Dilemma game structure observed in traffic flow at a 2-to-1 lane junction

Abstract

Using a cellular automaton traffic model based on the stochastic optimal velocity model with appropriate assumptions for both incoming and outgoing vehicle boundaries, the so-called bottleneck issue on a lane-closing section was investigated in terms of game theory. In the system, two classified driver agents coexist: C agents (cooperative strategy) always driving in the first lane and D agents (defective strategy) trying to drive in a lower-density lane whether the first or the second lane. In high-density flow, D agents' interruption into the first lane from the second just before the lane-closing section creates a heavier traffic jam, which reduces social efficiency. This particular event can be described with a prisoner's dilemma game structure.

Full text

Dilemma game structure observed in traffic flow at a 2-to-1 lane junction
Atsuo Yamauchi,*Jun Tanimoto, Aya Hagishima, and Hiroki Sagara
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University Kasuga-koen, Kasuga-shi,
Fukuoka 816-8580, Japan
共Received 24 October 2008; published 13 March 2009兲
Using a cellular automaton traffic model based on the stochastic optimal velocity model with appropriate
assumptions for both incoming and outgoing vehicle boundaries, the so-called bottleneck issue on a lane-
closing section was investigated in terms of game theory. In the system, two classified driver agents coexist: C
agents 共cooperative strategy兲always driving in the first lane and D agents 共defective strategy兲trying to drive
in a lower-density lane whether the first or the second lane. In high-density flow, D agents’ interruption into the
first lane from the second just before the lane-closing section creates a heavier traffic jam, which reduces social
efficiency. This particular event can be described with a prisoner’s dilemma game structure.
DOI: 10.1103/PhysRevE.79.036104 PACS number共s兲: 89.40.⫺a
I. INTRODUCTION
Recently, traffic flow models have been studied increas-
ingly. In order to understand significant traffic flow phenom-
ena, traffic models such as the kinetic gas theory 关1兴, fluid-
dynamical model 关2,3兴, car-following model 关4,5兴, and
cellular automaton 共CA兲model have been developed. In par-
ticular, since 1992, when the first CA traffic model, the
Nagel-Schreckenberg model 关6兴, was proposed, CA model
research has made extensive progress because of the model’s
simplicity and features that can reproduce real traffic flows
well. For example, the free-flow and jam phases can be ap-
proximately reproduced with the most basic CA model,
called the asymmetric simple exclusion process 共ASEP兲关7兴.
However, ASEP cannot reproduce an unstable flow phase,
called the metastable phase, which is commonly observed at
the critical density when real traffic flow switches from the
free-flow phase to congested flow. However, some advanced
CA models, e.g., the slow-start model 关8兴or stochastic opti-
mal velocity 共SOV兲model 关9兴, can reproduce this phenom-
enon.
Although numerous previous models have analyzed
simple systems without bottlenecks except at boundaries,
study on bottlenecks in real traffic has attracted many re-
searchers. For example, Li et al. 关10兴and Gao et al. 关11兴
introduced a model considering the effect of on-ramps. In
particular, the model introduced by Gao et al. 关11兴can repro-
duce complex traffic flow states discussed by Kerner 关12兴.
Moreover, a series of studies revealed that Burgers’ equa-
tion, which governs one-dimensional 共1D兲shock wave
propagation, is exactly equivalent to the asymptotic behavior
of elementary CA rule 184 关13兴when Cole-Hopf transforma-
tion is applied to the ultradiscrete diffusion equation 关14兴.
This theoretical consistency is one reason why CA traffic
models are regarded as a powerful tool for analyzing traffic
flow kinetics.
However, none of these previous studies provided a com-
prehensive understanding on real traffic flow phenomena be-
cause these models did not include the decision-making pro-
cess of the drivers. In other words, most of the studies focus
only on the kinetics of a self-driven many-particle system
and ignore the effect of drivers’ decisions on the entire sys-
tem. Beyond those backgrounds, the objectives of this paper
are to add a game theory framework as a rational decision
process to the traffic model, construct a new CA model, and
demonstrate that a bottleneck at a 2-to-1 lane junction has a
dilemma game structure.
This paper is organized as follows: in Sec. II, the traffic
model we use is explained along with basic simulation re-
sults of the SOV model with the open boundary condition. In
Sec. III, the 2-lane model that we assume is presented. In
Sec. IV, the results of the numerical experiment are shown
and discussed. Finally, a brief conclusion is provided in Sec.
V.
II. SOV MODEL
In this paper, we adopted the SOV model 关9兴. Considering
the work by Gao et al. 关11兴, SOV might be less plausible
when dealing with bottleneck effects. However, here we fo-
cus on whether the dilemma game structure exists in traffic
flow, not on the detailed physics of the traffic flow itself. On
the other hand, we should not adopt a simpler model which
cannot reproduce the metastable phase since bottleneck jams
relate to vehicles’ slow-start effect that is also bringing to
reproduce the metastable phase. This compels us to adopt a
relatively simple but acceptably accurate traffic model in this
study. Hence, we choose the SOV model because it is simple
to define a lane change by a driver using the model.
In the SOV model, the velocity vi
t+1 of vehicle iat time t
is defined by
vi
t+1 =共1−a兲vi
t+aVi共⌬xi
t兲,共1兲
where a共0ⱕaⱕ1兲is a parameter, ⌬xis the headway, and
function Viis the optimal velocity function, which is defined
as
V共⌬x兲=tanh共⌬x−c兲+ tanh c
1 + tanh c.共2兲
Here, cis a parameter. It is important to note that Viis
restricted to 关0,1兴at any ⌬xand c; thus, vi
tis also restricted to
*Corresponding author; es208181@s.kyushu-u.ac.jp
PHYSICAL REVIEW E 79, 036104 共2009兲
1539-3755/2009/79共3兲/036104共6兲©2009 The American Physical Society036104-1

关0,1兴if vi
0is less than 1. In other words, vi
texpresses the hop
probability 共i.e., a normalized velocity兲.
The SOV model encompasses two fundamental stochastic
submodels; i.e., when ain Eq. 共1兲is 0, this model is the same
as ASEP 关7兴, while in the case of a= 1, it becomes the zero-
range process 关15兴. In these two models, we can deduce the
exact probability distribution for the configuration of ve-
hicles in the stationary state by an analytical approach.
Moreover, the SOV model has the same structure as the dis-
crete OV model, which features coupled differential equa-
tions related to both inertia and the headway effect 关16,17兴.
These two points highlighting the SOV’s advantage in terms
of theoretical robustness led us to adopt it.
When modeling, the way we set boundary conditions is
important. In some previous models, a cyclical boundary
condition was applied because of its simplicity and plausibil-
ity for tracing basic traffic flow features. However, in a road
with a bottleneck, nonequilibrium flow can occur near the
bottleneck. To apply the cyclical boundary condition leads to
an idea that downstream of the nonequilibrium flow condi-
tion affects on the upstream flow of the bottleneck, which
seems unrealistic. To avoid this, an open boundary condition
is applied in this paper.
The scheme of the SOV model with the open boundary
condition is shown in Fig. 1. When the cell in front of ve-
hicle iis vacant, the vehicle can hop with probability vi
t+1.A
vehicle is created at the leftmost cell with probability
␣
and
deleted at the rightmost cell with probability
␤
. The hop
probability of the newly introduced vehicle is set as 1. Head-
way ⌬xof the furthest forward vehicle is set as ⬁. The up-
date rule is applied to parallel updates.
Figure 2shows fundamental diagrams of real traffic data
and our simulation result with a=0.01 in Eq. 共1兲and c
=3/2 in Eq. 共2兲. Note that the flux represented on the vertical
axis is defined by the product of the velocity and density.
Here, the system length is set as 200 cells. Figure 2共a兲关18兴
indicates that four traffic flow phases can be observed in the
real fundamental diagram. The most notable characteristic is
that a metastable phase, in which the flow condition drops to
that of the high-density phase even if a subtle disturbance is
added, emerges between the free-flow and high-density
phases. Some CA models cannot reproduce this feature, but
our SOV model with the open boundary condition can repro-
duce this particular characteristic, as shown in Fig. 2共b兲.
Note that the normalized flux and density in Fig. 2共b兲can be
transformed to the real scale, as shown in the top and right
axes, when we assume a cell size of 5 m and a maximum
velocity of 100 km/h.
III. 2-LANE MODEL INCLUDING BOTTLENECK
In this paper, we analyze the bottleneck effect when clos-
ing lanes from double to single by means of the SOV model
under the open boundary condition. The drivers’ decision-
making process is described by game theory; i.e., we assume
that drivers have a strategy that is either cooperative or de-
fective. Cooperative drivers 共C agents兲drive only in the first
lane. The rules of defective drivers 共D agents兲are as follows:
first, they are created in a lower-density lane; second, while
in an overtaking area and if the second lane has lower den-
sity than the first, they will move to the second lane; and
third, when they reach the interrupting area, they attempt to
cut into the first lane. The rules of changing lanes are as
follows: in the case of overtaking, drivers can change lanes
only when other vehicles are not present in both the adjacent
cell and one cell behind the adjacent cell; in the case of
interrupting, even if another vehicle appears one cell behind
the adjacent cell, they can change lanes with probability p.
The rules are illustrated in Fig. 3. Of course, a D agent can-
not be allowed to enter the first lane if the left adjacent cell is
occupied by another agent.
FIG. 1. Proposed SOV model considering both incoming and
outgoing open boundary conditions.
ja
ja
mph
m ph
a
a
s
s
e
e
m
m
e
e
ta
ta
st
st
ab
ab
l
l
e
e
p
p
h
h
as
as
e
e
hi
hi
gh-dens
gh-dens
i
i
t
t
y
y
pha
pha
s
s
e
e
free-fl
free-fl
ow pha
ow pha
s
s
e
e
Flux [1/5min]
(a)Observed
Density [1/km]
0
0.06
0.12
0.18
00.20.40.60.81
0
100
200
300
0100200
Flux
[
1/5min
]
Normalized Density
Density [1/km]
Norma
li
ze
d
F
l
ux
ja
ja
mph
m ph
a
a
s
s
e
e
hi
hi
gh-dens
gh-dens
i
i
t
t
y
y
pha
pha
s
s
e
e
free-fl
free-fl
ow pha
ow pha
s
s
e
e
m
m
e
e
ta
ta
st
st
ab
ab
l
l
e
e
p
p
h
h
as
as
e
e
(
b
)
Simulated
FIG. 2. Comparison of fundamental diagrams 共a兲field observed
data to 共b兲simulated one by the proposed SOV model.
YAMAUCHI et al. PHYSICAL REVIEW E 79, 036104 共2009兲
036104-2

The updated rules of the system within a single time step
are as follows:
共1兲A vehicle is created with probability
␣
. The strategy is
determined to be C or D according to the cooperation frac-
tion PC, which is one of the simulation parameters.
共2兲The velocity of all vehicles is calculated.
共3兲All vehicles’ moves are decided, i.e., hopping or not
for C agent and hopping or not or changing lanes for D
agent.
共4兲Update all vehicles.
共5兲If the newly created vehicle stays in the leftmost cell,
it is deleted.
The following results are drawn from each 30-trial en-
semble average observed from the 10 001th step to the
12 000th step. Lengths of L1= 400, L2=200, L3= 100, and
L4=20 cells are assumed. The density and flux are measured
for L2⬍x⬍L1because the flow efficiency should be evalu-
ated from amount the flux is in the downstream section of the
bottleneck. It is confirmed that the effect of changing L2,L3,
and L4is not very significant. We also confirm that there is a
certain dependence of L1−L2on the bottleneck effect, which
is not sufficient to invalidate what we are trying to demon-
strate below.
L4
L3
Overtaking-Area;
A D-agent can move
from the 1st lane to 2nd
lane.
p
Interrupting-Area;
A D-agent can move from
the 2nd lane to 1st lane with
probability p.
L2
L4
L3
L
1
1st lane
2nd lane
FIG. 3. Assumed model with both incoming and outgoing open
boundary conditions. L1, the length of the first lane; L2, the length
of the second lane; L3, the length of the overtaking area; and L4, the
length of the interrupting area. An open circle indicates a C agent
and a closed one is a D agent.
jam phase
metastable phase
high-density
Phase
free-flow phase
α
β
metasta
bl
ep
h
ase
α
β
(c)
0.110
Normalized Flux
(b)
(a)
FIG. 4. 共a兲Two-dimensional 共2D兲and 共b兲3D flow-
␣
-
␤
dia-
grams; gray-scale contour indicates normalized flux with an as-
sumption of cooperation fraction PC=1. 共c兲Indicates four represen-
tative traffic flow phases based on a sketch observation on 共a兲.
0.110 N
o
rmaliz
ed
Fl
u
x
●○
□
■
●○
□
■
●○
□
■
●○
□
■
●○
□
■
β
1
0
α
1
α
β
1
1
0
(e)
(d)
(c)
(b)
(a)
β
1
0
α
1
α
β
1
1
0
β
1
0
α
1
α
β
1
1
0
β
1
0
α
1
α
β
1
1
0
β
1
0
α
1
α
β
1
1
0
FIG. 5. Flow-
␣
-
␤
diagram; gray-scale contour indicates normal-
ized flux. Parameters are 共a兲Pc=0, 共b兲Pc= 0.3, 共c兲Pc=0.5, 共d兲
Pc=0.7, and 共e兲Pc=1. Four symbols 共open and closed circles and
open and closed squares兲are representative points of the free-flow
共
␣
=0.05,
␤
=0.65兲, jam 共
␣
=0.65,
␤
=0.05兲, metastable 共
␣
=0.10,
␤
=0.65兲, and high-density 共
␣
=0.65,
␤
=0.65兲phases
shown in Fig. 7.
DILEMMA GAME STRUCTURE OBSERVED IN TRAFFIC …PHYSICAL REVIEW E 79, 036104 共2009兲
036104-3

IV. RESULTS AND DISCUSSIONS
In this section, we analyze the flow state with changing
␣
−
␤
,
␤
, and the fraction of cooperators PC.
First, consider Fig. 4. Figures 4共a兲and 4共b兲show the
␣
-
␤
phase diagram in two and three dimensions with PC= 1, i.e.,
the system contains only C agents. Figure 4共c兲is a sketch of
the resulting classified flow phases. First, the area character-
ized by small
␣
and insensitivity to
␤
can be classified as the
free-flow phase. Second, the area characterized by small
␤
and insensitivity to
␣
can be classified as the jam phase.
Third, the area characterized by insensitivity to both
␣
and
␤
can be classified as the high-density phase. Finally, the area
with the highest flux can be classified as the metastable
phase.
Figure 5shows the
␣
-
␤
phase diagram with a changing
fraction of cooperators PC, where we can observe normalized
flux ahead of the bottleneck. Likewise, Fig. 6shows the pay-
off difference between D agents and C agents in several co-
operation fractions. In this diagram, if the payoff difference
is positive, defecting is more rational than cooperating. Fig-
ure 7shows payoff structure functions of four representative
points that indicate the free-flow 共
␣
=0.05,
␤
=0.65兲, jam
共
␣
=0.65,
␤
=0.05兲, metastable 共
␣
=0.1,
␤
=0.65兲, and
high-density 共
␣
=0.65,
␤
=0.65兲phases 共these four points
are also shown in both Figs. 5and 6兲. The payoffs of both
strategies are evaluated by hop probabilities measured within
the L2area. Note that employing the higher payoff strategy is
more rational than using the opposite strategy from the view-
point of individual benefit. We evaluate the social profit by
0.1-0.1 0
β
α
01
1
1
1





‐
‐
‐
■○
□
●■○
□
●
■○
□
●■○
□
●
■○
□
●
0.1-0.1 0
0.1-0.1 0 0.
5
-0.5 0
0
.
8
-
0
.
80
β
α
01
1
β
α
01
1
β
α
01
1
β
α
01
1
(d)
(e)
(c)
(b)(a)
FIG. 6. Payoff difference-
␣
-
␤
diagram. The payoff indicates
hop probability 共normalized velocity兲difference between D and C
agents. Each subgraph from 共a兲to 共e兲shows the result for PC
=0.1– 0.9. Four symbols 共open and closed circles and open and
closed squares兲are representative points of the free-flow 共
␣
=0.05,
␤
=0.65兲, metastable 共
␣
=0.10,
␤
=0.65兲, jam 共
␣
=0.65,
␤
=0.05兲, and high-density 共
␣
=0.65,
␤
=0.65兲phases
shown in Fig. 7. The + and − signs indicate positive and negative
areas, respectively.
0.6
0.7
0.8
0.9
1
0.06
0.07
0.08
0.09
0.1
0
0.1
0.2
0.3
0.4
0
0.01
0.02
0.03
0.04
0
0.2
0.4
0.6
0.8
00.51
0.05
0.06
0.07
0.08
0.09
0.6
0.7
0.8
0.9
1
0.01
0.02
0.03
0.04
0.05
Norma
li
ze
d
Ve
l
oc
i
tyNorma
li
ze
d
Ve
l
oc
i
tyNorma
li
ze
d
Ve
l
oc
i
ty
Norma
li
ze
d
F
l
ux
P
C
(A) free-flow
α
=0.05
β
=0.65
(B)jam
α
=0.65
β
=0.05
(D) high-density
α
=0.65
β
=0.65
Norma
li
ze
d
Ve
l
oc
i
ty
Norma
li
ze
d
F
l
ux
(C) metastable
α
=0.1
β
=0.65
Norma
li
ze
d
F
l
ux Norma
li
ze
d
F
l
ux
Normalized Flux
Normalized Velocity of D
Normalized Velocity of C
FIG. 7. Payoff structure functions of both C 共triangles兲and D
共gray circles兲agents with social average 共line兲. The payoff implies
averaged hop probability 共normalized velocity兲of agents. The so-
cial average indicates a normalized flux of the traffic. 共a兲Free-flow
共
␣
=0.05,
␤
=0.65兲,共b兲jam 共
␣
=0.65,
␤
=0.05兲,共c兲metastable 共
␣
=0.10,
␤
=0.65兲, and 共d兲high-density 共
␣
=0.65,
␤
=0.65兲phases.
Those four points are shown in both Figs. 5and 6by open and
closed circles and open and closed squares, respectively.
YAMAUCHI et al. PHYSICAL REVIEW E 79, 036104 共2009兲
036104-4

flux observed ahead of the bottleneck. We can observe both
individual and social benefits simultaneously in Fig. 7, which
indicates social dynamics. For example, if we have a situa-
tion where the D-agent payoff is always larger than the
C-agent payoff, this inevitably leads to fewer C agents in the
society and eventually no C agents 共PC=0兲. In addition, if
the social maximum payoff is not consistent with this situa-
tion, game theory defines this as a social dilemma 关19兴.
In the free-flow phase, there is no difference between C-
and D-agents’ payoffs, and social profit does not depend on
the fraction of cooperation, as indicated in Fig. 7共a兲. Thus,
this has a trivial game structure, where no social dilemma
occurs 关19兴.
Figure 7共b兲indicates the jam phase. We may consider this
as an n-player prisoner’s dilemma 共nPD兲since the D strategy
always dominates over the C strategy 关19兴because the D
agents’ profit is higher than that of C agents at all fractions of
cooperation. However, this structure can never be nPD since
the social profit is almost constant, regardless of the fraction
of cooperation. This means that the jam phase has a trivial
game structure, where the strategy dynamics do not affect the
social efficiency.
Figure 7共c兲shows the metastable phase. In PC⬎0.6, be-
cause the D agents’ payoff is higher than that of C agents,
there is an incentive for C agents to drive in the second lane
共i.e., to convert to the D strategy兲.InPC⬍0.6, however,
there is no velocity difference between strategies 共i.e., no
incentive for C agents to change their strategy兲. Therefore,
the strategy dynamics based on each individual’s payoff
leads to a midway cooperation fraction around PC= 0.6. The
society cannot avoid D agents’ incursions, but the equilib-
rium favors coexisting C agents and D agents. One should
note that this internal equilibrium fraction around PC= 0.6 is
consistent with the social maximum payoff point. In this
sense, this particular game class also does not contain any
social dilemmas. This implies that in the metastable phase,
driving in the second lane to a moderate extent can improve
flow efficiency rather than all driving in only the first lane
共PC=1兲.
Finally, Fig. 7共d兲shows the high-density phase. Here, the
strategy payoff structure is “D dominant over C” dynamics,
similar to the jam phase, but the social maximum point is at
the fraction of cooperation PC= 0.95. This situation can be
exactly the same as nPD, namely, the social maximum pay-
off appears around PC= 1. However, the society inherently
has a robust incentive to increase D agents since D agents
can obtain higher payoff than C agents at any cooperative
fraction. Thus, D agents always increase, finally reaching
absorbed equilibrium PC= 0, where no C agents can survive.
This structure can be recognized by observing the high-
density phase area in Figs. 5and 6. Figure 6explicitly shows
that in the high-density phase, the D agents’ payoff is always
higher than that of C agents. Moreover, Fig. 5explicitly
shows that the social payoff at PC= 0 is the lowest among all
PC.
In the high-density phase, traffic flow seems so fragile
that exogenous impacts can easily lead to congestion. The
best situation having high social efficiency from an egalitar-
ian point of view is that all drivers stay in the first lane. The
society, however, cannot avoid the entry of a defecting agent
who is willing to drive in the second lane because he can
enjoy a higher payoff than C agents by adopting the D strat-
egy.
V. CONCLUSION
To clarify the social dilemma structure in traffic flow that
is one of the pure physical processes, we built a CA model
based on the SOV model with the open boundary condition
and applied it to the bottleneck problem caused by reducing
lanes from double to single. The established model contains
a game theory framework to deal with drivers’ decision-
making processes.
We found that the four traffic flow phases have various
game structures. Precisely speaking, the nPD game structure
arises in high-density phase areas, which account for most
ranges of car-creating probability
␣
and car-deleting prob-
ability
␤
. On the other hand, no dilemma exists at other flow
phases. In the free-flow and jam phases, there is no incentive
to drive in the overtaking lane. In situations of quite high
flow, in the high-density phase for instance, selfish drivers
changing lanes can obtain a higher payoff than altruistic
drivers staying in the first lane, but they cause a remarkable
decrease in social efficiency. In contrast to the high-density
phase, in the metastable phase, the social efficiency is likely
to increase when drivers use the overtaking lane.
What we present here implies that the social dilemma
structure that game theorists use may underlie a traffic flow
phenomenon that is believed to be a typical physics problem,
which might be interesting.
In this paper, we analyzed the bottleneck effect at closing
lanes from double to single. However, there might be lots of
other application cases. Potential bottlenecks might be one of
the examples. For example, a question if frequent lane
changes in a 1D like homogenous road 共without any other
obvious bottlenecks such as lane-closing, uphill, or tunnel兲
may also bring another social dilemma might be interesting.
We are presuming that “changing lanes” itself could call the
dilemma in a traffic flow. To figure out this is our next chal-
lenge.
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