J EconDOI 10.1007/s00712-014-0393-5

Strategic delay and information cascades

Edward Cartwright

Received: 21 January 2013 / Accepted: 28 January 2014© Springer-Verlag Wien 2014

Abstract In a setting where agents must choose between two investments, Zhang(in RAND J Econ 28:188–205, 1997) proposed an equilibrium in which there is strate-gic delay. This equilibrium relied upon there being an information cascade. We shalldemonstrate that an information cascade need not generally occur. It will only occurif and only if the cost of investing takes relatively extreme values. Taking this intoaccount we derive a revised equilibrium that is still characterized by strategic delay.

Keywords Information cascade · Investment · Endogenous timing

JEL Classification D21 · C72 · L13

1 Introduction

A large literature, theoretical, experimental and empirical, has considered in somedetail the nature and prevalence of information cascades (c.f. Bikhchandani et al.1998 and Chamley 2004a). An important contribution to that literature is made byZhang (1997). Zhang analyzes a model of endogenous timing in which agents caninvest at any point in time in either of two projects. The cost of investing, and theprofitability of each project are not known ex-ante, but each agent receives a privatesignal of which project will be more profitable and is also told, crucially, the precisionof that signal. The equilibrium proposed by Zhang is intuitive and consists of a functionmapping signal precision into a time to invest. The less precise is an agent’s signal

I would like to thank two anonymous referees of this journal for their helpful comments on an earlierversion of the paper. And I would also like to thank Anna Stepanova for help with solving example 2.

E. Cartwright (B)Department of Economics, University of Kent, Canterbury CT2 7NP, UKe-mail: E.J.Cartwright@kent.ac.uk

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then the longer he will wait before investing. The first agent to invest will, therefore,be the agent with the most precise signal.

In deriving these results Zhang assumes that all agents will necessarily imitate thefirst investor (or no agent subsequently invests). This is called an investment cascade.The first contribution of the current paper, however, is to demonstrate that an investmentcascade need not occur in the model considered by Zhang. It will occur if and onlyif the cost of investing takes extreme values. From a welfare perspective this findingis important because investment cascades result in less revelation of information andtherefore less chance of agents choosing the optimal project. Avoiding an investmentcascade is thus a good thing. The fact that an investment cascade need not occur alsomeans that the equilibrium proposed by Zhang is incorrect. The second contributionof the current paper is to derive a revised equilibrium and show that it is qualitativelysimilar to that proposed by Zhang.

Most of the literature on endogenous timing has treated the decision of whether ornot to invest in a given project with different agents having different information on itsprofitability (e.g. Chamley 2004a). Zhang is a rare exception in treating the decision ofwhether or not to invest in a project A or project B with different agents having differentsignals about which option is profitable. A key contribution is to capture the strategicincentive to delay investment because of differences in information precision. Theseissues are still relatively unexplored in the literature but seem fundamental and canbe studied, as a first approximation, independent of how one ‘creates’ or ‘motivates’an information or investment cascade. This paper builds on Zhang’s contribution byfurthering our understanding of these important issues.

We proceed as follows: Sect. 2 explains the model, Sect. 3 looks at informationcascades, Sect. 4 at equilibrium behavior and Sect. 5 concludes.

2 The model

We shall follow closely the model introduced in Zhang (1997). There are n agents.Each agent can invest in one (and only one) of two projects a1 or a2. If the state of theworld is ω1 then project a1 yields a return of 2 and project a2 yields return 0 . If thestate of the world is ω2 then project a2 yields return 2 and project a1 yields return 0.Each state of the world is equally likely. There is a cost c to invest in a project with itbeing the same cost to invest in either project. The value of c is not known ex-ante andis drawn from interval [0, 2] according to some cumulative density function G, thusG(x) is the probability c ≤ x . Note that Zhang assumes G(c) = 0.5c. For reasonsthat will become clear we consider a more general setting.

Each agent i observes a signal θi ∈ {θ1, θ2} about the true state of the world. Theagent is also told the precision pi of his signal where pi = Pr(θ1|ω1) = Pr(θ2|ω2).The precision is independently drawn for each i from a distribution �(p) with support[p, p] where 1 ≥ p > p ≥ 0.5. The higher is the precision pi then the better qualityinformation agent i has about the true state of the world. For example, if agent ireceives signal θ1 then there is a pi probability that project a1 is profitable and a1 − pi probability that project a2 is profitable.

Time runs continuously starting from t = 0 and each agent can choose to investin a project at any time. Future payoffs are discounted with common discount rate δ.

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Strategic delay and information cascades

Once an investment is made it is irreversible and publicly observed. The return on theinvestment is not observed by others (or at least not observed until much later) but thecost c is observed by others. The history of play ht at time t details if any investmentshave been made and if so the choice and timing of the investment, as well as revealedcost c. An agent’s strategy consists of a function mapping possible histories of playinto a choice of wait, invest in project a1, or invest in project a2.

To briefly recap on the timing of events. Each agent i observes a private signal θi ofthe true state of the world and is told the precision pi of that signal. Agents can then,at any time, make an irreversible investment. When an investment is made the cost ofinvesting is observed but the return on the investment is not observed. This frameworkis motivated in length by Zhang (1997) and, while clearly stylized, is appealing. Itstwo principal features are that agents differ in the precision of their signal and thatthe investment of one agent reveals some but not all information to others about theprofitability of investment. Intuitively one would expect to find these features in manysettings of interest.

3 Information cascades

The optimal strategy is one that equates the benefits and costs of delay. The cost ofdelay is due to the discounting of future payoffs. The benefit to delay is that someother agent may invest first revealing both the signal that they received, and, moreimportantly, the cost of investing. Given, however, that the cost of investing will befully revealed by the first investment there is an important difference between first andsubsequent investments. In particular, the benefit to delay is much less once one agenthas invested because the cost of investing has been revealed. This points towards thefirst investment being pivotally important.

The potential importance of the first mover is captured by the notion of informationor investment cascade. As is standard, we say that there is an information cascade ifthe actions of agents other than the first mover reveal nothing about their signal andits precision. This will arise if the incentives of other agents to invest are the sameirrespective of the their signal and its precision. There are two types of informationcascade possible in this model: (1) having observed the first mover the optimal strategyof all other agents is to not invest (because cost c was revealed ‘too high’), or (2) theoptimal strategy of all other agents is to invest in the same project as the first mover(when c was revealed as ‘low enough’).1

Zhang (1997) implicitly assumed (see his Lemma 2) that if the first mover is knownto have the highest precision then an information cascade must occur. Our first resultshows, however, that an information cascade need not occur in general. As the informalexplanation for each type of information cascade, given above, might suggest, theprobability distribution of possible costs proves crucial. With a slight abuse of notationlet G(a, b) = G(b) − G(a) denote the probability of the cost being in the interval(a, b).

1 For reasons that will become clear as we proceed, it is not possible to have an information cascade whereall other agents invest in a different project to the first mover.

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Proposition 1 Suppose that the first movers choice of project and its timing alwaysreveals his private signal and its precision. Then, an information cascade will occurfor sure if and only if

G(1, c) = 0

where

c = 2p2

p2 + (1 − p)2.

Proof Suppose that the first mover is agent 1 , his signal is θ1 and its precision is q.An agent i �= 1 whose signal is θi with precision p, will form the updated belief2

Pr(ω|θ1, q, θi , p) = Pr (θ1, θi |ω)

Pr(θ1, θi |ω1

) + Pr(θ1, θi |ω2

) .

If agent i has the same signal as the first mover, θ1 = θi , his updated belief is

Pr(ω|θ1, q, θi , p)θ1=θi = pq

pq + (1 − p)(1 − q). (1)

If agent i has a different signal to the first mover, θ1 �= θi , his updated belief is

Pr(ω|θ1, q, θi , p)θ1 �=θi = q(1 − p)

q(1 − p) + p(1 − q). (2)

If there is an information cascade then, given the cost of delay and lack of infor-mation revealed by other’s choices, agents would want to act immediately. So, either(i) all agents should invest immediately, or (ii) all agents should never invest. Case (i)requires that

2 Pr(ω|θ1, q, θi , p)θ1 �=θi ≥ c

for all p because then the expected return on investing must be greater than the costfor all agents. The value of Pr(ω|θ1, q, θi , p)θ1 �=θi is minimized when p = q givingthat Pr(ω|θ1, q, θi , p)θ1 �=θi = 1

2 . So, if c is revealed as less than 1 there must be aninformation cascade. Case (ii) requires that

2 Pr(ω|θ1, q, θi , p)θ1=θi ≤ c

for all p < q because then the expected return on investing must be less than cost forall agents. The maximal value of Pr(ω|θ1, q, θi , p)θ1=θi is obtained when p = q = pgiving that

2 Recall that Pr(ω1) = Pr(ω2).

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Strategic delay and information cascades

Pr(ω|θ1, q, θi , p)θ1=θi = p2

p2 + (1 − p)2.

This means that an information cascade must occur if c takes value c or above.It remains to show that an information cascade need not occur if c ∈ (1, c). This,

however, follows from the above reasoning. Specifically, suppose that the first moverssignal is arbitrarily close to p. If agent i’s signal is the same as the first mover, θi = θ1,and has precision arbitrarily close to p, then, if there were to be an information cascade,agent i would invest if c < c. If, however, agent i’s signal is different to the first mover,θi �= θ1, and has precision arbitrarily close to p, then, if there were to be an informationcascade, agent i would not invest if c > 1. This, however, means that the actions of thesecond mover must reveal something about his signal when c ∈ (1, c). Consequently,an information cascade will not occur. ��

Recall that Zhang (1997) assumed G(c) = 0.5c. In this case the condition given inProposition 1 is violated and so the assumption of an information cascade is inappro-priate. The results of Zhang are, thus, flawed.3 More generally, Proposition 1 showsthat an information cascade will only occur for sure if the cost of investing lies outsidethe interval (1, c). This finding reaffirms the more general argument that informationcascades are less likely to occur when actions are chosen from a continuum (e.g. Gale1996). The requirement that the cost of investing takes extreme values essentiallymakes the setting not ‘fine enough’ for actions to reveal information. To illustratethese issues it may be instructive to work through an example.

Example 1 Suppose the first movers choice reveals that he had signal θ1 with precisionq = 0.8. Suppose also that p = 0.6. Given the information available to him the

expected payoff of any agent who also had signal θ1 is given by (1) and at least

2 × 0.8 × 0.6

0.8 × 0.6 + 0.2 × 0.4= 1.714.

Thus, any agent with signal θ1 would want to invest in project a1 if c < 1.714 becausehis expected payoff from investing is greater than the cost. Conversely, the expectedpayoff of any agent who has signal θ2 is given by (2) and at most

2 × 0.8 × 0.4

0.8 × 0.4 + 0.6 × 0.2= 1.45.

3 To directly apply the results of Zhang would require that

Pr(ω|θ1, q, θ2, p)θ1=θ2 = Pr(ω|θ1, q, θ2, p)θ1 �=θ2 = Pr(ω|θ1, q).

Basically this requires that only the agent with the most informative signal has any information about whichinvestment option is profitable. A related idea was used by Scharfstein and Stein (1990). It seems, however,a somewhat un-intuitive and unsatisfactory solution in this instance.

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Thus, any agent with signal θ2 would not want to invest if c > 1.45 because hisexpected payoff from investing is less than the cost. Suppose, that the cost is revealedto be c = 1.6. This means that the actions of a second mover will reveal somethingabout his signal because anyone who invests must have had signal θ1.

To see the consequences of this suppose that all agents act immediately havingobserved the choice of the first mover, i.e. suppose that there is an information cascade.This would mean that all those with signal θ1 invest immediately and all those withsignal θ2 do not invest. This, however, provides an incentive for an agent with signalθ1 to delay his choice. For example, let n = 12 and consider an agent 2 who has signalθ1 of precision 0.7. If no agent invests (but they would have done had they had signalθ1) agent 2 can infer that 10 agents received signal θ2. His belief that option a1 isprofitable can be bounded above by

0.8 × 0.7 × 0.410

0.8 × 0.7 × 0.410 + 0.2 × 0.3 × 0.610 = 0.139

and his belief that option a2 is profitable can be bounded from below by

0.2 × 0.3 × 0.610

0.8 × 0.7 × 0.410 + 0.2 × 0.3 × 0.610 = 0.861.

Clearly, agent 2 would now want to invest in project a2 rather than a1 and has gained inexpected payoff by waiting. More generally, given the negligible cost of waiting, agent2 does best to not invest immediately in option a1 but delay a little and see what otheragents do. This contradicts the assumption that there will be an information cascade.

Knowing whether or not an information cascade will occur is important from awelfare perspective. Information cascades mean a loss of information, because privatesignals become ignored, and this loss of information makes it more likely agentswill choose the non-profitable project. It is informative, therefore, to know that aninformation cascade only occurs if the cost to invest takes extreme values. Note,however, that the description of an information cascade one obtains in the model ismuch less subtle than that obtained elsewhere in the literature (e.g. Chamley 2004b).The more important insight one can get from the model concerns equilibrium behaviorand the incentive to delay investment. To this issue we now turn.

4 Equilibrium behavior

If G(1, c) = 0 and so an information cascade must occur we can characterize (follow-ing a similar approach to Zhang) a particular type of pure strategy Nash equilibrium.This equilibrium provides an optimal amount of time that an agent should wait beforeinvesting. The time will depend on the precision of his signal where those with a lessprecise signal wait longer. Once one agent has invested an information cascade willoccur and so all others will either immediately invest or will never invest.4 Before

4 Recall that Zhang assumes that G(c) = 0.5c and so we derive an equilibrium for a ‘different’ G function.

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Strategic delay and information cascades

stating the equilibrium some notation will be useful. Let E[c] denote the expectedvalue of c and E[c|c ≤ 1] denote the expected value of c conditional on it being lessthan or equal to one. Recall that � is the cumulative distribution function over theprecision p. Let φ be the corresponding probability density function.

Proposition 2 Suppose that G(1, c) = 0 (and G(1) > 0, G(c) < 1). There exists asymmetric Nash equilibrium in which an agent with signal precision p invests at timeτ ∗(p), where

τ ∗(p) = n − 1

δ

p∫

p

(E[c] − G(1)E[c|c ≤ 1] − 2x(1 − G(c))) φ(x)

(2x − E[c])�(x)dx, (3)

if no other agent has yet invested. Once one agent has invested, all agents shouldimmediately invest in the same project as the first mover if c ≤ 1 , or never invest ifc ≥ c.5

Proof Trivially, the first mover should invest in project a1 if his signal is θ1 and a2

if his signal is a2 (provided 2pi ≥ E [c]). This, together, with Proposition 1 and itsproof, demonstrates the last sentence in the statement of the Proposition, provided thatthe first mover is the agent with the most precise signal. It remains, therefore, to detailwhen the first mover should invest and show that (3) is consistent with equilibrium. Todo this we begin by assuming the existence of a one-to-one function q(s) that detailsthe signal precision of an agent who invests at time s (if no agent has previouslyinvested).

Consider agent i with signal of precision pi . Agent i will invest as a first moverat time ti (if no agent has previously invested) where pi = q(ti ) or, equivalently,ti = q−1(pi ). If i invests as first mover then his expected payoff is

2pi − E[c]. (4)

If agent i is not the first mover then i will invest if and only if c ≤ 1. If, therefore, thefirst mover is known to have signal of precision q the expected payoff of player i is

G(1)

(2pi q

pi q + (1 − pi )(1 − q)− E [c|c ≤ 1]

)

if he has the same signal as the first mover, and

G(1)

(2 max {pi , q} (1 − min {pi , q})

pi (1 − q) + q(1 − pi )− E [c|c ≤ 1]

).

5 Given that τ∗(p) must be positive we require that

E[c] − G(1)E[c|c < 1] − 2x(1 − G(c))

2x − E[c] > 0

for all x ∈ [p, p]. This imposes additional conditions on permissable probability distributions over cost.

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E. Cartwright

if he has a different signal to the first mover. The probability that they have the samesignal is pi q + (1 − pi )(1 − q) and the probability they have a different signal ispi (1 −q)+q(1 − pi ). So the expected payoff of agent i if the first mover has a signalof precision q is

G(1) (2 max {pi , q} − E [c|c ≤ 1]) . (5)

Given q(s) one can construct a probability density function f (s|t) over the timingof the first investment, conditional on no agent having invested before time t (moreon this shortly). Using (4) and (5 ) we see that if no agent has invested by time t thenagent i’s expected payoff from waiting until some future time u > t can be written as

Ui (u|t) = e−δu(2pi − E[c])∞∫

u

f (s|t)ds

+u∫

t

e−δs [G(1) (2 max {pi , q(s)} − E [c|c < 1])] f (s|t)ds.

The first term reflects the possibility that no other agent invests before time u and soagent i invests according to his signal and has expected payoff 2pi − E[c]. The secondterm reflects the possibility that another agent may invest before time u. In this caseagent i will have expected payoff G(1) (2 max {pi , q} − E [c|c < 1]). DifferentiatingUi (u|t) with respect to u and setting u = t gives a necessary condition for equilibrium

dUi (u|t)du

∣∣∣∣u=t

= e−δt {[G(1) (2 max {pi , q(t)} − E [c|c < 1])

− (2pi − E[c])] f (t |t) − δ (2pi − E[c])}= 0. (6)

We now derive an expression for f (t |t). Let F(s) be the probability the first moverinvests before time s, for any s ≥ 0. Note that F(s) is given by the probability thatti = q−1(pi ) < s for some agent i . Or, equivalently, is equal to the probability thatpi > q (s) for some i . Thus,

F(s) = 1 − [�(q(s))]n−1.

Next note that

F(s|t) = F(s) − F(t)

1 − F(t)

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Strategic delay and information cascades

for any s ≥ t where F(s|t) is the probability the first mover invests before time s ifno one has invested before time t . Thus,

F(s|t) = [�(q(t))]n−1 − [�(q(s))]n−1

[�(q(t))]n−1

and

f (s|t) = −(n − 1)

(�(q(s))

� (q(t))

)n−2φ (q(s))

� (q(t))

dq(s)

ds.

We, therefore, obtain

f (t |t) = −(n − 1)φ(q(t))

�(q(t))

dq(t)

dt. (7)

It is natural that q ′(t) < 0 and so f (t |t) > 0 as required.Combining (6) and (7) and using a natural symmetry condition that pi = q(t) gives

δ (2pi − E[c])= − [E[c] − G(1)E [c|c < 1] − 2pi (1 − G(1))] (n − 1)

φ(q(t))

�(q(t))

dq(t)

dt.

Thus,

dq(t)

dt= − δ (2pi − E[c]) �(q(t))

[E[c] − G(1)E [c|c < 1] − 2pi (1 − G(c))] (n − 1)φ(q(t))

Inverting this one can obtain the optimal τ ∗(pi ) that an agent with signal precision pi

should invest. Setting τ ∗(p) = 0 then

τ ∗(pi ) =p∫

pi

[E [c] − G(1)E [c|c < 1] − 2x(1 − G(c))] (n − 1)φ(x)

δ (2x − E[c]) �(x)dx

as desired. ��

In interpreting Proposition 2 note that G(1)E[c|c ≤ 1] is the expected cost thatwill be paid if the agent is not the first investor (because he will pay 0 cost if c ≥ 1).Given that E[c] is the expected cost if he is the first investor, E[c] − G(1)E[c|c ≤ 1]is the expected gain from waiting. Term 2pi (1 − G(c)) measures the expected lossfrom waiting in that the agent will not invest if c ≥ c but with probability pi knewthe most profitable project. In equilibrium the expected gain and loss are equated. Toillustrate further we provide a second example.

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E. Cartwright

Example 2 Suppose that c is 0 with probability η ∈ (0, 1) and is 2 with probability1 − η. Then, E [c] = 2(1 − η), E [c|c < 1] = 0 and G(1) = G(c) = η. Further,suppose that φ(x) is the uniform distribution over interval [0.5, p], so,

φ(x) = 2

2p − 1

and

�(x) = 2x − 1

2p − 1

for x ∈ [0.5, p]. Then

τ ∗(p) = n − 1

δ

p∫

p

2(1 − x)(1 − η)

(2x − 1) (x − 1 + η)dx . (8)

This expression will be positive if η ≥ 0.5.In order to simplify the expression for τ ∗(p), let A := (2x − 1) (x − 1 + η) =

2x2 − x (3 − 2η) + 1 − η . It is convenient to write

τ ∗ (p) = (n − 1) (1 − η)

δ

⎡

⎢⎣

1

2

p∫

p

4 − 3 + 2η

Adx − 1

2

p∫

p

4x − 3 + 2η

Adx

⎤

⎥⎦ .

Then6

1

2

p∫

p

4 − 3 + 2η

Adx = 1 + 2η

2(2η − 1)

[ln

(2p − 1

2(p + η − 1)

)− ln

(2p − 1

2(p + η − 1)

)].

Also,

p∫

p

4x−3 + 2η

Adx = ln

∣∣∣2p2− p (3−2η) + 1−η

∣∣∣−ln

∣∣∣2p2− p (3−2η) + 1−η

∣∣∣ .

6 Using

∫dx

ax2 + bx + e= 1

2√

b2 − aeln

∣∣∣∣∣ax + b −

√b2 − ae

ax + b +√

b2 − ae

∣∣∣∣∣

if b2 > ae.

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Strategic delay and information cascades

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

100

200

300

400

500

600

p

time

pbar = 1, eta = 0.75pbar = 1, eta = 0.6pbar = 1, eta = 0.9pbar = 0.8, eta = 0.75

Fig. 1 The value of τ∗ (p) for different values of p and η

This allows us to solve explicitly for τ ∗(p). For instance, suppose that η = 0.75 andp = 1. Then

τ ∗ (p) = (n − 1)

8δ

[5 ln

(2

3

)+ 5 ln

(4p − 2

4p − 1

)− ln

(3

4

)+ ln

(2p2 − 3

2p + 1

4

)].

Figure 1 illustrates τ ∗ (p) for different combinations of η and p. Note that n and δ

are merely scale factors in τ ∗ (p). An increase in the number of players or a decreasein the discount rate increases the equilibrium waiting time. We see in Fig. 1 that anincrease in η decreases the equilibrium waiting time. This is intuitive, because anincrease in η means that the cost of the project is more likely to be zero and so thereis less reason to delay. A decrease in p also leads to a like for like decrease in theequilibrium waiting time.

5 Conclusion

We have considered a setting, originally analyzed by Zhang (1997), in which agentscan invest in either of two projects and have differing precision of information about themost profitable project. The analysis of Zhang implicitly assumed that information orinvestment cascades must occur, but we have shown they will only occur if the cost ofinvesting takes extreme values. We have derived the equilibrium for this case and findthat it is qualitatively similar to that proposed by Zhang. In particular, agents shoulddelay investing by an amount that depends on the precision of their information.

It is interesting to conjecture what equilibrium behavior will be when informationcascades are not guaranteed to occur. That is, what happens when the cost of investing

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E. Cartwright

can take intermediate values. Intuitively one could obtain quite intricate equilibriumbehavior. For example, suppose an agent invests in a project A. Those agents whosesignal points towards project A will be inclined to invest, but those whose signal pointstowards project B will be inclined to delay. Suppose, therefore, all those who had arelatively precise signal of project A invest immediately while others wait. If manyinvest then an information cascade looks inevitable. Suppose, however, no-one invests.This signals that no-one, other than the first investor, had a relatively precise signal ofproject A. Plausibly we could then obtain a process of delay in which the next investorwill be an agent with a relatively precise signal of B or a less precise signal of A.Interesting though this issue is, addressing it leads to very different questions to thoseposed in this paper, and so we leave it for future work.

References

Bikhchandani S, Hirshleifer D, Welch I (1998) Learning from the behavior of others: conformity fads, andinformational cascades. J Econ Perspect 12:151–170

Chamley CP (2004a) Rational herds: economic models of social learning. Cambridge University Press,Cambridge

Chamley CP (2004b) Delays and equilibria with large and small information in social learning. Eur EconRev 48:477–501

Frisell L (2003) On the interplay of informational spillovers and payoff externalities. RAND J Econ 34:582–592

Gale D (1996) What have we learned from social learning? Eur Econ Rev 40:617–628Scharfstein DS, Stein JC (1990) Herd behavior and investment. Am Econ Rev 80:465–479Zhang J (1997) Strategic delay and the onset of investment cascades. RAND J Econ 28:188–205

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