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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Information Dynamics
Multiplicity in
of
and
Global
Equilibrium
Games
Regime Changes^
George-Marios Angeletos
Christian Hellwig
Alessandro Pavan
Working Paper 04-42
December 2004
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without c harge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=634381
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
PEB
1
2005
LIBRARIES
Information Dynamics and Equilibrium Multiplicity
in Global
Games
George-Marios Angeletos
MIT
and
.
Regime Change*
Christian Hellwig
Alessandro Pavan
UCLA
Northwestern University
chxis@econ.ucla.edu
alepavanOnorthwestern. edu
NBER
angeletSmit edu
of
December 2004
Abstract
Global games of regime change - that
which a status quo
is
been used to study
political change.
We
abandoned once a
crises
may
coordination games of incomplete information in
sufficiently large fraction of agents attacks
phenomena such
as currency attacks,
extend the static benchmark examined
to accumulate information over time
namics
is,
and take actions
lead to multiple equilibria under the
in the literature
many
in
bank runs, debt
periods.
it
- have
crises,
and
by allowing agents
It is
shown that dy-
same information assumptions that guarantee
uniqueness in the static benchmark. Multiplicity originates in the interaction between the arrival of information over
time and the endogenous change
that the regime survived past attacks.
in beliefs
induced by the knowledge
This interaction also generates interesting equilibrium
properties, such as the possibility that fundamentals predict the eventual regime
outcome but
not the timing or the number of attacks, or that dynamics alternate between crises and phases
of tranquillity without changes in fundamentals.
JEL
Codes: C7, D7, D8, F3.
Keywords: Global games, coordination, multiple
"For helpful comments, we thank
Andy
equilibria, information
dynamics,
crises.
Atkeson, Pierpaolo Battigalli, Alberto Bisin, V.V. Chari, Patrick Kehoe,
Alessandro Lizzeri, Kiminori Matsuyama, Stephen Morris, Ivan Werning, and seminar participants at Berkeley, MIT,
Northwestern,
NYU, UBC, UCLA,
Yale, the Minneapolis
FRB, and
the
SED
conference.
Introduction
1
Coordination games in which a status quo
abandoned once a
is
phenomena. Whereas the earher contributions
when
from
common
in the hterature focused
on the existence and im-
agents can perfectly coordinate with each other, recent work
on global games by Carlsson and van
has emphasized the
number of agents
have been used to study a variety of socioeconomic
takes an action that favors regime change
pHcations of multiple equilibria
sufficiently large
Damme
(1993) and Morris and Shin (1998, 2000, 2001)
fragility of this multiplicity to
perturbations of the information structure away
when
knowledge: a unique equilibrium often survives
agents observe a payoff-relevant
variable, such as the strength of the status quo, with small idiosjnacratic noise.
Variants of this uniqueness result have been established in the context of currency crises, bank
runs, debt crises,
and
Most
political change.^
abstracting from the possibility that agents
regime. For
many
In this paper,
we extend the
may have
may
regime change) or "not attack"
regime change in that period
static
benchmark examined
.
fundamentals - which
is
Attacking
is sufficiently
never
is
individually optimal
high.
The
game, thus
sufficiently
by allowing agents
a large
number
any given period, each
take an action that favors
and only
if
is
is
if
the probability of
abandoned
if
and only
G M, which parametrizes the strength of
commonly
referred to as the exogenous
Instead, as time passes,
the agents.
6.
tions that guarantee uniqueness in the static
is
is,
status quo, in turn,
common knowledge among
show that d3mamics may sustain multiple
private information
quo (that
either "attack" the status
agents receive noisy private signals about
We
alternative. In
the component of the payoff structure -
is
in the literature
and accumulate information over time. There
the fraction of agents attacking exceeds a critical value
the status quo. 6
static coordination
the option to take multiple shots against the
and two possible regimes, the status quo and an
agent has a binary choice: he
if
work assumes a
of the applications of interest, however, this possibility seems relevant.
to take actions in multiple periods
of agents
of this
equilibria
under the same information assump-
benchmark, namely that the precision of the agents'
high relative to the precision of the
common
initial
prior. Multi-
phcity originates in the interaction between two elements: the knowledge that the regime survived
past attacks and the
In the
first
arri\'al
of
new
period, agents find
private information over time.
it
dominant to attack
for sufficiently
low realizations of their
private signals. This ensures that, in any equilibrium, regime change takes place in the
for sufficiently
it
common
low
9.
At any future date, the observation that the status quo
certainty that
it is
not too weak,
for
otherwise
it
is still
first
period
in place
makes
would have collapsed under the
first
attack.
'See Morris and Shin (1998) for currency crises; Goldstein and Pauzner (2000) and Rochet and Vives (2004) for
bank runs; Morris and Shin (2004) and
regime switches; Atkeson (2000) and
Corsetti,
Edmond
Guimaraes and Roubini (2004)
(2004) for riots
1
for
and poUtical change.
debt
crises;
Chamley (1999)
for
This
effect implies
that no agent
wiUing to attack in any subsequent period
is
if
he expects no
other agent to attack. Hence, there always exists an equilibrium in which no attack occurs after
the
first
period. In fact,
equilibrium of the game.
if
no new information were
The
to arrive over time, this
would be the unique
possibility to take multiple shots against the regime
would then add
nothing to the static analysis.
However, as agents receive new private signals about
6,
the impact of the knowledge that the
regime survived past attacks on posterior beliefs eventually vanishes.
dependence of posterior
prior
is
beliefs
on the
initial
common
relatively aggressive, in the sense that
increase in the precision of private information
prior also diminishes over time.
induces a large attack in the
it
would induce
absence of past attacks. In this case, no action after the
In contrast,
when the prior
is
relatively lenient,
first
less aggressive
When
the
period, the
first
behavior even in the
period remains the unique equilibrium.
both the discounting of the prior and the discounting
of the knowledge that the regime survived past attacks contribute to
possible.
For the same reason, the
making new attacks eventually
Hence, there also exist equilibria where agents take multiple shots against the regime.
Indeed,
we show
number
of attacks can be sustained in equilibrium.
In the
that, unless the
benchmark model, we
game ends
for
exogenous reasons
assume away the
deliberately
of attack necessary for regime change varies over time.
in finite time,
any arbitrary
possibility that the critical size
This allows us to isolate the effects of
changes in information, as opposed to changes in fundamentals, on the dynamics of coordination.
Nevertheless,
we
also
show that the
multiplicity result
the fundamentals - in which case dominant actions
volatility of these
of the
shocks
is sufficiently
What
in posterior beliefs that follows
equilibrium
We
also
set,
exist in every period
- provided that the
any equilibriimi
by an equilibrium
arbitrarily closely
in the
sustains the multiplicity of equilibria and the corresponding dynamics
the combination of the arrival of
this shift takes the
may
robust to the introduction of shocks to
small. In the limit, as the volatility vanishes,
benchmark model can be approximated
with shocks.
is
new information
upward
again
shift
from the knowledge that the regime survived past attacks. That
form of a truncation
but
over time with the endogenous
is
game
in the
benchmark model
simplifies the construction of the
not essential for the multiplicity result.
is
emphasize that multiplicity does not originate from the presence of exogenous public
most
signals or the observation of the size of past attacks. Indeed, for
away any such source
of information
private signals over time.
What
and concentrate on the case
whatever the horizon of the game and
for
Finally, introducing
initial prior:
we assume
which agents receive only
the introduction of public news does,
dependence of the determinacy of equilibria on the
private information.
in
of the analysis
is
to "smooth out" the
multiple equilibria then exist
any prior mean and any relative precision of public and
endogenous signals about the
size of past attacks - a
simple form of (noisy) social learning - does not change the equilibrium dynamics, except for the
possibility that the
the arrival of
endogenous information revealed by the
new exogenous information and render
size of
an attack
may
substitute for
further attacks possible immediately after an
unsuccessful one.
The
either
existence of multiple equilibria leads to dynamics that might not have been possible with
common knowledge
the final regime outcome
(e.g.,
whether a currency
may
For example, fundamentals
or a unique equihbrium.
determine
eventually devalued) but not the timing and
is
number of attacks. Moreover, dynamics may take the form
of sequences of periods in
which attacks
can not occur and agents only accumulate information, followed by periods in which an attack
possible but does not take place, eventually resulting in a
from phases of tranquility to
new
attack.
An economy
is
can thus transit
any change in the underlying economic fundamentals.
crises without
Below, we discuss the relation of the paper to the pertinent literature. Section 2 then reviews the
static
benchmark and introduces the dynamic model. Section
equilibria. Section 4 establishes the multiplicity result
3 characterizes the set of
monotone
and discusses equilibrium dynamics. Section
5 examines robustness to the introduction of shocks in the fundamentals.
Section 6 introduces
public news and signals about past attacks. Section 7 concludes. Proofs omitted in the
main
text
are presented in the Appendix.
Related Literature.
global games. Heidhues
This paper contributes to a small but growing literature on dynamic
and Melissas (2003)
tarity that suiRces for uniqueness in
identify a condition of
an investment model;
dynamic
strategic
complemen-
this condition is also implicit in Gian-
nitsarou and Toxvaerd's (2003) uniqueness theorem for recursive binary-action games. Dasgupta
(2002) examines the role of social learning in a two-period investment
model with
irreversible ac-
Levin (2001) considers a global game with overlapping generations of players. Goldstein and
tions.
Pauzner (2001) and Goldstein (2002) consider models of contagion. Frankel and Pauzner (2000),
on the other hand, consider a complete-information dynamic coordination game, where uniqueness
is
obtained by assuming aggregate shocks and idiosyncratic
examines the
role of information
dynamics
in
games
inertia.
None
of these papers, however,
of regime change.
Closer to our analysis, Morris and Shin (1999) consider a dynamic model whose stage
is
similar to ours, but
game
where fundamentals follow a random walk and are commonly observed
the end of each period.
at
This reduces the analysis to a sequence of static games with a unique
equilibrium.
This paper departs from the above literature in that
information. In this respect,
it
considers dynamics as a natural source of
shares with Angeletos, Hellwig and Pavan (2003) - which considers
the signaling effects of policy in a static
may
it
game - the
idea that endogenous information structures
overturn uniqueness results in global games and lead to predictions that would have not been
possible with either
Finally,
common knowledge
or a unique equilibrium.
we share with Chari and Kehoe
(2003) the motivation that understanding the dynamics
of information
may help understand the dynamics
we examine the
A
2
role of information for coordination.
game
simple
Static
2.1
Model
of regime change
benchmark
There
set-up.
distributed over
The payoff
a continuum of agents of measure one, indexed by
is
take an action that favors regime change) or refrain from attacking.
(i.e.,
structure
is
illustrated in Table
whereas the payoff from attacking
least
c.
The
An
is
is
=
(a^
where
0),
optimal to attack
it
status quo
we denote by A,
quo.
=
otherwise {R
agent hence finds
and uniformly
i
Agents move simultaneously, choosing between two actions: they can either
[0, 1].
attack the status quo
and — c <
of crises. However, whereas they focus on herding,
in turn
The
1.
—
1) is 1
payoff from not attacking
>
c
if
the status quo
is
(a^
=
0) is zero,
abandoned
=
(i?
c G (0, 1) parametrizes the relative cost of attacking.
if
and only
abandoned
if
if
An
he expects regime change with probability
and only
if
1)
at
the measure of agents attacking, which
greater than or equal to 5 G M, which parametrizes the strength of the status
agent's incentive to attack thus increases with the aggregate size of attack, implying that
agents' actions are strategic complements.^
Regime Change {A >
Attack
(oi
=
Not Attack
1-c
1)
{ai
Status
6)
=
Quo {A <
9)
-c
0)
Table
1.
Payoffs
Agents have heterogeneous information about the strength of the status quo. Nature
9 from a
normal distribution
A/'(z, 1//3)
agent then receives a private signal Xi
and independent
of
of the
common
+ ^i,
where
z)
/3,
,
that
is,
~ J^ {0,
common
prior about
9.
Each
1/ct) is noise, i.i.d. across agents
allow to parametrize the information structure
the precision of private information and the precision and
Although the game presented above
is
highly stylized,
it
and possible applications. The most celebrated examples are
runs, currency attacks,
and debt
crises.
role of coordination
is
bank
In these contexts, regime change occurs, respectively,
when
most evident when 9
pure-strategy equilibria, one in which
all
is
its
payments, when a large speculative attack
commonly known by
agents attack and the status quo
which no agent attacks and the status quo
is
admits a variety
self-fulfilling
a large run forces the banking system to suspend
"The
^j
initial
draws
prior.
Interpretation.
of interpretations
9
which defines the
The Normality assumptions
9.
parsimoniously with (a,
mean
=
,
first
maintained
(.4
=
<
9).
is
all
agents: for 9 G (0,
abandoned {A
=
1
>
1],
6)
there exist two
and another
in
bank
forces the central
and
creditors
is
abandon the peg,
to
for
X
<
for
x
>
X,
where x
z,
Pr
solves
where x solves Pr
Indeed, suppose there
The measure
—
where $
and only
if
is
\{
9
is
decreasing in
low signals
for sufficiently
— namely
— namely
= c — and not to attack for sufficient high signals
< l\x) = c. It is thus natural to look at monotone Bayesian
<
{9
0\x)
is
the
is
M
a;
<
x.
<
x\9)
=
=
9 solves 9
It
follows that the status quo
is
if
and only
A (9) =
given by
is
of the standard Normal.
<9, where
if
such that each agent attacks
then decreasing in 9 and
c.d.f.
Nash
non-increasing in x.
a threshold x G
of agents attacking
9)),
abandoned
is
political change,
of an agent's posterior about 9
c.d.f.
dominant to attack
strictly
which the agents' strategy,
equilibria in
^{^ya{x
it is
(9
be interpreted as one of
Pr (x
j4(^), or equivalently
9^^{^f^{x-9)).
The
X <
(1)
posterior probability of regime change for an agent with signal x
<
Pr(0
X,
each agent finds
9\x^. Since the latter is decreasing in x,
where x
solves Pr(^
<
^|x)
=
c,
it
then simply Pr
is
optimal to attack
(i?
if
=
monotone equilibrium
always exists and
strictly
is
dominated
and only
Proposition
a > 13/
,
and
modify the
is
is
is
in place
static
the status quo
ait
G
{0, 1}
is
impHes
and only
that,
if o:
when
> 0^1
(x, ^) to (1)
(27r)
.
and
(2).
Such a solution
Moreover, iterated elimination of
the monotone equilibrium
is
unique, there
is
no
in
monotone
the static game, the equilibrium,
is
unique
if
and only
if
strategies.
game
discrete
and
if
benchmcirk) In
game reviewed above
quo repeatedly; second, we
Time
z
(2)
conclude that
1 (Static
Dyiicimic
2.2
We
(27r)
strategies
all
if
or equivalently
thus identified by a joint solution
unique for
We
other equilibrirmi.
is
=
l|x)
$(y^T^(9-^x-43.))=c.
A
its
establishment.'^
Equilibrium analysis. Note that the
his private signal x. Moreover,
also
coordinate
fails to
whether or not to take actions to subvert a repressive
citizens decide
some other poHtical
dictator or
The model can
forced to bankruptcy.
which a large number of
in
when a country/company
or
let
two ways:
first,
and indexed by
t
G {1,2,
is
...}.
The game
abandoned.
in place at the beginning of period
the action of agent
'For references, see footnote
1.
we
allow agents to attack the status
agents accumulate information over time.
over once the status quo
is still
in
i,
and with At £
[0, 1]
t,
We
continues as long as the status quo
denote with
with Rt
=
1
i?f
=
the event that
the alternative event, with
the measure of agents attacking in period
t.
Conditional on the regime being in place at the beginning of period
abandoned
of the status quo.
and
his payoff
=
period {Rt+i
in that
Agent
i's
1) if
and only
payoff in period
from the entire game
Kit
=
is IIj
=
At
if
>
9,
t
=
(Rt
the regime
0),
is
where 9 again represents the strength
t is
O-it
{Rt+\
-
c)
Yl'^i p'~^(l
,
~
where p £
Rt)T^it^
(0, 1) is
the discount
factor.
Like in the static model, 9
the
common
initial
X*
=
6,
where ^u
{xir}*,.^i
drawn
~ A/'(0,
t
>
l/?yt) is i.i.d.
denote agent
z's
information that the regime
own
is still
private signals i* and
own
oo
every agent
1,
across
i,
in
>
"*
to period
/(27r) V£
we
and
let at
lim at
—^oo
=
=
=
xn
+
6
serially uncorrelated.
^u
Let
Individual actions and the
t.
t
simply consists of the
whereas the private history of an agent
past actions. Finally,
> P
and
hence the public history in period
in place,
at
up
9,
Y1t=i
'7'^
'''^^
is
the sequence
assume that
oo.
the next section, at parametrizes the precision of private information in period
above ensure equilibrium uniqueness in the static game with precision
restrictions
which defines
1/13),
receives a private signal
i
t
As shown
game from Af {z,
independent of
history of private signals
size of past attacks are not observable,
of
at the beginning of the
and never becomes common knowledge. Private information, however,
In each period
evolves over time.
about
prior,
is
at,
t.
The
and that
private information becomes infinitely precise only in the limit.
Remark. While
this
dynamic game
that are absent in the static benchmark;
is
highly stylized,
first,
it
captures two important dimensions
the possibility of multiple attacks against the status
By assuming
quo; and, second, the evolution of beliefs about the probability of regime change.
that
per-period payoffs do not depend on past or future actions and by ignoring specific institutional
model may of course
details, the
such
as,
for
fail
to capture other interesting effects introduced
example, the role of wealth accumulation or liquidity in currency
by dynamics,
However,
crises.
abstracting from these other dimensions allows us to isolate information as the driving force for the
dynamics of coordination and
Equilibrium. In what
crises.
follows,
we
limit attention to
monotone
equilibria,
Perfect Bayesian equilibria in which the probability an agent attacks in period
abuse of notation we denote with aj(x*),
of his
own
past actions.^
is
t,
that
is,
symmetric
which with a
slight
non-increasing in his private signals x* and independent
Restricting attention to this class of equilibria suffices to establish our
multiplicity results.
The
possibility that agents observe noisy signals
°We do
not restrict the set of available strategies:
about aggregate past activity
we look
at equilibria in
is
considered in Section
which these properties are
6.
satisfied.
Equilibrium characterization
3
Since neither individual nor aggregate actions are observable, and
any strategy
is
in place. It
any
profile at
period and hence strategies are sequentially rational
and only
payoffs. Finally, define xt
and
ctj
xt_i
Xi and ai
=
rji;
if
and
Xt
H
cut
=
any other
or other players' actions in
if
they maximize period-by-period
=
at
+ m,
aj-i
Cat
and note that
Xt is a sufficient statistic for the agent's private information
{xrY^^i with respect to 6 and therefore with respect to regime change as weU.
that, in
quo
by
recursively
=
Xt
X*
as long as the status
foUows that behefs are pinned down by Bayes' rule in any relevant history of the game.
own
—
always compatible with
is
no agent can detect out-of-equilibrium play
i,^
Furthermore, payoffs in one period do not depend on
with xi
=
-Rt
We
conclude
any monotone equiUbrium,
e arg
at{i^)
max
=
{(Pr [Rt+i
l\xt,
Rt
=
0]
-
c)
a}
(3)
,
ae[o,i]
where Pr
[•]
is
pinned down by Bayes' rule using {at (Oli^i
•
This also implies that, in any equilibrium of the djmamic game, agents play in period
as in the static
game
his private signal
high.
The
Lemma
following
{±00} and
(ii)
The
9
>
which they can attack only
sufficiently
in that period:
low and the status quo
lemma shows
is
exactly
an agent attacks
if
and only
and only
is
sufficiently
maintained
if
if
6
if
that a similar property applies to subsequent periods.^
For any monotone equilibrium, there
1
(i) at
is
in
1
is
a sequence {xj,^^}^^!, where x^ £
E = RU
9l_i G (0, 1), such that:
any
t
>
l,an agent attacks
the status quo
is
in place at
fact that the status
9t_i- Since 9t_i
attacks in period
t
>
>
2,
9^
in
>
quo
if Xt
any
is
0, this
< x^and
t
>
2 if
does not attack
and only
in place in period
t
if Xt
if 9
>
df_-^.
>
2
makes
>
it
ar^;
common
certainty that
implies that there always exist equilibria in which
which case Xj
= — cx)
and
equilibrium in which an attack takes place in period
1
9^
=
9^_-y.
and never
nobody
In particular, there exists an
after.
If this
were always the
unique equilibrium, the possibility to take repeated actions against the regime would add nothing
to the static analysis
in the static
and the equilibrium outcome
in
the dynamic
game would
coincide with that
benchmark. In what follows we thus examine under what conditions there also
exist
equilibria with further attacks.
Indeed, the regime always survives any attack for 8
'To simplify the notation, we allow
for x*
= —00
>
1
and xj
and no realization of the private signal rules out 9
=
>
1.
+00, with which we denote the case where an agent
attacks for, respectively, none or every realization of his private information.
.
By Lemma
the size of the attack
1,
and the probabihty of regime change
in 9,
=
Pr
(^
any equihbrium
in
which an attack occurs
l\xt,Rt
=
0)
<
>
6t\xt,9
^*_i),
=
given by At{9)
is
an agent with
for
which
Pr {x <
in period
t,
and
6^
decreasing
is
if
O^
>
foUows that, in
It
6'j*_^.
Xj solve
9;^<!>i^t{x;-0;))
$ (yS7+?(^
Conditions
game:
and
(4)
if
and only
+
-.x;
game
the analogs in the dynamic
^^^
'
n
and
of conditions (1)
(2) in
X (0^
=
Solving (4) for Xj gives x^
if
,
and only
aj)
if
the static
equal to the critical size that triggers
is
the fundamentals are 9^, while (5) states that an agent
if
between attacking and not attacking
91,
0)
*
that the equilibrium size of an attack
(4) states
regime change
(5) are
(4)
at+P'
1-
his private information
and substituting
is
indifferent
is
Xj
this into (5) gives a single equation in
namely
U{ei9U,at,P,z)=0,
where
X
^ 1 and
x IR+
[0, 1]
:
=
X{9*,a)
9*
:
[0, 1]
x
1
x
IR2_
x
R
^
[-c, 1
-
c]
axe defined as follows:^
/Fv-l
1
+
C7
(6)
$
1-c
n/S
1-
=
U{e*,9^ua,P,z)
The
X
functions
and
U
c
v^
/a+0 ^-He')+-^i^-e') +V^+j3{e'-e^i)
have a simple interpretation: X{9*;a)
plements regime change for 9
threshold x* such that,
and only
ii
9
<
if
<
when the
agents attack
U {9*,9-i,a,
9*);
9*
j3,
z)
is
indifferent
is
since the regime has never
(6*2
,
— oo, ai, 13,
in the static
z)
=
0,
>
With a
if
r
> max{0,6l_i}
if
9*
<
>
9-1
<
x
9-1
the "marginal agent" that im-
x*, the status
is
quo
a
is
(that
is,
the
abandoned
if
9-i. Condition (6) thus simply requires that the
been challenged
in
period
i
>
2.
As
for
i
=
in the past, the corresponding indifference condition
U {9, — oo, a,
/3,
z) coincides
1,
is
with the payoff of the marginal agent
benchmark.
The next proposition then provides the complete
monotone
if
between attacking and not attacking
where
r=
the expected net payoff of the marginal agent from at-
tacking, conditional on the knowledge that 9
marginal agent
is
precision of private information
and only
if
if
•)+:^(--^')
v^+^
-
—c
U
=
sufficient statistic Xt is Pr(i?t_|_i
decreasing in xt
is
which
x^lO),
set of necessary
and
sufficient conditions for
equilibria.
slight
abuse of notation, we
let <J>(+oo)
=
1,
$(— oo) =
0,
$
'(1)
=
oo and
$
''(0)
= — oo.
Proposition 2 (Equilibrium characterization)
only
{ot {)}tZi
^*
monotone equilibrium
o.
if
and
there exists a sequence {x*,9f}^-^ such that:
if
for
(i)
all
for
(ii)
t
at
t,
=
t
>
2,
U {9t, 9;_^,at, ^,z)=0
An
<
1 if Xt
61 solves
1,
for any
(iii)
=
(•)
x* and at
U {91, -oo, qi,
=
either 9t
and
x*t
=
if Xt
z)
=
^
^ '^^^
/?,
^t-i
=
(•)
>
x*.
=
and x|
X(6'j aj).
,
~
-^t
~oo, or
6t
>
is
9^_-^
a solution to
X{9*t,at).
equilibrium always exists.
Proposition 2 pro\'ides a simple algorithm for constructing the entire set of monotone equilibria:
start
with
=
and
<
2
f
=
and
1
be the unique solution to
let 61
let either ^2
=
^^ ^ solution to
^i or ^2
[7 (0j,
(6);
— oo, qi, /?, z) =
0;
proceed to period
repeat the same step for
all f
>
the set
3;
of sequences {^t }J^i constructed this way, together with the associated sequences {x*}^-^
the set of monotone equilibria. Note that the above characterization
horizon
period
or infinite:
is finite
T<
it is
clearly valid even
gives
independent of whether the
game ends exogenously
the
if
is
,
an arbitrary
at
oo.
U {91, — oo, ai, /?, z) =
Existence of equilibrium follows immediately from the fact that
admits a solution and hence
6t
=
6^ for
lemma
are other equilibria, the next
always
always an equilibrium. To understand whether there
all t is
investigates the properties of
U
and the existence of solutions
to (6).
Lemma
2
U {9* ,9^i,a,
(i)
peaked in 9* when 9-i G
all
6-1
<
(ii)
9l_i
<
1
and
Let
6t
9t
and
6*
>
jS,
(0, 1),
z)
and
6-i, linia^oo
be the
U {9*, 9-1,
unique solution
to
•)
=
^oo
U{6t,— oo,
from above by
-
where 6oo
9*,
—
at, j3,z)
Q.
A
=
I
-
<
9*.
solution to (6) exists only
0-
(iv) If 6i_-^
<
9oo,
a solution to (6) necessarily exists for at sufficiently high.
(v) If 9*_-^
is
the highest solution to (6) for period
t
>
a.
solution to (6) does not exist for aj sufficiently high.
The non-monotonicity
<
1) is
of
U
t
such that
{9* ,9^_-^,at,P,z')
necessarily positive, but
<
1,
by the
>
6^_-^,
>
1.
His payoff
solid line in Figure
The dashed
1.
Any
is
there exists
a > at-i such
<
9*_-^,
on the other hand,
—
>
1,
that (6)
.
with respect to 6* in any period
converges to zero as 6*
attaches probability one to 6
a solution to (6).
a-r
~
a direct consequence of the fact that, for 9*
probability zero to regime change. For 9*
illustrated
if
9t.
0OO,
5J'_i
Furthermore, for
c.
>
admits no solution for any period t
is
and z for 9-i
strictly decreasing in 9-i
necessarily bounded
is
arguments, non-monotonic and single-
all its
If 9^_-^
(iii)
<
continuous in
is
for
then
t
>
2 (where
the marginal agent attaches
his belief
about regime change
X {6*,at)
—> oo and hence he
thus initially increasing and then decreasing, as
intersection with the horizontal axis corresponds to
line instead represents
the payoff of the marginal agent in the static
1-c
Figure
The
1:
game. Whereas the monotonicity of the
dynamic game leaves open the
The other properties
common
the
9,
latter ensures uniqueness, the
non-monotonicity in the
possibility for multiple equilibria.
identified in
Lemma
2 are also quite intuitive.
An
increase in the
mean
of
prior implies a first-order stochastic-dominance change in the posterior beliefs about
which explains
that the
payoff of the marginal agent.
U
virhy
decreases with
more aggressive the attacks
to have survived these attacks,
z.
U
To understand why
in the past, the stronger the
is
also decreasing in 0_i, note
regime must have been in order
and therefore the lower the expected payoff from attacking
in the
current period.
For the same reason, at any
U {6*, — oo, at, P, z)
is
,
that
This explains
at-
is,
why
>
i
2,
the payoff of the marginal agent
the payoff in the static
game where
the static-game threshold
the dashed line with the horizontal axis in Figure
Nevertheless, the impact of the information that 6
xt
>
9f_-^
as at -^ oo,
and U{9*, — oo,
as at
9^_-^
—
<
>
and hence,
for
at, P, z) also vanishes.
any
9*
>
the precision of private information
(which corresponds to the intersection of
9t
1) is
>
an upper bound for any solution to
9^_-^
on posterior
9*_j, the difference
Combined with the
where
O^d
is
Finally, to
understand
{v),
(i.e.,
0oo
=
(6).
any
U (^9*,9l_^,at, 0, z^
— oo, at, P, z)
admits a solution
the limit of the equilibrium threshold of the static
private information becomes infinite
beliefs vanishes for
between
fact that U{9*,
oo, this in turn implies that, for at sufficiently high, (6)
^ooi
always lower than
is
game
if
-^ ^oo
—
and only
9*
if
as the precision of
linaj^cx) 9t)-
note that any unsuccessful attack causes an upward
shift (a first-
order stochastic-doniinance change) in the posterior beliefs about the strength of the regime, which
other things equal reduces the expected payoff froin attacking.
attack (that
thereafter,
is,
the highest solution of (6))
no attack
is
is
follows that,
played in one period and no
possible in any subsequent period.
remain impossible as long as the change
It
By
if
the largest possible
new information
continuity then, further attacks
in the precision of private information is not large
10
arrives
enough.
1-c
Figure
4
Multiplicity
Part
[v) of
attacks to
This
prior.
Equilibria with multiple attacks.
2:
and dynamics
Lemma
2 highlights that the arrival of
become
possible after period
is
new
Whether
1.
private information
this is also sufficient
is
necessary for further
depends on the
initial
because an increase in the precision of private information leads agents to discount,
not only the information conveyed by the fact that the regime survived past attacks, but also their
initial prior beliefs.
When
z
is
low ("aggressive prior"), an increase in the precision of private information makes
the marginal agent
less aggressive in
the static
game
(that
decreases with at).
is, 6t
that the regime survived past attacks then only reinforces this
low, there exists a unique equilibrium
When
instead z
is
and
is
may
agent in period
is
illustrated in Figure 2.
Its intersection
1.
marginal agent in period 2
Clearly,
a
new
91
=
and
0:2 is
attack
=
is
is
t,
possible.
There thus
another in which 62
63 for all
i
>
3,
The dashed
represented by the dotted
where
63
is
z
if
is
and
^1
^3
period.
makes the
high enough so that 9^
is
line represents the payoff of the
line,
possible in period
and that
<
The
in period 3
In contrast,
2.
O^o-
cts is
and
6'^
=
6I3
for all
i
>
3,
and a third
<
9^,
sufficiently high.
marginal
payoff of the
by the
solid line.
high enough that
exist at least three equilibria in this example:
=
sufficiently
first
private information
with the horizontal axis defines 9^
low enough that no attack
0* for all
91
new
2 ensures that a second attack necessarily becomes possible once at
Such an example
;:
eventually offset the incentive not to attack induced by
the knowledge that the regime survived past attacks. Indeed,
Lemma
Therefore, for
such that no attack ever occurs after the
high ("lenient prior"), the arrival of
marginal agent more aggressive, and
efTect.
The knowledge
in
one
in
which
which
^2
=
^1
correspond to the two intersections of the solid line with
the horizontal axis.
11
In the example of Figure
becomes possible
also
to (6)
2,
both
some future
at
bounded from above by
is
becomes possible
any unsuccessful one.
after
<
most
If z E {z_,z), there are at
(ii)
number
such that no attack occurs after period
If z
(iii)
>
an equilibrium
is
=
Finally, z
z
=
when
z
Proof. Recall that
solution to U{B,
z<z<z
<
Bi
B^
— oo, at, (3,
c
if
first z
<
i
>
Next, consider z £
and
1;
U
—> oo, there exists
monotone
equilibria,
and
a
<
i'
there
all
T
an
>
t.
<
Bt
6i
B^o for all
=
when
>
c
oo
t
and N, there
1/2.
and only
t if
B^ for all
in
i
>
<
which case
t>
>
if
B\
=
9i
>
B^o for all
B^
>
is
> B^,
>
for
<
t
J,in which case BI
<
there exist (countably) infinitely
B^o-
thus
=
Bt
of
after period
all
t
z;
2, (6)
t.
>
i.
t
>
admits a
fact that
Moreover, since
finitely
many
i.
(iv) of
t'.
U {Bt, Bl,at,
equilibria,
<
but we can not rule out the
Then, by part
solves
Lemma
BI for all
Hence, there are at most
i.
many
(ii)
U {6* ,B\,at,P,z) =
(iii) of Lemma 2 and the
a solution for
Bt
B\ for all t if z
1.
oo such that (6) admits no solution for
most two solutions
<
and
6\
by part
t,
z
Bt
and hence, by part
i,
2 such that Bt
t,
is,
for any
z,
with the following properties:
U {9* ,Bl,at, P, z) = admits
equilibrium in which 6* = B^ for r <
That
<
exists i
t.
z<z<z
oo such that
is
>
equilibria; if in addition z
any equilibrium no attack occurs
in
(Hi) Finally, consider z
/3)
<
Bt
(^, ^),
single-peaked, (6) has at
is
and there
equilibria
The unique monotone equilibrium
2.
solution. Nevertheless, since Bl_-^
i
such that:
i.
and
possibility that there exists a period
at —> CO as
z<z<z
many monotone
finitely
1/2, whereas
Then,
z.
admits no solution at any
Hence, for z sufficiently high, not only there are
= Bi and, for all t > 2, B^ < Bt, where 6t = 6{at,l3,z) is the unique
z) = 0. As we show in Lemma Al in the Appendix, there exist thresholds
and only z > (<)
Consider
(ii)
<
any solution
z is sufficiently high,
if
which ensures that a new attack eventually
attacks occur after period
(possibly functions of a\
(>) ^oo
(z)
N
in which
then, a third attack
equilibrium and an attack occurs only in period one.
many
there are infinitely
z,
2,
of attacks can be sustained in equilibrium.
unique monotone
z, there is a
generally,
There exist thresholds
1 (Multiplicity)
If z
(i)
More
date.
By Lemma
are lower than Boo-
6'^
^oo in all periods,
multiple equilibria, but any arbitrary
Theorem
and
9'^
/3,
Lemma
2,
there exists
Hence, for any
z)
=
0,
and
B*
=
>
t
9^
t',
for
indexed by the time at which
the second attack occurs.
When
second attack
z G (z,z), the
attack might be impossible.
Lemma
2, Bt
<
B^o, for all
any unsuccessful one.
It
t.
If
however z
>
follows that, for any
and {B2,...,9n}, with
and so
The
following
is
lead to a threshold B*
z,
But then by part
{t2,...,t7v}
on.
may
ta
then an
>
*,
^
then
(iv),
>
1
such that
equilibrium: 9*
12
a
Bt
<
new
Boc for all
>
t,
Boo, in
which case a third
and hence by part
(ii)
of
attack eventually becomes possible after
and any
N
>
1,
there exist increasing sequences
U {B2,Blat._,l3,z) =
= B^ for r < t2, B* =
0,
U {B3,92,at:„ P, z) =0,
Bj for
r G
{tj,
...,
tj+\
—
1}
and
6
j
{2,
...,
an equilibrium
The
game
—
TV
—^
N
which
in
=
9*
Oj^i
>
any
equilibria in the case z
>'z
tf^.
many
continues forever as long as the status quo
cx)
as
f
M equilibria
—
would
oo, then, for
*•
ended
if it
That
for
for r
attacks occur after date
existence of infinitely
at a finite date, there
Qif
and
1},
exist only finitely
any M, there
>
z.
many
even
Finally, as
we
and any
N
>
1,
there exists
if
on the assumption that the
game ended
for
exogenous reasons
equilibria. Nevertheless, as long as z
T
T=
we show
>z
game would have
such that the
2,
02 high enough
suffices for the
and
at least
game
to
in Section 6.1, the introduction of public
any
signals ensures the existence of multiple equihbria for
In the remainder of this section,
1
relies
in place: if the
is
>
i
t.
exists a finite
at date T. Moreover,
have multiple equilibria when z
is,
T>
2
and any
z.
identify equilibrium properties that
may
turn useful in
understanding crises phenomena.
Corollary 1 Suppose 9
there exists
less,
<
t
>
doo
^"^ z >
^.
The status quo survives in any equilibrium. Neverthe-
00 such that, at any
>
t
an attack
t,
is possible,
yet need not take place.
Furthermore, any arbitrary number of attacks can occur.
This seems to square well with the
eventual outcomes
(e.g.,
common view
whether a currency
or whether attacks will cease.
is
On the contrary,
that economic fundamentals
eventually devalued) but not
this
version of the model, in which case fundamentals
view
fail
is
may help
when a
inconsistent with the
crisis
predict
wiU occm-
common-knowledge
to predict both the timing of attacks
and the
eventual regime outcome,^ as well as with unique-equihbriiun models like Morris and Shin (1999),
in
which both the timing of attacks and the ultimate fate of the regime are uniquely pinned down
by fundamentals.
Consider
now how
Corollary 2 After
the dynamics of attacks depend on the dynamics of information.
the
most aggressive attack for a given period
of tranquillity, during which no attack
This phase
is possible.
game
occurs, the
enters a phase
longer the slower the arrival of
is
private information.
Along with the property that
after
for 6
> 6^ and
z
>
2 a
new
attack eventually becomes possible
any unsuccessful one, the above result implies that dynamics
may
which the economy alternates from phases of tranquillity to periods of
into a
new
attack, without any change in the underlying fundamentals.
be possible either under common knowledge
^Of course,
this
outcome, as for 6
is
<
true provided 6 G
it is
[0, 1]. If
take the form of cycles in
crisis,
Once
eventually resulting
again, this would not
or with a unique equilibrium.
instead 8 ^
[0, 1],
dominant to attack immediately and
13
fundamentals predict both the timing and the
for 9
>
1 it is
dominant never to attack.
final
Unobservable shocks
5
we examine the
In this section,
possibility that the critical size of attack that triggers regime change
due to unobservable shocks. This
varies stochastically over time
is
not only a realistic scenario,
but also an appropriate perturbation. In the model without shocks, the knowledge that the regime
is
in place at
equilibria in
>
i
makes
2
it
common
which nobody attacks
certainty that 9
at
>
i
>
0,
which explains why there always
may
In contrast, shocks
2.
exist
guarantee that a positive
measure of agents attack in every period by "smoothing out" the information generated by the
knowledge that the regime survived past attacks and ensuring the existence of a lower dominance
region in every period.
We
modify the game as
follows.
benchmark model, but the regime
(5
>
and Wj
a real
is
random
Agents continue to receive private signals about 6 as
in the
+ Suit,
where
now abandoned
is
in period
variable with continuous
c.d.f.
and only
t if
=
Equilibrium characterization with
depends, not only on
monotone
9,
but also on
the strength of the
volatility
In the presence of shocks, the regime
outcome
the
{5)
in
0.
6
As a
ojt.
game
>
0.
result,
we
loose the ability to characterize the set of
equilibria in terms of a sequence of truncation points for
9.
Nevertheless,
we can
still
characterize strategies as sequences of thresholds {Sf }^i such that an agent attacks in period
and only
if
Xj
Consider
period
LJt [6]
t is
Xt, 6)
Xj,
first
At
=
<
{9)
in
which the
regime) and 6 as the volatility of these shocks. Denoting with
the baseline model can be nested as 5
F
(i.e.,
6
and the noise
F, independent of 9
the agents' signals. Wj can be interpreted as a shock to the fundamentals
is 5,
>
At
ii
t if
where Xt G M.
the regime outcomes induced by a given {xt}^^.
= $
{\/oit (^t
[$ (\/S( {xt
—
^))
—
^))
—
9] /S. It
and hence the status quo
The
size of
abandoned
is
the attack in
and only
if
<
if u>t
follows that the probability of regime change in period
monotone
conditional on 9 and given that agents use
strategies with threshold Xf
,
t
is
Pt{9;xt,5)=F{idt{9-xt,6)).
Next, for any
when
By
^
t
>
2, let ipt
[9;x^~^,5) denote the density of the
in previous periods agents followed
monotone
common
posterior about
=
strategies with thresholds x^^^
{xi,
...,
9,
Xf_i}.
Bayes' rule,
fg.^t-i ^^
where
V'l {9)
^
=
[l-pt-i{e-Xt-,,5)]xPt-i{e;x'-\S)
y/]i(j>{y/P [9
—
z)) is the density of the initial prior.
of the corresponding private posteriors with
Finally, consider payoffs.
nlz\[l -ps
^
For any
i
expected payoff from attacking in period
>
t
ipt
1,
for
{9\x] x^~^
,
5^ for i
x g M, and x* £
an agent with
14
We
>
K
2
,
{9; Xs, 5)] ^P, (9)
similarly denote the density
and
ipi
{9\x) for
t
=
1.
define Vt (x;x*,5) as the net
sufficient statistic
x when
all
other
,
<
agents attack in period r
t if
and only
/
t
their sufficient statistic in
if
is less
than or equal to Xr
:
+ 00
pt {9; xt, 6)
-
dO
{0\x; x^~'^,5)
ipt
c.
-oo
Note that
thresholds
depends on both the contemporaneous threshold Xj and the sequence of past
(x; i*, 5\
1;^
the former determines the probability of regime change conditional on
x*"-^;
the latter determines the posterior beliefs about
=
attacking in period
any
In
Lemma A2
x*
G
R
monotone
= +00
Vi (xt;x*,(5)
if
it
e
lim2:__oo
if
Xj
= —00
benchmark model:
in the
(iii)
An
let
M
represents the net payoff from
it
>
0, Vj (x*; J) is
continuous in x' for
equilibria.
()
all t, at
for
=
i
1,
>
5
monotone equilibrium for T
{at (Olt^i is a
0,
for any
(5
^j(+oo)
=
1 if
Xt
<
xi G
M
>
either Xj
t
2,
Xt
—
>
=
the shock
0.
1,
cut
Let
and
to
and
=
= —00
and
>
[5) if
and only
if there
=
maxT<t{0T
vanishes: for any Xj
<
Vt (x*; 5)
xj;
We
As
(xt)}.
and any 9
^
(5
— ^
9
^
i^t
^ ^t {9-x'-\Q)
9
=
>
Since the distribution of x given 6 satisfies the
linii_±oo Vt (x\ S',
(5)
vt
(x;5*,5)
exist for
any x' £
is
R
(y/at{xt
"
^
\ie>
'[
\
=
0.
{
—
9))
,
with
9t
(—00)
i
>
9t-i (x*""^)
^*
=
and
2 converges pointwise to the truncated
:
for
any x^~^ and any
if
v^^(Vff(9-.))
i-*(y?{9,_i{x'-i)-z))
MLRP
and pt
{d\xt, 5)
is
decreasing in
bounded
and therefore
Vt (x*, 5) is well-defined at xt
15
is
also
(x*-i)
9t-i
(8)
.
x.
vt
<
9,
otherwise
decreasing in
Since
(7)
J'^'J
9t (xt)
f
[
theorems (Milgrom, 1981),
Vt (x*; 6)
6t{xt)
posterior in any period
Normal generated by the knowledge that
{9-x'-\5)
and
the dependence of the regime outcome on
0,
[0
common
R
next consider the properties of the equilibrium
Pt{9-xt,5)^Vt{9;xt,Q)^\
follows that the
or Xj G
0,
0.
be the unique solution to
6t (xj)
>
if Xt
0;
benchmark game.
9t (x*)
=
at ()
(xi; 6)
and Vi
equilibrium exists for any 5
Convergence
'
,
x M, which we use to establish the existence, and complete the characterization, of
for
(ii)
It
M
€
x^.
any 5
for
5*
sequence {xt}^-^ such that:
exists a
set as
Vt (x; x*, ^)
Appendix, we prove that,
in the
and
1
xt
vt [x] x^, 5)
the marginal agent with threshold
for
t
Proposition 3 For any
(i)
U
analogue of the function
Vt is the
>
t
if
hmx-,+00
Vt{x':5)
Next, for any
0.
whereas
6,
S,
by standard representation
in [— c,
=
±00.
1
—
c],
it
follows that
Similarly, for the private posteriors,
The pointwise convergence
=
Vi {xi; 6) —> Vi {xi; 0)
To understand why
and
of pt
when
of the marginal agent except
f/ (^i (xj)
,
[6\x] x^~^
tpt
>
t
{0\x;x^~^,O) for any x.
ipt
in turn implies pointwise convergence of the payoff
ipt
2
—+
6)
,
and
= — oo.
xt
— oo, Qi, /?, ~)
In particular, for
and, for any
;
need not converge to Vt (x*;0)
Vt (x*;(5)
i
>
regime change in period
c to
When
this
>
t
2 even
if
may
2
=
and any
and
ensures that this dominance region vanishes as 5
generally,
we can prove
may
The
^
case where Vt (2;**;0) reaches
maximum
is
maximum
its
{x*}^^
equilibria
all
for
that,
<
5
all
5 (e,
T)
the following
,
= — oo and
Xt {5)
The above
<
is
ofT{5) such
result
Vi (xi
illustrated in Figures 3
is
of the
common
game without shocks
where x^
is
{6
may
((5)
;
Si
6)
((5)
=
it
and any
0,
for
that,
fail in
(5
>
small
the knife-edge
(^9f,6f_i,at,p,z^ reaches its
,
0;
satisfies (ii)
—
T>
<
all t
d) for
the
converge to
if
the
all
for
postei-ior in period 2
=
0).
0).
This
The
is
there exists 5
\,
T, either \x*
=
first
as
T
T —^
game ended
(e,
T)
>
all t
—
£ {2,...,T},
<
Xf(^)|
or
e,
for
periods of the infinite-
T =
2.
The
exogenous reasons at T,
solid line in Figure 3 repre-
generated by equilibrium play in period
initial prior
(or equivalently
truncated at 91
where
9^ is the
other two lines represent the equilibrium
{5
>
need not
oo, the result
periods of the finite-horizon game.
simply the
game with shocks
0),
where xi
[5) is
=
1
9i (x^),
unique solu-
common
poste-
the unique solution to
the dotted line corresponds to a relatively high 6 and the dashed one to a low
''Clearly, Proposition 3 applies also to the
only
and 4
the unique solution to Vi (x^; 0)
U {9y,—oo,ai, P, z) =
(^;
>
e
establishes convergence of equilibrium play in the
sents the p.d.f.
i/'2
may
(S) for
—1/e.
the above would immediately imply convergence in
riors
> — oo
true:
hold for an infinite number of periods. Of course,
tion to
note
benchmark game
,
horizon game, for any finite T. Since 6{e,T)
in the
any Xj
for
suchthatx* ^ argmax^Vj {x*^~^ X]Q) for
(0)
there exists an equilibrium {xj (5)}^j
The
= — oo,
at 6f).
For any equilibrium {xD^-j^ o/F
xl
U
(or equivalently
at Xj
in the
game T
needed because convergence
Theorein 2 (Shocks in fundamentals) For any
such
-oo.
0.
that "essentially"
qualification "essentially"
>
attack in every period in the perturbed
r(0) can be approximated by equilibria {xj (5)}^i in the perturbed
enough. ^^
Xj
xt
ii,
attach probability higher
game, unlike the benchmark model. Nevertheless, the pointwise convergence of Vt
More
and any
1
he expects no other agent to attack in that period.
the case, a positive measure of agents
is
x*"-'
= — c when t >
that, in the presence of shocks, an agent with sufficiently low xj
than
any
2,
t
{Hi) for 5
unperturbed game: {xj }^j
= 0.
16
is
a
monotone equilibrium
of
T
(0)
if
and
/
\
./-
^
-'\
•
«
y
«
Figure
^^
1
*
_
*"^\
f
^v^v^,^^
/
/
--^
3:
^^^^
Common
posteriors with
and without shocks.
1-c
X2
Figure
6.
4:
Since the support of Wf
any 6 and therefore
tp2
Payoff of the marginal agent with and without shocks.
is
the entire real
line,
the probability of regime change
assigns positive probability to
all 6.
However, as
converges to x\ and the probability of regime change converges to
By
implication, the
smooth common posterior
of the perturbed
1 for
game
9
5
is less
than
1 for
becomes smaller, xi
<
Q\
and to
for 6
>
(d)
9\.
in period 2 converges to the
truncated one of the benchmark model.
In Figure
2 for 5
=
0,
4,
the solid line represents the payoff V2 {^\-i^2]^) of the marginal agent in period
whereas the other two
X2 small enough, V2
attacking in period 2
is
is
negative
lines represent V2 (ii {5) ,X2]5) for 5
when
d
=
but positive when
part of an equilibrium in the
"In order to illustrate V^ over
its
entire
>
0,
0.^^
Note
that, for
which implies that nobody
benchmark model but not
domain, the figure depicts V2 {xi
17
(5
>
(S) ,X2\ 5)
against
in the
$ (0:2)
game with
rather than X2.
.
when
shocks. ^^ Moreover,
5
high (dotted Hne), V2
is
When,
in period 2.
instead, 5
and therefore has a
in X2
which case the equilibrium would be unique
intersection with the horizontal line, in
ended
monotonia
is
sufficiently small
is
(dashed
line),
V2
is
if
single
the game
non-monotonic and
has three intersections, which correspond to three different equilibria for the two-period
game with
shocks. Finally, the middle and the highest intersections approximate the two intersections of the
solid line, while the lowest intersection
with the fact that xi
{S)
arbitrarily small, thus approximating X2
is
=
—00. Along
converges to Xj, this implies that any equilibrium of the two-period
game
without shocks can be approximated by an equilibrium in the perturbed game.
In conclusion,
what sustains
multiplicity in the
dynamic game
the non-monotonicity of the
is
payoff of the marginal agent generated by the knowledge that the regime survived past attacks.
That
this
knowledge resulted in posterior
the benchmark model,
is
beliefs that assign zero
measure to
sufficiently
low 9
in
not essential.
Public news and signals about past attacks
6
In this section,
we
consider two extensions of the information structure assumed in the
benchmark
model.
Exogenous public
6.1
To capture the
signals
effect of public
we now modify the game
news,
signals, agents observe in each period
i
>
a public signal
1
normally distributed with zero mean and precision
and the agent's private
of 9
noise.
>
7y^
as follows. In addition to their private
Zt
=
9
-\-
0, serially
£t^
finite
We
These signals may represent,
also allow for the possibility that the
date and denote the horizon of the
The common
precision
/3t,
game with
for
posterior about 9 conditional on z'
A-l
??f = ~5-~t-l
+ IT-U
,
Pf
(zoi/^o)
=
(-z,/?)-
is
zt
n
Pt
=
Pt-1
is
the examples in Figures 3 and
4,
o
+
3, ...}
exogenous reasons
or
T=
at
a
00.
Normal with mean
Zf
and
z
,
7?t
now depends on
the realiza-
=
0.
thus allow
or equivalently on z'
=
in past periods
not a sufficient statistic conditional on Rt
agents to condition their actions on the entire sequence
^''In
{zt}\^i
for
announcements
Pt
However, since equilibrium play
tions of past public signals,
if
£ {2,
s
noise,
where
-t
with
T
common
example, the information
statistics, or
game ends
where
T,
£t is
uncorrelated, and independent
generated by news analysis in the media, publication of government
by policy makers.
where
an agent finds
it
2',
We
{zr}\-^i.
Apart
optimal to attack in period 2 for sufficiently low X2, even
he expects no other agent to attack. However, this need not be the case
small enough.
18
if ujt
had a bounded support and
5
were
from
monotone equihbria can be constructed
this modification, the set of
in
a similar way as in the
benchmark model.
Proposition 4
tions
{xj*,
(i)
Otjl^i
for
with xl
,
all t, at (•)
=
att
{ii)
{Hi) at
=
:
E*
1 if
^Rand 01
Xt
61 (zi) solves
l,
>
t
monotone equilibrium
{o; (•)}^q is a
R* ->
:
if
(0, 1)
,
and only
=0 if Xt > Xt (z*);
U {9\, -oo, 0:1,^1,21) =0 and x\ {z\) =
<
and
x^ (z*)
=
(2*)
sequence of func-
such that:
at ()
X{6\
(21)
,
a{)
either 0* (2*) solves
2,
u{eie*t_„at,pt,zt)
and X*
if there exists a
X{e*
(2*)
,
=
at), or 9* (2*)
o
and x*
(2*-!)
0*_.^
=
(9)
(z*)
=
-00.
Like in the benchmark model without pubhc news, there always exist equilibria in wliich attacks
cease after any arbitrary period. However, for any (^i_i, oit,Pt)
if zt
<
Zt,
where
=
Zt
an attack occurs
z {9f_-^,at, Pt) is
in period
t
for sufficiently
reinforces the multiplicity result of
Theorem
always
finite.
(9)
admits a solution
if
and only
Hence, there also exist equihbria in which
low realizations of
Theorem
,
zt-
The
following then extends and
1.
3 (Public news) In the game with public news, there always
exist m,ultiple equilibria,
thi prior, the precisions {at,/3t}j=i of private and public information, and
no matter the mean 2 of the
the horizon
T
Consider
of the game.
now how
the introduction of public news affects the ability of an "econometrician"
to predict the regime outcome and/or the occurrence of an attack in any given period.
i,
any 9 G
(0, 1),
and any
9l_-^
<
(9)
9,
enough, implying that, conditional on
given period
is strictly
can not observe
hand,
for
if
between
and
he also knows
some combinations
2*,
he
of 9
may be
and
U {9*,—oo,ai,Pi,zi) =
no equilibrium
in
2*,
and
which an attack occurs and one
')
1. It
follows that an econometriciaxi
is
if Zf
is
low
abandoned
in
any
who can
2, ii 9
9t (2*)
in
outcome
is
>
Ot (2')
>
O^ (21),
where
9i (21)
the highest solution to U{9*,9i (21)
abandoned
in period
t,
^j (^i) in
On
the other
in a given period
which no attack takes place
may
any equilibrium,
is
,
the lowest solution
aj,/3t, 24)
,^''
there
is
but there are both an equilibrium in
in that period.
The combination
of
help predict regime outcomes but not the occurrence of
benchmark model.
>
observe 9 but
without, however, being able to predict whether an attack will
fundamentals and public information
'''Since 9(_i (=*
and only
the probability that the status quo
able to perfectly predict the regime
>
t
which the status
attacks, like in the
if
necessarily faces uncertainty about the event of regime change.
2*,
take place. For example, at any
to
9,
admits a solution higher than 9
For any
9t (2') is
19
an upper bound
for 9^ [z').
Finally, note that the threshold
below which
Zt,
(9)
admits a solution, decreases with
^j_i.
Hence, an unsuccessful attack, other things equal, causes a discrete increase in the probability the
game
enters a phase of inaction during which no attack
is
possible. In this sense, the prediction of
the benchmark model that equilibrium dynamics are characterized by the alternation of phases of
tranquility
and phases of
one phase to another
now
is
the introduction of public news, but the transition from
crises, survives
stochastic, as
depends on the realization of
it
z*.
Signals about past attacks
6.2
We now
introduce an additional source of information by allowing agents to receive private and
public signals about the size of past attacks. These signals are, respectively,
Xit
for t
>
2,
where un
= 5 {At-l,u^t)
and
idiosyncratic noise, Vt
is
+ u, uu ^ M
= S {At-i,vt)
common
is
noise,
we adopt a
preserve Normality of the information structure,
,
and S
S{A,u)
$"1 {A)
may depend on exogenous/endogenous
we can again construct
The only
difference
where
It
is
l/7f)
sufficient statistics Xt
oit-1
+ Vt +
and
signals.
By
v^
x
K
To
M.
-^-
Dasgupta
.
{x^_-^
~
6^) -
where x*_j
1 if
is
3,
with precisions at and Pt such that
defined by
/3t
=
/3t_i
+
T]f
+
an attack occurrs at
and similarly the dynamics
endogenous, the equilibrium characterization
exogenous
now
an indicator variable that takes value
{xt, zt)
Zt
and
h-iott-ilt
Although the realizations of
and
,
[0, 1]
public signals - and following similar steps as in Section
that these statistics are
is
=
Qt
{{),
:
similar specification as
~ AA(0, I/t^)
Using the property that in any monotone equilibrium At-\ = ^{y/at-\
(2002):
=
Zt
It-iat-i^t^
t
{At
>
0)
and
of {at,l3t) are
otherwise.
now
partly
otherwise the same as in the model with only
implication, the multiplicity results extend to the
game with
past attacks. Similarly, the structure of equilibrium dynamics remains as in the
signals
about
benchmark model,
except for the property that an unsuccessful attack does not necessarily reduce the incentives for
further attacks. This
signals,
which
in
is
because an unsuccessful attack now also generates new private and public
some cases may
offset the
impact of the knowledge that the status quo
is still
in
place.
To
see this, consider the case
which case the only novel
where
effect is that
all
signals are private (7^
>
0,rjf
>
0, Jt
=
'7t^
=
0)
i
"^
an unsuccessful attack leads to an endogenous increase in
20
at.
A
game.
further attack
On
is
then possible only
when
the other hand,
becomes possible
this signal
if
We conclude that signals
if
the endogenous signal
low enough,
is
this increase is large
about the
like in
enough,
public (7^
is
like in
=
>
the
-yf),
a
benchmark
new
attack
the case with exogenous public news.
size of past attacks
may
substitute for the exogenous arrival
of private and/or public information, but do not otherwise affect the analysis.
7
Conclusion
This paper examined how the dynamics of information influences the dynamics of coordination in
We
a global game of regime change.
may
found that dynamics
sustain multiple equilibria under
the same informational assumptions that guarantee uniqueness in the static benchmark examined
in the literature.
Although the model
is
highly stylized and abstracts from
intertemporal payoff linkages,
it
all
sorts of institutional details
and
generates a few distinctive predictions for equilibrium dynamics,
such as the possibility that fundamentals predict regime outcomes but not the timing and number
of attacks, or the succession of phases of tranquility, during which agents accumulate information
and no attacks are
and periods of
possible,
necessarily take place. These predictions
may
crises,
during which attacks
occur but do no
help understand the dynamics of crises
such as currency attacks, financial crashes, or political change.
applications
may
phenomena
Extending the analysis to these
a promising line for future research.
is
Appendix
Proof of Proposition
(2) gives a single
Solving (1) for x gives x
1.
equation in 9
=^
9
+
a~^/-^^~^{6). Substituting this into
:
t/^*(^;a,^,z) =0,
(10)
where
U^' {9;a,P,z)
Note that
and
[/''*
lime_>i
[9;
[/^*
•)
{9)
is
^
1
-$
(^
[$-i
{9)
contiimous and differentiable in 9 e
= —c <
0.
A
+^{z- 9)])
(0, 1),
with
-
Iimfl_+o
for ?/'* to
[i-/4>
{^~^
be monotonic
(^))]
U^^ {9)
=
1
—
c
>
solution to (10) therefore always exists. Next, note that
1
Since mingg(o_i)
(11)
c.
~ V^^
in 9, in
the condition
a > 0^ j
(27r) is
both necessary and
which case the monotone equilibrium
21
§_
is
unique.
sufficient
Finally, for
the proof that only this equihbrium survives iterated deletion of strictly dominated strategies, see
Morris and Shin (2000, 2001).
Proof of
Lemma
We
1.
prove the claim by induction. At
since in any equilibrium, the agents' strategy for
t
>
i*,
At
Consider next any
non-increasing in
=
Rt^i
>
0) for all 6
and only
>
9
\i
2
and suppose that the
non-increasing in
(0) is
6*_-^,
in
which case
In the former case,
6^.
=
0j
=
i
1 is
=
i
the same as in the static benchmark.
implying that either At
9,
>
=
l\xt,
Rt
9;_^)
=
—
=
0)
t
—
(9)
<
<
any t
result holds for
9^_^, or there exists 9t
Pr(i?t-|-i
the result necessarily holds
1,
and hence x^
Xt
all
(•)
is
(and therefore
such that At[6)
9*__-^
Q for
Since aj
1.
<
9
\i
= — oo.
In the latter,
atxt+pz
=
Pr(i?t+i
l\xt,Rt
=
0)
-
Pr
<
(0
>
9;\xt,9
at+0
^
1
atxt+fiz
^{^at + p
which
M
that there exists x^ G
either case. At {9)
9^
G
(0, 1) for all
<
t,
ciency, take
and
Xj
-00, then Pr {Rt+i
is
= X (0*, at)
,
=
it is
Proof of
For
then,
satisfies
and to
> (<)c
9^_-^^)
as
>
9^
l\xt,
If
=
Rt
c>
=
0)
instead
Part
2.
U
(i)
Pr
>
9^
i?j
(0
<
=
if
=
9t
9t\xt, 9
in
9^_-^,
and
<
Xj
follows directly
< U
if
(iii);
and only
For (in), take any
9*
<
Q.
there exists
all
9*
a,
>
U
Since
a such
9^_-^
[/ (0*,
It
>
9oo-
Note that
— 00, a,/3, z)
that, for
{e*,9*_^,a,l3,z)
9l_-^.
<
xt
>
0,
91
>
9^_^)
let 9q
>
if
=
implying
{>)xt. In
implies that
U
any a
= -c <
follows that, for
any
>
a,
<
and only
(9*,-oo,at,P,z)
and
22
>
t
i
2, 9*_^
>
1;
>
and therefore
{9'[,9*_-^,at,P, s)
if
=
and
< {>)X{9l,at) and
xj
for Xt
>
<
lima^oo
[^J'_i, 1]
is
Xj
.
z)
for all 9*
£
U
{9* ,9*_^,a,p,z)
<
/3,
z)
<
for all 9*
>
at
f^ (6'*,
[^*_i, 1].
[/ (6**,
> -00) and
/J^/ (27r)). It
-00, a,/3, z)
compact,
/3,
a>a,U [9*, SJ'_i, a,
no solution.
9*_^, for all
for all 9* (since 9^_^
for all 9* £ [6'*_j,l],
9*_^
—00; suppose
by inspecting U.
U {9*, —00, a,
fox 9
suffi-
and the definitions of X{-) and
and not to attack
continuous in 9* and
is
=
for all Xf
U [9*, -00, at, P,z) = U'^'^ {9*,at,l3,z) is strictly decreasing in 9* (since
follows that U [9*, 9^_-^,at, /3, z) <
for all 6* > 9t, which gives the result.
—
-|-oo,
if
which case
^t_i, in
which case
[<)U{9*t,9*_^,at,l3,z)
[9*,9*_-^,at, l3,z)
(ii)
of the private posterior
indeed optimal to attack for xt
Lemma
conditions
that
9co
^
2:4
and only
if
which together with
which case
in
(i),
by the monotonicity
note that
(ii),
1,
>
for an individual agent. If
indeed optimal.
U{-),'Px{9<9;\xt,9>9l_-^) therefore
<
>
— — cx)
a;t
Necessity follows from the arguments in the main text. For
2.
any sequence {xj,0j }^^that
not attacking
9t\xt,9
9t
as
1
which completes the proof.
and consider the best response
Xj
and hence
1 for all 9
other agents follow strategies as in
=
<
such that Pr(0
Proof of Proposition
all
and converges to
strictly decreasing in xt
is
-^u]y
at+P
it
=
follows that
Moreover, for
-oo,a,/3, z) for
and therefore
(6)
admits
For
take any
(to),
(6't_i,^cx)),
U
a such
{0* ,9*_-^,a,f5,z) in 9*
>
Fix
-co) and suppose that
>
t
<0 for all 9* >
and
{9*, 9*_2,
is
non-increasing in 9-i, continuous in 0*, and equal to
A >
ists
U
ity of
<
which ar
and only
when
z
<
c
There
>
and only
(<)
and
z;
1/2 and
if
and only
and of
<
2
z
if
(>)
>
a>
z (where
=
=f=
>
for all
9i>
it
9q
=
—
t
which means that
1,
—c
<
for 9*
e
6-i, implies that there ex-
Furthermore, by continu-
[0,1].
U {9*,-,a,-)
is
(6)
U {9*,9^i, a, /?, z)
uniformly continuous over
U
<
(0*,^j_2,q,/3, z)
>
admits no solution in any period t
t
<
<
^oo for all
c
>
z
<
z such that: 9t
t
if
and only
<
>
if z
9i for all
t if
z
<
r;,
< (>) ^^o
9i
z='z =
These thresholds satisfy
'z.
if
z
1/2.
(27r))
,
=
let ^
^ (a,
j3,
be the unique solution to U{9,
z)
— oo, a, /3, z)
9t
z (a,/3)
z {a, P))
=
and
on the other hand,
,
is
such that 9
>
(<)0oo
defined so that 89 /da
is
To simplify notation, we henceforth suppress the dependence
1/2, in
which case
and 89 /da
9^^
z [a)
q > ai and therefore
>
0,
—
=
z [a)
9i
>
=
9t
= 9^
and hence
9\
<
any Q > aj, and therefore
<
13,
of 9
>
on
if
(<)
(/3,
z)
9i
>
9t
9t
1/2 for
a.
600 for all
for all
<
all
t.
When
t.
When
Finally,
^00 for all
t.
<
z
The
when
1/2, 9 [a)
instead z
z
>
>
=
9oo
1/2,
1/2, for
any
result thus holds with
1/2.
a and
a 6
z.
=
c
^oo for all a,
>
defined by U{9ao, ~oo, a,
p.
Next, consider
z (a)
9t
ai (> /3^/
is
for all
9no for
ai, 9 (a)
z
-c,
we use the convention
with the properties that
for all 9*
which proves that
[0, 1],
The threshold
z.
(<)
and z on
z
and d9/da <
=
G
z<'z<'z when
First, consider c
For
-A
<
at-i such that
exist thresholds 2
threshold z{a,P)
a >
9*
By
0.
the static equilibrium threshold) and
(i.e.,
9 (a)
all
=
9* 6
a.
Proof. For any
The
and
[at-i, q)
Lemma Al
2
a >
in {9*, a), there exists
a £
for all
if
9*_^. This, together
{9* ,9^_-^,at-i,l3,z)
>
[6' ,9l_-^,a,f3,z)
X [Qt_i,a]. This also implies that there exists a £ {at-i,a) such that
[0, 1]
for
U
such that
any
for
a.
U
at-i,P, z)
U
a,
>
6*
fact that lim0._i ^7 (9* ,9l_y,a,j3,z)
at-i, 0,
9^_2,
>
the highest solution to (6) for period
is
0^_j
2,
any a
that, for
and the
follows then that (6) admits a solution for at
Finally, consider (v).
— 6^ -
Since lima-.ooU {d* ,d^_-^^,a, P, z)
0oo-
G (^j*_i,^oo) and
6'
there exist
the continuity of
<
6'*_i
<
1/2, in
and converge to
therefore 9 (a)
9oo for all
t.
When
[Q:i,a'), ^ (a) is
becomes lower than
which case
9oo
is
z
G
z (a)
^oo as
and
^ 00.
a
z
(ct)
are both decreasing in a, satisfy z [a)
When
z
<
9oo,
then clearly z
<
z [a)
<
>
z {a)
always higher than ^oo and decreeising in a, which implies that
{9oo, z (ai)),
there are a"
>
a'
>
higher than ^00 and decreases with a.
and continues to decrease with
23
a.
ai such that z [a')
As soon
Once a > a",
as
a
G
=
z {a")
=
z.
(a', a"), 9 [a)
^ (a) starts increasing
with
,
,
.
>
a, but never exceeds ^oo- Hence, ^i
when
Finally,
all t.
>
z
z (ai)
<
that the result holds for c
>
Finally, consider c
<
z {a)
>
that 9i
>
9t
therefore 9 (a)
(ai,a'), ^ (a)
= 'z=^=
When
9i
For a e (ai,
a'), ^ (a) is
9t
therefore 9 (a)
is
9i
^oo-
and increases with
Hence, maxt>i
<
and
<
9t
z
=
[9t (xt)
is
[0, 1].
,
continuous
Vt
G
M
/
is
true:
all
t
>
Note that,
l*",
2
<
/
>
z) is
/3,
0.
We
t.
conclude
<
then clearly z
,
in a, satisfy
z (a)
<
z (a)
>
a'
is
ai such 5 (a')
=
a >
a', 9 [a]
decreases
>
>
9t
When
=
>
a'
9oo-
ai such that z (a')
a G
(a',
a"), ^ (a)
a and asymptotes
> z[a) >
then clearly z
^oo,
>
9i
z
=
is
z.
For
5(ai)
z (a")
=
z.
higher than
to 9oo from above.
a and
z (a) for all
with a, and asymptotes 9^0 from below. Hence,
>
1/2 with z
=
=
z (ai), z
z (ai)
2, Vt (x*; 6)
>
Note that,
2.
for all
continuous,
in
[0, 1].
continuous in
(6',
x M,
M."^^
= U {9i
(x; 0)
M
continuous in x* for any x* €
is
it
t,
(x)
9t (xj) is
U {9,
[0, 1]^. It
and
z)
/3,
Vj (x'; O)
M
continuous in xt G
follows that
Furthermore,
9_i) G
—00, ai,
,
9t (x*)
=
—00, a,
=
and
maXr<i{^T (^t)}
z) is
/3,
follows that, for
continuous
all t,
Vt (x*; O)
To simplify notation, we henceforth suppress the dependence
A
if
and only
(a) \xt
t:
(x')
—
Xr\
6 K, then
<
R
if,
t;
is
no
For
risk of confusion.
as follows. For any function f
for
if
\f (i')
any x' e
xt £ K;
-/
A
(b)
(i')
|
<
and any e
Xr
e;
<
—l/rj
(b') if
/
>
0,
if
Xt
(x')
:
A
>
all t
^
M.,
1,
where
of pj,
the function pt{9;xt)
^CR
and
t
>
1,
=
we say that
A such
>
= —00; (c) Xt > l/rj, if Xy = +00, the following
= -00, then / (£') < -1/e; (c') if / (x') = +00,
there exists
such that, for any x* €
?;
1/e.
if
/
:
and f [A)
continuous in
>
t
max operator is
on 5 whenever there
(a') if
then / (x')
BC
^co for all
9t for
.
continuous over
that for
when
which case Vi
0, in
~) for i
'^Continuity can be extended in
is
a" >
9 (a) decreases with
,
and takes values
U {9, 9^i,a,
in R
and
[0, 1]
=
ci'f,/3,
also continuous in x*
and
maxt>i
continuous in xi for any xi E
is
Since the
Consider next J
Vt
^oo), there are
,
^oo, increases
and any
first 5
9t-i {x^~^)
,
takes values in
G
z (ai)
conclude that the result holds for c
For any S >
Proof. Consider
in
9i. Finally,
We
and similarly Vi {xi;6)
is
<
z
z (qi)), there
,
follows that
It
z G (2 (cti)
always lower than
t.
= 9^ >
9i
^oo-
Lemma A2
U
>
z £ (z (ai)
And for a > a"
a.
,
z (cti)
lower than ^^o and increasing in a. For
> 9^ >
^oo for all
<
9t
z (qi)
higher than ^oo and increasing in a, whereas for
is
maxi>i 9t>
9oo
—
z
always higher than ^oo and decreasing in a, which implies
is
with a, converging to 9oo from above.
=
and therefore
a — oo.When
0oo as
When
t.
When
6t for all t.
which case z[a) and z{a) are both increasing
1/2, in
9ao for all
>
§1
for all a,
6'oo
1/2 with z
and converge to
^00,
a > ai and
for all
a e
<
z (a)
<
9 [a)
,
and
6*00
A
yl
—+
y.
g{B) C
R, p
B — R, and g C — K are
C Q l\ then the function «;
:
>
:
continuous, respectively, in A,
»
x B.
24
:
yl
x
B
-.
R
defined by
w
B
(x'.x'')
and C, where
=
j4
C K
g (/ (x') ,g (x'=))
,
is
.
F ([$
(yoj
(xt
-
-
^))
S\
continuous in {9,xt) G M^ and strictly decreasing in
is
/6)
9.
For
>
i
2,
J^^<P{V^tix-0'))A{O';x^'\S)de'
Ulz\
The
- and similarly the
p.d.f. ipt
xR.^^
(x*~^,Zt) £
Vi (xi)
is
xR.
Similarly, for
Vt
on
(J
whenever
thus continuous in
t
=
vi (x;xi)
1,
-
(^|a;;i*~"',(5)
=
is
vt
G
R
is
{^xt;
at+/3
and
£
{x, x''~^)
RxR
.
It
continuous in {x,x^^^,xt) G
x''"^
also continuous in
is
xt)
,
R
satisfies conditions (ii)
of vt (x;x*) with respect to x (see proof of
and any
R
x* G
that the strategies defined by
(i)
—
,
>
Uf (x; x*)
(<) Vt (x*)
{in) constitute a
Necessity. Conversely, suppose that {at(-)}t^i
is
^
^^
<
+00,
decreasing in
is
Xf,
= — oo,
if
xj
Xt
We next show
= +00, in which
9'
gR,
that, in
case pt
/+00
Xj
and
any equiHbrium,
=F
{9; -f-oo)
if
in the
above)
x
<
in
any such
(>) Xf
the probability
9,
implying that agents must follow
must be
if
and only
decreasing in
is
it
<
(iii)
Lemma A2
monotone equilibrimn. Since
For {xf}^^ to be equilibrium cutoffs,
Vt (x*)
if
and
Then, by standard representation theorems (Milgrom,
also decreasing in 9.
1981), the expected payoff from attacking
cut-off strategies.
vt
monotone equihbrium.
equilibrium the measure of agents attacking in every period
of regime change
pt,
this does not create confusion.)
guarantees that, for any x G
Xt
-
(To simphfy notation, we again suppress the dependence of
3.
The monotonicity
proposition.
— oo <
c
(e
continuous in {x,xi) G R^ and therefore
Consider a sequence {xt ((5)}^j that
Sufficiency.
It follows
(v/STT?
continuous in xi G R.
Proof of Proposition
and
is
<P
then immediate that Vt[x*~^,xt)
It is
R
^j -
{0; xs)]
Ps
= J_^ pt{9;xt)d''it
follows that vt (x;x*~-^,Xf)
RxR
c.d.f.
-
[1
(^ (1
>
Vt (x*)
< +co
xj
—
9))
,
=
for all
that, for all
Vt (x^)
t,
=
if
4-oo.
i
then, for any
>
and xi
1
t>2,
> — cx).
(x, x*"-')
R
G
x
Indeed,
R
,
if
and
F{\{\-9))iPt{9\x-,x'-\5)d9-c
•OO
r-\-oo
/6'
F{l{l-9))^t{9\x;x'-\6)d9+
oo
<
^
for
Note that
^t {9'\x;x'-\5)
Vt [x;x')
which p{6;xt)
>
y,
+F
(i
(1
-
continuous in
that
{x, x')
{1(1
d')) [I
- *t
£
is,
R
-
9))
{9'\x;x'-\6)]
can also be written as J^ Fp(y\x,x*)dy, where Fp[y\x,x^)
Fp{y;x,x^)
=
if
y
=
1
and Fp{y;x,x^)
p~' {y;xt) denotes the inverse oip(9;xt) with respect to
is
F
/
i^t
{9\x;x'-\5) d9
-
c
Je'
x
K
Dominated Convergence Theorem,
x R,
it
for
any y £
The
9.
[0, 1].
=
follows that J^ Fp{y;x,x^)dy
is
is
c,
the (probability) measure of 9
{p~^ (y',^t) |x;i'~^,i5)
continuities of p
Since Fp{y; )
25
"^t
is
-
also
and
"it
if
y £
[0,1),
where
then imply that Fp(y;x,x^)
bounded from above by
also continuous in (x,i').
1,
from the
where ^f
=
^,6)
(9'\x;x^
\ (5)
{0\x;x^
ipt
J_
knowledge that the
d0. Furthermore, since the
sta-
tus quo survived past attacks causes a first-order-stochastic-dominance change in posterior beUefs/'^
< $ (^/5H^
*t ie'\x;x'-\S)
0, this
implies that lima:^+oo
9'
it is
e M,
bounded from below by — c,
-F
F
V
this implies that
(
—
(j (1
—
j (1
[x^~^
,
—
c.
-^
0.
0'))
9')^
=
+00)
F (i i-O)) V'l
-
{d\^) rf0
c
-
{o'
^^))
argument
F (i
> *i
{9'\x)
vi (x;
—00) >
=
rules out xi
{-9'))
-
vt is
= —c <
Vt (x; 5;*^\ -t-00)
lim^^+oo
=
true for any
is
Together with the fact that
Xt
/+00
(V^7+^
Since the latter
= +00 can not be part of any equilibrium. A similar
suppose xj = —00. Then, for any x G M and any 9' € R,
and hence
Finally,
<
+00, in which case
>
Along with hm,_+oo *
^^f±|^)).
vt (x; x*~-^, -|-oo)
—
9'
also true for
-
[9'
-l-oo.
c,
-00
where ^1
is
= /_^ ipi
{9'\x)
-^ —00, and since vi
9'
true also for
limx-»-oo v\ (x; —00)
We
exist x',x" e M.
—
1
(i)
c
—
>
>
fact that
such that Vi
bounded from above by
is
implying that xj
0,
= —00
of x"
R
€
x'
M
e
t
>
we have
c,
the monotonicity of vi (x;xi) in xi along with
0,
limx^-00
>
(x')
(x,
''^i
vi (x',
—00) >
>
>
-00)
Since this
c-
(
that V\
(
— 00) =
can not be part of an equilibrium.
>
limx-,+cx)
>
(x',
Vi (x")
x") to Vi (xi; 5)
its
continuity in
implies that there
(x, -hoo),
'Vi
(x", -Foo)
i;i
Vi (xi) in x\ then ensures existence of a solution x\ {5) £
Next, consider
—
1
F (| — 0')) ~
any monotone equilibrium.
necessarily hold in
[iii)
For any J
any xi and the
for
=
conclude that
Existence.
X
and therefore \\xax^-^
{9\x) dO,
.
=
The
continuity of
0.
For any given x*~^, a similar argument as above ensures the existence
2.
<
such that Vt(x*~\x")
>
such that Vt (x*""\x')
0,
<
V( (x",x*~-^, -|-oo)
<
or Vt (x*"^,Xi)
Moreover, either there also exists
0.
e R. In the former case, the
for all Xj
continuity of Vt [x^~^,Xt) in xj ensures the existence of xt £ (x',x") such that Vt {^x*~^,xt)
<
In the latter case, vt (xjx*"-', —00)
at Xt
=
—00, Vt
(x*"-*-,— 00)
sequence {xj {5)}^q that
t
>
(x;x*^^,x)
lima;_,_oo
f't
2.
We
start
=
(x; x*~^,
satisfies conditions (ii)
Proof of Theorem
any
=
Vt
Vt (x*~^,x)
— 00) <
and
(iii)
0.
<
for
We
any x 6
by
(7)
and
(8),
0.
and therefore
conclude that there exists a
in the proposition.
by establishing pointwise convergence of
2 and any (x*"\x() e M*"'' x R,
R
=
we have
—
Vt as 5
>
For
0.
that^*
/-I- 00
pt {9; Xt, S) d'i't (9\xt;
x'-\S) -
c
=
*t
{9t {xt) |xt;
x'-\Q) -
c
-00
=
'
'
This can be seen by noting that the ratio of the densities
increasing in
'"For 5
>
Fp(y;x^,S)
0, let
°°
—
i-'t
(^l^'l
2;'
^
,
(5)
/\/at
+ P4'
(
\/cn
+P
8
—
'^'^'^t'
(
)
)
is
9.
and Fp{y;x'^ ,0)
Jg
U{9t{xt);et-i{x'-^),at,P,z)=Vt{x'-\xt;0).
>
Fp(j/;x*,(S)beasinfootnotel6. Similarly, for 5
=
if
y
=
1.
Since Fp
Fp{y]x*,0) as 5
limj^o Fp{y;
x',
S)dy
=
—
»
°°
Jg
0,
is
=
bounded from above by
0,
1
Fp{y;x\0)
=
>I't
and, for any y £
[Ot (x,)
[0, 1]
\xt;x*~\0)
and any
from standard Dominated Convergence Theorems, lim^-^o J^
Fp{y; x^ ,Q)dy, which gives the result.
26
°°
x'
e
ifj/
£
Fp{y;
R
x''
[0,1)
x R,
,
5)dy
=
—
Similarly, for
any xi £ R, by
lim Fi (xi;
=
Ut (x*)
games
*i
^ U {9^
c
(xj)
=
-cx>, ai, ^, z)
;
Vj (xi; 0)
unperturbed game
(S
=
>
0.
From
>
t/i
>
all i
and reserve the use of
0)
l,we henceforth
Vf for the
let
perturbed
0).
Consider
T=
first
and
1
an arbitrary £
fix
-
Ui (x|
By
-
{01 (xi) |xi)
prove the result by induction. To simplify the notation, for
Vt (x*;0) for the
>
{6
=
<5)
(7),
>0
o
We now
,
.
the convergence of Vi to
as 5
f/i
—
0,
>
>
£)
we can
the strict monotonicity of
(x^
find 6i{£)
+ £)
L''i(xi),^^
.
>
such that, for any 6
<
5i (e)
Vi{xl-s-5)>0>Vi{xl+s-6).
From the
continuity of Vi {xi;5) in xi for any 5
to Vi (xi; 5)
=
such that Xj
existence in Proposition
|xi [6)
—
x||
<
£.
3,
—
first
<
xj (5)
Xj
0, it
+
e.
follows that there exists a solution xi [5)
Following the same steps as in the proof of
we can then construct an equihbrium
T=
This proves the result for
Consider next an arbitrary
Take
<
e
>
T>
2, fix
>
£
{xt{6)}'^-^ for
T
such that
{6)
1.
and suppose the
0,
any equilibrium of r(0) such that x^
> — oo. By
result holds for
T-
1.
the (local) strict monotonicity of
around x^ implied by the assumption that x^ ^ aigmaxx Ut {x*^^^,x), there
exists
et
<
Ut
£ such
that either
Ut
Ut
or
(x*-^~-^,x^
—
<
et)
-
{x*^~\x*j,
< Ut
x'^~^
for
G
R
any 5
and the
<
5
there
1),
[CI] for all
i
< r-
1,
[C2] for all
t
< T-
1, |xj*
[C3] in period T,
Ut
is
{x^~^
xt{5)\
((5),
= -oo and
<
e'j.
<
6t G
(0, 5 [E'rp.T
—
1))
s
\i
x^ - zt) >
Next, by the convergence of Vt (x-^"\ xt) to
exists
1,
,
assume the
first
continuity of Ut{^^''^,xt) in
there exists
some
e'j-
S
(0,
£t) such that,
a sequence x^^^{5) satisfying the following three conditions:^"
either xj {5)
-
T—
+ £r)
loss of generality,
From the
identical.
is
fact that the result holds for
{e'^, T —
{x*^-\x*t
C/t
+ et) Without
{x*'^~^,x^
case - the argrunent for the other case
>
>
£t)
Vt (x^ [5))
such that, for any 8
0,
or Xj G
<
x; G M, and xj [5)
> Ut
Ut
<
(x^"^
(<5)
,
x^
(x-^~\ xy) for any
<
5t, there
is
R
and
-l/e'j,
K [x" {5) \5)=Q;
< -l/e
if
x*
=
-oo;
+ et)
(x-^~-^, X7-)
G
M
x R, there
x-^~^{S) that satisfies [C1]-[C2]
and
such that:
'"This follows from the the monotonicity of
and the monotonicity of
Continuity of
result holds for
U
T—
—00, ai,
)9,
z) in 9
- which in turn
is
impUed by oi >
/3^/\/27r
-
9i (xi) in xi.
implies existence of
1
(7(6';
e'-p
such that that [C3] holds for any x'''~^(S) that
then ensures that, for any S
<
S
(eT,T —
[C2].
27
1)
,
there exists
i"^~''((5)
satisfies [C2];
that the
that satisfies both [CI] and
.
[C3'] in
But
Vt
period T,
then, by the continuity of
-
with |xy
< et <
xt{5)\
x^ -
(^^"^(6),
Vt {x'^~^,xt)
that solves
s,
>
er, 5)
<
for all Xf
Xf,
where
> — oo
Xt
same
=
xlj.
+ ex; 5)
an xt[6) £ R,
0.
x^ = — oo. Recall
9t (if)
,
x^''^{8), there exists
=
Vt [x'^^^{6),xt{S);6)
solves
(6)
any
that, for
=
6t-i (i**~-')
result holds for
T—
1,
sequence x^~^[5) which
maXT-<i-i 9t
(a^*)
5
<
6t, there
for the
x^~^
is
(S)
<
same x
Vt
,xt) in
{5)
= — oo
satisfies
any 5 < 6{£,T) and
for all
<
t
if xj"
(x
Proof of Proposition
on
t
(/?t,
zt)
4.
and of
Proof of Theorem
continuous in 9*
= — oo,
then Xt
£
3.
(x*, 9^)
Consider
with
[0,1]
U{9l,—QO,aj,Pi,zi)
—
>
—
1
on
U
,
is
(0) for
< T
:
— —c
>
multiple equilibria even
dl{-)
if T =
(S)
<
xt G
for all
[e,
T) €
which
x^
if
set of x-^* that
5{s,T —
(0,
such that,
1))
^ argmax^t/f (x*~^*,x)
Xj
£ R, then \xt{5)
<
Vt (x* [6))
—
Xf\
<
£
Using the same
0.
part of an equilibrium {xj (5)}^^^
is
- the proof follows exactly the same steps as in the
thus omitted for brevity.
=
i
1.
(0j*_j,l),
U
is
and
[/ (1,
if
— oo,
•)
Next, consider any
[9*,9*_^,at,Pt,
Zt)
is
>
t
is
and only
if
ai
>
1.
28
01/2tt.
continuous in 9* G
£ [9^_^,1] as
Hence
2
and note
that,
strictly decreasing in zt
implying that necessarily maxg.g[g_j
monotonically for any 9*
unique
U {9* ,—oo,a\,l3i, zi) is
= — c. Hence a solution
For any (ai,/3i,2i),
This follows from the same argument used in the proof of Proposition
''Note that the function
5
3, x-^ {6) is
Furthermore, since U{9*,-,zt)
)
< xx
x'^^
and therefore the
t
< —l/e and
U {0,—oo,-) = 1 — c
=
always exists.^"
>
[9* 9^_-^,
z*
first
c>Oaszt — — oo,
for Zt sufficiently low.
since
[5)
t
F
then,
0,
0.^"^
in every
Hence, there
£ M, with
(^)
Vt (x^~^(5),xt;^) <
instead
— oo) <
all
an x^
>
(^),x!^;(5)
Apart from a notational adjustment - namely the dependence of
any {9^_-^^at,Pt) and any 9* £
U{9*,-,zt)
(5),
(0) is finite.
model with only private information, and
^'
such that, for
1)))
the proof.
,
in period
If
proof of existence in Proposition
for the
F {S) which completes
and
a
is
and such that:
e')
T—
(0, 5 {s',
such that Vt (x
also
is
equilibrium {xjj^j of
evei-y
=0; and
;(5)
arguments as
for
=0.
T, there exists x'^ {5) such that, for
Vt (x* [6)
9l{zi) to
— oo, x'^)
£ M, there
xj-
Vt
(
admits at most two solutions
can be part of an equilibrium of F
for
x^ £
-l/e, such that Vt [x^~'^{5),xt{5) ;^)
Finally, recall that (6)
and
to Ut, there also exists a 5t G
there exists an
(5),
(— cx), x^), then xt
for
there
1)),
that satisfies [Cl]-[C2] and such that:
by the continuity of Vt {x
x'j.
—
6 {s' ,T
fact that the
Vt{x'^-H^),^'t^S) <0.
[C4']
If,
<
above (replacing e^ with
satisfies conditions [Cl]-[C2]
the pointwise convergence of
any
£ (0,£) such that, for any 5
s'
Pick some
•
Ut{x'^-H5),x't) <0.
[C4]
By
some
there exists
(x**~^, Xj) =
>2,Ut
t
e (— oo,min{ — 1/e, xy}). From the continuity of Ut (x'^^^,xt) in x'^"^ and the
x'-p
U
{x^'^
in xt^ for the
Next, take any equihbrium of r(0) such that
—c <
> Vt
Q
zt
j]
[/ (0*,
[0j_^,
—^
1]
•,
for
zt)
and
>
any
zt,
+oo, from stan-
3.
for
a\
<
/3?/27r,
the
game
trivially
admits
U (9*,6*_i,-,zt)
dard Monotone Convergence Theorems, the function
as zt
—
strict
monotonicity of
»
maxe.g[5i_jj]
U
in zt
then guarantees that there exists a
U [9*, 6'j_j, at, ft, 2j) >
t
then an equihbrium:
for
i
=
(<)0
if
and only
>
9'^_-^{z'-~^)
1,
9l{zi)
that (9) admits a solution 9t(z^)
is
equilibrium, at any
and
{at, ft }j^ J
all t is
t
also
and any
>
2, 9^{z'')
>
if
if
z
:
for Zt sufficiently high.
and only
if zt
< 1
U {9^,
U {9*,9;_i{z^-^),at,
9*_i{z^-'^) for all ^t
<^
{9'^_^,at,Pt)
—oo^ai, Pi,
/3t,
{9;_i,at,Pt)
zt)
also
The
=
zi)
=0}. Note
Since 9^ (z*)
The
imphes
following
0;
for all
that, in this
=
9'[{zi) for
an equilibrium, we conclude that the game admits multiple equilibria
T>
—c
such that
finite 2 (5(_j, at,/?f)
< (>)3 (0*_^,at,/9t), which
any solution to
is
e {2,...,T}, 0^(2*) =max({e;_i(z«-i)}U{0*
all z*
<
+00, implying that ma.xg,^^g_^ijU [6*,6^_j,at,/3t,2t)
converges uniformly to
for
any
2.
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30
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ue
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