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MIT LIBRARIES
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DEWEY
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9080 03317 5917
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
AND
PERSISTENCE OF BAD GOVERNMENTS
POLITICAL SELECTION
Daron Acemoglu
Georgy Egorov
Konstantin Sonin
Working Paper 09-23
July 31, 2009
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Networl< Paper Collection at
http://ssrn.com/abstract = 1 444 1 45
and
Persistence of Bad Governments*
Political Selection
Daron Acemoglu
Georgy Egorov
MIT
Kellogg School of
Management
Konstantin Sonin
School
New Economic
July 31, 2009.
Abstract
We study dynamic selection of governments under different political institutions, with a special focus
on institutional "flexibility" A government consists of a subset of the individuals in the society. The
competence level of the government in office determines collective utilities (e.g., by determining the
amount and quality of public goods), and each individual derives additional utility from being part of
the government (e.g., corruption or rents from holding office). We characterize dynamic evolution of
governments and determine the structure of stable governments, which arise and persist in equilibrium.
Perfect democracy, where current members of the government do not have an incumbency ad\-antage or
special powers, always leads to the emergence of the most competent government. However, any deviation
from perfect democracy destroys this result. There is always at least one other, less competent government
that is also stable and can persist forever, and even the least competent government can persist forever
in office. Moreover, a greater degree of democracy may lead to worse governments. In contrast, in the
presence of stochastic shocks or changes in the environment, greater democracy corresponds to greater
flexibility and increases the probability that high competence governments will come to power. This
result suggests that a particular advantage of democratic regimes may be their greater adaptability to
changes rather than their performance under given conditions. Finally, we show that, in the presence
of stochastic shocks, "royalty-like" dictatorships may be more successful than "junta-like" dictatorships,
because they might also be more adaptable to change.
.
Keywords:
institutional flexibility, quality of governance, political economy, political transitions,
voting.
JEL
Classification: D71, D74, C71.
"Daron Acemoglu gratefully acknowledges
financial support from the National Science Foundation.
participants at California Institute of Technology Political
Economy Seminar,
in Moscow, NBER summer institute in Cambridge, Third
and University of Chicago theory seminar for helpful comments.
conference
Frontiers of Political
Game Theory World
Congress
in
We thank
Economics
Evanston,
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/politicalselectiOOacem
1
A
Introduction
central role of (successful) political institutions
competent, motivated) politicians.
to ensure the selection of the right (honest,
is
Beslej- (2005, p. 43), for
emphasize the importance of the selection of politicians
"The aim of every
men who
good of
political Constitution,
possess most
society:
and
wisdom
to discern,
example, quotes James Madison, to
for the success of a society:
or ought to be,
is
and most virtue
in the next place, to take the
most
to obtain for rulers
first
to pursue, the
common
effectual precautions for
keeping them virtuous whilst they continue to hold their public trust."
Equally important, but
ability to deal
less often stressed,
is
the "flexibihty" of institutions, meaning their
with shocks and changing situations,
government and changing the characteristics of those
environment.^
In this paper,
example by adapting the nature of the
for
in
power
in response to
we construct a simple dynamic model
to highlight the potential sources of inefficiency in the selection of
features of political processes that create "institutional
The "government"
may be
is
made up
a government, etc.).
determining the collective
Each
utility
flexibilitj'"
of
changes in the
government formation
governments and to identify
.'
of a subset of the citizens (e.g., each three-player
(potential)
it
government has a
provides to citizens
(e.g.,
different level of
group
competence,
the level of public goods).
Each
individual also receives rents from being part of the government (additional income, utiUty
of office, or rents from corruption).
"votes" from the citizens
New
governments are put
in
place by a combination of
and "consent" from current government members.
necessary consent of current government
members
is
The
extent of
a measure of the "degree of democracy".
For example, a perfect democracy can be thought of as a situation in which current incumbents
have no special power and no such consent
is
necessary.
Many
political institutions, in contrast,
provide additional decision-making or blocking power to current government members.
instance, in
in
many
power harder
Cox and Katz,
democracies, various sources of incumbency advantage
to oust
than instituting
it
anew would have been had
1996, for a discussion of such
'For instance, the
skills
it
incumbency advantage
make
the government
been out of power
in
For
(e.g.,
mature democracies).
necessary for successful wartime politicians and governments are very difTerent from
management of the economy during peacetime, as
Winston Churchill's political career.
"Even though we model changes in the underlying environment and the competences
those that are useful for the successful
illustrated
perhaps most
clearly by
of different governments
from deterministic changes in the nature of
the economy. For example, authoritarian regimes such as the rule of General Park in South Korea or Lee Kuan
Yew in Singapore may be beneficial or less damaging during the early stages of development, while a different
style of government, with greater participation, may be necessary as the economy develops and becomes more
complex. Aceraoglu, Aghion and Zilibotti (2006) suggest that "appropriate" institutions may be a function of the
distance of an economy to the world technology frontier, and Aghion, Alesina and Trebbi (2009) provide empirical
evidence consistent with this pattern.
as resulting from stochastic shocks, in practice these
may
also result
members
In nondemocratic societies, this advantage of current government
is
more pronounced,
even palpable: only new governments that include some members of pre-\aous governments might
be
feasible (as, unfortunately, illustrated
degree of incumbency advantage
is
by the recent events
in
Zimbabwe
While the
or Iran).
not the only characteristic of democracy, focusing on this
specific metric allows a simple parameterization of different regimes, ranging
from personalistic
dictatorship to representative democracy, and thus leads to informative comparisons between
these regimes both in stochastic and nonstochastic environments.
The
contribution of our paper
first
to provide a general
is
and tractable framework
for the
study of such dynamic political selection issues and to provide a detailed characterization of
the structure (and efficiency) of the selection of politicians under different political institutions.
Perfect democracies always ensure the emergence of the best (most competent) government.
In contrast, under any other arrangement, incompetent and bad governments can emerge and
persist despite the absence of information-related challenges to selecting
ample,
e-\^en
good
a small departiire from perfect democracy, whereb)' only one
politicians. For ex-
member
of the current
government needs to consent to a new government, may make the worst possible government
persist fore\'er.
The
intuitive explanation for wh}' even a small degree of incumbenc}'
might lead to such outcomes
is
as follows:
improvements away from a bad
government might lead to another potential government that
way
for a further
common members
have any
any of the
it
round of changes.
If this
with the
even the worst)
unstable and will open the
process ultimately leads to a government that does not
initial
government members. In
initial
is itself
(or
advantage
government, then
this case, the initial
it
may
fail
to get the support of
government survives even though
has a low, or even possibly the lowest, level of competence. This result provides a potential
explanation for
why many
power
in Iran. Russia,
in the
government
autocratic or semi-autocratic regimes, including those currently in
Venezuela and Zimbabwe,
resist
the inclusion of "competent technocrats"
—because they are afraid that these technocrats can
of further reform, ultimately unseating even the
is
democracy (and dictatorships). Any
in the long run.
show no
This result
clear-cut relationship
is
in political selection
of these different regimes
may
that,
lead to better governments
consistent with the empirical findings in the literature that
between democracy and economic performance
dictatorships, the
is
no obvious ranking among different shades of imperfect
and Limongi, 1997, Barro, 1997, Minier,
and extreme
become supporters
most powerful current incumbents.^
Another important implication of these dynamic interactions
beyond perfect democracy, there
later
1999).
(e.g.,
Przeworski
In fact, both mider imperfect democracies
competence of the equilibrium government and the success of
For example, on Iranian politics and resistance to the inclusion of technocrats during Khomeini's reign, see
Meneishri (2001), and more recently under Ahmadinejad's presidency, see Alfoneh (2008). On Russian politics
under Vladimir Putin, see Baker and Glasser (2007). On Zimbabwe under Mugabe, see Meredith (2007).
the society depend strongly on the identity of the
initial
consistent with the emphasis in the recent pohtical science
may
that leaders
Olken, 2004,
and
play under weak institutions
who show
this does not
(see, for
members
of the government.
and economics
literatures
This
is
on the role
example, Brooker, 2000, or Jones and
that the death of an autocrat leads to a significant change in growth,
happen with democratic
Our second contribution
leaders).
relates to the study of institutional fiexibilit}'.
For this purpose,
we enrich the above-mentioned framework with shocks that change the competences
of different
types of governments (thus capturing potential changes in the needs of the society for different
types of
of
skills
and
expertise).
dynamic games
is
Although the systematic and tractable
challenging,
we provide
analj'sis of this class
a characterization of the structure of equilibria
Using this characterization, we show
how
the quahty (competence level) of governments evolves in the presence of stochastic shocks
and
when
how
of
stochastic shocks are sufficiently infrequent.
this evolution
impacted by pohtical
is
institutions. While,
without shocks, a greater degree
democracy (beyond perfect democracy) does not necessarily guarantee a better government,
the pattern that emerges
when we turn
to institutional flexibility
different. In particular,
is
our
anah'sis shows that a greater degree of democracy leads to better outcomes in the long run; in
particular,
this
is
it
increases the probabihty that the best government will be in power.
Intuitively,
because a greater degree of democracy enables greater adaptability to changes
en\dronment (which
alter the relative ranking of
therefore formalizes the notion that
a shghtl}' more technical
governments
more democratic
level, this result reflects
in
in
the
terms of quahty). This result
institutions ensure greater flexibility."*
the fact that
when
At
the degree of democracy
is
high, there are "relatively few" other stable governments near a stable government, so a shock
that destabilizes the current government likely leads to a big
Finally,
we
also
show that
jump
in
competence.
nondemocratic regimes,
in the presence of shocks, "royalty-like"
where some individuals must always be
in
the government,
may
lead to better long-run outcomes
than "junta-like" regimes, where a subset of the current members of the junta can block change
(even though no speciflc
member
is
essential).
The
royalty-like regimes
greater adaptation to change because one (or more) of the
members
might sometimes allow
of the initial
government
is
secure in her position. In contrast, as discussed above, without such security the fear of further
changes might block
We now
'The
illustrate
all
competence-increasing reforms in government.
some
of the basic ideas with a simple example.
stochastic analysis also shows that
may sometimes
random shocks
to the identity of the
members
of the
government
lead to better governments in the long run because they destroy stable incompetent governments.
"History suggests that four main methods of selection to political office are available:
drawing lots, heredity, the use of force and voting." Our model suggests why, somewhat paradoxically, drawing
lots, which was used in Ancient Greece, might sometimes lead to better long-run outcomes than the alternatives.
Besley (2005) writes:
Example
>
Suppose that the society consists of n
1
A
individuals could form a government.
change
there
is
that any
/c
=
3
govermnent requires both the support of
in
the majority of the population and the consent of
Assume
6 of individuals.
=
/
1
member
of the government, so that
an imperfect democracy, with a "minimal" degree of incumbency advantage. Suppose
that individual j has a level of competence jj, and order the individuals, without loss of any
>
generaUty, in descending order according to their competence, so 71
competence of a government
sum
the
is
of the
72
>
>
...
The
7„.
competences of its three members. Each individual
obtains utility from the competence level of the government and also a large rent from being in
competence
government.
ofRce, so that each prefers to
be
Suppose
have a sufficiently high discount factor, so that the future matters
also that individuals
in office regardless of the
level of the
relative to the present.
It is
straightforward to determine the stable governments that will persist and remain in
power once formed. Evidently, {1,2,3}
a stable government, since
is
members
of competence, so neither a majority of outsiders nor
to initiate a change (some outsiders
would prefer government
and from the
{1, 2,
i,
j}
,
{i, 2,
j},
who would be
rest of the population,
{4,5,6}. This
=
is
also a stable
and
{i, j,
3} are unstable (for i,j
government must
or {i,j, 3}.
result in
5,
like
and
6
>
3),
which
all
exclude them.
of the initial
of the
of
of government
starts with the
government
the lowest competence government
a whole.
This
is
because any change in
one of the following three forms: {l,i,j}, {i,2,j},
of these types of governments are unstable.
more competent governments
members
for the society as
it is
member
more competent government.
and suppose that the society
anew government
But we know that
any of the members
6
willing to see a
government, even though
and thus the worst possible option
initial
to initiate a change: for example, 4,
3} will receive support from both one current
Consider next the case where n
the
government would
starting with these governments, there will necessarily be a change. In particular, in
each of these cases,
of the
has the highest level
but they do not have the power to enforce such a change). In
{4, 5, 6},
contrast, governments of the form {1,
means that
may want
of the
it
will ultimately
Therefore, any
take us to (1. 2,3}, which does not include
government. Since individuals are relatively patient, none of
government would support (consent
As a consequence, the
initial
change that
to) a
will ultimately
worst government persists forever.
Returning to
our discussion of the unwillingness of certain governments to include skilled technocrats, this
example shows why such a technocrat,
for exarnple individual
1,
will not
government {4,5,6}, even though he would potentially increase the
qualit}'
be included in the
and competence
of
the government substantially.
One can
this case
also verify easily that {4,5.6}
is
any change requires the support of
all
also a stable
three
government when
members
of government
/
=
3,
and none
since in
of
them
to a change. Therefore, a greater degree of democracjr (lower
would consent
better outcomes in the long run. In contrast, under
so the lower degree of
Now
consider the
democracy can improve the
same environment
=
{4,5,6}
2,
qualit)' of the
is
does not guarantee
not a stable government,
government.
as above, but with potential changes in the
of the agents. For example, individual 4
comes the third most competent agent
I
I)
may
(i.e.,
an increase
see
in his
competences
competence, so that he be-
74 G (73,72))- Suppose that shocks are sufficiently
infrequent so that stability of governments in periods without shocks
is
given by the same reason-
ing as for the nonstochastic case. Consider the situation starting with the government {4, 5, 6}
and
/
=
1.
Then, this government
will
remain
power
in
until the
shock occurs. Nevertheless, the
equilibrium government will eventually converge to {1,2,3}; at some point a shock will change
the relative competences of agents 3 and
4,
and the go^'ermxlent
{4, 5, 6}
would become unstable,
because now individual 4 would support the emergence of the government {1,2,4}, wliich
has the highest competence. In contrast, when
I
=
3,
now
the ruhng government will remain in power
even after the shock. This simple example thus illustrates how, even though a greater degree
of
democracy does not ensure better outcomes
in nonstochastic environments,
it
may
provide
on
political
greater flexibility and hence better long-run outcomes in the presence of shocks.
Our paper
economy
is
related to several different literatures. While
much
of the literature
focuses on the role of political institutions in providing (or failing to provide) the
right incentives to politicians (see,
among
others, Barro, 1973, Ferejohn, 1986, Besley
and Case,
1995, Persson, Roland and Tabellini, 1997, Niskanen, 1971, Shleifer and Vishny, 1993, Acemoglu,
Robinson and Verdier, 2004, Padro-i-Miquel, 2007), there
investigating the selection of politicians, most notably.
are not observed
by voters or outside
is
parties.
also a small (but growing) literature
Banks and Sundaram
The main
Keane, and Merlo (2005), and Besley (2005).
design of political institutions in these papers
is
(1998), Diermeier,
challenge facing the society and the
that the ability and motivations of politicians
Wliile such information-related selection issues
are undoubtedly important, our paper focuses on the difficulties in ensuring that the "right"
government
is
selected even
these literatures,
when information
we emphasize the importance
is
perfect
and common.
Also differently from
of institutional flexibihty in the face of shocks.
Besley and Coate (1997, 1998), Osborne and Slivinski (1996), Bueno de Mesquita et
(2003), CaselU
and Morelli (2004), Messner and Polborn
(2004), Mattozzi
and Besley and Kudamatsu (2009) provide alternative and complementary
ernments/politicians". For example,
Bueno de Mesquita
et al.
al.
and Merlo (2006),
"theories of
bad gov-
(2003) emphasize the composition
of the "selectorate," the group of players that can select governments, as an important factor in
leading to inefficient governments and policies.
CaseUi and Morelli (2004) suggest that voters
might be unwilling to replace the corrupt incumbent by a challenger
whom
they expect to be
equally corrupt. Mattozzi and Merlo (2006) argue that
opportunity costs of entering
politics.
Also notable
is
more competent
politicians
have higher
the recent work by Besley and
Kudamatsu
which
(2009), which relates the success of autocratic regimes to their ability to select politicians,
is
in turn related to the composition of the "selectorate" (as defined
by Bueno de Mesquita
2003). However, these papers do not develop the potential persistence in
bad governments
et
al.,
result-
ing from dynamics of government formation and do not focus on the importance of institutional
We
fiexibilit}'.
are also not aware of other papers providing a comparison of different political
regimes in terms of the selection of politicians under nonstochastic and stochastic conditions.
McKelvey and Reizman (1992) suggest that
seniority rules in the Senate
create an endogenous incumbency advantage, and current
incentive to introduce such seniority rules.
Our
members
of these bodies will have an
results are also related to recent
persistence of bad governments and inefficient institutions, including
(2008), Acemoglu, Ticchi,
More
and the House
work on the
Acemoglu and Robinson
and Vindigni (2007), and Egorov and Sonin (2004).
work are
closely related to our
prior analyses of
dynamic
political equilibria in the
context of club formation as in Roberts (1991) and Barbera, Maschler, and Shalev (2001), as
well as
dynamic analyses of choice
Acemoglu and Robinson
and Lagunoff
framework
paper
is a
for
(2006).
Our
(2006),
political institutions as in
Barbera and Jackson (2004), Matthias and Polborn (2004),
recent work, Acemoglu, Egorov, and Sonin (2008), provides a general
the analysis of the dynamics of constitutions, coahtions and clubs.
The
current
from our previous work
in a
number
continuation of this hne of research.
of important dimensions.
It differs
on the substantive questions concerning the
First, the focus here is
relationship between different political institutions
ments, which
and equilibrium
of constitutions
is
new, relatively unexplored, and
and the
selection of politicians
our view) important.
(in
and govern-
Second, this paper
extends our previous work by allowing for stochastic shocks and enables us to investigate issues
of institutional flexibility. Third,
results cannot
The
it
involves for a structure of preferences for which our previous
be directly applied.^
rest of the
paper
is
"
'
•
organized as follows.
-
Section 2 introduces the model.
Section 3
introduces the concept of (Markov) political equilibrium, which allows a general and tractable
characterization of equilibria in
tliis
comparison of different regimes
in
5
extends the analysis to allow
class of
games. Section 4 provides our main results on the
terms of selection of governments and politicians.
for stochastic
changes
in the
Section
competences of the members of
the society and presents a comparison of different regimes in the presence of stochastic shocks.
"Acemoglu (2008)
channel
also emphasizes the potential benefits of
democracy
— because the alternative, oligarchy, creates entry barriers and
in
the long run, but through a different
sclerosis.
""In particular, the results in Acemoglu, Egorov and Sonin (2008) apply under a set of acyclicity conditions.
Such acyclicity does not hold in the current paper (see Appendix B). This makes the general characterization of
the structure of equilibria both more challenging and of some methodological interest.
Appendix
Section 6 concludes.
while Appendix
B
A
contains
of the proofs of the results stated in the text,
all
considers an extensive-form
game with
procedures, and shows the equivalence between the
and voting
explicitly specified proposal
Markov
perfect equilibria of this
game and
the (simpler) notion of pohtical equilibrium used in the text.
Model
2
We
consider a dynamic
represented by the set
X
as coalitions
^ C C
game
J and
in discrete time
indexed by
n < oo
individuals.
consists of
and denote the
set of coalitions
size greater
than
k
= maxGsg
G
€ Q,
\G\
<
/cj
and
\G\, so k is
k. It is
less
is
bound
for
is
We
refer to
1.
is
non-empty subsets of
also designate a subset of coalitions
some
integer
/c2.
or
fco)
all
groups of individuals of
To simpUfy the
<
discussion,
we
i.e.,
define
for
any
n/2.
ruled by one of the feasible governments G* € Q.
given as part of the description of the
game and G^
in equilibrium as a result of the political process described below.
any date
The population
0,1,2
the size of any feasible goverrmient:
natural to presume that k
In each period, the society
government G^
(for
/cq
than some other integer
the upper
We
C.
=
For example, the set of feasible governments could
as the set of feasible governments.
consist of aU groups of individuals of size
by
i
for
i
>
is
The government
The
initial
determined
in
power
at
affects three aspects of the society:
It influences collective utilities (for
example, by providing public goods or influencing
how
competently the government functions).
2.
It
determines individual
utilities
because of rents of being in
3.
It
(members of the government may
receive additional utility
office or corruption).
indirectly influences the future evolution of governments
by shaping the distribution of
pohtical power in the society (for example, by creating incumbency advantage in democracies or providing greater decision-making
ernment under alternative pohtical
We now describe each
is
modeled via
its
of these in turn.
power or veto
as
members
of the gov-
institutions).
The
influence of the government on coUective utilities
competence. In particular, at each date
designating the competence of each feasible government
r^ € E
rights to
t,
G
there exists a function
G G (at that date).
We
refer to
government G's competence, with the convention that higher values correspond to
greater competence.
In Section
4,
we
will
assume that each individual has a certain
7
level of
competence or
ability,
members. For now,
r*
and the competence
assumption
this additional
depends on time. This generality
particular, changes in the relative
government
of a
is
a function of the abihties of
Note
not necessary.
is
also that the function
introduced to allow for changes in the en^dronment (in
is
competences of
different individuals
and governments).
in
power
herself in the government.
More
Individual utilities are determined by the competence of the government that
and by whether the individual
at that date
specifically,
each individual
£
i
X
/3
£
in question
is
by
= E5^;:^;3('-M,
the discount factor and u^
(0, 1) is
is
at time r has discounted (expected) utility given
t/f
where
is
(1)
individual's stage payoff, given by
uH^^(G^^*GO=^«.(G'),
where
this
in the
throughout unless special emphasis
Assumption
1.
(2)
second equality we suppress dependence on F^, to simplify notation; we
assumptions on
for each
The function Wi
satisfies the following properties:
T and any G.H
G
wi{H).
(E
Q such
F^ > F^;
that
E
i
if
G
or
i
^
H
then Wi [G)
,
•
;
„.
2.foranyG,HeGandanyieG\H,Wi{G)>Wi{H).
equal,
1
of this assumption
is
The same
G
is
give higher stage payoff.
and H, and
conclusion also holds
governments or when she
only a
when
latter.
to a
member
more competent one
G
is
of
in
G
is
:
is
a
this conflict of interest
payoffs from governments that include
competence
levels of the
member
member
is
prefers
of either of these two
Therefore, this part of the
less
competent
of the former, but not of the
income from
interesting interactions in our
assumption strengthens the
always present; that
is,
We
when
first
part
individuals receive higher
them than from those that exclude them
two governments).
else
an individual
office or additional
The
all
individuals prefer to be in the government even
this does not benefit the rest of the society. Part 2 of the
and imposes that
if
which an individual may prefer a
being in government due to higher salaries or corruption.
of the
not a
This simply captures the presence of rents from holding
setup result from the "conflict of interest"
imphes that
In particular,
(and not of H).
when she
is
It
:
more competent than H, then she
the individual
assumption implies that the only situation
government
;-
a relatively mild restriction on payoffs.
more competent governments
belongs to both governments
G.
>
.
;
,
Part
do
Throughout, we impose the following
necessary.
is
will
if,;.
1
i
its
impose both parts of
this
(regardless
assumption
throughout.
important to note that Assumption
It is
of the government, care about a one-dimensional
1
imphes that
voters,
all
government competence;
who
are not part
this featm-e simplifies
the analysis considerably. Nevertheless, the tractability of our framework makes
it
possible to
enrich this en\'ironment by allowing other sources of disagreement or conflict of interest
and we return to
voters,
We
this issue in the Conclusion.
next provide an example that makes some of these notions slightly more concrete.
Example
produced
2 Suppose that the competence of government G, Fq,
economy under
in the
Uj-
:
^R
from public good
being in
1
to
of
is
a strictly increasing function (for each
for individual
and ly
office,
Assumption
1.
is
In addition,
if
6,;
be a member of the government, even
Finally, the
ever consent of
if
this
(for
each
G
for
each
z,
i
froiri
e X, then (3) satisfies part
then each individual prefers
government has a very low
some current government members
necessary for change.
is
and government members) who can,
government by specifying the
G
>
obtains
level of
competence,
power influences the determination of future go^^ernments when-
in
of indi\'iduals (regular citizens
government
A'. If 6,
i
also satisfied.
1 is
government
G J) corresponding to the utility
i
sufHciently large for each
is
good
of public
(3)
a measure of the rents that individual
is
6,
the indicator of event
thus part 2 of Assumption
in
i,
amount
= vaTG) + b,l^^eG],
w^[G)
R
the
is
government G, and
feasible
':
where
among
winning
set of
G G). This
coalitions^
We
represent the set
collect! veh',
VVc which
is
induce a change
a function of current
an economical way of summarizing the relevant infor-
is
mation, since the set of winning coalitions
precisely the subsets of the society that are able to
is
We
force (or to block) a change in government.
only impose a minimal
amount
of structure on
the set of winning coalitions.
Assumption
2 For any feasible government
Wg = {X
e C
:
maximal
The
size of the
restrictions
government and n
imposed
in
government can be instituted
{ruG total votes) and
government
if it
is
<
,
V\)g ?s given by
Ig
<
\G\
<
k
G|
>
Ig)
,
< niG < n —
2 are intuitive. In particular,
receives a sufficient
receives support
{Ig of the current
Q
k (recall that k
is
the
the size of the society).
Assumption
if it
a
> mc and \X H
\X\
where Ig and tug are integers satisfying
G
number
they state that a
new
of votes from the entire society
from some subset of the members of the current
government members need to support such a change).
This
definition allows Ig to be any
2
that
is
it
and
One
|G|.
special feature of
Assumption
does not relate the number of veto players in the current government, Ic, to the
nmnber
total
number between
who wish
of individuals in the society
to
change the government, mo- This aspect
we return
of Assumption 2 can be relaxed without affecting our general characterization;
to a
discussion of this issue in the Conclusion.
Given
current
this notation, the case
members
where Iq
=
should be thought of as perfect democracy, where
and the case where
of the government have no special power,
extreme dictatorship, where unanimity among government members
Between these extremes are imperfect democracies
may
arise either
because there
is
some form
(or less strict
of (strong or weak)
is
Ig
=
\G\
as
necessary for any change.
forms of dictatorships), which
incumbency advantage
in
democ-
racy or because current government (junta) members are able to block the introduction of a new-
government. Note also that we imposed some mild assumptions on niG- In particular,
k individuals
is
insufficient for a
change to take place.
This ensures that a
cannot take power without any support from other individuals
mum
TTT-G
size of the
< n—
government, so the
< n/2 and mc =
[{n
among them.
+
1) /2J
In addition to Assumptions
which ensures that
the notation and
different
is
(i.e.,
1
majority
Assumption
3 For any
2,
we
also
>
£ Q).
we
of the
levels
and any G,
different
competences. This assumption simplifies
would restore
H
when
impose the following genericity assumption,
<E
Q such
it
were not satisfied
for a society,
it.
that
Political Equilibria in Nonstochastic
In this section,
G
i
members
rule).'^
loss of generality, since if
any small perturbation of competence
3
government must have no more than k members), and
governments have
without much
government
For example, these requirements are naturally met
and
than
denotes the maxi-
k individuals are sufficient to implement a change provided that Ig
current government are
k
rival
(recall that k
rival
less
G ^ H F^ ^
,
Fj^.
Environments
focus on nonstochastic environments, where F'
= F
(or
F^ = Tg
for all
For these environments, we introduce our equilibrium concept, (Markov) political
equilibrium,
and show that
equilibria have a simple recursive characterization.^
the more general stochastic environments in Section
Recall also that [x\ denotes the integer part of a real
We
return to
5.
number
x.
Throughout, we refer to this equilibrium concept as "political equilibrium'' or simply as "equilibrium".
do not use the acronym MPE, which will be used for Markov perfect equilibrium in Appendix B.
10
We
Political Equilibrium
3.1
Our equilibrium concept, (Markov)
new government
current go\'ernment to a
and
coalition will take place;
The
equiUbrium).
from the
onl)' transitions
that maximize the discounted utility of a winning
no such transition
if
imposes that
political equilibrium,
government
exists, the current
"Markov"
qualifier
added since
be stable
will
this definition
im-
(i.e., it
will persist in
plicitly
imposes that transitions from the current to a new government depend on the current
government
—not on the entire
To introduce
(j)
^
G
:
would emerge
in period
each indi\'idual
J
G
i
+
t
G
government
Given
we can
cf),
in
(G\(P)=
G
from
+
w, (G)
BVi
& G, individual
i
let
us
first
define the transition rule
power
at
time
t
to the
government that
write the discounted utility implied by (1) for
G
starting from current government
V,
Intuitivel}', starting
feasible
1.^
-
concept more formally,
this equilibrium
which maps each
G,
—
history.
is
(G)
i4>
for all
4>)
!
J
e
G G recursively as
G
Vj
{G
given
cb),
\
e G.
bj'
(4)
receives a current payoff of
u'j
Then
(G).
(p
(uniquely) determines next period's government 4){G), and thus the continuation value of this
individual, discounted to the current period,
A
government
G is
any (possibly
for
if
Ha = He then Ha = Hb =
(4),
if
infinite)
(ji
C G
chain Hi, H2,
if
Given
mapping
stable given
{4>{G)
is (3Vi
4>).
\
=
(G)
G. In addition, we say that
such that
i^fc+i
€
{Hj^),
<p
(p
acyclic
is
and any a <
b
<
c,
He-
the next definition introduces the notion of a political equilibrium, which will be
represented by the mapping d provided that two conditions are met.
Definition
G
G G,
1
A mapping
(p
—^
G
:
constitutes a (Markov) political equilibrium if for
G
the following two conditions are satisfied:
who prefer 4>{G)
either the set of players
(i)
winning
coalition,
5 =
i.e.,
£
{i
X
:
Vj {4>{G)
\
and
!5
n G| >
(ii)
.
tion
to
{iGl:
Wg
any
there
Ic); or else,
is
and
(p{G)
V,
no
(H\(P)>
cp
=
(G)
to
staying
V, [^
(alternatively, for
(G)
I
in
terms of discounted
(in
> Vi{G
4>)
G
(p)}
\
Wq,
forms a
utility)
(or equivalently \S\
> mo
G;
H
government
alternative
G
to
G
^)} G
any alternative
G
G
permanently,
Wg
H
,
and
S'Ij
that
there
i.e.,
=
either \S'^\
{i
preferred
is
G
I
< mc-
:
is
V,
or
{H
\S'fj
no
\
D
<P)
G\
both
H
such
>
w, (G) /
<
Ic, or
a
to
that
(1
transiS'^q
-
\S'lf\
/?)}
=
G
< mo,
or\S'lfnG\<lc).
"in principle,
(p
could be set-valued, mapping from Q into V (Q) (the power set of G), but our analysis be3, its image is always a singleton (i.e., it is a "function" rather than a
low shows that, thanks to Assumption
"correspondence," and also by implication,
it is
uniquely defined).
11
We
impose
this feature to simplify the notation.
This definition states that a mapping
equihbrium")
maps the
if it
to alternative
(G) that (unless
4>
members
not to be blocked). Note that in part
(so as
S=
as
not a winning coalition, then
(G)
cp
{i
£
I
= G
Vi{(l)
:
(G)
\
>
cf))
and thus Vi{G
\
Wi (G) /
=
4>)
to
(p
G
(G) and to stajdng in
number
H
H that
stage a
move
We
is
to
members. The
preferred to a transition to
H can be blocked.
is
section.
possible disadvantage
transitions are
is
made and
latter condition
is
The advantage
that
G
(G) but not to staying in
(j)
of this definition
is
does not explicitly specify
it
for
since
this set
,
if
Part
P)-
(ii)
of
that would have been a
1
and a
if
forever,
sufficient
there exists a
then at each
as a primitive
Markov
how
offers for different
MPE
it
simple and economical.
it is
'^
types of
Appendix B, we
In
and when transitions can
perfect equilibria
high discount factors,
and use
that
the exact sequences of events at each stage.
show that
(1
S could
^"^
take place exphcitly, and characterize the
We
—
imposed, since
specify the sequences in which offers are made, voting takes place,
environment.
/?) }
the set
preferred both to a transition
take the definition of political equilibrium given in Definition
and the next
in this
H that
there should be no
is,
(i),
forever by a sufficient majority of the population
of current government
subset
A
transition; that
—
(1
Wi (G) /
the definition requires that there does not exist another alternative
"more preferable"
coincides
it
G by a sufRcient majority of the population and a sufficient number
have alternatively been written
is
G
current government
with G) must be preferred to
of current government
constitutes a pohtical equilibrium ("is a political
cf)
(MPE)
extended
of this
coincide with our definition of
pohtical equilibria.
Genercd Characterization
3.2
We now
prove the existence and provide a characterization of political equilibria.
a recursive characterization of the mapping
elements of the set Q as {Gi, G2,
enumeration, Gi
is
described in Definition
4>
Gigj} such that Tq^
...,
Now, suppose that
for
some
q
>
3, this
1,
enumeration
we have defined
Mg = {j:l<j<q,{iel:w, (G,)
in (4),
'
this set
depends simply on stage payoffs
which are "endogenous" objects. This
The
explicit
>
game form
in
Appendix B
set
cp
is
well defined
for all
w^ (G,)} G
in (2), not
with
is
y.
With
this
the least competent
and unique.
Gj with
Wg,, and
start
Let us enumerate the
> ^Gy whenever x <
the most competent ("best") government, while Gigi
government. In view of Assumption
Note that
1.
We
j
<
Define the set
q.
(G,)
= G,}
on the discounted
(5)
.
utilities
defined
can thus be computed easily from the primitives
clarifies this further.
"'In this regard, our equilibrium concept
is
similar to the concept of
(1999).
12
Markov voting equilibrium
in
Roberts
of the
model
each
(for
Given
q).
mapping
this set, let the
''
rhtr
\
/
'^9
t
'^mm{je.M,}
•
Since the set .Mg
valued.
well defined, the
is
Theorems
and
1
2 next
mapping
show
be
ifMq =
0;
yWg
fe3.
II
fi
also well defined,
4> is
•,
and by construction
it is
that, for sufficiently high discount factors, this
,
single
mapping
constitutes the unique acyclic political equilibrium and that, under additional mild conditions,
it is
also the
Theorem
unique political equilibrium (even considering possible cyclic equilibria).
1 Suppose that Assumptions 1-3 hold and
there exists
/3q
<
let
such that for any discount factor
1
j3
cp
>
:
Q
^
Q
be as defined in (6).
4>
is
the unique acyclic political
j3q,
Then
equilibrium.
Proof. See Appendix A.
Let us
now
Recall that
Gi, since
Gi
all
illustrate the intuition for
is
the most competent
members
(from Assumption
to
G"
some
)
government.
of the population that are not in
move
alternative
to G'
.
may
HoweA'er, G'
of these
may
(p
constitutes a political equilibrium.
government G" E
But
G-
Gi
will prefer
= Gj
if
itself,
it
to
cp
(Gi )
=
any other G' G Q
and
it
may
we can apply
eventuallj'
this reasoning
Next suppose we start
applies.
G'
we must have
not be a winning coalition in
will
in this case,
power. The same argument apphes
in
that
It is clear
not persist
instead of G', and thus the conclusion (i>[Gi)
with government G2
One
"best"
mapping
the
Assumption 2 then ensures that there
1).
i&voi of a permanent
lead to
(
why
any one of G3, G4
is
G\g\-
eventually lead to Gi, thus for sufhciently high discount factors, a sufficient
majority of the population
may
support a transition to such a G' in order to eventuaUy reach
Gi. However, discounting also implies that in this case, a sufficient majority would also prefer
a direct transition to Gi to this dynamic path (recall part
choice for the society
sufficiently
many
is
between Gi and G2- In
supporters, that
coalition within G2, or
is,
more formally
if
this
(ii)
of Definition 1). So the relevant
comparison, Gi
will
the set of individuals preferring
be preferred
Gi
to
Go
is
if it
has
a winning
if
{iGl:u',(Gi) >k;, (G2)} e Wgj.
If this is
the case, (b{G2)
defined in (6) stipulates.
=
G\\ otherwise, d{G2)
Now
let
Gi
to G3.^' However,
G2. This
us start from government G3.
the choice between Gi, G2, and G3. To
prefers
=
move
to Gj,
whether the society
it
is
exactly
We
suffices that a
will transition to
what the function
4)
then only need to consider
winning coahtion within G3
G2 depends on
the stability
some winning coalition also prefers G2 to G3, then Gi should still be chosen over G2, because only members
Gj who do not belong to Gi prefer Go to Gi, and Assumption 2 ensures that those preferring Gi over G2
(starting in G3) also form a winning coalition. Then a transition to G2 is ruled out by part (ii) of Definition 1.
'"If
of
13
we may have a
of G2. In particular,
necessity,
to
(/>
{G2)
G3
move
=
Then a
G\.
to G2- In particular, there
may
coalition that prefers such a transition), even
move
is
not a stable government, which, by
G2
may be
a permanent transition
will lead to
non-desirable for
some
of those
who
winning coalition in Gs that prefers to stay
exist a
Gi (and
rather than transitioning permanently to
that would have preferred a permanent
G2
transition to
However, this sequence
in the next period.
Gi
prefer to
in
impUes that
situation in which
as a consequence, there
though there
also exists a
to G2. Writing this
more
no winning
is
winning coahtion in G3
explicitly,
we may have
that
{iGT:u.',(G2)>u.,(G3)}e Wg3,
but
{ieI:Wi{Gi)>w,{Gi)}iWG,.
If so,
the transition from
G3
to
G2 may be blocked with the
anticipation that
it
will lead to
which does not receive the support of a winning coalition within G3. This reasoning
Gi
illustrates
that for a transition to take place, not only should the target government be preferred to the
current one by a winning coahtion (starting from the current government), but also that the
target government should be "stable,"
will persist in
4>[G')
=
G'
.
This
exactly the requirement in (6).
is
Theorem
mapping
(p
and thus
equilibrium (will be stable)
if
there does not exist another stable government
In this light, the intuition for the
G
i.e.,
for
1
is
that a government
and the
receiving support from a winning coalition (a sufficient majority of the population
number
required
Theorem
1
of current
states that
members
in (6)
cp
We
not rule out cyclic equilibria.
Example
11).
is
-,
of government).
'
,,
'
the unique acyclic political equilibrium. However,
provide an example of a cyclic equilibrium in Appendix
Cyclic equilibria are unintuitive and "fragile".
We
it
B
does
(see
next show that they can also
be ruled out under a variety of relatively weak assumptions. The next theorem thus strengthens
Theorem
1
so that
(p
in (6)
is
the unique political equilibrium (among both cyclic and acychc
ones).
Theorem
light of
2 The mapping
Theorem
1,
any
<f>
defined in (6)
=
=
For any
G
2.
For any
G GQ. Ig>1-
3.
For any
collection of different feasible
i
,
\G\
£ I, we have Wi
the unique political equilibrium (and hence in the
political equilibrium is acyclic) if
1.
& Q
is
(i/i)
k, Iq
^
\2_]
I
and tuq
=
any of
m for som.e
governments Hi,
^^ (-^p)) /?•
14
k,
...
the following conditions holds:
and m.
I
,
Hq G Q
(for q
>
2)
and for
all
>
4-9
e
max{,gj
where
\g\,
„„rf
=
B
G,Hee; zeGnH}
{w')
rmu,^,^j ,„,^g,h^Q: i&G\H}
(G) -
u',
{^dG) - w,{H)] and
=
e
(iJ)}.
Proof. See Appendix A.
This theorem states four relatively mild conditions under which there are no cyclic equilibria
making
(thus
size,
/c,
unique equihbrium). First,
in (6) the
same degree
the
number
(p
of
incumbency advantage,
of total votes for change,
tti,
then
all
and the same threshold
/,
must be
equilibria
member
of the current
two results implj' that
there
is
government
Example
3
and
on preferences that also
Finallj',
it).^"^
does not care
much about
third part of the
under a very mild assumption. In
is
a slight strengthening of
did not hold, a small perturbation of
if it
In particular, this condition states that
being in govermnent (greater than
utility fi-om
6*)
if
and
the composition of the rest of the government (the difference in her
between any two governments including her
acyclic. In
The
B).
the fourth part of the theorem provides a condition
rules out cyclic equilibria.
each individual receives sufficiently high
4
political equilibria
(meaning that
also holds generically
payoff functions would restore
if
Appendix
11 in
the requirement on payoffs imposed in this part of the theorem
Assumption
be
some governments,
no incumbency advantage and either the degree of incumbency advantage or the vote
theorem proves that there are no acyclic
utility
in (6) is the
new government. These
starting from
if
(p
required
we always need the consent
for a transition to a
cyclic equilibria are only possible
threshold differs across governments (see
fact,
if
for the
and thus
acyclic
unique political equilibrium. Second, the same conclusion applies
of at least one
governments have the same
feasible
if all
Appendix B, we show (Example
neither of the four sufficient conditions in
11)
is
always
how a
Theorem
less
than
e),
then
all
equilibria
cyclic political equilibrium
is
must
possible
2 holds.
Characterization of Nonstochcistic Transitions
In this section,
we compare
ernments with high
different political regimes in
levels of
competence.
important interactions, we assume that
where k < n/2. More formally,
let
—
C^\
In addition,
simplify the exposition and focus
feasible
governments have the same
on the more
size,
fc
G N,
us define
&'
Then, Q
all
To
terms of their abihty to select gov-
= {Y eC:
we assume
\Y\
that for any
=
k].
G
S
C;,
i^j
=
/
g
N and mo =
m
e N, so
that the set of winning coalitions can be simply expressed as
Wg =
'"This requirement
is
exactly the
{A'
same
eC:\X\>m
as
Assumption
extensive form game.
15
and \X n G| >
3'
we impose
in
/}
(7)
,
Appendix B
in
the analysis of the
where
0<l<k<m<n —
k.
This specification impUes that given n,
corresponds to an inverse measure of democracy.
and there
is
Assumption
2'
We
then
0,
all
(6) is
of
1
government
of the
is
=
C^
given by
Theorem
and that there
,
individuals have equal weight
if
>
/
0,
necessary for a change, thus we have
Assumption
2,
exist integers
m
and
I
such that the set
(7).
2'
ensures that the acyclic political equilibrium
we
the unique equilibrium; naturally,
the rest of the analysis.
is
/
thus have strengthened Assumption 2 to the following.
have that Q
of winning coalitions
In view of part
members
We
an imperfect democracy.
given by
=
I
and m, the number
no incumbency advantage, thus we have a perfect democracy. In contrast,
the consent of some of the
(f)
If
k,
will focus
on
this equilibrium
still
mapping
(p
can be
ranked according to their
level of
In addition, given this additional structure, the
wTitten in a simpler form. Recall that governments are
throughout
competence, so that Gi denotes the most competent government. Then we have:
Mq = {j:l<j<
n
\Gj
q,
>
Gg\
I,
and
{Gj)
-
Gj}
(8)
,
and, as before,
iiMq =
Gg
'"'
Gr^;„(iaM.\
1
Naturally, the
,,
mapping
,.
this
0;
¥=0.
us also define
let
V={Geg:cP[G)=G]
governments (the fixed points of mapping
government
We now
Ma
again well defined and unique. Finally,
.._
as the set of stable
and
is
if
will persist forever if
the
it is
in^'estigate the structure of stable
initial
<^).
If
G
G P, then
government of the
governments and how
it
we assume that Assumptions
of the propositions to economize
Our
to
first
show that
and
1, 2'
highl)' incompetent/inefficient
governments
1
The
IfG^H
£
V
governments
governments.
stable, regardless of
Proposition
1.
this
qualifiers to
any
1).
It
then uses this result
democracy ensures the emergence of the best (most competent) government,
but any departure fi'om perfect democracy destro^'s
initial
Throughout
on space.
proposition provides an important technical result (part
perfect
G,
changes as a function of
and we do not add these
3 hold,
=
society.
the underlying political institutions, in particular, the degree of democracy.
section,
(l){G)
may
I,
have at most
I
their
governments
then
—
also shows that
how low
set of stable feasible
and \G D H\ >
It
this result
1
G =
H.
extreme dictatorship makes
competences
V
may
all
be.
satisfies the following properties.
hi other words, any two distinct stable
common members.
16
and enables the emergence of
2.
Suppose that
=
I
so that the society
0,
Then
a perfect democracy.
is
V =
In
{Gj}.
other words, starting from any initial government, the society will transition to the inost
competent government.
3.
Suppose
>
I
1,
so that the society
is
an imperfect democracy or
there are at least two stable governments,
governments may
4-
Suppose
I
=
government
\V\
i.e.,
>
Moreover, the
2.
Then
a dictatorship.
least
competent
be stable.
k, so that the society is
an extreme dictatorship. Then
V—
Q, so
any feasible
is stable.
Proof. See Appendix A.
Proposition
1
shows the fundamental contrast between perfect democracj', wliere there
incumbency advantage, and other pohtical
members
"insiders" (current
will necessarily
exist at least
With any
emerge.
With
no
which provide some additional power to
institutions,
of the government).
is
perfect democracy, the best
government
deviation from perfect democracy, there will necessarily
one other stable government (by definition
less
competent than the
best),
even the worst government might be stable. The next example supplements Example
1
and
from
the Introduction by showing a richer environment in which the least competent government
is
stable.
Example
3 Suppose
n =
9,
/c
=
3,
=
/
support from a simple majority of the
government. Suppose
Assume
{1, 2,
3}
includes
as
I =
{1, 2,
also that ^{^^^^2,^2]
is
1
—
.
.
.
^000
,
9},
—
1,
and
m=
is
is
lOOii
—
5
and
10*2
is
9 prefer the latter.
more competent must include
support from any of the members of {4,5,6}.
—
''s
(for zi
<
therefore stable.
?2
<
Any
of the current
(3) in
Example
2.
This imphes that
'a)-
other government that
However, government {4,5,6}
1
or 2 or 3,
and therefore
means that any such
Now, proceeding
government other than {1,2,3} and {4,5,6} that contains
unstable.
member
and that stage payoffs are given by
or will immediately transit to {1,2,3}, which
is
change in government requires
unstable. For example, the government {2,5,9} will transit to {1,2,3},
ah individuals except
government that
so that a
including at least one
societ)',
the most competent government, and
or 2 or 3
5,
Consequently, government {7,8,9}, which
is
stable:
any
either {1,2,3}
transition will not receive
inductiveh',
at least
is
is
we
find that
any
one individual 1,2,...,
the least competent government,
is
stable.
Proposition
1
establishes that with any regime other than perfect democracy, there will
necessarily exist stable inefficient/incompetent governments
17
and these may
in fact
have quite
low
le^'els
of competences.
incompetent governments
will
when highly
does not, howe\'er, provide a characterization of
It
be
stable.
We next provide a systematic answer to this question focusing on societies with large numbers
of individuals
(i.e.,
n
Before doing
large).
we introduce an assumption that
so,
the third part of the next proposition and in later results.
sometimes suppose that each individual
will
K+ and
by 7j 6
abilities of its
>
Ij-
e
I
members. This
is
be used
what follows we
more formally stated
a strictly increasing function of the
is
in the next assumption.
G G, and individuals i.j £
I
are such that
E G, j ^ G, and
i
Then Tq > ^ {G\{i])u{j]
The canonical form
of the
competence function consistent with Assumption 4
is
ra = E,,^7.
though
for
most of our
Assumption 4
This ranking
is
analysis,
7,;
is strict,
> 7 whenever
The next
k and
I),
we do not need
useful because
since
i
<
it
to
(10)
impose
this specific functional form.
enables us to rank individuals in terms of their "abilities"
Assumptions 3 and 4 together imply that
WTien we impose Assumption
that
in
has a level of ability (or competence) given
that the competence of the government
Assumption 4 Suppose G
li
i
In particular, in
will
4,
we
also
7^
^
7,
whenever
enumerate individuals according to their
i
i^ j.
abilities, so
j-
proposition shows that for societies above a certain threshold of size (as a function of
there always exist stable governments that contain no
and no member
of
any group of certain prespecified
sizes (thus,
member
of the ideal
government
no member of groups that would
generate a range of potentially high competence governments). Then, under Assumption
4, it
extends this result, providing a bound on the percentile of the ability distribution such that
there exist stable governments that do not include any individuals with competences above this
percentile.
Proposition 2 Suppose
1.
I
>
(and as
1
before, that
Assumptions
If
G
g
D
that contains no
ernment Gi
Take any x
2'
,
and 3
hold).
^
then there exists a stable government
2.
1,
eN.
^
'
If
.
18
members
of the ideal gov-
X
with \X\
<
no member of set
X
then for any set of individuals
such that
XnG=
(so
x, there exists a stable
G
G
T>
belongs to G).
Suppose in addition that Assumption 4 holds and
3.
government
let
'
'
1
I
Then
there exists a stable
government
+
G
€.
(13)
ik-iy.
(k
I) (,_jji(j._,).
does not include any of the [pn\ highest
T^ that
ability individuals.
Pi'oof. See
Appendix A.
Let us provide the intuition for part
2
members
G
imphes that
and therefore
as implied
of
is
stable, that
H nGi
by
(9).
G
Gi (such
exists, since
G
is,
n >
2/c
when
I
—
RecaU that Gi
I.
by assumption). In this
e V. The reason
is
that
if cp
(G)
=
Intuitively,
if
Z
=
1,
H ^ G,
member
will
stable, thus there are
Example
as in
G
will take us to
but
it
no threats that further rounds of transitions
in the Introduction,
1
G
itself
an unstable government.
becomes
the
do not
Th > Fc,
=
Gi,
member
consent to (support) a transition to
Gi, which will also receive the support of the population at large. She can do
is
then
then once the cm-rent government contains a
this
is
case, Proposition
contains at least one element by construction of G. But then d [H)
most competent government Gi,
of the
of Proposition 2
G be the most competent government among those that
most competent government. Let
include
1
stable,
harm
will
so,
because Gi
her.
But then,
because any reform away from
Part 2 of the proposition has a similar intuition,
states the stronger result that one can choose
any subset of the society with
size
not
exceeding the threshold defined in (12) such that there exist stable governments that do not
include any
member
governments).^^
of this subset (which
1;
it
shows that there
its
maximum
and
exist stable
at
/
2 of this proposition
and
=
k/2,
2
under Assumption
4,
and 4 of Proposition
also parts 3
governments that do not include a certain fraction of the
Interestingly, this fraction, given in (13),
highest abihty individuals.
reaching
1
taken to include several of the most competent
which follows immediately from part
Finally, part 3,
further strengthens both parts
may be
i.e.,
for
non-monotonic
is
in
/,
an "intermediate democracy". This partly anticipates
the results pertaining to the relative successes of different regimes in selecting more competent
governments, which we discuss in the next proposition.
We
next turn to a more systematic discussion of this question, that
with a greater degree of democracy "perform better". For
sumption 4 holds. The next example, which
^''Note that the upper
bound on
A' in
is
similar to
Part 2 of Proposition 2
tliis
Example
is
analysis,
1
we suppose that As-
from the Introduction, shows
0(x), meaning that increasing x does not
require an exponential increase in the size of population n for Proposition 2 to hold.
19
of whether societies
is,
same government, the long-run equihbrium government may be worse
that, starting with the
when
Example
=
{{4,5,6})
competent than
under
I
=
9,
=
/
=
/c
we
3,
there,
are not in a perfect democracy).
Z
=
Note that
either
fc
=
if
or
1
/
=
=
/c
and suppose
5),
is
unstable,
is
is
more
an example where the long-run equilibrium government
is
worse
=
/
democratic"
2 (with "less
would be stable again.
6}
the structure of stable governments
(Note that in this proposition, and
examples that follow,
in the
relatively straightforward.
is
denote the indices of
a, 6 or c
individuals, with our ranking that lower-ranked individuals have higher ability; thus 7^
whenever a <
If k
G
2.
=
=
and
I
when a
I
=
and
2,
=
1
is
I
I
=
I
Assumptions
=
(perfect democracy), in which case
k (extreme dictatorship), in which case
=
(perfect democracy), then
even and 6 [G]
is
odd and
any
G
£ Q.
6
=
=
{a
a
,
a-fl. If k
=
Proposition
3,
governments that
if
G—
(p
1}
2
and
I
=
.
.
is
will
—
added to the
I,
> To- However,
a} or
initial
these three cases,
(f)
(G)
G
(G)
{1} for any
G
e Q.
=
=G
{a
if
—
2
a}
I,
and only
— G
for
v
:
-
=
£ Q. If k
2 (extreme dictatorship), then (j){G)
be further exploited in the next section.
—
2
with rare exceptions, the quality of the
{a
we have
(p
=
for any
for any
odd; in particular,
and
/
=
1,
initial
initial
government
1}, so that at best
is
government. This means that,
will
improve to some degree,
G=
i.e.,
<
b,
the next highest ability individual
is
this increase is'generally limited;
0(G) — {a,a+
In the case of an iinperfect
the competence of the stable government
determined by the more able of the two members of the
=
b,
{Gi]
though simple, provides an important insight about the structure of stable
democrac}', highlighted here with k
typically r^(Q)
=G
{1. 2}
<
=
{G)
(j)
[G)
4>
= Gi =
(G)
{a, b) with a
when a
-t-
Proof. See Appendix A.
(p{G)
7;,
3 and 4 hold.
2' ,
1.
(imperfect democracy), then
a
if
that
then either
I.
G Q, or
If k
>
b.)
Proposition 3 Suppose
1.
and
stable,
Since {1,4,5}
3, {4, 5,
2,
=
2 instead. In that case, {4,5,6}
{1,4,5}; thus there will be a transition to {1,4,5}.
{4, 5, 6}, this is
m
and
1,
government {4,5,6}
(with "more democratic" institutions) than under
1
institutions).
When
=
As we showed
{4,5,6}.
is
Suppose, however, that
will therefore persist.
(j)
(as long as
4 Take the setup from Example 3 (n
that the initial government
and
more democratic
poUtical institutions are
when
{a,b} with a
government instead of the lower abihty member. Therefore, summarizing
we can say that with a
an extreme dictatorship, there
will
perfect democracy, the best government will arise; with
be no improvement
in the initial governxnent;
and with an
imperfect democracy, there will be some Umited improvements in the qualit}' of the government.
20
W'Tien
>
fc
the structure of stable governments
3,
more complex, though we can
is
still
de-
velop a number of results and insights about the structure of such governments. Naturally, the
extremes with
ward.
If
=
/
=
/
and the
1
democracy) and
(perfect
initial
government
members ranked above a —
that
Therefore, again with
ment
G=
fact,
d
a
<?C
{a,
b,
always a
is
6,
where a <
c},
member
when
<
b
c, (p
(G)
—
1
c,
of the stable
then we can show
government 0(G),
government
of the stable
G; instead
7^
In this case,
2.
when
=
I
{a,
6,
d},
2 allows the initial
initial
where d <
in the
where
in the case
(G)}^
cp
govern-
can be shown that
it
improvement
government (contrasting with the incremental improvements
speaking, the presence of two veto players
=
(G)
(p
possible. This implies a potentially very large
is
<
b
only incremental improvements in the quality of the
1,
where a +
c},
{a,
become members
This ceases to be the case
are possible.
whenever
=
/
of G, a,
(extreme dictatorship) are again straightfor-
3
G=
is
2 will never
and the most competent member
=
/
and
in
quahty of the
=
/
c
1).
Loosely
government to import
very high ability individuals without compromising stability. The next example illustrates this
feature,
which
at the root of the result highlighted in
is
Example
whereby greater democracy
4,
can lead to worse stable go\'ernments.
Example
5 Suppose
meaning that the
Then
of ability.
the
tliird
=
/c
member
take the case where
first
=
/
=
(G)
{100, 101. 102}, that
now
Suppose instead that
/
=
2.
in
terms
there will be improvements in the quahtj' of
is,
member (and
second highest ability member). More generally, recall
that only very limited improvements in the quality of the highest ability
that
{100, 101, 220},
of the government, but not in the quality of the highest ability
in this case, neither in the quality of the
in this case.
G=
Suppose
1.
government consists of individuals ranked 100, 101, and 220
initial
(pi-i
and
3,
Then
it
can be shown that
member
are possible
4)1^2 (G^) =" i^-' ^OO-
10^}) so
the stable government includes the most able individual in the society. Naturally
if
the
gaps in ability at the top of the distribution are larger, impljdng that highest ability individuals
ha\'e a disproportionate effect
on government competence,
this feature
becomes particularly
valuable.
The
following example extends the logic of
Example
under certain configurations, expected competence
is
5 to any distribution
higher under
/
=
2
than
and shows how,
I
=
1
under any
distribution over initial (feasible) governments.
Example
with
is
is
support
full
'°More
c
6 Suppose k
(i.e.,
specifically,
a multiple of
a multiple of
multiple of
3,
3.
(p
—
government
=
a (any) probability distribution over initial governments
fix
with a positive probability of picking any
Moreover,
(G)
and
3,
(a
—
G =
for
{a,
b,
c),
where
any government
1, a,
a
+
1}
if
a
+
1
a
G —
is
<
6
<
c,
is
a multiple of
3.
21
stable
{a,b,c} with a
3,
<
and
and only
if
b
cp
initial feasible
<
(G)
c,
=
if
(p(G)
{a,
a
government).
= 6 — 1 = c — 2,
= {a — 2, a — 1, a}
+ 1, a + 2} if a + 2
a
and
if
a
is
a
Assume
that of players
ij,
.
.
,i„,
.
the
(where g
first q
<
a multiple of 3 and 3
is
g
< n —
are "smart," while the rest are "incompetent," so that governments that include at least
players
ij,
.
.
,
.
and
also
among
4>i-2
(G) (under
at least one of the players
/
=
2) is greater
than that of
(G) (under
(pi^i
/
=
1)
.
—both expectations
intuitive in
is
the structure of stable governments under the two political regimes. In particular,
one of
.
.
,
i^
can be showoi that the expected competence of the stable government
it
evaluated according to the probability distribution fbced at the outset. This
at least
ii,
those that do not are small relative to the gap between the two groups of
Then
governments.
one of
have very high competence relative to governments that do not. Moreover,
i^ will
competence among governments that include
differences in
3)
ii,
.
.
.
so do (pi-i (G)
,iq,
not include them either, whereas
(pi-2
and
But
4>i-2 {G)-
if
G
view of
G
includes
(f^i-i
(G) will
if
does not, then
are
{G) will include one with positive probability, since the
presence of two veto players will allow the incorporation of one of the "smart" players without
destabiUzing the government.
Fq
Conversely, suppose that
is
very high
if all its
otherwise. In that case, the expected competence of
I
=
2.
Indeed,
of the players
if
is
/
=
from
1,
players are from {ii,
(f>
.
.
.
{G) will be higher under
I
the society will end up with a competent government
{ii,
.
,
.
.
iq},
while
if
/
=
because there are
2,
now two
and very low
,iq],
=
than under
1
if
at least
veto players, there
needs to be at least two "smart" players for a competent government to form (though,
I
=
2,
this
is
Examples
when
not sufficient to guarantee the emergence of a competent government either).
5
and
6 illustrate a
in these examples with
governments near the
in the
one
/
=
2,
initial
number
there are
of important ideas.
more
government
With a
and
"veto" players,
this
lower degree of democracy,
makes a greater number of
stable, leading to a higher probability of
competence of some of the members of the
higher degree of democracy, in these examples
=
/
initial
1,
government.
improA^ement
In contrast, with a
fewer governments near the
when
initial
one
are stable, thus incremental improvements are
more
high ability individuals in the government
very important, regimes with a lower degree of
democracy perform better
I
=
0,
is
(naturally, this effect
always performs best).
is
likely.
some
isolated instances.
including a few
non-monotonic, and the perfect democracy,
Another important implication of these examples
situations in which regimes with lower degrees of democrac)'
to
Consequently,
This feature applies
for a
may perform
broad
is
that the
better are not confined
class of configurations
and
expected competences, evaluated by taking uniform or nonuniform distributions over
feasible goverrmients.
Nevertheless,
we
will see that in stochastic
for
initial
environments, there will be
a distinct advantage to political regimes with greater degrees of democracy, a phenomenon the
intuition of which will also be illustrated using
Examples
22
5
and
6.
Equilibria in Stochastic Environments
5
we introduce
In this section,
stochastic shocks to competences of different coahtions (or different
study the fiexibihty of different pohtical institutions in their abiUt}' to
indiA'iduals) in order to
adapt the nature and the composition of the government to changes in the underlying environnature and structure of "high competence" governments
ment. Changes
in the
changes
in the
economic, pohtical, or social environment, which
t^'pes of
government
establish strong links between the degree of
democracy and the
result
from
in turn require different
Our main
with newly emerging problems.
to deal
may
may
results in this section
flexibility to
adapt to changing
environments.
Changes
Q
^
in the
in the function
P^
:
R, which determines the competence associated with each feasible government. Formally,
we assume that
probability
6,
at
distribution of
Tq
in this stochastic
when
each
there
assume that there
that
environment are modeled succinctly by allowng changes
i,
with probability
a shock and
is
Tq may
—
5,
there
change.
at
time
as functions of T'q
t
environment
is
.
is
no change
in
Fp (Tq
The
\
T^^)
this for illustrating the
T'q
,
and with
we
that gives the conditional
characterization of pohtical equilibria
a very challenging task in general.
is
Tq from
In particular, following such a shock
exists a set of distribution functions
S is sufficiently small, there
and exploit
1
However, we wiU show'
a characterization very similar to that in
main substantive implications
Theorem
1
of stochastic shocks on the
selection of governments.
In the rest of this section,
to this stochastic
we
first
generalize our definition of (Markov) political equilibrium
environment and generalize Theorems
1
and
2 (for
(5
small).
We
sj'stematic characterization of political transitions in this stochastic environment
the links between the degree of democracy and institutional
and
illustrate
flexibility.
Stochastic Political Equilibria
5.1
The
structure of stochastic political equilibria
need to consider the implications
when
have assumed here, then
political equilibria
in Section 3.
when
on future transitions under a variety of
the likelihood of stochastic shocks,
must follow a
6, is
sufficiently small, as
we
logic similar to that given in Definition
low probabihties of changes in the environment). Appendix
the discount factor
the exphcit-form
complicated in general because individuals
Motivated by this reasoning, we introduce a simple definition of stochastic
political equilibria (with
that
is
of current transitions
scenarios. Nevertheless,
1
then provide a
game
is
high and stochastic shocks are sufficiently infrequent,
B shows
MPE
of
outlined there and our notions of (stochastic) political equilibrium are
again equivalent.
To introduce the notion
of (stochastic
Markov)
23
political equilibriurn, let us first consider a
mappings
set of
^'
5
'
4>{rc]
G defined
mappings are indexed by {To}
of
competences of
now
as in (6), but
separately for each {^G}Geg-
to emphasize this dependence. Essentially,
These
the configuration
if
governments given by {Tcj^cg applied forever, we would be in a
different
nonstochastic environment and 4'irc} would be the equilibrium transition rule, or simply the
shown by Theorems
pohtical equilibrium, as
cpirn] will
sufRciently small (see
is
Appendix
determine political transitions, and
a stable government. However,
then political transitions
will
when
When
B).
if
set of
the set of players
(i)
be determined
mappings
winning
sition
that
to
I
:
no
is
=
prefer
|i G
J
'
(pirc]
G
—^
Q
^"''^
^'-'
G
I
©{Fg})
>
[G)
(f^iYn]
staying
in
{zeI:F, (i^l^^r^j) >
V, [i?
is
m (G) /
(1
-
/?)}
will
remain
and {'^GJceg changes
cpt^,
i
new government, G" =
{^cjceg^
power
in
to
and unless
,
is
{rclGGC'
(pi-p,
</)rp,
as
\
{G)
=
(G).
i
separate mapping for each configuration
(a.
When
competences
the configuration of
is
G Q:
G
to
V, \<t'{ra] (<^)
:
G
^^ then
the transition rule
bj'
H
government
alternative
(pirn} (^^
S'h
jz e
5=
coalition, i.e.,
there
(ii)
who
~
a stochastic shock hits
have that for any
'"-'^
the current configuration
(pirc} (*^)
{rc}g,ggj be defined by the following two conditions.
given by {^G}Geg'
that while the current configuration
is
G, following this shock, there wall be a transition to a
Definition 2 Let the
idea underlying our definition for
determine equilibrium behavior, because the probability of a change in
4>ifQ} will still
competences
The
2.
environment with infrequent changes
this stochastic
{Tq]q^q,
and
1
G
I
(in
^{rc]]
G
Q
terms of discounted
>yi{G\
that
is
permanently,
'P{Tc])
e
Wg
both
there
(alternatively, |5^|
< mo,
to
no
is
Wg
e
Vi((i){r^}(G)|(^{rG})}
forms a
^ ^G',
\
preferred
i.e.,
utility)
and
or |5^
n
a
tran-
H
such
=
S'fj
G|
<
Ig,
or\S'ff\<mG, or\S'l^nG\<lG).
Then
a set of
mappings
(Pstq}
G ^> Q constitutes a (stochastic Markov)
'
for an environment with sufficiently infrequent changes
Gt+i
at time
t
(starting with
government Gt)
if there is
and only
if
if
political equilibrium
a transition to
government
=
{^g}g'C ~ {^G}Geg ""^ Gt+i
-"
^{fg} (Gt)-
-
:•
"
•
Therefore, a political equilibrium with sufficiently infrequent changes involves the
ical transitions (or
same
polit-
the stabihty of governments) as those implied by the mappings (pirc] defined
in (6), applied separately for each configuration
The next theorem provides the
{Tg}-
general characterization of stochastic political equilibria in
environments with sufficiently infrequent changes.
Theorem
3 Suppose that Assumptions 1-3 hold, and
by (6) applied separately for each configuration {Tg}-
any discount factor
j3
>
/3q,
(ptrn]
*•'
let (psrc)
Then
'
G
—•
G
be the
there exists
/Sg
*^^ unique acyclic political equilibrium.
24
<
mapping defined
1
such that for
Proof. See Appendix A.
The
same
all
intuition for this
'.
theorem
_
is
When
straightforward.
shocks are sufficiently infrequent, the
calculus that applied in the nonstochastic environment
determines preferences because
still
agents put most weight on the events that will happen before such a change. Consequently, a
stable government will arise
and
remain
will
in place until a stochastic shock arrives
and changes
new
the configuration of competences. Following such a shock, the stable government for this
configuration of competences emerges. Therefore,
Theorem
we use
of characterizing stochastic transitions. In the next subsection,
links
between
difi'erent political
The Structure
5.2
regimes and institutional
this result to
we compare
(adaptability to stochastic shocks).
flexibility.
difi'erent political
Our main
regimes in terms of their fiexibihty
though a greater degree
results will be that, even
democracy does not guarantee the emergence of more competent governments
stochastic en^'ironment (nor does
guarantee greater expected competence),
it
shocks institutional structures with a greater degree of democracy
wiU be more
"flexible"
study the
of Stochastic Transitions
In the rest of this section,
of
way
3 provides us with a tractable
("will
always impose Assumptions
(less
in
in the
non-
the presence of
incumbency advantage)
perform better in the long run"). In the results that follow, we
1,
3
2',
and 4 (which again ensure uniqueness of
acyclic political
equilibria).
We
impose some additional structure on the distribution
also
that any shock corresponds to a rearrangement
Put
individuals.
time
t.
that ai
s
7j
>
individual
>
,
>
a2
is
we assume throughout
differently,
=
of abilities, say a
is
...
{ai,...,a„},
On.
rest of this section,
G.{Tg}) be the
regime characterized by
k,
we
will
We
adopt the convention
shock
will
change which
the difference in flexibility implied by different political regimes.
our measure of
"flexibility" is
the best government will be in power (either at given
government
a fixed vector
abilities across individuals at
of this vector a.
tp^
is
best placed to solve certain tasks and thus most effective in government functions.
Throughout the
\
T'q^) by assuming
\
of the abilities of different
Intuitively, this captures the notion that a
The next proposition shows
Trt{l.k.n
)
(Tq
this subsection that there
and the actual distribution of
given by some permutation
>
"permutation"
(
F-p
G
I
i
or as
probability that in a society with
(for
given
fc),
a conflguration of
G Q, the most competent government
think of a regime characterized by
I'
as
will
more
£
n
^
the probability with which
oo).'-''
More
indi^'iduals
formally, let
under a pohtical
competence given by {Fg}, and current
be
in
power
flexible
at the
time
t.
Given n and
than one characterized by
I
if
This IS a natural metric of flexibility in the context of our model; since we have not introduced any cardinal
comparisons between the abilities of individuals, "expect competence" would not be a meaningful measure (see
also footnote 18). Note also that we would obtain similar results if we related flexibility to the probability that
one of the best two or three governments comes to power, etc.
25
jTt {I',
k,n
shock.
Similarly,
limt—oo "{
>
G, {Tg})
\
{!',
k,n
TTf {I,
k,n\ G, {To})
we can think
G, {Fg})
\
for all
G and
{Tq} and
of the regime as asymptotically
>
limt_oo
T^t
(h
k,n\ G, {Fg})
l.lfl
=
more
for all
these limits are well defined). Clearly, "being more flexible"
Proposition 4
for all
G
t
following a stochastic
than another,
flexible
if
and {To} (provided that
a partial order.
is
perfect democracy), then a shock immediately leads to the
(i.e.,
replacement of the current government by the new most competent government.
2.
If
I
=
1
imperfect democracy), the competence of the government following a shock
(i.e.,
never decreases further; instead,
{k
^
increases with probability no less than
it
~
ly.
in
-
ky.
fn-r^"'^
_
[n-iy
Starting with any
ultimately
come
G
and {To},
power
to
\k-i,
as a result of a shock
is
3.
fixed k as
If
=
I
k
>
2
n
^ oo,
(i.e.,
tt (/, /c,
—
n)
t) after the
^
'
G
is
in
"'.'''"
:= /c,/c,n
,•)
=
than
iTf (l
=
to a
power
.
I
for any
1.
'
is
is strictly less
<
J
(
will
'
0.
>
TTf (/
This probability
1
extreme dictatorship), then a shock never leads
shock
•'
n — k\ /n^
t
ment. The probability that the most competent government
(any
,
'
lim7Tt{Lk,n\G,{TG})^7ril,k,n) = lFor
most competent government
the probability that the
change in governat
any given period
,
f
^
0,k,n
\
G, {Tg}) and
tt; (/
=
1, k,
n
\
G, {Tg})
and {Tg}.
Proof. See Appendix A.
"
Proposition 4 contains a number of important results.
A
perfect
democracy
{I
=
0)
does not
create any barriers against the installation of the best government at anj^ point in time. Hence,
under a perfect democracy every shock
is
"flexibly"
met by a change
in
government according to
the wishes of the population at large (which here means that the most competent government
will
come
to power).
As we know from the
analysis in Section 4, this
is
no longer true as
soon as members of the governments have incuinbency advantage. In particular, we know that
without stochastic shocks, arbitrarily incompetent governments
in
power. However,
in
may come
to
power and remain
the presence of shocks the evolution of equilibrium governments becomes
more complex.
Next consider the case with
/
>
1.
Now, even though the immediate
effect of a
shock
may be
a deterioration in govermnent competence, there are forces that increase government competence
26
in the long run. This
institutions, there
is
is
most
clearl)' illustrated in
1=1. With
this set of political
zero probability that there will be a further decrease in government
tence following a shock. Moreover, there
and
the case where
a positive probability that competence will improve
is
in fact a positive probability that, following a shock, the
most competent government
be instituted. In particular, a shock may make the current government unstable, and
case, there will be a transition to a
new
stable govermnent.
A
transition to a less
government would never receive support from the population. The change
be such that the only stable government
may
after the shock, starting
can deteriorate following shocks
Most importantly, part
degree of
flexibility
will
(in fact,
in this
competent
may
competences
political institutions take the
never be any transition, thus the current government
can do so significantly).
it
3 of Proposition 4
shows that imperfect democracy has a higher
than dictatorship, ensuring better long-run outcomes (and naturally perfect
democracy has the highest degree
democracy and dictatorship
of flexibility). This
unambiguous ranking between imperfect
shocks (and
in the presence of stochastic
in the next proposition) contrasts with the results in Section 4,
stronger version stated
its
which showed that general
comparisons between imperfect democracy and dictatorship are not possible
case.
in
will
with the current government,
be the best government.^' Proposition 4 also shows that when
form of an extreme dictatorship, there
compe-
in the
Thus, a distinct advantage of more democratic regimes might be their
nonstochastic
flexibility in
the
face of changing environments.
An
informal intuition for the greater flexibility of more democratic regimes in the presence
Examples
of stochastic shocks can be obtained from
from these examples that an advantage of the
of
two veto players makes a large number
imphes that
move
if
the initial govermuent
to a nearby government.
is
of
less
5
and
6 in the previous section.
democratic regime,
governments near the
Z
=
2, is
initial
that the presence
one stable. But this
destabilized because of a shock, there will only be a
In contrast, the
more democratic regime,
1
=
1,
often
highly incompetent governments stable because there are no nearby stable governments
example, part 2 of Proposition
for
3).
Recall
But
this also implies that
if
makes
(recall,
a shock destabilizes the
current government, a significant improvement in the quality of the government becomes
hkel}'.
Thus, at a broad
level,
more
regimes with lower degrees of democracy "create more stability,"
facihtating small or moderate-sized improvements in initial government quality, but do not
create a large "basin of attraction'' for the highest competence governments (in particular, the
best government).
In contrast, in regimes with higher degrees of democracy, low competence
governments are often made stable by the
instability can
'
instability of
nearby alternative governments;
be a disadvantage in deterministic environments as illustrated
in the previous
Nevertheless, the probabiUty of the most competent government coming to power, though positive,
arbitrarily low.
27
this
may be
section, but turns into a significant flexibility advantage in the presence of stochastic shocks
because
may be
creates the possibihty that, after a shoclv, there
it
competence government
a
jump
to a very high
government, which now has a larger "basin
(in particular, to the best
of attraction" than in less democratic regimes).
The next
proposition strengthens the conclusions of Proposition
lishes that the probability of
having the most competent goverrunent in power
the degree of democracy more generally
Proposition 5 The
any
(for
t),
lit
(i.e., it is
probability of having the
{l,k,n
G, {Tg}),
j
democratic regimes are more
is
I
in
G
for any k, n,
-
'
.
and
and {Tc}.
after a shock
That
more
is,
produce the most competent government.
likely to
results of Propositions 4
increasing in
is
most competent government in power
decreasing in
estab-
it
decreasing in l)}^
Proof. See Appendix A.
The
In particular,
4.
5
-
can be strengthened further, when shocks are "limited"
>
the sense that only the abilities of two (or in the second part of the proposition, of x
The next proposition contains
individuals in the society are swapped.
Proposition 6 Suppose
1.
=
that any shock
permutes the
abilities of
If
X
1,
then the competence of the government in power
TTt {I,
(so that the abilities of two individuals are
2
k,n
\
Moreover,
G, {Tc})
if
is
nondecreasing in
t
for any
then the most competent government will be in power as
2.
If
X
>
2,
G
the probability of swapping of abilities between
limt_,oo "f {l,k,n
\
G, {Tg})
=
1
then the results in part
(for
1
any
I.
k, n,
G
t
and {Tg} such
I
that
any two individuals
<
I
<k
—
that
\x/2\.
I
<
k
—
<
I
is
^> oo with probability
and {Tg} such
hold provided that
and
at a time)
k
nondecreasing over time, that
k, n,
I,
these results.
x individuals in the society.
swapped
is
2)
k
—
—
is,
I.
positive,
I,
that
is,
I).
..
Proof. See Appendix A.
An
interesting application of Proposition 6
is
that
when shocks
are (relatively) rare
and
limited in their scope, relatively democratic regimes will gradually improve over time and install
the most competent government in the long run.
This
is
not true for the most autocratic
governments, however. This proposition, therefore, strengthens the conclusions of Propositions
4 and 5 in highlighting the flexibility benefits of more democratic regimes.
This conclusion need not be true
for
"expected competence" of the government, since we have not
made
some player is not a member of any stable
government for some
and becomes part of a stable government for some I' < /. If this player has very low
ability, then expected competence under V may be lower. In Appendix B, we provide an example (Example 10)
illustrating this point, and we also show that expected competence of government is monotone in
when is close
"cardinal" assumptions on abilities. In particular,
it
is
possible that
I
I
to
or to k.
28
I
The
political institutions considered so far are "junta-like"
member
is
members
no specific
in the sense that
indispensable. Incumbencj' advantage takes the form of the requirement that
of the current government
must consent to change. The alternative
is
a "royaltj^-like"
environment where one or several members of the govermnent are irreplaceable. All
this can
be conjectured
to be a negative force, since
it
some
else equal,
would mean that a potentially low
ability
person must alwa^-s be part of the government. However, because such an irreplaceable meniber
(the
member
of the "roj'alty")
likely to arise
resist certain
More
Wg
changes because of the further transitions that these
formally,
we change Assumption
We
Wg = {X
The next
like
may
be more
We
will unleash.
of the set of
assume that there are
royalty individuals
be implemented (regardless of whether they
denote the set of these individuals by Y. So, the
all
e C
:
royal individuals are
|.Y[
>
m
members
and
YC
A'}
new
set
.
of the initial government, that
proposition characterizes the structure of equilibrium in this
and junta-like
/
winning coalitions
becomes
of winning coalitions
assume that
and the structure
for a transition to
are current government members).
also
2
"royalty-like" situations.
whose votes are always necessary
We
governments
also unafraid of changes, better
under certain circumstances, whereas, as we have seen, junta members would
accommodate
to
is
Ccise
is,
YC
G°.
and compares royalty-
competence of the equilibrium (stable)
institutions in terms of the expected
go\'ernment
Proposition 7 Suppose
that
governments are given by
// {ai,
...,
a„}
is
we have a
(10).
royalty-systern with
so that the
sufficiently "convex"
I
meaning
royals are never
that
°'~°^
competence of the government under the royalty system,
like system,
1
=
(with the
same
I).
1
I
<
k and cornpetences of
removed from, the govem.ment.
sufficiently large, then the expected
is
is
<
greater than under the original, junta-
The opposite conclusion holds
°^~°"~^
is
if
sufficiently low
and
1.
Proof. See Appendix A.
Proposition 7 shows that royalty-like regimes perform better in the face of shocks than
junta-like regimes in terms of generating higher expected
that {ai,...,a„}
is
competence governments, provided
highly "convex" (such convexity implies that the benefit to society from
having the highest ability indiAddual
in
government
is
relatively high).
As discussed above,
juntas are unlikely to lead to such high quality governments because of the fear of a change
leading to a fmther round of changes, excluding
like
regimes avoid this
fear.
all
initial
members
of the junta.
Royalty-
Nevertheless, royalty-like regimes have a disad\'antage in that the
29
may be
ability of royals
verj'
low or
may change
at
some point and become very
and the
low,
royals will always be part of the government.
In this sense, royalty- like regimes create a clear
disadvantage. However, this result shows that
when
outweigh the
royal),
loss of
it
is
(so as to
may have
is
distinct implications
government and performance, and regimes that provide security to certain
incumbent government may be better
of the
convex
sufficiently
nonetheless higher under the royaltj'-like system. This result
suggests that different types of dictatorships
for long-run quality of
members
is
expected competence because of the presence of a potentially low ability
expected competence
interesting because
{ai, ...,a„}
relatively high-quality
governments
at dealing
with changes and in ensuring
in the long run.
Conclusion
6
we provided a tractable dynamic model
In this paper,
to the selection of
good
politicians
and
The main
of political selection.
to the formation of
good governments
in our
barrier
model
is
not the difficulty of identifying competent or honest pohticians, but the incumbency advantage
of current governments.
Our framework shows how
can lead to the persistence of highly
inefficient
a small degree of
incumbency advantage
and incompetent governments. This
is
because
incumbency advantage implies that one of (potentially many) members of the government needs
to consent to a
go\'ernment
change
may
in the
composition of government. However,
may
recognize that any change
all
current
members
unleash a further round of changes, ultimately
im.seating themselves. In this case, they will all oppose any change in government, even
changes can improve welfare significantly
governments can remain
Using
this
in power.
for the rest of the society,
;
if
such
and highly incompetent
-
:
,
framework, we studj' the implications of different political institutions
A
lection of governments both in nonstochastic
and stochastic environments.
corresponds to a situation in which there
no incumbency advantage, thus
is
of the
,
for the se-
perfect
democracy
citizens
can nomi-
nate alternative governments and vote them to power without the need for the consent of any
member
ment
of the
will
incumbent government. In
this case,
we show
that the most competent govern-
always come to power. However, interestingly, any deviation from perfect democracy
breaks this result and the long-run equilibrium government can be arbitrarily incompetent
ative to the best possible government).
In extreme dictatorship, where any single
the current government has a veto power on any change, the
in
power and
this
can be arbitrarily costly
for anj' political institution other
for the societ}'.
initial
More
government
than perfect democracy. Moreover, there
is
member
alwa5's
surprisingly, the
(rel-
of
remains
same
is
true
no obvious rank-
ing between different sets of political institutions (other than perfect democracy and extreme
dictatorship) in terms of
what they
will
imply
for the quality of long
30
run government. In
fact.
regimes with lower degrees of democrac}' (greater incumbency advantage) can lead to higher
and
quality governments both in specific instances
in
expectation (with uniform or non-uniform
distribution over the set of feasible initial governments).
pohtical institutions
is
possible,
stable governments in our
we provide
and competences
"fiexibiUtj'"
A
.
distinct
In particular, in stochastic environments,
ei-
of individuals or the needs of government functions change,
governments
shuffling the ranking of different possible
fectiveness.
society.
more democratic pohtical regimes have a
In contrast, in stochastic environments,
ther the abilities
a fairly tight characterization of the structure of
benchmark nonstochastic
advantage because of their greater
Even though no such ranking across
in
terms of their competences and
ef-
greater degree of democracy then ensures greater "adaptability" or flexibility.
Perfect democracy
is
most
flexible
and immediately adjusts to any shock by
government that has the greatest competence
after the shock.
opposite and again leads to no change in the
initial
installing a
Extreme dictatorship
is
new
the polar
government. Therefore, shocks that reduce
the competence of the individuals currently in power can lead to significant deterioration in the
quality of government.
Most
more democratic
interestingly,
political institutions allow a greater
degree of institutional flexibility in response to shocks. In particular, we show that shocks cannot lead to the emergence of a worse government (relative to the competence of the government
in
They may, however,
power following the shock).
destabilize the current
induce the emergence of a more competent government.
We
show that
government and
political institutions
with a greater degree of democracy have higher probability of improving the competence of the
government following a shock and ultimately instalhng the most competent government.
Finally,
mark
we
society,
members
also
compare
where change
and
"junta-like"
in
"royalty-like" regimes.
The former
is
our bench-
government requires the consent or support of one or multiple
The second corresponds
of the current government.
to situations in
which one or
multiple individuals are special and must always be part of the government (hence the
title
"royalty"). If royal individuals have low ability, royalty-like regimes can lead to the persistence
of highly incompetent governments.
However, we
royalty-like regimes ma}' lead to the
emergence of
than junta-like regimes. This
is
because
also
show that
liigher qualit)'
in stochastic environments,
governments
may
increase goA-ernment competence or quality because such changes
round of changes, ultimately excluding
An
all
members
important contribution of our paper
is
For example,
to provide a tractable
it is
may
may
resist
changes
lead to another
of the initial government.
analysis of the selection of politicians. This tractability
in various fruitful directions.
run
individuals are not afraid of changes in govern-
roj^al
ments, because their powers are absolute. In contrast, members of the junta
that
in the long
makes
it
framework
for the
dynamic
possible to extend the analysis
possible to introduce conflict of interest
31
among
voters, for example,
by having each government be represented by two
tence and ideological leaning. In this case, not
government.
The
all
characteristics:
compe-
voters will simply prefer the most competent
general approach developed here remains applicable in this case.
Another
generahzation would allow the strength of the preferences of voters towards a particular govern-
ment
to influence the
number
of veto players, so that a transition
a few insiders, but more unified opposition from government
government can be blocked
bj'
members would be necessary
to block a transition
An open
ronment when shocks occur
approach here,
in this
is
frequently.
emphasized
this
of the government.
equilibria
ability of
is
again feasible using the general
an individual
is
not observed until she becomes part
In this case, to install liigh quality governments,
become a
guarantee.
difficult to
For example, we can generalize the environment
"experiment" with different types of governments.
analysis will then
becomes
framework to incorporate the asymmetric information
in the previous literature.
paper such that the
struc-
This case introduces a number of nontrivial technical
direction for future research, which
an extension of
is
more
the characterization of equilibria in the stochastic envi-
and even existence of pure strategy
The most important
issues
away from a highly incompetent government.
question, which would be interesting to investigate perhaps by placing
ture on preferences and institutions,
difficulties,
away from a semi-incompetent
The dynamic
barrier to such experimentation.
institutions that will ensure high quality governments
it
is
necessary to
first
interactions highlighted by our
In this case, the set of political
must exhibit a
different type of flexibility,
whereby some degree of "churning" of governments can be guaranteed even without shocks.
Another interesting area
is
to introduce additional instruments, so that
can provide incentives to pohticians to take actions in
large. In that case, successful political institutions
line
some
political regimes
with the interests of the society at
must ensure both the
selection of high ability
individuals and the provision of incentives to these individuals once they are in government.
Finally, as hinted
to develop a
by the discussion
in this
paragraph, an interesting and challenging extension
more general "mechanism design" approach
in this context,
is
whereby certain aspects
of political institutions are designed to facilitate the appropriate future changes in government.
We
view these directions as interesting and important areas
32
for future research.
A
Appendix
we use
In the appendix,
on the
governments
set of feasible
H
G
of
> G
have higher stage payoff in
than
V=
The next two lemmas summarize
Lemma
Suppose that
1
e I, w, (G)
1.
If for
2.
H ^G.
3.
i{2
i
I
e
>
w, (G)
:
Proof of Lemma
i
is
E
G
or
i
G.H
<
u),
[E)}\
u;;
(G)
ruled out by Assumption
Assumption
Part
Wg, and
We
hence,
\{i
We
Lemma
I
e
:
<
have \H \G\
H ^ G by
Pai-t 3.
all
members
as
(j).
Then:
i£H\G.
k.
Fg >
w, [H)
F//.
mapping
F/f then, by
is
Assumption
possible only
At the same time,
ii
e
i
H\G
€
i
H G
\
Wj (G)
1,
>
w, (H) whenever
(note that Wi (G)
implies
ib\
(G)
<
=
U',
(H)
Wi [H) by
hence the equivalence.
1.
2.
such that
the properties of payoff fimctions and
> n/2 >
3).
D
G. Let us also define set
G and To >
1. If
<
in
(Al)
{GeG:(j>{G) = G].
w^ [H], then
Part
1.
^ H. Hence,
(E
G
there exists a winning coalition in
if
H
Q we write
E
a {ie I: w,{H)>Wi{G)} eWo-
only
and only
\i
H
any G,
Q. For
H yGii&nd
In other words,
we introduce the following binary relation
the following notation. First,
\H\
I
have
{i
G
w^ (G)
>
w^ {H)]\
cp{G)=G
Either
2.
V^(G)
3.
If(p{G)
4.
(/>((/.(G))
k
< mo;
since by
Assumption
= I
€
:
w^ (G)
(and then
>
w, {H)}
G
=
and
H
>-
\ {i
I
>n-k>n-n/2 = n/2
cp
defined in (6)
G eV)
or
4,
(G)
> Tg>-
2
fc
< mo, then
H \G
^
definition (Al).
2 Consider the mapping
1.
<
G. then T^^Q)
> ^H-
0(G).
33
and
>-
:
>
w, (G)
L
letG,H€G.
G.
< w, {H)} D I
Then:
\
{H
\
G),
Proof of
Lemma
The proof
1.
Proof of Theorem
As
1.
y.
straightforward and
is
we prove that the function
First,
constitutes a (Markov) political equilibrium. Take any Gq,
Gq
Gq
(i.e.,
Since by definition
(5).
(Gq)
(p
0-stable,
is
of players. Hence, in either case condition
Now, suppose,
X,Y
Vi
Wg^
e
{H) >
hence the
we have
Vj
y
G
set of players
who have
This
Wi
is
{4>
>
[H))
Wi
impossible
if
[4>
a government
mc,
<p
{H) which
players
at least Iq
except perhaps
(all
members
of
Gq - those
then means they belong to
and
r^i^Gg)
>
(p-stable
is
(p
(6).
{H) >
are the
'^(i>(H)
1,
=
Vi
{cp
r^(G
>
)•
=
^(h{G
>
for the current period.
=
G.
exist
If this
G eg
would not be
p
the case F^///)
is
close to
^ci>(G
)>
4>
{H) would
however, we would get
members
of
Gq
>
w^ {Gq) /
(1
—
{cp
by
{H))
>
{H))
(5),
and
and thus would be
,
at least Iq
Vj {4>
1,
y^
Gq by
preferred to
Wj {Gq)) as
(p>
(indeed,
which
/?),
{H)
is
>
but in that case r^(^H)
at
stable
^4>{Gg)
{Gq)) for
all
i
G
=
3 implies cp{H)
as the instantaneous utilities
1',
Since this includes at least viq^ players,
(6), so
>'^h>
T^^h)
156
satisfies
(p{Gq).
we must
r^iG,)- which contradicts
both conditions of Definition
is
another acychc political equilibrium
all
we must have
satisfied for large
transitions
x{G)
is
that F^(G)
^'-stable for all G.
^ Tg;
ip.
We
For each
prove the
indeed, otherwise condition
/?.
must take place
in
one step,
x {G)
i.e.,
=
ip
{G) for
were not the case, then, due to finiteness of any chain of transitions there would
such that
Th > Fg and
and
G
i
and
Mq
{G); due to acycUcity,
tp
Second, we prove that
all
X
(only players in
j
>
first
Y
and
political equilibrium.
£ Q, define x{G)
1
Gq by
violated,
for all
/3)
as
1 is
is
This contradiction proves that mapping
and thus forms a
of Definition
{cp
But T^^h) > Th by
F^(g,).
following sequence of claims. First,
(i)
-
/ {I
which by Assumption
)>
Wi
To prove uniqueness, suppose that there
G
of Definition
and which
had
=
.
{Gq)) impUes Wi {H)
same except
have that F//
in coalition A' -
cpiGg)
cp
{H), and hence must have Wi
Finally, consider the case ^q,{H)
Vi
T^i^q
If ^q,{H}
of Gq)
Tg,- This would imply that (p{H) G
would contradict
Now
(ii)
£ Q. Consider
<
by construction of
members
(6), either
1 is satisfied.
{Gq)) would include
F^j^/)
possibly prefer (p{H), and they are less than mCg)-
(p
higher stage payoff under
Vi{H) > Vi{cp{Gq)) would imply Wi{d){H)) > Wi{(p{Gq))
a winning coalition in Gq.
least
H
and some alternative
}
defined in (6)
(Gq) for a winning coalition
Vt
V,{H) > Wi {G q)
are winning coalitions such that
i
>
{Gq))
{4>
of Definition
(i)
By
\G\-
G\g\
...,
(Gq))) form a winning coalition in
to obtain a contradiction, that condition
Vi {4>[Gq)) for all
t^tien
^^(G,)!
> Wi
those with Wi {(p{Gq)
<q
I
<
who obtain
or (f>{Gq) £ M.q. In the latter case, the set of plaj^ers
(p{Gq) than under
omitted.
is
enumerate elements of Q as {Gi, G2.
in the text, let us
> Tcy whenever x <
such that Fg^
lemma
of this
ip
{H)
ip
(G)
= H.
7^
For
x {G) but
/?
ip'^
(G)
= x (G).
sufficiently close to
34
1,
Take
H
= x
{G).
the condition Vi {H)
Then F^ >
>
Wi {G) /
(1
F^(G),
—
/3) is
who had
automatically satisfied for the winning coalition of players
We
>
next prove that Vi [H)
condition
{(f)
Fh >
be members of H;
naturally',
The
Wi {(p{G)).
H
they prefer
to
(p
we prove that
Finally,
(p{G)
Without
i'{G).
r"
loss of generality,
/
government that has (p{G)
we have
that F^^c)
As
treated similarly).
Definition
1,
for if
Wi {(p{G])
>
ib'i
Now, we
=
ip{G),
(G). Consequently, condition
=
(G) in particular. Since
(G), and thus Vi
winning coahtion of players:
{H
two
\
>
cp)
H
and
)/'
>
^^,^G)
facts
{G)
cp
must hold.
from the
this follows
(1
]/,
—
G
H
fact that
{H
it')
\
>
Vi
By Assmuption
^G-
H leads to
of such
(G) are 0-stable, and the former
is
covered
bj'
>
(b{Hq)
Take
=
H
Part
2.
1.
We
{H
>
V)
j
w^ (G) /
(G)
ip) for
|
the
same
is
that
i/'-stable as
a winning coahtion
preferred to the latter (in terms
^^{G]
The
=
I
The
0.
governments Hi,
....
4>{H2}
V^
for all
(if
(H) >
g
V
>
then
2
>
Fi/.(g)
existence
case
/
>
1
=
let
Hj,
a cycle: this imphes that so that there
Hi be the
and
=
that d>{Hj)
if
g
=
least
2
V,
[H) > w, [Hi]
[H) >
/ (1
-
V
/?)
//j+i for
all
<
1
<
j
q,
and
competent of these governments.
then
{H2) holds for a winning coalition in
governments, so
1, V^
H
Hg such
is
H
=
As
Hi).
(p
is
a political
But winning coahtions are
ffo-
{H2) holds for a winning coahtion in Hi. Moreover,
for all pla3'ers except,
perhaps,
members
the least competent government). However, the existence of such alternative
<p is
for a
/9)
Part 2 of the theorem.
by Assumption
Hi
—
(1
and thus
(p-st&h\e
prove the statement for the case
Hi. Without loss of generality,
equilibrium,
of
(ii)
a contradiction which completes the proof.
2 different
=
is
must be that
it
cp
To obtain a contradiction, suppose that there
are g
r^(G)
satisfy condition
mc players and at least Ig government members, as
of H and G contains at least Iq members (since H =
(G)).
Proof of Theorem
is
<
a winning coaUtion of players.
is
(•!/'
that
i.e.,
the most competent
is
Th >
must
3), for
First, V,
to
must also
Suppose not,
(6).
r^(G) (the case r^,^)
it
H
of Definition
(ii)
a pohtical equilibrium,
is
G
of
of states payoffs) by at least
and the intersection
/?).
G.
all
ip{H) whenever
w, (G) /
fi-om the choice of G. Second,
of players; indeed,
=
4>{H)
i.e.,
{G) forms a political equihbrium,
li-'
see that the following
Th > Fg and
(G) for
ip
we may assume that
Suppose that
r^(G)-
¥"
—
/ (1
players prefer
members
at least Iq
{G) coincides with (p{G) defined in
tl'
mc
fact that at least
x (G) = H,
F^f^). Moreover, since
violated. This contradiction proves that x (G)
1 is
> Wi (G)
(G))
(G)) for a winning coalition of players in G. Note that this
>
equivalent to Wi {H)
is
(p{G) follows from
3,
Vi
V-j {(p
a political equilibrium, as the condition
(ii)
of Definition
1 is
of
Hi
(as
H contradicts
violated for government
Hi. This contradiction completes the proof.
Part
different
Without
2.
Suppose
to obtain a contradiction that there
governments Hi,
.
loss of generality,
.
,Hq such that 4>{Hj)
.
let
=
H-j+i for
Hi be the most competent
35
a cycle, so that there are q
is
all 1
<
j
<
q,
and
(f>{Hq)
of these governments.
=
>
2
Hi.
Take any
i
>
E Hi. In that case, Vi {Hi)
we have
Wi {Hi)
+ pVi
of
Hi and
Vi
{H2)
>
>
>
is
of mappings
/?
can only consider
>
as
/?)
gets the highest utihty under
i
most preferred government
is i's
>
{Hi)
V,
4),13)
is
(p
=
there exists
finite,
is
which
> Vi{G
arbitrarily close to
mapping
1.
As
this inequality
limit as
/?
^
^
i
<
9
Without
G
cycle.
Consider
first
H2 may
=
for
cyclic political
,
.
.
Hq
.
.
,Hq], the inequality
coalition of players for
(1
—
/?)
Vi {(p{G)
|
some
0,
/?),
/3
we
<Ui = -'^Wj {Hp)
,
-
We
2.
.
Hi, so that the sequence
€ {Hi,.
<
we
is finite,
that
suppose that H2
the case q
and hence w^ {Hi) < Wi {Ho),
of
same winning
the quality defines Ui.
loss of generality,
=
and 4>{Hq)
equivalent to Wi {G)
is
1)
Wi (G)
last
As the
supported by the same coalitions in players. Let Hi,
satisfied for the
is
which forms a
d)
Moreover, since the number of coalitions
1.
-^j + i for all 1
(p,P)
\
must have (by taking the
where the
Hi the condition
This contradiction completes the proof.
Q —^ G
in
hence
members
to obtain a contradiction that the statement does not hold.
2 satisfs' 'P{Hj)
\
(formally,
V, (i?2))- Since this holds for all
constitutes a cycle for 0. This means, in particular, that for any
Vi {4>{G)
Hi
in the cycle,
impossible that for a winning coalition of players in
arbitrarily close to
(3
Hi
which means
satisfied.
Suppose
3.
—
(1
Vi {H2),
it is
1,
{Hi)
equilibrium for
with q
Wi [Hi) /
{H2)
///^
Vi
Part
number
<
Vi {H2)
Vi {H2), as player
is
the least competent of the governments in the
must have that
,.
,
a winning coalition of players in Hi. However, only members
satisfy these inequalities,
and
number
their
is
less
than
vih-^
We
get an
immediate
contradiction.
Consider now the more complicated case
governments.
for current
We
take
H =
H^ and show
government Hi and alternative
>
(?
3.
In this case, Hi,H2,H'i are three different
that the condition
H=
Vi {H3)
>
Wi {H2)
<
This
Vi {H2) for a
Vj (i/3).
The
last inequality
7n//j of
of Hi.
is
.
Note that
satisfied for sufficiently large
holds for aU players except, perhaps,
However, we know that Wi{Hi)
Assumption
1 for
Hi and
Fj/j
<
<
/3
Ui for at least
<
this
is
equivalent to
provided that Wi {H2)
members
them. Therefore, we just have to prove that Wi {Ho)
over, since they belong to
violated
is
"
winning coalition of players.
latter
1
H^. In particular, we need to check that the
following two conditions are satisfied:
1.
of Definition
(ii)
of i/o,
i.e.,
members
^Hi, we must have that
u-'j
{H2)
each of these players. But this immediately implies u'i{H2)
36
members
of Hi.
<
Ui.
for at least
u, for at least Ifj^
Ifi^
<
More-
Wi {Hi] by
<
Ui.
>
Vi (Hs)
2.
must
—
w, (Hi) / (1
exist player
i
a winning coalition of players.
for
/?)
such that
—
(1
/3) V,,
<
{H^)
u),
<
(Hi)
{I
—
Suppose not; then there
(3)
we
inequality holds for a winning coalition of players). Taking the limit,
=
and hence w, (Hi)
Ui,
Consequently, we found
get u,
<
un (Hi)
<
However, this contradicts the assumption.
u,.
H=
(Ho) (since the latter
Vi
H^
which the condition
for
of Definition
(ii)
1 is
violated.
This
contradiction completes the proof.
Part
The proof
4.
checking the condition
it
of this part follows the proof of Part
Vi
(H^)
>
Wi (Hi) /
—
(1
for a
/?)
3,
except the steps im^olved in
winning coahtion of players (indeed,
only in this step that the assumption in Part 3 was used). To check that last condition,
is
again suppose that
<
Wi {Hi)
(1
Taking the
this
is
—
Vi
/3)
we
limit,
did not hold.
it
only possible
Then
there must exist player
i
such that
<
get u,
if
player
i
<
w, [Hi)
is
member
either a
them (otherwise Wi [Hi) >
and hence w, (Hi) =
u,,
of
all
u,.
V, (i^s)
<
Given the assumption,
go\'ernments Hi.
<
G Hi and w^ (Hi)
.
none
of
IV,
{H2)
<
1.
This contradicts the double inequality above, which proves that Vj {H^)
for a
- P)
[Ho) (since the latter inequahty holds for a winning coalition of players).
of
w, (Hi),
(1
u^ if
and thus w, {Hn) <
i
But then
u,.
>
V; (i/3)
u,
{H2)
V,
.
,Hq or a member
.
^ Hi). In both cases,
if i
if
sufficiently close to
is
>
w, {Hi) / {1
—
j3)
winning coalition.
Consequently,
we found
H = H^ for which the condition
of Definition
(ii)
1 is
violated. This
contradiction completes the proof.
Proof of Proposition
but
G^
H
and
by
(5),
Gq =
for
some
(t){Gq) 7^
i^
;^
G
Suppose, to obtain a contradiction, that \G
1.
to
have 7(^
Lemma
by
<
j such that j
q.
H
is
>
stable,
However,
(6).
>
7/^ or ^'q
since \G r\H\
1,
Since
Gq, as follows from
<
(p
-)fj\
I.
H\ >
I,
w-ithout loss of generahty,
Note that
{Gj)
(1
=
G—
Gq
Gj, but then
for
Mq
some
7^
this contradicts the hypothesis that
€ V, and thus completes the proof.
Part
since |G
Now,
= Gj
and so
G
Part
H. By Assumption 3 we need
assume the former. Then
q
1.
for
2.
By
definition of
n Gij >
=
/,
(Gi)
=
Gi by part
1.
mapping
we have
G=
(p,
4>
any G, 6 (G) e P, and thus 4){G)
Pai-t 3.
As
before, the
=
Gi, so Gi e P. Take any government
Consequently,
V=
{Gi], so
P
is
G
£ P;
a singleton.
Gi.
most competent government, Gi,
is
the set of governments which intersect with Gi by fewer than
stable,
/
i.e.
Gi £ P. Now consider
members:
B = {Ge^:lGnGi| </}.
This set
is
non-empty, because n >
intersect with G\\ obviously,
it is
2/c
in B.
implies that there exists a government which does not
Now
take the most competent government from B, Gj
37
where
j
We
have Gj
Gq such
j^
=mm{q:l <q <
Gi, because Gi ^ B. Let us show that Gj
> Jq
that 7(^
.
=
1.
Now we
= Gi
=
violated
is
Gj, so Gj
elements. Finally, note that this boundary
Part
H
y^
4.
If
/
=
then for any Gq S Q,
k,
Gq, as there will exist player
for all q,
— Gq
and thus 4>{Gq)
Proposition
1
Part
whom
Consequently,
(6).
if
I
impossible that
ioi
unstable, except for
is
1
>
Wi (H)
C=
^,
H
<
and
I,
if
V
/
<
q
But
<
j
this
contains at least two
=
1
y Gq
and n <
for
some alternative
Hence,
Wi [Gq).
3k.
Mq =
this completes the
proof of
1.
Proof of Proposition
of part
by
H
Gq\
E
i
it is
common members with
or the second one (otherwise).
1)
achieved: for example,
is
/
each government Gq with
for
This proves that
stable.
is
=
g
(if
Note that any government
and therefore Gq
empty:
is
stable.
is
therefore has at least
(see (6)),
Mj
observe that set
either the first condition in (8)
implies that (p{G^)
B and
does not belong to
stable government Gi. Hence, (j^iGq)
the case g
and Gg E B}
\G\
will
Part
2.
We
1.
be a corollary: to obtain
prove the more general part
2,
one only needs to substitute x
(11),
then the statement
=
k into (12).
Let us prove the existence of such stable government. Define a set-^'alued function
2.
X:C' ^C'^-'u{0} by
^,o.\ G\S
..
'^^
\
'
if
In v/ords, for any coalition of
such that their union
set
when
does not
it
is
ifGeDandScG;
there exists no G 6 D such that
individuals, function
/
(part
1
exist.
Note that x (S)
1), as
ii,.
.
.
,ii-i.
We
/
now add
will
we wiU denote the intermediate
requirements. Let X;_i
=
X
,
X,
Intuitively,
we take the
coalition oi k
G
H
and
—I
/c
members from
individuals such that ATil/_i
-
/
+
1
individuals
coalitions
and
/
which contain 5,
by
Yi,
.
.
.
ii,.
,Yk,
.
.
=
for this
A'(_i
=
in the
A'.
individuals
empty
would violate
0; denote these individuals
,ik to this
coaUtion one by one and
and then prove that Y^
satisfies
the
let
= (XUl;_i)ufu,Y(^;-iU{i})J.
set of individuals
same government with
Now
I
S.
all
take some individual
-
;,
(A3)
which are either forbidden to join the government under
construction by our requirements (A') or are already there {Yi^i), and add
can be
—
'
a well defined single valued function: indeed, there
is
they intersect by at least
Let y;_i be some coalition of
by
^
a stable government whenever such other coalition exists or an
carmot be two different stable governments
Proposition
x assigns a
S CG.
ii
all
individuals which
individuals from l;_i and at least one individual from
E
I
\ A';
38
(below we show that such individual exists)
and
—
let Y]
Y(_i
U
At each subsequent step
{i;}.
the government under construction as follows.
z,
We
I
+
first
<
1
<
z
we choose zth individual
k,
for
define
and then take
eI\X,
i,
(A5)
(we prove that we can do that later) and denote Y^
obtained
in this
We now
By
way be denoted by
show that (p{Y}
we must have that
(6)
>
added
Now
go\'ernment
and
A'
(p
n 1' =
(Y)
0),
side of (A4) for ~
picked
(i>
(Y)
ij
n
It
=
—
0, so
j,
t
ij
(p
we
let S"
nY\ >
will
is
take the individual
I;
could not be a
ij
the last government
{i})
(p
\ {ij}
such that
(V). Consequentl)',
if
immediately get that
as well as
</>
We
according to (A5).
(F)
c
i
A'^-,
(1>{Y)
n X.
and was
of the initial Yi-i
X
S
e
i
with the highest j of such
ij
(E
entire
=
individual
is
member
be a subset of (0 [Y] n Y)
and contains the
T\Xj,
[Y)
Suppose not, then there
0.
we must have \ (5 U
such that
A'
=
U {u}. Let
Y^-i
Y'k.
so individual
I,
stable
is
X
\(t>{Y)
individuals. Clearly, j
at a later stage.
Ci
Y=
=
(and
|5t
^
i
= —
/
Since
1.
S because S C
Y
we consider the right-hand
and therefore
ij
£
But we
A'j.
get to a contradiction, wliich proves that
a stable government which contains no
member
of A^.
remains to show that we can always pick such individual; we need to show that the
number
of individuals in A';
is less
than n
for
any
z
:
I
<
z
<
Note that the union
k.
the inner
in
parentheses of (A4) consists of at most
individuals, while z
—
I
<
—
k
n
Because we are dealing with
Part
3.
1.
Therefore,
it is
sufficient to require that
>
x
+ k-l + {k-l)(
=
x
+ /.-l+.(.-/)
]x
J^li^.
integers, this implies (12),
which completes the proof.
This follows immediately by using Assumption 4 and setting p
=
x in
which
(12),
gives (13).
Proof of Proposition
/
=
0,
for all
then Proposition
G
1
€ Q, where Gi
3.
Part
1.
By Assumption
2,
<
/
<
/c,
so either
(part 2) implies that the only stable government
=
{ii}.
If
/
=
1,
then Proposition
stable.
39
1
is
/
=
Gi, so
or
cp
/
=
(G)
(part 4) implies that any
If
1.
=
Gi
G
is
Part
In this case, either
2.
that of part
/
=
0,
/
=
and foUows from Proposition
1
=
/
If
2.
(part
1
=
Z
and
(parts 2
1
By
competent, and hence stable, government.
12 is
or
1,
or
4). li
I
=
^
=
2,
<
the construction of
then
b,
cp
and
is
because
ia,
either
cp,
(G) will include either
state which includes
exists
mapping
ia
or
cp
(G)
=G
Now
ifc.
evident that
it is
3.
The proof
Theorem
follows immediately from
that changes are sufficiently infrequent. Indeed, in the latter case,
1
and
Part
[n/k\
is
2.
< n/k
4.
Part
1. If
=
/
Suppose
=
/
1,
then Proposition
Each
stable governments.
stable.
<
j
'^72.
h} with
{ia,
(G) will be the stable
the latter
15 if
The government
the strict inequalities in
all
then by Proposition
•
1
•
1
any G,
for
provides a
full
=
l/(^)
is
change to a more competent one
The most competent government
at least
1
of the k
contain any of these equals
k most competent
("^^
will
—
1
<
[n/k\ /(^)
is
'
"
.
if
=
'^7
^
and only
if it is
The
members divided by the
random
=
"~(„_i)T
^/(fcZi)-
happens
unstable, which
."
l/()!lj).
be installed
(this
G\,
probability
';
^
if
and only
if
after the shock, the govern-
most competent members. The probability that
")/(]!)
=
There are
characterization.
,
will
0* (G)
i]i.}.
,
coincides with any given stable government
with probabihty greater than or equal to
ment contains
and the assumption
1
consists of k individuals, so the probabihty that a
coincides with any stable government
it
0,
the most competent government {^i,
new government
that
G=
If
I.
<
1
or
2 are preserved.
Proof of Proposition
where G\
cf)
foi'
12, is
ii
different.
Proof of Theorem
Definitions
=
or [^ [G] r\G\
i2j}
or
ii
more competent than the one which includes
it is
most
the
is
any other government containing
1),
unstable. Hence, {13,14}, the most competent government not containing
By
similar to
is
then {11,12}
I,
Proceeding hkewise, we find that the only stable governments are {i2j-i,
a
the proof
is
the
total
number
number
(n-
it
does not
of combinations that do not include
of combinations).
We
have
-
ky.kl [n -k\)
(n-2/c)! n!
(n--k)\ (n- k\)
(n- 2ky.
n
k\
I
n
TJ'
-'-'-7=1
Since each of the k factors tends to
1
as
n
—
>
00, so
that the most competent government wiU arise,
as
n
^
Part
change.
lit
-
-k-
n—
J
j
does the product. Hence, the probabihty
(l,k,n
j
G, {Fg})
=
1
—
("^
')/(fc)'
tends to
oc.
3.
If
It will
/
=
k,
then
(p^
(G)
= G
be the most competent
pens with probabihty l/(^)-
This
is
if it
less
for
any
t
and G.
Hence, the government
will
not
contains k most competent individuals, which hap-
than
40
1,
which
is
the corresponding probability for
/
=
so
0,
=
TTt {I
k,k,n\ G, {Tg}) <
1=1:
the corresponding probability for
ments which
TTt {I
=
I
0,
G,{rG}) > 2/(^) >
Proof of Proposition
n
|
G, {To})-
in the latter case,
=
7rt(/
specific
most competent government G\
that at least
members
members
of G' are
than
also less
is
it
2,
there are at least two govern-
and
,i\}
{i\,
>
fc
{*!,
•
i.e.,
ifc_i, 'ij^+i},
,
•
I
Given the
5.
If
G,{Tg}). This completes the proof.
0,/c,n
of ha\-ing the
/
fc,
the most competent one:
will lead to
0,k,n
=
tt^ (/
of
changes in {To} in this case, the probability
(for
G\
.
any
initial
G
and {Fg})
the probability
is
This probability equals (from hypergeometric
distribution):
eU
n{l,k,n\G,{TG})
and
strictly decreasing in
is
Proof of Proposition
a.
where a
/,
Part
6.
F^ =
G
1 all
>
—
>
—
F!-
G
F*,^,.
p{G)
hence,
1,
p{G)
F^
G <
If
which
Hence,
is F'^^^,.
cj)^
(G)
is
is (;/)^-stable.
G
at least
p{G)
either
or a
competence of government cannot decrease. However,
competent members. Indeed,
j e G.
F^ >
0.
if
<£'*"'
has) induces a one-to-
and only
=
(G)
a
if
G G.
{i)
0* {p{G}) for all
with
G*""'
is (i)'~''^-stable.
>
/c
—
Consider two cases.
>
—
If
G
But
/.
F^ >
F*,^,, since there
P[G)
common members and
/
>
1
is
F',^,,
is
a
rf)*-
the competence
more competent government. Hence, the
it
may
in that case there exist z,j £
increase, unless
T
with
i
<
G
contains k most
j such that
i
^
G
and
Obviously, swapping the abilities of these individuals increases the competence of G:
F^
,
and thus the stable government that
improve. Since there
is
then with probability
1
2.
a finite
thus \{G\p (G))
U
more than x
.
number
of possible values of current government's competence,
from an argument of part
k
—
(p (G) \ G)|
>
{p(G)\G) changed
T^q
non-zero, eventually the competence of government will
X individuals changed, then \Gr\p[G)\
(1
> F^ >
the most competent government will emerge.
Tills follows
we would have \G
will e\'oh'e will satisfy '^\iq:.
is
Since the probability of this swapping
Part
£ p (G)
F' ^,. then again
r*,,„,
b
p(G)<p'{G)
Stable government p (G) which has with
of
now
i
any transposition
triggers a period of instability, then
where government
t
generallj',
only two individuals are swapped, then \Gr]p{G)\
(^'"^-stable with probability
<p'{G)
and a shock
i
mapping
r*,^,, and, by construction of
shocks arrive at times
If abilities of
then F',,^,
government p (G):
to
transitions occur in one stage,
probability
more
(or,
the indi\'idual whose former competence individual
(i) is
B}' construction,
If all
Any such swapping
1.
one mapping that maps government
G.
iU)
p {G)\
<
[(i
+
2 [{x
>
k
-
[x/2J
1) /2J since
+
1)
/2J
>
1,
>
taking into account that
I.
Indeed,
if
\Gr\p{G)\ <
abilities of
k-
[x/2\,
the numbers of both sides are integers, and
x.
However, aU individuals
their abilities, so the last inequality contradicts the
individuals did. This contradiction completes the proof.
41
if
in
(G
\
p (G))
U
assumption that no
Proof of Proposition
under the royalty system
I
irreplaceable
The
7.
is 1.
Indeed, government
members and k —
the most competent player.
probability of having the most able player in the government
Now
is
i.e.,
k, this
is
always includes
is
a positive probability
is
sufficiently large,
if
and only
if
the irreplaceable
The
not.
i^, is
any
a part of
member
is
with probability 1/n. In a junta-like system, the probabihty that the
is
government immediately
that any given plaj^er
If
<
and any G* consist of
consider the probability that the least competent player,
most competent government
in the
stable.
°'~°^
t
more competent that a government that does
is
i\
i\
the government. In a royalty system, this can happen
the least competent,
I
In the case of a junta-Uke system, there
government that includes player
part follows.
(G*) will for any
most competent members. Since
I
that a goverrmient that does not include player
first
(/>*
among
installed
is
the probability that one of the players
This probability
after the shock.
i\,
.
,
.
.
i\,
which
is
42
is
ij,
.
,
.
.
i^ is
higher than the probability
k/n > 2/n. This completes the
proof.
B
Appendix
This Appendix presents an extensive-form game in wliich individuals make proposals for alternative governments and vote over alternatives.
of this game, establishes their existence
the notions of
MPE
also provide a
number
and
It
defines the
their properties,
and that of the (stochastic)
Markov
perfect equilibria
(MPE)
and shows the equivalence between
We
political equilibrium defined in the text.
MPE
of examples referred to in the text (e.g., on possibility of cyclic
or
pohtical equilibria, and on changes in expected competence).
Dynamic Game
We now
present a dynamic
ment change. This game
then
game
that captures certain saUent features of the process of govern-
individuals (including current
all
and
involves different individuals proposing alternative governments
For reasons that will become clear
members
the
belo'v\',
of the government) voting over these proposals.
game
also involves
an additional state variable,
which governs when new proposals and transitions can be made.
Let us
first
introduce this additional state variable, denoted by v\ which determines whether
the current government can be changed. In particular, v' takes two
\'alues;
u'
to a "sheltered" pohtical situation (or "stable" political situation, although
we
"stable" for go\-ernments which persist over time)
The government can
bilizes
and
u'
=
=
s
corresponds
reserve the term
u designates an unstable situation.
A sheltered political situation destaeach period, that is, P (u* = u v^~^ = s) = r.
we also assume that v*^ = u. An unstable situa-
only be changed during unstable times.
(becomes unstable) with probabihty
r in
These events are independent across periods and
tion becomes sheltered
\
when an incumbent government
sur\'ives a challenge or is not challenged
at all (as explained below).
We
next describe the procedure for challenging an incumbent government.
some government G^
at
time
t.
If at
time
ordered according to some sequence
governments A^
to
C G
\
{G^} that
t
r)Qt.
will
the situation
Individuals, in turn,
in
^
as follows.
some
0, then
AU
all
= {G eg\{G*}
alternatives in
order as the protocol, ^^t
.
The
individuals
i
G
with
X
are
An
individual
=
may
choose not
0. All nominated
set .4', so
-.G e A, for
A* take part
start
nominate subsets of alternative
in the
some
i
e l}
(Al)
.
primaries at time
of the alternatives in .4' are ordered ^Qt (1) .^Qt (2)
pre-specified order (depending
all
which case he may choose A*
governments (except the incumbent) make up the
If .4'
unstable, then
be part of the primaries.
nominate any alternative government,
A^
is
We
,
.
.
.
t.
The primaries work
,E,Qt (|-^*[)
on A^ and the current government G*).
We
according to
refer to tliis
primaries are then used to determine the challenging government
43
G" t ^*. In particular, we start with G\ given by the
At the second
step, G'l
Appendix that the sequence
in the
not have any affect on the outcome).
G2 =
^Qt (2); otherwise G'2
and G'
=
G'
and only
we say
there
If
becomes
sheltered, that
new government, but
is,
G*"*"^
it
votings
which votes take place does
in
1
.
.
and
.,
then
are determined,
G',_^ti
This ends the primary.
•
a coalition that belpngs to
(i.e.,
that the incumbent government G* wins.
the incumbent goverrmient wins,
all
of individuals support the latter,
G'^^t
no challenger and the incumbent government
is
assume that
(1).
voted against the incumbent government G*. G' wins
a winning coalition of individuals
if
Otherwise,
.
is
We
(2).
Gj. Proceeding in order, Gj, G4,
After the primary, the challenger G'
if
more than n/2
If
equal to the last element of the sequence,
is
element of the protocol ^^t, ^^t
voted against the second element, ^Qt
is
show
are sequential (and
first
is
=
u*+-'
^*
=
to start with, then
again the winner.
and moreover the
stays in power,
= G* and
If
Wnt) supports
political situation
Otherwise, the challenger becomes the
s.
the situation remains unstable, that
is,
individuals receive stage payoff m, (G*) (we assume that the
G*"^^
=
G' and
new government
v^'^^
—
v*
=
All
u.
starts acting
from
the next period on).
More
•
formally, the exact procedure
Period
=
t
0,1,2,... begins with
sheltered, v^
=
period,
=
G^~^'^
s,
G*,
as follows.
f*"*"^
=
=
u*
is
unstable, vt
s
I
£
i
—
feasible
in power.
If
the political situation
is
receives stage payoff uj (G'); in the next
with probability
1
—
r
and v^^^
=
u with probability
r.
then the following events take place:
u,
Individuals are ordered according to
nominates a subset of
'
,
government G*
then each individual
• If the poUtical situation
1.
is
and
i]Qt,
governments A^
in this sequence, each individual
C Q
\
{G^]
for the primaries.
i
These
determine the set of alternatives A^ as in (Al).
2.
If .4'
=
0, then we say that the incumbent government wins,
and each individual
in A*^ are
3.
If
^'
the
7^
ordered according to protocol
If
=
7^
G*,
u*"*"^
=
s,
0, then the alternatives
<J^t.
is
Gj =
^^' (1).
voted against
If |y4*|
<^^, (j).
>
1,
then
for 2
<
S^Qt (j) if
majority
and only
(i.e., if
y^
>
if
j
<
\A*\,
a.t
step
j,
alternative
Voting in the primary takes place as follows:
individuals vote yes or no sequentially according to
G'
A^
=
0, then the alternatives in A^ are voted against each other. In particular, at
first step,
G'._i
receives states payoff u* (G*).
G*"*""^
the set of the individuals
n/2); otherwise, G^
44
=
some
pre-specified order,
who voted
yes, y'-, is
Gj_j. The challenger
is
all
and
a simple
determined as
Government G' challenges the incumbent government
4.
takes place.
some
pre-specified order,
voted
If
5.
In particular,
3^"',
yes,
G' wins, then G'+i
and each
case,
individuals vote yes or no sequentially according to
all
and G' wins
=
G*, v^+^
=
for
if
the set of the individuals
who
£ ^g'^' otherwise, G* wins.
(i.e., if 3^*
G' wins, then G^+^
s; if
=
G' u*+i
,
=
u. In either
indi\'idual obtains stage payoff u^ (G').
There are several important features about
winning coalitions,
First, the set of
and only
if
a winning coalition in G'
is
G', and voting in the election
W^t when
game
this dj-namic
the government
govermnental change are accepted. Second, to
is
that are worth emphasizing.
G^
determines which proposals
specif}' a well defined
game we had
to introduce
the pre-specified order tjq in which individuals nominate alternatives for the primaries, the
protocol ^Q for the order in which alternatives are considered, and also the order in which votes
are cast.
we would
Ideally
an essential part of the economic environment and we do not have
equilibria, since they are not
a
good way
of
orders not to have a major influence on the structure of
like these
mapping the
We
specific orders to reality.
in the equilibria of interest.
Finally, the rate at
which
will see that this
political situations
is
indeed the case
become unstable,
r,
has an important influence on payoffs by determining the rate at which opportunities to change
the government
In
arise.
what
situations are not unstable
we assume that
follows,
most of the time. Here,
it is
r
is
also
relatively small, so that political
important that political instabiUty
ceases after the incumbent government withstands a challenge (or
if
there
no challenge). This
is
can be interpreted as the government having survived a "no-confidence" motion. In addition,
we
as in the text,
focus on situations in which the discount factor
/3
large.
is
Strategies and Definition of Equilibrium
We
define strategies and equilibria in the usual fashion. In particular, let
of the
game up
to period
This history includes
The
We
set of histories
can wTite
h'-'^
is
=
all
t
and stage Q*
governments,
denoted by
in period
.
A
events that have taken place up to period
\^dthin the time instant
A
when
when
strategy for individual
i
i
there
is
t
-
for
alternatives
is
is
I
Ti'''^
and Q*
is
= H^
the
list
x Q*
=
u).
to this time.
where
,
h^
summarizes
all
an opportunity to change the government.
g I, denoted by
is
v'
of events that have taken place
at,
(i.e.,
maps
Ti.''-^
(for all
done according to a protocol).
and Q*)
into a proposal
i
where
v'
=
u)
(recall that the ordering of
A Subgame
a strategy profile {o'i},^j such that the strategy of each
45
t
at the first stage of the period
each possible proposal at each possible decision node
automatic and
up
t if
historj'
can also be decomposed into two parts.
history h*'^
nominates an alternative government
and a vote
(SPE)
t
(there are several stages in period
and correspondingly,
(/]',(5*)
denote the
proposals, votes, and stochastic events
all
'H*'*^
t
/i*''^
is
Perfect Equilibrium
the best response to the
strategies of
all
other individuals for
many supported by complex
games,
Appendix, we
Since there can be several
all histories.
dj^namic
in
which are not our focus here,
trigger strategies,
We
our attention to the Markovian subset of SPEs.
will limit
SPE
in this
next introduce the
standard definition of Markov Perfect Equilibrium (MPE) in pure strategies:
Definition 3
a* for each
period
A Markov
Perfect Equilibrium
in each period
i
is
an
depends only on C*.
t
SPE
such that
profile of strategies {o'*}jgj
W*, and Q* (previous actions taken
F*,
in
t).
MPEs
dynamic games,
are natural in such
since they enable individuals to condition
on
all
of the payoff-relevant information, but rule out complicated trigger-like strategies, which are
not our focus in this paper.
behavior.
MPEs
For this reason,
is
it
initial
characterize, motivating our
_
Qti
_^
An
Qt2
MPE
a*
Q^iQfig ifiQ
MPEs
Cyclical
main focus on
it
at
is
fold.
acyclic
MPEs.
equilibrium path
is positive.
i
depend on the order
= {ic
S'S
\
\S,]
lead to the
We
orem
'
if
for any
same
acycUc
Formally,
we
An MPE
a*
MPEs
is
more
^'
if
cycles
difficult to
have:
<
acychc
^2
<
such that
^3
if it is
not cyclic.
or simply order-independent
These equiUbria require that
which certain events
(e.g.,
proposal-making) un-
slightly modify) their definition for our present context.
denote the above-described game when the set of protocols
Definition 5 Consider
GAME
in
is
some future date) do not take place
cyclic if the probability that there exist ij
is
Here we generalize (and
this purpose, let us
in particular, acyclic
an equilibrium
introduced by Moldovanu and Winter (1995).
strategies should not
MPEs,
are both less realistic and also
Another relevant subset of MPEs, order-independent
equihbria,
potentially lead to a very rich set of
in the text,
government but then reinstalling
along the equilibrium path.
Definition 4
MPEs
also useful to consider subsets of
and order-independent MPEs. As discussed
(changing the
Qti
turns out that even
It
GAME
GAME
[^]
6 M, there
[<J]
and denote the
Then a'
.
exists
an
an equilibrium
distributions of equilibrium
will establish the relationship
is
set of feasible protocols
order-independent
a'* of
governments C^
between
a'cyclic
[
GAME
\^'\
G* for t >
is
hy
For
given by
X
equiUbrium for
such that a* and a'*
t.
and order-independent
equilibria in
The-
5."^^
One could
also require order
verified that the equilibria
independence with respect to 77 as well as with respect to n. It can be easily
satisfy this property and hence, this is not added as a requirement
we focus on already
of "order independence" in Definition
5.
46
Markov
Characterization of
mapping defined hy
Recall the
theorem
Q -^ Q
and r/
Q -^ Q he the mapping defined by
:
to establish the equivalence
Theorem
:
4>
Perfect Equilibria
political equilibria
4 Consider the game described above.
/3)
>
MPE in the dynamic game.
and
Then
there exists £
>
such that
MPE
an acyclic
1.
There
2.
Take an acyclic
(G^)
MPE
=
in pure or
if
•
otherwise, with probability
there are no transitions, so
Proof of Theorem
We
take
(l-^l^l)w,
G
Part
4.
/3q
mixed strategies a*
1
Then we have
.
there exists a period
=
I/*
u.
game by
finite
t
where the government
H
=
The proof
1.
Lemma
I
(G')+/3l^'lu,v(G)
<
(l-p''^^')w,{H')+p\^^w,{H).
X Now
^ ^
.
mtake
'
,
^
Xg
€
^
;
After that,
.
w, (G)
<
introduced
1
Pq the following inequalities are
i
is
>t.
theorem rehes on
of this
>
t
(G°)
satisfied:
w^ {H) implies
(A2)
S-
liH ^G
iiH = G
4j[H)
^
some node
of the
game
in the beginning of
some period
Consider the stages of the dynamic game that take place in this period as a
assigning the following payoffs to the terminal nodes:
C^'^^ is the
any such period
is
all
g and
-
t,
take a
government that
SPE
in
,iH^G
w.{H) + ^^w,{0{H))
is
government that defeated the incumbent G^
SPE
(G^) for
<S
f
where
<p
E Q, define the following mapping
Take any protocol
when
—e
1
that
H,H'
G',
Xg
t
G^ =
such that for any P
any G,
for
For each
P <
G°, then there are no transitions;
•
Appendix A.
if
let
in pure strategies a*
proposed, wins the primaries, and wins the power struggle against G^
in
use the next
then for any protocol ^ e A':
e
exists
We
Suppose that Assumptions 1-3 hold and
be the political equilibrium given by (6).
—
(1
between
(6).
scheduled to be in power in period
if it
was defeated and G*
pure strategies ctq
=
ct^,
itself if it
t
+
I,
was
i.e.,
not.
the
For
of the truncated game, such that this
the same for any two nodes with the same incumbent government; the latter requirement
ensures that once
profile will
we map
these
SPEs
to a strategy profile a* of the entire
be Markovian. In what follows, we prove that
47
for
any
G
game GAME{£^],
& Q,
this
(a) if a*Q is played,
then there
no transition
is
if (p
= G and
(C)
actions in profile a" are best responses
if
there
with part
start
subgame where some
First, consider the
a*Q.
G
take any government
(a);
complete the proof of part
will
and consider the
H will be accepted
imphes, in particular, from the construction of mapping
4>,
that
We
proved by backward induction: assuming that
evident:
makes
its
prove that
(we drop the subscript
v{G,H)
either get
where
waj^ to jth round,
q
(p
for player to refer to
H
for
population, and thus
cp
^
if
=
[G)
if (b
[H) > G. This
G, then no alternative
<
w
j
<
\A\,
then
and v
(p
it
(p{G)
will
win
>~
this round.
= Xg
=
(G))
(4'
as vectors of payoffs), while
from gth round,
starts
=
(f>{cl)[G'))
(G))
Clearly, voting for (p{G)
(p{G).
made and the
(p{G) £ A, then 4>{G) wins the
(p
G. This
let
is
show
us
The base
jztr''^' {<P
is
(G))
they
if it loses,
better for a majority of the
is
(G) wins the primaries and defeats G.
subgame which
hence, in the
and only
(G) has number q in the protocol,
(G) wins in the last round, players will get v (G,
ii (p
if
if (p
primaries regardless of ^ (and subsequently wins against G, as
if it
of the truncated gam.e
Second, consider the subgame where nominations have been
players are voting according to protocol ^q.
that
SPE
1.
H has won the primaries and challenges
alternative
the incumbent government G. Clearly, proposal
H may be accepted.
{G) otherwise and (b)
4)
continuation payoffs are taken from profile a* rather
than assumed to be given by (A3). These two results
We
a transition to
is
The
step
is
proven
similarlj',
(G) defeats the incumbent government.
Since this holds irrespective of what happens in previous rounds, this concludes the second step.
Third, consider the stage where nominations are made, and suppose, to obtain a contradiction,
that
H
7^
<p
(G)
is
not proposed. Then, in the equilibrium, players get a payoff vector v (G, H), where
But then,
4>{G).
nominate
<p
clearly,
member
any
of (p{G) has a profitable deviation, which
(G) instead of or in addition to what he
is
nominating
Part (b)
H
is
straightforward. Suppose that the incumbent government
(a).
is
G.
If
some
alternative
defeats government G, then from part (a), the payoffs that players get starting from next
period are given by
the payoff
is
jzrg'^i
i^)
becomes
given by
Wi (G)
Vi
+
(p
(G)
ii^g)n-Rii_T))
(G, H), which
''^i
i'P
some node on
and only
if
no alternative defeats government G, then
government
G
stays until the situation
G=H
an acyclic
and
G
z^'"^-^
=
s
becomes
in all periods ever since; this again gives the payoff
if
in the entire
game
profile a*
is
played, no player has a
1.
MPE. Take
any government
or off the equilibrium path. Define binary relation -^
either
in either case,
{G))- This implies that the continuation payoffs are indeed
means that
is
If
power
in
is
H and w^ {H) + —011)1 {(p{H)) otherwise;
after that,
profitable deviation. This proves part
Pcirt 2. Suppose a*
=
{G,H).
and
stable),
unstable, and government
i_g,'[_^\
''^^4>{H)
exactly' equal to Vi
(the situation
if
SPE
in profile a*Q. Since in a
there should be no profitable deviations, this competes the proof of part
to
is
G=
on
G*
set
Q
at
some period
in
-^
H
when G^ =
G
as follows:
has a positive probability of staying in power
48
t
G
and
u^
=
G
u, or
H
y^
and
another binary relation h^ on
empty) of
H^
H.
different
G
from
to
Hq such
set
[H
€ Q
G
:
'-^
H)
H
.
.
Hq such
.
any only
if
G ^
that
Hi
^
H2
and stabiUzation of pohtical situation
is
H2
there
-^
G
Hq
—^
-^
H
Hq =
-^ Hi.
Suppose that
From
{H' € Q
set,
H
:
E Q
1-^
:
H']
it
since a*
G
one
call
cTq.
it
a positive probability,
The way we picked
and
profile a' does not exhibit
government H'
j^
G
G
easy to derive the
is
G
^-*
is
a singleton.
H}
contains at least
G
may defeat G
any such government
for
any uncertainty, and the
= H)
(but perhaps H'
H
H
such that
between two terminal nodes of
governments
H'l
G
V'
{G)
G
>-
proof,
alternative
of
is
0(G)
(otherwise
G
it
—
\Q\
2 steps.
no player
is
on the equilibrium
last player to act
must mix, and therefore be
H
G
such that
H
>—*
But
overthrowm.
is
is
.
This implies that
must win the primaries;
it
government
is
indifferent.
well defined.
cp
(G)
for all
G.
Suppose not; then,
is
V'
=
(G)
if
G=
if
some
easily
who
prefers cp{G) to
V
1-^ cp
G.
By construction
of
A
we
(G)
is
no such
If
(G) (any
deviating and nominating ^(G).
(t>{G) for all
(p
H such that H
shown by backward induction.
a player
off
alternative
to stay in G),
member
deviation
mapping
ip,
this
[G) and one or more transitions ultimately leading
(G) otherwise. This completes the proof.
The most important
result
from
by the same mapping
defined in Definition
equihbrium used
is
would be better
not possible in equihbrium. so
sitions given
this
nominated, then, since there
does), such player
=
would not be defeated as players would prefer
imphes that there are no transitions
to
no more than
and one where
stays
we must show that V (G)
must have that T^jg) > r^(G)
nominated,
in
unique
government by V (G).
this
To complete the
since
H'
small, this implies that
is
This contradiction proves that for any G, government
Denote
with
which ultimately lead to two different
possible to get different outcomes
it is still
^^
means that the
players act sequentially, one at a time, which
path when
r
this period
and H^, or between one where
in
is
the subsequent evolution prescribed by
political situation will stabilize at the
Given our assumption (A2) and the assumption that
indifferent
some government
implies that
is
& Q, the
Consider the restriction of profile a* on the part of the game where government
power, and
and
governments
different
for at least
acyclicity
{H
Now,
H.
at
two
at least
with the following properties:
H of this
Define
u.
an on-equilibrium path that involves a sequence
is
no sequence that contains
—*
—
u*
there exists a sequence (perhaps
if
H
ii
= G and
C'
>—*
any element
for
.
iJ
contains at least two elements.
existence of government
two elements, but
G h^
if
G
-^
Hi
that
with positive probability
as follows:
cj
an acyclic equilibrium, there
Hi
H
governments H\,
In other words.
of transitions
=
G*"^^
in
1.
this
(p^
theorem
is
that acychc
MPE
lead to equilibrium tran-
defined in (6), which characterize political equilibria as
This result thus provides further justification
our main analysis.
49
for the
notion of political
The hypothesis
that r
ciently "stable," so that
some
for
in
nontrivial
sufficiently small ensures that stable political situations are suffi-
is
if
the government passes a "no-confidence" voting,
amount
Such a requirement
of time.
is
stays in
important to ensure that an
power
MPE
pure strategies exists (which in turn allows us to obtain a characterization of equilibria),
and
is
related in spirit to the second requirement in part 2 of Definition
Example
1.
presented next, illustrates the potential for nonexistence of pure strategy
is
it
MPE
7,
which
without this
assumption.
Example
7 Suppose that the society consists of
each government consists of two members, so
=
(/
1),
m=
and suppose
competence
member
a
if
One can
is
"almost perfect" democracy
There
2.
30
— min
is
is
if
if
in equilibrium there
is
challenged by G, both
{1,2} and {3, 4}.
no
MPE in pure strategies if u' =
no alternative may win
1
and
2
for all
t
show that any
G ^
and {2,5})
of the governments that include
lose the contest for
the primaries,
it
power to {1,2}
must be the winner
for otherwise individual
1
{^,'^},
then in the
off rejecting
lead to {1, 2} which
for 3
and 4
is
We now
or 2
(intuitively, this
is
stable:
(i.e.,
it
as well; therefore,
if
there
is
some
4,
the alternative and postponing
would be better
off
less. It is also
not hard
Indeed,
if
{1,2}
happens because {1,2}
must always be included
is
is
1
included in
the Condorcet
in the primaries,
We
can
now
or 2 will immediately
while {3, 5} and {4, 5} are worse than {3, 4}
transition in equilibrium, then 3
and
4 are better
an extra period, which makes a profitable deviation.
from {3,5} the individuals move
when
2}: indeed,
{1,3}, {1,4}, {1, 5}, {2,3}, {2,4},
consider the governments {3,5} and {4.5}. First,
the last voting
incmnbent
when {1,2}
last voting,
any government which includes
undesirable for both 3 and
off staying at {3, 4} for
1
is {1,
would have a profitable deviation and nominate {1,2}.
conclude that government {3,4}
being
such equilibrium for some protocol
in equilibrium.
winner in simple majority votings). Given that,
(so that the
the incumbent government
if
would be better
is
u
transition to the government (or a chain of governments) that they like
to
utility of
almost indifferent. In this environment, there are
d>:
a transition to some
is
.
an individual compares
contested in each period). Suppose that there
easily see that
is
— 5max{i,j}
{i.j}
they are not in the government; however,
two fixed points of mapping
^.
=
of two different governments, she
government
=
/c
5).
individuals care a lot about being in the government, and about the
all
Let us show that there
Suppose
1,2,3,4,5 (n
Assume
3.
F/jji
Moreover, assume that
=
five individuals
to {4,5}
the government
is
and vice
{3, 5}
blocking this transition
we
versa. Indeed,
and the alternative
(i.e.,
50
rule out the possibility that
postponing
it
if
this
is {4,
for
was the
case,
5}, individuals
then in
1, 2, 3,
5
one period). Hence, either
one of governments
in
two
We
steps.
{3, 5}
and
{4, 5} is stable or
may be
is
a
in
was the
case,
an equilibrium.
then
1.2.3.4 would support
it.
won
5).
This
than {3,5}
for a
individual 4
is
the primaries,
cannot be stable. Indeed,
its
way through the primaries
if it
get a contradiction, since
accepted, even
must
or
case to consider
,3 is
if
close to 1). Similarly,
we proved
is
nominated,
individuals
latter).
make
is
that in this case, {3, 4}
off
which proves that
cyclic
it is
if
better
for, say,
{3, 5}
were
{3. 5}.
if it
won
the primaries: indeed, individuals
is
also better off
if
{1,2}
any alternative which would lead to {1,2} in the next period
leads to {1,2})
for
This imphes, however, that such alternative ({1,2}
must necessarily win the primaries
nominated (by
if
not be a stable government, and hence the only choice the
means that {1,2} must be nominated,
no protocol
and acyclic
MPEs
1
there exists a
do not
exist
is
MPE
in
the
for otherwise, say, individual 1
Hence, we have come to a contradiction in
,f
the)' prefer
all
possible cases,
pure strategies. Thus, the proof that
complete.
and Order- Independent Equilibria
acyclicity requirement in
Theorem
may
doing that.
Cycles, Acyclicity,
librimn in
Because of
must be nominated,
whether to move ultimately to {3,4} or to {1.2}, of which
This, in turn,
would be better
for
would
where from {3,5} the individuals transit to {3,4}. Note
also be accepted in the last voting.
the previous discussion, {4,5}
The
if
this
the former grants him an extra period of staying in power (as the discount
some other one which
both
leads to {3,4}.
2 prefer {1,2} over {3,4} for obvious reasons, but individual 5
factor
to {1,2}
were, since he prefers {3, 4} to {3. 5}. Consequently,
that in this case, alternative {1,2} would be accepted
is
it
if
as individuals
simple majority of individuals. But then {3,4} must be nominated,
The remaining
and
the intermediate step to
would be accepted,
it
win the primaries and take over the incumbent government
1
if
this transition,
At the same time, any alternative that would lead
would make
better off
we would
stable,
would block
a profitable deviation which cannot
is
not be accepted, and neither will be alternative {4,5}, unless
that, alternative {3,4}
5
more period (even
also trivial to check that {3, 5}
It is
alternative {3,4}
if
and
last voting, individuals 3
government that includes either 3 or
happen
{1, 2}
considered similarljf and also leads to a contradiction.
since they are better off staying in {3,5} for one
{1,2}
one step or
to see that a transition to {1,2} (in one or two steps) cannot be an equilibrium.
were the case, then in the
If this
{3, 4} in
consider these three possibilities for the government {3,5} and arrive to a
contradiction; the case of {4, 5}
It is trivial
one of them leads to
Theorem
in the text)
is
4 (similar to the requirement of acyclic pohtical equi-
not redundant.
We
next provide an example of a cyclic
MPE.
Example
8 Consider a society consisting of five individuals (n
51
=
5).
The only
feasible govern-
ments are {1} {2} {3} {4}
,
for
G
,
,
,
Suppose that there
{5}.
& Q^ and that voting takes the form of a simple majority
competences of
also that the
=
so {1}
is
also that stage payoffs are given as in
utilities
Finally,
,
we
{5}
and
{2})
,
we
^fij
^
again define
(^^^
<^^
^2}
Cg
A^
for
Example
as follows. If
({3}
,
{4}
in A: for example,
^
define ^«J>'W,{4},{5}}
^{{j},{2},{3},{5}}
A
i,
,
{3}
,
{5})
and
{5}
,
is
^
{2};
{2})
is
obtained from
~
^^^
({^}
'
{^}
^
^^3}
,
{3}
,
^jj2}.{3},{4},{5}}
^
by dropping
^{^^''"'^
^
{4}
^^^^
,
{1}
,
,
{2}
and
{!}),
{4}),
,
other
for
by dropping the governments absent in A. Then there exists an equilibrium
{!},
—
{4}
>
—+
{3} -^ {2} -^ {1}
^
two individuals nominate {2} and the other three nominate {5};
if it is
{4},
{!};
if it is
{3}.
if it is
.
Let us next turn to voting strategies. Here we appeal to
which individuals always vote
in this class of
games,
Lemma
it is
if it is
{5), two
the
{2},
nominate
_
;,
.
from Acemoglu, Egorov
1
sufficient to focus
for the alternative yielding the highest payoff for
stage. In equilibrium, any alternative
If
two nominate {4} and three
two nominate {5} and three nominate {3};
and Sonin (2008), which shows that
—>••.
{5}
nomination strategies by the individuals.
{1} and three nominate {4}.
in
=
{2})- For other govern-
>
^3^^^
^{g}.{2},{3},{4}}
two individuals nominate {3} and three nominate
nominate
{!}, then ^^^^^^
Cm
verify this claim, consider the following
government
G=
^{{i},{2},{4},(5}}
^-^
(^^^
,
where the governments follow a cycle of the form {5}
To
In particular.
2.
from government competence.
define the protocols
governments which are not
ments,
aU G.
3 for
imply that each individual receives a high value from being part of the government
relative to the utility she receives
{4}
-
mc = m =
=
I
the best government.
Assume
These
5
=
Ig
i.e.
governments are given by
different feasible
Tfi}
,
rule, i.e
,
Suppose
({3}
"perfect democracy,"
is
government which wins the primaries, on or
on strategies
them
off
at
each
equilibrium
path, subsequently wins against the incumbent government. In particular, in such an equilibrium
supporting the incumbent government breaks a cycle, but only one person (the
incumbent government)
is
in favor of
it.
nomination stage, then next government
current government
is
We
next show
in the cycle still
last voting in the primaries is
wins in the primaries. Suppose that the
may
Then by
construction, governments
either be
nominated or
between {2} and {4}, then {2} wins:
indeed,
that both alternatives can take over the incumbent government, but {2}
individuals
1, 2,
and
5 (because they
would want
52
to
of the
only one individual deviates at the
{3} (other cases are treated similarly).
{2} and {4} are necessarily nominated, and {1} or {5}
know
if
member
all
is
not. If the
individuals
preferred by
be government members earlier rather than
later).
however, the
If,
last
stage involves voting between {4} on the one
Now,
or {5} on the other, then {4} wins for similar reason.
then
in the first voting
voted against {2}.
is
it
if
hand and
either {1} or {5}
All individuals
know
either {1}
nominated,
is
that accepting {2} will
ultimately lead to a transition to {2}, whereas supporting {1} or (5} will lead to {4}. Because
of that, at least three individuals (1, 2, 5) will support {2}.
This proves that {2}
will
win
against the incumbent government {3}, provided that {2} and {4} participate in the primaries,
which
is
necessarily the case
if
no more than one individual deviates. This,
nomination strategies are also optimal
for
We
in the sense that there
no profitable one-shot deviation
that this holds for other incumbent governments as well.
any
indi^'idual.
We
have thus proved that the strategies we constructed form an SPE; since they are also
Markovian.
^
it
is
can
MPE
a
easilj' verify
Along the equihbrium path, the governments follow a cycle
as well.
{4} -^ {3} -^ {2}
—
{1} -^ {5}
the other direction: {1}
—
{2}
{5}
is
in turn, implies that
>
»
different protocols). Hence, for
Intuitivelj', a cycle
government
makes
is
it
can similarly construct a cycle that moves in
{3} -^ {4} -^ {5} -^ {1} -^
»
some
(though this would require
protocols, cyclic equilibria are possible.
to enjoy the benefits of being in power. This example,
when
there
from the statement
clear
clear that
is
structure. In the text,
we showed how
and
4,
intuition
also
artificial
from Theorem
and
5).
we suggest,
still
exists.
Example
8 also
Moreover, as em-
less robust.
an intuitive and economically meaningful
certain natural restrictions rule out cyclic political equi-
Here we take a complementary approach and show that the refinement of
(Theorem
MPE
introduced above, order independence,
2).
and
acyclic equilibria have
libria
without the conditions in Theorem
that with order-independent
in the
1,
cycUc equilibria are somewhat
1
and the
a cyclic equilibrium, an acyclic equilibrium
Theorem
in
phasized in Theorems
shown
We
.
enables different individuals that will not be part of the limiting (stable)
also highlight that even
(This
—
—^
2).
MPE,
This
is
is
also sufficient to rule out cyclic equilibria (even
established in the next theorem, which also shows
multi-step transitions, which are possible under
MPE
as
next example, will also be ruled out.
Example
9 Take the setup of Example
individual
1 is
8,
with the exception that Zni
needed to change the government when the government
check that the strategy profile constructed in Example 8
since individual
1
will vote against
is
a
MPE in
is
=
1
(so that consent of
{!}). It
is
then easy to
this case as well.
However,
any alternative which wins the primaries, the difference
is
that alternative {5} wiU not be accepted in equilibrium and government {1} will persist. Hence,
in equilibrium, the transitions are as follows: {5} -^ {4}
We now establish
—
>
{3} -^ {2} -^ {1}.
that order-independent equilibria always exist, are always acyclic, and lead
to rapid (one-step) equilibrium transitions.
As
such, this theorem will be a strong
53
complement
to
Theorem
though
2 in the text,
proof requires a shghtly stronger version of Assumption
its
3,
which we now introduce.
Assumption
(for q
For any
3'
>
2),
G
i
1 and any
,Hq € Q
sequence of feasible governments, Hi,H2,
we have
Wi{Hi)
^ -^
.
q-
1
same
Recall that Assumption 3 imposed that no two feasible governments have exactly the
competence. Assumption
strengthens this and requires that the competence of any government
3'
should not be the average of the competences of other feasible governments. Like Assumption
3,
Assumption
3' is satisfied
perturbation of competence levels would restore
anj' small
Theorem
5
and
let
Q -^ Q
if (3
<
(j)
:
—
1
e
Consider the game described above.
and r/
(1
—
>
/?)
£•
2.
Any
3.
In any order-independent
order-independent
•
if
4>
(G°)
= C^
•
if
4>
{G^)
r G^
there are no
MPE
'^-
MPE a*
more
MPE in pure
n?
MPE
in
5.
is
MPE a*
•
Proof of Theorem
,
transitions:
-•
there exists a
Suppose that Assumptions
Then
S,
G
Part
X
>
3'
hold
such that
we have
.'.,''-
is
acyclic.
that:
'
'
'
'-
'
,
a transition from,
G^
=
<p (G"*^)
for
G^
all t
to
>
fj
[G'^)
.
the payoff of each individual
= ^.(G°) + ^i..(^(G°)).
1.
In part
1
Theorem
of
4,
= G^
for each
in period
t
=
0,
t;
and
1.
i
G
1
is
given by
;•
-
we proved that
pure strategies, and from part 2 of Theorem 4
constructed for different
2 and
1,
there exists e
strategies a*
in pure str^ategies a*
then there
!
for a society,
it.
then there are no transitions and government G*
,
In any order-independent
4-
were not satisfied
for any protocol ^ E X:
an order-independent
There
exists
if it
be the political equilibrium defined by (6).
1.
.-
-"
"generically," in the sense that
it
for
X
MPE
any ^ &
follows that these
have the same equilibrium path of governments. The existence
of an order-independent equilibrium follows.
Part
a*
is
2.
Suppose, to obtain a contradiction, that order-independent
cychc. Define
mapping x
•
5 ~*
path which starts with government G^
the equilibrium
is
in
t?
x (G)
as follows:
=G
and
u'
pure strategies, this mapping
54
=
is
u,
=
i?
if
for
MPE
pure strategies
any node on equihbrium
the next government
well defined
in
G*"*"^
= H.
Since
and unique. The assumption
that equilibrium a*
,Hg (where
Hi.Ho..
=
>
2)
such that x {Hj)
H2 cannot
implies that
H2
those in
= Hj+i
>
defeat
Hi even
we onh' need
is
=
and x {Hq)
q
Hi. Without
governments Hi,H2-
all
,
Hq.
the worse government. However, this
wins the primaries, since
to consider the case g
we ma}' assume that the protocol
definition,
H2
<
j
competence of
least
>
then, by the choice of H2, Fj/j
3,
H3
(if
if it
<
for 1
players, except, perhaps,
all
Hi, prefer Hi to an eternal cycle of Hi and H2- This immediate contradiction
\
implies that
the end
a sequence of pair-wise different governments
is
then the cycle has two elements, of which
2,
If g
q
assume that H2 has the
loss of generality,
If q
acyclic implies that there
is
is
nominated); this
is
such that
>
3.
r//2
if
the incumbent government
possible since
is
we must have that proposal H2
is
and Fh^ > ^H2- Without
cr* is
loss of generalit}',
is
Hi Hs
put at
is
,
By
an order-independent equilibrium.
nominated and accepted
in this equilibrimii along
the equilibrium path.
Let us
first
prove that alternative
the primaries. Consider a player
government, to win over Hi
if
Hs
will defeat
the incumbent government
who would have weakly
i
H2 won
utiUty of
be
strictlj'
prefers
Hi
defeat Hi;
to defeat
governments
all
better off
Hi
if
H2
Now
Hi.
H2 won
defeat
The
the primaries.
then, as implied by
1,
Hi
also support H^.
or that
H2
^ H2, then if2 brings
i
would be willing
suppose
e H2.
i
Hi on the equilibrium
defeats
i
the lowest
to skip H2', hence, such
in addition,
If,
i
would
& Hi, then he
i
weakly prefers that H^ does not
i
last case to consider
Assumption
which proves that
Ht,
is
would be accepted
H'j,
must defeat
if
i
H2 and
€
^ Hi.
i
all
players
if2 in this voting.
/?
is
who would support
nominated.
primaries: indeed, in the last voting,
Hn may
If
wiU either prefer that both
not nominated. Suppose the opposite,
H3, and since, as we showed, only members of
next government,
is i
3', plaj'er
none of them does. Consequently,
Then H2 cannot win the
nominated.
If
to obtain a contradiction, that
Let us prove that in equilibrium if 3
is
i
wins
then easy to see that since he prefers Hi to H2, he would strictly prefer H2 not
sufficiently close to
Ho and H^
H'i defeated
if
Assume,
to H2.
it is
in the cycle; hence,
if it
preferred H2: the next equilibrium
the primaries; since
path, such players must form a winning coalition in Hi.
Hi
prefer that
Hn
This means that
in
i.e.,
ii"2
H3
that
must face
rather than
H^
is
the
equilibrium
H^
is
not
nominated.
Consider, however, what would happen
some government
Hi: indeed,
iii).
and
if
this
G
then wins the primaries.
instead
Hi would
imphes that
is
at least
one
It
alternatives were nominated.
G
7^
H3 (we know
in the last voting of the primaries,
plaj'er
i
gets
with Vi{G)
>
if
Vi
Suppose that
must necessarily be the case that
stay in power, then
the continuation utility that player
there
if all
Hs would
some government
H
that
G
defeats
H3 would
defeat
defeat G. Let us denote
comes
to
power
as
i;,
(H).
If
(H2), then this pla^^er has a profitable deviation
55
during nominations: he can nominate
Otherwise,
defeats Hi.
for a
if v^
[G]
<
Vi
and ensure that
alternatives
all
{H2) for
players,
all
imphes that there are no
Part
The proof
3.
chose,
G
d,-
<
{G)
Vj
(H^)
cannot win the primaries. This
H2 cannot be
the next government, and this
cyclic order-independent equihbria in pure strategies.
is
similar to the proof of part
H such that
and choose government
we
wins the primaries and
we must have that
winning coalition of players, which again means that
contradiction proves that for the protocol
G
x ix {H))
We
2.
mapping x
define
X {H), but x [x {x i^)))
7^
take a protocol which puts government x {x {H)) at the end whenever
= xix
it is
same way
in the
i^})-
We
then
nominated and come
to a similar contradiction.
Part
at period
t
=
Stochastic
We
This follows straightforwardly from part
4.
since the only transition
3,
0.
Markov
Perfect Equilibria
next characterize the structure of (order-independent) stochastic
independent
MPE
in the presence of stochastic shocks)
order-independent (or acyclic)
Once
may happen
again, the
MPE
and our notion of
most important conclusion from
this
MPE
(that
is,
order-
and establish the equivalence between
(acyclic stochastic) pohtical equilibrium.
theorem
is
that
MPE of the dynamic game
discussed here under stochastic shocks lead to the same behavior as our notion of stochastic
pohtical equilibrium introduced in Definition
Theorem
2, 3',
1,
and 4
Let
hold.
>
:
(f)'
such that
if
Q ^> Q
P <
1.
There exists
2.
Suppose that between periods
-
.
1
—
Suppose that Assumptions
be the political equilibrium defined by (6) for
r/
e,
[1
—
/S)
>
and
e
5
>
e then for
F^.
any protocol ^
Then
<E
X
the following results.
MPE m pure
.
:•.
,
6 Consider the above-described stochastic environment.
there exists c
we have
2.
aii
order-independent
ti
MPE in pure
and
t2 there
strategies.
are no shocks.
.,
;
Then
in.
any order- independent
strategies, the following results hold:
~
*
if
4>
{G^^)
•
if
4>
(G'*') 7^ G^\-
G^*^'
then there are no transitions between
then alternative
cp
(C^)
is
ti
and
^2/
accepted during the first period of insta-
bility (after ti).
Proof of Theorem
6.
Part
1.
If 5 is sufficiently
not change the ordering of continuation
utilities at
small, then the possibility of shocks does
the end of any period any for any player.
Hence, the equilibrium constructed in the proof of part
as well.
56
1
of
Theorem
4 proves this statement
Part
Theorem
the proof of
If S is sufficiently small,
2.
here with minimal changes, which are omitted.
and
5 (parts 2
3) maj'
be applied
.
Examples
Example
10 This example shows that in stochastic environments, even though likeUhood of
the best government coming to power
competence
higher under more democratic institutions, the expected
is
=
governments maj' be lower. Suppose n
of stable
9,
the individuals be denoted 1,2,3,4,5,6,7,8,9, with decreasing ability.
given by
abilities of individuals 1,...,8 are
=
7,
2^~',
governments, in the order of decreasing competence, are given
,
,
1234
2358
1256
2367
1278
2457
1357
2468
1368
3456
1458
3478
1467
5678
=
and 79
m =
= 4,/i=3,
fc
5.
Let
Namely, suppose that
—10^.
Then
the 14 stable
as:
,
(Note that this would be the
=
I2
2.
Then
governments
of stable
may become than
except for that, say, rnsggi
but take
list
rri458i.)
evolves
is
=
j|g
of
jg:
any decreasing sequence {7,},_i,
Now
consider the
same parameters,
there are three stable governments 1234, 1567, and 2589. For a
government, the probability that individual 9
initial
for
=
(^)
random
be a part of the stable government that
will
126 feasible governments there are 9 governments that lead to
2589. which are 2589, 2689, 2789, 3589, 3689, 3789, 4589, 4689, and 4789. Clearly, the expected
competence
of
government
government includes the
Excimple 11 There
B =
=
are n
players' utilities
G
A
B
h
C
The
1,
say,
2
negative, whereas for
is
C =
>
/3
and
19 players
/i
=
The
0.999.
3
it is
positive, as
no stable
9.
A =
3 feasible go^'ernments:
{13,14,15,16,17,18,19} (so k
from them are given
=
7).
{1,2,3,4,5,6,7},
The discount
institutional parameters of these
factor
in the following table.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
10
90
90
90
90
90
90
90
60
60
30
30
30
30
30
30
30
30
30
45
11
60
20
20
20
20
20
80
80
80
80
80
80
80
20
20
20
20
20
20
12
10
10
10
10
10
10
10
10
10
10
10
10
70
70
70
70
70
70
70
constructed as follows.
who
receives 45.
receive 10, except for 1,2,3
that the condition 7
=
/c
<
m
< n—
for all go\-ernments, w'hich
is
governments and
1
receive 30, except for 19,
> n/2
=
VI
utilities are
non-members
Zt
competent individual
least
{7.8,9,10.11,12,13},
sufficiently close to
m.Q
for
A:
=
Members
Members
who
12
is
of
A
of government
B
receive 80, except for 7
receive 60.
who
Note that Assumption
satisfied for all feasible
means that any two winning
57
receive 90, non- members
1
receives 30,
holds,
and
governments. Moreover,
coalitions intersect.
We claim
Let us
that
4)
given by
check property
first
(j)
=
(-4)
0,4) (B)
of Definition
(ii)
{10, 11, 12, 13, 14, 15, 16, 17, 18, 19} (as
for player
i is
whether Wi (A)
to verifj'
is
/3
is
=
A,
(C)
cp
The
1.
close to
1,
=B
is
& (cyclic) political equilibrium.
set of players
the simplest
with Vi{C)
way
greater than or less than the average of Wi [A], wi {B),
The
example). These ten players form a winning coalition in A.
(B)
The
is
{2, 3, 4, 5, 6, 14, 15, 16, 17, 18, 19}; these eleven players
set of players
with
V,,
form a winning coalition
Let us
the only
is
now check
H
{B)
V^ (C)
is
to consider
B.
is
ii
H
Indeed,
=
1.
H=
B, condition
(ii)
ii
H = C
A, then the
and any two winning
of Definition
1
is
close to
1,
as 45
only government
and
this
is
thus
H
this is not a
(p
is
the average of 20 and 70).
H=C
If
= A
twelve players
needs to be checked. But
winning coalition
in
>
58
=
Vi{4>{A))
Vi {(p{A})
>
Vi {A) holds for a
is
Vi
>
Wi {A) /
>
Vi [B],
but
Vi
[B)
the current government
if
V^
(1
—
/?)
(C)
for a
winning
But
for
holds for
not a winning coahtion, as there are only
>
K (A)
is
the current government
(A)
>
Vi
>
w^ {A) /
(1
-
/?)
B, then, as before,
holds for ten players only,
is
C, then again,
{B) holds for only eleven players,
C. So, both conditions of Definition
a cychc political equilibrium.
A; then
is
(H) > Vi{(p{A}) cannot hold
needs to be considered. But Vi (C)
not a winning coalition in B. Finally,
only the case
and
is
V,
then Vi{H)
also satisfied, as Vi (B)
nine players (we used the fact that player 19 has V^ (A)
P
>
form a winning coalition in B.
coalitions intersect in this example.
players {10, 11, 12, 13, 14, 15, 16, 17, 18} only, which
for
with Vi (A)
Suppose the current government
winning coalition of players, as the opposite inequality
(i)),
set of players
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}; these
of Definition
(ii)
impossible for any player, and
coahtion (condition
critical for this
is
in C.
condition
we need
>
is
to check this condition
Wi (C); the case where these are equal deserves more detailed study, and
Vi
> Vi{A)
1
are satisfied,
and
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