Dynamics of Economic Choices
Involving Pollution
Gérard Weisbuch
Howard Gutowitz
Guillemette Duchateau-Nguyen
SFI WORKING PAPER: 1994-04-018
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the
views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or
proceedings volumes, but not papers that have already appeared in print. Except for papers by our external
faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or
funded by an SFI grant.
©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure
timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights
therein are maintained by the author(s). It is understood that all persons copying this information will
adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only
with the explicit permission of the copyright holder.
www.santafe.edu
SANTA FE INSTITUTE
1
DYNAMICS OF ECONOMIC CHOICES INVOLVING POLLUTION
Gerard Weisbuch+t, Howard Gutowitz:j:t, Guillemette Duchateau-Nguyen+*t
tSanta Fe Institute suite A, 1660 Old Pecos Trail Santa Fe NM87501, USA
+Laboratoire de Physique Statistique de I'Ecole Normale Superieure, 24 rue Lhomond, F
75231 Paris Cedex 5, France.
* Dassault Aviation. DGTI DEAl IA 78, Quai Marcel Dassault. 92214 Saint Cloud Cedex
:j:ESPCI Laboratoire d'Electronique 10 rue Vauquelin 75005 Paris, France
Abstract
Economic choices involving pollution, like those concerning common resources, relate to
the emergence of cooperation among actors. Since pollution propagates in space, the temporal
dynamics of economic choices is coupled to the spatial dynamics of pollution. We start from
a simple description of the internal representations of the agents proposed by Arthur and
Lane (1993) to describe information contagion. The simulations done in this paper allow us
to discuss the maximum price that the agents agree to pay for non-polluting devices as a
function of pollution, propagation of information and memory characteristics of the agents.
We also characterize the spatio-temporal dynamics of choices, market shares and pollution.
1. INTRODUCTION.
The purpose of the present study is to determine under which conditions economic
agents would agree to pay the extra cost of devices that prevent polluting the environment,
such as catalytic converters for cars. We depart from the conventional view of one perfect
rational decider with full information, able to evaluate the cost of pollution and to impose the
rational choice on all agents. In real life, agents have incomplete information and they decide
in function of their own interests. Choices then depend on available information, readiness
to cooperate to avoid pollution and possibly government intervention. We will concentrate
in this paper on the role of information in controlling the dynamics of individual choices
and suppose that the agents decide egoistically without government intervention (although
generalizations of the model including some kind of government intervention are immediate).
This approximation is not unrealistic, since in real life agents only make cooperative decisions,
including compliance to government decisions, if these decisions are not against their long
term interests.
The role of information exchange and imitation processes in economy has already been
described by Myerson (1990), Sharfstein and Stein (1990), Blume (1991), Banerjee (1992),
Kirman (1992, 1993) and Ellison (1993). The studies on cooperation are often based on
2
the iterated prisoner's dilemma, Lindgren (1991) and Nowak and Sigmund (1992), or on
cost-benefit analysis, Glance and Huberman (1993, 1994).
A simple example that we use throughout the discussion is that of two brands of otherwise equivalent cars, except for the fact that one brand is polluting and the other one is
non-polluting thanks to its catalytic converter. The a priori utility of the non-polluting car
is lower than that of the polluting car, due to the cost of the catalytic converter. If the
agents were fully rational (e.g. if they were fully informed) and cooperative (no cheaters),
they would agree to pay for the converter a price equivalent to the cost of pollution generated
by each polluting car. We propose here a model based on bounded rationality which shows
that agents only agree to pay a fraction of the cost of pollution. In this model, derived from
Arthur and Lane (1993), agents compute the utilities of both brands according to their a priori expectations, the opinion of their neighbors about the brand they have chosen, their risk
aversion, and the cost of local pollution. The model supposes no government intervention,
except perhaps, a tax on polluting cars, and that the agents have no global view of the real
cost of pollution. They simply experience the decrease in utility of the brands due to present
local pollution.
Due to the locality of information exchange and the diffusion of pollution, agent opinions depend on time and space. The model takes into account several coupled dynamics:
those of pollution, of agent internal representations and of their choices. We are ultimately
interested in the time evolution of the market shares of polluting and non polluting cars, and
how it depends on the parameters governing the economic variables, pollution diffusion and
especially the dynamics of the internal representation of the agents.
2. THE MODEL
At each time step, agents set up an internal probabilistic representation of the utilities of
the two brands of cars based on prior information and on polling their neighbors. They buy a
car according to the computed utilities. Meanwhile pollution is changing because of polluting
cars emissions and due to spatial diffusion. The presence of pollution in a neighborhodd
decreases car utilities locally. The internal representation updating, car choice and pollution
processes are iterated in parallel for all agents until an attractor of the dynamics is reached,
such as domination of the market by one brand of cars.
2.1. Computation of the utilities
The agents choose cars according to their" internal representation" of the characteristics
of the products. These internal representations are probability distributions of the utility of
each product that are computed according to the following procedure.
3
- Agents access some public information about the performance of each brand i, represented by a normal prior probability distribution, with mean !-'i and standard deviation
(]'i·
- They take advice from n other purchasers, among which ni bought brand i. Only those
ni purchaser that bought brand i contribute to the information about brand i by sending a
measure of its utility:
(1)
Xi = !-'o (i) - P + e ,
where !-'o( i) is the average utility of brand i in the absence of pollution. Since the sampled
purchasers already possess the brand, their opinion about the brand includes the negative
effects of pollution. We suppose here that they don't know the origin of pollution, in such
a way that the decrease in the car utility is proportional to local pollution, irrespective of
which brand they have purchased themselves. Utility is then decreased by the presence of
pollution P expressed in the convenient cost units. e, representing measurement error, is a
normally distributed random variable, but with mean 0 and standard deviation 0" ob invariant
through the iteration process.
The agents process this information to obtain a posterior distribution of performances.
This processing is done by taking the convolution products of Gaussian integrals corresponding to the prior and to the information obtained from other purchasers. The average expected
posterior utility !-,post,i is then:
1
!-,post,i =
nj
+ (l<' ["
Xi,j + (l<i!-'i]
n'
L.J
t
Z
,
(2)
j=l
where index j refers to the pooled agent. The mean utilities are averaged with a weighting
factor that is inversely proportional to the variance of the distribution:
(3)
In other words the posterior utilities average the different polled opinions weighted inversely to the variance of the distributions. The computation of posterior utilities is illustrated
for a particular case on figure 1.
Economic agents are adverse to risk. The greater the uncertainty in the value of a
product, the less they are likely to buy it. To take into account risk aversion of the agents,
we use a classical mean variance utility function: one more term, proportional to the variance
of the posterior distribution, is added to !-,post,i to compute the effective utility function U i
used by the agents to choose their car:
Ui
=
f-lpostJi -
\
2
A(7post i = tJpost,i I
2
0" ob
\
A
ni
+ (Xi '
(4)
1.0
4
----
0.8
prior
neighbor
neighbor
posterior
0.6
0.4
0.2
0.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Fig. 1. An example of updating the internal representation of an agent according to equations
2 and 3. The prior distribution of utilities for a given brand, with mean 5 and
standard deviation 0.5 (intermediate linewidth Gaussian) is updated to the posterior
distribution with mean 4.83 and standard deviation 0.41 (larger linewidth Gaussian)
after taking into account information obtained from neighbors with means 3 and 6 and
standard deviation 2 (smaller linewidth Gaussians). In this example, the posterior
distribution is shifted and sharpened with respect to the prior.
where A is the risk aversion parameter and a;ost,i is updated by narrowing the variance
according to the number of sampled purchasers. The complete expression for Ui is then:
(5)
Upon computing Ui for each product, purchasers choose the brand with the highest
expected utility.
2.2. Pollution dynamics
The dynamics of the agent representations and choices interacts with the diffusion dynamics of emitted pollution. Agents occupy the cells of a two dimensional grid that represent
space. Polluting cars generate pollution at a constant rate s. This pollution diffuses in
5
space according to Fick's law, and part of it simply evaporates locally ( in the sense that it
disappears). Pollution, P(r,t) in r at time t, obeys the following partial differential equation:
dP(r, t)
dt
= Df:"P(r, t) -
evP(r, t)
+s
,
(6)
where D is the diffusion constant, f:"P the laplacian of P, ev the evaporation term.
s, the source term, is only present when the occupant of the cell bought a polluting car.
The s term strongly couples the dynamics of pollution to that of the agents' choices. The
range of possible pollution levels correspond to fixed configuration of choices. Pollution is
zero when all agents always bought non-polluting cars and maximum and equal to s/ev when
they all have always bought polluting cars. s/ev, the maximum pollution, reflects the cost
of pollution, and is the natural scale on which to compare differences in the prior utilities of
the agents.
However, since the agents only use local information, the gradient of pollution times
the range of polling is the relevant parameter to compare differences in prior utilities. The
agents base their choices on the information provided by neighbors. In homogeneous regions
of either brand purchasers, the pollution gradient is small and the risk aversion parameter
makes the agent choose the same brand as their neighbors. A pollution gradient is present
in frontier regions. Pollution is higher on the side of the boundary where polluting cars
were bought which lowers the utilities of the polluting cars. On the other side pollution is
lower, and the utility of non-polluting cars is thus increased. The agents at the boundary
get their information on polluting (resp. non-polluting) cars from purchasers of polluting
(resp. non-polluting) cars in the polluted (resp. non-polluted) region. The posterior utilities
are thus split according to the difference in pollution betweeen the two regions. Given this
type of information, the agent at the boundary might agree to pay the extra cost of a nonpolluting car that compensates for the difference in pollution he has been able to poll. The
maximum gradient is obtained when one supposes that a half plane has always been occupied
by polluting cars and the other half by non-polluting cars. In this case the gradient of pollution
IS:
VP =
'!-j
1 ,
(7)
2 Dev
The difference in pollution experienced by neighboring agents is simply the product of the
gradient by the range of polling. For reasonable parameter values, this product is much less
than saturation pollution, which explains why the agents never agree to pay an extra cost for
the anti-polluting device equal to saturation pollution.
2.3. Updating the priors: the dynamics of agent internal representations
Every time the agent is facing a new choice, he polls his neighbors and gets some
information. This succession of information can be used in different ways.
6
One way is to forget the information obtained at previous time steps and recompute at
the next time step the utility function from the same initial priors. In this case the priors are
not updated.
Another possibility is to memorize all the information acquired at each time step by
using as priors for fl.i, ai and O"i the posteriors derived from equation 2 at the previous
time step. We call this choice maximum updating. But ai increases in time as the total
number of polled opinions. The importance of the risk aversion parameter is then rapidly
decreasing. Furthermore, agents using maximum updating would give the same importance
to old and new information and would tend to be overconfident because the variance of the
utility probability distribution is constantly decreased. An image of such an agent would be
an obstinate old man that does not care any more about the present situation and refers
mostly to the past. Maximum updating is thus fairly insensitive to the variable changes in
time and does not fit well with dynamical processes.
An intermediate possibility is to decrease the relative importance of older information.
This is simply achieved by limiting the ai to a maximum value of a max . To see why choosing
an a max is equivalent to a short term memory, let us suppose that several iterations have
been made with a = a max , in a neighborhood where the fraction ni of purchasers have not
changed. By iterating the utility expression (2) towards the past one obtains:
fl.(t) =
n
n
+ Cl: max
[fl.o - P(t) ] +
a max
n
+ G'max
fl. ( t - 1) ,
(8)
(9)
where I is the time difference between the past step and the present. Utilities from the
past are then discounted by a factor n ~ma.
at each time step, and since a max > n, this
O'rna:c
corresponds to a decay time for memory of~. In other words limiting ai to a max is
somehow equivalent to limit the averaging process to the a max most recent information.
The prior updating process is only performed on the internal representation of the
agents. It does not apply to the information transmitted between the agents, when the main
source of uncertainty is the polling process itself. In equation 1, fl.o( i) always represent the
initial average utility of brand i. It is, of course, a constant of the simulation.
7
3. SIMULATION RESULTS
3.1. Computer implementation of the model
The agents occupy the nodes of a square lattice (the grid). At each time step the agents
update their posterior distribution of utilities according to their prior and the information
they receive from their neighbors. They buy the brand that maximize their expected utility. Pollution dynamics occurs in parallel, the Laplacian operator being discretized by the
standard method.
We used two kinds of initial distributions of agents:
-One can start from initial conditions where the grid is divided in two regions, an upper
half plan where all agents have chosen polluting cars and a lower half plan where all agents
have chosen non-polluting cars (see fig.2, 3 and 4). The gradient at the boundary between
the two regions is maximum. In the range of parameters we use, we expect any change in
the agent choices to start close to this boundary.
-In the other cases, we start from random distributions of polluters and non-polluters,
see figures 6 and 7.
We monitor the configurations of purchases on the grid and the time evolutions of
market shares and local pollution. The simulations are run iteratively until some attractor is
reached. Examples of attractors are all agents choosing polluting cars or all agents choosing
non-polluting cars.
3.2. Parameter set-up
The general philosophy of agent based simulations such as this one, is to correlate
observed dynamical regimes with the range of parameters where they are observed. The
dynamical regimes are characterised by the attractor they reach. We have then searched
the transition parameters separating in the parameter space the regions corresponding to
different attractors, such as domination of the market by polluting or non-polluting cars.
We kept invariant the following parameters:
The 32x32 grid. We used circular boundary conditions: the upper row (resp. left
colummn) of the grid is connected to the lower (resp. the right colummn).
Initial standard deviations
set to 5.
(7 i
and
(7ob
were set to 1. The risk aversion parameter A was
8
In most simulations, 5 neighbors were polled to obtain information. This number was
sometimes changed to 12.
The pollution diffusion rate was set to 0.5 and the evaporation rate was 0.2.
At a given pollution cost, the most influential parameter with respect to which attractor
is reached is the difference in prior utilities of both brands. A priori utilities for polluting cars,
f.lo(l) were set to 10. A priori utilities for non-polluting cars, f.lo(O), were varied downward
from 10. When f.lo(l) is close to f.lo(O), the decrease in posterior utilities due to pollution is
sufficient to make the agents prefer non-polluting cars. When the differences in prior utilities
is increased, some threshold is reached where an inverse process is observed: the region of
non-polluting cars buyers is invaded by polluting cars, and partial domination of polluting
cars is achieved. The agents don't agree any more to pay for the extra cost of the catalytic
converter. The difference in prior utilities at the threshold, represents the maximum price
that the agent would agree to pay to avoid the pollution costs. We have systematically
searched these thresholds for the above range of parameters: a max was varied between 1 and
1024 which corresponds to memory decay times of 1 and 200. Pollution rates were 5, 10 and
20, which correspond to maximum pollution costs of 20,40 and 80 (these values are slightly
less than eSv because of time and space discretization on the grid).
3.3. Description of some typical simulations
Half-plane initial conditions
When f.lo(l) is close to f.lo(O) the region of polluting car buyers shrinks in time (see
fig.2). The fraction of agents choosing polluting cars decreases to 0 as does pollution. In
this figure, f.lo(0)=9.5, which represents a small difference with f.lo(l) as compared to the
maximum pollution cost of 40. The total domination of the market is achieved in less than
100 steps.
The dynamics of invasion of the grid by non-polluters differs according to the difference
in prior utilities. For small differences (see figure 2), the invasion proceeds directly from the
non-polluted region. For larger differences, spots of non polluting purchasers appear in the
polluted regions, while the initially non polluted region is invaded by checkerboard patterns
(see figure 3).
9
Pollution
Market Share
<. '"',-r----,---.:====--r-----,
:.
...
! :
\ ..
:
!
'.:
,,',:
!
:
:
j
!
j
t/
•
.\,
! :
!
\
1
Legend
mR-""---!------I-----j'----j- minimum
i,---,
;
M
:,'
_~~
!
!
20
\
Legend
i
H f - ' \ - - - - t - - - - - - t - - - - - - 1 - minimum
U
I
_~~
-·..·maximum
11
.
!
···-·maximum
\
~
\
i
l
i
••
l
<.
20
Time
60
•
Time
Fig. 2. Fast evolution towards market domination by non-polluting cars in case of a small
difference in prior utilities (fLo(0)=9.5), and intermediate term memory (a max =32),
S = 10, all the other parameters being defined in the text.
The eight patterns
represent the spatial distribution of car choices at times t=O , t=l , t=2 and 5 from
left to right for the upper patterns and at times t=8 , t=l1 , t=15 and 50 from left to
right for the lower patterns. The time 0 pattern corresponds to the initial conditions
for simulation. The 32x32 grid of agents is half filled with buyers of polluting cars,
the black squares, and half filled with buyers of non-polluting cars, the light grey
squares. Circular boundary conditions are used. The two lower figures represent the
time evolution of the market share of polluting cars and of pollution. The three lines
in the pollution graph correspond to minimum, average, and maximum pollution on
the grid. At time t=l, an inversion of car choices occurs in the polluted region (see
the Don Juan effect defined in the text). Invasion by non-polluters then occurs in a
straightforward manner from the non-polluted region.
-Market Share
40
;
,
\.
! .I, ." .....
i
i
Pollution
...
.?",
.. ..
\......
.
0.5
V
,j
I
o
h~
o
,
.......................
!
~
20
Time.
Legend
minimum
- average
.. ..·maximum
.
\,
'\.
\
,..-'
I
,.....,
.
i
;
20
o
Legend
f---!'f----\-7,L.--=="---~1_-....:"'::_--------_Jminimum
- average
..··-maximum
o
20
40
Time
Fig. 3. Evolution towards market domination by non-polluting cars in case of an intermediate
difference in prior utilities (/Lo(0)=8.5), and short term memory (a<max=8), S = 10,
all the other parameters being defined in the text. The six upper patterns represent
the spatial distribution of car choices at times t=O , t=6 , t=10 from left to right for
the upper patterns and at times t=13 , t=15 , t=36 from left to riijht for the lower
patterns. Note the apparition and growth of islets of non-polluters from the polluted
region at times 13 and 15. The two lower figures represent the time evolution of
the market share of polluting cars and of pollution. The three lines in the pollution
graph correspond to minimum, average, and maximum pollution on the grid. The
early oscillations on these lines correspond to an inversion of choices in the polluted
region obtained during the strong initial variation of pollution (refer to the Don Juan
effect in the text) The systems evolves towards complete domination of the market
by non-polluters at t=100, as in the last pattern of figure 2 (not represented in this
figure).
11
--
Fig. 4. Slow evolution towards a coexistence distribution among polluting and non-polluting
cars in the case of a large difference in prior utilities (flo(0)=7.5), and long term
memory (a max =256), S = 10, all the other parameters being defined in the text. The
three patterns represent the spatial distribution of car choices at times t=O , t=32
and t=1100, from left to right. Note the regular metastable patterns corresponding
to 43 pollution at the lower half of pattern at time 1100, and the persistence of some
memory of the initial conditions at the upper half of the pattern.
:v
In the beginning of the simulation, when pollution builds up rapidly, a few period 2
oscillations of the market share in the region of polluting car buyers are observed (figures 2 and
3). This is because the agents only update the utility of the car bought in their neighborhood,
while keeping the former utilities for the other brand. Since pollution is increasing, the former
utilities always appear higher, which make the agents switch their choice at each time step.
We called this effect the Don Juan effect in reference to the classical myth; Don Juan has high
prior expectations about Woman, which makes him constantly change his female partners
because none can meet his expectations.
When the differences in prior utilities are increased, the convergence time increases until
the transition is observed: the region of non-polluting cars buyers is invaded by polluting cars,
and partial domination of polluting cars is achieved.
One striking observation is that total domination of the polluting brand is not achieved
unless the difference in prior utilities get close to the total cost of pollution (corresponding to
flo(O) = -30, for s = 10). Rather, one observe a transition towards a mixed configuration of
polluters and non polluters (see figure 4). Changing the memory range (by changing a max )
has some influence on the final pattern. With long term memory, some memory of initial
conditions is kept, and the upper half plane is nearly filled with polluters, even after 1100
time steps although the memory range is of the order of 50. With short term memory, a
uniformly random final pattern is reached, closely resembling the one reached from random
initial distribution of purchasers (see figure 7).
The existence of stable mixed configurations was a surprise to us, since we expected
the risk aversion parameter to give rise to homogeneous attractors with eventual local defects
due to noise terms. A close examination of local conditions show that polluters enjoy both
12
.1111
111111
111111
Fig. 5. Regular metastable and stable patterns corresponding to
and to 43 pollution (right pattern).
:v
2:v pollution (left pattern)
a higher prior utility and a decrease in pollution due to the neighborhood of non-polluters.
Non-polluters surrounded by polluters are screened from any up-to-date information about
the non-polluting cars. The posterior utility they compute for the non-polluting cars does
not include pollution, or when it does, it is at a lower level (old information). They thus stick
to their previous choices.
The transition to market domination by the polluters is discontinuous, the final market
share of polluting cars varying from 0 to more than 0.5 (0.75 is both a simulation estimation
and theoretical conjecture related to observed regular patterns, see below). But due to the
probabilistic character of information transmission and the finite size of the grid, there are
some uncertainties on which attractor is reached in the neighborhood of the transition. We
have done some statistics, by doing several simulations with the same set of parameters with
different random seeds. We then found that the width in f.lo(O) of the transition is of the
order of 0.18 for a probability of a zero market share of polluting cars varying between 1/3
and 2/3. In the coexistence region, the final market share varies continuously from 0.75 when
f.lo(O) is decreased, until total domination of polluters is achieved at negative values of f.lo(O).
Regular patterns appear, on both sides of the transition:
- The checkerboard pattern is built of alternating black and white squares corresponding
respectively to buyers of polluting and non polluting cars. The corresponding pollution is
since the relative density of polluting cars is 1/2. (See figure 5 left). This pattern is only
metastable and it is eventually destroyed by the noise terms. It can be observed on figures
3,4 and 6.
2:v
- The other pattern is made of alternating lines of black squares separating lines of
alternating black and white squares, with pollution in this region being 43
(See figure 5
right). This pattern is apparently stable in the coexistence region for very long times (see
figures 4 and 7).
:v'
Pollution
40,--------=..:=r:=-----,
Market Share
;__\
/';
".r-
\.
i 1:-\./\,;\1 "'.--
:
'f}
--L
-
0.5
~---_--+-
Legend
1_minimum
-average
\
:
I.
Legend
f=-";T~~L\----+',,----T----l-minimum
!
-average
1
..... maximum
20 It"
i
---·-maximum
\
\
\
o
o
o
50
100
\ .•.•....
o
50
100
Fig. 6. Evolution towards market domination by non-polluting cars. The initial conditions
are a random distribution of polluters and non-polluters. The conditions are a small
difference in prior utilities (1-'0(0)=8.8), short term memory (a max =8), s = 10, all
the other parameters being defined in the text. The six upper patterns represent the
spatial distribution of car choices at times t=O , t=30 , t=36 from left to right for
the upper patterns and at times t=41 , t=51 , t=71 from left to right for the lower
patterns. Note the apparition and growth of an islet of non-polluters at times 36, 41
and 51.
These translational symmetric patterns are local attraetors of the dynamics. They are
a compromise between the lower cost of polluting cars and the lower pollution due to the
presence of non-polluting cars. They represent metastable and stable configurations of nonpolluters and cheaters that benefit from the presence of the non-polluters. This coexistence
of cooperators and cheaters is similarly observed in many simulations of the iterated prisoner
dilemma, as observed by Lindgren (1991) and Nowak and Sigmund (1992).
14
Pollution
4Or--------::=:.r==------,
Market Share
v'..!~.-. ('-r-
.~:.jf.~..
..~.._..._...w.:..~'~~
._.~_.~N
.."..,
· -.--.--·.-
~
_
lI!
l~'
Legend
minimum
-average
o.5
V"
20
Legend
ft---------+---------j-minimum
-average
---·maximum
-.-.- maximum
o
o
o
500
Time
1000
o
500
1000
Time
Fig. 7. Slow evolution towards a coexistence distribution among polluting and non-polluting
cars. The initial conditions are a random distribution of polluters and non-polluters.
The conditions are a large difference in prior utilities (/'0(0)=8), longer term memory
(a max =64), S = 10, all the other parameters being defined in the text. The three
patterns represent the spatial distribution of car choices at times t=O , t=39 and
t=980, from left to right. Note the regular metastable patterns corresponding to 43:.
pollution. The two lower figures represent the time evolution of the market share of
polluting cars and of pollution.
The characteristic length in these patterns is determined by the characteristic length
of information propagation. The wavelength 2 patterns of figures 5 are observed when 5
neighbors are polled, but 2x2 square patterns are observed when 12 neighbors are polled.
15
10.0
9.5
9.0
•
8.5
•
8.0
7.5
7.0
0.0
•
2.0
4.0
6.0
8.0
10.0
Fig. 8. Transition lines in the [og2(Ci max ), po(O) plane, (the memory range scale is logarithmic), for homogeneous half planes initial conditions as in figures 2, 3 and 4. The
upper line is the lower boundary of complete domination by non-polluters and the
lower line the upper boundary of mixed configurations with a market share of polluting cars larger than 0.8. Pollution rate is 10. The upper transition is discontinuous,
with market shares of polluters varying from 0 to 0.75, while the lower transition is
continuous. At low memory ranges, the two transitions are simultaneous within simulation uncertainties. Also note the slight minimum of p(O) as a function of [Og2( Ci max )
around Ci max =32. The squares represent the parameters of simulation for figures 2,
3 and 4.
Initial random distribution of polluters and non-polluters generate final spatia temporal
patterns, (see figures 6 and 7), similar to those generated by homogeneous half plans initial
configurations.
3.4. The regime diagrams
Figure 8 and 9 represent the transition lines in the plane po(O) versus the base 2
logarithm of Ci max in the case of initial homogeneous half-planes of purchasers and of random
16
10.0
9.5
9.0
•
8.51:::----1
•
8.0
7.5
7.0
0.0
2.0
4.0
6.0
8.0
10.0
Fig. 9. Transition lines in the log2(a max ), 1'0(0) plane, (the memory range scale is logarithmic), for random initial distributions of polluters and non-polluters as in figures 6
and 7. 1'0(1) the prior utility of polluting cars is set to 10, and 1'0(0) the prior utility
of non-polluting cars varies between 10 and 7. The upper line is the lower boundary
of complete domination by non-polluters and the lower line the upper boundary of
mixed configurations with a market share of polluting cars larger than 0.8. The upper
transition is discontinuous, with market shares of polluters varying from a to 0.75,
while the lower transition is continuous. Pollution rate is 10. The squares represent
the parameters of simulation for figures 6 and 7.
initial distribution of purchasers. The upper lines correspond to the lower boundary of nonpolluter domination. The lower lines are the upper boundary of polluter partial domination
with a final market share larger than 0.8. Since transients are very long in the neighborhood
of the transition, simulation times were restricted to 300 time steps. Convergence towards
polluter (resp. non-polluters) domination was considered as certain when the market share
of polluters was above 0.8 (resp. below 0.1). Although each transition point is obtained
by averaging over nine simulations, the determination is' not extremely precise, due to the
probabilistic character of information transmission.
Homogeneous initial conditions (figure 8) favor cooperation and the choice of nonpolluting cars with respect to random distribution because the pollution gradient is large
at the boundary between the two regions, thus giving clear information to the agent.s in
17
this neighborhood. An interesting feature is the existence of an optimum memory range
(a max ~ 32) corresponding to a maximum difference in prior utilities at the transition. This
weak optimum was observed consistently for all simulations. The agents perform better by
having an intermediate memory range, allowing them to average information on several time
steps while still being able to cope with change.
When the agents are in a random environement (figure 9), they have to rely on pollution
inhomogeneities to get useful information. The pollution gradients are much smaller, which
explains why the upper transition line rests higher in J.lo(O) than for the homogeneous initial
distribution (figure 8). Rather than averaging on several time steps to separate signal from
noise, they better follow time fluctuations to get the meaningful signal: long term memory is
thus a handicap, hence the positive slope of the upper transition line.
The lower transition lines are very similar for both kind of initial conditions.
This systematic search of transition J.lo(O) when a max is varied was also done for other
pollution rates, s = 20, and 5, and give similar results, with the transition lines scaling in
J.lo(O) in proportion to s.
4. CONCLUSIONS
This model shows that agents with imperfect knowledge are able to switch to more
expensive non-polluting devices by comparing information about pollution costs obtained
from neighboring agents. When information sampling is local, the maximum accepted extra
cost for the non-polluting device scales as the gradient of the pollution cost times the range
of polling. Unless the pollution spatial distribution has very abrupt changes from maximum
to minimum pollution (on distances lower than the range of polling), the accepted cost is less
than the cost of pollution. FUrthermore, except when the cost of the polluting device is very
small with respect to sampled cost of pollution, the dynamics of invasion by the non-polluting
devices is rather slow, much slower than the maximum velocity on the grid.
A surprising result of the simulations is the importance of the coexistence regions with
regular patterns of cheaters and cooperators.
The memory range of the agents plays an important role in the dynamics of market
domination: when a strong pollution gradient is established from the initial conditions, an
intermediate memory range is optimum to favor the adoption of the non-polluting brand. In
the presence of strong spatial (and thus temporal fluctuations), short term memories are an
advantage allowing the agents to use the information generated by rapidly vanishing spatial
fluctuations.
Another interesting result is the fact that invasion of the polluted region by the nonpolluting devices does not always proceed from the non-polluted region. When mixed
metastable attractors are reached, the polluted mixed region can be invaded from islets of
18
non-polluting devices that started as fluctuations inside this region. Revolutions don't start
in the most advanced countries, but rather in the most retrograde.
Some generalization are implemented simply in the present model. The role of constant
taxes per polluting device imposed by a government simply decreases the prior utility of
polluting devices. Changing the densities of agents between cities and country-side is also
easily accomplished by having variable densities of agents on the grid.
ACKNOWLEDGMENTS
We thank B. Derrida, F. Grau, V. Gremillion, A. Kirman, J.P. Nadal and J. Vannimenus
for helpful discussions. SWARM software, thanks to Chris Langton and David Hiebeler,
was used for numerical simulations. The Laboratoire de Physique Statistique is associated
with CNRS (URA 1306) and we acknowledge financial support from NATO CRG 900998,
Curie Foundation external grant program and the John D. and Catherine T. Mac Arthur
Foundation, through the New Century Fund, the sponsor of Project 2050.
References
ARTHUR W.E. and LANE D.A. (1993), 'Information Contagion', Structural Changes
and Economic Dynamics, 4, 81-104.
BANERJEE A. (1992) "A simple model of heird behaviour", Quarterly Journal Of
Economics ", 108, 797-817.
BLUME L. (1991), "The statistical Mechanics of Social Interaction", Mimeo, Cornell
University, Ithaca, N.Y.
ELLISON G., (1993) "Learning, Local Interaction and Coordination", Econometrica,
forthcoming.
GLANCE N. and HUBERMAN B. A., (1993), "The outbreak of cooperation" Journal
of Mathematical Sociology, 17, 281-302.
GLANCE N. and HUBERMAN B. A., (1994), "Organizational Fluidity and Sustainable
Cooperation" , in Computational Organization Theory, edited by K. Carley and M. Prietula,
Lawrence Erlabaum Associates, page 123.
KIRMAN A.P. (1992), "Whom or what does the representative individual represent?",
Journal of Economic Perspectives ,6, 117-136.
KIRMAN A.P. (1993), "Ants, Rationality and Recruitement", forthcoming, Quarterly
Journal Of Economics", February.
19
LINDGREN K. (1991) "Evolutionary Phenomena in Simple Dynamics", in Artificial
Life II, edited by C. langton, C. Taylor, J. Farmer and S. Rasmussen, 295-312.
MYERSON J. (1990), " Graphs and Cooperation in games" Mathematics of Operations
Research, 2, 225-229.
NOWAK M. and SIGMUND K. (1992), " Tit-for-tat in heterogeneous populations",
Nature 355,250-253.
SHARFSTEIN D. S. and J. C. STEIN (1990), "Herd behaviour and investment", American Economic Review, 80 , 465-479.