MIT LIBRARIES
3 9080 03130 2299
y^.d'
o^-'^L
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
EQUILIBRIUM REFINEMENT
IN
DYNAMIC VOTING GAMES
Daron Acemoglu
Georgy Egorov
Konstantin Sonin
Working Paper 09-26
October 6, 2009
1
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
httD://ssrn.com/abstract =1490164
Equilibrium Refinement in Dynamic Voting Games*
Daron Acemoglu
MIT
Georgy Egorov
Northwestern University
Konstantin Sonin
New Economic School
October 2009.
Abstract
We propose two related equilibrium refinements for voting and agenda-setting games, Sequentially
Weakly Undominated Equilibrium (SWUE) and Markov Trembling Hand Perfect Equilibrium (MTHPE),
and show how these equilibrium concepts eliminate non-intuitive equilibria that arise naturally in dynamic
voting games and games in which random or deterministic sequences of agenda-setters make offers to
several players. We establish existence of these equilibria in finite and infinite (for MTHPE) games,
provide a characterization of the structure of equilibria, and clarify the relationship between the two
concepts. Finally, we show how these concepts can be applied in a dynamic model of endogenous club
formation.
Keywords:
JEL
*
voting, agenda-setting games,
Markov trembling-hand
perfect equilibrium.
Classification: D72, C73.
Daron Acemoglu
gratefully acknowledges financial support from the National Science Foundation.
Introduction
1
In
many
political
economy problems, including models
of legislative bargaining
(e.g.,
Baron,
Diermeier, and Fong, 2008, Battaglini and Coate, 2007, 2008, Diermeier and Fong, 2009, Dug-
gan and Kalandrakis, 2007, Romer and Rosenthal, 1978) and models of
mation
(e.g.,
Acemoglu, Egorov, and Sonin, 2008), agents participate
proposal making and strategic voting.
which proposals are made and the voting protocols
These
.
in particular, the
whether voting
"details" as well as the exact notion of equilibrium often
have a major impact
and may introduce "non-intuitive"
This
a significant challenge for applied analysis in this area.
game
procedures through
sequential or simula-
set of equilibria
of the
rounds of
is
(e.g.,
on the
is
in multiple
Noncooperative game-theoretic formulations of these
models typically specify the extensive form of the game,
taneous)
political coalition for-
that appear unimportant and difficult to
map
many
equilibria in
It
of these games.
would be desirable
for features
to reality (such as the exact order of
proposals and the details of the voting protocol) not to have a major impact on predictions, and
for non-intuitive equilibria not to
political
economy theory
of political situations;
emerge as predictions. Therefore,
to have equilibrium concepts that
(ii)
(i)
appears important for
can be applied to a wide variety
and
rule out non-intuitive equilibria;
it
(iii)
make broad
predictions
that are independent of the procedural details specified in the extensive form of the political
game. The next example
political
illustrates
how
games.
Example
1 Consider three individuals, each strictly preferring option a to
options will be implemented by voting.
majority will be implemented.
which
non-intuitive equilibria can emerge in even the simplest
all
It is
Voting
is
not think that
The
ample
is
all
a, is
a Nash equilibrium. However, the non-intuitive
is
also a
Nash
three individuals supporting option 6
is
When
any two
a reasonable prediction.
Nash equilibrium
generally ignored, either by using one of two refinements,
strategies or trembling-hand perfect equilibria
so that instead of simultaneous voting there
approach would work as voting
approach also works
equilibrium.
a weak best response for the third one to do so as well. Naturally, we do
non-intuitive prediction implied by the notion of
undominated
of these two
simultaneous; the option that receives the
possibility that all three individuals vote for option b
b, it is
One
straightforward to verify that the natural equilibrium, in
three individuals vote for option
players vote for
b.
for 6
is
is
Nash
(THPE),
sequential voting.
or
It is
in the previous ex-
equilibria in (weakly)
by changing the game
evident
why
a weakly dominated strategy for each player.
for relatively intuitive reasons:
the
first
The second
with sequential voting, the game becomes
Digitized by tine Internet Archive
in
2011 with funding from
IVIIT
Libraries
http://www.archive.org/details/equilibriumrefin0926acem
dynamic, and
it is
easy to show that in any subgame-perfect equilibrium (SPE) at least two play-
ers vote for a, so option a
is
chosen. Moreover, this result applies regardless of the order in which
the three individuals vote. Nevertheless, sequential voting has the potential disadvantage that
most votes are not
in practice
cast sequentially, but simultaneously (and with secret ballots).
Therefore, this approach "works," though
is
it
does so by changing the environment to one that
a worse approximation to the situation being modeled. Moreover, this discussion highlights
that the set of predictions might crucially depend on seemingly unimportant procedural details
such as whether voting
is
sequential or simultaneous.
In addition, while these two refinements eliminate non-intuitive equilibria in static voting
games,
many
involve
dynamic proposal making and
interesting political
economy problems, such
rule out non-intuitive equilibria. This
Example
at
this option
Example
1
If
2.
In addition,
option b in both dates
2, all
illustrated in the next example.
subgame
is
Example
in
1,
except that voting takes place
a unanimity of voters prefers one of the two options in period
is
we assume that delay
the relevant equilibrium concept
in the
is
implemented. Otherwise, there
is
1.
and
do not necessarily
voting. In this case, these refinements
2 Consider the same environment as
two dates,
as those mentioned at the beginning,
is
SPE.
an SPE. This
starting from date
2.
is
It
a majority vote at date
is
costly.
Now
because
all
all
b
players would be best responding by voting for b at date
is
multi-stage,
three individuals voting for
three players voting for b
Then, expecting option
then
identical to that in
2,
because the game
can be verified that
1,
is
an equilibrium
being implemented at the date
1.
It
can also be verified that
eliminating weakly dominated strategies does not change this outcome, since voting for option
date
b at
not weakly dominated because of costly delay.
1 is
Example
2
shows that simply eliminating weakly dominated strategies
resolve the problems that arise in
dynamic voting games.
Nevertheless,
is
it
to illustrate that the sequential elimination of weakly dominated strategies
powerful.
For example, the strategy "vote for b in both periods"
strategy "vote for b in period
1,
for 6 in
period 2"
is
1
,
vote for a in period 2"
weakly dominated by "vote
for
,
for
^See
Duggan
We
is
typically
more
weakly dominated by the
and the strategy "vote
for a in period
for b in period 1
and a
in period 2"
a in both periods" in the reduced game. Note that this
process of sequential elimination requires that
simultaneously.^
can also be used
a in both periods". After these weakly
dominated strategies are eliminated, the strategy "vote
becomes weakly dominated by "vote
is
not sufficient to
all
weakly dominated strategies are eliminated
can strengthen the point even further and show that even allowing
(2003) for a discussion of problems with this equilibrium refinement.
for
sequential elimination of weakly dominated strategies
Example
equilibria (see
may
not always rule out non-intuitive
5 in the Appendix).
As already mentioned above, a powerful equilibrium concept
intuitive predictions
is
trembling-hand perfect equilibria (THPE). However, the use of trembling-
hand perfection concept
trembles
is
if
to focus
in
dynamic games requires some caution
One way
1994, pp. 246-252).
THPE
to define the
in
(e.g.,
dynamic games
Osborne and Rubinstein,
to
is
aUow
for correlated
a player has a move at different stages of the game; a more restrictive generalization
on uncorrelated trembles. Example 6
generalizations
is
in the
this
(SWUE) and Markov
perfect-information
game
as
the Sequentially Weakly Undominated Equi-
Hand
Trembling
Perfect Equilibrium.
an agenda- setting game
of the players (or, possibly. Nature) moves, or there
is
dynamic voting games.
paper introduces two simple, easy-to-use and intuitive
equilibrium concepts for agenda-setting games:
formal definition
Appendix shows that neither of these
sufficient to eliminate non-intuitive equilibria in certain
Motivated by these problems,
librium
that often eliminates non-
if
is
(MTHPE). We
game
at each stage of this
a voting
among a
refer to
either one
subset of players (a
provided below). Dynamic voting games, in which there are several stages
of voting over proposals, are a special case of agenda-setting games. In addition, most
multilateral bargaining
an
games can be
cast as agenda-setting games.
intuitive generalization of the idea of eliminating
games.
One disadvantage
of
SWUE
games. Another, and a more serious one,
equilibria, as
Example
5 in the
is
is
that
that
it
it is
We
does not necessarily rule out
alternative notion,
cuses on Markovian trembles, has three desirable features. First,
larger class of games, including infinite
it
it
games with
finite
number
several "details," such as whether voting
not affect the predictions in these games. In particular, we
agenda-setting games with simultaneous voting
is
though admittedly
we
show
will
^We
of
MTHPEs)
less realistic,
all
will
is
is
in
non-intuitive
MTHPE,
eliminates
all
which
fo-
non-intuitive
for
a
of non-equivalent subgames.
MTHPE
in agenda-setting
simultaneous or sequential, do
show that the
identical to the set of
of the equivalent sequential voting games. This result
voting,
is
apply to infinite
can be applied easily and straightforwardly
Third, we will provide a tight characterization of the structure of
games and show that
SWUE
SWUE
prove existence of
difficult to generalize or
Appendix shows. Our
equilibria in agenda-setting games. Second,
In this context,
dynamic
weakly dominated strategies, and uses this
procedure iteratively starting from the terminal subgames.
finite
a
also useful, since
set of
MTHPE
MTHPE of
(and of SPE)
games with sequential
allow a straightforward characterization of
SPEs (and
by backward induction."
are not aware of any other equilibrium concept or refinement that has these three desirable features.
We
will first establish the existence of
MTHPE
a broad class of games, including
for
agenda-setting games, and then provide a characterization of the set of
While
setting games.
MTHPE
that for agenda-setting
SWUE
libria in
SWUE
games
for
MTHPE
is
for finite
MTHPE
games) an
agenda-
we show
are not subsets of one another in general,
games (and, more broadly,
(so within this class of
Our main
and
MTHPE
all
always a
is
a stronger concept).
characterization result as well as our result on the equivalence of the set of equi-
simultaneous and sequential voting games rely on the notion of regularization which we
introduce.
A regularization replaces each stage of a dynamic game where multiple players vote si-
multaneously by a sequence of actions (stages) in which one player moves at a time. This implies
that whenever several players
move
simultaneously, there are
many
the resulting games have the same set of equilibrium payoffs.
that the set of
result
is
set of
SPE
is
of
MTHPE
A
regularizations. However, all
key part of our characterization
of the initial (simultaneous voting)
game
is
equivalent to the
any regularization. This result both implies that which regularization
MTHPE
immaterial and shows that computing the set of
is
is
chosen
generally quite straightforward
and can be done by choosing an arbitrary regularization (an arbitrary sequence of voting stages
corresponding to the original game). Note, however, that the Markovian trembles restriction
in
MTHPE
critical for these results:
is
Example
Appendix shows that the above
6 in the
respondence between the equilibria of simulataneous voting game and
down when we
Finally,
use the standard
we show how
proposed and analyzed
game can be
in
THPE
either or
concept instead of
its
regularization breaks
MTHPE.
both concepts can be applied to a dynamic voting model
Acemoglu, Egorov and Sonin (2008) under sequential voting. This
interpreted as one of elimination (as in Soviet Politburo) or as a
coalition formation, similar to previous pioneering
bera, Maschler,
cor-
and Shalev
work
in this area,
(2001). In Roberts (1999), the club
game
of
dynamic
Roberts (1999) and Bar-
membership changes subject to
approval by a majority of the current members; in Barbera, Maschler, and Shalev (2001), any
member
of the club might unilaterally admit a
applies our
also
main
results to a
game
of
dynamic
be straightforwardly applied to dynamic
interesting
new member
legislative bargaining
work by Diermeier and Fong (2009) studies
in their
model would
Although Section 4
coalition formation, our equilibrium concept
agenda-setting power and uses a solution concept that
MTHPE
to the club.^
games. For example, recent
legislative bargaining
is
can
a special case of
with persistent
MTHPE.
Applying
give identical results.
^See also Jack and Lagunoff (2006) and Granot, Maschler, and Shalev (2002), where unanimity
admission to a club.
is
required for
,
Although the hterature on voting games
modern treatment), the notions
of
is
MTHPE
and Banks, 1999,
vast (see Austen-Smith
and
SWUE
Two
are new.
a
previous contributions
Moulin (1979) pioneered application of dominance
are particularly noteworthy.
for
solvability to
games. Duggan (2003) discusses the possibility of non-intuitive equilibria in political
political
games, suggests several solutions and also contains a
The
rest of the
paper
of related
list
open problems.
organized as follows. Section 2 introduces the concepts of Markov
is
Trembling-Hand Perfect Equilibrium and Sequentially Weakly Undominated Equilibrium. Section 3 contains the
main
an application of
results, while Section 4 provides
MTHPE.
Section 5
concludes and the Appendix contains two additional examples.
Setup
2
Setup and Notation
2.1
T
Consider a general n-person T-stage game (where
a natural number or oo), where each
is
individual can take an action at every stage. Let the action profile of each individual
=
a*
with
t
al
S Al and
G A'
a*
(not including stage
game, and
histories
let
t),
Ht be the
up to date
t
(a*!
= nf=i ^twhere a^ =
,
.
a^) for
,
.
.
(a^,
.
=
IJs=o
a'-*
=
.
•
.
t
.
.
with the set of continuation action profiles
.
.
,
.
:
<
,
.
.
.
n,
be the history of play up to stage
at)
is
,
the history at the beginning of the
<T~1. We
t
be
denote the set of
all
potential
i-continuation action profiles be
^« ^^^
(aj, aj+i,
(ai,
1
a"), so ^o
,
.
set of histories ht for
by if *
=
Let ht
=
i
i
,
a^) for
for player
i
=
i
1,
.
.
.
n.
denoted by
A*'*.
Symmetrically, define
t-truncated action profiles as
a
L-t
[a\,a\,...,a\_i) for
with the
set of i-truncated action profiles for player
notation
a'
and a~'
and A~* are used
depend only on
actions,
i.e.,
assume that even
of histories.
We
if
T
is infinite,
player
the
1,...,5
denoted by A*'~*.
i
i's
payoff
is
We
also use the standard
and the action
for the sets of actions).
.i
We
=
to denote the action profiles for player
players (similarly, A^
players
i
i
The
profiles of all other
payoflF functions for the
given by
('^1
(a\...,a").
game admits only a
finite set of payoff-relevant classes
also define the restriction of the payoff function u^ to a continuation play
asuMai'-*,...,a"'-*
(ai'*,...,a"'*)
the utihty of player
then,
when
a^i,...,af
the action profile
(aj,...,a") at time
is
it
i
:
t,
a^'^+i, ...
:
is (a-^'~*,
and then
finally,
.
,
.
.
it
,a"'*+M
restricted to be a^•^...,a"'*
onwards. Symmetrically, this payoff function can also be read as the
action profile
profile
A
f
(a^'*"*"^,
a^'~*,
.
a"'~'
.
,
.
given that up to time
ja"'*"*"^)
.
aj,
:
.
.
.
.
,a^
A {X)
(corresponding to strategy
where //*"' \ //*~^
is
is
u^
0-1'*,
.
(
.
,
.
at time
t
(t"'*
continuation
:
time
S''~*.
^-truncated strategy for player
(including time
t
A
t
t), i.e..
i-continuation strategy for player
t
(including time
i
t), i.e.,
or longer.
will also use the
same
and
ct*''
(7^'*+\
.
payoff
.
.
,
to
other
,G
il
utility function defined over
)
\
player
players
i
after
use
Similarly,
for the payoff to player
i
We
cr"''*.
when he
h*~^
history
uses
(ct^-*,
the
notation
the
use
also
/i*-M as the payoff from strategy profile
ct"'*+i
.
.
.
,
ct"'*)
I
restricted to the continuation strategy profile [a^'*'^^,
/i*"-^.
(corre-
i
and write
strategy
given history
Q-i,t
the play has consisted of the action
specifies plays only after time
ct*)
U UJ
continuation
A
S*.
the set of histories until time
strategies (as actions)
the
by
denoted by
abuse of notation, we
slight
i
a^) specifies plays only until
set of truncated strategies
denote
from continuation
i is
set of strategies for player
sponding to strategy
to
t
denotes the set of probability distributions defined over the set X.
Denote the
With a
t,
utility
from
\.
(possibly mixed) strategy for player
where
The
.
This function specifies
up to and including time t—1;
a"'~*)
is
.
.
.
,
.
cr"'*+^)
from
t
+
1
onwards,
we use the notation
when he chooses the continuation
action profile
a*'*
and others choose
given history h^~^.
We
start
by providing the standard definitions of Nash equilibria and subgame perfect Nash
equilibria.
6
Definition
A
1
strategy profile
A
Definition 2
u'
>
.
.
(ct^,
a Nash Equilibrium
ct") is
,
.
{a\ a'')
strategy profile
h'-^)
u' (<T^'*,d-^'*
>
{a\ a-')
u'
((T^,
...,a^)
is
Towards introducing weakly dominant
one stage game with actions
We
Definition 3
.
,
.
n, for
1,
.
.
.
,
n.
for all
t,
if
a~' G E~* and for alH
=
1,
.
.
,
.
n.
Weakly Undominated Equilibrium
Sequentially
.
alH =
H'-\
h*'^ G
all
I
for all
1,
for
a Subgame Perfect Nash Equilibrium
h*-^) for
u' (a''*,a-^'*
I
G £* and
for all a'
if
any
cr*
say that
strategies, let us take a small digression
and consider a
(a^, ...,a").
(<t^,
.
.
,(t")
.
a weakly undominated equilibrium
is
if
for
each
i
=
G S' we have
u'{a\a-') >u'{a\a-')
(i.e., it is
a Nash equilibrium) and for each
i
=
1,
.
.
,
.
n, there does not exist
cr'
G E' such that
u'{a\a-') >u'{a\a-')
G S~* with
for all (T~'
at least one strict inequality.
Naturally, such equilibrium always exists in static finite games. Let us next extend
general T-stage
of the
game
game
for
T finite. A
defined analogously to Definition
is
Definition 4 Take any
t
<
I
:
weakly undominated equilibrium
librium for any
u
I
a
'
,a
'
/i*
(b'\
a-*'*
for all (T~*'*
G
that
may
< T+
t
if
(ct^,
;
exist
G
ct*'*
ai'*+\
.
.
Strategy profile (a\
I.
.
.
.
.
,
£*'*
\
ti
^u
.
.
.
a h^~^ -sequentially
,(t") is
a /i*-sequentially weakly undominated equi-
,(t") is
j
last stage
3.
occur after h*~^ and for each
.CT,...,a
and there does not
u'
weakly undominated strategy equilibrium in this
to the
it
\
a
'
,a
=
i
'
1,
.
.
,
.
.
a
'
n, for
any
G
a*'*
,...,(T'
\
n
E*'*,
i
\\.)
such that
a^'*+i
|
h'-^\
>
u'
U''\
E~*'* with at least one strict inequality.
a"*'*
\
a''*+\
.
.
.
,
a^'*+i
|
h^'A
(2)
Definition 4
inductive, but this induction
is
terminal node, then the
condition
first
recorded any further. In words, we
for a
weakly dominant strategy
Definition 5 Strategy
profile
Weakly Undominated Equilibrium
.
.
it
will
be played from time
(SWUE)
+
t
1
not be
will
onwards and then look
game.
game
of a finite extensive-form
,(t")
.
leads to a
/i*""^
hypothesize that there exists a /i*-sequentially weakly
first
profile at stage t of the
((j\
history
if
because the history
satisfied automatically,
is
undominated equilibrium, impose that
Indeed,
finite.
is
is
a Sequentially
a /i^-sequentially weakly undominated equi-
if it is
librium.
Markov Trembling-Hand
We now introduce the
we need
to define a
Definition 6
for
aU /i^"\
A
Perfect Equilibria
notion of
Markov Trembling-Hand
Markovian strategy (Maskin and
continuation strategy
e H*~'^ such that
/i*~^
ct*'*
any
for
is
>
e
(MTHPE). For
this,
Tirole, 2001).
Markovian
a*'*, d*-*
/i*-i)
u' (a''*,a-^'*
Perfect Equilibria
A''*
if
and any
u' (a^'*,a-*'*
a"'-*
G A~^'*
h'-^)
|
I
implies that
U^\a-'''
h'-^)
>
u' (a^'Sa"'-*
h''^
|
I
Definition 7
form
strategic
We
is
say that a strategy profile
continuation strategy
(m)
u*
,
.
,
.
.
ct"
a^''^
(m)) -^
{a\a~^ (m)) >
We say
u*
{m)
((T^,
(cr*,
.
is
.
,
.
ct~'
profiles
Markovian
(j") as
an extensive-form game
...,0"")
Trembling- Hand Perfect Equilibrium
all a'
(a^ (m)
{
for all
m ^ oo
(m)) for
that a strategy profile (a^,
strategic
of
Markov Trembling-Hand Perfect Equilibrium (MTHPE)
quence of totally mixed Markovian strategy
((7^
(<j^, ...,ct")
z
=
,
.
,
.
.
ct"
if
in agent-
there exists a se-
(m)) }^^^ (meaning that
l,...,n and
all
m
for
alH =
£ N) such that
and
£ E\
for all
meN
of an extensive-form
(MTHPE)
if it
is
and
game
MTHPE
in strategic
in the
1,
.
form
.
.
is
,
n.
Markov
corresponding agent-
form game.
Note that
problems that
MTHPE is defined directly
arise
when trembling hand
in the agent-strategic
perfection
8
is
form
in order to avoid
defined on the strategic form
standard
(e.g.,
Selten,
.
MTHPE
Osborne and Rubinstein, 1994, pp. 246-252). Naturally,
1975,
than THPE.^ Nevertheless, we establish
In a class of political games, which
MTHPE
we
its
a stronger concept
is
existence under general assumptions
refer to as agenda-setting
MTHPE
(Theorem
1).
games, we prove that a pure-
these two equihbrium concepts do not always coincide.
SWUE (Theorem 2). Nevertheless,
First, a MTHPE always exists, while
SWUE
exist
strategy
may
exists,
and moreover every
not. Second, in
an
infinite
a
is
game, there may
SWUEs
that are not
MTHPE.
Agenda- Setting Games
2.2
Let us
first
define general agenda-setting games, a class of extensive form
games with perfect
information that includes most voting games as a special case.^ In Section
existence of
To
MTHPE
and
SWUE
for these
3,
we
establish the
games.
define a general agenda-setting game,
we first need a definition of a stage.
Inside one stage,
information sets are possible as the definition should be general enough to nest simultaneous
voting.
Definition 8
game
1.
A game
if it satisfies
Game F
F
in extensive
form with a
set of players
N
is
called
an agenda- setting
the following conditions.
consists of a (possibly infinite)
(a) a "proposing stage",
i.e.,
number
of stages,
where each stage
is
either:
a decision node and a number of possible actions of a single
player (possibly Nature), the agenda-setter at stage Q.
(h)
a "voting stage",
i.e.,
a connected subgraph of F, in which each player
most one decision node and two actions
2.
e
N has
and a" (0)
at
.
For each voting stage 0, there are only two equivalence classes of continuation-payofiidentical
3.
at this node, say a^ (0)
i
subgames following terminal nodes of
In each voting stage 0, for each player
i
£
in
game
F, say y (0)
and n (0).
N and for any other players' actions held fixed,
action of (0) does not decrease the probability of moving into a
subgame that belongs
to
the equivalence class y (0)
*
Another trembling hand refinement used
of
MTHPE. A
in
some neighborhood
in the literature, truly perfect equilibrium,
is
stronger than our notion
mixed profiles
any one sequence of profiles in the standard notion of trembling-hand
perfection and to one sequence of Markovian profiles in the case of MTHPE. However, this equilibrium concept
fails to exist in many games, including our dynamic game of coalition formation (except in some special cases).
"The study of agenda-setting games was pioneered by Romer and Rosenthal (1978) and Baron and Ferejohn
(1989).
truly perfect equilibrium requires strategies from
of
a rather than
to
a
to be best responses to all fully
.
For any two nodes
4-
£,
and
of
^'
exists a voting stage 8' of
game
game F
F,
if
.
^ and ^ belong to one information set, than there
that contains both ^ and
^'
This definition states that any game in which one of the agents makes a proposal and others
vote in favor or against this proposal
restriction that
more
all
dynamic voting games
and then subsequently, a subset
and then
of,
or
in
less likely to succeed).
intuitive
Agenda-setting
which at some stages proposals are made,
players vote in favor or against these proposals.
all,
dynamic bargaining games, where a
also include several
stage,
do not make a proposal
"yes" votes
games naturally cover
an agenda-setting game (and imposes the
is
division of a pie
They
offered at a certain
is
needs to be accepted by the other participants acting simultaneously or in
this
sequence, as special cases.
Characterization
3
Existence Results
3.1
We start
is,
in
a very broad class of games.
nonetheless, a sufficiently strong equilibrium refinement to rule out
Theorem
MTHPE
1
Any
player
N
£
i
finite or infinite extensive-form
(possibly in
Proof. Suppose
is
first
mixed
that
game
with a finite
(a^
Markovian strategies
(rj)
,
.
.
.
,a"
(77)).
T
is finite.
non-intuitive equilibria
number
of stages has a
Consider a perturbation of the original game where each
By
t]
>
where
0,
the standard fixed point theorem argument, this perturbed
only.
Therefore, the perturbed
Because the action space has
can choose a sequence
limit.
it
strategies).
has a Nash equilibrium; moreover, the fixed point theorem applies
to
later prove that
restricted to play each action at each stage with probability t]\^>
sufficiently small.
is
all
We
games.
in agenda-setting
rj
MTHPE exists
by observing that
77^,77^, ..
.
we
restrict
our attention
game has a Markov
Perfect Equilibria
dimensions and
thus compact, we
finite
which converges to
if
game
such that (a^
(rj^)
is
,
.
.
.
,a"
(t?'""))
This limit would be a trembling-hand perfect equihbrium in Markovian strategies,
has a
i.e.,
an
MTHPE.
If
T
is
infinite,
the previous reasoning applies straightforwardly
if
instead of "stages"
"classes of stages with payoff-equivalent histories"
Theorem
2.
2
1.
In any
finite
Any
(finite)
game, a
agenda-setting
MTHPE
is
a
game has
SWUE.
10
a
MTHPE
in pure strategies.
we use
.
3.
Any
Proof.
We
(1)
SWUE in pure
agenda- setting game has a
finite
strategies.
proceed by induction on the number of stages.
consider a one-shot agenda-setting game.
stage
If this
is
perfect. If the single stage
of the outcomes to another.
outcome
MTHPE,
We
MTHPE
agenda-setting
there
subgames there
same
ctJ,
MTHPE
a^,
.
.
.
for
in all agenda-setting
i
if
player
each j G
{1,
.
actions for infinitely
(at
.
MTHPE,
in the corresponding
MTHPE), and we
and take any m.
[i)
i
which
is
ct^, cr^,
.
.
.
i.
we can proceed
MTHPEs
i
first
for
a^.
In each of k corresponding
MTHPE; we
j„ such that Cj^
It is
is
now
is
=
for infinite
.
Take another
.
.
for
any
n.
action aj that
is
weakly better than other
is
of the whole game.
a voting stage. Then there
MTHPE
number
a^
1.
Now
i
MTHPE.
11
.
are played, respectively, then
There
is
an action
consider two subsequences of
m
for
which a
find action a{i).
number
It is
{i)
is
(z)
weakly
will get action
of players
now
the stages starting from the second one and actions a
stage form a pure strategy
.
Consider one of the players
n, to another one.
of m's.
profiles cr^, cr^,
and repeat the procedure; then we
any player
are, essentially,
in each (if they are isomorphic, take
which are formed by values of
for
Now, consider
straightforward to prove that aj, along with
MTHPE
player, j,
and
Moreover, we can require
cr",
of the previous subsequences. Since the
and
can choose the
weakly better than other actions.
has two actions, y and n, in stage
ct^,
where
MTHPE for each such j
1-stage games.
the subgames strategies a"^ and
in this way,
chosen
.
can similarly construct two sequences of strategy
and u^,
and two subsequences
at the
T—
of actions, there
values of n.
weakly preferred
better for player
(j)
If in
,
.
a pure strategy
weakly prefers to choose one of the actions, y or
sequences
a
is
forms a pure strategy
and (j\,a\,... Each player
a
n, there
number
many
.
j\,J2) are isomorphic, then ct"
any
start with the case
which converge to a pure strategy
/c}
=
ai,
than T; take an
less
Therefore, there exist sequences of strategy profiles
most) two subgames; take a pure-strategy
player
weakly prefers one
i
We
its first stage.
consider the case where the initial stage
the same
(i)
,
.
(for j
Since there are a finite
Now
Consider
stages.
isomorphic subgames.
for
at first stage. For
the chosen
games with number of stages
exists (by the induction hypothesis)
two subgames
i
Markovian
Suppose that we have proved the existence of
making a choice between k actions
which are Nash equilibria
that
T
game with
one player
is
voting, then each player
it is
straightforward to verify that voting for a weakly preferred
It is
next proceed with the induction step.
pure-strategy
since
MTHPE.
an
is
is
indeed,
is trivial;
one-player move, then, evidently, the
action which maximizes his utility constitutes a pure-strategy
and trembling-hand
The base
is finite,
evident that the
for
each player
i
.
(2) Take any strategy profile
on the number of stages
with
it.
<
q'
By
Consider
This
MTHPE, when
moves
i
by
at first stage
>
u°.
action a*
If
possible over the set of feasible actions. Hence, this
consider the other situation where the
mixed
on how other players
let
/x"*", iJ,~
,
it
votes,
i
>
fi^
is
small then
yields fx'^uf
it is
it is
is
i,
and
is
a
this utility
is
if
implies
uf >
how he
a
a small
for
(/i"*" -I-
votes,
By
Depending
ry.
is
played. Thus,
if u'^'
<
u~'
it is
if
u^'
>
and player
definition,
n^) u^'
pf) u~' where u^' and u~' are
profile a'
maximum
the
a voting stage. Consider a profile a'
probabilities of these events.
+ {fx~ +
i's utilities
u~' and thus by definition of
,
MTHPE
is
i
in this
take any player
MTHPE.
This
who
i
a weakly dominant strategy
is
himself and other players)
weakly dominant.
If,
and thus
this
any
MTHPE
Similarly,
.
= u~
u^
inated. Therefore, for
participates in voting. If
is
if
uf > u~ then he
for
,
him
,
is
a
SWUE.
1,
from acceptance
If
player
rj
i
i's
is
sole best
sufficiently
must support
uf < u~
votes for the proposal
plays in this
never pivotal, then any strategy
player, the strategy he plays in this
=
(given continuation strategies of
uf < u~ then the strategy he
or the player
pivotal;
in expectation,
u~', then player
it.
accepted
^'^+^~+fi'P
+ A*~'"i~'
voting against
is
the proposal in equilibrium with probability one. Similar reasoning applies to the case
Now
with a
SWUE.
?7-close to
rejected regardless of
voting for the proposal, and
u^ > u~
stage
Voting for the proposal yields
0.
and rejection of the proposal
response
first
MTHPE
vote, three mutually exclusive situations are possible: proposal
player
and by assumption
an action played with a
is
actions played in a
all
MTHPE
and suppose that
strategies
and ^^ be the respective
voting against
in
uf for any other feasible action a (otherwise
non-zero probability yield the same expected utility for player
how
MTHPE
at this stage; denote his expected utility (in this
there would exist a payoff-improving deviation). Hence,
regardless of
games
its first stage.
from taking action a
consisting of fully
for
truncated to any of the game's proper subgames, forms a
non-zero probability in equilibrium then u^'
Now
proved by backward induction
is
game. Suppose that the Theorem has been proved
Suppose that only one player
MTHPE)
MTHPE.
Consider an agenda-setting game with q stages and take any
q stages.
induction, this
SWUE.
in the
a that forms a
MTHPE is
is
MTHPE
is
weakly undom-
weakly undominated,
This completes the induction step.
(3) This follows from (1) and (2) of this Theorem.
Existence of an
will
MTHPE
be exploited below
for
SWUE
in
(existence of an
procedure,
is
a
equilibrium in pure strategies
full
is
at first a surprising result
characterization of these equilibria in agenda-setting
and
games
pure strategies, which can be computed by a backward-induction
obvious).
12
(0,0)
(1,1)
Figure
A Game
3:
Relationship Between
3.2
SWUE
However,
MTHPE
and
(1,1)
With
SWUE
which
MTHPE
not
is
MTHPE and SWUE
concepts are not equivalent, and neither
in general stronger
is
than the other (though see Theorem 4 below).
Example 3 There
exists
SWUE is not
an agenda-setting game where an
a game of two players with extensive form depicted on Figure
because at each stage only one player has a (non-trivial) move.
{R,r). However, there are two
there
SWUEs:
a non-zero chance that player 2 will play
is
As an
alternative to the concept of
The
{R,r) and {L,r).
SWUE,
player
I,
This
3.
1 is
It
is
an
is
Consider
an agenda-setting game,
game has
latter
MTHPE.
MTHPE
exactly one
not
MTHPE,
because
if
better off choosing R.
one might consider Sequentially Weakly Dominant
Equilibria, defined as follows:
Definition 9 Take any
t:l<t<T+l.
weakly dominant equilibrium
any
h} that
a
'
may
,a
occur after
a
:
,
.
.
.
if ((t\
/i*~^
we use
,a
(ct^,
.
.
.
,17")
.
,(t") is
ti
I
\
the convention that
Strategy profile
.
is
(ct^,
.
.
.
,ct")
is
a h^~^ -sequentially
a h* -sequentially weakly dominant equilibrium for
and
for
(Here,
.
Strategy profile
all
any
>
u
G
i;*'*,
CT*'*
profile is
[
a
,a
for
a
.
all cr'''^
G
,...,(7'
'
S"^'*,
\
and for
=
all i
h^ -sequentially weakly dominant
a Sequentially Weakly
n
I,.
.
.
,N.
equilibrium.)
Dominant Equilibrium (SWDE)
if it is
a
h^ -sequentially weakly dominant equilibrium.
Note that we do not require that the inequality holds
order to ensure existence in situations where one player
Clearly,
SWDE
shown that
is
much
we can
games
indifferent
stronger equilibrium concept than
for agenda-setting
for general finite
precisely,
a
is
games the notions
SWDE
is
of
between
SWUE.
13
(7''*,cr~*'*
different
Nevertheless,
SWDE and SWUE coincide,
a "strictly" stronger concept and
establish the following result.
one
strictly for at least
may
in
outcomes.
it
can be
though naturally
fail
to exist.
More
Theorem
a
is
2.
3
SWUE is a SWDE (and thus a MTHPE
SWDE in pure strategies.
In any finite agenda- setting game, a
1.
SWDE).
In particular, such games have
In any finite game, a
SWDE is
a
SWUE
(so for agenda- setting
games
these notions coin-
cide).
game where
There exists a
4-
There exists an agenda-setting game where a
Proof. (1) This
stage;
finite
easily
is
may be proved
tially, it
voting; take
stitute
a
i
given that a
i
i.
If
he
assume that
SWUE
<T,
if
SWDE
not a
is
SWUE
MTHPE.
The base
a.
SWDE
and
in profile a.
the case where
accepted and
is
uf > u~ must hold (according
indeed, voting "against"
is
Now assume
action a. Finally,
if
if
must be that u^
.
a
is
would con-
pivotal with
votes "for" the proposal. Denotes his
i
rejected by u'^
and
u'^
respectively, then
,
to Definition 4; otherwise voting "for"
a prescribes a mixed strategy (and
= u~ But
that he
is
then again, any player
i is
would be
weakly dominant
also a
the only alternative action, and hence u'l
is
is
plays a pure strategy; then without loss
i
prescribes that player
the proposal
take any
allow for any action that
a dominated strategy at this stage). But in that case, voting "for"
strategy:
Step:
played from that stage on. Suppose the stage involves
one he plays
first
SWUE
is
SWUE
(essen-
is trivial
pivotal with probability zero, then any strategy
is
in particular, the
continuation payoffs
then both
acts,
a positive probability. Consider
of generality
SWDE.
not a
with exactly the same words as the induction step).
some player
SWDE,
a
proven by induction. Take any
only a single player
if
best response of player
in
MTHPE is
3.
>
uf for any
pivotal with a positive probability),
strategy at this stage forms a
i's
SWDE
after the corresponding history. This proves the induction step.
(2) Again, this
trivial.
is
proved by induction. Take any
Step: take any player
i
finite
game and
and suppose that he plays strategy
SWDE a.
ct*'*
G
The base
!]*•'.
is
again
This implies, by
definition, that
Ui{a''',a-''') >u^{a'^',a-''')
(we drop conditioning on past and future actions
immediately implies that condition
contradiction, that there
strategies
ct~*'*
G
S"'-*
is ct''*
G
(1)
(3)
for brevity) for
holds for any
a''*
G
S*'*.
Now
a^'^
and
u"*'*.
This
assume, to obtain a
such that for at least one combination of other players'
S'''
we have
Ui
[cr
'
,a
'
)
>
Ui
{^a
'
,
a
)
(recall that Definition
4 requires that one of the inequalities
condition violates
This contradiction completes the proof.
(3).
any
14
(2)
be
strict).
However,
this last
(3) Consider a one-stage
game with two
players
making simultaneous moves with payoff
matrix
SWDE,
This game does not have
I
r
L
(1,1)
(0,0)
R
(0,0)
(1,1)
because
players has a weakly dominant strategy.
and {R,
r) are
MTHPEs
SWUE is SWDE in
SWDE is not MTHPE.
MTHPE
3.3
in
In this subsection,
We
games.
It is
one-stage and in that only stage neither of the
straightforward to check, however, that both (X,
Example
3.
SWUE
Indeed, in this example the
an agenda-setting game. Hence, there
Agenda-Setting
is
also a
is
one player takes an action,
it
of the structure of
MTHPE
into
an alternative game
in
stages where there are simultaneous
i.e.,
as
an agenda-setting game where
in agenda-setting
define the notion of a regularization of an agenda-setting game.
game transforms
MTHPE,
Games
we provide a characterization
of an agenda setting
I)
of this game.
(4) This follows from
any
it is
.
which
A
at
regularization
each stage only
moves are replaced by a
sequence of stages corresponding to each one of these moves.
Definition 10 Suppose that T
tion of
game V
if it is
is
among
i
&
X
move
players
X
C
N
i
has the same two actions that he had at stage
^^
after {cj (C)}iex ^^ played at stage
played
if
and only
game
is itself
more
players at the
game
F'.
same
is itself).
We
a
abuse of notation, we say that u
Notice that the converse of the statement
is
15
Crucially,
next show that
of
game T can be
to a strategy profile a' of a regularization F', so that players play the
slight
game T
set of equilibrium payoffs to all players.
establish this result, first note that any strategy profile
With a
game
the
that has simultaneous
an agenda-setting game, with the unique regularization that
as in profile a.
and
same stage (and the regularization of an agenda-second
different regularizations will also lead to the
mapped
which players
if the original
however, each of these regularizations has the same set of terminal payoffs.
To
^,
If there is
£,.
Clearly there are several regularizations of an agenda-setting
actions by two or
an action.
that each stage only one player takes
proceeds with subgame y (^) after profile {at (Oliex
{£,)
F' a regulariza-
at stage ^, then ^ is substituted by \X\ stages in
sequentially. Player
proceeds with y
game
call
obtained from T by substituting every stage where more than one player
moves by a sequence of stages such
voting
We
a (finite) agenda-setting game.
is
not true, since a'
naturally
same actions
a strategy profile of
may
game
involve a pattern of
.
.
actions that cannot be captured by
cr,
example, when later actions depend on earlier actions.
for
MTHPE
Nevertheless, the next theorem shows a strong equivalence between the set of
agenda-second game and the
Theorem
1.
2.
set of
MTHPE
is
a finite agenda-setting game.
MTHPE
of
game
any regularization
T'
a
a
is
Conversely,
MTHPE
a
if a' is
F, then
Proof.
by
F'
Take
(1)
MTHPE
MTHPE).
Since the
game
.
,(7")
.
we may, without
is finite,
and a SPE)
MTHPE a
then there exists an
fully
mixed
—
as r
at every stage or
i
SWUE
Denote the corresponding strategy
F.
profile (cr^,.
every player
,
(and thus an
in
m
T
payoffs as in a'
Consider a sequence of
which converge to the equilibrium
(r) for
MTHPE
an
in regularization T'
a of game
a' to avoid confusion.
best response to a
it is
same
in which all players obtain the
an
(and SPE) of any regularization of the same game.
4 Suppose T
If
of
profile in
profiles (g^ {r)
,. ..
oo and such that the
>
node ^
,a^
(r))
ct*'*
is
a
by the definition
(this exists
assume that
loss of generahty,
game
at every
voting stage every player that participates in the voting has the same preferences over the two
and n
alternatives y
sequence
for all r's in the
(i.e.,
This implies that at voting stage, each player either
between the two
indifferent
to
(cr'^
(r)
,
.
,
.
.
{a'^ (r)
mixed and converge to
a'^ (r)) for every player
a single action by some player in
game
F,
(cr^ (r)
well, for the
game
,
.
.
.
F.
,(t" (r))
If
is
same reason
then he must have picked
i
.
.
—
(ct'^
it is
(r)
,
.
game
F'; obvi-
oo. Let us verify that a' is a best response
»•
any node ^ and
then
of the regularized
,(t'" (r))
.
any
for
r.
If
node
corresponds to
£,
obvious, since the payoffs of
.
,
.
cr'"
and
(r))
(ct^ (r)
,
.
.
.
,
all
actions are
ct" (r))
have the
were indifferent about the outcomes of the voting in F (provided
played after that voting), then he
as before.
it
in
If
game F
strategies were fully mixed). In the
he
strictly preferred
(since he
game
indiflferent in
is
this voting,
the
one of the outcomes
game
F' as
(say, y(C))i
was pivotal with a positive probability since
F', if profile (tr'^ (r)
,
.
.
.
,
a'" (r))
of action af (^) cannot decrease the likelihood that the corresponding
is
played, his choice
subgame
is
reached after
because other players' strategies in this (now sequential) voting do not depend on
his action. Consequently, choosing af (^)
i
is
distribution over the payoffs at terminal nodes. Consider the case where ^ corresponded
to a voting in
that
,
a' as r
at
i
the same, as the continuations of profiles
same
strictly prefers y, or strictly prefers n, or
for all r.
Consider the corresponding profiles
ously, they are fully
we can always take such subsequence).
strictly preferred
outcome n
{£,) is
is still
a best response for him.
The
case where player
treated similarly. This establishes that profile a' consists of
16
best responses to
(r)
(cr'^
,
.
.
.
,
and
(r)) for all players, at all stages,
cr'^
for all r's,
a' is
i.e.,
a
MTHPE.
Take any
(2)
(^a'^ (r)
,
.
.
that the
part
1
,
.
MTHPE
which converge to the equilibrium
(t'" (r))
ct"'*
a best response to
is
we may, without
,
profile (<7'\
a' (r) for every player
assume that
loss of generality,
.
.
a'") as r
,
.
^ oo and such
node
at every stage or
i
As
^.
at every voting stage every player
same preferences over the two
participates in the voting has the
in the
Consider a sequence of fully mixed profiles
a' in regularization F'.
and n
alternatives y
in
who
for all r's
sequence (note that these preferences are well-defined, since a' must consist of Markovian
strategies).
For any voting node ^ in game F, take the probability p
in the
corresponding voting in game F' under profile
reaching this node; note that this probability
all
votings in F' that
One can
players
if
they
y
{^) is
who
who
strictly prefer
voted af
not sufficient for n
indifferent
now
Let us
decision, let
p
(^)
af
(^).
players
players
Finally,
if
p (^) >
who
p{^) G
taken conditional on
is
0,
and n
in F'. If
it is
then the set of players
{^)
subgames
a
as follows.
if
p(0 ^
the
same
for
then
(0, 1),
(Oil)>
n
is
players
and
(^),
p
{£,)
n
strictly prefer
then there
i^
game
with p (0
and
=
is
(^)
to
a player
who
(this
is
who were
makes a
F' (there,
0, let all
and
strictly preferred
those
let
i
let
single
he chooses
players
the rest choose of
let
or are indifferent choose a^ {^)
n choose a"
then the set of
to be reached even
(^)
At a node where player
as in corresponding nodes of
let all
probability of acceptance
who
1,
for all r.
At a voting node
prefer y to
<
p(C)
not sufficient for y
be reached. Moreover,
Markovian).
strictly preferred
way that the
with
(^)
about votings
indifferent (for all r above) choose a" (^),
1, let all
who
if
him choose the same action
is
probability
well defined in the sense that
(^) (for all r's) is
define strategy profile
n to y or were
=
n
(^) to
(^) to
between y
the same action as a'
prefer
y
Similarly,
(^).
The
correspond to node ^ in F since a' consists of Markovian strategies.
easily check the following statements
all
is
may
is
a'.
that the proposal will be accepted
{£,)
who
(^). If
the rest choose
y choose af
indifferent
mix
possible, since neither of the
in
(^), all
such a
two groups
strict preferences is sufficient to enforce either of the decisions).
Let us verify that
(fj^ [r) ,.
.
.
,a^
(r))
ct is
a
MTHPE.
Observe that
which converges to a and that
for all players as profiles (a'^ (r)
best responses, due to the
,
.
.
.
,
cr'"
(r))
way we defined
for
any
for every
r
we can take a
subgame
in
fully
mixed
profile
F have the same payoffs
chosen above. Then at each node of F players play
profile a.
This completes the proof.
This result establishes the equivalence of agenda-setting games with simultaneous and
quential votings once a proper equilibrium concept
also highlights the importance of
MTHPE
as
(MTHPE)
is
used.
More
importantly,
seit
an equilibrium refinement. Example 6 shows that
17
part (1) of
Theorem
4 would not hold
Appendix shows, among other
in the
we considered
if
things, that
it
THPE
would
MTHPE. Example
instead of
also not hold
we considered
if
5
se-
quential elimination of weakly-dominated strategies (and in particular, with this notion certain
non-intuitive equilibria
theorem estabhshes a "generic uniqueness" of
Finally, the next
games. Example 6 in the Appendix again shows that
Theorem
SWUE
MTHPE
survive). Thus, the equivalence result for
still
5 Suppose agenda-setting game T
it
would not hold
does not hold.
in agenda-setting
THPE.
for
such. that for any two terminal nodes,
is
one
if
player obtains the same payoffs in these two nodes, then each player obtains the same payoffs
Then
two
We
ui and
payoffs,
and
prove the following statement by induction by the number of stages:
(a) for
any
in
same
these payoffs are the
Proof.
MTHPEs
any two
same
in both nodes.
MTHPEs
(Ti
and
as those in
fj2 all
any
players get the
SPE
all
(72,
players obtain the
of any regularization ofT'.
same
payoffs,
and
(b)
if
at
MTHPE
some
a players
get a certain expected payoff vector, then there exists a terminal node where players get these
payoffs. Let us prove the stage (the base
i
making a
the
first
node of the game involves player
decision, then of all available actions he will choose one
subgames are
gets the
same
realized at
payoff,
some terminal nodes
and by assumption
of the statement; part (b)
induction, either
all
which maximizes
In the
strict preference.
same. Again, part (b)
To complete the
SPE
of
move one
or
in
first case, all
MTHPE
is trivial.
proof,
and where
MTHPEs
his utility.
proof
may be done by
to
show that these
4, all
he
stage involves voting.
By
same payoffs
all
players have
for trivial reasons,
and hence the payoffs are the
payoffs coincide with the payoffs in
of the game). It remains to prove that in a
in
indifferent,
(a)
players get these payoffs in any
any two terminal nodes either
all
all
MTHPE
of the
game where
players
same
payoffs
players get the
players in any two
any
SPEs
are the same. This
induction, similarly to the one in the beginning of this proof.
Application: the Elimination
To demonstrate the
involve the
is
This proves part
payoff.
first
i
This proves the induction step.
players get different payoffs, the payoffs of
all
same
strategies of players are unique,
we need
some SPE
at a time
players get the
Consider the case where the
is trivial.
any regularization. By Theorem
game (and thus
all
of the subgames. Since player
players are indifferent between the alternatives y and n, or
while in the second, the
4
If
there are several such actions then, by induction, the payoff vectors of the corresponding
If
a
is trivial).
usefulness of the
endogenous club formation game
MTHPE
Game
and
SWUE
concepts,
(or "pohtical elimination"
18
we now use a
game) studied
in
version of the
Acemoglu, Egorov,
and Sonin
is
an
(2008). This
is
an infinite-horizon dynamic game with the following structure: there
initial collection of individuals,
power
political
>
7^
(for
of
We
total
assume that powers
Ix +
.
.
,n}, with each individual
.
Xt C
A^,
and members of
members, thus determining a new ruling
its
Xq = N. The
have
{1,
example, representing individual
there exists a ruling coalition
some
N=
power of any coalition
X
C
this ruling coalition
coalition
A^
Xt+i
C
N
endowed with
At each date
"guns").
z's
€
may
Xt-
t
>
1,
vote to exclude
At time
t
=
we
0,
is
{ijjjgyv ^^^ generic in the sense that
if
X,Y c
and
A^
X^Y
,
then
7y-
In any period
>
i
Nature at random
1,
the voting proceeds as follows.
Alternative
An
becomes the next ruling
member
votes.
If
who
coalition,
an absolute majority of the available ''weighted
of each
agenda-setter, which
is
determined by
At^i
which
is
a
are entitled to do so vote for or against
i.e.,
Xt^\
=
Aj^j, if
and only
if it
receives
where votes are weighted by the power
votes,^^
of the junta, so that an individual with a greater 7j has proportionately
more
the proposal At^t does not receive an absolute majority, then the next agenda-setter
nominates a proposal and so on. In case no proposal
The
An
through any fixed protocol) proposes an alternative coalition
(or
subcoalition of the current coalition. All individuals
At^i-
i
preferences (for each
accepted, Xt+i
is
£ N) consist of two parts. The
i
=
Xt-
first is utility
from power,
oo
[/+
= (l-/3)]B,^/3--*;^I,ex..
^^t
T=t
Here
/3
G
(0, 1)
is
function for individual j being a
expectations at time
t.
form
is
members
member
The term 7j/7xt
ruling coalition members.
coalition
It
member
Formally, the extensive form of the
picked randomly by nature; yet
(with
=
0, 1,
individuals, li^Xr
.
all
.
.,
t,
is
the indicator
and Et denotes
represents the power of the individual relative to other
in proportion to their power.
stage, j
all
of the ruling coalition at time
that each player obtains greater utility
At each
across
can be motivated from the division of a unit
Therefore, each prefers to be a
1.
common
the discount factor
The important
when
size pie
among
the
implication of this functional
the power of the ruling coalition
is
smaller.
of a smaller ruling coalition.^
game
is
as follows.
We
assume that an agenda-setter
is
the results go through for any fixed protocol of agenda-setters.
the
game
starts with
an intermediary coalitions by Nj
C
N
No = N).
^ These payoffs are a special
case of the more general payoffs introduced in Acemoglu, Egorov and Sonin (2008)
and are adopted to simplify the notation here (without affecting any of the results).
19
,
=
2.
Nature randomly picks agenda
3.
Agenda
4.
All players in Xj^q simultaneously vote over this proposal; let
setter
e Nj
ij^g
G
Xj^q. Let
YesjXj,,}
=
{z
G X^,,
ior q
=
If Xj^g
Nj
|
{i,Xo,q)
Vj^q
li q
<
b,9+i
=
Then,
y}).
with Nj+i
1
\Nj\,
is
game proceeds
otherwise the
G
{y,
n} be the
in favor of this proposal
if
to step
=
from Xj^q add
—e
by
1).
Xj^q (and j increases
then next agenda setter
7^ ^j,r ioT 1
to step
Nj
Xo,q)
Vj^q {i,
i&Nj
Nj, then we proceed to step 7 and the
stage j) and the
7.
5;
\ Xj^q are eliminated, players
to step
6.
a member of the coalition
Xj^q has the absolute majority of the weighted votes within Nj, then the
if
proceeds to step
5.
(i.e.,
Yes {Xj^q} be the subset of Xj^g voting
i£Yes{A'j,,}
i.e.,
1
P (Nj).
proposes a coalition Xj^q G
i
vote of player
(i.e.
ij^q
setter
<
r
<
q
(i.e., it is
game proceeds
G Nj
ij,q+i
picked
among
is
game
6.
game
Otherwise players from
ends.
to their payoff,
and the game proceeds
randomly picked by nature such that
those
who have not made a proposal
to step 3 (with q increased by
1).
at
Otherwise, we proceed
7.
becomes the ruling
coalition of this terminal node,
and each player
i
G Nj adds
Wi {Nj) to his payoff.
game form a sequence Nq D Ni D
Coalitions that emerge during the
the
number
.
Summing
disutility cost equal to e
>
over the payoffs at each node, the payoff of each player
•
Ui
=
Wi (Nj)
-eY,
In,
.
D Nj where
j is
Whenever there
of coalitions (except initial one) that emerges during the game.
a non-trivial transition, each player incurs a
small).
.
(e
i is
is
can be arbitrarily
given by
(i)
(4)
i<j<J
where Ix
(•) is
the indicator (characteristic) function of set X. This payoff function captures the
fact that individuals' overall utility in the
game
is
of rounds of elimination in which the individual
number
With a
of players ehminated equals
slight
|A'^|
abuse of terminology, we
-
|A'j|,
refer to j
20
related to their share Wi
is
and
to the
involved in (the second term in
and there are
totally J
number
(4)).
The
rounds of elimination.
above as "the stage of voting."
.
The next two examples show
game
that, as in
Examples
and
1
2 in the Introduction, in this
there are also several "non-intuitive" SPEs, and in fact, there are several "non-intuitive"
Markovian SPEs.
Example 4
suppose that a
coalition
(and
the
is
The
rest).
=
it is
A, B, and
clearly.
SPE, where
all
C
make a
{A^B,C}
{A,B,C,D} which
should be the ultimate ruling coalition
is
7^,
is
10,
and
rejected by the
is
any two-
intuitive:
its
members. And
exists another
and everybody votes
to a pre-fixed protocol
C} and
Consequently, both {A, B,
We
is
{A, B, C,
D}
can emerge as
not a reasonable outcome, since A, B, and
can also note that
not intuitive in the sense that
is
=
indeed an SPE, because no deviation by a single player can lead
a SPE, even though the former
Yet, this equilibrium
all
it is
C have
players are using Markovian strategies.
resulting from
an
artificial
coordination
among A,B, and C.
The main
result for this application
application of
MTHPE
Proposition
1 Let the
4>
is
given in the following proposition and shows
mapping
(X)
—
(p
min
SWUE (MTHPE)
most one stage of elimination and coincides with
of each
i
is
7y.
(5)
where the ultimate ruling coalition
after at
N
the
be defined as
arg
there exists a pure strategy
&
how
removes the non-intuitive equilibrium leading to {A, B, C, D}.
Y€{ZCN: iz<ix; lz>lx\z; 4>{Z)=Z}U{X}
Then
and
5
a
proposal, he proposes
make proposals according
players
enough votes to eliminate D.
failure
7^ =
would prefer {A, B, C} to {A, B, C, D}. Nevertheless, there
to acceptance of any proposal.
in
4,
"unstable" and would lead to the elimination of one of
against any proposal. This
outcomes
=
MTHPE where A, B, or C proposes the
B and C vote for in favor of this coalition
is
following this proposal. A,
to
first
fact that
person subset of
{A,B,C,D}, with 7^ = 3,7^
1/2. It can be verified that there
{A,B,C}, and
D
if
N =
Consider
(p
{N) with probability
1,
is
and
reached
the payoff
given by
Ui (N)
Vice versa, in any
SWUE
=
Wi {X)
- el^N)
(i) I{<f>{N):^N}-
in pure or mixed strategies (and in
strategies) the ultimate ruling coalition is
(p
{N), and
it is
any
(6)
MTHPE
in pure or
necessarily reached after at
mixed
most one
stage of elimination, and players payoffs are given by (6)
'
Proof. Acemoglu, Egorov, and Sonin (2008) considered a version of
voting and established that the unique
SPE
is
characterized by
21
this
mapping
game with
(5)
,
sequential
with payoffs
(6) as
given in the proposition. Given the notion of regularization introduced here, this impHes that
they considered a regularization of the game here. Then Theorem 5 immediately applies and
establishes that the unique
MTHPE
is
given by (5) and
(6),
completing the proof.
This proposition provides an illustration of the usefulness of the concept of
of considering a
Simon
all
5
In
more complicated game with sequential voting
(2008), one can consider a simpler
game and obtain
Instead
Acemoglu, Egorov, and
as in
exactly the
MTHPE.
same
results (eliminating
non-intuitive equilibria).
Conclusion
many
political
generally, in
economy models, which
what we defined
equilibria are "non-intuitive"
involve voting or multilateral bargaining, or
as agenda-setting games, several
.
Nash
In addition, details concerning voting
cedures often have a major effect on the set of equilibria. In
many
or
subgame
more
perfect (Nash)
and proposal making pro-
applied models, non-intuitive
equilibria are eliminated (sometimes without specifying the exact notion of equilibrium),
and
modelers often have to make sure that the "right assumptions" are imposed so that unimportant
details
do not influence conclusions.
In this paper,
we proposed two
related equilibrium refinements for voting
games. Sequentially Weakly Undominated Equilibrium
Perfect Equilibrium
(MTHPE). These
or deterministic sequence of agenda setters
MTHPE
(SWUE) and Markov Trembling Hand
concepts are widely apphcable, easy to use and eliminate
non-intuitive equilibria that arise naturally in
tight characterization of
and agenda-setting
dynamic voting games and games
make
offers to several players.
in agenda-setting
in
We
games and showed that the
does not depend on features such as whether voting
is
which random
also provided a
set of
simultaneous or sequential.
that these concepts can be applied in a wide variety of political
MTHPE
We
believe
economy models and can both
eliminate the need to devise ad hoc equilibrium refinements to remove non-intuitive equilibria
and ensure that major predictions do not depend auxiliary assumptions that are
to reality (such as whether voting
is
simultaneous or sequential).
22
difficult to
map
References
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in
Non-
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A
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A
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-
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Observable Actions", Jour-
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-
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23
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24
4:25-55
,0)(0,0,0)
(1 ,1
(1 ,1
,0)(0,0,0)
(1 ,1
,0)(0,0,0)
(2,2,0)
Appendix
Example
5 Consider the following extensive-form game. In this example, player 3 gets payoff
However, his choice of action
in all terminal nodes.
2.
First, players 1
and 2 simultaneously choose
a simultaneous vote.)
2.
Otherwise, player
players
R and r,
MHTPEs
(There are multiple
and
r,
suppose that players
is
indifferent
and 2 choose
1
If
is
or
indeed what happens in any
(I/,
/,
01,62,63),
Then, neither
L
nor
I
which
(iJ, r),
both get the payoff of
MTHPE.
2,
SPE
However, the following unintuitive
an SPE. None of player
is
that
between the outcomes).
eliminated by sequential elimination of weakly dominated strategies.
profile
this as
One would expect
0.
and
both get the payoff of
respectively, then
1
1
their actions sequentially (which corresponds to
they play
arguably the most intuitive outcome.
the payoffs of players
(We can think about
their strategies.
respectively. This
as player 3
sequential process of casting votes).
is
R
aflfect
choice determine whether they both get
3'
and 2 should play
1
Now
they play strategies
If
may
and
this
cannot be
Consider the strategy-
3's strategies is
weakly dominated.
can be eliminated.
Example
6 Consider a game of three players with extensive form and payoffs as shown on
Figure
The
4.
first-best;
and
if
'bad'.
first
two players vote, and
one of them votes
if
for the 'left'
All players receive the
same
both vote
for the 'right', all three players receive
then the third player chooses between 'moderate'
in all terminal nodes, so there
is
no strategic
conflict
between them.
Equilibrium
efficiency
is
(i?, r, (ai,
L
(01,62,63)). Indeed, take
=
((1
-
is
trembling-hand perfect, but so
is
(Z/,Z, (ai,
not achieved because of 'herding' in voting (note that neither
strategies: for instance,
a"
02,03))
7?3)
L+
best response to second player playing
some
rj^R, (l
Evidently, player 3 (and
is
-
not
I
are dominated
and third player playing
and consider
t]
rj^)
all his
/
L
02,03)) where
I
+ rj^r,
((l
-
i?^)
oi
+ 77^61,
(1
-
77)
02
+ 7762,
agents in agent-strategic form) are better
25
(1
oflF
-
r?)
03
+ 7763))
choosing oi over
.
(5,5,5) (0,0,0)
Figure
6i,
a2 over 62,
L, he obtains
he obtains
1
weight to
The
I.
A Game With
and 03 over
=
u/,
ur =
should put
4:
all
5 ((l
b ((l
—
(5,5,5) (0,0,0)
(5,5,5) (0,0,0)
63.
-
Now
(1
—
in
Trembling-Hand Perfect Equilibrium.
consider payoffs of player
-
7?^) (l
77'^)
Herding
77))
(7,7,7)
1
choosing
+rf {I - r])) = 5 - bif + 777^ = 5 — 577 + 277^ + 677^.
77^)
+
Sr;^
L
or R. If he chooses
brf.
If
he chooses R,
Hence, For small
player
77,
weight to L, and a similar argument would show that player 2 should put
This proves that (L,
effect that
Example
/,
(ai, 02, 03)) is also
6 emphasizes
required to be Markovian, which
is
a trembling-hand perfect equilibrium.
would not be the case
what our
all
definition of
if
fully
MTHPE
mixed
imposes.
were
profiles cr"
Indeed,
it
is
a
natural restriction to require that in the three subgames where player 3 moves and payoffs are
identical, his
mixed action
utility of player 1
profile cr"
due to the
by worse development
in the
should lead to identical place. In that case, the increase of
possibility of player 2 playing r instead of
subgame
if
he
still
plays
26
/.
I
would be not be
offset