Precise Signals and Multiple Equilibria in
Coordination Games: an Illustration
Y. Stephen Chiu∗†
June 2003
Abstract: There is a growing literature using the global game approach to study
coordination games. Several papers show a unique equilibrium as long as signals are not
too noisy. In this paper, we provide an example in the context of currency attack in which
multiple equilibria exist no matter how small the noises are.
JEL classification code: C72, F30, G21
Key words: Currency Crises, Coordination games, Global games, Multiple Equilibria
∗ Financial support from an HKRGC earmarked grant (HKU4011/00H) is acknowledged. Any remaining errors are the author’s.
† School of Economics and Finance, 918 K. K. Leung Building, University of Hong Kong, Pokfulam
Road, HONG KONG. Phone: 2859-1056. Fax: 2548-1152. Email: [email protected].
1
1
Introduction
Multiple equilibria usually arise in coordination games with strategic complementarity
(e.g., Diamond and Dybvig 1983, and Obstfeld, 1996). Based on the pioneering work of
Carlsson and van Damme (1993) on global games, Morris and Shin (1998) eliminate multiplicity of equilibria in a standard second generation currency attack model by assuming
that agents observe fundamentals with small errors. This very strong result is weakened
by more recent work such as Morris and Shin (2001) and Heinemann and Illing (2002)
in which multiple equilibria may reemerge when observed signals are noisy enough. The
present paper provides an example, also in a currency crisis framework, where multiple
equilibria exist no matter how precise the noisy signals are. It thus complements the
existing literature in delineating the exact conditions for the global approach to yield a
unique equilibrium.
2
Model
The model closely follows Morris and Shin (1998); the reader is referred to their paper
for more details. There are a government and a continuum of atomistic speculators. Let
the measure of all speculators be unity. The state of fundamentals is denoted by θ ∈ [0, 1]
so that a greater θ means stronger fundamentals. The country’s fixed exchange rate is
overvalued even for the strongest state θ = 1. The state θ is distributed with prior density
function f (θ), which equals 2−h for θ ∈ [0.25, 0.75] and equals h for θ ∈ [0, 0.25)∪(0.75, 1];
as will be clear, the discontinuities in the density function at θ = 0.25 and 0.75 for h 6= 1
play a crucial role in triggering multiple equilibria.
The following game is played. Upon the realization of θ, the government observes θ and
1
the speculators each observe only an iid signal x uniformly distributed in the [θ − ε, θ + ε]
interval, where ε is small and known. The speculators then each simultaneously and
independently decides whether to short sell one unit of domestic currency. After observing
the size of attack (i.e., the proportion of speculators who have short sold), the government
decides whether to defend or abandon its currency peg. We assume that it will defend as
long as the size of attack is less than a(θ), where a(θ) ≡ 0 for θ < 0.2 and ≡ −0.2 + θ
for θ ≥ 0.2. In this case, the peg is maintained. In case the government chooses not
to defend, the peg will be abandoned and the currency will simply go afloat. Note that
for θ < θ ≡ 0.2 the government will abandon the peg even in the absence of attack. A
speculator has a zero payoff if he has not attacked, regardless of what happens to the peg
subsequently. He will have a payoff of −t if he has attacked but the peg remains stable,
and a payoff of R(θ) − t = 1/(1 + θ) − t if he has attacked and the peg is subsequently
abandoned, where t ≡ 5/9 is the transaction costs. It is easy to show that, for θ > θ ≡ 0.8,
the net gain to a speculator from even a successful attack will be negative. All of the
above is commonly known.
Remark 1 To simplify the presentation, we assume that ε ≤ 0.01 throughout the paper.
Since we are interesting in the outcome for sufficiently small ε, setting such an upper
bound is without loss of generality.
Denote by u(x, Ix ) as a speculator’s expected utility from short selling when he observes
a signal x and each other speculator j follows Ix , where Ix (y) prescribes j to attack if
his signal y < x, and not to attack otherwise. Given that Ix is played by every other
speculator j, one can compute the size of attack as a function of the true state. By
comparing it to a(θ) defined earlier, one can easily show that there exists a critical state
φ(x) such that currency peg will be abandoned if and only if the true state θ ≤ φ(x). In
2
our model, it can be shown that
φ(x) =



 x+ε



if
1.4ε+x
1+2ε
if
x < 0.2 − ε
(1)
x ≥ 0.2 − ε.
Since the true state cannot fall below x − ε, we have
u(x, Ix ) =
Z
φ(x)
x−ε
R(θ)g(θ|x)dθ − t,
(2)
where g(θ|x) is the posterior density function of θ conditional on the signal x.
3
Analysis
It is now well known that if u(x, Ix ) is not monotonic in x, multiple equilibria may become
possible. Let x1 and x2 be the smallest and largest x such that u(x, Ix ) = 0. Then for
θ ∈ [φ(x1 ), φ(x2 )], multiple equilibria exist, and both maintenance and abandonment of
the currency peg can be sustained as equilibrium outcomes.
Define A1 ≡ (0.25 − ε, 0.25 + ε) and A2 ≡ (0.75 − ε, 0.75 + ε). The following two
lemmas are immediate.
Lemma 1 For x ∈
/ A1 ∪ A2 , the value u(x, Ix ) is invariant with h. In other words, the
function u(x, Ix ) for the case of h = h1 completely coincides with its counterpart function
for the case of h = h2 6= h1 , except possibly for the region where x ∈ A1 ∪ A2 .
0
Lemma 2 Suppose 0 < h < h00 ≤ 1. Denote by u0 (., .) and u00 (., .) as the corresponding
u(x, Ix ) for the two cases, respectively. (i) For x ∈ A1 , u0 (x, Ix ) < u00 (x, Ix ). (ii) For
x ∈ A2 , u0 (x, Ix ) > u00 (x, Ix ).
The intuition of Lemma 2 is as follows. Suppose speculator i observes a signal slightly
greater than 0.25 − ε, i.e. his signal x = 0.25 − ε + ∆ where 0 < ∆ < ε. For the case
3
where h = 1, his posterior density function is uniformly distributed in [x − ε, x + ε].
On the contrary, for the case where h < 1, the posterior density g(θ|x) equals Kh for
θ ∈ [x − ε, 0.25) and K(2 − h) for θ ∈ [0.25, x + ε] (where K is a normalizing coefficient
so that
R x+ε
x−ε
g(θ|x)dθ = 1). Then the signal x is a biased estimator of the true state; it
is smaller than the expected value of the true state. Therefore, from the perspective of
speculator, a slight increase in the observed signal from 0.25 − ε to 0.25 − ε + ∆ can lead
to an accelerated improvement of the fundamentals (hence, a sharp drop in the chance of
a successful attack, holding other speculators’ strategies constant) that is not exhibited
when h = 1. This thus explains part (i) of Lemma 2. Part (ii) can be explained likewise.
–––––––––––––Figure 1
–––––––––––––—
Figure 1 consists of 3 panels depicting the corresponding u(x, Ix ) function for different
h and ε. Panel a corresponds to the case where ε equals 0.01 and h equals one. The
function is strictly decreasing, cutting the horizontal axis at x∗ = 0.411. This is also the
benchmark scenario studied in Morris and Shin (1998). Panel b depicts the case where,
while ε remains to equal 0.01, h is reduced to 0.01. There are a U-curve for x ∈ A1 and
an inverted U-curve for x ∈ A2 . The curves are so drastic that there are altogether four
points of intersection on the horizontal axis around x = 0.25 and x = 0.75. Denote by
x1 and x2 as the smallest and largest x that make u(x, Ix ) = 0. Hence there are multiple
equilibria for θ ∈ [φ(x1 ), φ(x2 )] = [φ(0.240), φ(0.758)] = [0.249, 0.757]. Panel c depicts
the case where, while ε remains the same, h is increased to 0.2. The inverted U-curve for
x ∈ A2 now is too weak to make u(x, Ix ) = 0. The range of θ that allows for multiple
equilibria is [φ(x1 ), φ(x∗ )] = [φ(0.241), φ(0.411)] = [0.249, 0.417].
While Figure 1 illustrates the role of discontinuous prior density function in trigger-
4
ing multiple equilibria, the following Proposition shows that such multiplicity cannot be
eliminated by increasing signal precision.
Proposition 1 (i) Suppose h ≤ 0.204. No matter how small (but positive) ε is, there
always exists a point x1 ≤ 0.25 + ε such that ui (x1 , Ix1 ) = 0. (ii) Suppose h ≤ 0.0105. No
matter how small (but positive) ε is, there always exists a point x2 ≥ 0.75 − ε such that
ui (x2 , Ix2 ) = 0.
Proof. We show part (i) first. Consider an observation x0 = 0.25 − 0.9ε ∈ A1 . For
this value of x0 , according to (1), φ(x0 ) equals 0.25. Coupled with the assumption that
ε ≤ 0.01, we have
0
u(x , Ix0 ) =
Z
0.25
0
x0 −ε
R(θ)g(θ|x )dθ − t <
Z
0.25
x0 −ε
R(0.23)g(θ|x0 )dθ − t = R(0.23)p0 − t
where
0
p =
Z
0.25
x0 −ε
g(θ|x0 )dθ =
h × (0.25 − (x0 − ε))
.
h × (0.25 − (x0 − ε)) + (2 − h) × (x0 + ε − 0.25)
(3)
(To see (3), note that for any x ∈ A1 , g(θ|x) = Kh if θ ∈ [x−ε, x+ε] and = K(2−h) otherwise, where K is a normalizing factor that satisfies the condition that
R x+ε
x−ε
g(θ|x)dθ = 1.)
Substituting x0 = 0.25 − 0.9ε into (3), we have p0 = 1.9h/(1.9h + 0.1(2 − h)), which is
independent of ε and can be chosen arbitrarily small by reducing h sufficiently. Hence,
R(0.23)p0 −t < 0 if and only if p0 < t/R(0.23), which with some manipulation is equivalent
to h ≤ h1 ≡ 2t/ (1.9R(0.23) − 1.8t) = 0.204. Hence, for all h ≤ h1 , no matter how small
ε > 0 is, there exists some x0 (ε) = 0.25 − 0.9ε ∈ A1 such that u(x0 , Ix0 ) ≤ 0. Since u(x, Ix )
is positive for x < 0.25 − ε and is continuous in x, it follows that another point x1 exists
such that x1 ≤ x0 ≤ 0.25 + ε < x∗ and u(x1 , Ix1 ) = 0. We thus obtain the first part of the
proposition. Proof of Part (ii) is similar fashion and is omitted.
5
The key assumption in our model is discontinuity of the prior density function. Imagine
that the discontinuity at θ = 0.25, say, is replaced by a continuous, steep change in prior
density function. Then for large ε, the drastic change in the value of u(x, Ix ) is still
conceivable, and multiple equilibria will result as well. However, as ε approaches zero,
the prior density function around θ = 0.25 becomes flatter and flatter, and multiple
equilibrium will no longer exist. In this respect, our result is consistent with previous
work (Morris and Shin 2001, Heinemann and Illing) that increasing precision of signals
makes unique equilibrium more likely.1
References
[1] Carlsson, Hans and Eric E. van Damme (1993), “Global Games and Equilibrium
Selection,” Econometrica, 61(5), 989-1018.
[2] Chan, Kenneth S. and Y. Stephen Chiu (2002), “The Role of (Non)transparency in a
Currency Crisis Model,” European Economic Review 46(2), pp397-416.
[3] Diamond, Douglas and P.H. Dybvig (1983), “Bank runs, deposit insurance, and liquidity,” Journal of Political Economy 91: 401-19.
[4] Heinemann, F. and G. Illing (2002), “Speculative Attacks: Unique Sunspot Equilibrium and Transparency,” Journal of International Economics 58(2): 429-450.
[5] Morris, S. and H. S. Shin (1998), “Unique Equilibrium in a Model of Self-Fulfilling
Currency Attacks,” American Economic Review, 88 (3), 587-597.
[6] Morris, S. and H. S. Shin (2001), “Rethinking Multiple Equilibria in Macroeconomics,”
in NBER Macroeconomics Annual 2000, 139-161. Cambridge: M.I.T. Press.
1 Note that our result gives rise to a limiting result presented in Chan and Chiu (2002) who argue that
multiple equilibria might occur if there is a zero prior probability associated with the event that θ ≤ θ or
θ ≥ θ.
6
[7] Obstfeld, M (1996), Models of Currency Crises with Self-fulling Features, European
Economic Review 40, 1037-47.
7