In his "Simple model of herd behaviour", Banerjee (1992) shows that - in a sequential game - if the first two players have chosen the same action, player 3 and all subsequent players will ignore his/her own information and start a herd, an irreversible one. In this paper we analyse the role played by the tie-breaking assumptions in reaching the equilibrium. We showed that: players' strategies are parameter dependent; an incorrect herd could be reversed; a correct herd is irreversible.Herd behaviour; Run

Andrea Morone

Eleni Samanidou

English

Research Papers in Economics

Università degli Studi di Bari
Dipartimento di Scienze Economiche e Metodi
Matematici
Southern Europe Research in Economic Studies
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WORKING PAPER NO.
Andrea Morone and Eleni Samanidou
A simple note on Herd Behaviour
0013  1
 
 
 
A simple note on Herd Behaviour
*
 
 
 
 
 
Andrea Morone  Eleni Samanidou 
University of Bari 
a.morone@gmail.com 
University of Kiel  
samanidou@gmx.de 
 
 
Abstract 
 
In his ‘Simple model of herd behaviour’, Banerjee (1992) shows that – in a sequential game – if the 
first two players have chosen the same action, player 3 and all subsequent players will ignore 
his/her own information and start a herd, an irreversible one. In this paper we analyse the role 
played by the tie-breaking assumptions in reaching the equilibrium. We showed that: players' 
strategies are parameter dependent; an incorrect herd could be reversed; a correct herd is 
irreversible. 
 
Keywords: Herd behaviour, Run. 
JEL classification: D8 
                                                 
* An earlier version of this paper was presented at the XXIV Annual Colloquium International 
Association of Research in Economics and Psychology (IAREP) “Inquiries into the Nature and 
Causes of Behaviour”, June 30 – July 3, 1999 Belgirate, Italy. 
   2
 
1. Introduction 
In the last decade, studies on ‘herding’ were abundant. Herd behaviour refers to the 
phenomenon according to which people follow the example of other people ignoring their private 
information. This kind of behaviour was first pointed out by Becker (1991) and was first formally 
analysed by Banerjee (1992) and Bikhchandeni et al. (1992). 
Studying herd behaviour could be useful to explain countless social and economic issues. In the 
real world (usually) people make their decision sequentially. In the process of decision making, 
agents observe the decisions taken by previous agents, and may be influenced by that information.  
The basic idea of herd behaviour is very simple: ignoring private information and joining the 
queue. The assumption that agents possess rational expectations is usually being used to assert that 
an agent, although not familiar with the true model, could reach an efficient outcome drawing on all 
available information including the one acquired observing other people’s actions. 
Banerjee in his analysis tries to demonstrate the hypothesis of ‘herd behaviour’: everyone is 
doing what everyone else is doing, even when his or her private information suggests acting 
differently. The model is a sequential decision-making game with one wining action and three 
possible information states of the subjects: correct, wrong or no information. Banerjee shows that as 
soon as the first two players choose the same action, all subsequent players will follow them, 
independently of their private information. Such an irreversible queue (“lock-in”) – caused by the 
individual rational behaviour to gain a better information position by looking at what other agents 
are doing – may result in a Pareto non-optimal equilibrium since the imitated action may be a non-
winning one.  
Banerjee’s model has two important features: it has both a very simple and intuitive structure 
and it gives very strong results. 
Subjects are rational, and sometimes they are indifferent among several possible actions to 
break this an pass a set of tie-breaking rules is needed, indicating what he/she does in any occasion 
in which he/she is indifferent between two or more actions. Using Banerjee’s words, “each of these 
assumptions is made to minimize the possibility of herding” (Banerjee, 1992: 803) . 
Assumption A. Whenever a decision-maker has no signal and everyone else has chosen zero, 
he/she always chooses zero. 
Assumption B. When decision-makers are indifferent between following their own signal or 
someone else’s choice, they always follow their own signal.   3
Assumption C. When a decision-maker is indifferent between two or more of the choices 
made by the previous decision-makers, he/she chooses to follow the one which has the highest 
value. 
It can easily be seen that the rules chosen by Banerjee are not the only plausible ones. In what 
follows, we will therefore examine if there is any critical relation between the tie-breaking rules and 
the overall dynamics of the model. In particular, we will show that by choosing an alternative 
decision rule in case that the first player has no signal, the possibility of an irreversible queue 
starting from the first player is generally smaller. 
Developing a model of herd behaviour which draws on Banerjee’s model has some pros and 
cons. On the one hand, herding models à la Banerjee are more general compared to Informational 
cascades one (see Bikhchandeni et al. 1992). For this reason a generalization of it can produce a 
wide-ranging conclusions
1. On the other hand, even a minor modification of Banerjee’s model 
would produce a more complex analytical solution.  
The main goal of this paper is to present some possible extensions to Banerjee’s model, which 
will explore the robustness of the original results, as well as the presence of analytical complexities 
in the extended models. 
 
2. “A simple model of herd behaviour” under different assumption sets 
Let A = [0, 1] ∈ R be the set of all possible investments, where only a* ∈ A pays a positive 
pay-off. Let S = [0, 1] ∈ R be the set of all possible signals, where only s* ∈ S signals to invests in 
a*. The aim of the game is to invest in a*. The pay-off is one when action a* is chosen and zero 
otherwise. There is a population of N players who take their decision sequentially and in a fixed 
order. Each player knows the choices made by those before him/her but is not aware of the 
information on which these choices were based. Let indicate with α the probability of receiving a 
signal and with β the probability that the signal is correct. 
The decision-making process is based on the following three tie-breaking rules (we will follow 
the notation adopted by Banerjee and refer to these rules as assumptions): 
Assumption A1. Whenever the first decision-maker has no signal, he/she chooses randomly an 
action from the set of all possible actions. 
Assumption B1. When decision-makers are indifferent between following their own signal or 
someone else’s choice, he/she chooses randomly an action from the set of all possible actions.   4
Assumption C1. When a decision-maker is indifferent between two or more of the choices 
made by the previous decision-makers, she/he chooses randomly among the actions that leave 
his/her utility unchanged. 
 
2.1 Assumption A vs. Assumption A1 
We can demonstrate
2, however, that the dynamics of the model may change if we replace 
Assumption A with Assumption A1. 
The expected pay-off for the Subject is zero under both assumptions. Under Assumption A, 
however, an action equal to zero is a clear signal for the subsequent players that the decision-maker 
did not receive a signal. Thus, the first player in the sequence who chooses an action different from 
zero must have received a signal. In the case that the subsequent player does not receive a signal 
he/she will follow the choice of his/her predecessor. Since two identical actions are more likely to 
be correct than incorrect, the next player will follow the queue even if he/she receives a signal 
suggesting acting differently, and consequently, all the next players will follow this action. Under 
Assumption A1 however, the first player will choose an action different from zero in case he/she 
has not received a signal. Thus, a queue starting from the first player is less informative for the next 
decision-makers than under Assumption A.  
 
2.2 Assumption B vs. Assumption B1 
Note that replacing Assumption B with assumption B1 does not affect the results of the game, 
in fact whatever subject i will play a herd will start. 
 
2.3 Assumption C vs. Assumption C1 
Note that, compared to Assumption C, the information that can be deduced out of an observed 
sequence of actions would now be generally less precise. Under Assumption C, two non-consequent 
actions, which are not the option with the highest value, have to be based on a correct signal (since 
the probability that two wrong signals match is zero, and since in the case of no-signal the option 
with the highest value is chosen). However, under Assumption C1, two non-consequent actions may 
be either both correct or both incorrect. Two scenarios are plausible: in the first scenario both 
players receive the same signal, which means that it is the wining one and, thus, the action is 
                                                                                                                                                                   
1 For an application of Bikhchandeni et al.  (1992) model in a market contest see Hey and Morone (2004), and Morone 
(2005).  
2 For a proof see Morone (1999), and Morone and Samanidou (2006)   5
correct; in the second scenario only the first among the two players receive a signal while the 
second chooses, randomly, to follow the action of the first player (in this case the action chosen will 
be correct only if the signal of the first player was the correct one). 
Though under Assumption C1 the sequence of actions observed by the decision-makers may be 
less informative than under Assumption C, the overall decision of the model is not affected. Note 
that under Assumption C1 the probability that two non-consequent identical actions are correct is 
smaller than under Assumption C, nonetheless, it is still higher than the probability that two non-
consequent identical actions are wrong. Thus, the subsequent agents who do not receive a signal or 
do receive a signal which, does not match with the options chosen by previous players, will decide 
to follow (as they would have done anyway under Assumption C).  
 
3.  The simulation 
For different combinations of α  and β  we simulated 1000 runs with 100 agents per run for Set 
1 (i.e. the original model of Banerjee) and Set 2 (i.e. the revised model under Assumption A1). The 
simulations were run in GAUSS. For each agent we drew three numbers from a uniform random 
distribution between 0 and 1. If the first number drawn was less than or equal to α , we assumed 
that the player had received a signal. This signal was the correct one if the second number drawn 
was less than or equal to β , and wrong otherwise. Finally, the third number drawn represented the 
wrong signal. We used the same random numbers for Set 1 and Set 2, so that different results may 
emerge solely due to the variation of Assumption A from Assumption A1.  
 
4. Results 
Figure 1 shows the probability of herding for different α  and β  in both sets. More precisely, 
it illustrates the proportion of ‘herd decisions’ out of the total number of decisions taken (i.e. 
100*1000 decisions per each combination of α  and β ). ‘Herd decisions’ are those taken in the 
case where the player did get a signal but chose to ignore it. Even though for each combination the 
probability of herding is nearly the same for Set 1 and Set 2, the probability in the case of Set 2 will 
always be smaller.  
In Figure 2 and 3, on the other hand, we report the probability of observing irreversible runs. In 
other words, the proportion of irreversible runs out of the total number of 1000 runs in each 
combination of the parameters. In the case of Set 1, a run is irreversible and correct if at least two 
subjects played the correct action. A run is irreversible and incorrect if at least two subjects played   6
the same incorrect action and no correct action was played before. In the case of Set 2, a run is 
irreversible and correct in two cases: first, when the correct action is played by all the first players 
and the length of the critical queue is reached; secondly, when at least two subjects played the 
correct action, which has been played at least once before the length of the critical queue was 
reached. Conversely, a run is irreversible and incorrect in two cases: first, when all the players 
choose the same incorrect action and the length of the critical queue is reached; secondly, when at 
least two subjects played the same incorrect action and no correct action has occurred before. Table 
1 shows that under Assumption A1 the probability of a correct irreversible run is always slightly 
higher than under Assumption A. The probability of an incorrect irreversible run will therefore be 
always smaller under Assumption A1 than under Assumption A. It is worth noting that, by using 
100 agents, we reduced the probability of observing a reversible run under both assumptions to 
nearly zero, for all the parameter sets we considered. Under Assumption A this result is quite 
intuitive: as soon as two agents play the same action, an irreversible run will start. Under 
Assumption A1, on the other hand, this scenario is delayed by one lag. This delay allows for the 
aggregation of some additional information, so discovering the correct action is more likely.  
Herd decisions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
α=0.25,
β=0.25
α=0.5,
β=0.25
α=0.75,
β=0.25
α=0.25,
β=0.5
α=0.5,
β=0.5
α=0.75,
β=0.5
α=0.25,
β=0.75
α=0.5,
β=0.75
α=0.75,
β=0.75
F
r
e
q
u
e
n
c
i
e
s
Set 1
Set 2
Irreversible correct herd
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α=0.25,
β=0.25
α=0.5,
β=0.25
α=0.75,
β=0.25
α=0.25,
β=0.5
α=0.5,
β=0.5
α=0.75,
β=0.5
α=0.25,
β=0.75
α=0.5,
β=0.75
α=0.75,
β=0.75
F
r
e
q
u
e
n
c
i
e
s
Set 1
Set 2
 
    F i g u r e   1         F i g u r e   2  
Irreversible incorrect herd
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
α=0.25,
β=0.25
α=0.5,
β=0.25
α=0.75,
β=0.25
α=0.25,
β=0.5
α=0.5,
β=0.5
α=0.75,
β=0.5
α=0.25,
β=0.75
α=0.5,
β=0.75
α=0.75,
β=0.75
F
r
e
q
u
e
n
c
i
e
s
Set 1
Set 2
Correct action
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α=0.25,
β=0.25
α=0.5,
β=0.25
α=0.75,
β=0.25
α=0.25,
β=0.5
α=0.5,
β=0.5
α=0.75,
β=0.5
α=0.25,
β=0.75
α=0.5,
β=0.75
α=0.75,
β=0.75
F
r
e
q
u
e
n
c
i
e
s
Set 1
Set 2
 
    F i g u r e   3         F i g u r e   4  
It is evident that irreversible herding entails some very undesirable features. In fact, when an 
irreversible queue starts, agents are trapped in it and cannot do anything else but following. For this 
reason, it would be appropriate to study the composition of irreversible herding. As can be seen in   7
Figure 4, the proportion of correct actions is generally bigger under Assumption A1 (Set 2) than 
under Assumption A (Set 1). Set 2 is therefore more efficient than Set 1, or in other words, Set 1 is 
Pareto-dominated by Set 2.  
Incorrect signals/Correct actions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
α=0.25,
β=0.25
α=0.5,
β=0.25
α=0.75,
β=0.25
α=0.25,
β=0.5
α=0.5,
β=0.5
α=0.75,
β=0.5
α=0.25,
β=0.75
α=0.5,
β=0.75
α=0.75,
β=0.75
F
r
e
q
u
e
n
c
i
e
s
Set 1
Set 2
Correct signals/Incorrect actions
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
α=0.25,
β=0.25
α=0.5,
β=0.25
α=0.75,
β=0.25
α=0.25,
β=0.5
α=0.5,
β=0.5
α=0.75,
β=0.5
α=0.25,
β=0.75
α=0.5,
β=0.75
α=0.75,
β=0.75
F
r
e
q
u
e
n
c
i
e
s
Set 1
Set 2
 
    F i g u r e   5         F i g u r e   6  
Figure 5 and 6 demonstrates that the ratio of incorrect signals to correct actions is always 
higher in Set 2 than in Set 1, while the ratio of correct signals to incorrect actions is always lower in 
Set 2. In consequence, the proportion of incorrect signals being ‘wasted’ in favour of the correct 
action is bigger in Set 2, whereas the proportion of correct signals being ‘‘wasted’ in favour of an 
incorrect action is smaller in Set 2.  
 
5. Conclusion 
In this work we extended Banerjee’s model of herd behaviour replacing one of its fundamental 
and ‘innocuous’ assumptions. More precisely, we replaced Assumption A (whenever a decision-
maker has no signal and everyone else has chosen zero, he/she will always choose zero), with 
Assumption A1 (whenever a decision-maker has no signal and everyone else has chosen zero, 
he/she will always choose randomly among all possible actions). 
The consequence of this slight change in the assumptions’ set leads to an important alteration 
in the players’ strategy. On the one hand, in Banerjee’s model two identical actions are enough to 
generate an irreversible queue. This fact results in a loss of information with the consequence of a 
non optimal aggregated result. On the other hand, in the modified model players’ strategies are 
parameter dependent: private information will not be systematically ignored in the presence of a 
queue. Our results are extremely important if compared to various Bikhchandeni’s generalizations. 
Since we can conclude that changes in the apparently innocuous tie-breaking rules produces a 
change in agents strategy. 
To corroborate our findings, we developed a simulation programme able to replicate the 
subjects’ behaviour under both assumption sets. Although the simulation results showed rather 
small differences between the two models (e.g. the probability of herding differed by   8
approximately 5%), the main result of our study is given by the fundamental change in the agents’ 
strategies. Furthermore, we believe that another important improvement that our model has offered 
to Banerjee’s findings is the introduction of the possibility of agents to decide not to join the crowd 
at an early stage of the cascade. 
 
 
References 
Becker, G. S., (1991). A Note on Restaurant Pricing and Other Examples of Social Influences on 
Price. The Journal of Political Economy, Vol. 99, No. 5., pp. 1109-1116.  
Banerjee, A. (1992). A Simple Model of Herd Behaviour. Quarterly Journal of Economics, 
CVII:797-817. 
Bikhchandeni S., Hirshleifer, D., and Welch, I. (1992). A Theory of Fads, Fashion, Custom and 
Cultural Change as Informational Cascades. Journal of Political Economy, 100:992-1026 
Fiore, A., and Morone, A., (2005). Is playing alone in the darkness sufficient to prevent 
informational cascades?  Discussion Papers on Strategic Interaction from Max Planck 
Institute of Economics, Strategic Interaction Group 2005-09 
Hey, D. H., and Morone, A., (2004). Do Markets Drive Out Lemmings—or Vice Versa? 
Economica Vol. 71 Issue 284 
Morone A. (2000) To herd or not to herd, that is the question?" Economics and Complexity Vol. 2 
N. 3 Wimter 1999-2000 pp. 17-30. 
Morone A. (2005) Financial Market in the Laboratory, an Experimental Analysis of some Stylized 
Facts, Discussion Papers on Strategic Interaction from Max Planck Institute of Economics, 
Strategic Interaction Group 2005-27 
 
 

A Note on Restaurant Pricing and Other Examples of Social Influences on Price.

A Simple Model of Herd Behaviour.

A Theory of Fads, Fashion, Custom and Cultural Change as Informational Cascades.

Financial Market in the Laboratory, an Experimental Analysis of some Stylized Facts,

A simple note on Herd Behaviour

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A simple note on Herd Behaviour

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