self-organising behaviour in the presence of negative externalities: a conceptual model of commuter...
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European Journal of Operational Research 157 (2004) 501–513
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O.R. Applications
Self-organising behaviour in the presence of negativeexternalities: A conceptual model of commuter choice
Ann van Ackere a, Erik R. Larsen b,*
a Department of Management, HEC Lausanne, BFSH1, University of Lausanne, 1015 Lausanne, Switzerlandb Faculty of Management, Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK
Received 5 October 2001; accepted 15 April 2002
Abstract
We use a model with local interaction (a one-dimensional cellular automaton) to study how commuters choose
among alternative roads. Commuters have information about their neighbours� most recent experience (local inter-
action) and they remember their own experiences (memory). We illustrate how a simple, self-organizing system, based
on local information and locally rational agents can in some cases outperform the Nash equilibrium. While the social
optimum is unenforceable without a central planner, due to the variations in individual travel times (i.e. the social
optimum is not individually rational), variations across commuters in the steady-state of our self-organising system are
at least equivalent to, and mostly significantly larger than those required for the social equilibrium. Increasing the
neighbourhood size illustrates that more information without co-ordination leads to worse overall performance.
� 2003 Elsevier B.V. All rights reserved.
Keywords: Simulation; Collective choice; Traffic; Cellular automata
1. Introduction
In this paper we focus on how commuters
choose between alternative roads, based on their
expectations about travel times. Specifically, weconsider a group of commuters, all of whom travel
daily from the same origin to the same destination,
during the same time-slot. Although we cast the
model in terms of commuters travelling daily, a
* Corresponding author. Tel.: +44-20-7040-8374; fax: +44-
20-7040-8328.
E-mail addresses: [email protected] (A. van Ack-
ere), [email protected] (E.R. Larsen).
0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv
doi:10.1016/S0377-2217(03)00237-6
similar analysis applies to any situation where a
large number of people regularly perform the same
journey at the same point in time, for instance
people returning to the city on Sunday evenings,
having spend the week-end at the sea-side.Traffic related research covers many different
aspects and methodologies, ranging from experi-
mental study of simple systems to mathematical
analysis of more complicated ones, and simulation
of real-life systems. Attention has been devoted to
studying equilibrium conditions, as well as analy-
sing transient behaviour. A significant part of the
literature in the traffic and congestion area focuseson the choice of an optimal starting time. For in-
stance, Mahmassani and Herman (1987) conduct a
ed.
502 A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513
controlled experiment involving real commuters ina hypothetical computer simulated traffic system.
There is no diffusion of information, i.e. com-
muters rely solely on their own experience. Other
references include Alfa (1986), Arnott et al. (1993),
and Mahmassani and Chang (1986). Another
problem, addressed by numerous papers, is that of
a single bottleneck, see among others Arnott et al.
(1993) and Newell (1988).Ben-Akiva et al. (1991) present a mathematical
model aimed at analysing driver decision making
in the presence of real-time information systems.
The authors point out that drivers� reaction to
information can cause congestion to transfer to
another road, and could lead to oscillations. This
occurs if too many drivers react simultaneously to
the same information, without anticipating theresponse of other drivers.
Other papers, such as Emmerink et al. (1995)
use simulation to study the issue of information
provision, and the potential to market information
systems commercially. They deviate from the op-
timisation principle present in many papers by
using a utility-based satisficing principle. Several
forms of information (own experience, after-trip,real-time pre-trip, real-time, and combinations of
these) are considered. They show for instance that
after-trip information is only beneficial to all
drivers if this information is available to less than
20% of the driver population, which supports Ben-
Akiva et al.�s arguments. Real-time information
needs to be of high quality to be beneficial, else
over-reaction occurs, making matters worse.Hu and Mahmassani (1995, 1997) consider real-
time information in a dynamic context, combined
with traffic control policies. In their 1995 paper
signal control parameters are assumed fixed, while
in their 1997 paper they consider both the adjust-
ment of signal timing parameters to the previous
day�s traffic conditions and to real-time informa-
tion.Several authors have focussed more specifically
on how drivers process their information. Alfa
(1989) assumes that commuters have global in-
formation and a one-period memory (i.e. they as-
sume that travel times are about the same as they
were the previous day), a rather unrealistic set of
assumptions. Mahmassani and Chang (1986) as-
sume that commuters update their estimatesthrough a weighted average of past experience, but
points out that this may not be a good way of
learning, because measurements on different days
are not repeated measurements of the same
thing as the other commuters vary their departure
times.
Iida et al. (1992) carry out an experimental
study where participants choose between two hy-pothetical routes, and update their expectations
based on their own experiences. Data is collected
on expected and actual travel times and used to
estimate how much weight users give to their most
recent experience versus their previous experi-
ences. The experiments indicate that in this simple
context route choices can get fairly close to a user-
equilibrium. They also state that, when drivers caninfer conditions on other roads (which is possible
in this example) they predict other driver�s reac-
tions, and act accordingly. It is drivers� inability to
do so in more complex settings which can create
the oscillations discussed by for instance Ben-
Akiva et al.
Recently a number of models based on spatial
representation of roads and microlevel interactionbetween boundedly rational agents have been de-
veloped (Nagatani, 1993; Nagel and Rasmussen,
1994; Rickert et al., 1995).
Our work focuses on the choice of road, with
special attention to the transient behaviour and the
learning process (rather than the full information
equilibrium analysis). We compare the resulting
equilibrium to the socially optimal solution (re-sulting from central planning) and to the full in-
formation user equilibrium (Nash equilibrium).
These two optimality concepts are discussed in
Section 2.
As mentioned, we also study the transient be-
haviour. This enables us to investigate the impor-
tance of the initial conditions (commuters� initialassessment of congestion on the various roads andon the train, as well as the initial road allocation).
Both the resulting steady-state, and the time to
reach it, vary widely depending on these initial
conditions, everything else being equal. We also
study the way in which commuters learn and up-
date their expectations about travel times, which
enables us to speculate on what kind of traffic
A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513 503
information should be provided to commuters,and how it should be made available.
It is worth emphasising that we deal with a
captive group of commuters; i.e. not travelling is
not an option. This differentiates our work from
most of the literature on congestion and negative
externalities, where it is commonly assumed that a
high expected level of congestion does deter po-
tential users. See for instance Dewan and Men-delson (1990) for an application to service
facilities.
At a more theoretical level we can view the
question posed here as a typical micro–macro-
problem which is well known in disciplines ranging
from physics to sociology. Given very clear mi-
crorules for how the single agent behaves, it is in
most cases impossible to predict how the aggre-gated system will behave. On the other hand given
the aggregated outcome from a microprocess it is
in most cases impossible to deduct the microrules
which created the aggregated behaviour (see Lomi
and Larsen, 1996; Kari, 1990).
More information is not necessarily better. For
instance Arnott et al. (1989) show that perfect in-
formation can be worse than no information. Ar-nott et al. (1996) extend these results more
generally to free-access congestible facilities facing
fluctuating demand and capacity.
Our approach enables us to look more specifi-
cally at the impact of information diffusion. In a
different context (choice between two actions, no
externalities), Bala and Goyal (1998) use a mathe-
matical model to show that more informationlinks can increase the probability that a society
gets locked into a sub-optimal action.
Our main results are as follows. We illustrate
that for certain initial conditions a simple self-
organizing system based on local information and
locally rational agents outperforms the Nash
equilibrium (which assumes full information and
co-ordination), but when averaging performanceover many different initial conditions, the Nash
equilibrium is superior.Our model outperforms the
Nash equilibrium mainly when commuters update
their memory reasonably fast, i.e. there is relatively
little inertia in the system. In those cases we ob-
serve huge variations between the resulting equi-
libria for the different runs. We also illustrate that,
while the social optimum is generally considered tobe unenforceable without a central planner, due to
the variations in individual travel times (i.e. the
social optimum is not individually rational), the
variations across commuters in the steady-state of
our self-organizing system often vastly exceed
those required for the social equilibrium. Finally,
by increasing the neighbourhood size, we illustrate
that more information without co-ordination leadsto a worsening of the overall performance. Look-
ing at the transient behaviour, we point out the
impact of the speed at which memory is adjusted
on the length of time required to reach a steady-
state.
The paper is organised in the following way.
After this brief introduction and motivation we
introduce the model used to analyse the choice ofroad, and the underlying modelling framework,
based on cellular automata. Then we describe the
parameters used for the simulation experiments
conducted with the model. This is followed by the
simulation results. Finally we discuss the implica-
tions of our findings and outline future work.
2. The model
As pointed out previously, instead of having a
spatial model of the roads and modelling the traffic
flow we start with a spatial representation of the
commuters and we attempt to model their (re-
peated) choice of which road to take, given their
past experience and local information.The motivation for this choice of model is that
most people are faced with a decision of which
road to take when they are travelling daily from A
to B. Assuming that all leave around the same
time, there is little information available about
which road would be best this given morning. It
could be argued that not all would start the jour-
ney at the same time. However, if we focus onthose individuals who wish to arrive at approxi-
mately the same time at point B (e.g. 9 am), it is
reasonable to assume that a significant number of
them will leave at about the same time. As long as
a sufficiently large number of commuters does
leave around the same time, the question is rele-
vant: how do they choose between alternative
504 A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513
roads when making this decision simultaneouslywith many others?
We assume that commuters learn both from
their own experience and from communicating
with their neighbours. The term �neighbours� canbe interpreted either in a physical location sense
(people living next door) or in a communication
network sense (e.g., colleagues, friends doing the
same commute, etc.). They estimate an averagetravel time for each road based on their own past
experience and information from neighbours, i.e.
we assume rational decision making based on ex-
perience and local information. Each commuter,
independently of everybody else, then has to make
his or her choice of which road to take on a given
day.
We use a simple model with local interaction,based on a one-dimensional cellular automata
(Wolfram, 1983, 1984; Gutowitz, 1991). The spa-
tial dimension comes into the model in the way the
agents are positioned, i.e. each commuter has ex-
actly two neighbours, one on each side. The model
is organised on a ring, which avoids any boundary
problems, i.e. that some commuters would only
have neighbours on one side. 1 The number ofneighbours taken into account when information
is gathered is given by K. A value K ¼ 1 indicates
that there is only communication between a given
cell Ci and its two nearest neighbours, located in
cells Ci�1 and Ciþ1. A value K ¼ 2 means that there
is communication with the two nearest neighbours
on each side of a given cell Ci, i.e. Ci�1, Ci�2, Ciþ1,
Ciþ2, and so forth. The set of neighbours a cellcommunicates with is referred to as the K-neigh-bourhood.
For simulation purposes we consider a popu-
lation of 120 commuters choosing between three
alternative roads. All commuters, each represented
in the model as a cell, are identical except for their
initial estimates of congestion on the different
roads and the comfort level of rail travel. We as-sume that each commuter has a memory which
1 This is one of the two ways of dealing with the boundary
problem in cellular automata, the other is to assume fixed
boundaries, i.e. some cells always keep the same value (Wolf-
ram, 1983).
contains the expected travel time for each of thethree roads.
Following several authors (e.g. Horowitz, 1984;
Iida et al., 1992; Emmerink et al., 1995), this
memory is then updated after the daily travel with
the new experience from the present day: the ac-
tually chosen road gets updated using an expo-
nentially weighted average with weight k (i.e.
adaptive expectations). If k ¼ 0, full weight isgiven to the most recent experience, while k ¼ 1
implies no updating of expectations. A low value
of k implies that commuters consider only their
more recent experiences to be relevant (i.e. cir-
cumstances change rapidly) while a high value of kimplies stickiness of expectations and thus a high
level of inertia in the system (i.e. a significant
amount of evidence is required before expecta-tions, and behaviour are adjusted).
Iida et al. (1992) derive from experimental data
that the weight assigned by commuters to their
most recent experience lies generally between 0.3
and 0.7. Horowitz (1984) shows that in his model
the equilibrium may not be stable for extreme
weights (i.e. close to 0 or 1).
When commuters decide which road to selecton a given day, they compare these estimates to the
latest experience of other commuters in the K-neighbourhood, i.e. the travel times the commuters
in their K-neighbourhood achieved yesterday, and
pick the most attractive alternative. The memory
of each commuter is initialised by using a uniform
distribution of expected travel times in the range
60� 7.5 minutes.The three roads are ranked according to ca-
pacity. The expected travel time as a function of
the number of commuters is shown in Fig. 1. The
travel time is calculated according to:
TTR1 ¼ 20þMaxðCR1 � 15; 0Þ � 3
þ ðMaxðCR1 � 30; 0ÞÞ1:5; ð1Þ
TTR2 ¼ 40þMaxðCR2 � 30; 0Þ � 2
þ ðMaxðCR2 � 60; 0ÞÞ1:5; ð2Þ
TTR3 ¼ 60þMaxðCR3 � 60; 0Þ � 1
1:5
þ ðMaxðCR3 � 120; 0ÞÞ ; ð3Þ0
20
40
60
80
100
120
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
People
Tra
velt
ime
Road 1
Road 3
Road 2
Fig. 1. Travel time on the three roads as a function of the
numbers of people.
A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513 505
where TTRi is the travel time for road i at time t,and CRi is the number of commuters selecting
road i at time t. These functions are piece-wise
linear approximations to the MTC curve, whichaccording to R. Singh (1999) is an improvement
on the BPR function. While the choice of these
functional forms may seem arbitrary, we have
experimented with various configurations, as well
as with a variations of the model containing up to
10 roads. The qualitative results hold across these
various models. We have elected to present the
three-road version for reasons of clarity and con-ciseness.
It is worth emphasising that we focus only on
average travel times, ignoring variations between
commuters on a given road for a given traffic in-
tensity. We justify this simplification on the
ground that, as traffic intensity increases, average
travel time increases exponentially, while the rel-
ative variation of travel time decreases signifi-cantly (see for instance Nagel and Rasmussen,
1994). Our analysis is focused on roads with high
traffic intensity, as this is the situation where
choice of road is most relevant.
To summarise, the model evolves in the fol-
lowing way: the first day, i.e. at time equal 1, the
commuters are allocated randomly to one of the
roads, with equal probability for each alternative.Each commuter then updates his or her memory
with the new information, i.e. the travel time ex-
perienced on the road used. The next period, the
commuter compares his neighbours� travel time at
time t � 1 with his expectations for the various
alternatives (based on memory) and selects the
road with the shortest expected travel time.
An alternative would be to assume that com-muters only switch roads if their expected gain
exceeds a certain threshold, which would cause the
system to reach equilibrium faster. This concept of
a threshold is in line with the �indifference band� oftolerable delay in the context of desired arrival
times, as first introduced in Mahmassani and
Herman (1987), and used in Mahmassani and
Chang (1987) and Hu and Mahmassani (1995).To evaluate commuters� choice of road, we need
a benchmark of what constitutes a ‘‘good’’ choice.
We will compare our results to the following two
definitions of optimality: the Nash equilibrium
(often referred to as a Wardrop User Equilibrium
in the traffic literature) and the social optimum.
The Nash equilibrium is optimal from an indi-
vidual point of view: no single commuter can re-duce his travel time by selecting a different road.
The Nash equilibrium assumes full information,
i.e., commuters know exactly how many com-
muters choose each road. This implies that in the
Nash equilibrium, travel times are equalised across
roads.
Mahmassani and Peeta (1993) illustrate that the
gain from achieving a social optimum rather thana Nash Equilibrium is non-monotonic in traffic
density. In their example, the gains are almost
non-existent for low traffic density (free-flowing
traffic for all decisions), peak at about 20% as
density increases, and start falling as saturation is
approached (grid-lock for all decisions).
Mahmassani and Chang (1987) use the concept
of a boundedly rational user equilibrium (BRUE),�a state of the system in which all users are satisfied
with their current choices, and thus do not intend
to switch� (p. 91). The difference between a BRUE
and the Nash equilibrium is that the former does
not assume full information. The equilibria
reached in our model are BRUE.
Table 1 shows the resulting allocation for our
simulated example, with travel times rounded tothe nearest minute. The requirement that the
number of commuters on any one road be an in-
teger causes the minor variations in travel time
between roads.
The concept of social optimum implies the
presence of a central planner who allocates com-
muters to roads. This results in minimizing the
Table 1
Nash equilibrium and social optimum
Route Nash equilibrium Social optimum
People Timea People Time
1 28 59 22 41
2 40 60 33 46
3 52 60 65 65
Travel time
Weighted average 59.8 55.4
Maximum 60.0 65.0
Minimum 59.0 41.0
a The slight difference between the roads is a consequence of the requirement that people per road be integer valued.
506 A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513
average travel time of all commuters, and equalizes
the marginal change in travel time resulting from achange in the number of commuters, weighted by
the number of commuters, across roads. The re-
sulting allocation is also shown in Table 1. Note
that the social optimum is not sustainable without
a central planner: commuters on 3 would be very
tempted to switch to the less congested roads, at
the expense of commuters presently allocated to
these.
3. The base case
Figs. 2–6 illustrate how the model behaves for
K ¼ 1 (i.e. commuters only communicate with
their direct neighbours) and k ¼ 0:3 (i.e. a 30%
weight to memory and a 70% weight to the mostrecent experience when updating memory).
Fig. 2 shows the average travel time for three 40
day simulation runs, as well as the maximum and
minimum travel times once the system has stabi-
lized. The average travel time of the commuters
stabilizes fairly fast. The major fluctuations dis-
appear within 10 days. For these three runs, the
average travel times after stabilization respectivelyequal 58, 76 and 93 minutes (Table 2).
Note that for the first run, the average travel
time lies below the Nash equilibrium of 60 minutes
(average travel time¼ 58). Roads 1 and 2 have a
high degree of uncertainty associated with them in
steady-state, with the maximum being about 50%
higher than the minimum travel time, while on
road 3 minimum and maximum travel time differ
by less than 5%. For roads 1 and 2 the minimum is
well below the Nash and social equilibrium values,while the maximum is above both. Road 3 on the
other hand is always above the Nash equilibrium.
On any one day, there is a difference of about 25
minutes (i.e. over 50%) between the experience of
the best off and worst off commuters.
In the second and third simulations, the differ-
ences between minimum and maximum equilib-
rium travel times are even larger. In the secondrun, the average travel time is 76 minutes, and
road 1 is overlooked by commuters. Similarly, in
the third run (93 minutes average travel time) road
2 is overlooked. In these cases we observe varia-
tions of up to 100% and 300% respectively.
Fig. 3 illustrates how commuters allocate
themselves to the different roads over time. The
fluctuations during the transient period are ex-tremely high. The precise realisation of this tran-
sient period (and the resulting equilibrium)
depends very much on the random seed (i.e. the
initial memory and road allocation) and the speed
at which people update their memory ðkÞ. This willbe elaborated upon in the next section.
While in the first example commuters spread
themselves across the three roads, the next twoexamples show that a road can be �forgotten�, i.e.customers are clustered on only two of the three
roads (roads 2 and 3 in the second example and
roads 1 and 3 in the third one). It is this phe-
nomenon which causes the large average travel
times to arise. This results from a situation where a
bad experience causes everybody to be convinced
that a road is congested, thus avoiding it, resulting
Average travel time = 58 minutes in steady state
Average travel time = 76 minutes in steady state
Average travel time = 93 minutes in steady state
Average travel time
Average travel time
Average travel time
Maximum travel time
Maximum travel time
Maximum travel time
Minimum travel time
Minimum travel time
Minimum travel time
Fig. 2. Three examples of average travel time for k ¼ 0:30
with different initial values of expected travel time (i.e. initial
memory).
Average travel time in steady state 58 minutes
Average travel time in steady state 76 minutes
Average travel time in steady state 93 minutes
R o a d 1
R o a d 2R o a d 3
0
2 0
4 0
6 0
8 0
1 0 0
1 2 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
Num
ber
of c
omm
uter
s
R o a d 1 R o a d 2
R o a d 3
0
2 0
4 0
6 0
8 0
1 0 0
1 2 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
Num
ber
of c
omm
uter
s
R o a d 1
R o a d 2
R o a d 3
0
2 0
4 0
6 0
8 0
1 0 0
1 2 0
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
Num
ber
of c
omm
uter
s
Time
Time
Time
Fig. 3. The distribution of commuters on roads for l ¼ 0:30
depending on initial values of expected travel time.
A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513 507
in the road being empty. Consequently, as no one
dares try out that road, there is no opportunity fornew experience to correct this wrong perception.
This can be interpreted as a reputation effect. If a
commuter has bad experience, he will remember
this. As he will not return to this road on his own
initiative, the memory will remain unchanged until
such time as a neighbour uses that road. This
implies that if all commuters have a bad memory
about a road, no one will return to this road, and
so this perception of congestion will never be up-dated.
This type of behaviour is typical of self-orga-
nizing systems where, from an initial random state,
an organized system emerges without any form of
central planning or outside intervention (Mose-
kilde, 1996). Given the initial conditions, it is not
possible to analytically derive what steady-state
Fig. 4. Distribution of average travel time for three different
values of k (based on 1000 runs with each value of k).
Maximum travel time
Minimum travel time
Maximum travel time
Minimum travel time
Average travel time = 58 minutes in steady state
Average travel time = 76 minutes in steady state
Average travel time 93 minutes in steady state
Minimum travel time
Maximum travel time
Fig. 5. Minimum and maximum travel time for commuters in
steady-state for k ¼ 0:30 with different initial conditions.
508 A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513
will materialize. Equally, given a steady-state, one
cannot infer what initial conditions, nor rules of
interaction, yielded this steady-state. Different
initial conditions can lead to the same steady-state,and two sets of almost identical initial conditions
can lead to a very different organisation at the
system level, and thus very different equilibria.
Once the system has stabilised, the number of
commuters on any one road still fluctuates con-
siderably. Considering for instance the third ex-
ample, the number of commuters on road 1 varies
from 16 to 38 (compared to a social optimum of 22and a Nash equilibrium of 28.
Fig. 4 shows the distribution of travel time for
k ¼ 0:05, 0.3 and 0.95, based on 1000 runs for each
case. This graph provides insight into the fre-
quency with which certain roads are forgotten, and
illustrates that the three selected scenarios are
representative of the system�s behaviour for
k ¼ 0:3. For this case the first peak (around 60minutes) represents scenarios where all three roads
are in use, the second peak (around 75 minutes)
represents cases where road 1 is forgotten, and the
third peak (around 90 minutes) scenarios where
road 2 is forgotten. The peak at the end (around
121 minutes) represents scenarios where all the
commuters are using road 3.
As k increases, memory plays an increasing role,and the frequency with which a road is forgotten
decreases. While a large k does not allow the
commuters to achieve the best results (the first
peak is around 65 minutes rather than 60 minutes)
it does eliminate the worst scenarios.
Fig. 5 provides more insight into the variations
across commuters, as well as the daily fluctuations
commuters face. The horizontal axis represents the
commuters (recall that commuter 120 is a neigh-
Average travel time = 58 minutes in steady state
Average travel time = 76 minutes in steady state
Average travel time = 93 minutes in steady state
Fig. 6. Commuters average travel time in steady-state with
k ¼ 0:30 and three different initial conditions. Dashed line in-
dicate Nash equilibrium and social optimum respectively.
A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513 509
bour of commuter 1) and the vertical axis showstravel time. The dashed line and the solid line re-
spectively represent the minimum and maximum
travel time faced by commuters once the system has
stabilised. There are no commuters for whom thesetwo lines coincide. Comparing this to Fig. 3, we can
conclude that while the total number of commuters
per road does not fluctuate particularly widely in
equilibrium, i.e. not that many commuters change
their behaviour on a daily basis, those who do may
significantly affect the travel times experienced by
nearly all commuters. For instance, in the first ex-
ample the worst affected commuters experiencevariations between 40 minutes and 1 hour. They
might describe this journey by saying that it�s typ-ically a good 45 minute drive. On the other hand,
certain commuters in the third example face travel
times ranging from less than half an hour to nearly
two hours, and would probably describe their
journey to work as a real nightmare: half an hour
on a good day, but it can take hours!Next, consider the commuters on road 2 in the
second simulation. Having experienced a travel
time of 81 minutes, 40% of these commuters (20
people) switch to road 3 in the next period. This
results in a 41 minutes travel time for those staying
on road 2. Consequently, the 20 commuters who
switched to road 3, come back the following day.
Another interesting phenomenon is the occur-rence of clusters of commuters with identical be-
havior. Consider for instance the group of
commuters 13–23 or 97–107 in the simulation with
an average of 76 minutes. They all behave identi-
cally (they stay on the same road once the system
has stabilized). The fluctuations in travel time they
face are solely the consequence of other commut-
ers switching roads. Such commuters stop anydiffusion of information.
Fig. 6 shows the average travel time faced by
each of the 120 commuters in equilibrium, and
compares this to the Nash equilibrium and social
optimum. In the first example, about a third of the
commuters outperform the Nash equilibrium. In
the second example about a third get close to the
Nash equilibrium, while in the third example theexperience is quite abysmal.
4. Variations of the base case
In this section we consider a number of varia-
tions of the base case. Specifically, we carry out a
Table 2
Average travel time in steady-state
Average travel time¼ 58 Average travel time¼ 76 Average travel time¼ 93
Road 1 Min # of people 22 0 16
Max # of people 30 0 38
Min travel time 42 21a 24
Max travel time 66 21a 112
Road 2 Min # of people 27 30 0
Max # of people 41 50 0
Min travel time 41 41 41a
Max travel time 63 81 41a
Road 3 Min # of people 57 70 82
Max # of people 63 90 104
Min travel time 61 71 83
Max travel time 64 91 105
aNote that these times are theoretical (i.e. the time a commuter would need should he be alone on this road) as the road is empty.
Fig. 7. Sensitivity of travel time with respect to k in the case of
K ¼ 1.
510 A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513
sensitivity analysis with respect to k and K. We let
k range from 0 to 0.99 (in steps of 0.01) for K ¼ 1,
2 and 3. The average travel times are based on the
average of 1000 simulation runs of 1500 days for
each ðk;KÞ combination. Average travel times are
computed over the last 500 days of each run. For
k ¼ 1, the average travel times are much higherthan for the other values. In this extreme case,
commuters do not update their memory at all.
Consequently they do not learn from experience
and are much more likely to stabilize at a sub-
optimal solution.
Fig. 7 shows the travel time averaged over the
1000 runs for K ¼ 1 (solid line) as well as the 90%
prediction interval for the daily average traveltime, and the 90% confidence interval for the
sample average (referred to as the grand mean
travel time).
As k increases, average travel time decreases up
to approximately k ¼ 0:8, and increases thereafter.
For larger values of k, commuters update their
memory more slowly, i.e. there is more inertia in
the system. This figure illustrates that to achieve areasonable level of performance, a significant de-
gree of inertia is required, but too much inertia is
damaging. Referring back to the experimental
work by Iida et al. (1992), we notice that the values
they derived for k (0.3–0.7) are slightly below the
optimum (i.e. insufficient weight to memory, too
much to the more recent experience).
Note the sharp drop in the upperbound of theprediction interval around k ¼ 0:7. As inertia in-
creases, the system has more memory. Conse-
quently the commuters become less likely to
�forget� a road. Referring back to Fig. 4, we ob-
serve that the very high travel times (i.e. the cases
where a road is forgotten) become significantly less
likely as k increases.
It is also worth noting that the transient periodis non-monotonic in k. For values of k close to 0,
the transient period is considerable longer than for
values of k around 0.5. Intuitively, as k increases,
there is more inertia in the system (people update
Fig. 9. Percentages of runs below the Nash equilibrium as a
function of k and K.
A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513 511
their memory more slowly), resulting in less vola-tility, so a steady-state is reached sooner. At the
other extreme, for very large values of k the in-
formation obtained from experience affects mem-
ory so gradually, that this basically creates a delay
in the system: evidence obtained today will only be
acted upon much later, when sufficient additional
evidence has been gathered.
Fig. 8 shows the average travel time of the 100runs for neighbourhood sizes K ¼ 1, 2 and 3. As Kincreases (i.e. more communication), the average
travel time increases. Thus, in the presence of
negative externalities (congestion), more informa-
tion is actually worse. Letting K increase does not
move the system closer to the full information in-
dividually rational equilibrium, due to the lack of
co-ordination. It is also worth noticing that thedecrease in average travel time gained from going
from K ¼ 2 to 1 is almost twice the decrease re-
sulting from going from K ¼ 3 to 2.
As K increases, the transient period shortens
considerably. Intuitively, this is due to information
flowing faster through the system. Unfortunately,
this does not result in better performance.
Fig. 9 shows the percentage of runs yielding anaverage travel time below the Nash equilibrium
value of 60 minutes. First, let us consider large
values of k. For these we observe little difference
for the different values of K. As k increases, the
percentage of runs outperforming the Nash equi-
librium decreases to zero. A large value of k means
Fig. 8. Average travel time as a function of k and K.
that a lot of weight is given to memory, and
memory is updated slowly. Therefore the proba-
bility of forgetting a road is low, independently of
whether or not communication occurs, as observed
in Fig. 8 where increasing k decreases the average
travel time. Next let us consider lower values of kFor K ¼ 1, the percentage of runs below the Nash
equilibrium is fairly constant for values of k in therange [0; 0:4], and then decreases. This might at
first seem surprising as the average travel time
decreases over this range. The intuition is as fol-
lows: while an increasing k results in fewer �excel-lent� scenarios (i.e. better than Nash), there are
also fewer �disaster� scenarios, so the overall per-
formance improves.
For K ¼ 2 or 3, the percentage of runs belowthe Nash equilibrium initially increases in k, as
would be expected given the decrease in average
travel time over this range.
5. Summary of results and conclusions
In this paper we have presented a simple, self-organizing system, with local information and
locally rational agents. Commuters obtain infor-
mation through their own experience, and through
experience from neighbours. They update their
memory based on their own experience, and follow
a simple decision rule.
512 A. van Ackere, E.R. Larsen / European Journal of Operational Research 157 (2004) 501–513
We have illustrated that for certain initial con-ditions, the resulting steady-state can be an im-
provement over the Nash equilibrium (which
requires full information and co-ordination), in
that the average travel time of commuters is re-
duced. The resulting individual travel times exhibit
significant variation, in contrast to the Nash
equilibrium where all commuters have identical
travel times.The social optimum is generally considered to
be non-sustainable without a central planner
having the power to enforce decisions, due to the
variations in individual travel times (i.e. the social
optimum is not individually rational when agents
have full information). Still, the steady-state re-
sulting from our simple self-organizing system
exhibits wider variations in travel time betweencommuters than those required for the social op-
timum.
Performance is optimal for values of k around
0.8. For lower values, the system lacks memory,
and a road is more likely to be forgotten. Large
values of k indicate that more weight is placed on
memory, i.e. commuters react more slowly to new
experiences, and are thus more likely to stick tosub-optimal behavior.
We also considered the impact of increased
access to information by enlarging the neigh-
bourhood. This resulted in a significant worsening,
as the average travel time increases in neighbour-
hood size, illustrating that an increase in infor-
mation can make matters worse rather than better
in an environment characterized by negative ex-ternalities and lack of co-ordination.
The present paper assumed that, except for the
initial conditions, everything is deterministic.
Further analysis indicates that introducing even a
slight amount of randomness (e.g. each period one
commuter selects a road at random) is sufficient to
eliminate any form of stable behaviour for most
values of k. The presence of one such commuter issufficient to eliminate equilibria where one (or
more) roads are forgotten.
Further work focusing on a more detailed study
of behaviour at the individual level should enable
us to throw some light on this issue. We also in-
tend to study models with different road charac-
teristics to see to what extend our specific
assumptions influence the qualitative results de-rived in this paper.
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