Analytical Solutions of Fukui-Ishibashi (FI) Model and Quick-Start (QS) Model

Full text

Title
Social dilemma structures hidden behind a traffic flow with lane changes
Author names and affiliations
Jun Tanimoto, Shinji Kukida, and Aya Hagishima
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University Kasuga-koen,
Kasuga-shi, Fukuoka 816-8580, Japan
Corresponding author
Jun Tanimoto
tanimoto@cm.kyushu-u.ac.jp
Abstract
With aiming to dovetailing traffic flow analysis with evolutionary game theory, we investigate
a question: whether or not such structures can be formed from frequent lane changes in a
usual traffic flow without any explicit bottlenecks. In our model system, two classes of
driver-agents coexist: C-agents (cooperative strategy) always remain in the lane they are
initially assigned, while D-agents (defective strategy) try to change lanes to move ahead. In
relatively high-density flows, such as the metastable and high-density phases, we found
structures that correspond to either n-person Prisoner’s Dilemma (n-PD) games or to
quasi-PD games. In these situations, lane changes by D-agents create heavy traffic jams that
reduce social efficiency.
Keywords
Cellular automaton, Traffic flow, Lane change, Dilemma game, N-person Prisoner’s Dilemma
PACS: 89.40.-a

1. Introduction
Stimulated by growing social concern with traffic problems, several previous studies have
dealt with the so-called traffic model (e.g., [1–5]). In particular, many statistical physicists
consider these problems interesting because a traffic flow can be interpreted as a self-driven
multiparticle system. In that sense, among several traffic models, such as the kinetic theory of
gases (e.g., [6]), the fluid dynamical model (e.g., [7]), and the car-following model (e.g., [8]),
the cellular automaton (CA) model (e.g., [9]) has been most heavily investigated because of
its flexibility and expandability. After considering the basic question of whether each model
can successfully reproduce real traffic flows, which has been pursued in most previous studies,
maximizing traffic flux in an urban context might be a greater general concern. This kind of
pragmatic question has prompted studies on how much fore-seeing (or predicting) the
downstream flow situation, probably supported by sophisticated devices based on an ITS
(Intelligent Transportation System), contributes to improving traffic efficiency relative to the
usual flow (e.g., [10]).
Our interests slightly differ; we question whether a social dilemma structure used by game
theorists may underlie traffic flow phenomena that are believed to be typical physics problems.
Although no previous studies have been reported, the decision-making processes of drivers
unequivocally affect traffic flows. If we recognize that the traffic flow results from
competition for a finite resource (“road”) among many drivers who are seeking shorter
driving times and more comfortable driving, it is natural to ask whether or not a social
dilemma originates from the intentions of drivers. If so, we question the class of dilemma,
such as the Prisoner’s Dilemma (PD) game, Chicken (Snow Drift) game, or something
additional, lies hidden beneath the observed traffic flow and performs the game change when
the flow changes from one kind of phase to another. With this background, we detected that
several social dilemma structures, represented by n-person Prisoner’s Dilemma (n-PD) games,
appear in certain traffic flow phases at a bottleneck caused by a lane closing [11, 12]. We
confirmed that an n-PD game structure appears in the high-density phase area, but no social
dilemma exists in the free-flow and jam phases. It seems plausible for a social dilemma to
underlie such traffic flows because closing a lane creates an obvious bottleneck. Thus, our
next challenge is whether a social dilemma still lies beneath traffic flow that does not involve
any explicit bottleneck like a lane closing, on-ramp (merging), off-ramp (exit), or uphill travel.
Although there have been several reports of work addressing dilemmas in traffic flow [13, 14],
none have considered the issue we raise here.
This paper addresses whether or not only lane-changing actions by drivers can give rise to a
social dilemma in an ordinal two-lane road system with cyclic boundaries. The paper is
organized as follows. Section 2 describes our model and the simulation procedure, Section 3

presents and discusses the results, and Section 4 draws conclusions.
2. Model setup
2.1 Revised S-NFS model
We applied the revised S-NFS model [15] for driving vehicles forward. The S-NFS model,
developed by Sakai et al. [9], takes into account motions that are commonly observed in real
vehicles: slow-to-start (S2S), quick start (QS), and random braking (RB). S2S implies an
inertial effect, which is important for producing metastable states in fundamental diagrams.
QS results from an acceleration or deceleration by a driver who is anticipating the intentions
of both the precedent vehicle and several further precedent vehicles. Kokubo et al. [15]
modified this model by refining RB to improve reproducibility of the so-called three-phase
theory by Kerner [5]. The updating rules of the revised S-NFS model can be written as
follows.
Rule 1. “Acceleration”




1,min 0
max
)1(  ii vVv (1)
(only if




0
1
0

 iii vvGg then Rule 1 is applied).
Rule 2. “Slow-to-start”




i
t
i
t
siii sxxvv i-- --

11
1
)2( ,min (2)
(only if rand() ≤ q then Rule 2 is applied) and (if rand() ≤ r then si = S else si = 1).
Rule 3. “Perspective (Quick start)”




i
t
i
t
siii sxxvv i-- 
,min 2
)3( (3)
Rule 4. “Random brake”




1,1max 3
)4( - ii vv (4)
(only if rand() <1−pi then Rule 4 is applied).
)( Ggif i
pi = P1 (5-1)
)( Ggif i
pi = P2 for )0(
1
)0(

ii vv (5-2)
pi = P3 for )0(
1
)0(

ii vv (5-3)

pi = P4 for )0(
1
)0(

ii vv (5-4)
Rule 5. “Avoid collision”






4
11
4
)5( 1,min  -- i
t
i
t
iii vxxvv (6)
Rule 6. “Moving forward”
)5(1
i
t
i
t
ivxx 
 (7)
where
t
i
x
is the position of vehicle i at time t, )0(
i
v is the velocity )5(
i
v at the previous time
step t − 1, defined by
1-
-
t
i
t
i
xx
, si is the number of precedent vehicles from the ith driver’s
perspective, i
gis the gap between vehicle i and vehicle i+1 (thus, t
i
t
ii xxg - 1), and Vmax is
the maximum velocity. The notation rand() represents a random number drawn from the
uniform distribution on [0, 1]. The quantities G, q, r, S, P1, P2, P3, and P4 are model
parameters. The probability of random braking is given by 1–pi. We presume P1 > P2 > P3 >
P4.
2.2 Lane-changing rule
We applied the lane-changing rule used by Kukida et al. [16] in the CA model. That rule is
defined as follows:
Incentive criterion: )(
1
)()(
1
)( n
i
p
i
f
n
p
i
p
i
f
pvvgapvvgap  -- , (8)
Safe criterion: )()(
1
p
i
n
i
b
nvvgap - -. (9)
Here,
f
p
gap
is the number of unoccupied sites in front of the focal vehicle (agent i) in the
same lane,
f
n
gap
is the number of unoccupied sites in front of the focal vehicle in the
opposite lane, and
b
n
gap
is the number of unoccupied sites behind the focal vehicle in the
opposite lane. As schematically depicted in Figure 1, )( p
i
v and )(
1
p
i
v indicate the velocities of
the focal and the precedent vehicles in the same lane of the focal vehicle, while )(
1
n
i
v- and )(
1
n
i
v
mean the velocities of the following and the precedent vehicles in the opposite lane of the
focal vehicle. If a vehicle meets the two criteria (8) and (9), an actual lane change occurs with
probability PLC. This lane-change rule applies symmetrically in the two lanes.

2.3 Agent and simulation flow
In the system there are two types of agents: cooperators (C-agents) remain in the lane
initially assigned without making any lane changes, and defectors (D-agents) change lanes
according to the rule in Section 2.2. We use cyclic boundary conditions to keep the vehicle
density constant during a single simulation episode. The procedure in a single simulation
episode, repeats the following steps (i) to (v) in which steps (ii) to (v) correspond to one time
step.
(i) NS vehicles are generated and placed at random positions in the system. The
C-agent fraction among NS is Pc.
(ii) Only D-agents decide whether or not to change lanes, based on (8) and (9).
(iii) The random brake probability, 1 − pi, of all agents is determined according
to (5-1)–(5-4).
(iv) The next step velocity of all agents is determined from (1)–(4) and (6).
(v) All agents update their positions in the system by (7).
2.4 Simulation setting
In the actual numerical experiments, we set the system length to L = 500 and assumed the
following values for model parameters: q = 0.99, r = 0.99, S = 2, Vmax = 5, P1 = 0.999, P2 =
0.99, P3 = 0.98, P4 = 0.01, G = 15, and PLC = 1. We basically varied NS from 50 to 950 in
increments of 10 vehicles, although for cases in the middle density region ( 35.0012



), we
used NS increments of 1 due to the high sensitivity in that region because it contains a
metastable phase. We also varied Pc from 0 to 1 in increments of 0.1.
All results were drawn from 100 independent realizations. We evaluated the average
velocity of each agent by L/(travel time), and then averaged those over all C-agents
(D-agents) for the average payoff of each strategy. For the social payoff, we used the
time-averaged traffic flux.
Each realization (a single simulation episode) involved a run-up period (RU-period) to
attain a fully developed flow, followed by an observation period (O-period) in which the
above simulation results were evaluated. Because a developed flow may contain unsteady
features, the run-up and observation periods differed depending on traffic density.
For 65.0


, RU-period = 10 000 time steps and O-period = 500 time steps.
For 82.065.0



, RU-period = 25 000 time steps and O-period = 2500 time steps.
For 95.082.0



, RU-period = 50 000 time steps and O-period = 5000 time steps.

3. Results and discussion
Figure 2 shows fundamental diagrams for (a) Pc = 1 and (b) Pc = 0 in which each dilemma
class discussed below is identified by a different color. Figure 2(a) shows that flows of all
cooperators can exhibit the so-called metastable phase, while Figure 2(b) shows that no
metastable phase occurs in flows of all defectors. This seems plausible because a flow in
relatively high-density regions can be stable with high traffic flux so long as none of the
vehicles change lanes. In contrast, a flow with lane changes becomes volatile, since
turbulence caused by frequent lane changes promotes traffic jams. Behaviors of the observed
dilemma classes are explicitly discussed below; here, we merely note that only the Prisoner’s
Dilemma (including quasi-PD and quasi-little PD) class appears in the middle density region
with relatively high traffic fluxes. The Trivial game and Neutral game also appear there, but
these are not categories of social dilemmas.
Again, we deliberately discuss what kind of game structure, or dilemma class appearing in
the following section. With respect to the classification, we basically obey the general
definition of multiplayer’s games that refers to payoff functions of defector and cooperator as
well as social payoff. Whenever a peak of social payoff function (maximum social payoff,
MSP) entirely accords with Nash equilibriums (NE), we should call a game ‘Trivial’, meaning
none of dilemma. Otherwise, we should call it is a dilemma game. Meanwhile, if cooperator’s
payoff is always larger than that of defector for any cooperation fractions, a game should be
called cooperation-dominate (C-dominate), which implies the NE appears at all-defectors
state. If defector’s payoff is always larger than that of cooperator for any cooperation fractions,
a game should be called defection-dominate (D-dominate), which implies the NE appears at
all-cooperators state. If cooperator’s payoff is larger than that of defector for smaller
cooperation fraction and vice-versa for larger cooperation fraction (inevitably meaning both
payoff functions for cooperator and defector have an intersection), a game should be
classified as Polymorphic, which implies the NE is attracted to a certain cooperation fraction
between 0 and 1 (that is called an interior equilibrium). If cooperator’s payoff is smaller than
that of defector for smaller cooperation fraction and vice-versa for larger cooperation fraction,
a game should be classified as Bi-stable, which implies the NE is attracted to either
cooperation fraction of 0 or 1 depending on the initial cooperation fraction of an evolutionary
episode. If a game is D-dominate and has dilemma, it must be a Prisoner’s Dilemma (PD)
game. If a game is Polymorphic and has dilemma, it must belong to Chicken game. If a game
is Bi-stable and has dilemma, it must be a Stag Hunt game.
3.1 Effects of vehicle density
Figure 3 shows the payoff functions and velocity frequencies for Case A in Figure 2(a)

(1.0


), which is in the free-flow phase. Figure 3(a) shows that all payoffs for Case A are
insensitive to the cooperation fraction; this implies a kind of gameless situation. So we denote
this as a Neutral game class. This is not surprising because most of the vehicles in Case A run
at maximum velocity (see Figure 3(b)), so lane changes in the system are rare.
Figures 4 to 9 show counterparts of Figure 3for the other cases explicitly marked in Figure
2(a). The situation in Figure 4 ( 141.0


) can be called a Trivial game because Nash
equilibrium (NE) accords with the Maximum Social Payoff (MSP) at Pc = 0. This game is
dominated by defection, since the defector’s payoff is always larger than that of the
cooperator. However, the maximum social payoff also appears at all defector states. In a
nutshell, we call this a D-dominate Trivial game, which implies that more frequent lane
changing is preferable in this density region from both social and individual points of view.
Figures 6 ( 179.0


) and 10 ( 6.0


) show the same tendencies as in Figure 4. Thus, all
these should be classified as D-dominate Trivial games. The fact that the jam phase belongs to
the D-dominate Trivial game (Fig. 10) seems reasonable because lane changes into even a
slightly small vacant space between jamming vehicles brings benefits for not only the focal
vehicle who changes lanes but also for the society as a whole, even if its frequency is low.
Figure 5 ( 155.0


) suggests a weak Prisoner’s Dilemma (PD). This is confirmed by the
following facts. At Pc = 0, NE is trapped because the defector’s payoff is always greater than
that of the cooperator. MSP appears at Pc = 1 because the social payoff increases with
increasing cooperation fraction, although the effect is subtle. The same tendency appears in
Figure 8 ( 211.0


), although the extent of this dilemma (discussed in Sec. 3.2) seems more
severe than that in Figure 5. In Figure 8, the social payoff function does not monotonically
increase with the increase in the cooperation fraction, as observed in Figure 5; rather, it shows
an N-character shape, in which a local peak (much smaller than MSP at Pc = 1) appears at a
lower cooperation fraction. This point is carefully discussed in Sec. 3.2.
Figure 7 ( 194.0


) differs slightly from the simple PD because MSP is not observed at
Pc = 1, although NE is trapped at Pc = 0. At any rate, MSP is largely inconsistent with NE
since MSP, which is the peak of social payoff, appears above Pc = 0.5. Therefore, we call this
game structure a D-dominate quasi-Prisoner’s Dilemma game.
Figure 9 ( 244.0


) seems odd; it looks analogous to a D-dominate quasi-PD Game (Fig.
7), but it differs. EPO defined by the peak of social payoff appears below Pc = 0.5 and is
relatively close to NE found at Pc = 0. Therefore, we call this a D-dominate quasi-light PD
game.
Figure 11 shows the effects of vehicle density on the strength of dilemma,

, defined by
Nakata et al. [12] and expressed by

EPO
NEEPO
q
qq -


(10)
Here EPO
q and NE
q are the fluxes at MSP and NE, respectively. Figure 11 shows that the
density at severe dilemma strength is consistent with the density observed in the high-flux
region, including the metastable phase (Fig.2 (a)). This seems physically plausible because, in
this density region, a driver has a strong incentive for changing lanes to exploit other drivers
and ensure his own benefit is maximized (smaller travel time). However, when one driver
changes lanes, others might follow. Therefore, states with high flux, say in the metastable
phase, collapse with the phase shifting to the jam phase.
3.2 Multiple game structures at one vehicle density
As discussed above, Figure 8 ( 211.0


) shows the general tendency of the Prisoner’s
Dilemma game class, although the social payoff function has an N-character shape rather than
a monotonic increase. We discuss this point later. Figure 2(c) shows that, even when the same
traffic density is presumed (

= 0.211), the equilibrium points are scattered. Obviously, there
are three different subphases in this particular density region, denoted by I, II, and III in
Figure 2(c). Figure 12 shows all 100 realizations for 211.0


, sorted by the fluxes, for Pc =
1. The inset panels in Figure 12 indicate the typical payoff functions for the subphases I, II,
and III when we vary Pc. Note that the subphase with the higher flux (I) shows an obvious PD
tendency; however, for the subphase with the lower flux (III), the three payoff functions are
insensitive to the cooperation fraction, so the presence of PD behavior is ambiguous. This
implies that the N-character appears in Figure 8 because the payoff functions in Figure 8 were
drawn from averages over all the data shown in Figure 12. This fact lets us infer that this
particular traffic density region leads to different flow fields that realize different game
structures such as obvious PD ((I) and (II)) and almost Neutral Game (III). This bifurcation is
a stochastic process brought by a subtle difference of initial random allocation.
Figure 13 shows spatiotemporal diagrams for both lanes at Pc = 0 and Pc = 1 for subphases
I, II, and III. At Pc = 0 in which all drivers are defectors, a huge stop-and-go wave—a
jam—occurs in all realizations. However, at Pc = 1, some realizations successfully avoid
forming a jam; these correspond to high fluxes and, consequently, appear in subphase I. In
some other realizations, a jam only happens in one of the two lanes; these correspond to
reasonable fluxes and appear in subphase II. In other realizations, both lanes suffer jams; this
significantly reduces the flux and appear in subphase III. This particular bifurcation into the
three subphases I, II, and III is caused by the initial random allocation of vehicles between C
and D.

In short, we can say that in the density region near

= 0.211, where scattered points occur
in Figure 1(c) (and which is almost consistent with the density region found in Figure 11 for
large dilemma strength), the flow field potentially contains several different fully developed
states or equilibrium states. This means that several dilemma games may form with slightly
different structures. This is why the N-character shape appears in Figure 8, and why the
dilemma class underlying that shape is not obvious. Nevertheless, we can identify the game
class and dilemma strength, as a whole, by referring to Figure 11 at any arbitrary density.
4. Conclusions
For ordinal traffic flows, we have successfully demonstrated that there are hidden
social-dilemma structures evoked by drivers’ decisions whether or not they should change
lanes. This was confirmed by a series of numerical simulations using the revised S-NFS
cellular automaton model combined with a lane-changing model that we developed and
applied with cyclic boundary conditions.
Interestingly, social dilemmas, as classified by the Prisoner’s Dilemma game or its
variants, were only observed in situations of middle vehicle density; these situations
correspond to the region on the fundamental diagram, including the metastable phase, in
which data are scattered. This seems plausible because, when a driver is surrounded by other
vehicles, that driver has a serious incentive to change lanes. However, if all drivers make the
same decision, social efficiency declines phenomenally and huge traffic jams emerge. We also
evaluated the relation between dilemma strength and density of vehicles.
Our results imply that social-dilemma structures used by game theorists may underlie
traffic flow phenomena that are commonly believed to be mere physics problems. Although
the current model assumes symmetric lane-changing rules and makes no distinctions among
vehicles, asymmetric lane-changing behaviors and mixed-flow situations should be
considered if we want to model what happens in real traffic flows. These kinds of realistic
cases should be investigated in future work.
Acknowledgment
This study was partially supported by a Grant-in-Aid for Scientific Research by JSPS,
awarded to Prof. Tanimoto (#25560165), Tateishi Science & Technology Foundation. We
would like to express our gratitude to these funding sources.

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Fig
ure 1
Schematic explanation on how the lane change sub-model is defined.
The green granular indicates the focal vehicle agent i which is trying lane
change. Pink one expresses the vehicle who might be influenced by the focal
agent’s lane change.
0
00
00
0
0
0
0
0
000
0
0
0
0
00
00
0
0
0
0
0
000
0
0
0
f
p
gap
f
n
gap
Present lane
Neighbor lane
Present lane
Neighbor lane
)(
1
n
i
v-
b
n
gap
)( p
i
v)(
1
p
i
v
)(
1
n
i
v

Figure 2. Fundamental diagrams (normalized flux vs normalized density) for situations with
(a) all cooperators (Pc = 1) and (b) all defectors (Pc =0). Colored symbols identify (A)
Neutral game, (B) Prisoner’s Dilemma game at two densities in Panels (a) and (b), (C)
D-dominate Trivial game at three densities in Panels (a) and (b), (D) D-dominate
quasi-PD game, and (E) D-dominate quasi-light-PD game (see the main text). Panel (c)
shows an enlarged portion of Panel (a).
(b) Pc=0
0
1.2
0
Normalized density
(a) Pc=1
1
1
Normalized Flux
(A)
(B)
(E)
(C)
0.8
0.4
0.5 0.5
(D)
(A)
(B)
(E)
(C)
(D)
Enlarged in Panel (c)
0.15
Normalized density
0.35
0.211
(Ⅰ)
(Ⅱ)
(Ⅲ)
0.6
Normalized Flux
0.8
0.9
1.0
1.1
0.7
(c) Partially enlarged the case of Pc=1

Figure 3. Results for 1.0


at point A in Figure 1(a). (a) Effect of fraction of cooperators
(Pc) on payoff functions (velocity and flux). Red closed circles are average payoffs of
defectors, and blue triangle are average payoffs of cooperators. Green bold line indicates
traffic flux as a social payoff. (b) Effect of fraction of cooperators (Pc) on velocity
frequency. This behavior corresponds to a Neutral game.
Figure 4. Same as in Figure 3, except at 141.0


, which corresponds to one of the three
points C in Figure 2(a). This behavior corresponds to a D-dominate Trivial game.
4.5
4.6
4.7
4.8
4.9
5
0 0.5 1
0.65
0.67
0.69
0.71
0.73
0.75
PC
Normalized Velocity
Normalized flux
PC
0.5 0 1
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.141
0
Velocity frequency
4.5
4.6
4.7
4.8
4.9
5
0 0.5 1
0.45
0.47
0.49
0.51
0.53
0.55
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
C-agent’s velocity
D-agent’s velocity
Flux
△
●
1
0
(a) Payoff functions (b) Velocity frequency

Figure 5. Same as in Figure 3, except at 155.0


, which corresponds to one of the two
points B in Figure 2(a). This behavior corresponds to a weak Prisoner’s Dilemma game.
Figure 6. Same as in Figure 3, except at 179.0


, which corresponds to one of the three
points C in Figure 2(a). This behavior corresponds to a D-dominate Trivial game.
4.5
4.6
4.7
4.8
4.9
5
0 0.5 1
0.8
0.82
0.84
0.86
0.88
0.9
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.179
0
4.5
4.6
4.7
4.8
4.9
5
0 0.5 1
0.7
0.72
0.74
0.76
0.78
0.8
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.155
0

Figure 7. Same as in Figure 3, except at 194.0


, which corresponds to point D in Figure
2(a). This behavior corresponds to a D-dominate quasi-Prisoner’s Dilemma game.
Figure 8. Same as in Figure 3, except at 211.0


, which corresponds to one of the two
points B in Figure 2(a). This behavior corresponds to a weak Prisoner’s Dilemma game.
3.7
3.8
3.9
4
4.1
4.2
0 0.5 1
0.78
0.8
0.82
0.84
0.86
0.88
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.211
0
4.2
4.3
4.4
4.5
4.6
4.7
0 0.5 1
0.82
0.84
0.86
0.88
0.9
0.92
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.194
0

Figure 9. Same as in Figure 3, except at 244.0


, which corresponds to point E in Figure
2(a). This behavior corresponds to a D-dominate quasi-light PD game.
Figure 10. Same as in Figure 3, except at 6.0


, which corresponds to one of the three
points C in Figure 2(a). This behavior corresponds to a D-dominate Trivial game.
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0 0.5 1
0.4
0.42
0.44
0.46
0.48
0.5
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.6
0
3
3.1
3.2
3.3
3.4
3.5
0 0.5 1
0.73
0.75
0.77
0.79
0.81
0.83
PC
Normalized Velocity
Normalized flux
Velocity frequency
PC
0.5 0 1
(a) Payoff functions (b) Velocity frequency
V=5
V=4
V=2
V=3
V=1
V=0
V=5
V=4
V=2
V=3
V=1
V=0
0.244
0

Figure 11. Effects of vehicle density on dilemma strength,

. Each color identifies one of the
dilemma classses shown in Figure 2(a): (A) Neutral game, (B) Prisoner’s Dilemma game,
(C) D-dominate Trivial game, (D) D-dominate quasi-PD game, and (E) D-dominate
quasi-light-PD game.
Normalized Density
η
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 0.2 0.4 0.6 0.8 1
D-dominate q-light PDG
D-dominate q-PDG
D
-
dominate Trivial Game
Neutral G
ame
Prisoner’s Dilemma Game

Figure 12. Upper panel shows results from all 100 realizations for Pc = 1 and 211.0


. The
results are sorted by fluxes into the three subphases I, II, and III. Lower panels show the
corresponding payoff functions for subphases I, II, and III.
(Ⅰ)
(Ⅲ)
(Ⅱ)
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
3.7
4
4.3
4.6
4.9
0 0.5 1
0.7
0.8
0.9
1
(Ⅰ)
3.7
4
4.3
4.6
4.9
0 0.5 1
0.7
0.8
0.9
1
(Ⅱ)
3.7
4
4.3
4.6
4.9
0 0.5 1
0.7
0.8
0.9
1
(Ⅲ)
Normalized flux
PC PC PC
Normalized Velocity Normalized Flux
Order

Figure 13. Spatiotemporal diagrams at 211.0


for both lanes at (left) Pc = 0 and (right)
Pc = 1. Results are shown for each of the three subphases identified in Figure 12: (top
row) subphase I, (middle row) subphase II, and (bottom row) subphase III.
Left lane Right lane Right lane Left lane
Pc=0 Pc=1
(Ⅰ)
(Ⅱ)
(Ⅲ)