes for the outcome that he or she strictly prefers and
outcome that
that in any (mixed- or pure-strategy) equilibrium the
many
players (a "winning coalition") will be implemented.
Lemma
and
N
Consider the following I-player extensive-form game G/v with perfect information
1
stages. In each stage k, one player
The payoff vector
action a^ e {y,n}.
ft
gW
=
=n
a-k, ihe vote a^
outcome y
=n
if
v^^,
<
may
a^)
,
.
.
MPE
y whenever
=
some
for
—
ai^
y for
.
,
.
.
a/v)
=
first
N
:
all i^
fi
= V,
n,a^k)
,
=
y
:
>
G
Then
3^.
whenever a^
h with
if Vi^ [y)
in
=n
probability
>
(n)
v^^,
•,
is
'
large enough, in the sense that
SPE
any
for
and
under the payoff
i
,
(n)}
vi
,
r-.
.
.
vi (y)
make
'
'
Similarly, if the set of players
1.
=
v (a^
denote the payoffs of player
fj (tI)
{i
takes
{y,n}, where y,
>
k,
's)
y; this ensures that action y does not
,
y=
—
{y,n}
:
different k
for any profile of actions other than that at stage
•,
Gjv, the equilibrium payoff vector will be
Suppose that the
if
same for
(and SPE) in Gyv where a^
and
vi (y)
set of players
large enough, so that v (aj,
(Hi)
if
n, respectively).
=
be the
given by the mapping v
is
(i.e.,
y
payoff vector will be y with probability
is
player
the following results hold.
(where
v,^ (h)
Suppose that the
(ii)
.
[y)
and
vectors y
V {ai,
(this
leads to y, then so does a^
Then
less likely).
=
y, a-fc)
There exists a pure-strategy
(i)
ah
if;
are the two possible vectors of payoffs. Suppose that
then we also have v [a^
k,
preferred by sufficiently
is
all
if.
of G/v, the equilibrium
M=
{i
:
Vi {y)
<
G Af, then in any
Vi
(n)}
SPE
of
1.
stages of a finite or infinite extensive-form
game
payments
perfect information satisfy the requirements above, except that instead of
Gj\!'
at
with
terminal
nodes taking two values only, we have that there are two classes of isomorphic subgames, Sy and
Sfi,
with payoff vectors y and
and Af
If V (ai,
=
.
{i
.
.
,
:
Vi
(Sg)
=
a/v)
reached after
N
versely, if V [ai,
game reached
<
,
.
.
after
respectively.
{Sn)}, where
y whenever a^
stages
.
Vi
fi
is
ai\f)
N
=
f,
Take any
(Sy)
y for
all i^
Sy and the expected
=
and
is
32
let
y=
{i
:
v^
(Sy)
>
v,
the equilibrium continuation
players receive
all
Sn and the expected
and
{Sn)}
(Sn) ire continuation payoffs of player
€ y, then
utility
n whenever a^ = n for
stages
v^
MPE a
if.
G
utility
J\f,
m
this
MPE is v (Sy).
i.
game
Con-
then the equilibrium continuation
players receive
m
this
MPE is v (Sn).
Proof
ttk
=
Vif.
(n)),
y
Lemma
of
if v^i^
>
(y)
there
(Part 1)
1
and a^
(n)
Vif^
We
= n
need to show that for the profile of strategies in which
if v^^
no profitable deviation
is
MPE
existence of a pure-strategy
suppose that he plays
=
ai^
he switches to
Similarly, for player ik with
profitable.
f,,.
a'^.
(y)
ik,
=
{y)
Vi^.
ij-
We
Base:
=
0.1
SPE
y
take
<
Vi^ (n),
Suppose that
1.
set 3^
may be
the equilibrium payoff vector
is
probabihty
would then receive
1,
since he
deviation from
game with
Case
=
h.
>
suppose that
[h) for all 2
Vi_^
SPE
any
k stages. Suppose that set 3^
1:
>
Vi^ (y)
in this
game the
<
j
subgame
Vi-^
is
<
Vi-^
ij
better off
is
{y),
n with
y with probability
stage 2
suppose that
2:
all
players for
vector will be
is
y.
By
The two
The
v^^ [y)
whom
1.
J\f is
(n).
large enough.
In this case,
=^ y,
M
is
But
large
of
first
if for
1
v,
<
y with probability
by choosing
We
>
v,
ai
is
y with
an equilibrium.
in
have thus proved the base.
for
alH <
—
fc
player
if
is
SPE
in
in stage
?'i
true:
1;
consider the
1
takes action
players for
if all
Now
1.
if
for
=
y.
<
By
some SPE
i\
whom
induction,
of the entire
could ensure that the
y and would thus have a profitable
must be y with probability
(n) for all 2
j
<
if
same holds
in
the
k choose aj
any subgame starting
this implies that the
enough
which
in
=
he chooses action aj
Then, by assumption,
v^^ (n).
[y)
choosing action
ii
then the payoff vector will be
1.
subgame
=
at stage
then the paj'off
y,
1,
starting at
the payoff vector
for the entire /c-stage
game.
3^ is large
enough.
analogous. This argument completes the proof of part
(Part 3) This immediately foUows from part
game
if
cases together complete the induction step for the case where
case where
y can
k.
positive probability, then player
induction, for any
y with probability
=
1.
which cannot be the case
deviation. Therefore, in this case, the payoff vector
Case
a'f.
not
large enough. Consider two cases.
is
k choose aj
will lead to
payoff vector were
payoff vector
to
from y with a positive probability,
suppose that we have proved the result
to k:
1
n
is
To obtain a contradiction, suppose that in a
y.
then in the subgame starting at stage 2 the following
y,
Vi^ [y)
~
=
a;;
large enough, so that player
different
can similarly consider the case where set
Step from k
aj
be
But then player
case the payoff vector
We
is
sufficient for the payoff vector to
is
and
v^^ (n)
not profitable for such player. Finally,
is
the
then any outcome yields the same payoff and thus this player does
Vi^ (n),
=
>
v,^ {y)
=
would not change the outcome of the
prove this by induction on the number of stages
/c
[y)
(this will establish
such that
not have a profitable deviation, which completes the proof of part
(Part 2)
if v^^
would not change the action of
this
7Z,
n
or
j/
from y to n. In both cases this deviation
it
change the payoff vector from n to y only, which
player
=
this either
the payoff vector) or will change
(i.e.,
either
is
any player at any stage
for
any of the subsequent voters, and therefore
voting
(and a^
Vi^ (n)
and SPE). Consider player
If
y.
<
(y)
2,
since a
MPE induces a SPE
in the
k stages with payoffs given by continuation payoffs of the original game.
33
2.
truncated
Proof of Theorem 2
7.2
Proof of Theorem
any
for
There
is
a
J and
6
i
/\
/
w^
2 (Part 1) First, suppose that
G
,.
.
(-A-1)
iS,
^i<?i
more than
(not
Wi
>
w^ (s) imphes 0^^^
number
finite
N
/
<
(s')
s, s'
=
St
(j)
construct a
We
(s(-i)-
MPE
x
\I\
q
and
(f>{q)
payoff of player
on
q).
—
x {\S\
(s)
< -^-^
all /?
following property:
i
—
(s')
1(;,
-^.
of conditions in (Al).
1))
such that for
(0, 1)
In the equilibrium
G J and
we construct below,
>
each period
for
drop the time indices.
For each
s
E S, take
>
A's
|5|
i
(i)
either k
= Kg
>
1,
^^'^
,
—
Take
1.
tt^
be the continuation
Vt {s,q) will
'
,
.
..•
.,
..
() such that
tt^
Given the focus
is q.
-
=
(A's)
•.
•
/
f
•,
£ X votes
7^
,
if
and only
(we are at the last stage of voting), P^'s
=
4>
and
(*)
s;
Consider the
is satisfied).
proposal P^ (says yes)
for
.
if (f>(s)
(f){s)
strategy profile a* constructed as follows:
i
is
s,q e S, let
(making sure that Assumption 4
(k) arbitrarily
Each player player
It
(Al) holds.
/Jq,
and accepted proposal
as a function of the current state s
tTj
Wj
/?
given by one of the four expressions in (A2) depending on whether
MPE, we
otherwise, take
-
-—1
if.^,ri^V(<^(g)) if^(g)7^9r
Vj (5, q) is
=
z
and
introduce the following notation: for
(This means that
=
,
'
\S\
game with the
of the
/'^'''^-j
s
(s')
\
straightforward to verify that there exists Pq S
We
the following conditions:
satisfies
/3
,
,
if:
V, (s,
cp
(s))
>
Vi (s, s);
(ii)orK(5,Pfc)>K(5,0(5)).
In addition,
The
if tt^ (/c)
e T
strategy profile a*
First,
we
will
if (j)[s)
whom
>
(j)
some
(s))
=
We
Markovian.
If 4>[s)
s.
Vi (s, s) is a
P^
this player chooses proposal
/c,
show that under the strategy
and no transition
Vi {s,
is
for
^
s,
will
show that
profile a*
,
there
then Axiom
winning coahtion
1
in s, that
an
it is
is
arbitrarily.
MPE
in three steps.
a transition to
(p
(s) if 4>{s)
^
s
implies that the set of players for
is,
Xs = {i:wi{(f){s))> Wi{s)] gWsTo
see this, observe that (Al)
and the
fact that
^ >
(5q
imply that
fiwi
{<p
(s))
>
Wi
(s) for all
'
i
G Xs- Therefore,
for all
i
G Xs, we have
•
Viis,cP{s))=Pw,{(P{s))>il-P)w,{s)+p''w,{<P{s))
Next, we can similarly show that there exists no X'^ G
all
i
G
ATj,
that
is,
the set of players for
whom
Ws
V, (5, P/^)
34
>
such that
Vj (s,
cp
=
Vis,s).
Vi (s,
(s))
Pk)
>
V, (s,
(p
i^)) for
does not form a winning
coalition in
and
To obtain a
s.
=
(j){<p{s))
we would have
(p{s),
Then
contradiction, suppose there exists such a X'^.
that
,
^
Pj^
s
,,,,,,
,,,
.
since
-
Pw,{cp{Pk))>V\{s,Pk)>V,{s,<P{s))>pwUi>{s))ioial\ieX'^,
and thus
foralHeX^
wU(l>{Pk))>w,{(l){s))
Then
Wj
the fact that .Y^ £
implies 4>{Pk) >~s 4>{s), which, given that (p{s)
4>{Pk) >-s s by
Assumption 2(b)*. But 4>{Pk) ^s 0(s),
Axiom
yields a contradiction to our hypothesis that X'^ G
and
3
players with Vj
>
P^)
(s,
K
(5,
=
established that under a*, P^^
^s
and 4>(Pk)
s,
Wj.
4>{s) if 4'{s)
accepted and
s is
7^
=
yields
s,
Pk contradicts
Therefore, the set of
does not form a winning coalition in
(s))
(j>
<t>{Pk)
>~s
all
s.
We
have therefore
other proposals are
rejected.
Second, we verify that given a*, continuation payoffs after acceptance of proposal q are given
by (A2).
proposal g
If
accepted, then there
s is
7^
another transition next period in case
there
is
no transition
in addition,
if
in the current period,
=
4>{s)
^
4>{q)
=
q
an immediate transition, and there
no proposal
and each player
then there
s,
If
(q) 7^ q.
is
is
i
is
accepted, so that q
receives stage utility (1
—
=
s,
/3)
is
then
w^
(s);
a transition next period. In either case, the
continuation payoffs are given by (A2).
we show that there are no
Third,
setter this holds because
this follows
from
Lemma
no proposal that an agenda-setter makes
then in the
the proposal
receives
game with two
I',
(s,
Pk)
receive V^
(s,
(p
last
proposal P^
if
(s))
if
=
(s) 7^ s
a best response to
accepted and
is
and no proposal
Vt {s, s) if
Since
we have already
are given by V,
the
first
itself for
{s, q)
any
i
Lemma
For a voter
1(a) then establishes
compares continuation payoff
Vi (s, s) if it is rejected.
(s)
given the continuation payoffs in
s
possible outcomes.
voting stage, each player
accepted and
is
eventually accepted
(7).
accepted.
always a best response for a voter to vote for the option that he (weakly) prefers.
it is
(f>{s) 7^ s,
if
is
the continuation strategies are Markovian, and therefore each
1(a):
voting stage constitutes a finite
that
profitable deviations from a* at any stage. For an agenda-
=
G
will
(s))
all
if it is
be accepted,
(s))
other voting stages, player
rejected (because
in
(p
(s) will
which case each player
<S
i
be
will
Therefore, there are no profitable deviations from a*
This argument establishes that the strategy profile a*
(7).
5
s).
V, (s,
In
V, {s,
If
in the
truncated
game
is
given the continuation payoffs in
established that under a*, the continuation payoffs starting in state
in (A2), a*
is
a
MPE
of the entire game,
which completes the proof of
part of the Theorem.
(Part 2)
We
in part 2 of the
first
prove that a
MPE
exists.
We
Theorem.
35
then show that
it
has the properties stated
Let us
=
If for
fj.-^.
>
/c
Mk —
we have
2
satisfying
(j)
Axioms
0, then
cp (fif.)
=
/j-i.;
Take a sequence of states
1-3.
Then, follow the procedure described
satisfying (7).
j /ixj, ..., /i|_5| >
(j){Hi)
construct a specific niapping
first
Theorem
in
C
otherwise, let Zk
we
First,
1.
Mk
set
be defined
as
The
Zk
set
nonempty by Assumption
is
Zk may be chosen
We
as 0(m/c)- Proceeding inductively, a specific
q
accepted, Vi
is
at a given period, each player
can be viewed as a
induction,
=
TTs{k*)
we can construct
(j){s)
game with terminal
SPE
a
>
Wi (s) (such
deviations from a' and thus a'
and on the stage
requirement
SPE
is
is
it
because
an
exists). Therefore,
SPE
we have
(p
equilibrium path
=
x{s)
any n
q.
>
1,
First,
x"
Suppose there
MPE
note that x
exists
proposal and whether
strategies are Markovian).
to X
(s).
Case
last
Now
(i):
it
all
voting stage
and that he
(s)
By
MPE satisfies.
s \i all
cannot be accepted
in
any
—
require that
I
<
(•?)
(x
etc.).
(*'))
=
s,
but x
the definition of the
two
/3) it's (i)
MPE,
>
i
G X
(j){s) is
k*
.
Given
only depend on proposals
a' to
be Markovian (the only
indifferent; clearly,
such
MPE.
Take any
set of
sequences
{ttj
()}sg5
accepted along the
is
proposals are rejected) and
if
us denote
let
x
it
then
(s) ¥^ ^
by
for
This can be established by contradiction.
Denote by Jj C {!,..., Ks] the
(s) 7^ s-
mapping
is
accepted
/c,
since
x, the first voting stage in J^ leads
cases.
Here each player knows that he
will receive (1
a'
the proposal q that
s,
voting stages in Js lead to cycles.
.
=
vr^ (/c*)
accepted do not depend on the play before current stage
suffices to consider
k'
accepted
k* be such that
where a proposal Pk made along the equilibrium path
it is
s
By backward
(s, q).
let
We
has "no cycles," in the sense that
=X
n such that x"
set of voting stages in state s
(this
»
(where x^
(s) 7^ s
any
=
5 — 5
:
1).
is
is
payoffs, each period
accepted at any stage
established the existence of an
well-defined (let q
is
is
game, we can choose
For any state
u.
Axiom
satisfies
where each player votes no when
We now establish the properties that
and any pure-strategy
no alternative
stage where
first
indeed a SPE. Since actions in
ct',
the current state
straightforward to verify that there are no profitable
is
of this finite truncated
to choose
if
payoffs given by Vj
k* be the
let
,
the continuation paj'offs in (A2),
is
truncated game as follows:
a' of this
exists,
i
if
i
of
obtained.
(f)
Given these continuation
(s, s)).
such k* exists; otherwise,
if
tOj (<^(s))
receives Vj
i
(truncated)
finite
given by (A2) (in particular,
(s, q), is
proposed and accepted at stage k* and that no proposal
an
mapping
construct an equilibrium in which continuation payoff of player
and proposal
where
and according to the procedure, any element
2(b)*,
if
PkJ
is
Suppose
this
is
the case and consider the
will receive zero utility if Pk' is accepted,
accepted.
Then Lemma
thus yielding the desired contradiction.
36
1(c) implies that Pk'
Case
not
(ii):
voting stages in Jj lead to cycles. In this case, denote the voting stages
all
C
that do not lead to cycles by J^
first
voting stage in
in
any
MPE,
This
define
!/)
=
result in turn implies that
Pfc',
= min{nG NU{0}
m (s)
<
Evidently,
s).
(s)
x"
= X
~
<
<
any state
5,
\S\
—
:x"(5)
and
1,
=
involves showing that (1)
is
m (s) =
equivalent to showing that \ (s)
Then we prove
we prove
that
1/'
Proof that
To
ip{Pki)for
that
satisfies
if
Axiom
proposals
establish
s for
(A3)
if
and only
tliis,
for
have, again by
Lemma
since Wi
{ip
fact that
/3
>
<
Wi
=
=
x{s)
.
is
1(c),
(?/'
MPE
s.
(A4)
(A5)
in state s alternative q is
The
a) follows.
and then
(so) for all
Axioms
(2) a.'(s)
>
t
1).
We
Then we prove
1-2.
are proposed
Pk,
s
=
implemented,
proof
rest of the
'0 (s)
(this
second
start with
an intermediate
—
that x (s)
Finally,
4' (•s)-
accepted, so
P/,.,
and accepted in
state
s,
then
tp (P^.
")
~
consider again the set of voting stages J such that
J =
accepted. Let
Since each
only players
(Pt,))
(s)
Finally, let us also define
ip).
if
i
1-3,
for 1
tl'
<
J
{/ci,
—
{Pk-^)
/
<
.
.
.
,
fciji},
where
(this implies
j^
i'is)
|J| is
x
<
(s) 7^ s
and m{Pk-^)
accepted
kj
=
ki for
<
j
I
and m{s) >
m..{s)
—
in this equilibrium,
1
(we
1).
(which
we must
that
V^(Pfc,)^. 0(Pfc,+J for
(in particular,
i/'
the dynamically stable state reached with zero
is
=x
ct, st
each state
P^. is
to the
convenience), and suppose that
ip {s) 7^ s).
if
(This order makes the proof simpler).
3.
and
Pf;
In equilibrium, proposal P^j
implies that
MPE
(s) satisfies
il'
each k e J, the proposal
drop index
mapping
Axioms
(s) satisfies
V'
or one transition, so that in the
result.
Then,
1.
^(5)},
i'{x{s))=i^{s)
a) gives the continuation paj'off of player
and subsequently equilibrium play (according
statement
—
|5j
= {('-^^"''(^^ ;[:;j}+/3'"(')-V(^(.)).
K(.,.)
•
(s,
=
x{^P{s))
as follows from the definition of
.
Clearly, V^
=
ij{i^{s))
'
n >
for all
1 (•s)
Moreover,
for
each
leads to zero utility to
'
and
x''^'"'' (s),
=
\'° (s)
fc',
leads to a positive payoff. Therefore, proposal Pk/ cannot be accepted
it
?^^(s)
(with
at
again yielding a contradiction and establishing the "no cycle" result.
''no cycle"
(s)
made
Accepting the proposal
J^.
player, while rejecting
Consider the last voting stage k in Jg that precedes the
Jj.
who weakly
(Pfc,^J)
is
prefer
V'(P/,-,
sufficient to
/3o).
37
</<
1
)
imply
to
|J|-1
4' {-^ki^i)
Vj (s, P^,)
<
could vote
for
VJ (s, Pfc,+i) in
acceptance,
view of the
we
In addition,
also
have
This can be seen as follows.
p
(1
V
Wi
'"i-'i/
(ip
while
(P^.j.)],
accepted, each player
is
if
it
(v
<
(Pfc,,,))
P^'^'^+'w^
accepted, the set of players for
[i'
whom
(PfcJ),
Wiiipi
and hence
Pk,j,
~
{Pkj)
for all
1
tp iPi-.j.]
4>{Pki) for
<
I
<
J|
will receive Vi (s,Pic.j.
player
receive
will
ui,
<
V, (5, P,,,,)
(0
(Pfej ))
=
]
=
Vi(s,s)
then we would have
t7,
must be
Since
(s, s).
P,,,^,
sufficiently small,
we must have
This estabhshes that
i^
<
)
i
< w^i^iPkJ),
[Pk^j^))
)
and, more precisely,
each
rejected,
is
- p)w^ {s)+P"'^'^+^w,{i>iPkJ). Now iiw, (V
l3H''^wl)+'uj,
is
If P^.
—
all
1.
1
<
j
Indeed,
<
if
Now,
'4'{Pki)-
>:s
<
Z
since
were not the
this
we prove
In addition,
\J\.
Assumption 2(b)*
holds,
<
that m{Pk,)
we
ha,ve
''^{Pki^-^)
each player would receive a strictly
case,
I
higher payoff
Proof that
Pk^
if
ip
was rejected
satisfies
1.
If Pfc.j. is rejected,
V-(5,s)
=
have established, however, that
+1. Since
Ki (s, s) for
P/c
for
possible. If
P/,.
.
is
accepted, each player receives
(1-/3)«;,(s)+/?-(^)+1iz;,(t^(5)).
u;,-
f
V-'
(
Pku, ))
accepted in equilibrium, by
is
a winning coalition of players in
s,
=
Wi{il} (s))
Lemma
that
,
and that
.
/3™(^)«;, (V' (s))
>
m (Pk^) +
that V,
,.
(
(1/'
(s))
>
(1
-
/3) tu,
(s)
'
+ /3"'(^)+iu;, (i^ (5))
'
,
{i>
(s))
>
w,
38
(s)
for all
i
£ X^,
s, P^.
•
.
,
;9"(^'|'l) + 'u;^
/S'^f'^ty,
—
.
'
•
,
,
alH e Xs, we must have
This implies that
m (s)
we must have
1(c)
is,
-'
\-
Xs = {iei:K(s,Pfc|,|) > v;(s,s)}e w..
-;
Then
is
/ciji,
each player receives
-:
Pk.j:
so P^, could not be accepted in the equilibrium.
/c/,
Consider the set J introduced above and consider stage
the last stage where acceptance
i.e.,
We
Axiom
at stage
,,
1
<
j
>
P >
which, in view of the fact that
if
some
for
m{s) >
and
1
=
e Xs, Wi{ip{s))
i
{s)
ip
imphes that w,
{iel
> w,
{ip (s))
we would have
Wi{s), then
> w^
Consequently, Wi (0 (s))
s.
"/=
Pq,
(s) for all
<
(3'^^^'w^{ip{s))
(s) for all
£ Xg. However,
i
Wi{s), because
G Xs, which imphes that
i
eW,,
.UH{i>{s))>w^(s)}
.
thus establishing that
0
and therefore Axiom
Proof that
any
s for
^s
G S with
s
holds.
1
satisfies
ip
(s)
ip (s) 7^ s,
'
,-
Axiom
This
2.
is
',
.
:
,
straightforward in view of the fact that
(s))
ip [ip
=
'
i>{s).
.
_
Proof that x
=
(s)
place in one step,
i.e.,
Let us prove that
4' (?)•
that
0
Pk
for
=
(s)
(s) (or, equivalently, in
X
then transition to state
(s) 7^ s,
if
(A3) in
=
(s)
1
-i'
(*)
whenever x
takes
{s) 7^ s).
Consider two cases.
Case
is
=
some
j
:
<
1
Case
and therefore
0,
=
0' (s)
(ii):
=
in (s)
happens on
if
for
some
or off equilibrium path), then
with the following argument: take any player
(s))
Vi (s,
=
/3wi {ip (s)), while
if it is
=
V- (s, Pk,)
for
some
/
if
/c
<
V, (5, s)
/c
>
/c|j|.
/ciji
=
In the
(s))
>
0 (s)
is
(1
-
(3)
w,
(s)
first case, all
^ J we have P^
/c
become agenda-setter
(s))
utility V, (s. Pi;,) for
for
therefore, m{x{s'))
2
=
If
i.
>
=
1
and
(s) (regardless of
^0
(s) 7^
whether
=
P/,.
accepted, this player will receive
(5) is
rejected, he will receive
/?™(^'=')+^u;, (0. (s))
>
Wj
/?"^(^''i)+i«;,
+
(0
<
/3\;, (V (s))
(s) will
have
/3ui,
if
(0
<
(1
Pfc
(s))
-
/3) «;, (s)
+ /^^u',
(0' (s))
accepted, while in the second case,
> w,
(s) since
/?
>
,/3q,
and therefore
such players form a winning coalition, we conclude that Pt
some
I
(s))
players prefer to have
k.
k
<
this k
^
J,
k\ji
and
k.
V^
it
By Assumption
But then take any player
agenda-setter at stage
some
/,
Axiom
since
should be accepted. This can be established
P/^
never considered on equiUbrium path.
is
in
=
This implies that m{Pk^)
j.
be accepted. Since we know that
suppose that he
weakly increasing
)
•
Vj (s, s). Since
will necessarily
m [Pk
and
each player with w^ (0
Vj (s,
is
does not hold for any
P^
In this case,
|J|.
1.
First observe that in this case,
(s).
this
if
<
j
proven to hold. But we proved that m{Pk^)
m [Pk-i) =
X
0(5)
(i):
i
=
0' (s)
must be the case that proposal
4, it
must be that each player
such that Wj (0
(s))
>
Wi
(s)
and
This player's equilibrium proposal will give him
(.s,
s) if
39
/c
>
fciji.
However, proposing
(s) will
give
him a
shown above. Therefore, player
strictly higher utility Vi{s,4>{s)), as
where
deviation. This contradiction shows that the case
=
(s)
ip
impossible, and thus finishes the proof that transition to state
that
Ip {s)
z
^
for
and z ^s
some
4'
Case
(i);
Case
(ii):
>
takes place in one step, so
ip (s)
S
ip
i^)
^
such that
''/'(s))-
As
i)
=
(z)
=
any player
/3wi (s) for
we can prove
before,
(which implies
that
if
=
Pk
z
are:
< pwi
Vi (s, Pk,)
=
V, (s, s)
-
(1
{ip
/3)
some
(s)) for
w^
(s)
ii
I
J
+ /S'^^'^+^w,
y^
fact that (by hypothesis) z
In case
and k <
j^
<
{4> (s))
(s)
ip
-
1
(
implies that {i
from (A5), we have
(i),
,_..,
In case
/ciji;
/3) «;«
+ pw, {-p [s])
(s)
if
=
J
because
(ii),
>
/3
(s)
Wi
[ip
and therefore
Vi (s, c)
>
=
form a winning coalition in
^'
=
/3q (recall
pw^
{z)
(Al)),
{z)
~
is
ip (s).
agenda-setter for some
Consider player
If
> Pwi
{iP (s))
>
V (S, Pk,).
''•
we have
'
,
in
both cases
Lemma
<
.,
:
and implies that pro-
1(c) applies
mapped
(by ip)to equivalent states. Hence,
Since ip[z)
i
for
k.
no ip (s), it
either each state
The former
whom
>
Wi [z)
clearly cannot
Wi
{ip {s))
if
must be the case that
arid
is
z
z is ever
is
not proposed.
be the
case, so
suppose that he
is
suppose the
latter applies.
the agenda-setter at
some
he makes his equilibrium proposal, he receives either
/
<
1J|,
.:
K {s, 5) =
-
P) w,
is)
+ r^'^+'w,
[iP
40
•
'
.
;
'
(1
.
_
'
'
or
'
.
[s))
<
we
proposed,
proposed or that each player becomes
V{s,Pk,)<(iw,{iP{s)),
1
s).
(s, s).
V',
Assumption 4 now imphes that
/c.
is,
must be accepted. However, we have already shown that any proposals that are
z
proposed and accepted
must have
(that
i,
have therefore established that
posal Pk
Ws
S
(s))}
-
V, (5, z)
.
ip
>
Wi (^)
:
Pw,{z)>{l-P)w,{s)+Pw,{iP{s)),
where
s
while rejecting can lead to one of two possible payoffs.
i,
players that obtain higher stage payoff from z than from
stage
^g
z
z,
/cij|.
The
We
is
Suppose, to obtain a contradiction, that Axiom 3 does not
3.
(which imphes
{s)
These possible payoffs
/c
does not hold for any j
Pk
then proposal Pk must be accepted. In particular, accepting proposal z will lead to
fc,
utility Vi (s, z)
or
Axiom
satisfies
ip
This implies that there exists state s,z G
s),
has a profitable
= X (s).
Proof that
hold.
i
(1
-
/3)
w,
(s)
+
M
{ip is))
,
depending on
<
fc
or k
/c|j|
> kpy
Instead,
= pw^{z)> max {Pwi{iP{s)),
V^ {s, z)
where the inequality follows from Wj
>
{z)
=
he proposes Pk
if
wi
will receive
-P)w^(s) + pw,{ij{s))},
{I
and
{ip (s))
he
z,
>
/3
This implies that player
(Sq.
a profitable deviation, yielding a contradiction. This establishes that
0
Axiom
satisfies
has
i
and
3,
thus completes the proof of part 2 of the Theorem.
(Part 3) This
7.3
The
result
immediately follows from Theorem
step
first
again a key lemma, which
is
median voters
to the properties of
of potential independent interest.
is
in
Lemma
2 Given
{^s] s^S
satisfying
J C
R,
C
<S
Assumption
Ms
ffi,
For each
2.
If the single- crossing property
the set
(SC) in Definition 5 holds, then for any states x,y,
>
{"T's}sg5 of players such that rus £
y, rux
Proof.
all
is
<
>
Wi [y) for
I — oc <
:
j
Let
1)
<
6}
^ Ws-
a be such that
A=
be
b
such
£
{j
I
:
a
<
quasi-median voters must be both in
is
also true.
nected" coahtion
for
X
To obtain a
~
{j
that
such
Intuitively,
the "rightmost" winning coalition.
Z C Ms
Ms
^
all s
G
z
<S,
€ M^, and
i
all
i
G M^.
holds, then there exists a sequence
S and whenever
states
x,y £
<
e
S
satisfy
TUy.
(Part
larly, let
A
(y) for all
ujj
monotonic median voter condition in Definition 8
<
and winning coalition
nonempty.
is
X hz y <^ Wi (x)
X
Ms
s.
payoff functions {wi{s)]-^j^^^,
X >, y <^ Wi [x)
If
political institutions (parallel
the following are true.
1,
1.
s,
more general
lemma
This
majoritarian elections). For this lemma, recall that
denotes the set of quasi-median voters in state
{j 6
Theorem.
2 of this
Proof of Theorem 3
characterizes properties of quasi-median voters under
3.
and part
1
G I
:
j
<
B
B
=
is
oo} S
Ws and
and B, we
1
is
:
{j
e I
also have
<
a
:
Ws
Since A'
is
41
j
b)
the inclusion
Ms C
is
Z C
not in
winning, coalition
j
<
Ws and
f)
B
Z. Next,
Then
j^
0.
Y =
Since
we show that
for
some "con-
A' does not hold.
A'.
Simi-
oo} ^ Ws, so that
Z = A
contradiction, assume the opposite.
x < j < y} e
the case.
I -oo <
implies that
evidently, either the lowest or the highest quasi-median voter
of generality, the latter
e
the "leftmost" winning coalition.
Assumption
A
{j
Then,
Suppose, without loss
{j
G
I — oo <
:
J
<
y]
(where y
X)
the highest player in
is
the highest quasi-median voter
is
contradiction. This proves that
Ms = Z ^
X
neither in
nor
Y
in
E I
{i
Wx
:
> Wy
{i)
and therefore includes
i
Now SC
S Mz-
Part
>
that Wi (x)
implies that
^
proof of the results for the
(Part 3) By part
1
M,
G
i
ii
x <
the opposite; then
vis
S
Mx
m.s
<
minnie_;\/j
Ms
lies
then
y,
all
m, which
proves that vis G
Ms
and there
is
Take a monotonic sequence
for each s
= {X
W's
if
x
ioi
^
>~z
1
<
G C
<
I
:
lUg
—
Without
then ms2
<
and w^^^
(S2)
in si.
(•Si);
,si
.
nonempty
is
each
for
some x <
Mg
We
1
^^'^
m^
and
sj
>-'
^si
Now
exists.
loss of generality,
> Wm,^
rvs^.
But
(si).
- Wm,^
we conclude
<
all
last
statement, assume
nig
^ Mg, then either
elements in Ms. In the
(S2)
that
>
there
x
z,
)^z z,
Suppose that
<
1
<
A;
vis
—
/
was a cycle
Since S3
This impUes that the hnk S2
(ss)
y'g^
s^ ioi
.
1
.
m^
> Wz
<
k
s;-
y-si
<
I
Now
{vriz).
the dictator,
is
—
Sfc+i
and
I
i.e.,
we have that
>-',
such that
,s;
s\
part of any connected
is
then Wx {viz)
si,.
and
1
does not
it
sj
)~s^.
s^
>-(.
s;;
take the shortest cycle for y' (this need not be a cycle
<
0, i.e.,
Wm,
case,
second case, we find that
2(a).
^s^ Sk for
Sfc+i
some x and
if
first
increasing, part 3 follows.
is
Assumption
suppose that S2
S2
The
for this case.
G 5. Let
s
we assumed
in (A6). In the
then we have s^+i
s;,
G Mz-
i
median voter property. This contradiction
start with
Consequently,
z.
>
j
all
(A6)
Since
s.
or greater than
such that
for
if
y.
Wi [y) for
min m.
median voters {"^s}sg5- Recall that
therefore,
m^j and msn
implies Wm^^ (ss)
Wms
1)
,
This establishes
G X}. Denoting the induced relation between states by
therefore, a cycle for
for >-).
.
G W,.
i}
consider an alternative set of winning coalitions where
then x
z,
k
S
s,
of
.
>
such that j
G 5. Since the sequence (A6)
a cycle si,
'winning coalition in
Ms
violating the monotonic
for all s
>
y-
connected,
is
whenever
for player j
and completes the proof
>-, y,
violates the definition of
Proof of Theorem 3 (Part
hold,
Mz
= max
for
elements in
Mx,
to the left of
this coalition
Moreover, vis G Mg. To prove this
m^ = min^gMi
than
is less
m^ < my.
G
the set
rus
Evidently,
But by SC,
z).
Suppose x
analogous.
is
Lemma,
impossible and thus yields a
is
Conversely, suppose that Wi {x)
3i
:
implies x
relation
of this
1
G
{j
that
this implies
treated similarly).
\s
same inequality holds
implies that the
{y) for all
iDj
winning in
(is
But
.
0.
players from Mj.
all
Lemma
of the
1
Wz
G
(i)}
which
,
(Part 2) Consider the ca^e x > y (the case x < y
Then
Z c Y
winning, and therefore
is
y'^^ S2
S3
and
Wm,^
>
'Wm,
may be
is
and
tUs^
(ss)
the lowest state (so S2
S2
<
sj,
we have Wm,^
(52).
But then
si
(S3)
and
> Wm^^
(S3)
-
Combining
this
with Wm,^
S3 y'
si,
since lUs^
<
S2
Wm,^
vis^, hence,
> Wm.j
(si).
y-'g^
<
Wm.,^ (S2)
is
S3;
(§2)
>
(S2)
>
the dictator
skipped in the cycle, which contradicts the assumption
42
that the cycle
was the shortest one. This contradiction establishes that Assumption
{sfc}fc=i
2(a) holds.
To show that Assumption
follows that
it
some x,y we have x
for
if
contradiction, that there
Without
si,
.
.
a cycle
si,
.
.
This means that Wms
> Wms
(s/)
state
(si),
s;
>
(x)
7J7,„^
such that Sk+i
s/
,
.
then
>~s y,
we may assume that
loss of generality,
Sj.
,
.
is
S and some
2(b) holds, take any s G
€ Ms- From
771^
Sk ioi
<
< —
k
I
maximizes the payoff of
to^
which implies
1
and
Si >-
among
5 s;.
st ates
and thus contradicts the
si )fg si
existence of a cycle. This shows that Assumption 2(b) holds and completes the proof of part
(Part 2) Note that
Q
to any subset
5
of
preferences of player
if
maximum
We
for
<
1
Assumption
start with
<
/c
6,^ ({si,
.
.
,
.
/
—
and
1
yg^
si
nondecreasing in
s/}) is
winning coalitions, even
k. It is
1
<
<
fc
X=
and
Z
£
{ij
I
<
ip
:
ij
favorite state of quasi-median voter
assume
of generality,
>
s^
pick any of these).
a cycle
is
5i,
.
.
<
coalition,
I
X
for
Then
By
s/..
which
Sfc
as i-i,...,iij\ so that
straightforward to use the assumption that any two
iq] for
i-m-
such that s^+i ^s^
s/
,
.
and prove that
any
some
X
i.p,
iq
G C that
for this order there exists
satisfies A'
=
g J. Let z
y,
.
5;})
,
.
.
be the
Without
z.
players
all
who
the choice of i^, players
G Wj^ for some
6,^ ({si,
there exists s^ such that s^
Because preferences are single-peaked,
~.
including i^, (weakly) prefer z to
do not form a winning
we
for different states, intersect,
a quasi-median voter im such that im G
€ I, let
i
(A7)
Let us re-enumerate players in
s;.
and
axgm&x Wi{s)
Suppose there
2(a).
QCS
nonempty subset
achieved at multiple states,
is
1.
are single-peaked, then his preferences' restricted
are also single-peaked. For any
bi{Q) €
(in case this
i
2
Suppose, to obtain a
w-m^ (y)-
I
>~ s
Lemma
i^
loss
< m,
with p
strictly prefer Sk to z
a contradiction proving that Assumption 2(a) holds
is
in this case.
Now, suppose that Assumption
such that
so that
bii_
im G AT
z
s/;
=
({si,
for
6j^ ({si,
>
r.
^s
s'^._|_j
.
,
.
,
.
X
any
.
.
Sfc
.
s/}) is
G
s;});
But then
for
Ws
I
<
k
<
2(b)
I
—
violated,
is
and
1
S]
nondecreasing for this
such that
X =
{ij
&
I
i.e.,
^s
s/-
new
:
ip
there exist s £
<
ij
then there exists Sk such that Sk >s
s^ >-s
::
is
impossible, since
all
<
a cycle
we re-enumerate
Again,
cycle si,
S and
.
.
.
,si
iq} for
some
- Without
players ip with p
and choose
<
.
.
.
,
s;
players in I,
i,„
G X.
ip, iq
Sj,
such that
Now
loss of generality,
take
assume
rn (weakly) prefer z to x.
This proves that Assumption 2(a,b) holds.
(Part 3) The case of part
Assumption
^ hz
Wi (y)
2(b),
y-
Finally,
>
Wi {x) >
making use
if
li),
x,j/
(s),
G
S
of
1:
The
first
Assumption
part of Assumption 2(b)*
5 to rule
are such that x >-s s
43
s.
shown
similarly to
out indifferences between x and y
and y
which, in turn, implies y ^g
is
>-s
x,
then for any
i
G
Ms we
when
have
The
case of part
2,
that x
>-s
rris
and y ys
s
G Ms-
We
bm^{{x,y,s])
=
if
y
I would not hold.
Without
(y)
>
This implies y ^s
s.
Now
Wi {x)
s.
Assumption 2(b)*
>"s s,
we must
5.
is
proved with an argument analogous
To prove the
y (with
6
and only
if 6,
y, in
which case x ^s
{{x, y, s})
=
y, so
Second, suppose that either
Now,
if
if 6„i^
therefore have y ^g
s.
s
mote
=
then
> y
< x <
x<y<soTS<y<x.
In both cases,
see this
{{x,y,s})
implies y
s,
s,
or x,s
<
y.
Now, we
s.
Without loss
he must also prefer y to
we have y ^s
.•,'.
s,
of
given
s;
which completes the
~
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46
,
Appendix B: Limited Transitions (Not
Modeling Limited Transitions
7.4
In the main body of the paper
is
for Publication)
we assumed that any
transition (from any state to any other st ate)
may be possible.
possible. In certain apphcations, not all transitions across states
Example
in
discussed in the Introduction,
1
possible from constitutional
it
may be
monarchy (and not
that a transition to democracy
directly
more substantial example highlighting the importance
Acemoglu, Egorov, and Sonin (2008), also discussed
current
members
of the
3, is
of limited transitions
is
The key
1
Q c
and
is
to denote that the transition to
of the
we
and generalize the
the binary relation -^ on the set of states S.
any state
Q
; in
is
possible and x -^
is
For
Q
for
possible, provided that these
positions are supported by a winning coalition in x (similarly, Qi -^ Q2).
main body
B
Appendix
2.
to the analysis in this section
<5
in
Assumptions and the Axioms of Section
any x,y G S, we write x -^ y to denote that a transition from x to y
some
model
ruhng coalition can be part of future ruling coalitions and thus transitions
analysis, after proper reformulation of the
Theorems
the
In that model, only
in subsection 6.4.
applicable to the case where only certain state transitions are allowed
results in
only
is
from absolutist monarchy). Another
to states that include individuals previously eliminated are ruled out. Here in
show that our
For example,
paper thus corresponds to the special case where
S
~^
The
S
ior
analysis in the
We
any x G S.
adopt the following natural assumption on the transition relation.
Assumption
6 (Feasible Transitions) Relation -^
(a) (reflexivity) \/x
e
(h) (transitivity) \/x,
S
:
satisfies the following properties:
x -^ x;
y,z £
S
:
x -^ y and y -^ z imply x -^
Part (b) Assumption 6 requires that
if
some
z.
indirect transition
.
from x to
2 is feasible,
so
is
a direct transition between the states. Without requiring transitivity, there would be additional
technical details to take care
y
onl}',
then
it
is
of,
only possible
coalition in y prefers z to
y}^
because, for instance,
if
if
transition from x to z
both a winning coalition
in
is
possible through
x prefers z to x and a winning
Nevertheless, this assumption can be dispensed with, and
could assume instead that whenever x -^ y and y -^ z but x
-/^
z,
then
Wx =
we
VVy (or a weaker
version of this assumption). ^^
'
See
Chwe
(1994) for another model where different transitions require different winning coahtions,
6 fails to hold includes economic games in
determined as a result of the actions in the
current state (for example, capital accumulation, which might depend on the current enforcement of property
rights). Since our game does not involve such dynamic linkages. Assumption 6 is natural here. In particular,
'^One set
which there
of economically interesting cases in
is
which Assumption
a capital-stock-like variable, such as capital, that
47
is
We
next consider slightly weaker versions of Assumption 2 and Assumption
when some
the fact that only certain transitions are feasible (since
it
becomes
incorporating
3,
transitions are not feasible,
easier to rule out cycles).
Assumption
(Payoffs with Limited Transitions) Payoff functions
2'
(s)},gisg5 sat-
{^'Ji
isfy the following properties:
For any sequence of
(a)
states 5i,
.
,
.
.
S
in
s/j
with Sj ^^ Sj^i for
<
all 1
<
j
—
k
1
and
Sk -^ 51,
Sj+i ysj Sj for
For any sequence of
(b)
and
for
Sj >~s s
<
all I
states
<
j
s, si,
(b)* For any sequence of states
some
for
I
<
j
<
I
<
k,
and
Moreover,
Assumption
if
3'
for x,y,
s
s,
y
unchanged, though we state
1'
^x
y
-
- "
.
Axiom
3'
(Rationality)
there
is
this
(p{x).
new
no reason
.
.
for
for
.
,
in
s/j
all 1
all 1
s ~-> Sj
Sfc.
for
all
<
1
<
j
<
~
k
S
<
with
<
j
==>
1
k
<
J
s^
/c,
If
S
S
either
s ~~> Sj
An
si,
for
1
x
<
all I
<
j
k, Sj r^ si
,'
,
==>
>~s s
si
^^
Sfc.
;,,.,,
;
and y ^s
y^sXarx^s
for this slightly
are such that y
then y
s,
y.
,.,,
S
\
=
=
4>{3:),
that s -^
Axiom
-
then y
are such, that x -^ z, z
I
.
=
^x
3
;
(p{x), then either y
-
..
-
.
,,
.
_
-.•
)fs x.
modified set up (note that
-;
-
are such that y
x,y,z £
=
x or x -^ y and
-
_
,
-\„
'
.,
.
(f){y).
=
x, z
(p{z),
and
z
y^
y,
-,.,;,,,.
Axioms, a
for a sufficiently
~~+
Sfc.
k,
—
y,
^s
si
.
set of
the continuation game.
oo y,
.;
(Stability) If x,y €
With
s, Sj,
:
2'
^
<
with
again for completeness)
it
Axiom
then y
j
<
all I
Axioms 1-3
(Desirability) If x,y G
X.
S
have s -^ x, s -^
and x
>-s s,
Finally, let us reformulate
Axiom
in
S] ^^^^
(Comparability with Limited Transitions) For x,y,s & S such
X, s -^ y, X >~s
is
for
Sj
S we
£
Sfc
,
.
.
Sj >-s s
hs
Sj+i
.
<k-l =^
j
k,
>-s Sj
Sj-|_i
<
all 1
slightly modified version of
Theorem
1
holds:
powerful coalition not to be able to implement a change that
interesting
changes agents' preferences over time
model
is
provided
in
is
feasible in
dynamic enfranchisement where capital accumulation
Jack and Lagunoff (2006).
of a gradual
48
-
-
Theorem
4 (Dynamically Stable States with Limited Transitions) Suppose that binary
relation
satisfies
~~»
Assumption
1.
There exists mapping
2.
Any mapping
struct sequence
or/j,i^^
^i^i
< yUj, ...,
=
IJ-j.Let 4>{iJ.i)
Mk =
>
Axioms
Axioms
can
=
For each k
fi.i.
be recursively constructed as follows.
2,
.
.
.
s
\
y^^
e
Mk
(pi
Mk
:$~ e
and
with
and
Axioms
that satisfy
<j)2
-^ z
/i^.
Assumption
and for any
is
mapping (sequence)
s
G
iS,
3'
holds,
jj,
To construct mapping
</ij
(s)
:
{I,
.
ft,
.
.
~
,\S\} ^^
cj)2
S
<.?<'<
6 guarantees that
if
<^->
i>S|
,
then
and only
/*
/x^
-/^
/j,;
and
^fc,
=
s}
Mk
7^
(/)(s)
z >-^^ s
if
'
map-
induces relation
--+
ii
=
Axioms
that satisfy
1
or
//,
idea of the proof
<
/c
<
jiS|
to construct
is
we have
that
'
/,/,;
(Bl)
/.t^.
^^^
a
x -^ y and y -^
x.
,
{y G
<5
in equivalence classes
:
X
«—> y}
intersect.
The binary
by letting Ex -^ Ey
well-defined in the sense that
it
.
.
if
and only
is
ii
on elements
x -^
y;
note
acyclical in the sense that
,£' such that E^ -^ E^'^^ for 1
49
relation —^
does not depend on the elements x and
y picked from Ex and Ey, respectively. Furthermore, this relation
there do not exist distinct classes E^,.
(po
an equivalence relation, inducing equivalence classes
is
'^x
and
4>-^
is
relation «-^ defined as
and £y either coincide or do not
is
The
.
.
that this relation
1.
such that for any
{^a:]x^S defined as
S
jjl^
{s).
we introduce a binary
X *~^ y
of
\S\, either
then the m.apping that satisfies Axioms l'-3'
an extension of that of Theorem
if 1
to be such that E^
<
l'-3' the stable states of these
"payoff-unique" in the sense that for any two mappings
Assumption
I
.
in addition,
Proof. The proof
<
j
,\S\, let
iJ.1,-^ s, s
:
<
I
Con-
'
pings coincide.
l'-3'
Then:
hold.
l'-3'.
l'-3'
{s e {/Zj,...,/Xfc_i}
For any two mappings
If,
2'
.
^^^1^'
4.
and
1
with the property that for any
and define
3.
Assumptions
that
satisfying
4>
that satisfies
cf)
and
6,
,
<
j
<
/
and
E^ -^ E^
Consequently, we can form a sequence of
number of
classes)
enumerate
its
such that
elements as
due to Assumption
£1
numbers
take
1
members
the one of
/x
Theorem
/if,^,
.
.
,
<
1
so that
(a),(b)
1.
and 4
and
1
is
Theorem
hold.
reaches
For any
Then
(p
(j>
there exists
S
:
S
-^
MPE
(sq) after
£
/3o
[0,
--^
j
<
as follows:
sequence
mapping
in addition,
Axioms
satisfying
if
we
=
(
give
^]
(this
is
feasible
members
of class
\£'^\
,
.
.
.
and so on.
fi^,
The
fi^
<
I
(p
one period and stays in
:
S
any
,
fiLi
It is
,
,
j
then we
easy to
show
rest of the proof closely follows
Assumption
-^
S
is
Axioms
that
the discount factor
(i
Assumptions
>
(3q,
1,
then:
St
—
4>{so) for
any
>
t
1; that is,
Therefore, s
this state thereafter.
=
(j)
the
(sq)
'
'
'
,
'
,
^ (b)*
satisfying
3
6,
'
initial state sq
Assumption
satisfies
Proof. The proof
holds.
G S,
Axioms
Then:
St
l'-3'
=
MPE. Any
s°° for all
such that
t
s"^
=
>
such
MPE a
has
Moreover, there
1.
0(so).
Therefore,
all
stable.
MPE is essentially unique in the sense that
any pure-strategy MPE a induces st ~
(so) .for allt > 1,
holds, then the
l'-3'.
-
essentially identical to that of
These theorems therefore show that the
to an environment with hmited transitions.
Theorem
2
and
essential results of
The
is
omitted.
Theorems
1
intuition for these results
and 2 generalize
and the recursive
characterization of dynamically stable states are essentially identical to those in
and
we
,
l'-3' there exists a set of protocols [t^s^ s<^s
a of the game such that
for any set of protocols {ns}^^^,
(f)
satisfies
such
1)
dynamically stable states are axiomatically
where
jj,
<
1
set of protocols {''^s]s^s ^^^re exists a pure-strategy
the property that for
If,
ioi
S'^
2 again applies.
Moreover, suppose that Assmnption
3.
fi'^
within each class
omitted.
a dynamically stable state.
exists
k
constructed in this way satisfies (Bl).
and a pure-strategy
2.
fc
as they are listed in the sequence
£'2
For any mapping
is
< m. Now,
<
the
is
(Noncooperative Foundations of Dynamically Stable States with Lim-
5
game
)/-
jif
ited Transitions) Suppose that binary relation
S!
7
Next, construct the sequence
Similarly, an equivalent of
Theorem
.
any
S'^ for
in the order they are listed in the
|
of class
that the sequence
/xj',
2'(a)).
to |^i
7^
f-'
m
equivalence classes (S\...,£'"^ (where
all
2.
50
Theorems
1
Generalization of Proposition 4
7.5
We now
show how Proposition
4 can be
generahzed by allowing only certain types of transitions.
This generahzation makes the political elimination model of Acemoglu, Egorov, and Sonin (2O08)
(and
its
extension to infinite horizon) also a special case of the analysis here.
Proposition 5 Consider
1.
Assumptions
^ 3 and
1,
and Sonin (2008). Then:
the environment in Acemoglu, Egorov,
,
6 are satisfied (provided that
,
X
-^
Y
feasible if
is
and only
if
Y CX).
an arbitrarily small perturbation of payoffs such that Assumption
2.
There
3.
If only eliminations are possible, there exists a
exists
Axioms
unique outcome mapping
2!
(b)* holds.
4>eiim ^^'^^ satisfies
This mapping yields the same equilibrium (dynamically stable) states as in
l'-3'.
Acemoglu. Egorov, and Sonin (2008).
4-
In the case where any transition
unique outcome mapping
from
Assumptions
2'
the latter two,
all
X
transitions
and
Axioms
1 is
~-+
1'
Y
and
Z
~-+
is
Z
-^
which makes the
are feasible, then
is
text), there exists a
potentially different
1),
since
feasible, so
this result
is
Assumption 6
has not changed.
it
and
3,
because
in
set of potential cycles larger. Finally,
Y C
A'
and Z C
i',
hence
Z C
A',
and
satisfied.
is
very similar to the proof of part 2 of Proposition 4 and
omitted.
(Part 3) Existence and uniqueness of a mapping
from part
is
1,
since
Theorem
1
applicable.
is
Axiom 1-4
't'elim
=
it
(Inclusion):
cases, 0e/i^
X^
suffices to
Take any
{X)
0g,,„
'Pelzmi-^)
C X.
(p^^^^
show that mapping
A';
Axiom
For
1'
Axiom
{X) and 0^,,^ (X)
S ^A'
exists a
of that paper This correspondence
assumption, so below, we will treat
that satisfies
(p^i^j^^
Similarly, in
shown that under these assumptions, there
that satisfy
or
This mapping
1-3.
by Proposition 4 (part
satisfied
transitions are allowed,
(Part 2) The proof of
it
Proposition 4 in the
are in fact weaker than corresponding Assumptions 2
3'
therefore transition A'
is
that satisfies
(p
m
(?)^,„„.
Proof. (Part 1) Assumption
if
feasible (as
is
unique outcome correspondence
(p^^^^ is
single- valued
satisfies
Axiom 1-4
2 (Power): Again,
X. In both
in the notation of that paper.
51
Axiom
cases,
Axiom
1'
<?^,,j„
=
To prove that
of that paper. For
^ or -^
Axiom
C
X
1
"^ 4>ehTn {^)- ^^ both
implies that either ^g;j„, (X)
(X)
(f)^^^
due to genericity
as a mapping, not a correspondence.
4>f,iim
l'-3' follows
Acemoglu, Egorov, and Sonin (2008)
implies that either (peUm {^)
^x
Axioms
=X
and 7^^„„(a') > "7a-> so
3 (Self-Enforcement):
Take any
X
and
let
Y =
Axiom
4 (Rationality):
X
that
Z C
Y=
4>i,n^
if
Axiom
(j)^i,^{X).
Axiom
=
Iz < 7y, then 7^
and
Z ^x
2',
^^"^
and
for
7y,
^—
F
=
(PeHm{<PeiimW)
'Pehm i^)- If
)-x Z.
Z—
Y
(p^n^n
7^
Axiom
{^)- This proves that
satisfied, hence, (/>,;j^
=
=
3'
7g =
4,
Jq =
4>{{A, B,
(f)
{X)
<?!>e/t„
5,
and
cj)
7o =
but
(f)
{X)
may be different,
that
X
-^
(/"eiijn
but (j){{A,B,C,D,E,F})
E.F} with
not ^-stable (0({A,
Y
>-x
-'^j
(j)^i^^^
7^, together with genericity, implies
Axioms 1-4
Z =
Y, and
and Sonin (2008) are
of Acemoglu, Egorov,
it is
possible that for
Z
some
coalition
the least power;
B,F}) = {A,
power
it
In
also
fact,
(l)^ii„,{{A,
{A,B,F}
X,
=
{A, B, C}, but
<?^ehm ('^")
C
(in
{A, 5, C, D,
52
£",
F}).
X
F
and
with
B,C,D,E, F}) = {A,B,F},
is
the winning coalition in
happens to be ^g^^^-stable.
i?,C}), and in this case
= 3,
with powers 7^
(X): this would be the case for six players A, B, C, D, E,
= {D,E,F}).
coalition with the least total
To show that the two
4 (part 1).
consider four individuals, A, B, C,
powers 100,101,103,107,115,131, respectively (here,
{A, B,C, D,
-^ Y,
(meaning
Z ^ (peiim (^)- Conversely,
Z, Z yx X, Z =
(Z),
Y = 0e/im(^)i ^^ Axiom 3'
straightforward to verify that (p^n^ ({^i B, C})
4.5. It is
^
Wx
e
X
Y- Finally,
c})^^^.
C]) = {A, B, Z}. Moreover,
C X,
Z
=
then implies that
(Part 4) Existence and uniqueness follow from Proposition
mappings,
4>eHm{X)
7y < 7Z' then we have
7^ < 7y would imply
But 7^
4'elimi'^)-
=
(t>eHm{^)^ ^^i^ suppose that
^i in which case we would get a contradiction with
would imply
hence
Y =
Take any X,
and 7^ > aj^)
[Y) by
implies <t>eHm'J)
2'
{D,E,F}
is
the
However,
</)-stable
it
is
winning
Appendix C: Examples, Proofs From Section
and Additioned
6
Results (Not for Publication)
Definition of
MPE
Consider a general n-person infinite-stage game, where each individual can take an action at
every stage. Let the action profile of each individual be
a-
e Al and
e A,
Oj
including stage
and
let if*
We
t),
=
YYt^i ^i- Let
=
where a*
(af,
be the set of histories
denote the set of
.
.
for player
i
(aj
,
denoted by
profiles for player
i
aj,
.
(a\
a'~
,
.
.
/4,,_;.
We
and the action
i
t
aj
<
:
=
j
puie strategy
for player
i
i
=
7i,
?'
!,...,
t
(including time
H^
We
\ i/(_2 is
i
=
1,
.
with
n,
,
.
.
(not
t
.
.)
by
for
i
=
1,
.
,
.
.
n,
with the
con-
set of
Symmetrically, define i-truncated action
with the set of t-truncated action profiles
and a_, to denote the action
player
payoff
i's
is
Ai and A-i).
The payoff
given by u, {a}
,
.
,
.
.
a").
is
'•
^oo
*
^i'
(corresponding to strategy a^) specifies plays only after
i
t), i.e.,
CTi.t
where
.
f
profiles of all other players (similarly,
i.e.,
for
.)
.
.
1.
to date
(a',a''''^,
a^,
,
the history at the beginning of the game,
< T—
up
(a^
also use the standard notation ai
t-continuation strategy for player
time
is
=
be the history of play up to stage
a')
Aj.t.
^i
A
,
/i°
functions for the players depend only on actions,
A
.
denoted by
for
)
.
.
,a^), so
potential histories
all
tinuation action profiles for player
=
.
for
/i*
Let f-continuation action profiles be
profiles as Oj _t
=
/i'
a^
:
Hoc
\
Ht^2 —>
Ai^t,
the set of histories starting at time
(.
then have;
Definition 9 (Markovian Strategies)
for all T
>
t,
whenever
/i(_i,/it-_i
£
//oo
A
continuation strategy
'^'re
such that for any
a^^t
ai^t,(ii,T
A-^,t,
Ui
ht-i)
[a^^f. o--i.t
>
u, (aj,r, a-i,t
>
Ui (di^r,a-.i^t
I
I
^r-i)
implies
Ui iai^t,a-i,t
I
/if-ij
53
\
is
/^T-lj
Markovian
S
Ai^t
if
<md any
a-^^t
e
Markov
perfect equilibria in pure strategies are defined formally as follows:
(MPE) A
Definition 10
rium (MPE)
(in
Ui
pure strategy profile a
pure strategies)
{(Ti,
d-i)
>
—
each strategy Oi
if
Ui {di, a-.i) for all Oi
(cti,
is
...,
Markov
is
(7„)
perfect equilib-
Markovian and
£ Ej and for
all
i
=
1,
.
.
,
.
n.
Examples
Example
MPE
a
in
pure strategies
are 8 states
Wa = {X
=
W/r
[6],
(Nonexistence without Transaction Costs) In
3
S =
e C
:
the dictator, so
W2{-)
u,5(.)
=
=
[i]
to exist
fail
n
y^E
=
>
A'l
y^G = W// =
= {X
e C
:
i
(here,
[7]
=
u)3(-)
=
(0,0,0,0,1,0,0,0), wq{-)
Evidently, state
that satisfies
H
\i\
2
1,
and 2b* are
H
C
is
the only state where dictator 4 receives a positive
E
and
F
1
H
.
Moving
to
H
immediately
()
um(-)
=
=
=
to B,
i
is
(0,0,1,0,0,0,0,0),
(0,0,0,0,0,0,0,1).
satisfied
(5],
(0, 30, 0, 0, 20, 0, 0, 1),
(it
is
It
is
helpful to notice
v
and similarly any of the states
will
immediately lead to C, because
utility; similarl}',
move from
D
immediately leads
state
A
not possible in an equilibrium either:
B
and
C
the state were to change to C, then players
then to E, followed by H). Finally,
if
to F, so as to follow the path to
G
moving
Wd =
can form a
which would then lead to
for
C
Then
off'
players
and only then to
H
each of these players (recall that the
close to 1).
Let us consider possible moves to
If
is
would be higher
since the average payoff' of this path
is
[4],
impossible to stay in A: players 1,2,3 would be better
it is
and 3 would rather deviate and move
discount factor
B
also evident that
Let us prove that no
immediately leads to G.
pure-strategy equilibrium. First,
moving to
It is
Wb =
are:
'
G will immediately
.
liJi
A\ss = H).
F,
lead to
winning coahtions
'
£',
to
follows:
stable (dictator 7 will never deviate),
is
There
denotes the set of winning coalitions where
(0,0,0,0,0,0,1,9), wj {)
>-^
s
the transaction cost.
set of
(0,0,20,0,0,30,0,1),
straightforward to show that Assumptions
s
The
7 players.
e X}). The payoffs are as
(0,0,0,30,0,0,20,1),
that the only state
we assume away
example, we show that
majority voting between 1,2,3),
(i.e.,
2}
if
H} and
{A, B,C, D, E, F,G,
|{1,2,3}
Wc =
may
this
to B, followed by
C
(the
1
moves
G are considered similarly).
deviate and move to D (and
to Z), E, F,
and 2 would rather
the state were to change to B, then 2 and 3 could deviate
and
H
after that; this
is
better for these players than
and H. So, without imposing a transaction cost
a pure-strategy equilibrium does not exist.
54
it is
possible that
Example
we show
4 (Cycles without Transaction Costs) In this example,
that in
the
absence of transaction cost, an equihbrium maj' involve a cycle even though Assumptions
1, 2
and 2b*
=
Wi {A, B, C)
(4, 5, 10),
{X
e C
A
state
Wa = {X
Then one can
5,6 G X}.
:
to state B, from
B
e C
A
move
or
to C,
C
and from
B
dislikes state
while for player
this
true because
is
this
1
But staying
to C.
is
:
A
in
in
terms of discounted present value
To
:
3,4 e X},
see this, because of the
the current state
hurts both player
same time, moving
evident). At the
is
S C
Wc
=
an equilibrium which involves moving from
to A.
if
Wb = {X
and win-
(4, 10, 5),
The
A.
is
and player
1
symmetry
C
to
it
alternatives are to
who
2 (for player 2
postpones the move to C, the state that he
it
C
is
=
Play-
B,C) =
(5,4, 10), wj, [A,
B, C)
(10, 4, 5), wj, {A,
1,2 £ X],
see that there
suffices to see that the players will not deviate
stay in
=
-
W2 {A,B,C)
10,4),
(5,
S, C)
(10, 5, 4), u-g (A,
ning coahtions are defined by
S = {A,B,C}.
{1,2,3,4,5,6}, and 3 states,
B,C) =
preferences are given by wi (A,
ers'
=
There are 6 players, J
hold.
hurts player
likes best,
because state
1,
the worst of the three states for him not only in terms of instantaneous payoff, but also
the cycle continues, as
(if
should due to the one-stage
it
deviation principle). So, this cycle constitutes a (Markov Perfect) equilibrium.
It is also
easy to see that in this example, Assumptions
are no two states
cycle
in
is
s,
C
Example
5
A
and 3
whereas player 2
states,
(10,8,5),
= {X
Assumption
To
if
B
will
and
C
=
C
e C
:
3
G
(5,10,8),
A'},
have dictators
2(a)
is
u's
Ws =
1, 2, 3,
(C)
>
u's
(A)
W3{A,B,C) =
{A G C
1
:
violated.
>
We
will
aforementioned
then have
is
B
stable, will
three states are
:
A yb
{1,2,3},
> wj
coalitions are given
2 G
A}
B,
B ^c
(in
other words,
C,
C ^a
-
is
.
C
which
is
is
stable: player 2,
game
is
not, since player
who
is
the dictator in C, knows
worse than C. However, then player
have an incentive to move from
55
A
in this case.
B
dynamically stable, then state
will lead to A,
A, so
•
,
A
(C),
wi{A,B,C) =
example,
G C
J=
> wi{B)
easy to see that there are no dynamically stable states in the dynamic
that a transition to
also arise
always block transition
Winning
= {A
may
are 3 players,
(for
(8,5,10)).
all
there
to B).
Wi[A)
u'3 (i?)
G A}, W,4
respectively).
enforce transition to A. Therefore, state
C
1
A
Players' preferences satisfy
u'2iA), and
see this, suppose that state
that
are stable, then players
always block transition from
S = {A,B,C).
W2 {A, B,C)
states A, B,
It is
satisfied: in fact,
Finally, notice that the
sq.
(Nonexistence without Assumption 2(a)) There
woiB) > W2{C) >
by W.4
^^^
and 2b* are
because of symmetry), and situation where
(this holds
also possible (indeed,
is
to
from
s
2
not the only equilibrium. In particular, the cycle in the opposite direction
an equilibrium
stable
G {A, B, C} such that
sq
1,
to C.
3,
1
would
knowing
In equilibrium this deviation
should not be profitable, but
the transition costs, there
hence, there
it is;
MPE
no
is
in
is
A
no equilibrium where
pure strategies, since
no state
if
Now, given
stable.
is
dynamically
is
sta,ble,
the players would benefit from blocking every single transition in every single state.
now formally show that
Let us
that there
such mapping
is
Without
stable).
not stable:
stable,
then either
who
a
is
member
and by Axiom
that
4>
mapping
cf)
Example
and 4
=A
(C)
or
(j)
no mapping
there
2,
suppose that
we would obtain
A
is
is
=
(C)
The
B.
=
a stable state
such a state:
first is
W2 (C) >
Axioms
that satisfies
states,
=
iS
Winning
{{1,2}
,
{1,3}
A yo D and A yo
Kd =
and
3,
{2,3}
,
B,
tto
there
is
majority voting rule).
were, then
if it
it is
not dynamically stable,
A
are accepted in equihbrium. Suppose that
two players,
players, 2
and
would prefer
A
1
and
3, strictly
3, strictly
to
have
must be rejected
we would obtain a
Hence, there
is
2,
B,
Axiom
3,
all
no
it
prefer
prefer
C
A
to B),
to A).
in
=
A,B,C
there
A
then have
violated.
if
pure strategies
=
{1}
must vote
we must have
and therefore
C
that
B may
>-
d D, A
who
it
is
for
some
D,
2,3,
(1,
easy to see that
off'
from
any such proposal in
of proposals C, S,
A
not be accepted (because
must be accepted (because two
as players 2
A
is
.,
and 3
Lemma
1(c)
accepted,
and
we assumed that some other proposal
'
y^
in addition,
receives 10
accepted in the next period, and by
in this case.
(C) >
unanimity
three players would be better
accepted, then
56
twi
{{1,2,3}},
is
Assume,
This contradicts our assertion that
similar contradiction
MPE
is
{1,2,3},
W3{A,B,C,D) =
But then A may not be accepted,
C
rejected so that
in the equilibrium.
is
T=
> ws{B) > w^ (D)
Consider state D;
transition to either of the three other states A, B, C, so they
that
is
"t^C
are dynamically stable: evidently, player
not dynamically stable. Indeed,
Now
We
2(b)
are 3 players,
(5,10,8,4),
other words, in states
(in
respectively) will block transition to any other state.
equilibrium.
=
Therefore, 4){C)
wj,{A)
Wa = Wb =
B yp C, C yp A., so Assumption
(1) = C, -np (2) = S, tt^ (3) = A.
C
>
(C*)
W2{A,B,C,D) =
(10,8,5,4),
{1,2,3}}
,
D
In this case, states A, B,
it is
not
is
since player
1,
wi{A) > wi{B) >
Players' preferences satisfy
coalitions are given by
voting rule, while in state
that
Axiom
is
1-3.
{j4,B,C, D}.
wi{A,B,C,D) =
(8,5,10,4)).
C
>^a A- If
is
C
state
A; this means, by
(Nonexistence without Assumption 2(b)) There
6
example,
Wo =
=
Then
A.
C
since
3,
102 {A).
(p{A)
—
4>{s)
s,
cannot hold. This contradiction shows that with these preferences, there is no
wi{D), W2{B) > W2{C) > W2{A) > W2{D), and W3
(for
Axiom
{A)
(j>
Assume
1-3.
any state
(for
impossible by
A yg B and
B. But we have
Axioms
that satisfies
(p
a contradiction with
of any winning coalition in C, has
2, (t>{B)
= B
[B)
(f)
is
By Axiom
(p.
loss of generality,
were,
if it
there
is
accepted.
We now
show that there
such mapping
state D. If
and
C
Axiom
=
{A)
(p
satisfies
=
(D)
(f)
D,
this
A. Hence,
C ^d
C
^i
4>
would violate Axiom
^ D;
{D)
C
there
=
A,
implies that <p{A)
1
Axioms
that satisfies
(j)
Since for each of the states A^ B,
(j).
three players, then
all
no mapping
is
=
By Axioum
C.
mapping
Example
Assumption
and
3
7 (Multiple Equilibria without
(for
Winning
by W/i
unanimity voting rule
Assumption 3
is
One can easily
Let
1-3.
Mappings
(Pi
(A)
one maps state
and
(C)
(Pi
= A
differ in
(p2
C
and
to state A. It
these two mappings satisfy V^^
lui
=
Let
(p2
satisfies these
=
[A,
B] =
dt-^
and
=
{1,2},
> wi(C),
wi (B)
(3,5,1)).
C
4>i
which
4>2
(B)
to state A,
and 02
a
B
>~c A).
satisfy
Axioms
nor
=
=
(C)
<j>2
B.
and the second
Axioms
satisfy
Theorem
1-3,
under
sets of stable states
they should according to
is
while
satisfied,
A yc B
(p2,
= A and
(A)
Axioms. Note that the
T>^^, as
1-3.
and 2(a,b) are
1
one maps state
first
A. This
{{1,2}} (in other words, there
two mappings,
B.
=
W2{A,B,C) =
(5,3,1),
straightforward to verify that
is
>
A y jj D
A. But then state
3) There are 2 players, /
(A)
is
C. Consider
satisfies
Axioms
are preferred to C, but neither
(B)
cp^
=
{1}
A
=
(D)
that satisfies
Then Assumptions
only that the
no other mapping
also that
A and B
=
VVc
see that in this case there exist
=
and
(/)j
= Wb =
=
(t){C)
we cannot have 4>{D)
3
wi{A,B,C) =
example,
in all states A, B, C).
violated (both
4>
ip
that there
preferred to it by
is
instance, state
preferences satisfy
Players'
W2{B) > W2{A) > W2{C)
coalitions are given
B, and
without loss of generality assume
>~d D, and (f){C)
S = {A,B,C}.
=
(S)
contradiction proves that there does not exist
states,
no state that
is
3, since, for
Assume
1-3.
1.
Proofs of Propositions From Section 6
Proof of Proposition
1
(Part
1)
Take ms^
even. Evidently, for any of the rules Wsl"''
,
=
+
(k
VVJ^^'',
or
l)/2
Wi^ where
a quasi-median voter and, moreover, the sequence {ws;,};.^!
(Part 2) In
holds.
From
Finally,
all
part
cases W]""-', W^"'^
1 it
follows that
(Part 3) In an odd-sized club
we have
si >-s^
Sk,
Sfc
5,
Wi^ where
or
Theorem
Assumption 2b* follows from
of majority voting,
,
as
3 (part 1)
median voter
if
single-crossing condition. In either case,
and only
if s/
not indifferent between them by Assumption
for
any
Sj
and
s;,
is
Theorem
and
5.
which completes the proof.
57
/c/2
is
/c/2
<
l^
<
=
k for
k
is
m^,
is
/c/2 if
all
/c,
monotonically increasing.
k,
Assumption
1
trivially
apphcable, so Assumption 2(a,b) holds.
is
applicable in this case.
a single person [k
if ii'(jr._|_jW2 (si)
Sj
odd and m^
is
< k <
3 (part 3)
is
k
if
>
+
1) /2,
''^{k+i)/2 (Sfc)
and
.s/
the case
because of the
are two different clubs, player
This implies that either
in
)-s^ Sj
(/c -|-
1)
/2
of Sj >-5^
is
s;
Proof
rule
s^ for
>-5^.
si
>
I
because
k,
to exist will break at
take any club
=
k/2 <
I
<
s
k.
The
s^.
s/,
G
{1,
Sm with
.
.
.
m
<
I
s;
yg^
Let us now take
m>k.
m
If
who belong
that satisfy
that Assumption 2(b) holds. Likewise,
<
to
y-g^
S(
Sfc
and Sm ^s^
but not to Sm prefer
s;
Sfc-
then
all
players in
This means k/2
who
then the set of members of club s^
/c/2,
constitute at least half of the club in
5^
s;,
Finally, to
Sfc,
!^5t
(i.e.,
5/
sj.,
so
Sm ^s^
a majority) will prefer
(Part
k
is
Si
s;
E
s^,
impossible
si is
impossible, so the
hs,^ s; is
(/c
with k
< Wj
s;,
s; is
i
(sfc);
+
1)
in
is
odd; clearly, {tti^^J^^j
Suppose Wi
I.
^
and therefore
Sk,
this
(s;)
means that
>
s^,
I
j S s^,
m): those
,
to
s„,,
But they do not
5;.
s/
>
k.
But
We
have
/
and Sm such that
< m. But then
J^^^
s;
Sk,
y^k ^m, since
s;
all
be player k/2
to
rngf.
an increasing sequence of quasi-median
is
is
But then
s;.
or
This proves that Assumption 3 holds.
possible
But then Wj
j ^ s^.
.
s;-
s/,-.
This
u', (sfc).
.
< k/2
not belong to either of Sm and
I
that the single-crossing condition holds, take i,j €
<
.
and therefore Sm ^s^
—
holds, take s
and they form a majority
/2
{1,
m
Sm- Without loss of generality assume
prefer
To show
S
^
to
m
either
Consider the second case,
to s„i,
s;
s^
strictly prefer
s/-
and
k,
Monotonic median voter property holds, since we can take
2)
In either case,
(si)
s;
even and
voters.
Wj
we
^k
>-sk
will prefer s; to
:
show that Assumption 3
and
players from
Sk,
and Sm
Sk
>-s^.
<
I
prefer
who do
while those
Sm may
<
proved that Assumption 2b* holds.
if
s;
which form a simple majority
,1}
are indifferent. So, players only players in
Sm
the set of clubs that satisfy
^fc is
part of Assumption 2b* holds as well.
first
S(
Therefore, any cycle that
s;.
both clubs, but prefer the smaller one. Therefore, Sm ^Sk
< m, which proves
/
i
impossible to
it is
smaUest club. To show that Assumption 2(b) holds,
its
Hence, for any clubs
as they are included in
for
of s^ prefer s^ to
set of clubs {s;} that satisfy
s^: indeed, players
si >-5^
members
all
we hypothesize
have
holds in each club Sk, because the voting
1
simple majority. To show that Assumption 2(a) holds, we notice that
is
have
of Proposition 2. (Part 1) Assumption
i
G
(s;)
Sfc, s;,
if
S
i
> Wj
s;
such that
but
(s^).
i
<
i
^ Sk or
^
i
j
and
Sf^, s;.
Suppose now that
and therefore Wi
(s;)
<
Wi
(sfc).
This estabhshes that the single-crossing condition holds.
(Part 3) Notice that
Club
smaller clubs.
now
consider club
si
is
it is
never possible that
and
stable
log2
1
=
2°.
^ Z, then club
Sj for
If
than half members of
s^..
we know
do not contain more than
to be stable
Hence,
induction step.
(Part 4)
then
S2[iog2J:j
Sfc
.
e Z, then
is
the only club which
is
—
=
/c,
if
members,
/c/2
therefore start with
is
log2
/c
so s^
stable
58
Sfc
<
k and
and contains more
G Z, then the only clubs
is
stable.
This proves the
.
and the statement follows from part
preferred to
for j
,
.
2l-'°S2*=J
log2
j
2'-'°S2'^'J
unstable. Conversely,
,
If
fc
is
ys^Skiik<l. We can
Suppose we proved the statement
Sfc.
A;
s/
3.
If logj
k ^ Z,
by a majority (other stable clubs are either
.
than st or
larger
The
at least twice as small as Sjtiogj/cj
Theorem
(Part 3)
We
(Part 1) Assumption
3.
and Assumption 2(a,b) and 2b* are
3 applies
(Part 2) By part
1,
Theorem
if
E
\i
T
:
myopically stable state
is
> Wi
which
<
{a)\
b for all feasible a'
Q
we only need
to prove that
which
(a, 6)
is
nonempty. Note that
most preferred by
is
i
are several of these).
Consider state
Second, this state
(/)^-stable:
(a", b") )^(a',iV)
is
among
G Q, then
(a'
N) e
,
indeed,
if
First, since
Q.
{a'
weakly prefers
(a', 6')
(and therefore
,
N)
stable.
and
is (p^-stable
is
Axiom
7y > 7y,
so any cycle
4.
(Part 1) Assumption
would break
2(a) holds,
Z
prefer
>~x
Z
to
One can
then,
case
by
Z
then jy > 7^ implies
1',
and they form a winning
similarly
Z ^x Y
X
means that any
'^
Y
it
1
us prove that
{a',b') '^ia,b) i'^^^)]
Q
e
(a', A'')
(because
one of such states
lies in
{a'
Q,
,
N)
if
there
^(a,6) (c^,'')-
(a', A^),
is
the
which of course
But there
same
as far as
is
player
i
immediate
{a",b") does not exist,
cannot equal
(a, 6), since
state
immediately foUows from (18) and that
coalition in A', for
if
implies
Hence, Assumption 3
would break
holds:
is
satisfied.
59
)/-x
Z:
(which
^
is
indeed,
they did not,
is
impossible
if
Y
all
Z ^x
if
Y
>~x A'
players in
A'
>^x A' and
loss of generality, 7^-
if
unique because
Z
would be
at the least powerful coalition in
proved likewise:
7y ^ j^- Without
is
it
>~x
holds, one can notice that
Y
cycle
Y
suffices to notice that
)^x ^1 and thus
show that Assumption 3
genericity,
(or
means that such
Assumption 2(b)
A',
impossible. Again, this
let
is
i
at the least powerful coalition in
of genericity). Similarly, to prove that
and
any
Now take some player and
so {a",b") £ Q.
3 then implies that 4>^{a,b)
To prove that Assumption
1/2.
1,
preferred to {a,b). This completes the proof.
Proof of Proposition
a >
it
N), which
payoffs are concerned) to any other element in Q. This
,
Corollary
were not the case, we would have some other
it
who
is
Q
the states within
(a, 6),
and state {a',N)
By
such that
{{a',b')}
if (a', b')
implies that at least a players prefer {a",b") to
at least
equivalent to
is
any dynamically stable state
This means that each player prefers (a", b") to
(O'', A'').
this
not myopically stable;
is
Q=
Consider the set of constitutions
unstable, this set
that
de-
terminology of Bar-
But
.
the second part of the pair of rules does not enter the utility directly).
€
By
dynamically stable, but not vice versa, which establishes the result.
fight of part 3
[{a,b)] 7^ (a, 6).
(a', b')
Therefore,
immediately follows.
the definition of myopic stability.
is
myopically stable. Take any constitution
is
result
(a, 6) is self-stable (or self-stable, in
w-i (a')
{a',b') ^{a,b) (a, ^) for all (a, ^),
since (a, b)
> N/2.
follows from b
prove this result for constitutions and voting rules simultaneously.
bera and Jackson)
(Part 4) In
1
satisfied.
The
applicable.
1 is
a voting rule (constitution)
finition,
{a'
Sf-).
result follows.
Proof of Proposition
(pf.
more than two times smaller than
i.e.,
,
Z ^x
> 7^, and
it.
AT,
in this
(Part 2)
Let.
us perturb players' payoffs so that
small. After this perturbation,
still
The
valid.
Assumptions
and either 7^ < 07 y or 7^ > 7,v
in
Z
^ll
because of the perturbation we made, for
do not form a winning coalition
where 7^ > 7x; then
Z
and thus
)/-x
Y
Tliis
.
players in
all
Y
in this case,
Y
prefer
e
from part
X. This imphes ajx < 7y
who
players
obviously true for the part that belongs to Y, while
this is true
where
>
is
1
are
a corresponding >:-cycle existed, then
X and Z ^x
)~x
7z — ^Ixi
If
hold, as the proofs
still
if
e-fx
ruled out by Assumption 2(b). To show that the
is
Y
second part of Assumption 2b* holds, take
is
3
part of Assumption 2b* follows, for
first
by genericity we would get a ;^-cycle which
this
and
2
1,
=
^ X, then Wi {X)
if i
a player
if
we have Z
Z
)/-x
jy <
Y
neither in
is
Y
prefer
7^ > ajx ^ 7z-
in this case
to Z, since
are not in
<
7x
to Z:
nor in Z,
Since players
Y. Consider the second case
I'z-
Y yx Z
This means that
proves that Assumption 2b* holds, which completes the proof.
The Relationship Between P, von Neuniann-Morgenstern Stable
and
Set,
Chwe's Largest Consistent Set
The
Chwe
following definitions are from
(1994) and von
S and any
Definition 11 (Consistent Sets) For any x,y E
X -^X y
1.
if
We
only
c-'n-d
2.
We
either x
say that state x
such that X
for
if
all
£
i
.
>_\'
.
.
A
set
dominated
we
e Wx-
X
(1944).
—>a'
G C, define relation
by
'
'
'
by y (and write x
<
y)
there exists
if
write x -<x y os a shorthand for Wi {x)
<
X
d C
w-i (y)
X
x
is
,Xm € S such
Xj -^Sj 2;j+i and Xj
3.
directly
X
y and
^
y or x
y and x ~<x y, where
that state
sa.y
xo,Xi,
—
is
=
Neumaim and Morgenstern
SC S
there exists z
^5
is called
dominated
indirectly
=
that xq
y for j
=
consistent
& S, where y
=
x and
0, 1,
.
x £
if
z or y
^
.
.
S
z,
,
by y (and write x <^ y)
Xm — y and
tti — 1.
Xq, Xi,
stable
(Internal stability) For any x,y G
2.
(External stability) For any x E
Proposition 6 Suppose Assumptions
1
X
S\
such that x -^x
and 2
)/-x
hold.
60
that
Then:
E
x ~^x y
~-
set of states
X
C S
x;
there exists y
that
,
following properties:
we have y
X
,Xm-i S C such
and only ifiy € S,'iX e C such
if it satisfies the
1.
.
.
Definition 12 (von Neumann-Morgenstern's Stable Set) A
Neumann- Morgenstern
.
there exist
-
.
if
.
if
X
such that y y^ x.
is
von
The
1.
V
set of stable states
2.
T> is the largest consistent set;
3.
Any
consistent set
set;
'
.
V
either
is
von Neumann-Morgenstem stable
the unique
is
^
or any subset of the set of exogenously stable states (and
vice versa, all such sets are consistent).
We
Proof. (Part 1)
X
of states
since
V
n
yU;.
{/xj,
Theorem
that
for
,
.
Aik
some
follows that
1 it
<
k such that
A'n{/Ji,
But by
.
.
.
(7),
=
/x;._i }
,
fii
and therefore
^/-n^
^L).
us prove that
let
2;
prove that
= XO
{/ij,
e X.
Hence,
^ X.
Now
.
.
.
e
/x^.
i^ij^
>
/
k.
the opposite
It is
Wy
[i).
By
also true, so
Hence, for any
fif
X
^
X
€
ij,i
x < y
if
and only
if
x
<
y;
is
x
=
Now
satisfied.
impossible that z
mapping
Now
(f)),
X=
Vz G
X
:
V
is
consistent, consider
<
x <^
y implies
Note that
z
>y
y,
for
(x)
it
is
From
G V.
^u^.
=
/i^.j }
0- Suppose
first
y^^
/i;
ji^
would
k and
fij
>~^^
jif..
we have
/.t;
^^^
f.if.,
y.
To
see this, suppose that x <^y\ take a
S and
any A' G C such that x
y E V, so z
=
y
suppose that x
;^x x, since
any
z
(2),
G
.
Let
=
>
x)
/c
be lowest number
and Vi G Xk
Wx
Axioms
1-3.
that satisfies
y or y <^
< Wi
1 1
(f>
=
1,
<C y. In our setup, however,
a;
some mapping
take some x ^ V.
ti^i
,
V.
Xk+i-
T>
Since
^
y;
then
X
:
z
=
is
Wa-.
On
> Wi
Axiom
and
\.
P
=
{i
A
:
1
(0
> w^
But
as
In
(?,)
it is
for
{z), too.
=
G I
:
y or
Wi (x)
<
j/
S and
•C z^
X
we
Wi (y)} G
Wx\ then x
z, for
Therefore, any z G
V
such that either 2
then y
€ C such that
necessarily have
<^
61
w^^
would be violated
we have y
impossible that for some 2 G
which would violate Axiom
X
Take
= (p{y); then,
= y and x ^ y.
the other hand, z G V.
are stable (otherwise,
<
{i)
Let z
nonempty, property 3i G
satisfies that either z
4>{x)
V-
consider two possibilities: x
X
X G
w^ (x)
—>x
need to show that there exist y G
which
Take y
Now
x.
both x and
We
z.
=
hence, in this case, 3i e
^ ~>X y and
k
^
either z
Axiom
first case,
^
I
<
/
note also that A'^ G Wx, since x
any x e V, and then take any y £
the
.
This means that
0-
such that
such that
This means that x -^Xk ^k+\ (because Xk
j^ x.
that set
follows from
.
.
G X,
jj,-^
stability property
Mk =
consider the case where
obvious that for any x,y G S, x
definition,
To show
if
Mk =
if
G X, then internal
sequence of states and a sequence of coalitions as in Definition
such that Xk+\
{/ij,
set
G X, for otherwise external stability condition would be violated. This proves
fii^
is
Xn
by construction, we have that
,iii._i}
if /x^,
V. Clearly,
and only
li
and only
Af if
the induction step, a,nd therefore completes the proof that
(Part 2)
X
e
ij.f.
0, and therefore there does not exist
whenever
=
us prove that Pd
let
Suppose that
satisfying (7).
>
suppose that we have proved that
suffices to
/x;
be violated, and therefore
>
k
0; then, since Mlk
7^
/
it
Now
/j.^..
some
^^._i} for
.
.
any state
for
//j
ft], ..., fii^i
<
von Neumann-Morgenstern stable;
is
)/-^^
take the sequence of states
<
z,
—>x
y.
and therefore
=
y or y
<§C
2
must
V
that
=
satisfy z
To
But then, by our choice of X, we have
y.
indeed a consistent
is
V
sliow that
largest consistent set
< /Ltj, ...,
the smallest number,
such that
i.e.,
D
Since
and among
satisfying (7),
>
^|5|
S ^V.
\s
Take some mapping
consistent.
X=
£
{i
=
that y
2
fii
<
Wj (x)
:
fox
<
I
which implies z
=
impossible that
y, z
either 2
=
Now
for
jj..
E
y or y >C
"D
r,
some
and
<
j
a state which
<
and
z
X
:
X
Wi {x)
If
y
G
and
vS
possibilities. If state
2
with z
x
E
G 5 we had y <^
^
y\ at
X
is
this
is
2,
same reasoning
X
.
P\
yy y
P
\
with
that, according
=
and
implies
(7)
and hence
z,
P
G
(x)
(f>
which by
x,
<
then y
Axiom
z,
Therefore,
1.
we do have
yy
z
G
\/i
X
some
for
if
<
Wi {x)
:
y.
it is
z
E S
Wi
(2),
is
=
G
5
5
4>(x)
if
that
S ^ V, but S
—>x
2/1
y
not exogenously stable and y G
is
E.
S where
z,
=
either z
y ot y <^
—
then x
In that case,
y.
Now
—>>-
X=J
2 for
impossible, as 2 G
for
for y, z
This proves that
X =
some
{i
2
z
=
S*
y or y
is
E
T
:
Wi {x)
G 5 either 2
G V. But
S
z,
which implies, as before,
if
Take any x e 5.
we can take
some
and y
for
such
2,
< Wi
=
Y
=
=
x;
then x
G C, which
62
(y)}
is
^x
V-
If for
incompatible
=
\/i
E
2,
X
then
:
z
for
y should not
(p
satisfying
G Wx', then x —^x
y or y <C
set,
S and
x ^ S. This means that
and therefore x
we have
indeed a consistent
y E
take any x ^ S. Consider two
and y
<g; 2,
=
z
not exogenously stable. Again, consider mapping
and
P \ 5;
E V. This contradiction proves that
y, z
Wi (2) trivially holds.
is
includes
not exogenously stable.
such that either
is
set.
and hence y -^
5,
then, in particular, y
as before,
y <^ z would imply z
construction of
=
z,
y
consists of exogenously stable states only.
>
Wi [x)
suppose that x
Axioms 1-3 and take y
5
^^ G
show
let
^V,y ^
S C V. Suppose
impossible, since
S which
:
Now
the largest consistent
is
But y G
(z).
the same time, 2
in S. Finally,
the
such
for
exogenously stable, then take
this y there does not exist 2
be
since x
violate
a consistent set, then
£ C are such that x
find that condition 3i
some
that T>
not include any state which
X
=
^ S\V, and therefore z e V. However,
z
would
But
y.
sequence
a contradiction, since by definition of a consistent set x ^ S, while
> Wi
Consider, however, any
This proves
(z).
S. Consider
We now
k.
1-3.
G Wx- Since x G S, there exists z
However,
2 >~y y.
S ^V, S may
=
then in fact z
2, if 5" is
then x -^x y for some
y
But then
k.
>
/
G 5 we have y <^
2:
VC
pick state x
7^
Axioms
not exogenously stable. Suppose x G
is
such that 3i £
<
I
we picked x € S \V. This proves
(Part 3) By part
Wi
which would contradict the assertion that state S
that satisfies
4>
P
\
6 S \'D, then
if fi;
z )~y y, as this
.
5
states in
all
some
for
if
X We get
by construction of
<
Wi {x)
we must have
Wx; then x -^x V and,
Wi (y)} G
k.
:
suppose, to obtain a contradiction, that the
set,
consistent,
is
to the definition of a consistent set, x ^ S,
is
X
£
set.
the largest consistent
is
"ii
=
Wi (x)
By
because
y,
<
V-
Wi
(2)
which completes the proof.
by