The Onset of Speculative Currency Attacks1

Yulia Rodionova2 and Jay Surti3

December 6, 2004

1We are grateful to participants in seminars at Boston University and NEUDC, and especially toChristophe Chamley, Hsueh-Ling Huynh, Michael Manove, Jeff Miron, Dilip Mookherjee and BobRosenthal for helpful comments and discussions. The usual caveat applies.

2EBRD. E-mail: RodionoY@ebrd.com3IMF. E-mail: jsurti@imf.org The views expressed in this paper are those of the authors and do

not necessarily represent those of the EBRD and the IMF.

1

Contents

1 The Onset of Speculative Currency Attacks 21.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Market Outcomes Under Complete Information . . . . . . . . . . . . . . . . 101.3 Speculation under Aggregate Uncertainty . . . . . . . . . . . . . . . . . . . 191.4 Other Dynamic Models Of Currency Crises . . . . . . . . . . . . . . . . . . 311.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.6 Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Bibliography 59

2

1 The Onset of Speculative Currency Attacks

Following speculative attacks on a number of European currencies in 1992, and episodes

of massive capital flight from financial markets in South-East Asia about half a decade

later, a popular interpretation of the ensuing exchange rate collapses has emerged. This

literature seeks to explain speculative activity in these markets as the outcome of a static

coordination game.1 A currency crisis with an accompanying devaluation is, therefore, one

of the multiple rational expectations equilibrium outcomes. Most papers in this area have

stressed a switch of expectations as critical in moving the market from one equilibrium, (a

stable exchange parity with relatively little speculation) to another (a speculative frenzy

with an accompanying devaluation). In the case of the 1992 ERM crisis, this view seems to

be particularly appealing. The empirical work of (for example) Eichengreen and Wyplosz

(1993), seems to suggest that the relevant economic fundamentals in countries such as

Denmark and Belgium were stable for significant periods, both before and also after the

attacks. Obstfeld (1994) has presented similar evidence for Sweden for 1991-92. The major

methodological weakness shared by these papers is their inability to explain - within the

setting of the models - the switch of expectations that drives the economy away from one

equilibrium and toward the other; i.e., none of these papers presents a satisfactory theory

of the onset of a speculative attack. This in turn has made a meaningful discussion of a

central bank defence strategy notoriously difficult.2

1This view of speculative attacks against fixed exchange parities was first proposed by Obstfeld (1986),and developed subsequently by many others. See Jeanne (1999) for an excellent survey of the theoreticalliterature on currency crises.

2See for example the debate between Eichengreen et al (1995) and Garber and Taylor (1995) on thebenefits of capital controls in the aftermath of the ERM crisis.

3

One recent attempt at making headway in this direction is the application of the global

game method pioneered by Carlsson and van Damme (1993). They consider a class of

parametrized, static coordination games which include Obstfeld’s (1996) speculative attack

model. In their setting, players receive accurate, slightly noisy signals concerning a payoff

relevant parameter. In the case of currency markets, this parameter would be some measure

of the relevant economic fundamentals which pin down the floating exchange rate. It will be

convenient to think of this parameter as some real valued index of the strength of the central

bank: for example, the amount of reserves the bank is willing and able to commit in defence

of the peg. The parameter space is, therefore, an interval. The two tails of this interval are

characterized by unique equilibria in dominant strategies. For example, a very weak central

bank (the left tail of the interval) would be unable to withstand any speculative pressure,

and hence, speculators would always attack (successfully). Correspondingly, the right tail

represents a very strong bank that is willing to defend the peg under any circumstance.

In the absence of noise, when the fundamentals are common knowledge, the intermediate

region of this parameter space allows two pure-strategy equilibria: in the first one, the

entire market shorts the domestic currency inducing a devaluation, whereas in the other,

no one speculates and the parity survives. If the signals are noisy, however, the lack of

common knowledge of fundamentals generates a unique equilibrium via iterative elimination

of strictly dominated strategies. This equilibrium is characterized by a threshold parameter

value (θ∗) such that a speculator attacks iff he receives a signal whose value is below θ∗.

Each speculator, therefore, behaves in accordance with his private information. Since the

result is valid in the limit; i.e., as noise becomes vanishingly small, the conclusion is that the

4

economic fundamentals do indeed determine equilibrium play in static coordination games

when common knowledge of payoff relevant parameters is absent. Morris and Shin (1998)

have applied this model to study equilibrium behavior of speculators in a noisy version of

the Obstfeld model. Their major prediction is that in this noisy, static coordination game,

whenever the coordinated attack equilibrium is risk dominant, it is the unique equilibrium

selected by iterative dominance. Hence, on an open set of parameter values, aggregate

uncertainty aids the selection of the devaluation equilibrium.

While the global game model selects a unique equilibrium as a function of the underlying

economic fundamentals, it does not incorporate some key elements of speculation in real

world currency markets. Rather disturbingly, these also seem to be of central importance

to a satisfactory analysis of the nature of strategic interaction in most financial markets.

First, traders in these markets interact strategically over time, and are free to choose the

timing of their trades. Static games - or ones with a fixed sequential order of decision making

- possess a strategic structure that is very different from games with endogenous choice

of timing. One cannot help but wonder, therefore, whether global game models without

endogenous timing might generate misleading insights about the resolution of coordination

problems under different informational constraints. This becomes especially important if

one wishes to use these insights to make general statements about the desirability of specific

government policies.

Second, in trading environments such as currency markets with fixed exchange rates, there

arises a trade-off between the incentive to wait and observe market activity, and a desire to

5

bail out right away. The central bank will not have sufficient foreign exchange to cover the

sales orders of all traders in the event of a devaluation. This implies that the opportunity to

avoid capital losses (or to make capital gains) vanishes, if one is too late. Formally, in real

investment models, strategic complementarities alone affect player payoffs, and thereby, the

incentive to participate. In financial markets, however, the desire to wait and observe the

way the market is moving is countered by the incentive to preempt the market in case of a

general movement against an asset. Thus, preemption emerges as an important and influ-

ential strategic variable in determining the players’ optimal timing decisions. In currency

markets, this could be succinctly captured by introducing sequential service constraints that

operate at the time of a devaluation. The manner in which these two conflicting incentives

play each other out is a central and open question in the context of these markets.

Finally, speculators may have imperfect information concerning critical payoff relevant vari-

ables such as the ratio of the central bank’s foreign exchange reserves to aggregate spec-

ulative capital. The dynamic setting then provides an avenue through which speculators

can learn more by delaying their action, and observing the actions of other participants

(speculators or the central bank). This is not possible in the static model of Morris and

Shin (1998).3

This paper presents a model that incorporates all these elements4. We approximate the

3Learning by observing the actions of other participants is possible in the models of Morris and Shin(1999) and Corsetti et al (2001). See section 3.4 for a comparison of these papers with the model of thischapter.

4Surti (2004) incorporates market size effect into a similar set up. He finds that information imperfectionsprevent traders from exploiting profitable opportunities, and shows how large traders help alleviate thisproblem by undertaking risky arbitrage opportunities early in the investment process in return for higherprofits if the attack is successful. Similarly to our paper, he finds that credible monetary authorities canprovide a better defense of an exchange rate regime by obfuscating the value of reserves that they committedto use for this purpose.

6

continuous time trading environment of real world currency markets by using a discrete

time model and taking the limit as the period length converges to zero. By reducing the

period length sufficiently, a unique equilibrium is generated. In this equilibrium, the desire

to preempt completely determines players’ timing of sales. A reduction in period length

lowers the cost of selling earlier than the market but does not affect the probability of

not being able to convert successfully if one sells late. For period length sufficiently small,

therefore, all speculators attempt to sell their domestic currency in the first period itself

making it too costly for the central bank to protect the peg (Proposition 1). The assumption

of a finite number of pivotal players is of critical importance to this result. With players

of negligible size, the coordination failure equilibrium always exists. The addition of the

preemption element, therefore, resolves the coordination problem for small reaction lags.

What happens when we put informational imperfections back in? The model is extended to

allow for the possibility that the market may lack sufficient liquidity to pull together a suc-

cessful attack. The strong results of the benchmark model are sensitive to the introduction

of this aggregate uncertainty. In particular, if speculators are sufficiently pessimistic about

the possibility of pulling together a successful attack, then the outcome where all specula-

tors rush to sell the domestic currency will no longer be an equilibrium. (Proposition 3)

This is because aggregate uncertainty lowers the expected payoff from selling, by creating

the possibility of a “bad” state in which even a fully coordinated attack will not generate

positive returns.

Further, we show that pure-strategy, symmetric equilibria fail to exist, implying that an

equilibrium will - with positive probability - be characterized by a delay in the decision to

7

sell by some speculators. Delay of sales in equilibrium reflects the trade-off between the

incentive to sell early, (therefore ensuring oneself of the profit accruing from an attack), and

the incentive to delay one’s action in the hope of acting on more precise information later,

having first observed the actions of other participants.

Our main result is the construction of a symmetric mixed equilibrium that terminates

speculation if there is even a single period in which there is no short sale of domestic assets

(Proposition 5). The intuition for this result is that the absence of speculation in any period

generates so much pessimism, that if no one else wishes to speculate in the future, then it

is a best response to not do so as well.

Since this equilibrium is robust to smaller reaction lags - it exists for all discount factors

sufficiently high - our result has strong implications for the impact of aggregate uncertainty

on the set of equilibria. If the initial beliefs are not very optimistic, then the pure strategy

profile where every remaining trader in the market sells at once is no longer a pure-strategy

equilibrium. Since at best, equilibrium behavior is the outcome of a strictly mixed strategy,

there is a positive probability that the realized amount of speculation is zero, in which case

there is no further speculation.

Our results, therefore, tug in a very different direction from the global games literature. The

latter predicts that as long as the payoff dominant and risk dominant equilibrium outcomes

coincide, injecting a little aggregate uncertainty will resolve the coordination problem. In

our case, aggregate uncertainty takes us away from the coordinated action outcome, which

may be payoff dominant. Indeed, a central bank may gain by withholding - or obfuscating

8

- its commitment to the fixed exchange parity precisely because it makes the speculators

more cautious.

The paper is organized as follows: section 3.1 sets out the basic model of the dynamic spec-

ulation game, which is analyzed in section 3.2. There we relate the possibility of equilibrium

delay to aggregate market liquidity; in section 3.3, we introduce the incomplete information

framework, and characterize the properties of symmetric, Markov perfect equilibria of this

game. Connections to the relevant existing literature are examined in detail in section 3.4.

Section 3.5 summarizes our conclusions and provides some thoughts about future direction.

Some of the longer and more technical proofs are relegated to an appendix.

1.1 The Model

A central bank establishes a fixed exchange rate e.5 The market exchange rate, if the bank

decides to free float the domestic currency is given by θ. We will assume that e > θ, and

that θ is constant over time.6

In defending the position of the domestic currency against any selling pressure, the central

bank runs down its dollar reserves. It is well known that the consequences of doing this

are very costly beyond a point, and therefore, the bank will not continue to protect the

exchange rate forever. In order to incorporate this, we will assume that the central bank

sets aside a fixed maximum amount of dollars, R <∞ to protect the peg. Once the bank’s

reserves run out, it will devalue by free floating the domestic currency.

The strategic actors are N identical speculators with N < ∞. Each speculator has an5By which we mean the number of dollars that can be exchanged per unit of the domestic currency.6The constant θ is quite appropriate since we are concerned with explaining the onset of crises in the

absence of any striking change in the economic fundamentals.

9

identical short-sale constraint, which, denominated in domestic currency units, is denoted

by ωe > 0.7

Time is discrete. In each period, a speculator who has not yet sold his reserves of domestic

currency can take either of two decisions: he can choose to sell domestic currency and

demand foreign exchange from the central bank right away, or he can choose delay sales

for at least one period more. We will assume that the decision to sell is indivisible and

irreversible, and that it involves a fixed transaction cost of c > 0.8 Speculators discount the

future at a common per period value δ ∈ [0, 1] .

Formally, let xIt = 0 if speculator I has not sold by the end of period t, and let xIt = 1

if he has. The state of the game at the end of period t is represented by the vector

xt = (xAt, ..., xNt) . The history of the game at the beginning of period t is ht = {x1, ..., xt−1}

with h1 = {φ} . The set of possible t-histories is Ht and H = ∪Ht denotes the set of all

paths of play. A behavioral strategy for speculator I is a function sI : H → [0, 1] , where

sI (h) is the probability that he sells at a history h.

A speculator who sells short t periods prior to a devaluation receives a net present value

payoff

δtω

e[e− θ]− c = δtωπ − c (12)

where π = 1− θe . Here e− θ denotes the per dollar demanded profit accruing from partici-

pation in a successful attack.7Hence, each speculator can demand at most ω dollars from the central bank.8All costs and revenues in this chapter are denominated in dollars.

10

If however, he sells exactly at the time of the devaluation, then his return will be a proba-

bilistic function of the number that sell at that date. This is a consequence of the fact that

the central bank will not swap currencies with anyone at the rate e, once its dollar reserves

are exhausted. Hence, in the period in which the peg falls, speculators demanding foreign

exchange will be rationed by the central bank. Let n be the number that wish to sell in this

period, and k ≤ n denote the actual number whose demand the central bank can satisfy.

Then his expected present value return is

k

n(ωπ − c) (13)

kn represents the sequential service constraint signifying that only k clients will be served

at the rate e.

If he sells and the fixed exchange regime survives, then he loses c. If he never sells, he

receives zero.

Finally, the per period discount factor δ is defined in exactly the same way as in the previous

chapter. In particular, let period length be τ > 0, and the instantaneous discount rate be

ρ > 0. Then δ (τ , ρ) = e−ρτ . Hence, for any ρ > 0, as τ ↓ 0, δ ↑ 1.

1.2 Market Outcomes Under Complete Information

In this section, we analyze the properties of subgame perfect equilibria (SPE) of the spec-

ulation game. Two properties will be emphasized. The main result of this section shows

that as period length becomes sufficiently small, a unique SPE emerges. If total market

11

liquidity (Nω) exceeds the bank’s reserves, then all speculators attack in the first period

itself, yielding an immediate devaluation. On the other hand, if it is insufficient, then there

is no speculation.

The second result characterizes the graph of the pure strategy SPE outcomes for alternate

values of the discount factor δ. The characterization is sought in terms of the number of

periods that can elapse prior to a devaluation in equilibrium. We demonstrate a striking

“discontinuity” in the equilibrium set as a function of δ. In particular, there exists a value

δ∗ (ω) ∈ (0, 1) with the property that if δ > δ∗ (ω) , then there exists a unique pure strategy

SPE with a devaluation occurring in period 1. If δ < δ∗ (ω) , then a devaluation can occur

in any period t ∈ N; i.e., any amount of delay is feasible in equilibrium.

Denote by

n∗ (ω) :=»R

ω

¼(14)

the minimum number of speculators that need to sell before the bank’s reserves are depleted

and a devaluation ensues. We will assume that ω > cπ , n

∗ (ω) < N, and that n∗ (ω) ∈ N.9

Since we will at a later stage restrict attention to pure strategy SPE, it is necessary to

introduce some notation specific to this case. Let E (ω) denote the set of pure strategy SPE

outcomes of the game corresponding to the common liquidity constraint ω, and by T (ω) ,

the supremum, taken over E (ω) , of the number of periods elapsing prior to a devaluation.9These assumptions can be dropped from the analysis without affecting any of the results. However, this

will open up a lot of if and but clauses which we wish to avoid.

12

We prove first that a SPE exists. Since ωπ > c, there is a trivial SPE with everyone

attempting to sell the domestic currency in the first period.10 We also remind the reader

that if ω ≥ R, then this is also the unique SPE of the game. This is because if each

speculator can defeat the central bank on his own, then it is a strictly dominant strategy

to attack at once.

We will need the following notation. By Γk, we mean a subgame where sales by k more

speculators will exhaust the central bank’s reserves.

We now develop an apparatus that enables us to make use of the sequential service con-

straint. Consider a speculator who is yet to sell in subgame Γk. He faces a strategy profile

of the remaining speculators that implies that they will all delay in the current period and

sell in the next, independent of his current period course of action. The question is: should

he sell right away, or delay for one more period and coordinate his timing precisely with

the others?

For each Γk, define a set of values {ω}k such that if his wealth is exactly ωe units of domestic

currency for ω ∈ {ω}k , then he is indifferent between preempting the rest of the traders or

coordinating with them. Clearly such ω should satisfy the following condition

δωπ − c =k

N − n∗ (ω) + k[ωπ − c]

where the left-hand side represents the payoff from preemption and the right-hand side, the

payoff from coordination.10A complete description of the corresponding strategy profile is the following: at any history h, the

strategy adopted by each remaining speculator is s (h) = 1.

13

⇔ ω =c

π

N − n∗ (ω)δ (N − n∗ (ω) + k)− k

(15)

Since the right-hand side of (15) is increasing in n∗ (ω) , whereas n∗ is non-increasing in

ω, the sets {ω}k are singletons. We will henceforth denote their unique element by ωk.

Clearly also, ωk < ωk+1. The higher is a trader’s speculative capital, ω, the greater is his

incentive to preempt and avoid the sequential service constraint as compared to his aversion

to investing strictly earlier than the devaluation and have his profits being reduced owing

to the discount. If ω > ωk, then conditional upon a devaluation in the next period of Γk,

any speculator would strictly prefer to sell in the current period. Conversely, if ω < ωk,

then this speculator would instead strictly prefer to sell in the next period.

Proposition 1 For each ω, there exists a value τ (ω) > 0 such that if τ < τ (ω) , then

there exists a unique SPE. In this equilibrium, for each I ∈ N, and each subgame Γk;

k ∈ {1, · · ·, n∗ (ω)} , sI ({φ}) = 1; i.e., if period length is sufficiently short, then the spec-

ulation game has a unique subgame perfect equilibrium in which everyone attempts to sell

their domestic currency in the first period of any subgame. This SPE therefore yields a

devaluation in the first period of the game.

We present the analysis in two steps. In the first step, we demonstrate that if ω > ωn∗(ω),

then there exists a unique pure strategy SPE with all the players attempting a sale in

the first period. From (15) it is clear that ωn∗(ω) is a decreasing function of δ, and that

limδ↑1 ωn∗(ω) = cπ . Therefore this result in turn establishes that the unique pure strategy

SPE that survives at sufficiently high values of δ is one where all speculators attempt a sale

14

of their domestic currency in the first period. The second step proves the non-existence of

a strictly mixed equilibrium for high values of the discount factor.

Lemma 1 If ω ∈ ¡ωn∗(ω),∞¢ , then in each subgame Γk; k ∈ {1, · · ·, n∗ (ω)} , there existsa unique pure-strategy SPE in which all speculators attempt to sell the domestic currency

in the first period; i.e., in which sI ({φ}) = 1.

Proof: We have already argued that (∀I) (∀Γk) (sI ({φ}) = 1) is a SPE. So it remains to

show that this is the only one in pure strategies. We prove by contradiction. Suppose there

exists another pure-strategy SPE.

In Γ1, there is a unique equilibrium in which all the remaining players sell in the first period.

We begin by proving that there cannot be a pure-strategy SPE of Γ2 where no speculator

ever sells. We prove this by contradiction. So suppose that there is a pure-strategy SPE

profile that yields this outcome. Since ω > ωn∗(ω), it follows from (15) that δωπ − c >

n∗(ω)N (ωπ − c) > 0. If trader I were to deviate and sell in some period of Γ2, then he would

be the only speculator to sell in that period and consequently, he would generate Γ1 at the

end of the period. From the previous paragraph, this implies that all remaining speculators

will sell in the next period. This means that I 0s deviation payoff in this case is δωπ− c > 0.

Since the proposed SPE payoff is zero, this is a contradiction.

It follows that if a pure-strategy profile is a SPE profile of Γ2, then there must be some period

t ≥ 1 of this subgame and a speculator I with sI (ht) = 1. Now consider the best response of

speculator J 6= I in period t. If he delays, then at best, he will find himself in subgame Γ1;

i.e., at best, I would be the only trader to sell in period t. In period t+1, J and all remaining

15

speculators sell, yielding a payoff of 1N−n∗(ω)+1 (ωπ − c) < n∗(ω)

N (ωπ − c) < δωπ − c. If he

sold in period t instead, then I and J would force a devaluation at the end of period t and

collect a payoff of ωπ − c. In this case, therefore, J 0s best response is to sell in period t.

Finally, if there exists an agent K /∈ {I, J} and sK (ht) = 1, then J would be clearly better

off by selling in period t than delaying for one period more. From the preceding argument,

it follows that any pure strategy SPE profile of Γ2 must recommend an identical period

t ≥ 1 for speculative sales for all speculators still in the game. In turn, this implies that

if for some SPE in pure-strategies, this t > 1, then for all periods s < t, the payoff to a

speculator I from deviating and selling at s is δωπ − c. This must, therefore, be the lower

bound on the expected SPE payoff in all periods s < t.

Since at t, all traders will sell together, this implies that the expected SPE payoff in any

period s < t is

2

N − n∗ + 2[ωπ − c]

<n∗

N[ωπ − c] < δωπ − c

Since the last expression is the deviation payoff at any s < t, the proposed SPE strategy

profile is not deviation proof and hence, cannot be a SPE unless t = 1. This proves that in

the unique pure-strategy SPE of Γ2, (∀I) (sI ({φ}) = 1).

We can now extend this argument to cover all subgames Γk; k ∈ {3, · · ·, n∗} by backward

inducting on the subgame structure in the following manner. Suppose that for each m ≤ k,

the unique pure-strategy spe of Γm, is (∀I) (sI ({φ}) = 1). Then in any pure-strategy spe

16

of Γk+1 wherein the devaluation occurs with a delay, the payoff to a speculator is bounded

above by

k + 1

N − n∗ + k + 1[ωπ − c]

< δωπ − c

the deviation payoff to a speculator who sells in the first period of Γk. This finishes the

proof of the lemma. ♦

Lemma 2 There exists δM (ω) ∈ (0, 1) such that if δ ∈ (δM (ω) , 1) , then there is no SPE

in mixed strategies in any subgame Γk.

This lemma is proved in the appendix.

Proof of Proposition 1: Notice that as δ ↑ 1, ωn∗(ω) ↓ cπ . From lemma 1, this implies that

for each ω > cπ , there exists δP (ω) < 1 such that the unique pure-strategy SPE profile in any

subgame Γk is (∀I) (sI ({φ}) = 1) .11 Lemma 2 implies that there exists δM (ω) ∈ (δP (ω) , 1)

such that there is no strictly mixed spe provided δ > δM (ω). So, for any δ > δM (ω) , in

any subgame Γk, there exists a unique SPE in which any remanining speculator sells at

once. This concludes the proof of the proposition. k

The argument developed in lemma 1 may now be fruitfully applied to provide a character-

ization of the graph of the set of pure strategy SPE outcomes as a function of the discount

factor. From lemma 1, we know that if δ > δP (ω) , then there exists unique pure strategy

SPE, with all speculators attempting to sell in the first period. We will now show that if

11The required δP (ω) is the unique value of δ < 1 such that ω = ωn∗(ω) (δ) .

17

δ < δP (ω) , then there exists infinitely many pure strategy SPE in which a devaluation can

come in any period, because for each such period, there exists a SPE with all speculators

selling in that period.

Proposition 2 If δ < δP (ω) , then for each t ∈ N, there exists a pure strategy SPE with

the property that all speculators attempt to sell in period t.

The proof is in the appendix.

Discussion

1. The impact of the preemption motive upon the equilibrium set was the main concern

of this section. The complete information game sets up a benchmark against which to

compare outcomes generated by corresponding strategic market games where the quality of

information possessed by the traders is imprecise. Proposition 1 is reminiscent of arguments

invoked in defence of the efficient markets hypothesis. These arguments it may be recalled

assume a final period in which the asset price is corrected for exogenous reasons (for example,

due to an earnings announcement). In this case, the informed traders have a dominant

strategy to take appropriate short-long positions in the penultimate period. This means

that in the absence of a constraint on short-sales, the mispricing will be corrected in the

penultimate period itself. This generates a backward induction argument that precludes the

persistence of mispricing. In our game, as the friction generated by period length disappears,

the strategic structure of the game approaches that of these games where the bubble bursts

in finite time due to exogenous reasons. An important distinction nevertheless remains

between these finite horizon models and our infinite horizon model. If our speculators were

18

negligible, then without an exogenous shock that creates a devaluation in finite time and,

therefore, a dominant strategy to liquidate and move to dollars, there will always be an

equilibrium with coordination failure where there is no speculation. Our results show that

in a market composed of speculators whose size relative to the market is non-negligible, there

is a unique equilibium even without any exogenous shock generating a dominant strategy.

2. Proposition 1 also demonstrates a significant departure from dynamic investment models

without strategic substitutabilities. For example, in the Gale (1995) model, there is a

countable set of N investors, each of whom is endowed with an indivisible, irreversible

option to undertake an investment costing c, and where the choice of time is endogenous.

The flow payoff to an agent who has exercised her option, is in any subsequent period,

an increasing function υ (α) , of the proportion of agents α, who have acted in, or prior

to, that period. Assumptions on the payoffs guarantee that υ (1) > (1− δ) c > υ (0) . In

the Gale model, the only “penalty” from acting later than the others is the loss of the

payoff from the intervening dates. He proves that as τ ↓ 0, D (τ) ↓ 0, where D (τ) is the

supremum, taken over the set of all SPE, of the real time delay preceding the termination

of the game through successful coordination. However, since agent payoff functions do not

reflect strategic substitutability generated by sequential service constraints, the reduction of

period length serves to reduce the cost of investing earlier relative to investing later than the

rest of the agents. Hence, as τ ↓ 0 the infinite delay equilibrium no longer exists. However,

since there is no cost to delay, there are always multiple pure-strategy SPE, with outcomes

corresponding to alternate periods in which the investment process is completed.12

12Gale (Theorem 2, p.8) shows that there are at least n∗ (ω) equilibria for any δ ∈ (0, 1). See also thediscussion in section 2.5.2 and footnote 10.

19

1.3 Speculation under Aggregate Uncertainty

The Incomplete Information Game

In most situations involving coordination issues, participants probably do not know whether

they possess an adequate amount of resources or whether there are a sufficient number

of them to bring about a profitable change from the status quo; i.e., there is aggregate

structural uncertainty concerning the profitability of action, in addition to the problem of

strategic uncertainty which operates at the level of other people’s intended action.

In our model, a natural way to incorporate this would be to assume away common knowledge

of the set and the number of players in the game.13 In the current section, we explore the

implications of adding this uncertainty structure to the dynamic game described of section

3.1.

In particular, the number of agents N, is assumed to be a Poisson random variable with

parameter λ.14 A particular example of how such an uncertainty structure may be generated

is described here. We consider an infinite population from which players are selected to

participate in the game. The number of players K is a Poisson random variable with mean

γ. Suppose there are two types of players in the economy: well endowed, (H) and poorly

endowed, (L) . Type H is described by a capital endowment ωH > cπ , whereas type L is

described by a capital endowment ωL < cπ . Let the probability - conditional on being selected

13An alternative approach would be to assume that the central bank’s reserves were private information,possessed by that agency alone. Players would then receive signals about the actual volume of reserves thatthe CB is willing to commit in defending the peg. The qualitative nature of the results of this section alsoapply in this alternate setting.14We are grateful to Bob Rosenthal for having referred us to Myerson (1997), which prompted us to try

this Poisson formulation of uncertainty.

20

to play - of being of type H be r. Then we know, ( from Myerson (1997), for example),

that the number of players of type H is a Poisson random variable with parameter γr.

The specification of the game with respect to the timing of moves, the choice sets of each

speculator, and speculator payoffs is identical to that in section 3.1. Clearly, since the type

L players each have a strictly dominant strategy ; viz., to not sell their endowments, they

are non-participants in the game. Now simply denote the number of type H players by N

and redefine λ = γr. We will assume that at least two speculators have to sell in order to

induce a devaluation; i.e., n∗ (ωH) > 1. For the remainder of the section, we will suppress

the argument from n∗ (ω) .

We will assume that the game is anonymous; viz., that player names do not matter. A

t−history may then be defined as a list of the number of players who have sold at each of

the dates prior to t : for each t ≥ 2, ht := {x1, · · ·, xt−1} , and for t = 1, h1 := {φ} . xt

denotes the number of speculators who sell at date t. As in section 3.1, Ht is the set of all

t−histories, and H = ∪Ht is the set of all paths of play.

By a belief B induced by a history of play h, we mean the infinite vector (Bh (0) , Bh (1) ,

· · ·, Bh (n) , · · ·) where Bh (n) denotes the posterior probability that a speculator attaches

to the number of other speculators remaining in the game being n.

In general, a strategy for a player is a mapping h 7→ [0, 1] where h ∈ H, with s (h) be-

ing a probability of sales for a remaining speculator at a history h. Given anonymity of

players however, we will restrict attention to those strategies that constitute a mapping

(B× {0, 1, · · ·, n∗}) 7→ [0, 1] with the interpretation that sI (B, k) is the probability with

21

which speculator I sells, given that his posterior beliefs about the number of remaining

speculators (other than himself), is given by the vector B ∈ B, and given that the num-

ber of additional speculators required in order to induce a devaluation is k. The following

equivalent representation of a strategy is much more convenient and will be used in all that

follows. In any subgame, Γk, a behavioral strategy, s (k) , is a collection of action functions,

sm (B) , one for each subgame m ∈ {1, · · ·, k} .

A Markov perfect Bayesian equilibrium (MPE) for this game is such a strategy profile

(sI (k))n∗k=1 and a system of beliefs B such that in each subgame Γk, and for each I, sI (k)

represents a best response of player I to the strategies of the other agents given B, and

where B is updated according to Bayes’ rule when possible. A symmetric MPE is a MPE

with the property (∀I) (∀k) (sI (k) = s) .We use the symmetric MPE as our solution concept

in this chapter.

We begin by recalling the convenient property that the Poisson distribution does not possess

memory on symmetric equilibrium paths. This facilitates a particularly simple translation

of current period beliefs into the next in any symmetric MPE. Let B denote the current

probability distribution on the number of other players remaining in the game; i.e., B (n)

is the probability that n other players are in the game in the current period. Assume that

B is Poisson with parameter λ; i.e.,

B (n) =e−λλn

n!

for some parameter λ > 0. Suppose that each speculator still in the game sells with proba-

22

bility s ∈ (0, 1) . Let k0 be the number of speculators who sell in the current period. Then,

the updated posterior belief that the number of remaining speculators still in the game is

n, is given by

∀n ≥ 0, B (n) =

µn+ k0

k0

¶sk

0(1− s)n e−λλn+k

0

(n+k0)!P∞m=0

µm+ k0

k0

¶sk0 (1− s)m e−λλm+k0

(m+k0)!

=e−λ(1−s) (λ (1− s))n

n!

which is a Poisson p.m.f. with parameter λ (1− s) < λ.

Hence, if the initial belief about the number of other players in the game is given by a

Poisson distribution, the updated posterior beliefs will also be Poisson.

Since beliefs along a symmetric equilibrium path can be summarized by the “current” belief

parameter λ, we may equivalently redefine symmetric equilibrium strategies by sk (λ) . We

note that a consequence of defining strategies in this manner is that in any symmetric MPE,

if the path of play reaches (λ, k) such that sk (λ) = 0, then the game stops forever because

at the next date,¡λ0, k0

¢= (λ, k) .

In the subgame Γ1, it is once more strictly dominant for a speculator to sell.

In each subgame Γk; k ∈ {2, · · ·, n∗} , we may distinguish between three types of equilibria.

The first type of equilibrium is one where all the remaining speculators sell at once. This

will be called a rush equilibrium. The second type of equilibrium involves a termination

of further sales by the remaining speculators, yielding an end to the game. This will

23

be referred to as a termination equilibrium. The third type of equilibrium is a mixed

equilibrium, which may produce behavior corresponding to either of the two pure-strategy

outcomes (immediate devaluation or a termination), or may produce a new kind of outcome:

a gradually increasing pressure on central bank reserves as more and more speculators sell

over a number of periods. This may ultimately culminate in a full blown attack after a

few periods, or if inadequate sales register, may lead to undue pessimism, and terminate

further speculation. It is precisely these latter equilibria, in which speculators who delay,

refine their information about the true state, and are thus able to act on the basis of this

refined knowledge in subsequent periods. In what follows, we provide sufficient conditions

for the existence of a symmetric MPE. In doing so, we will find that for small period lengths,

pure-strategy equilibria do not exist for generic sets of parameter values.

The analysis of the game revolves around the comparison that a speculator makes - each

time that he is called upon to act - between the expected payoff to selling in the current

period, and the expected payoff to delay. In equilibrium, he compares, in each subgame,

the two expected values, given the revised (common) beliefs and the strategies of the other

speculators. Let λ denote the revised common mean of the Poisson distribution, and sk (λ)

denote the common strategy adopted by all remaining speculators other than some I, where

k denotes the minimum number of speculators that will induce a devaluation. In what

follows, we will often abuse notation by referring to sk (λ) as s. Denote by V Sk (λ, s (k)) (alt:

by V Pk (λ, s (k))) the expected payoff from selling (alt: delaying) in the current period in

subgame Γk.

We first show that

24

Proposition 3 (a) In any subgame Γk, there exists λk ∈ (0,∞) such that a rush equilibrium

exists iff λ ≥ λk; (b) λk > λk−1; i.e., as more speculators sell, the minimum level of

optimism required to ensure the existence of a rush equilibrium decreases.

Proofs of all the results in this section are in the appendix.

Owing to strategic substitutability, the marginal expected payoff (with respect to one’s own

probability of action) displays non-monotonicity in the (probability of) actions of other

players. This phenomenon is absent in games with strategic complementarities. This re-

sult is, therefore, not an immediate consequence of the payoff structure of the model. At

sufficiently high values of λ, the expected payoff to selling when everyone else is doing so,

will decline in value as λ increases. Therefore, if selling with everyone else is to continue

to be a better response than delaying for one more period, then the expected payoff to

delay must fall even faster than the expected payoff to selling, as λ increases. In the ap-

pendix, we show that this is true by proving that V Sk (λ, 1) − V P

k (λ, 1) ≤ 0 implies that∂∂λ

¡V Sk (λ, 1)− V P

k (λ, 1)¢> 0.

The analysis of symmetric equilibrium behavior for the case of high values of δ will be the

focus of our inquiry for the remainder of the chapter. We first provide a detailed analysis

for the subgame Γ2, and then extend its analysis, by induction, to all other subgames.

We proceed in this staggered fashion because the analysis of this subgame (k = 2) uses

techniques quite distinct from the case of k ≥ 3. The reason for this is the fact that there are

only two possible transitions from this subgame that can be induced owing to speculation.

If more than one trader sells, the game ends with a devaluation, whereas if a single agent

25

sells, then the game moves to Γ1, where the remaining agents have a dominant strategy.

So the analysis of behavior in future subgames is independent of beliefs and therefore, the

backward induction analysis is particularly simple. Nevertheless, the proof of the case where

k = 2 generates important insights on the nature of restrictions on the parameters required

to extend the analysis to k ≥ 3 using similar techniques.

Analysis of Γ2.

Rushes

From proposition 3, we know that there exists λ2 ∈ (0,∞) such that a rush equilibrium

exists if and only if λ ≥ λ2.

Terminations

In Γ2, a termination equilibrium exists iff the expected payoff from selling is non-positive,

(since the payoff to not selling when no one else does, is zero). This is equivalent to:

δωπ∞Xk=1

e−λλk

k!≤ c

Since δωπ ≥ c, a termination equilibrium exists if and only if λ ≤ λ2 < ∞, where λ2 is

defined as the value of λ that solves

δωπ∞Xk=1

e−λλk

k!= c

So a termination equilibrium exists iff:

26

⇔ λ ≤ lnµ

δωπ

δωπ − c

¶= λ2 (16)

Equilibria With Delay

For δ sufficiently high we have λ2 < λ2, and so any symmetric MPE must involve a strictly

mixed action for beliefs λ ∈ ¡λ2, λ2¢. The following result proves that in the subgame Γ2,a mixed, symmetric MPE exists.

Proposition 4 There exists a symmetric MPE in Γ2. In this equilibrium, the realization

of no sales in any period ends the game in a termination equilibrium in the next period.

Moreover, there exists M2 ∈ (0,∞) such that ωπc > M2 implies that the equilibrium strategy

is an ordered pair of two lipschitz continuous action functions. In Γ2, the equilibrium action

function, s∗2 (λ) is a non-decreasing, lipschitz continuous function which is continuously

differentiable everywhere except at λ2 and λ2.

It is useful to give an intuitive presentation of the construction of the symmetric equilibrium

strategy. The first step restricts behavior in the interval λ ∈ [0, λ2] to correspond to a

termination equilibrium; i.e., s∗2 (λ) = 0. For λ > λ2, define a critical probability es2 (λ)by λ (1− es2 (λ)) = λ2. Observe that if in the current period, all remaining speculators

are using a probability of action s ≥ es2 (λ), then no sales in this period ends the gamein the next period. This is because λNEXT ≤ λ2, and therefore, s

∗2 (λNEXT ) = 0. Since

s∗1 (λ) = 1, this enables us to compute a precise algebraic expression for the relevant value

functions, V S2 and V P

2 . We then show that V S2 − V P

2 > 0 at es2 (λ) for λ ∈ (λ2,∞). Sincethe difference of these two functions is continuously differentiable in the current period

27

action probability s, (for fixed λ), and is negative at s = 1, the intermediate value theorem

automatically implies the existence of an s ∈ (es2 (λ) , 1) such that V S2 − V P

2 |s= 0. This

proves the existence of a symmetric equilibrium in which no sales in any period ends the

game with no further speculation. To prove the continuity and monotonicity of the selected

equilibrium action function for Γ2, we prove that V S2 − V P

2 is decreasing in s over the

set [es2 (λ) , 1] for λ ∈ [λ2, λ2]. This implies that for each λ in this range, there is at most

one s ∈ [es2 (λ) , 1] that can be an equilibrium action probability. Finally, we show that

for λ ∈ [λ2, λ2], V S2 − V P

2 is increasing in λ for fixed s. This establishes that s∗2 (λ) is an

increasing function in this range, and the implicit function theorem guarantees that the

continuity properties are valid also.

Aggregate Uncertainty and Coordination Failure: A Characterization

We are now in a position to state and prove the main result of this section. We construct

a symmetric MPE for high values of the discount factor δ, which has the following prop-

erties. In each subgame Γk, equilibrium behavior is governed by a lipschitz continuous,

non-decreasing action function sk (λ) . Moreover, in any subgame, and for any λ, sk (λ) is

chosen in a manner such that no sales in a single period ends further speculation.

The basic idea in the construction is similar to the proof of proposition 4. We restrict the

probability of action in each period to be sufficiently high so that if there are no sales, the

posterior beliefs imply the existence of a termination equilibrium. The proof is built upon

backward induction but now since the constructed equilibrium action functions for future

subgames are arguments in the value functions, we can no longer rely on using explicit

28

expressions of the latter. The proof of proposition 4 suggests, however, that a key condition

in ensuring these properties is the size of the profits ωπ, relative to the cost, c. We show

that if the profit to cost ratio is sufficiently high, then a symmetric equilibrium with these

properties does exist. Given the markets with which we are concerned, this assumption is

justified more often than not.

Proposition 5 Fix the critical number of players n∗ < ∞. There exists δ∗ ∈ (0, 1) and

M (n∗) ∈ (0,∞) , such that if δ ∈ (δ∗, 1) and ωπc > M (n∗) , then there exists a symmetric

MPE for the speculation game. In each subgame, Γk, the corresponding action function

s∗k (λ) is continuous and nondecreasing. There exist two values, λk < λk in each subgame

satisfying s∗k (λ) = 0 iff λ ≤ λk and s∗k (λ) = 1 iff λ ≥ λk. Finally, if there is no speculation

in any period of any subgame, then there is no further speculation thereafter. Therefore, the

game lasts for at most n∗ periods.

Discussion

1. As already hinted at in the introduction, therefore, our results indicate the existence

of an inverse relationship between the quality of public information about payoff relevant

variables, and the likelihood of successful speculation. Our predictions are at odds with

corresponding results in global game models of coordination in financial markets (see for

example, the extended discussion in Morris and Shin (1998, section 3a) concerning the

desirability of transparency of central bank foreign exchange reserves). In the Morris and

Shin type models, transparency of the reserves position of a central bank can “create” an

equilibrium which entails coordination failure. In light of the results of this chapter, this

29

is clearly seen to be an implication of the static nature of their analysis. In our model,

transparency destroys the fixed exchange rate regime because of a desire to preempt an

attack, whereas lack of transparency may in fact save the day for the central banker.

2. (a) Our results also provide an interesting contrast between the predictive content of

our model vis-a-vis that of coordination models with purely strategic complementarities. In

the model of Gale (1995), adding aggregate uncertainty in a manner similar to this section

would - for δ sufficiently close to 1 - purge it of pure strategy rush equilibria. This is because

there is no penalty (corresponding to complete exclusion from profits that exists in our case)

for players who invest late. Therefore, if a player is sufficiently patient, he would always

like to wait one more period, and then invest (if at all), after becoming perfectly informed.

(b) In related work, Matsumura and Ueda (1996) have introduced incomplete information in

a Gale type model, but with a continuum of negligible investors. Specifically, the actual mass

of investors could be high (µ) or low¡µ¢with individual payoffs being υ (µ) > (1− δ) c >

υ¡µ¢. In this environment, they show that for δ close to 1, there is a unique symmetric

termination equilibrium. The intuition is quite straightforward. Provided the law of large

numbers holds for this continuum, any mixed symmetric equilibrium (including the rush)

would perfectly reveal the state (the true value of µ) whereupon for δ sufficiently large, the

best response is to wait for one period more, contradicting the existence of the proposed

mixed equilibrium.

30

(c) The contrast between these games and the one explored in this chapter is that in the

earlier papers, the main cost was that of going too early and have discounting eat away

your returns. However, the penalty of going late is simply some amount of profit foregone.

As the discount factor comes close to unity, this latter cost becomes negligible, converging

to zero in the limit. In our game, however, late comers continue to be severely penalized

- through the loss of the entire profit - independent of the discount factor. So, in order to

learn the true realization of the state, one may have to forego the opportunity to capitalize

on this information. This explains the robustness of rush equilibria in our model.

3. Our main result suggests that if certain natural parametric conditions are satisfied15,

the speculation game can last for at most n∗ periods because if there is a period with no

sales, this will result in posterior beliefs so pessimistic that a termination equilibrium will

arise. Chamley and Gale (1994) present an investment problem with similar informational

assumptions but with purely informational externalities. The major qualitative conclusion

of that paper is similar to ours. The injection of structural uncertainty enhances the prob-

ability of coordination failure. In that paper, under the unique symmetric perfect Bayesian

equilibrium, the game lasted for at most n∗ periods (proposition 6, p.1075) and ended in a

termination if there was no investment in any one period along the way (proposition 1, p.

1070). In the analysis leading up to proposition 5 above, we construct an equilibrium which

implies a similar property for behavior along symmetric equilibrium paths of play. However,

the results in the two papers derive from considerations quite distinct from each other. In

the Chamley-Gale paper, the construction relies on the fact that lower realizations of the

15Sufficiently small period length and sufficiently high profit-to-cost ratio.

31

investment rate in the previous period, (for fixed probabilities of investment), necessarily

lower the expected payoff from investment in the current period, independent of the current

period probability of investment of other players. This property follows from the fact that

in the Chamley-Gale model, there are no pecuniary externalities. Therefore, in any equi-

librium, if at some history of play, no investment is realized, the game terminates. If not,

then the investor could have done better by investing earlier. In our case, the existence of

pecuniary externalities forces us to select an equilibrium that possesses the corresponding

property. We obviously do not claim that all equilibria must share this property.

1.4 Other Dynamic Models Of Currency Crises

We have attempted to model, within a well-defined market setting, the interaction of two

kinds of externalities: informational externalities which have been extensively studied in

the macroeconomic literature on social learning and herding; and payoff externalities, which

have typically been analyzed using static coordination models, with multiple equilibria often

used to model market failure or regime switches. While the choice of the payoff function

in this chapter has been guided by the specific application at hand, we wish to emphasize

that strategic substitutability with respect to the timing of action is a critical component

of several other dynamic coordination games in financial markets: examples include bank

runs where sequential service constraints à la Diamond and Dybvig (1983) imply that it

is better to preempt a run rather than be part of it, and bubbles in asset markets where

the incentive to ride the bubble persists until moments before the crash, whereupon it is

better to short sell to avoid capital losses. Other examples of coordination problems that

may possess a payoff structure akin to the one in this chapter include academic research on

32

potentially promising ideas, which require costly investment of research time, adoption and

development of the idea by the profession at large (the coordination aspect), and conditional

upon successful adoption, it is better of course to have been an early contributor. Here we

compare our model to some related recent literature on currency crises.16

Related work in the global games tradition includes the paper of Morris and Shin (1999),

which dynamizes their previous one (1998). This model presents a sequence of one period

economies, in which (i) the fundamental parameter follows a Brownian motion process; (ii)

the value of the fundamental parameter at date t becomes commonly known at date t+ 1,

and (iii) where in a given period t+ 1, a fresh set (of a continuum of speculators of equal

measure) replaces the set that traded in period t. These assumptions guarantee that the

iterative elimination procedure of Carlsson and van Damme can be applied period by period.

They demonstrate the existence of delayed speculative attacks in equilibrium: attacks which

may begin or reach their zenith after some time has elapsed. In their paper, however, the

reason why attacks may develop after some delay is unrelated to the optimal strategic

timing decisions made by speculators. This is because each speculator is in the market for

one period only. Finally, both the Morris-Shin papers preclude strategic substitutability

in the payoff functions and are, therefore, not amenable to an analysis of the resolution of

conflicting incentives generated by preemption and coordination.

A recent paper of Broner (2000) addresses issues similar to ours but within a framework

reminiscent of the first generation style models of BOP crises. (Krugman (1979) is the classic

reference here) Broner demonstrates that with a secularly declining floating exchange rate,16The papers of Corsetti et al (2001) and Chamley (2002) were discussed in chapter 1 and so a comparison

with this model will not be attempted here.

33

a BOP crisis may involve a devaluation if there is private information about the short-sale

constraints of the speculators. Moreover, imperfect information combined with interest

rate policy that makes it more attractive to postpone a sale of domestic assets can delay

this devaluation. However, the Broner paper is situated within a context of domestic credit

policy that is in the long-run, fundamentally inconsistent with a fixed exchange rate regime.

Finally, market interaction in that paper is non-strategic as players ar atomistic, whereas

this chapter is concerned with speculation in a market with large players, whose action

visibly influence market parameters.

Abreu and Brunnermeier (2003) analyze a dynamic game between arbitrageurs trying

to find an optimal time for a short sale of a bubble asset. In their paper, a coordination

problem arises because agents are temporarily unable to coordinate their selling strategies

due to lack of common knowledge of the time, at which the bubble will be corrected ex-

ogenously. Since the price of the asset increases over time before the bubble is ultimately

being corrected, this generates an incentive to wait before selling the asset. Rochon (2004)

considers a model of speculative attack on a fixed exchange rate, which builds on that by

Abreu and Brunnermeier, where she studies the delay strategies of the speculators that

may trigger an endogenous devaluation before it occurs exogenously. She finds that higher

uncertainty about the reserves of the Central Bank increases the delay and extends the

ex ante mean delay between the exogenous shock and devaluation. However, both models

preclude strategic arbitrage since they assumed that a market is composed of a continuum

of agents. Also, the existence of an exogenous deadline on the bubble (or on the exchange

rate) has a considerable effect on their analysis. Such an effect is absent in our model, owing

34

to the infinite horizon.

1.5 Concluding Remarks

In this paper, we analyzed currency markets as dynamic coordination models with a se-

quential service constraint. Adding this competitive element was seen to yield fairly sharp

results concerning the implications of alternate uncertainty structures upon the range of

cooperative possibilities. In particular, the presence of aggregate uncertainty considerably

enhanced the ex-ante probability of a coordination failure, in comparison to a world in which

complete information of all payoff relevant variables is available to the speculators. This

might be interpreted as a rationalization of central bank’s reluctance to disseminate infor-

mation about payoff relevant variables like the commitment capability to a fixed exchange

rate as measured by the amount of dollars it is willing to sacrifice in its defence.

1.6 Appendix: Proofs

Proof of Lemma 2: The proof uses backward induction. Suppose that for some k ∈

{2, · · ·, n∗ (ω)} and for each m ≤ k − 1, there is a unique spe in Γm, with each speculator’s

behavior governed by the strategy sI ({φ}) = 1.

Denote by δP (ω) that value of δ ∈ (0, 1) such that ω = ωn∗ (δ) . If δ > δP (ω) , then from

lemma 1, there exists a unique SPE in pure strategies in which every speculator sells in

period 1 of any subgame Γk. We will assume henceforward that δ > δP (ω) . This implies

that the pure strategy profile in which for every period of Γk, and every trader I, sI (ht) = 0

is not a SPE strategy profile. This means that if (σ∗I)NI=A is a mixed strategy SPE of Γk,

then there exists t ≥ 1 and a speculator I such that σ∗I,t > 0. Let J be another speculator.

35

If J sells in period t of Γk, then his expected period t payoff is:

(ωπ − c)

" Pk−2s=1

P{U⊂S\{j}:|U |=s} α (δ)

QI∈U σ∗I (ht)

QI /∈U (1− σ∗I (ht))

+PN−n∗+k−1

s=k−1k

s+1

P{U⊂S\{j}:|U |=s}

QI∈U σ∗I (ht)

QI /∈U (1− σ∗I (ht))

#+(δωπ − c)

YI 6=J

(1− σ∗I (ht))

where S is the set of remaining players in Γk. α (δ) = δωπ−cωπ−c → 1 as δ → 1. The first term

inside the square bracket represents the payoff if (k − 2) or lesser number of traders sell in

the current period. This uses the induction hypothesis which claims that in Γm; m ≤ k−1,

the strategy of the traders is to sell in period 1. So the devaluation occurs in period t+ 1.

The second term in the square bracket represents the expected payoff if at least (k − 1)

other traders sell in period t, thereby inducing a devaluation in period t. The final term

represents the payoff under the induction hypothesis if J is the only trader to sell in period

t. If J delays in period t, his period t expected payoff is

(ωπ − c)k−2Xs=1

k − s

N − n∗ + k − s

X{U⊂S\{j}:|U |=s}

YI∈U

σ∗I (ht)YI /∈U

(1− σ∗I (ht))

+YI 6=j

(1− σ∗I (ht)) υJ (σ∗ | ht+1)

Comparing the two expressions above, we note that if the number of speculators selling in

period t, st ≥ 1, then J would do strictly better by selling in period t than by delaying.

For J to be indifferent between selling and delaying at t (or to prefer the latter), it is

necessary that his payoff from delaying when st = 0, υJ (σ∗ | ht+1) > δπ − c. As δ → 1,

the latter payoff converges to ωπ − c. Since this is also the highest possible payoff in the

36

game, it follows that as δ → 1, for J to be weakly prefer delay at t, it must be the case that

υJ (σ∗ | ht+1) → ωπ − c. However, since limδ%1 (υJ (σ∗ | ht+1)− δωπ − c) = 0, it follows

that J is willing to delay at sufficiently high δ iff the probability that st = 0 given σ∗ is

sufficiently high. In fact in the limit, the probability that st = 0 must converge to 1. For

this to be the case, it follows that limδ%1 σ∗I,t = 0 for each trader I still in the game.

Since σ∗I,t → 0 as δ → 1, for δ sufficiently high, the probability that more than one agent

invests in any single future period of Γk is also close to zero, whereas the probability that

exactly one speculator will eventually invest in the future and move the game to Γk−1 is

close to unity. But given that |S| > 2, it follows that there is a player J such that the

probability conditional on σ∗ | ht+1 that J is this player is strictly less than 12 . In this case,

υJ (σ∗ | ht+1) < δωπ − c for δ sufficiently close to 1. This means that given t and a player

I with σ∗I,t > 0, this player J will strictly prefer to sell in period t, which is only possible if

σ∗I,t = 1 for all I 6= J. Since t was arbitrary, this contradiction proves the lemma. ♦

Proof of Proposition 2: Fix an arbitrary period t. Consider the following strategy profile.

For each Γk; k ∈ {1, · · ·, n∗ (ω)− 1} , and for all I ∈ N, let

sI (h) = 1

In Γn∗(ω), let

sI (hτ ) =

½0, if τ ≤ t− 11, if τ > t− 1

¾

We claim that this symmetric strategy profile is a SPE profile. From the proof of lemma

1, the claim is valid for the subgames Γk; k ≥ 1. So it only remains to verify that the

37

recommendation for Γn∗(ω) is deviation proof. Clearly, a deviation to sell later than t yields

a strictly lower payoff. So consider a deviation to sell at any period τ < t. Since this

deviation leads to subgame Γn∗(ω)−1, it yields a payoff

δωπ − c

which is strictly dominated by the expected payoff under the proposed strategy profile

n∗ (ω)N

(ωπ − c)

because δ < δP (ω) implies that ω < ωn∗(ω). This proves that the proposed strategy profile

is a SPE profile. k

Proof of Proposition 3: (a) For brevity, we will simply denote the two value functions

described in the text by V S and V P . In the subgame Γk,

V S − V P

=

"(ωπ − c)

à ∞XN=n∗−k−1

k

N + 1

e−λλN

N !

!− c

Ãk−2XN=0

e−λλN

N !

!#− (ωπ − c)

e−λλk−1

(k − 1)!

where the term inside the square bracket is V S and the last term is V P . This can be

simplified to yield

(ωπ − c)

µk

λ

¶µ1− Γ (k + 1, λ)

Γ (k + 1)

¶− cΓ (k − 1, λ)Γ (k − 1)

38

where for any pair (n, λ) , Γ (n, λ) =R∞λ zn−1e−zdz with Γ (n, 0) = Γ (n) . Notice that¡

V S − V P¢ |λ=0= −c, and limλ→∞ V S − V P = 0.

We next prove that there exists a value λ̂ such that λ > λ̂ implies that ∂∂λ

¡V S − V P

¢< 0.

This derivative can be directly computed as

(ωπ − c)

"e−λλk−1

Γ (k + 1)k −

µk

λ2

¶µ1− Γ (k + 1, λ)

Γ (k + 1)

¶#+

c

Γ (k − 1)λk−2e−λ

which has the same sign as

(ωπ − c)

"e−λλk+1

Γ (k + 1)k − k

µ1− Γ (k + 1, λ)

Γ (k + 1)

¶#+

c

Γ (k − 1)λke−λ

whose limit as λ → ∞ is − (ωπ − c) k < 0. Since ∂∂λ

¡V S − V P

¢is continuously differen-

tiable, this verifies the existence of the required λ̂.

We will next demonstrate that the set©λ > 0 : ∂

∂λ

¡V S − V P

¢= 0

ªis a singleton. This

will complete the proof of part (a) , because (i) ∂∂λ

¡V S − V P

¢is a continuous function; (ii)

V S − V P declines as λ increases for all λ sufficiently large; (iii) this unique critical point

is, therefore, a local (in fact a global) maximizer of¡V S − V P

¢; (iv) moreover, no local

minimizers can exist and, therefore,¡V S − V P

¢(λ) ≥ 0⇒ ¡∀λ0 > λ

¢ ¡V S − V P

¢ ¡λ0¢ ≥ 0.

∂∂λ

¡V S − V P

¢= 0 is equivalent to

(ωπ − c)

"µk

λ2

¶µ1− Γ (k + 1, λ)

Γ (k + 1)

¶− e−λλk−1

Γ (k + 1)k

#=

c

Γ (k − 1)λk−2e−λ

39

⇔ (ωπ − c)h(Γ (k + 1)− Γ (k + 1, λ))− e−λλk+1

i= c (k − 1)λke−λ (17)

Now define by

hk (λ) = Γ (k + 1)− Γ (k + 1, λ)− e−λλk+1

and by

ik (λ) = (k − 1)λke−λ

We may therefore rewrite (17) as (ωπ − c)hk (λ) = cik (λ) .

Observe that ik (0) = 0; i0k (λ) ≥ 0 iff λ ≤ k and that limλ→∞ i0k (λ) = 0 = limλ→∞ ik (λ) .

Also hk (0) = 0 and limλ→∞ hk (λ) = k! > 0.

Therefore,

hk (0) (ωπ − c) = ik (0) c;

limλ→∞

(ωπ − c)hk (λ) > limλ→∞

cik (λ)

Furthermore

h0k (λ) = −e−λλk (k + 2− λ)

This implies that for λ > 0

40

h0k (λ) < 0 iff λ < k + 2

Since hk (0) = 0, this in turn implies the existence of a value λk > k + 2 with the property

that for λ > 0,

(ωπ − c)hk (λ) < 0 iff λ < λk

Since cik (λ) is a strictly concave function positive in the interior of its domain, and with

a unique maximizer at λ = k, the properties of hk that are established above prove that

(ωπ − c)hk becomes positive (and thereafter continues to increase) over a range where cik

is positive decreasing. This guarantees that (18) has a unique solution in λ ∈ (0,∞) as

desired. This completes part (a) of the Proposition.

(b) Take any fixed but arbitrary λ. Define ∆ (λ, 1, k) =¡V S − V P

¢(λ, 1, k) . Then

∆ (λ, 1, k)−∆ (λ, 1, k + 1)

= (ωπ − c)e−λλk

(k + 1)!

"k −

à ∞Xl=0

1Qln=0 k + 2 + n

!#+ c

e−λλk−1

(k − 1)!

> (ωπ − c)e−λλk

(n∗ − k + 1)!

·(k + 1)− k + 2

k + 1

¸> (ωπ − c)

e−λλk

(k + 1)!

·3− 4

3

¸=5

3(ωπ − c)

e−λλk

(k + 1)!

> 0

41

where the penultimate inequality arises because we are considering k ≥ 2. Therefore, λk+1 >

λk. This proves part (b). k

Proof of Proposition 4: Let s∗2 (λ) denote our symmetric MPE action function for the

subgame Γ2. Attached to the constant function, s∗1 (λ) = 1 for the subgame Γ1, these two

functions constitute a symmetric MPE for a game where n∗ = 2. We begin by assuming

that if λ ≤ λ2, then s∗2 (λ) = 0; i.e., players will coordinate on a termination equilibrium in

Γ2 if λ ≤ λ2. Given any λ ∈ (λ2,∞), define by

es2 (λ) ≡ λ− λ2λ

Hence, for any probability of sales s ≥ es2 (λ) , we may now write the two value functions as

V S2 (λ, s; s

∗2, s

∗1)

= (ωπ − c)XN≥1

e−λλN

N !

NXk=1

µ2

k + 1

¶µNk

¶sk (1− s)N−k

+(δωπ − c)XN≥1

e−λλN

N !(1− s)N − ce−λ

and

V P2 (λ, s; s

∗2, s

∗1)

= (ωπ − c)XN≥1

e−λλN

N !

1

N

µN1

¶s (1− s)N−1

= (ωπ − c)XN≥1

e−λλN

N !s (1− s)N−1

42

To save on notation, we will suppress dependence of the value functions upon the continua-

tion strategies s∗2, s∗1.When comparing V S2 (λ, s) and V

P2 (λ, s) , we notice that at s = es2 (λ) ,

V S2 (λ, s)− V P

2 (λ, s) is positive if

(δωπ − c)XN≥1

e−λλN

N !(1− s)N − ce−λ ≥ 0

⇔ e−λes2(λ) ³1− e−λ(1−es2(λ))´ (δωπ − c)− ce−λ

= e−(λ−λ2)³1− e−λ2

´(δωπ − c)− ce−λ

= eλ2³1− e−λ2

´(δωπ − c)− c

= c− c ≥ 0

where the last equality follows from the definition of λ2 given in (16) .

So we conclude that V S2 (λ, s)−V P

2 (λ, s) |s=es2(λ)> 0. By hypothesis, V S2 (λ, 1)−V P

2 (λ, 1) <

0. The value functions defined above are clearly continuously differentiable in s over the set

[es2 (λ) , 1] . The intermediate value theorem, therefore, ensures the existence of s∗ (λ) ∈(es2 (λ) , 1) such that V S

2 (λ, s)− V P2 (λ, s) |s=s∗(λ)= 0. We will take as our symmetric MPE

action function, s∗2 (λ), for Γ2 as follows. For λ ≤ λ2, s∗2 (λ) = 0. For λ ≥ λ2, s

∗2 (λ) = 1.

Finally, for λ ∈ ¡λ2, λ2¢ , s∗2 (λ) = s∗ (λ) .

In order to prove the monotonicity and continuity properties of the function s∗ (λ) defined

above, we need a little more work. Let us begin by defining the difference function:

43

∆2 (λ, s) ≡ V S2 (λ, s)− V P

2 (λ, s)

Using the expressions for V S2 (λ, s) and V P

2 (λ, s) given above, we may write this function

as

(ωπ − c)XN≥1

e−λλN

N !(N − 1) s (1− s)N−1

+(ωπ − c)XN≥1

e−λλN

N !

NXk=2

2

k + 1

µNk

¶sk (1− s)N−k χ (N, k) (18)

(δωπ − c)XN≥1

e−λλN

N !(1− s)N − ce−λ

where χ (N, k) = 1 provided k ≤ N and is zero otherwise. Defining by

α (δ) ≡ δωπ − c

ωπ − c→ 1 as δ → 1

we may rewrite (18) as

(ωπ − c)α (δ) e−λλ (1− s)− ce−λ +

(ωπ − c)XN≥2

e−λλN

N !

µα (δ) (1− s)N +

µN1

¶µN − 1N

¶s (1− s)N−1

¶

+(ωπ − c)XN≥2

e−λλN

N !

NXk=2

2

k + 1

µNk

¶sk (1− s)N−k (19)

We first establish

44

Lemma 3 λ2 < 1

Proof: The proof follows from direct computation. Set M2 = 1 +1

2(e−2.5) ∼ 3. 2906. From

the value functions defined in proposition 3, it follows that ∆2 (λ, 1) is

(ωπ − c)2

λ

·1− e−λ

µ1 + λ+

λ2

2

¶¸− ce−λ

Setting λ = 1 in the above expression gives us

1

e[2 (ωπ − c) (e− 2.5)− c]

> 0

as ωπc > M2 by assumption. This implies from part (a) of Proposition 3 that λ2 < 1 as

desired. ♦

Lemma 4 There exists ε > 0 such that for each λ ∈ £λ2, λ2 + ε¤and for any s > es2 (λ) ,

∂∂λ∆2 (λ, s) > 0. Furthermore, there exists uniform B2 <∞ such that ∂

∂λ∆2 (λ, s) < B2.

Proof: Using (18) , we can directly compute this partial derivative:

(δωπ − c)XN≥1

e−λλN−1 (N − λ)

N !(1− s)N + ce−λ

+(ωπ − c)XN≥1

e−λλN−1 (N − λ)

N !×

×Ã(N − 1) s (1− s)N−1 +

NXk=2

2

k + 1

µNk

¶sk (1− s)N−k χ (N, k)

!

45

From lemma 3, λ2 < 1 and thus, for some positive ε sufficiently small, λ2+ε < 1 also. Since

λ ≤ λ2 and in the previous expression, N ≥ 1, positivity of the partial derivative follows.

For the second part, note that the last expression is strictly and uniformly bounded from

above by the finite number:

2 (ωπ − c) + c = 2ωπ − c = B2

It follows that for (λ, s) ∈ £λ2, λ2 + ε¤× (es2 (λ) , 1) , ∂

∂λ∆2 (λ, s) ∈ (0, B2) as desired. ♦

Lemma 5 Fix λ ∈ £λ2, λ2¤ . Then for any s > es2 (λ) , ∂∂s∆2 (λ, s) < 0. Furthermore, there

exists a uniform b2 > 0 such that ∂∂s∆2 (λ, s) < −b2.

Proof: From (19) , we can compute:

∂∂s∆2 (λ, s)

ωπ − c

= −α (δ) e−λλ+XN≥2

e−λλN

N !

∂

∂s

ÃNXk=0

α (k)

µNk

¶sk (1− s)N−k

!

where α (k) is defined in the following manner.

α (k) =

2

k+1 ; for N ≥ k ≥ 2N−1N ; for k = 1

α (δ) ; for k = 0

Let us define by N∗ (δ) ∈ N such that N−1

N < α (δ) iff N ≤ N∗ (δ) . Since limδ→1 α (δ) = 1,

limδ→1N∗ (δ) =∞. For N ≤ N∗ (δ) , it is clearly the case that α (k) > α (k + 1) , for each

k ∈ {0, 1, · · ·, N − 1} . Therefore, for each such N,

46

∂

∂s

ÃNXk=0

α (k)

µNk

¶sk (1− s)N−k

!< 0

Since limδ→1N∗ (δ) =∞ and limN→∞ e−λλNN ! = 0, it follows that by pushing up δ sufficiently

close to 1, we can guarantee that

¯̄̄̄¯̄ XN≥N∗(δ)+1

e−λλN

N !

∂

∂s

ÃNXk=0

α (k)

µNk

¶sk (1− s)N−k

!¯̄̄̄¯̄

<

¯̄̄̄¯̄N∗(δ)X

N=2

e−λλN

N !

∂

∂s

ÃNXk=0

α (k)

µNk

¶sk (1− s)N−k

!¯̄̄̄¯̄

implying that

∂

∂s∆2 (λ, s) < − (ωπ − c)α (δ) e−λλ < 0

From lemma 3, λ ∈ (0, 1) and hence, e−λλ is minimized at λ = λ2 in the range£λ2, λ2

¤.

Therefore, if we now define by

b2 = (ωπ − c)α (δ) e−λ2λ2

it follows that ∂∂s∆2 (λ, s) < −b2 as desired. The proof of the lemma is complete. ♦

Lemma 6 Fix any λ ∈ £λ2, λ2¤ . The set { s ∈ (es2 (λ) , 1) : ∆2 (λ, s) = 0} is a singleton.Proof: We already know that the set is non-empty. The cardinality of the set follows from

lemma 5. ♦

47

From lemma 6, it follows that there is a unique symmetric MPE action function s∗2 (λ) that

has the one-period property. From lemmas 4 and 5, the following properties of this function

emerge:

Lemma 7 Over the range [0, λ2),¡λ2, λ2

¢,¡λ2,∞

¢, the function s∗2 (λ) is continuously

differentiable. Furthermore, s∗2 (λ) is strictly increasing over the range¡λ2, λ2

¢. Finally,

limλ↓λ2 s∗2 (λ) = 0 and limλ↑λ2 s

∗2 (λ) = 1.

Proof: Since s∗2 (λ) is constant over [0, λ2) and¡λ2,∞

¢, the differentiability property

claimed follows trivially. For the range¡λ2, λ2

¢, it follows from the implicit function theo-

rem, which applies by virtue of lemmas 4 and 5. These properties also imply the limiting

properties of this function. For monotonicity, notice that

ds∗2 (λ)dλ

= −∂∂λ∆2 (λ, s)∂∂s∆2 (λ, s)

> 0 (20)

The lemma is proved. ♦

Finally, the reader may note that in (20) , ds∗2(λ)dλ < B2

b2<∞. This combined with lemma 7

imply that the function s∗2 (λ) is lipschitz with constant l2 =B2b2. This completes the proof

of proposition 4. k

Proof of Proposition 5:17 Suppose that for some N ≥ 3, and for each k ∈ {1, · · ·, N − 1} ,

there exist continuous, non-decreasing functions s∗k : R+ → [0, 1] satisfying the following

conditions: (1) ∆k

¡λ, s∗k (λ) ; s

∗k−1, · · ·, s∗1

¢is nonpositive if s∗k (λ) = 0, zero if s∗k (λ) ∈

17Hsueh-Ling Huynh suggested the strategy of proof used to establish this result.

48

(0, 1) , and non-negative if s∗k (λ) = 1; (2) There exist two values 0 < λk < λk satis-

fying (∀λ ≤ λk) (s∗k (λ) = 0) ,

¡∀λ ≥ λk¢(s∗k (λ) = 1) , and

¡∀λ ∈ ¡λk, λk¢¢ (s∗k (λ) ∈ (0, 1)) ;(3) λ ≥ λk implies that λ (1− s∗k (λ)) < λk; (4) λ ∈ ¡

λk, λk¢ ⇒ V I

k (λ; s∗k, · · ·, s∗1) <

V Ik

¡λ, s = 0; s∗k−1, · · ·, s∗1

¢. This constitutes our backward induction hypothesis. While parts

(1) − (3) are pretty easy to understand, part (5) begs some explanation. Part (4) implies

that in any subgame Γk, for k = 1, · · ·, N − 1, and belief λ that falls within the range for

which s∗k (λ) ∈ (0, 1) , a speculator’s expected investment payoff is lower in equilibrium than

would be the case if the other agents deviated from their equilibrium strategies and decided

to delay at λ in Γk. Proposition 4 establishes the basis of induction as parts (1)− (4) of the

induction hypothesis are valid for the case of N = 3. We will demonstrate that there exists

a non-decreasing, lipschitz continuous action function, s∗N for ΓN , that when combined with

the sequence of functions©s∗N−1, · · ·, s∗1

ª, satisfies properties (1)− (4) for ΓN .

Step 1: Existence of λN and λN .

The existence of finite and positive λN follows from part (a) of Proposition 3. In order

to establish existence of the first threshold, notice that for λ ≤ maxk=1,···,N−1 λk, the best

response to s∗N (λ) = 0 is s∗N = 0. This is because if a speculator sells at such a λ in

ΓN , he does so unilaterally, thereby generating ΓN−1 at an unchanged λ. Since λ ≤ λN−1

by assumption, it follows that in this subgame, s∗N−1 (λ) = 0. So a deviant’s payoff from

speculating against the fixed exchange rate is −c, which is clearly worse than the zero payoff

that results from not speculating. Now fix the strategy of speculators other than I in ΓN

to be s (λ) = 0. Their behavior in subgames Γk, for k ∈ {1, · · ·, N − 1} is dictated by the

induction hypothesis. Notice that speculator I 0s payoff from not selling is zero (as λ does

49

not change in this case, and the other speculators’ common strategy is s (λ) = 0). If he

sells then he generates the subgame ΓN−1 at common belief λ. It is trivial to show that

as λ increases, the probability that I assesses on the event that there are at least N − 1

other speculators increases also. Since part (4) of the induction hypothesis implies that

conditional on this event, the probability of a successful attack in equilibrium increases

with λ, this also implies that as λ increases, the expected payoff to selling increases for

player I. But in this case, there will be a λN <∞ such that in response to s (λ) = 0, player

I 0s best response is to sell if λ > λN , not sell if λ < λN , and either if λ = λN . ♦

Step 2: λN < λN .

From Proposition 3 (b) , we know that λk > λk−1. If λN > λN , then the decision by player I

to sell in subgame ΓN at common belief λN in the case where other speculators are adopting

the strategy s (λN ) = 0 will generate a N −1 player subgame with belief λN > λN > λN−1.

In the generated subgame, therefore, part (2) of the induction hypothesis implies that any

remaining speculator will sell in the first period. Hence, the deviation by I to sell in the

current period ends the game with a rush in the next period. By doing so, player I 0s

expected payoff is:

δωπ∞X

M=N−1

e−λN (λN)M

M !− c

> ωπ∞X

M=N−1

N

M + 1

e−λN (λN )M

M !− c

> 0

where the first inequality is obviously valid for δ sufficiently close to 1, and the second

50

follows from the fact that for all λ ≥ λN , the expected payoff from participating in a rush is

positive. Since the expected payoff from selling when all others are delaying is positive, this

contradicts the hypothesis that a termination equilibrium exists at λ ≥ λN . This establish

the inequality claimed above. ♦

Step 3: Establishing pure-strategy thresholds for s∗N (λ) .

From step 2, we can now begin constructing the symmetric equilibrium action function for

the subgame ΓN , which when appended to the sequence (s∗k)Nk=1 will constitute a symmetric

MPE strategy for ΓN . For λ ≤ λN , set s∗N (λ) = 0, while for λ ≥ λN , set s∗N (λ) = 1.

From step 2, this leaves an intermediate range of beliefs¡λN , λN

¢for which the symmetric

equilibrium action is necessarily mixed. We will now prove that such a mixed action exists

and that it is generated by a lipschitz continuous function. Thus step 3 verifies the existence

of an action function satisfying part (2) of the induction hypothesis for ΓN . ♦

Step 4: No Sales Implies Termination.

Define for λ > λN , a value esN (λ) by λ (1− esN (λ)) = λN . If at any λ > λN , other

speculators are selling with probability esN (λ) , the next period belief is λN . We will nowshow that given the induction hypothesis,

¡V SN − V P

N

¢ ³λ, esN (λ) ; (s∗k)Nk=1´ > 0. In defining

the arguments of the function¡V SN − V P

N

¢, by the function s∗N is meant its restriction to the

interval [0, λN ] . From step 3, we know that over this range of values, s∗N (λ) = 0. The proof

while quite simple, demonstrates very clearly how each part of the induction hypothesis is

utilized in the construction of s∗N . From step 1, it is clearly the case that λk > λk−1, for

N ≥ k ≥ 2. Let the number of other speculators who sell in the current period be denoted

S ≥ 0. From speculator I 0s perspective, the following events are relevant:

51

S 0 1 2 · · · ≥ Ns∗NEXT 0 > 0 > 0 > 0 game ends

(21)

In (21) , the first row entries are the possible realization of sales in the current period, while

the second row tabulates the corresponding action probability in the next period. The

reason why s∗NEXT > 0 for S ≥ 1 is because the next period belief λN > λk for k ≤ N − 1,

and hence, speculators will sell with positive probability in the mixed symmetric equilibrium

of the next period’s subgame. Observe that for the case S ≥ N − 1, it is strictly better for

I to sell in the current period rather than delay until the next. Similarly, if the realization

of S were positive but less than N −1, the equilibrium strategy leaves I indifferent between

selling and delaying in the next period (if λN ≤ λk), or strictly preferring sales (if λN > λk).

In the latter case, from part (b) of Proposition 3 it immediately follows that I would have

preferred to sell in the current period rather than delay. In the former case, part (5) of the

induction hypothesis implies that the expected investment payoff is higher if I invests in

the current period rather than delay and play the mixed equilibrium strategy in the next

period. So we conslude that for any realization of S ≥ 1, I would strictly prefer to invest

rather than delay in this period. This leaves us with the possibility S = 0. In this case, the

next period action of the other speculators is s∗N (λN ) = 0. At λN , I is indifferent between

speculating and delaying in the next period. In either case, his expected payoff is zero. But

in the event that S = 0, his expected payoff to investing in the current period would be

identical to delaying in the current and investing in the next period because in either case,

this generates the subgame ΓN−1 at a common belief λN . This implies that for this case,

I is indifferent between investing and delaying. Since esN (λ) > 0, this last event has ex-

52

ante probability strictly less than one, implying that¡V SN − V P

N

¢ ³λ, esN (λ) ; (s∗k)Nk=1´ > 0

as desired. ♦

The argument of the previous paragraph implies that if at λ > λN , other speculators are

selling with the probability esN (λ) , then speculator I will strictly prefer to sell in the currentperiod, because under all possible realizations of current period sales, he weakly prefers to

sell than to delay, and for a subset of such realizations which have positive likelihood ex-

ante, he strictly prefers to sell. This proves that¡V SN − V P

N

¢ ³λ, esN (λ) ; (s∗k)Nk=1´ > 0 as

desired.

Finally, since the functions s∗k are lipschitz continuous on [0, λN ] for all k ≤ N − 1,

and the from step 3, so also is the function s∗N ,¡V SN − V P

N

¢ ³λ, esN (λ) ; (s∗k)Nk=1´ > 0 >¡

V SN − V P

N

¢ ³λ, 1; (s∗k)

Nk=1

´for λ ∈ ¡λN , λN¢ implies from the intermediate value theorem,

the existence of sλ ∈ (esN (λ) , 1) that satisfies ¡V SN − V P

N

¢ ³λ, sλ; (s

∗k)

Nk=1

´= 0. Step (4)

establishes that there exists an action function s∗N (λ) which satisfies properties (1) − (3)

of the induction hypothesis for the subgame ΓN .

Notice that the construction of this preliminary part of the proof of Proposition 5 relies

crucially on the continuity of the functions (s∗k)N−1k=1 . In order to complete the proof, we

therefore need to establish that the function s (λ) = sλ is continuous on¡λN , λN

¢, and also

prove two limiting properties of this function: (i) limλ&λNsλ = 0; (ii) limλ%λN

sλ = 1. We

begin with the following lemma which mirrors lemma 3 for the case of general N ≤ n∗.

Lemma 8 For each N < ∞, there exists MN < ∞ such that ωπc > MN implies that

λN < (N − 1) .

53

Proof: From the proof of part (a) of Proposition 3, the function V SN (λ, 1) − V P

N (λ, 1) is

defined as

(ωπ − c)

µN

λ

¶µ1− Γ (N + 1, λ)

Γ (N + 1)

¶− cΓ (N − 1, λ)Γ (N − 1)

Evaluating this expression at λ = N − 1, we get

(ωπ − c)

µN

N − 1¶µ

1− Γ (N + 1, N − 1)Γ (N + 1)

¶− cΓ (N − 1, N − 1)Γ (N − 1)

which is positive iff

ωπ

c> 1 +

(N − 1)2 Γ (N − 1, N − 1)Γ (N + 1)− Γ (N + 1,N − 1)

whereupon setting MN = 1 +(N−1)2Γ(N−1,N−1)Γ(N+1)−Γ(N+1,N−1) gives the lemma. ♦

For a game where n∗ is the critical number of speculators, define

M (n∗) := maxN∈{2,···,n∗}

MN

By hypothesis, we know that ωπc > M (n∗) , and therefore, using lemma 8, λk < k − 1 in

each subgame, Γk, for k ≤ n∗. Using this fact, we can prove the following result:

Lemma 9 Fix λ ∈ £λN , λN¤ and let sλ ∈ (esN (λ) , 1) satisfy ∆N

¡λ, sλ; s

∗N−1, · · ·, s∗1

¢= 0.

Then there exists ε > 0 sufficiently small such that if λ0 ∈ (λ, λ+ ε) , then ∆N

¡λ0, sλ; s∗N−1, · · ·, s∗1

¢>

0.

54

Proof: For our purposes, it is sufficient to define ε to be that value at which (λ+ ε) (1− sλ) =

λN . Let the actual number of speculators still left in the game (including speculator I who

will be our representative speculator), be denoted by A.

Observe that as λ increases, the likelihood that I assesses on A = n, for each n ≥ N

increases. To see this, notice that Pr (A = n ≥ N) = e−λλn−1(n−1)! , whose derivative with respect

to λ is e−λλn−2(n−1−λ)(n−1)! . Applying lemma 8, we have λ ≤ λN < N − 1, and thus, n ≥ N

implies that this derivative is positive. The important implication of this is that the function

∆N

¡λ, sλ; s

∗N−1, · · ·, s∗1

¢, conditional on some realization of A, is positive only if A ≥ N.

To see that this is true, consider any A < N. If I delays in the current period, then there

is positive probability that the realization of current period sales, S, is so poor that he will

not sell in the future; i.e., there is positive probability of obtaining zero payoff by delaying.

In contrast had I speculated in the current period, he would have lost c dollars. The first

impact of an increase in λ, therefore, is to transfer some probability mass from the states

A < N, to each of the states A ≥ N. However, A ≥ N is a sufficient and not necessary

condition for ∆N ≥ 0. We will now show that for each Ai ≥ N, such that conditional on

Ai, ∆N < 0, the actual value of this function is higher than for any Aj < N. First consider

current period realizations of S, which have positive probability of occurence for both Ai

and Aj . If speculator I delays sales until the next period, then he will behave identically in

the next period in either case. If s∗NEXT (λ, S) = 0 for example, then ∆N < 0 in both cases.

However, since by selling in the current period, he can (weakly) increase the likelihood of

success if Ai ≥ N, ∆N | Ai ≥ ∆N | Aj . An identical argument proves a similar inequality

for realizations S such that s∗NEXT (λ, S) > 0, and for S ≥ N, which can obtain only under

55

Ai. We now summarize what we have proved so far. An increase in λ pushes up (i) the

relative likelihood of all states A ≥ N such that ∆N > 0 at the expense of states A < N ; (ii)

the relative likelihood of states A ≥ N such that ∆N ≤ 0 at the expense of states A < N,

and where ∆N | (A ≥ N) is greater than ∆N | (A < N) . Finally, observe that over the set

A ∈ {1, · · ·, N − 1} , as A increases, the likelihood that given a fixed sλ, speculator I will

sell in the next period (after delaying in the current) is higher. This is because, an increase

in A increases the likelihood of higher realizations of S thereby increasing the equilibrium

probability of sales in the next period. But if so, this decreases the difference between the

payoff from selling in the current period (−c) , and delay −c∗Pr (I sells in the next period) .

This completes the proof of the lemma. ♦

Lemma 10 Fix λ ∈ ¡λN , λN¢ and let sλ ∈ (esN (λ) , 1) satisfy ∆N

¡λ, sλ; s

∗N−1, · · ·, s∗1

¢= 0.

Consider any s0 > sλ. Then ∆N

¡λ0, sλ; s∗N−1, · · ·, s∗1

¢> 0.

Proof: We need to show that ∂∂s∆N

¡λ, sλ; s

∗N−1, · · ·, s∗1

¢< 0. Let us first establish the result

for δ = 1.

We know that

∆N

¡λ, sλ; s

∗N−1, · · ·, s∗1

¢= ωπPr(game ends in a devaluation)− c = 0

If we can show that for ds sufficiently small, ∂∂s∆N

¡λ, sλ; s

∗N−1, · · ·, s∗1

¢< 0, we are done.

Consider an infinitesimal increase in s from sλ to sλ + ds. Then, his expected payoff if he

sells in the current period is

P (x = 0) ∗ {ωπPr(game ends in a devaluation with x = 0)− c}

56

+∞Xx=1

P (x) ∗ {ωπPr(game ends in a devaluation with x other agents selling)− c}

= P (x = 0) ∗ {ωπPr(game ends in a devaluation with x = 0)− c}

+P (x = 1) ∗ {ωπPr(game ends in a devaluation with x = 1)− c}

+ · · · ·

The last equality simplifies into

n (sλ + ds) (1− s)n−1 ∗ {ωπPr(game ends in a devaluation with x = 0)− c}+

n (n− 1) (sλ + ds)2 (1− s)n−2 ∗ {ωπPr(game ends in a devaluation with x = 1)− c}

+ · · · ·

where the later parts of the series involve third and higher order terms in s. Noting that

{ωπPr(game ends in a devaluation with x = 0)− c} < 0

owing to monotonicity of strategies in the continuation game, and dividing through by

(sλ + ds) , we obtain negativity of the series as ds→ 0. ♦

Lemma 11 There exists a continuous, non-decreasing action function s∗N (λ) which satisfies

properties (1)− (3) of the induction hypothesis for the subgame ΓN . This function assumes

the value 1 if λ ≥ λN , 0 if λ ≤ λN , and equals sλ for λ ∈ ¡λN , λN¢ .

57

Proof: We first establish that the function described by the lemma is continuous. This fact

rests upon the following observation: take a continuous function f (x, y) defined on a product

space of two compact intervals, X × Y. Define by y (x) ∈ Y, that value which satisfies (i)

f (x, y (x)) = 0; (ii) for some εx > 0 sufficiently small, x0 ∈ (x, x+ εx) ⇒ f (x0, y (x)) > 0;

(iii) y > y (x) ⇒ f (x, y) < 0. (ii) and (iii) jointly imply that y (x) is a strictly increasing

function. To see this suppose x < x0 and y (x) ≥ y (x0) . By continuity of f, this implies

that there exist two values x ≤ x1 < x2 ≤ x0, with x2 ∈ (x1, x1 + εx1) and y (x1) ≥ y (x2) .

From (iii) , it follows that f (x2, y (x1)) ≤ f (x2, y (x2)) = 0, whereupon from (ii) it follows

that f (x1, y (x1)) < 0, which contradicts (i) . We now prove that y (x) is a continuous

function. Suppose not; i.e., let X 3 x∗ = limn%∞ xn such that for limn%∞ y (xn) =

limn%∞ yn = y∗, f (x∗, y∗) 6= 0. Wlog, let f (x∗, y∗) > 0. Using the monotonicity of f

in y, this implies that y (x∗) > y∗, and for each y ∈ (y∗, y (x∗)) , f (x∗, y) > 0. Wlog,

extract an increasing subsequence (xnk , ynk)→ (x∗, y∗) . For each point on this subsequence,

ynk < y∗, and from (iii) , y > ynk implies that f (xnk , y) < 0. Consequently, f (xnk , y) < 0

for y ∈ (y∗, y (x∗)) . Since f (x∗, y) > 0, however, this can happen only if the function f is

discontinuous on the set {x∗} × (y∗, y (x∗)) . To apply this argument to our problem, let

us focus on the subgame ΓN and take X =£λN , λN

¤, Y = [0, 1] . For each λ ∈ ¡λN , λN¢ ,

let y (λ) = sλ, and let y (λN ) = 0 and y¡λN¢= 1. The function f is represented by the

function ∆N

¡·, ·; s∗N−1, · · ·, s∗1¢ . From Step (4) , property (i) is valid, while lemmas 9 and

10 establish properties (ii) and (iii) . This proves that the function sλ is strictly increasing

and continuous over¡λN , λN

¢, that limλ&λN

sλ = 0, and limλ%λNsλ = 1. This completes

the proof of the lemma. ♦

58

This completes the proof of Proposition 5. k

59

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