We extend the Bikhchandani, Hirshleifer and Welch (1992) informational cascade framework to allow for asymmetric signal accuracy. Simulations demonstrate that even small departures from symmetry may lead to non-monotonic effects of signal accuracy on the likelihood of an inefficient cascade.Learning, Herding, Signal Precision

Ivan Pastine

Tuvana Pastine

English

Research Papers in Economics

 
 
 
 
 
 
UCD CENTRE  FOR  ECONOMIC  RESEARCH 
 
 
 
WORKING  PAPER  SERIES  
 
 2006 
 
 
  Signal Accuracy 
and 
Informational Cascades  
                    
                      Ivan Pastine, University College Dublin and  
              Tuvana Pastine, National University of Ireland Maynooth 
 
 
WP06/20 
 
November 2006 
 
 
 
 
 
 
UCD SCHOOL OF ECONOMICS 
UNIVERSITY COLLEGE DUBLIN 
BELFIELD  DUBLIN  4 
 
 
 
 
 
 
 
 
 Signal Accuracy
 and
Informational Cascades
July 24, 2006
Ivan Pastine
University College Dublin
and
CEPR
Tuvana Pastine
National University of Ireland Maynooth
and
CEPR
Abstract
We extend the Bikhchandani, Hirshleifer and Welch (1992) informational cascade
framework to allow for asymmetric signal accuracy. Simulations demonstrate that even
small departures from symmetry may lead to non-monotonic effects of signal accuracy
on the likelihood of an inefficient cascade.
JEL: D80
Keywords: Learning, Herding, Signal Precision 
Corresponding Author: 
Ivan Pastine, School of Economics, University College Dublin, Belfield Dublin 4,
Ireland.  Email: Ivan.Pastine@ucd.ie- 1 -
I. Introduction
Banerjee (1992) and Bikhchandani, Hirshleifer and Welch (1992), henceforth BHW,
show that it may be optimal for a rational agent to herd, i.e. to follow the actions of his
predecessors when his own private signal suggests the opposite. If early movers’ signals
are incorrect, followers will be mislead, yielding an inefficient informational cascade.
This paper studies the effect of signal accuracy on the probability of an inefficient
cascade. 
Analysis of factors that affect the likelihood of an inefficient cascade may be of
interest in helping to reduce the probability of such events. For instance, the 1933
Securities Act and 1934 Securities Exchange Act were enacted to prevent crashes like
Black Thursday in 1929. They require reporting of financial information concerning
traded securities to improve signal accuracy. Depending on the market, signal accuracy
may be affected by factors such as accounting standards, technological advancement in
information dissemination and advertising.
In BHW, signals have symmetric accuracy in both states. An increase in signal
accuracy leads to a decrease in the probability of inefficient herding since early movers
are more likely to take the correct action. We extend the BHW framework to allow for
asymmetric signal accuracy.  The signal need not have the same accuracy in high and
low states. For instance, a good candidate may come to a job interview on time with 95
% probability and a bad candidate may be on time with 85% probability. As long as the
probabilities are different, promptness may be a useful signal of candidate quality. The
symmetric case restricts the probability of the bad candidate sending the correct signal
(being late) to 95%. We show that even small departures from symmetry may lead to- 2 -
non-monotonic results. An increase in signal accuracy may result in a higher likelihood
of an inefficient cascade.
II. Symmetric Accuracy
The following is equivalent to the BHW framework: Each risk-neutral agent chooses
between two investment projects. The risky project yields either 1 or 0 and the safe
project yields ½. The payoff matrix is:
      Table 1
Risky Safe 
High 1  ½ 
Low 0   ½ 
Prob(High State)=0.5
Each agent receives a private, conditionally independent signal about the value of the
risky project, either h or R. The signal is correct with probability p. The sequence of
moves is predetermined and agents observe the actions of those ahead of them. Agents
follow Bayes’ Rule. When indifferent an agent randomizes with even probabilities.
Equivalently, there are two urns; H and L. Each urn has some balls marked h and
some marked R. Urn L has a higher percentage of balls marked R than urn H. The
percentile of correct balls in each urn, ph and pR, is symmetric. Nature draws one urn with
even probabilities. All agents privately draw one ball with replacement from the same
urn. The agent’s problem is to determine which  urn the ball comes from.
BHW show that at some point public information overwhelms the informational
content of a single signal. If most early agents happened to receive signal R, all
newcomers may choose L even when the state is H. An L cascade when the true state is
H is called an inefficient negative cascade. An H cascade when the true state is L is an
inefficient positive cascade. The inefficient cascade probability is given by the
probability of these weighted by the ex ante probabilities of states H and L. 1When signal accuracy is symmetric the recursive nature of the framework allows
closed-form solutions for the probabilities. When we depart from symmetry the recursive
nature breaks down. All simulations use 10 million runs per data point. In all cases 99%
confidence intervals are less than the width of the symbols used to represent data points.
- 3 -
Figure 1
Figure 1 summarizes our replication of BHW’s results.
1 An increase in signal
accuracy always leads to a decrease in the probability of inefficient herding.2See Anderson and Holt (1997) for experiments using asymmetric accuracies.
- 4 -
III. Asymmetric Accuracy
The BHW framework is symmetric because the signal has the same accuracy in both
states and the probability of each state is even. However even small departures from
symmetry of signal accuracy across  states may lead to violation of the monotonicity
result.
BHW’s Result 1 is closely related. It shows that, in a symmetric setting, all agents
after the second are better off when the first agent’s signal accuracy (expertise) is slightly
decreased. This results in more information  for later individuals. Here all agents have
the same expertise but right around the point of symmetry of signal accuracy across
states, we will have a similar story. Building on the intuition gained from this we analyze
cases where the probability of inefficient cascades is non-monotonic in signal accuracy.
 Asymmetric signal accuracy translates into asymmetric percentile of correct balls
in each urn.
2  For the signals to be informative pR…1-ph. Without loss of generality take
pR+ph>1.  Increasing either pR or ph increases the informativeness  of the signal in the
Blackwell sense.
Figure 2 reports simulation results right around the point of symmetry for signal
accuracy in state H fixed at 70% and varying the accuracy in state L. Fixing ph at
different levels does not change the spirit of the results.  The payoff matrix and
probability of state H  are as in Table 1.- 5 -
Figure 2
 The probability of an inefficient cascade is monotonic. However, the probability
of an inefficient positive cascade (“Prob. H when L”) is not. It decreases with an
improvement in signal accuracy until the point of symmetry. Then it jumps from 0.12 to
0.38. The probability of an inefficient negative cascade jumps down.
When ph and pR are symmetric, if the first and second signals are different the
second simply cancels the first since they have equal accuracy. When there is asymmetry,
however slight, signal h and signal R do not cancel each other out because they have
different weights in the updating process. Therefore herding can start earlier. 3An agent receiving signal h updates his belief that the state is high from 0.5 to
Prob(H|h)=ph/(1+ph-pR). An agent receiving signal R updates his belief that the state is low
from 0.5 to Prob(L|R)= pR/(1+pR-ph). As long as pR<ph, Prob(L|R)> Prob(H|h). 
- 6 -
 • When pR is just below ph, the second agent always herds when the first agent
chooses L.
3 Hence the probability of an inefficient L cascade is high. 
 • When pR is just above ph, the second agent always herds when the first agent
chooses H. Hence the probability of an inefficient H cascade is high. 
And at the point of symmetry the inefficient negative and positive cascade probabilities
are equal. Therefore the inefficient positive cascade probability jumps up and the
inefficient negative cascade  probability jumps down.
While our primary purpose is to analyze the probability of an inefficient cascade,
in many markets a planner may place greater weight on inefficient negative cascades than
on inefficient positive cascades due to externalities from the market to society at large;
Bank panics, capital flight and market crashes may have drastic external consequences.
In the IPO market, however, companies may simply try to increase the probability of a
positive cascade.
3.2. Inefficient Cascade Probability
The inefficient cascade probability may be non-monotonic in signal accuracy with
uneven ex ante probabilities.  Here is a payoff matrix with a riskier project but the same
expected value:
Table 2
Risky       Safe 
High 2   1/2
Low 0   1/2
Prob(High State)= 0.25- 7 -
Figure 3
ph=pl yields the same monotonicity result as in BHW. Now fix the signal accuracy in
state H, but vary the signal accuracy in state L. Figure 3 summarizes the simulation
results for ph=0.7. Fixing ph at other levels does not change the qualitative results. The
probability of an inefficient cascade is non-monotonic in signal accuracy. It jumps up at
three levels: At 0.505, at 0.7 (the point of symmetry), and at 0.9275.
The jumps are due to the binary nature of the problem. The agent decides whether
to follow his own signal or not. As signal accuracy improves  the expected value of each
of these options changes continuously, but the agent’s decision switches between them
in a discrete jump.- 8 -
The point of symmetry has the same incentives as before.  However, since the ex
ante probability of L is now 0.75, the positive cascade probability has a higher weight
in the ex ante inefficient cascade probability.
   Examine the jump at 0.505. When pR=0.5, just below 0.505, consider the sequence
of actions: H,L,H,H. At this level of signal accuracy these actions reflect the private
signals. Having observed this sequence, it is optimal for an agent to follow his own
signal. But when  pR =0.51, just above 0.505, having observed the same sequence (at this
level of accuracy the actions still reflect the private signals) it is optimal for the agent to
herd to H. Therefore, just past pR=0.505 the probability of an inefficient H cascade jumps
up. And the probability of an inefficient L cascade jumps down. The weighted average,
the probability of an inefficient cascade, jumps up from 0.32 to 0.335. 
Many alternative sequences of signals could occur before herding starts. The
discontinuities in the probabilities arise at points where small changes in parameters
switch agents in some sequence from one action to the other. The size of the
discontinuity is related to the likelihood of that sequence.    
3.3. Changing Both Signal Accuracies
This exercise is in the same spirit as in BHW where a single parameter represents the
signal accuracy in both states. Figure 4 gives results for payoff Table 2, starting with
asymmetric signal accuracies ph=0.5 and pR = 0.8. Then both accuracies are changed
together. The effect of changes in signal accuracies on the probability of an inefficient
cascade is non-monotonic.4See Hung and Plott (2001). 
- 9 -
Figure 4
IV. An Example
Consider an example from the labor market. The safe alternative is to hire an adjunct
professor with payoff ½. The risky alternative is to hire a tenure-track professor with
payoff of either 2 or 0 (Table 2). The candidate for the tenure-track position presents
himself in private office meetings to each hiring committee member. Committee
members then vote sequentially.
4- 10 -
A good candidate has a higher probability of successfully presenting himself than
a bad candidate. Now imagine that schools stop training bad candidates for  presentation
skills. This leads to a decline in the probability of bad candidates successfully presenting
themselves ( signal accuracy in the Low state increases).
There are two forces at work. Observing a good presentation by a bad candidate
is now less likely. But if the first voter happens to have seen a good presentation, herding
may start early since all followers would put more weight on that good report. This tends
to increase in the probability of hiring a bad candidate. The second effect may
overwhelm the first depending on the initial levels of signal accuracy (Figure 3).- 11 -
References
Anderson, L. R. and C. A. Holt, 1997, “Information Cascades in the Laboratory”,
American Economic Review, December, 847-862.
Banerjee, V. A., 1992, “A Simple Model of Herd Behavior,” Quarterly Journal of
Economics, 107:3, 797-818.
Bikhchandani, S., D. Hirshleifer and I. Welch, 1992, “A Theory of Fads, Fashion,
Custom, and Cultural Change as Informational Cascades,” Journal of Political
Economy,100:5, 992-1026.
Hung, A. and C. R. Plott, 2001, Information Cascades: Replication and Extension to
Majority Rule and Conformity Rewarding Institutions,” American Economic
Review, 91,1508-1520.

Information Cascades: Replication and Extension to Majority Rule and Conformity Rewarding Institutions,”

Signal Accuracy and Informational Cascades

http://www.ucd.ie/economics/research/papers/2006/WP06.20.pdf

Signal Accuracy and Informational Cascades

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics