self-organising behaviour in the presence of negative externalities: a conceptual model of commuter...

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O.R. Applications Self-organising behaviour in the presence of negative externalities: 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, Switzerland b 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 neighboursÕ 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. Ó 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, we consider 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 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 focuses on the choice of an optimal starting time. For in- stance, Mahmassani and Herman (1987) conduct 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 Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00237-6 European Journal of Operational Research 157 (2004) 501–513 www.elsevier.com/locate/dsw

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Page 1: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

European Journal of Operational Research 157 (2004) 501–513

www.elsevier.com/locate/dsw

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.

Page 2: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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

Page 3: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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

Page 4: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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 ; ð3Þ
Page 5: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

0

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

Page 6: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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

Page 7: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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

Page 8: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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-

Page 9: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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

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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

Page 11: Self-organising behaviour in the presence of negative externalities: A conceptual model of commuter choice

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.

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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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