Mathematical Social Sciences 40 2000 197–213 www.elsevier.nl locate econbase
Imitation and the dynamics of norms
John Conlisk , Jyh-Chyi Gong, Ching H. Tong
Department of Economics , University of California-San Diego UCSD, 9500 Gilman Drive, La Jolla,
CA 92093-0508, USA
Received August 1998; received in revised form August 1999; accepted October 1999
Abstract
In the model, each person in a large population has a probability of adhering to some behavior, such as a social norm. At random, people observe each other and adjust their adherence
probabilities in an imitative direction. In a main case of the model, there are high and low stable equilibria; the distribution of adherence probabilities evolves upward or downward, depending on
initial conditions, until the population is in strong conformity with the norm or in strong rejection of the norm. In other cases of the model, there is a single stable equilibrium. Equilibria may be
fragile. Small differences in initial conditions may lead otherwise similar populations to opposite equilibria, and social shocks may tip a population from one equilibrium to another or change the
number of equilibria.
2000 Elsevier Science B.V. All rights reserved.
Keywords : Social norm; Equilibrium patterns; Imitative influences; Tipping models
1. Introduction
People often make choices which are socially beneficial even though at odds with narrow self interest – stopping to help a stranger, refraining from undetectable cheating,
voting despite the negligible probability of making a difference, rewinding a rented videotape, and so on. The list is long. We will call these ‘norms.’ Of course, people often
do the opposite – reject social benefit in favor of narrow self interest. At first glance, individualistic models of choice seem adequate for both outcomes. Standard utility
models cover narrow self interest, and minor modifications allow altruism. However, the issue is more complicated. Whole populations often cluster at or near one of the two
extremes – the socially beneficial choice or its opposite – as if everyone were pushed by
Corresponding author. Tel.: 11-858-534-3832; fax: 11-858-534-7040. E-mail address
: jconliskucsd.edu J. Conlisk 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 4 9 - 9
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. Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
a collective force. More striking, the population may, for no clear reason, tip from one extreme to the other. In one community, people may routinely turn in lost objects;
whereas, in a seemingly similar community not far away, finders–keepers may be the rule. In time, one or both communities might unpredictably tip to the opposite extreme.
Strong interdependence among individual choices is suggested. We study imitative dynamics which can lead to these patterns. Separated from the norm interpretation, the
model is about the stochastic binary choices of individuals in a population, and about how their choice probabilities evolve through imitation. The binary choice need not be
acceptance or rejection of a norm, but rather of an alternative technology, a novel idea or product, a political belief, a smoking habit, a fad or fashion, and so on.
Relative to other ‘tipping’ models in the literature, the model here is distinctive in that each person has only a probability of adhering to the norm, rather than an all-or-nothing
commitment. Among the related models are diffusion models Bartholomew, 1982, threshold models Schelling, 1978; pp. 102–110; Granovetter, 1978; Granovetter and
Soong, 1986, consumer choice models Smallwood and Conlisk, 1979, technical choice models Arthur, 1988, 1989, cascade models Banerjee, 1992; Bikhchandani et
al., 1992, game models Young, 1993, 1996; Weibull, 1995; Sections 4.4 and 5.3, Bicchieri et al., 1997; Bowles and Gintis, 1998; Bowles, 1998b, and various other
tipping models Kirman, 1993; Ellison and Fudenberg, 1995; Brock and Durlauf, 1995. Elster 1989a,b and Bowles 1998a discuss norms and related issues from a broader
perspective.
2. The model
Assumptions are intended to be as spare as possible. Norm terminology is used, though there are other interpretations.
1. Setting. Time is discrete labelled t 5 1, 2, . . . . There is an infinite population a continuum. Each person in each period chooses whether or not to adhere to a social
norm. 2. Stochastic choice. Each person in each period has a probability of adhering to the
norm. People’s realized choices are determined by independent draws, each person according to his or her own probability.
3. Discreteness of adherence probabilities. Each person’s adherence probability in each period must equal one of the n values q , q , . . . , q . The set q , q , . . . , q is
1 2
n 1
2 n
assumed to contain at least three distinct q values and to be ordered so that
i
0 q q ? ? ? q 1.
1 2
n
4. Adjustment of adherence probabilities. Each person in each period randomly observes the behavior of one other person, whose identity is determined purely at random.
Between periods, the first person adjusts his or her adherence probability up or down one step on the ladder q , q , . . . , q from q to q
or from q to q according to
1 2
n i
i 11 i
i 21
whether the second person did or did not conform to the norm. However, if the adjustment is impossible because the person is already at the top or bottom of the
ladder, then the person stays put.
J . Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
199
Let p t be the fraction of the population with adherence probability q in period t.
i i
Thus, p t, p t, . . . , p t is the distribution of the population over adherence
1 2
n
probabilities, and the p t sum to one. When densities are fat to the left or right of p t,
i 1
p t, . . . , p t, the population is in lesser or greater adherence to the norm. Given
2 n
parameters q , q . . . , q , and given initial values p 0, p 0, . . . , p 0, the model
1 2
n 1
2 n
determines the evolution of p t, p t, . . . , p t. Assumption 1 sets a context.
1 2
n
Assumptions 2 and 3 reflect the critical feature of the model that a person’s state is a probability of adherence to the norm, a matter of degree which might take one of n
different values q , q . . . , q . Nearly all other tipping models in effect assume just two
1 2
n
values for q , usually q 5 0 and q 5 1. Although these models generate tipping
i 1
2
behavior for the population as a whole, the model here has more realistic variability at the individual level.
Assumption 4 is the imitation rule. If the norm is against littering, for example, Assumption 4 says that observing another person carry litter to the trash will reinforce
the observer in similar good citizenship, whereas observing another throw trash on the ground will work the other way. Assumption 4 is tailored to situations in which a person
does not observe population data, but only a small sample of information each period, set for simplicity at one observation per period. Under Assumption 4, people do not
make subtle statistical inferences from their limited information, but rather react in sensible adaptive ways, perhaps unconsciously so, or perhaps recognizing that the cost
of detailed deliberation would exceed the personal stakes. Imitation, not preference, is the behavioral primitive. Bowles 1998a,b argues persuasively that in some contexts
preferences are best seen as endogenous results of more fundamental behavioral forces, such as imitation. He cites a body of supporting evidence, including studies of the
powerful human urge to imitate and conform briefly surveyed in Ross and Nisbett, 1991; Chapter 2, and Boyd and Richerson, 1985; pp. 223–227.
Now turn the assumptions into a compactly stated dynamic system. Let q and pt be n 3 1 vectors defined by q 5 [q , q , . . . , q ]9 and pt 5 [p t, p t, . . . , p t]9,
1 2
n 1
2 n
where primes denote transposes. The inner product ft ; q9pt is then the probability
that a randomly selected person from the population will obey the norm in period t. By law of large number considerations, ft is also the fraction f for fraction of the
population adhering to the norm. At a point in time, a single person will have one of the n possible adherence probabilities q , q , . . . , q . Let p t be the probability that a
1 2
n ij
person at q in period t will be at q in period t 1 1. By Assumption 4, the matrix
i j
P[pt] 5 [ p t] of these transition probabilities is
ij
1 2 ft ft
? ? ? 1 2 ft
ft ? ? ?
1 2 ft ? ? ?
P[pt] 5 , where ft 5 q9pt.
: :
: :
:
? ? ? ft
? ? ? 1 2 ft
ft
1 Since the same transition matrix applies to every person, the product P[pt]9pt equals
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. Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
the expectation of the next period’s population distribution. That is, E[pt 1 1] 5 P[pt]9pt. Since the population is a continuum, however, the expectation E[pt 1 1]
of the frequencies vector pt 1 1 equals the frequencies vector itself. Thus, the basic equation of motion for the model is
pt 1 1 5 P[pt]9pt. 2
A simple example illustrates the dynamic 2. Suppose n 5 3. Since the three population frequencies pt 5 [p t, p t, p t]9 sum to unity, one is redundant. Hence a value of
1 2
3
pt can be represented on a two-dimensional graph, with p t horizontally, p t
1 3
vertically, and p t excluded. The relevant region of the graph is the right triangle with
2
unit sides and vertex at the origin. Fig. 1 illustrates. It is like a ‘probability triangle’ from decision theory. The points on and inside the triangle represent the possible values of
pt. Points to the upper left correspond to higher values of the average adherence rate ft 5 q9pt. Given parameters q , q , q and an initial value p0, the model
1 2
3
generates the path of points p1, p2, p3, . . . Consider the particular parameter setting q , q , q 5 0.06, 0.55, 0.92. It says that
1 2
3
a person may be as low as 0.06 likely to obey the norm, as high as 0.92 likely, or at the middle value 0.55. For this setting, computations reveal that there are three equilibria for
pt, indicated by the points HIGH, MID, and LOW on the figure. At HIGH, MID and LOW, the average adherence rates are approximately 0.83, 0.43, and 0.17. The dashed
line through the MID equilibrium splits the triangle into regions of attraction. The model will send pt to the HIGH, MID, or LOW equilibrium depending on whether the initial
distribution p0 is to the upper left, on, or to the lower right of the dashed line. The intuition is that, at HIGH, most people adhere to the norm; hence imitation attracts
Fig. 1. Illustration for n 5 3 with q , q , q 5 0.06, 0.55, 0.92.
1 2
3
J . Conlisk et al. Mathematical Social Sciences 40 2000 197 –213
201
people to HIGH. Similarly, at LOW, most people reject the norm; hence imitation attracts people to LOW. The MID equilibrium involves a balance of opposite attractions;
hence a push off the dashed line in either direction tips the scale and thus sends pt to either HIGH or LOW. Although the population distribution pt thus converges,
individuals in the population continue to move among the adherence levels q , q , q ,
1 2
3
with two important exceptions. If the example had q 5 1 instead of q , 1, then the
3 3
HIGH equilibrium would be at the upper left corner of the triangle, where everyone would have the single adherence rate q 5 1. Similarly, if the example had q 5 0 instead
3 1
of q . 0, the LOW equilibrium would be at the lower right corner of the triangle.
1
Two adjustment paths for pt are shown on Fig. 1: a path from p0 5 0.1, 0.7, 0.29 heading toward the HIGH equilibrium, and a path from p0 5 0.5, 0.45, 0.059
heading toward the LOW equilibrium. Intuitively, Fig. 1 invites us to visualize a ‘turnpike’ running through all three equilibria. For each of the two sample paths shown,
the model’s initial movements are as if a search for the turnpike, with repeated, partial overshooting in the process the initially jagged pattern. Once near the turnpike, the
model moves more smoothly along it toward the relevant equilibrium. If the initial point p0 were on the dashed line itself, but away from the MID equilibrium, then pt
would stay on the dashed line, jumping back and forth from one side of MID to the other. Despite the overshooting, the jumps would get smaller, and pt would converge
to MID.
Fig. 1 suggests fragility in social behavior. Two societies with identical parameters and very similar starting points can display similarly erratic initial behavior, but
nonetheless tip to opposite equilibria if their starting points are on opposite sides of the dashed line. A one-time social shock which perturbs a society from the HIGH or LOW
equilibrium to the opposite side of the dashed line can tip the society to the opposite equilibrium, with some erratic movement on route. The HIGH or LOW equilibrium is
more or less fragile according to whether it is close or far from the dashed line. Further, as discussed below, a perturbation of parameters can change the number of equilibria.
Since a central concern of the model is that people may have different degrees of commitment to a norm, the n 5 3 case is not adequate. We think of n as sizable. Thus,
we need results for any n 3. The general results to follow can be previewed relative to Fig. 1. So long as q , q , . . . , q have sufficient spread, there will be exactly three
1 2
n
equilibria, as on Fig. 1. Further, the regions of attraction for the high and low equilibria will extend all the way to the middle equilibrium, which is thus a tipping point. There
will never be more than three equilibria, but there may be two or one. The number of equilibria will be found by counting real roots of an nth degree polynomial.
3. Equilibria and stability
