Analysis of Price of Anarchy in Traffic Networks With

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015
2227
Analysis of Price of Anarchy in Traffic Networks
With Heterogeneous Price-Sensitivity Populations
Xuehe Wang, Nan Xiao, Lihua Xie, Fellow, IEEE, Emilio Frazzoli, Senior Member, IEEE,
and Daniela Rus, Fellow, IEEE
Abstract— In this paper, we investigate how the scaled
marginal-cost road pricing improves the price of anarchy (POA)
in a traffic network with one origin–destination pair, where
each edge in the network is associated with a latency function.
The POA is defined as the worst possible ratio between the
total latency of Nash flow and that of the socially optimal flow.
All players in the noncooperative congestion game are divided
into groups based on their price sensitivities. First, we consider
the case where all players are partitioned into two groups in a
network with two routes. In this case, it is shown that the total
latency of the Nash flow can always reach the total latency of
the socially optimal flow if the designed road price is charged
on each link. We then analyze the POA for general case. For a
distribution of price sensitivities satisfying certain conditions, a
road pricing scheme is designed such that the unique Nash flow
can achieve the social optimal flow, i.e., POA = 1. An algorithm
is proposed to find the price scheme that optimizes the POA for
any distribution of price sensitivities and any traffic network with
one origin–destination pair. Finally, the results are applied to a
traffic routing problem.
Index Terms— Noncooperative congestion game, price of
anarchy (POA), price sensitivity, road pricing, traffic networks.
V
E
r
fr
F
e
fe
le ( f e )
lr ( fˆ)
L( f )
ρe ( fe )
N OMENCLATURE
Vertices representing road intersections.
Edges representing road segments.
Route connecting the origin and destination
points.
Flow on route r .
Total flow on the network.
An edge.
Flow on edge e.
Latency of edge e.
Latency of route r .
Total latency of the network.
Road pricing on edge e.
Manuscript received September 5, 2014; revised January 12, 2015; accepted
February 15, 2015. Date of publication March 24, 2015; date of current
version October 12, 2015. Manuscript received in final form March 3, 2015.
This work was supported by the National Research Foundation of Singapore
through the Singapore-MIT Alliance for Research and Technology, Singapore,
within the Future Urban Mobility Interdisciplinary Research Group Research
Programme. Recommended by Associate Editor C. Canudas-de-Wit.
X. Wang and L. Xie are with Exquisitus, Centre for E-City, School of
Electrical and Electronic Engineering, Nanyang Technological University,
Singapore 639798 (e-mail: [email protected]; [email protected]).
N. Xiao is with the Singapore-MIT Alliance for Research and Technology
Centre, Singapore 138602 (e-mail: [email protected]).
E. Frazzoli and D. Rus are with the Massachusetts Institute of Technology,
Cambridge, MA 02139 USA (e-mail: [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCST.2015.2410762
β
Fj
fr,β
Jr,β
F̄ j0
F̂ j0
p̄ j0
p̂ j0
Price sensitivity.
Total flow for group with price sensitivity β j .
Flow on route r contributed by group β.
Cost of route r for group β.
Total flow with β > β j0 .
Total flow with β < β j0 .
Percentage of players with β > β j0 .
Percentage of players with β < β j0 .
I. I NTRODUCTION
A
S THE technologies in transportation, communication,
control, and information rapidly advanced in recent
decades, the problem of how to create systems with high
efficiency has attracted a lot of attention. Especially, for
systems with noncooperative agents, each agent behaves
selfishly without considering the overall payoff of the whole
system, which may cause significant efficiency loss. This
situation occurs in a variety of fields, such as
communication network [1], operating temperature [2], market
mechanisms [3], and traffic congestion [4].
Congestion game is a branch of game theory [5], in which
the payoff of each player depends on the resources it chooses
and the number of players choosing the same resource. Like
all types of games, every player in a congestion game tries to
minimize his/her own cost and the equilibrium point yielded
in this way is known as Nash equilibrium, which is defined
as the action profile of all players where none of the players
can reduce his/her individual cost by a unilateral move. It is
shown in [6] that any congestion game is a potential game,
and the converse is proved in [7]: for any potential game,
there is a congestion game with the same potential function.
Therefore, congestion game inherits the desirable property of
potential game—the existence of at least one pure strategy
Nash equilibrium. However, it is widely known that Nash
equilibria often exhibit suboptimal behavior compared with
the socially optimal assignment. In the fast-developing society,
it becomes increasingly crucial to improve the efficiency of
the Nash equilibrium. There has been a lot of research on
the inefficiency of the Nash equilibrium. Early work studying
the Pareto inefficiency of the Nash equilibrium can be found
in [8]. It is shown in [9] that the price of anarchy (POA) [10],
which measures how the efficiency of a system degrades
due to selfish behavior of its agents, can be arbitrarily large.
For certain class of latency functions in networks of parallel
links, an upper bound and a lower bound of the POA are
given in [11]. To make the Nash equilibrium achieve the social
1063-6536 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
2228
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015
optimum, a distributed method is proposed in [12] to help
players find their paths. In [13], two models (latency model
and pricing model) are introduced in a routing network to
discuss the efficiency of the Nash equilibrium. In the latency
model, it is proved that the price of stability (POS), which is
defined as the best ratio between the cost of a Nash equilibrium
and the socially optimal cost, is unbounded, while in the
pricing model, all Nash equilibria have optimal flow under
certain conditions.
To improve the efficiency of the Nash equilibrium and
the performance of a network, Stackelberg game [14] is
introduced, in which a fraction of the players (leaders) are
assumed to be centrally controlled, while the rest of the
players (followers) are considered to react selfishly based on
the actions of the leaders. The main idea of Stackelberg game
is to find a leader strategy that induces the followers to react
in a way that minimizes the total cost of the system. The
existence and uniqueness of maximally efficient Stackelberg
strategy that leads the system to the global optimum is
investigated in [15]. For Stackelberg routing games on parallel
networks, an algorithm is introduced to compute the best
Nash equilibrium in [16], and the POS of this network is
analyzed, which is proved to be sensitive to demand change
when link flows are close to their capacities. In [17], the largest
latency first strategy is proposed to ensure that the POA of
the network be no more than 1/θ , where θ is the proportion
of leaders. In [18], a game theory controller is constructed
based on the feedback Stackelberg equilibrium framework to
reduce fuel consumption and oxides of nitrogen emissions.
However, Stackelberg game generally deals with networks
with homogeneous players. To promote the efficiency of the
Nash equilibrium in networks with heterogeneous players,
road pricing is introduced.
Road pricing has been implemented to many modern cities
all over the world. A case study on the traffic system in
California, USA, documented in [19] shows that transportation
pricing, such as congestion pricing, parking pricing, fuel-tax
pricing, vehicle miles of travel fees, and emissions fees, can
better manage the transportation systems to a great extent.
As another example, the electronic road pricing system in
Singapore is designed to charge motorists when they use the
road during peak hours, and it is effective in maintaining
an optimal speed range for both expressways and arterial
roads [20], [21]. In general, road price on each link of the
traffic network is a function of the flow on this link. Due
to individual user’s failure to share the cost he/she imposes
on other users, marginal cost pricing (first-best pricing) is
introduced, which illustrates that road user using congested
road should pay a fee that is equal to the difference between
the marginal social cost and the marginal private cost [22].
The marginal cost pricing is established in the case with
homogeneous players and the Nash equilibrium can achieve
the social optimum by charging a marginal-cost price on each
link in the network [23]. For the case with heterogeneous
players, many road pricing schemes are designed to improve
the network performance. In [24], the existence of optimal
static tolls that optimize the behavior of a single
commodity network is shown. For multicommodity networks,
Fleischer et al. [25] show that there exist static tolls making
selfish users act in a way that minimizes the average latency.
To get a desired equilibrium flow for traffic networks, the
influence of static tolls on drivers’ decisions is discussed
in [26]. In [27], a dynamic road pricing is established to
achieve socially beneficial trip timing in average strategy
fictitious play. To spread out players’ choices, weighted
entropy is applied to the dynamic road pricing [28].
A pricing-based energy control strategy is proposed in [29]
to remove the peak load for smart grid. However, we cannot
ensure that there always exists a road pricing which can
eliminate all the efficiency losses resulting from selfish
routing. In this paper, we quantify the effect of road pricing by
the POA for the case when players have heterogeneous price
sensitivities.
The main contribution of this paper is summarized as
follows. First of all, we formulate the routing problem on a
traffic network with one origin–destination pair as a congestion
game, in which the scaled marginal-cost pricing is charged on
each link to affect road users’ routing choices. To be close to
reality, road users are assumed to have heterogeneous price
sensitivities and are divided into groups accordingly. Second,
we analyze the Nash flow and the POA for the network.
Some preliminary work about the conditions for the POA
to reach 1 can be found in [30], while the present paper
includes refined results, additional case studies on two group
players, and an algorithm for finding the best POA. For two
groups case, we show that the POA can always achieve 1 by
an appropriate scaled marginal-cost pricing on each link. For
general case with more than two groups, the existence and
uniqueness of Nash flow is discussed. If the distribution of
price sensitivity satisfies certain conditions, we prove that
there exists an optimal road pricing such that POA = 1.
Otherwise, the POA cannot achieve 1. For any traffic network
and distribution of price sensitivity, an algorithm is given
to calculate a scaled marginal-cost that minimizes the POA.
Finally, we apply the results to the traffic routing problem,
and numerical examples and real data simulation verify that
the designed scaled marginal-cost pricing indeed reduces the
total latency of the network and the optimal POA depends on
both the distribution of price sensitivities and the topology and
parameters of the traffic network.
The rest of this paper is organized as follows. In Section II,
the problem of POA of a traffic network with scaled
marginal-cost pricing on each link is formulated. How to
design the optimal road pricing is discussed in Section III.
In Section IV, we present numerical examples and real
data simulations. The conclusion and future work are stated
in Section V.
II. P ROBLEM F ORMULATION
Consider a traffic network (V, E) with one
origin–destination pair (Fig. 1). Each player chooses
his/her route from a common set of routes R = {r1 , . . . , r N },
where each route r ∈ R consists of one or several links.
Define the vector of route flows asfˆ = ( fr1 , . . . , fr N ) and
the total flow of all routes as F = r∈R fr .
WANG et al.: ANALYSIS OF POA IN TRAFFIC NETWORKS
2229
Fig. 1.
Simple traffic network (V, E) with V = {v 1 , v 2 , v 3 } and
E = {e1 , e2 , e3 , e4 }. In this example, R = {r1 , r2 , r3 } is described by
r1 = {e1 , e2 }, r2 = {e1 , e3 }, and r3 = {e4 }.
Note that a link e ∈ E, e.g., e1 in Fig. 1, can be contained
by more than one route. The latency of one link e ∈ E is
associated with the total flow on this link as
l e ( f e ) = d e f e + ce
(1)
where f e =
r∈R:e∈r f r is the total flow on link e, and
de ≥ 0 and ce ≥ 0 are known constants.
Note that a given flow on a road can correspond to either
a high density of vehicles on a congested road, in which
case the latency is large, or a low density of vehicles on a
free-flow road, in which case the latency is small [31]. Due
to this phenomenon, the latency is not uniquely determined
by the flow, and depends on the congestion state of the road.
However, in our formulation, we try to grasp the main features
of the relationship between the traffic flow and travel time
before congestion. Therefore, we adopt the linear relationship
between delay and flow used in [9] and [32] to formulate our
problem.
The latency on route r ∈ R is the sum of the latencies of
links on this route
le ( fe ).
(2)
lr ( fˆ) =
e∈r
Suppose that the road manager charges a toll ρe ( fe ) on
each link e ∈ E to affect road users’ routing behaviors, and
each road user has a price sensitivity β > 0 which may be
different for different road users. We consider the case where
there are finite possible values for β and all road users are
classified into groups according to their price sensitivities. Let
B = {β1 , . . . , β M } denote the set of price sensitivities and
P = { p1, . . . , p M } be the corresponding distribution.
Therefore, the total flow for group with β j ∈ B is
F j = p j F. Denote fr,β by the flow on route r ∈ R
contributed by group with β ∈ B and f = ( fr1 ,β1 , . . . ,
N
=
fr1 ,β M , . . . , fr N ,β1 , . . . , fr N ,β M ). Note that
i=1 f ri ,β j
M
f
=
f
for
all
p j F = F j for all β j ∈ B, and
ri
j =1 ri ,β j
ri ∈ R. By incorporating the price sensitivity, we define the
cost of route r ∈ R for group with β ∈ B as
Jr,β ( f ) =
(le ( f e ) + βρe ( f e )).
(3)
e∈r
For the noncooperative congestion game formulated above,
we assume that each player selfishly chooses to travel on
route with minimal individual cost (3). A Nash flow is defined
as follows.
Definition 2.1: A flow f ne is called a Nash flow if for any
β j ∈ B and r1 , r2 ∈ R
frne
> 0, frne
> 0 ⇒ Jr1 ,β j ( f ne ) = Jr2 ,β j ( f ne )
1 ,β j
2 ,β j
frne
1 ,β j
> 0,
frne
2 ,β j
= 0 ⇒ Jr1 ,β j ( f
ne
) ≤ Jr2 ,β j ( f
ne
).
(4)
(5)
Note that Nash flows always exist for congestion games of
the type considered in [6] and [7].
In [23], it is shown that games with homogeneous players,
i.e., all players have the same price sensitivity (M = 1),
can achieve the social optimum by charging all players a
marginal-cost price. The marginal cost for the latency function
given in (1) is de f e . However, in our general formulation
with M > 1, the social optimum cannot be achieved by
charging de f e on each link e ∈ E, since each group of drivers
have different price sensitivity. Therefore, we redefine the
pricing function as the scaled marginal-cost toll as
ρe ( f e ) = μde f e
(6)
where μ ≥ 0 is a parameter to be designed. Note that Nash
flow is a function of the toll ρe ( f e ) which is a function of μ.
Therefore, Nash flow is also a function of μ. In this paper,
we will use f ne (μ) and f ne interchangeably when no
confusion is caused.
The total latency of the network is given by
fr · lr ( fˆ).
(7)
L( f ) =
r∈R
Then, the socially optimal flow f ∗ , which is defined as
the flow that minimizes the total latency of the network, is
described by
f ∗ = arg inf L( f )
f
with
(8)
fr = F.
r∈R
The POA, which is the worst possible ratio between the
total latency of a Nash flow and that of the optimal flow, is
defined as
L( f ne )
.
(9)
POA = sup
∗
f ne L( f )
The POA characterizes the worst case efficiency loss of all
possible Nash flows, and the larger the POA, the larger the
efficiency loss. Obviously, POA is always larger or equal
to 1 and POA = 1 indicates that the social optimum is
achieved.
In this paper, we seek to design the pricing function to
minimize the POA for a given distribution of price sensitivities.
Namely, our goal is to find μ∗ for (6) such that
L( f ne (μ∗ )) = inf L( f ne (μ)).
μ≥0
(10)
In other words, the Nash flow f ne obtained after introducing
the designed road pricing ρe ( fe ) = μ∗ de f e leads to the
minimal worst case efficiency loss.
III. D ESIGN OF O PTIMAL ROAD P RICING
In this section, we aim to design the optimal scaled
marginal-cost pricing that minimizes the POA for networks
with heterogeneous players. Without loss of generality, we
assume β1 > β2 > · · · > β M . Denote F̄ j0 = F1 + · · · + F j0 −1
by the total flow with β > β j0 and F̂ j0 = F j0 +1 + · · · + FM
2230
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015
by the total flow with β < β j0 . Let p̄ j0 = F̄ j0 /F and
p̂ j0 = F̂ j0 /F. For route ri ∈ R, define Vi = e∈ri ce . For
any ri , r j ∈ R, denote
di j =
de .
(11)
e∈ri
A. Nash Flow Analysis for Case With
Two Groups and Two Routes
First, we consider the special case where there are two
routes R = {r1 , r2 } in the network with two different groups
B = {β1 , β2 }. We will show that, for any distribution of B,
there always exists μ∗ such that POA = 1.
In the following lemma, the existence and uniqueness of
Nash flow for the special case is guaranteed.
Lemma 3.1: Under Assumptions 3.1 and 3.2, the Nash
flow of the two groups case always exists and is uniquely
determined as follows.
V −V
frne
= 0,
1 ,β1
frne
=
1 ,β2
frne
=
2 ,β2
2 1
F (d11 −d12 )− 1+μβ
2
F (d11 +d22 −2d12 )
V2 −V1
1+μβ2
d11 + d22 − 2d12
p2 F(d11 − d12 ) − p1 F(d22 − d12 ) −
V2 −V1
1+μβ2
frne
= 0,
1 ,β1
3) If
V2 −V1
F (d11 −d12 )− 1+μβ
1
F (d11 +d22 −2d12 )
frne
1 ,β1
frne
2 ,β1
=
=
2
frne
(d1k − d2k ) =
k
k=1
V2 − V1
.
1 + μβ2
(13)
Since β1 > β2 and V1 < V2 , (V2 − V1 )/(1 + μβ2 ) >
(V2 − V1 )/(1 + μβ1 ). Therefore
(1 + μβ1 )
2
d1k frne
+ V1 > (1 + μβ1 )
k
k=1
2
d2k frne
+V2 . (14)
k
k=1
= 0 and frne
Thus, frne
1 ,β1
2 ,β1
ne
and fr2 ,β1 + frne
=
2 ,β2
ne
ne
fr1 ,β2 + fr2 ,β2 = p2 F, we
= p1 F. Insert frne
+ frne
= frne
1
1 ,β1
1 ,β2
ne
fr2 into (13) and note that
obtain frne
= ( p1 F(d22 − d12 ) +
1 ,β2
p2 F(d22 − d12 ) + (V2 −V1/1 + μβ2 ))/(d11 + d22 − 2d12 )
= ( p2 F(d11 − d12 ) − p1 F(d22 − d12 ) − (V2 − V1 /
and frne
2 ,β2
> 0.
1 + μβ2 ))/(d11 + d22 − 2d12 ). Obviously, frne
1 ,β2
ne
ne
ne
Under Assumption 3.2, fr2 = fr2 ,β1 + fr2 ,β2 > 0 when
μ = 0. Thus, (d11 − d12 )F > V2 − V1 . Therefore,
>
0. Since
(d11 − d12 )F − (V2 − V1 /1 + μβ2 )
frne
>
0,
0
≤
p
<
(F(d
−
d12 ) −
1
11
2 ,β2
The
Nash
(V2 − V1 /1 + μβ2 ))/(F(d11 + d22 − 2d12 )) .
flow 1) is proved.
Similarly, we can show 2) and 3).
Based on Lemma 3.1, we give the total latency of the
Nash flow.
Lemma 3.2: Under Assumptions 3.1 and 3.2, the total
latency of the Nash flow for the two groups case is as follows.
frne
= p2 F,
1 ,β2
μβ2
(V2 −V1 )2
d11 + d22 − 2d12 (1 + μβ2 )2
L F (F) =
.
(15)
(d11 − d12 )(d22 − d12 ) 2
F
d11 + d22 − 2d12
(d11 − d12 )V2 + (d22 − d12 )V1
+
F. (16)
d11 + d22 − 2d12
V −V
frne
= 0.
2 ,β2
< p1 ≤ 1
V2 −V1
1+μβ1
d11 + d22 − 2d12
p1 F(d11 − d12 ) + p2 F(d11 − d12 ) −
V2 −V1
1+μβ1
d11 + d22 − 2d12
frne
= 0.
2 ,β2
2 1
F (d11 −d12 )− 1+μβ
2
F (d11 +d22 −2d12 )
L( f ne ) = L F (F)−
V −V
p1 F(d22 − d12 ) − p2 F(d22 − d12 ) +
frne
= p2 F,
1 ,β2
By the definition of the Nash flow, if frne
> 0, frne
>0
1 ,β2
2 ,β2
2 1
F (d11 −d12 )− 1+μβ
1
F (d11 +d22 −2d12 )
frne
= p1 F,
2 ,β1
(12)
with
d11 + d22 − 2d12
≤ p1 ≤
rk :e∈rk
frne
dik + Vi .
k
V −V
p1 F(d22 − d12 ) + p2 F(d22 − d12 ) +
2 1
F (d11 −d12 )− 1+μβ
2
F (d11 +d22 −2d12 )
r k ∈R
1) If 0 ≤ p1 <
frne
= p1 F
2 ,β1
V −V
2) If
e∈ri
= (1 + μβ j )
rj
If Vi = V j for some ri = r j , then there exist infinite Nash
flows as defined in Definition 2.1. However, the total latency
is identical for all these Nash flows, and so is the POA. For the
special case with Vi = V j for all ri = r j , there exist infinite
Nash flows for any distribution of B, but the POA can always
= fr∗k , rk ∈ R. In most
achieve 1 for any μ ≥ 0 since frne
k
practical cases, Vi are different for different routes. Therefore,
we make the following assumption.
Assumption 3.1: Vi = V j for all i = j .
To further ensure the positivity property of the Nash flows
with road pricing, we give the following assumption.
Assumption 3.2: For the untolled situation with μ = 0, any
Nash flow f ne has frne > 0 for all r ∈ R.
Under Assumption 3.1, we can let V1 < V2 < · · · < VN .
Assumption 3.2 indicates that every route will be in use when
all routes are free of charge.
1) If 0 ≤ p1 <
Proof: First, we prove
the Nash flow 1). By the price
function (6) and fe = r∈R:e∈r fr , we rewrite (3) as
Jri ,β j ( f ne ) =
frne
+
c
(1 + μβ j )de
e
k
2) If
2 1
F (d11 −d12 )− 1+μβ
2
F (d11 +d22 −2d12 )
V −V
≤ p1 ≤
2 1
F (d11 −d12 )− 1+μβ
1
F (d11 +d22 −2d12 )
L( f ne )
= (d11 − d12 )F 2 + (V2 − V1 − 2(d11 − d12 )F)F p1
+ (d11 + d22 − 2d12 )F 2 p12 .
(17)
V −V
3) If
2 1
F (d11 −d12 )− 1+μβ
1
F (d11 +d22 −2d12 )
< p1 ≤ 1
L( f ne ) = L F (F) −
μβ1
(V2 − V1 )2
.
d11 + d22 − 2d12 (1 + μβ1 )2
(18)
WANG et al.: ANALYSIS OF POA IN TRAFFIC NETWORKS
2231
ne in Lemma 3.1 into (7) and
Proof: Substitute all fr,β
j
ne
simplify to obtain L( f ).
In the following theorem, we give the main result of this
section.
Theorem 3.1: Under Assumptions 3.1 and 3.2, for any
given p1, there always exists μ∗ satisfies L( f ne (μ∗ )) =
inf μ≥0 L( f ne (μ)) = L( f ∗ ), i.e., POA = 1. Furthermore
L( f ne (μ∗ )) = L( f ∗ ) = L F (F) −
1 (V2 − V1 )2
4 d11 + d22 − 2d12
(19)
where L F (F) is as defined in (16).
Proof: The optimal flow of this network is:
=
(V2 − V1 + 2F(d22 − d12 )/2(d11 + d22 − 2d12 ))
fr∗1
and fr∗2 = (V2 − V1 + 2F(d11 − d12 )/2(d11 + d22 − 2d12 )).
Thus, L( f ∗ ) = L F (F)−(1/4)((V2 − V1 )2 /d11 + d22 − 2d12 ).
For 0 ≤ p1 < (F(d11 − d12 ) − (V2 − V1 /2))/(F(d11 +
d22 − 2d12 )), if we set μ∗ = (1/β2 ), the Nash flow is as
in Lemma 3.1 1). Based on the total latency (15), it is easy
to verify that the social optimum is reached at μ∗ = 1/β2 ,
i.e., L( f ne (μ∗ )) = L( f ∗ ) and POA = 1. Similarly, for
(F(d11 −d12 )−(V2 − V1 /2))/(F(d11 + d22 −2d12)) < p1 ≤ 1,
the social optimum can be achieved at μ∗ = (1/β1 ) with
the Nash flow as shown in Lemma 3.1 (iii). If p1 =
(F(d11 − d12 ) − (V2 −V1 /2))/(F(d11 + d22 −2d12 )), for any
μ∗ ∈ [(1/β1 ), (1/β2 )], the Nash flow is as in Lemma 3.1 2),
substitute p1 into (17), we can check POA = 1.
B. Nash Flow Analysis for General Case
In this section, we will analyze the POA for the case
with several groups and routes connecting one origin and one
destination.
Let {A1 , . . . , A N } and {B1 , . . . , B N } be the set of solutions
to the following equations:
Ak = 1
(20)
r k ∈R
Bk = 0
r k ∈R
r k ∈R
Ak dik =
r k ∈R
Bk dik + Vi =
(21)
Ak d j k ∀ri = r j
(22)
Bk d j k + V j ∀ri = r j .
(23)
r k ∈R
r k ∈R
We can see from the proof of Lemma 3.3 that {A1 , . . . , A N }
and {B1 , . . . , B N } always exist. The following lemma, whose
proof is given in the Appendix, shows that under certain
conditions, a Nash flow always exists and it is unique.
Lemma 3.3: Under Assumptions 3.1 and 3.2, for any μ ≥ 0
and a given distribution of B, if there exists a group with
β j0 ∈ B satisfying
⎧
B1
⎪
⎪p̂ j0 < A1 +
⎨
(1 + μβ j0 )F
(24)
BN
⎪
⎪
⎩ p̄ j0 < A N +
(1 + μβ j0 )F
then a Nash flow always exists and it is uniquely determined
as follows:
1) the Nash flow for group β j0 is given by
⎧
1
⎪
⎪
A1 F − F̂ j0 +
B1 ,
if k = 1
⎪
⎪
⎪
1 + μβ j0
⎪
⎨
1
B N , if k = N
frne
(25)
= A N F − F̄ j0 +
k ,β j0
1 + μβ j0
⎪
⎪
⎪
⎪
1
⎪
⎪
Bk ,
otherwise;
⎩Ak F +
1 + μβ j0
2) all players with β < β j0 choose route r1 ,
i.e., frne
= F j0 +1 , . . . , frne
= FM ;
1 ,β j0 +1
1 ,β M
3) all players with β > β j0 choose route r N ,
i.e., frne
= F1 , . . . , frne
= F j0 −1 .
N ,β1
N ,β j0 −1
Based on the proof of Lemma 3.3, the properties of the
Nash flows can be concluded as follows.
1) If there is a group choosing more than one routes at
Nash flow, it must choose several successive routes,
e.g., {r1 , r2 } and {r2 , r3 , r4 }.
2) If players with β j ∈ B choose routes {r j1 , . . . , r j2 },
then players with β > β j choose some routes r ∈
{r j2 , . . . , r N } and players with β < β j choose some
routes r ∈ {r1 , . . . , r j1 }.
Consider the case where players with β j0 ∈ B choose routes
{rk1 , . . . , rk2 } with k1 ≤ k2 and {k : 1 ≤ k < k1 } ∪ {k :
k2 < k ≤ N} = ∅, and all other players choose only one
route r ∈ R at Nash flow. Based on the properties of the
Nash flows described above, all players with β < β j0 only
choose a route r ∈ {r1 , . . . , rk1 } and all players with β > β j0
only choose a route r ∈ {rk2 , . . . , r N }. Therefore, for route
rk ∈
/ {rk1 , . . . , rk2 }, frne
is the sum of certain groups’ total
k
flow.
k1 −1 ne
/F, Fk1 =
Denote p̃k = frne
k=1 frk , Fk2 =
k
N
ne
ˆ∗
k=k2 +1 f rk , ρk1 = Fk1 /F, and ρk2 = Fk2 /F. Let f be
the vector of route flows corresponding to the socially optimal
flow. A distribution of B is called a critical point, if it satisfies
⎧
∗
⎪
⎪ p̃k = frk /F, for 1 ≤ k < k1 or k2 < k ≤ N
⎪
⎪
⎨
Bk
p̂ j0 − ρk1 < Ak1 + 1
(26)
2F
⎪
⎪
⎪
B
⎪
⎩ p̄ j − ρk < Ak + k2
0
2
2
2F
where Ak1 , Ak2 , Bk1 , and Bk2 are the k1 th and k2 th elements
of the sets {A1 , . . . , A N } and {B1 , . . . , B N }, respectively.
Note that (24) and (26) are exclusive with each other. The
next theorem, as the main result of this paper, shows how the
road pricing and the distribution of price sensitivities influence
the POA.
Theorem 3.2: Under Assumptions 3.1 and 3.2, for a given
distribution of B, if:
1) there exists a group β j0 ∈ B satisfying
⎧
B
⎪
⎨p̂ j0 < A1 + 1
2F
(27)
⎪
⎩ p̄ j < A N + B N ;
0
2F
or
2232
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015
2) the distribution of B is a critical point with
⎧
∗
⎪
⎪ p̃k = frk /F, for 1 ≤ k < k1 or k2 < k ≤ N
⎪
⎨
Bk
p̂ j0 − ρk1 < Ak1 + 1
2F
⎪
⎪
⎪
⎩ p̄ − ρ < A + Bk2
j0
k2
k2
2F
then μ∗ = 1/β j0 satisfies
(28)
L( f ne (μ∗ )) = L( f ∗ )
N
N
N
Bk Vk
=
Ak dk1 F 2 +
Ak Vk F +
4
k=1
k=1
k=1
(29)
i.e., POA = 1. Otherwise, POA > 1.
Proof: First, we consider case 1). From Lemma 3.3 and
the definition of L( f ) in (7), if group β j0 ∈ B satisfies (24),
then the total latency of the Nash flow with road pricing is
L( f ne ) =
N
Ak dk1 F 2 +
k=1
N
Ak Vk F +
k=1
N
Bk Vk μβ j0
. (30)
(1 + μβ j0 )2
k=1
Taking the partial derivative of (30) with respect to μ,
we can see that the minimum of L( f ne ) is achieved at
μ∗ = 1/β j0 . Substitute μ∗ into (30), we have
L( f ne (μ∗ )) =
N
Ak dk1 F 2 +
k=1
N
Ak Vk F +
k=1
N
Bk Vk
. (31)
4
k=1
Furthermore, we can check that the total latency of the Nash
flow is equal to the total latency of the socially optimal flow,
i.e., POA = 1.
For case 2), only players with β j0 ∈ B choose more than
one routes. Based on the definition of critical point, the Nash
flow is determined as
⎧ ∗
f rk ,
if 1 ≤ k < k1 or k2 < k ≤ N
⎪
⎪
⎪
⎨F̂ j0 − Fk1 + f ne , if k = k1
rk1,β j0
=
(32)
frne
k
F̄ j0 − Fk2 + frne
, if k = k2
⎪
⎪
k2,β j0
⎪
⎩ f ne ,
otherwise
rk ,β j
0
where
k2
frne
k ,β j
0
>
0 for all k1
≤
≤
k
k2 and
frne
= F j0 .
k ,β j0
Similar to case 1), under Assumptions 3.1 and 3.2, we can
show that μ∗ = 1/β j0 can lead to L( f ne (μ∗ )) = L( f ∗ ),
i.e., POA = 1.
If the distribution of B does not satisfy (i) or (ii), there are
at least two groups choosing more than one routes at any Nash
flow. Assuming that group β j1 chooses ri1 and ri2 , and group
β j2 chooses ri1 and ri2 , by the definition of a Nash flow, we
have
Vi − Vi1
frne
(di1 k − di2 k ) = 2
(33)
k
1 + μβ j1
k=k1
r k ∈R
r k ∈R
frne
(di1 k − di2 k ) =
k
Vi2 − Vi1
1 + μβ j2
.
(34)
The total latency of a Nash flow L( f ne ) is a function of both
1/(1 + μβ j1 ) and 1/(1 + μβ j2 ). It follows from β j1 = β j2 that
POA > 1.
Remark 3.1: Note that for critical point, under
Assumptions 3.1 and 3.2, if we set μ∗ = 1/β j0 , then
the Nash flow is uniquely determined as follows:
1) for group β j0
⎧
0,
if 1 ≤ k < k1 or k2 < k ≤ N
⎪
⎪
⎪
⎪
1
⎪
⎪
⎪
⎨ Ak1 F − F̂ j0 + Fk1 + 2 Bk1 , if k = k1
frne
=
1
k ,β j0
⎪
Ak2 F − F̄ j0 + Fk2 + Bk2 ,
if k = k2
⎪
⎪
⎪
2
⎪
⎪
1
⎪
⎩ A k F + Bk ,
otherwise;
2
(35)
2) all players with β < β j0 only choose one route
= fr∗k for 1 ≤ k < k1 ;
r ∈ {r1 , . . . , rk1 } and frne
k
3) all players with β > β j0 only choose one
route r ∈ {rk2 , . . . , r N } and frne
= fr∗k for
k
k2 < k ≤ N.
From above, we conclude that, if the distribution of
B satisfies certain conditions, we can always find a μ∗ ≥ 0
such that the POA can achieve 1 by charging the designed
toll ρe ( f e ) = μ∗ de f e on each link e ∈ E. Moreover, in
the following theorem [9], it is shown that the POA can be
bounded by 4/3 if the network has linear latency functions.
Theorem 3.3 [9]: For networks with multiple origin–
destination pairs, if the latency functions are linear, then
POA ≤ (4/3).
The game we considered is a congestion game and it is
well known that any congestion game is a potential game [6],
which possesses a desirable property—the existence of a pure
Nash equilibrium [7]. Therefore, for any distribution of B
and any given μ ≥ 0, we can always find a Nash flow.
Under Assumptions 3.1 and 3.2, we provide a method to
design the optimal μ∗ that minimizes the POA as summarized
in Algorithm 1.
IV. N UMERICAL R ESULTS AND R EAL DATA S IMULATIONS
In this section, we first use numerical examples to illustrate
that whether the POA can achieve 1 for any given network
depends on the distribution of B. Then, based on the real traffic
data in Singapore, we analyze the POA for two different traffic
networks with identical distribution of B, which indicates that
whether the POA can reach 1 also depends on the topology
and parameters of a traffic network.
A. Numerical Examples
Consider the road network with three routes, as shown
in Fig. 1, and suppose that there are five groups of drivers with
a total flow F = 2. The parameters of the network are given by
{de1 = 25, de2 = 68, de3 = 47, de4 = 86}, {ce1 = 10, ce2 = 21,
ce3 = 85, ce4 = 98}, and B = {100, 1, 0.35, 0.18, 0.01}.
The total latency of the Nash flow without tolls is
L( f ne (0)) = 290.7, and the socially optimal flow on each
route is fr∗1 = 0.8277, f r∗2 = 0.5166, f r∗3 = 0.6557 with
L( f ∗ ) = 280.3. Next, we analyze the POA for three different
distribution of B.
WANG et al.: ANALYSIS OF POA IN TRAFFIC NETWORKS
Algorithm 1 Best POA
For the first case, the distribution of B is given by
P = {0.25, 0.46, 0.04, 0.20, 0.05}. Based on Algorithm 2,
condition 1) in Theorem 3.2 is satisfied. Therefore, we can
set μ∗ = 1/β2 = 1 to obtain L( f ne (μ∗ )) = L( f ∗ ), i.e.,
POA = 1. As expected, the road pricing decreases the total
latency.
For the second case, the distribution of B is
P = {0.1225, 0.2053, 0.2972, 0.2735, 0.1015}. There exists
a group β3 such that p̂3 < fr∗1 /F = 0.4138, p̃3 = p1 + p2 =
fr∗3 /F = 0.3278, i.e., condition 2) in Theorem 3.2 is satisfied,
which can be checked by Algorithm 2. Thus, we can set
μ∗ = 1/β3 = 2.857 to achieve POA = 1.
For the third case, the distribution of B is
P = {0.35, 0.40, 0.11, 0.09, 0.05}. In this case, according
to Algorithm 2, the conditions in Theorem 3.2 are not
satisfied. Using Algorithm 1, we get μ∗ = 0.9729 and
L( f ne (μ∗ )) = 281.0. The road pricing reduces the total
latency, but POA = 1.003 > 1.
Note that the results in this paper can be applied to any road
network with one origin–destination pair and to the case with
more user groups.
B. Real Data Simulations
For real data analysis, we consider two road networks in
Singapore. One road network is in the east of Singapore
(Fig. 2), and the other one is in central business
2233
Algorithm 2 Condition Checking of Theorem 3.2
Fig. 2.
Traffic network in the east of Singapore.
district (CBD) (Fig. 3). Assume that road users are
divided into groups according to their vehicle modes—car,
motorcycle, taxi, and bus. According to the 2004 stated
preference survey data, the price sensitivities (min/cent) for
these four groups are 0.25, 0.36, 0.20, and 0, respectively.
The vehicle distribution of these four vehicle types
provided by Singapore land transport statistics in brief 2005 is
{0.7239, 0.1884, 0.0281, 0.0596}. Assume that the total flow
of the network is 500.
In the following simulation, we will show that for the
network, as shown in Fig. 2, the POA cannot achieve 1
since both conditions 1) and 2) in Theorem 3.2 are not
satisfied. While for the CBD road network, as shown in Fig. 3,
the distribution of B satisfies condition 1) in Theorem 3.2.
Therefore, we can find μ∗ such that POA = 1.
1) East Road Network of Singapore: To see the road
network in Fig. 2 clearly, we extract its structure and show
2234
Fig. 3.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015
Fig. 6.
Relationship between μ and POA for road network in Fig. 2.
Fig. 7.
Relationship between μ and POA for road network in Fig. 3.
Traffic network in the CBD of Singapore.
Fig. 4. Structure of the east road network of Fig. 2. In this traffic network,
R = {r1 , r2 , r3 , r4 } is described by r1 = {e8 , e9 }, r2 = {e1 , e2 , e3 , e10 },
r3 = {e4 , e5 , e10 }, and r4 = {e4 , e6 , e7 , e3 , e10 }.
Fig. 5. Relationship between the traffic flow and travel time for edge e1
(i.e., le = 0.0932 fe + 5.5409) in Fig. 4.
it in Fig. 4. For each edge e in Fig. 4, we fit the latency
function le ( f e ) to real traffic data (e.g., the loop count data
to record the traffic flow and the taxi data to record the
average speed). Then, we get the value of de and ce for
each edge e. For example, as shown in Fig. 5, de = 0.0932
and ce = 5.5409 for edge e1 in Fig. 4. The socially optimal
flow for this network is fr∗1 = 176.7981, f r∗2 = 198.0003,
fr∗3 = 114.2368, and fr∗4 = 10.9648 with L( f ∗ ) = 35922.807.
We can check that this network does not satisfy either
condition 1) or 2) in Theorem 3.2. Through Algorithm 1,
we get μ∗ = 3.394 and the corresponding Nash flow is
frne
= 177.028, frne
= 197.966, frne
= 113.710, and
1
2
3
frne
=
11.296.
We
can
further
check
that
motorcycle
group
4
chooses routes r3 and r4 , car group chooses routes r1 , r2 ,
and r3 , and both taxi and bus groups choose
route r1 . The total latency of the Nash flow without
toll is L( f ne (0)) = 35 950.711 and with toll is
L( f ne (μ∗ )) = 35 922.899. Compare L( f ne (μ∗ )) and L( f ∗ ),
we have POA > 1. As shown in Fig. 6, the POA achieves
its minimal point at μ∗ = 3.394, where the POA slightly
deviates from 1. It is also verified that the POA is bounded
by 4/3.
2) CBD of Singapore: In the CBD network, there are
three routes R = {r1 , r2 , r3 }. Similar to the east road network
case, we fit the latency function le ( f e ) to real data for each
edge e in Fig. 3 and get the value of de and ce . Then, we
can calculate the socially optimal flow for this network is
fr∗1 = 160.9258, f r∗2 = 226.7071, and fr∗3 = 112.3671 with
L( f ∗ ) = 12 532.291. It is easy to check that this network
satisfies condition 1) in Theorem 3.2. Therefore, we can find
μ∗ = 4 such that POA=1, which is shown in Fig. 7. At the
Nash flow, only car group chooses more than one routes,
i.e., r1 , r2 , and r3 . The motorcycle group chooses route r3
and both taxi and bus groups choose route r1 . The total
WANG et al.: ANALYSIS OF POA IN TRAFFIC NETWORKS
latency without toll is L( f ne (0)) = 12 547.216 and with toll
is L( f ne (μ∗ )) = 12 532.291. Obviously, the designed toll
improves the social welfare.
Remark 4.1: The simulation results coincide with conditions in Theorem 3.2 since Ak , Bk , k = 1, and N are related
with the traffic network parameters de , ce , e ∈ E. To be precise,
both the distribution of B and the topology and parameters of
traffic networks affect the POA.
V. C ONCLUSION
In this paper, we have analyzed the Nash flow and the
POA for traffic networks with one origin–destination pair.
The scaled marginal-cost road pricing has been introduced
to optimize the POA for the case where all players in the
noncooperative congestion games have heterogeneous price
sensitivities. It has been shown that for two groups and
two routes case, the social optimum can always be achieved
after charging the designed toll on each link. For general
case, if the distribution of price sensitivities satisfies certain
conditions, the designed toll can guarantee the unique
Nash flow approaches the optimal flow, i.e., POA = 1.
However, the optimal POA cannot always achieve 1. For
any traffic network with one origin–destination pair and any
distribution of price sensitivities, an algorithm is introduced to
find a road price that minimizes the POA.
Note that our model assumes that the road manager has
perfect information on the distribution of price sensitivities.
However, in real traffic systems, the distribution of players’
price sensitivities may be unknown to the road manager.
Therefore, one of our next steps is to estimate the distribution
of price sensitivities and analyze the robustness of the POA.
In addition, nonlinear latency functions and pricing schemes
will be considered.
2235
For groups with β j > β j0 , (VN − Vi /1 + μβ j0 ) >
(VN − Vi /1 + μβ j ), for all i < N. Therefore, Jr N ,β j ( f ne ) <
= 0, frne
= F j , and
Jri ,β j ( f ne ) for all i < N, i.e., frne
i ,β j
N ,β j
ne
ne
fr N = F̄ j0 + fr N ,β j . Similarly, for groups with β j < β j0 ,
0
(V1 − Vi /1 + μβ j0 ) < (V1 − Vi /1 + μβ j ), for all i > 1.
Thus, Jr1 ,β j ( f ne ) < Jri ,β j ( f ne ) for all i > 1, i.e.,
frne
= 0, frne
= F j , and frne
= F̂ j0 + frne
. In conclusion,
1
i ,β j
1 ,β j
1 ,β j0
the Nash flow is given by
⎧
ne
⎪
⎨ F̂ j0 + fr1 ,β j0, if k = 1
, if k = N
frne
= F̄ j0 + frne
(37)
N ,β j0
k
⎪
⎩ f ne ,
otherwise.
rk ,β j
0
From (36), each {i, j } forms an equation.
Take any distinct
= F j0 ,
N − 1 equations and combine them with rk ∈R frne
k ,β j0
we obtain a N-dimension linear equation. The solution of this
system uniquely exists and any solution frne
must be a linear
k ,β j
0
combination of F j0 , F̄ j0 , F̂ j0 , and 1/(1 + μβ j0 ), and we denote
them as
1
= A k F j0 +
Bk + Ck F̂ j0 + Dk F̄ j0 (38)
frne
k ,β j0
1 + μβ j0
where Ak , Bk , Ck , and Dk are independent of μ, β, F j0 , F̄ j0 ,
and F̂ j0 .
Since potential game guarantees the existence of a pure
Nash equilibrium, the solution
to the above system always
= F j0 , we have
exists. Substituting (38) into rk ∈R frne
k ,β j0
⎞
⎛
⎞
⎛
1
⎝
A k − 1⎠ F j0 + ⎝
Bk ⎠
1 + μβ j0
r k ∈R
r k ∈R
⎛
⎛
⎞
⎞
+⎝
Ck ⎠ F̂ j0 + ⎝
Dk ⎠ F̄ j0 = 0.
(39)
r k ∈R
A PPENDIX
Proof of Lemma 3.3: We first show the construction of a
Nash flow. Consider the group with the price sensitivity β j0 .
Assume frne
> 0 for k = 1, N for any μ ≥ 0. We will
k ,β j0
show later that under (24), frne
> 0 and frne
> 0 always
1 ,β j0
N ,β j0
ne
hold. If frk ,β j = 0 for some rk = r1 or r N , by the definition
0
of the Nash flow, we have Jri ,β j0 ( f ne ) ≤ Jrk ,β j0 ( f ne ), for
all i = k. Note that β j and V j are ordered as β1 >
β2 > · · · > β M and Vr1 < Vr2 < · · · < Vr N , respectively. For
groups with β j > β j0 , i.e., β1 , . . . , β j0 −1 , since frne
> 0,
N ,β j0
we have Jr N ,β j ( f ne ) ≤ Jri ,β j ( f ne ), for all i = N, which
= 0 for all β j > β j0 . For groups with
indicates that frne
k ,β j
β j < β j0 , i.e., β j0 +1 , . . . , β M , since frne
> 0 we have
1 ,β j0
Jr1 ,β j ( f ne ) ≤ Jri ,β j ( f ne ), for all i = 1, which also indicates
that frne
= 0 for all β j < β j0 . Thus, we can conclude that
k ,β j
frne
=
0,
which
implies frne
= 0 when μ = 0. This contradicts
k
k
Assumption 3.2.
Therefore, frne
> 0 for all rk ∈ R.
k ,β j0
By the definition of the Nash flow, for any ri = r j , we have
V j − Vi
frne
(dik − d j k ) =
.
(36)
k
1 + μβ j0
r k ∈R
r k ∈R
Since (39) holds for any F j0 , F̄ j0 , F̂ j0 , and μ, we obtain
Ak = 1
(40)
r k ∈R
Bk = 0
(41)
Ck = 0
(42)
Dk = 0.
(43)
r k ∈R
r k ∈R
r k ∈R
Substituting (37) and (38) into (36), similarly to the above,
we obtain
Ak dik =
Ak d j k
(44)
r k ∈R
Bk dik + Vi =
r k ∈R
Ck dik + di1 =
r k ∈R
r k ∈R
Dk dik + di N =
r k ∈R
Bk d j k + V j
(45)
Ck d j k + d j 1
(46)
Dk d j k + d j N .
(47)
r k ∈R
r k ∈R
r k ∈R
2236
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015
For β j1 ∈ B
By (42) and (44)
N
Ci
i=1
N
Ak dik =
N
k=1
Ak d1k
N
k=1
Ci = 0.
(48)
i=1
By (40) and (46)
N
i=1
Ai
N
Ck dik =
k=1
=
=
N
i=1
N
k=1
N
Ai
N
Ck d1k + d11 − di1
1
k=1
Ck d1k + d11 −
Ck dik + di1 −
N
i=1
N
k=1
(50)
0
Thus
F j0 = F − F̂ j0 − F̄ j0 ≥ F − (A1 + A N )F − (B1 + B N ) > 0
F − A N F − B N ≤ F̂ j0 ≤ A1 F + B1
(Ak − Ck )dik = 0.
F ≤ (A1 + A N )F + (B1 + B N )
F > (A1 + A N )F + (B1 + B N ).
that is
(A1 − C1 − 1)di1 +
0
that is
Ak dik
k=1
N
1
F̄ j1 = F − F̂ j0 ≤ A N F + B N .
(49)
N
N
N
N
Ci k=1
Ak dik = i=1
Ai k=1
Ck dik , combinSince i=1
ing (48) and (49), for all ri ∈ R
N
(56)
F̄ j0 ≤ A N F + B N
k=1
Ck dik + di1 =
if k = N
otherwise.
F̂ j0 ≤ A1 F + B1
Ai di1
Ak dki .
if k = 1
Without loss of generality, we assume that group β j1 is next
to group β j0 and β j1 < β j0 . Thus, F̄ j1 = F − F̂ j0 . Since
frne
≥ 0, frne
≥ 0, frne
≥ 0, and frne
≥ 0, we have
1 ,β j
N ,β j
1 ,β j
N ,β j
k=1
N
frne
k ,β j1
⎧
⎪
⎨ A1 F − F̂ j1 + B1 ,
= A N F − F̄ j1 + B N ,
⎪
⎩A F + B ,
k
k
(51)
This is a contradiction. Therefore, the Nash flow is unique for
a given distribution of B satisfying (24).
k=2
Since (51) holds for all ri ∈ R, we have
A 1 = C1 + 1
Ak = Ck for all k = 1.
Similarly to the above, we have
Ak = Dk for all k = N
A N = D N + 1.
R EFERENCES
(52)
(53)
Insert (52) and (53) into (38), we obtain (25). According
> 0 for k = 1, N is guaranteed.
to (24), frne
k ,β j0
By Assumption 3.2, frne
> 0, rk ∈ R for μ > 0 is
k ,β j
0
guaranteed. For the special case when F̂ j0 = 0 or F̄ j0 = 0,
the results are the same.
Next, we show the uniqueness of the above Nash flow. For
the given distribution of B, we assume that there exists another
> 0 for all rk ∈ R, thus
group β j1 ∈ B satisfying frne
k , j1
⎧
1
⎪
⎪
A1 F − F̂ j1 +
B1 ,
if k = 1
⎪
⎪
1
+
μβ j1
⎪
⎪
⎨
1
frne
(54)
A N F − F̄ j1 +
=
B N , if k = N
,β
k j1
⎪
1
+
μβ j1
⎪
⎪
⎪
1
⎪
⎪
Bk ,
otherwise.
⎩Ak F +
1 + μβ j1
Since (25) and (54) also holds for μ = 0, we consider the case
when the routes are free of charge. Therefore, for β j0 ∈ B
⎧
⎨ A1 F − F̂ j0 + B1 , if k = 1
=
frne
(55)
A F − F̄ j0 + B N , if k = N
k ,β j0
⎩ N
otherwise.
A k F + Bk ,
[1] R. Johari and J. N. Tsitsiklis, “Efficiency loss in a network resource
allocation game,” Math. Oper. Res., vol. 29, no. 3, pp. 407–435, 2004.
[2] B. J. Brinkworth and M. Sandberg, “Design procedure for cooling ducts
to minimise efficiency loss due to temperature rise in PV arrays,” Solar
Energy, vol. 80, no. 1, pp. 89–103, 2006.
[3] R. Johari, “Efficiency loss in market mechanisms for resource
allocation,” Ph.D. dissertation, Dept. Elect. Eng. Comput. Sci., Massachusetts Inst. Technol., Cambridge, MA, USA, 2004.
[4] A. Downs, Still Stuck in Traffic: Coping With Peak-Hour Traffic
Congestion. Washington, DC, USA: Brookings Institution Press, 2004.
[5] R. Gibbons, Primer in Game Theory. New York, NY, USA:
Harvester Wheatsheaf, 1992.
[6] R. W. Rosenthal, “A class of games possessing pure-strategy
Nash equilibria,” Int. J. Game Theory, vol. 2, no. 1, pp. 65–67, 1973.
[7] D. Monderer and L. S. Shapley, “Potential games,” Games Econ.
Behavior, vol. 14, no. 1, pp. 124–143, 1996.
[8] P. Dubey, “Inefficiency of Nash equilibria,” Math. Oper. Res., vol. 11,
no. 1, pp. 1–8, 1986.
[9] T. Roughgarden and É. Tardos, “How bad is selfish routing?” J. ACM,
vol. 49, no. 2, pp. 236–259, 2002.
[10] E. Koutsoupias and C. Papadimitriou, “Worst-case equilibria,” in Proc.
16th Annu. STACS, 1999, pp. 404–413.
[11] T. Roughgarden, Selfish Routing and the Price of Anarchy, vol. 174.
Cambridge, MA, USA: MIT Press, 2005.
[12] S. Lim and D. Rus, “Stochastic distributed multi-agent planning and
applications to traffic,” in Proc. IEEE Int. Conf. Robot. Autom.,
May 2012, pp. 2873–2879.
[13] C. Papadimitriou and G. Valiant, “A new look at selfish routing,” in
Proc. Innov. Comput. Sci., 2010, pp. 178–187.
[14] H. Von Stackelberg, D. Bazin, R. Hill, and L. Urch, Market Structure
and Equilibrium. New York, NY, USA: Springer-Verlag, 2010.
[15] Y. A. Korilis, A. A. Lazar, and A. Orda, “Achieving network optima
using Stackelberg routing strategies,” IEEE/ACM Trans. Netw., vol. 5,
no. 1, pp. 161–173, Feb. 1997.
[16] W. Krichene, J. Reilly, S. Amin, and A. Bayen, “On the characterization
and computation of Nash equilibria on parallel networks with horizontal queues,” in Proc. 51st IEEE Conf. Decision Control, Dec. 2012,
pp. 7119–7125.
[17] T. Roughgarden, “Stackelberg scheduling strategies,” SIAM J. Comput.,
vol. 33, no. 2, pp. 332–350, 2004.
WANG et al.: ANALYSIS OF POA IN TRAFFIC NETWORKS
[18] C. Dextreit and I. V. Kolmanovsky, “Game theory controller for hybrid
electric vehicles,” IEEE Trans. Control Syst. Technol., vol. 22, no. 2,
pp. 652–663, Mar. 2014.
[19] E. Deakin, G. Harvey, R. Pozdena, and G. Yarema, “Transportation
pricing strategies for California: An assessment of congestion,
emissions, energy, and equity impacts,” Univ. California Transp. Center,
Berkeley, CA, USA, Tech. Rep., 1996.
[20] A. P. G. Menon, “ERP in Singapore: A perspective one year on,” Traffic
Eng. Control, vol. 41, no. 2, pp. 40–45, 2000.
[21] P. Olszewski and L. Xie, “Modelling the effects of road pricing on traffic
in Singapore,” Transp. Res. A, Policy Pract., vol. 39, no. 7, pp. 755–772,
2005.
[22] P. Patriksson, The Traffic Assignment Problem: Models and Methods.
Utrecht, The Netherlands: VSP Publishers, 1994.
[23] H. Yang and H.-J. Huang, Mathematical and Economic Theory of Road
Pricing. Amsterdam, The Netherlands: Elsevier, 2005.
[24] R. Cole, Y. Dodis, and T. Roughgarden, “Pricing network edges for
heterogeneous selfish users,” in Proc. 35th ACM Symp. Theory Comput.,
2003, pp. 521–530.
[25] L. Fleischer, K. Jain, and M. Mahdian, “Tolls for heterogeneous selfish
users in multicommodity networks and generalized congestion games,”
in Proc. 45th IEEE Symp. Found. Comput. Sci., Oct. 2004, pp. 277–285.
[26] G. Como, K. Savla, D. Acemoglu, M. A. Dahleh, and E. Frazzoli,
“Robust distributed routing in dynamical networks—Part II: Strong
resilience, equilibrium selection and cascaded failures,” IEEE Trans.
Autom. Control, vol. 58, no. 2, pp. 333–348, Feb. 2013.
[27] N. Xiao et al., “Average strategy fictitious play with application to road
pricing,” in Proc. Amer. Control Conf., 2013, pp. 1920–1925.
[28] X. Wang, N. Xiao, T. Wongpiromsarn, L. Xie, E. Frazzoli, and D. Rus,
“Distributed consensus in noncooperative congestion games: An application to road pricing,” in Proc. 10th IEEE Int. Conf. Control Autom.,
Jun. 2013, pp. 1668–1673.
[29] K. Ma, G. Hu, and C. J. Spanos, “Distributed energy consumption
control via real-time pricing feedback in smart grid,” IEEE Trans.
Control Syst. Technol., vol. 22, no. 5, pp. 1907–1914, Sep. 2014.
[30] X. Wang, N. Xiao, L. Xie, E. Frazzoli, and D. Rus, “Analysis of price
of anarchy in heterogeneous price-sensitive populations,” in Proc. IEEE
53rd Annu. Conf. Decision Control (CDC), Dec. 2014, pp. 6478–6483.
[31] M. J. Lighthill and G. B. Whitham, “On kinematic waves. II. A theory
of traffic flow on long crowded roads,” Proc. Roy. Soc. London, Ser. A,
Math. Phys. Sci., vol. 229, no. 1178, pp. 317–345, 1955.
[32] T. Roughgarden, “The price of anarchy is independent of the network
topology,” J. Comput. Syst. Sci., vol. 67, no. 2, pp. 341–364, 2003.
Xuehe Wang received the bachelor’s degree
in mathematics from Sun Yat-sen University,
Guangzhou, China, in 2011. She is currently
pursuing the Ph.D. degree with the School of
Electrical and Electronic Engineering, Nanyang
Technological University, Singapore.
Her current research interests include game theory,
cooperative control theory, and road pricing design.
Nan Xiao received the B.E. and M.E. degrees in
electrical engineering and automation from Tianjin
University, Tianjin, China, in 2005 and 2007,
respectively, and the Ph.D. degree in electrical and
electronic engineering from Nanyang Technological
University, Singapore, in 2012.
He held visiting appointments with the Hong Kong
University of Science and Technology, Hong Kong,
and the Massachusetts Institute of Technology,
Cambridge, MA, USA. He was a Research Associate
and a Research Fellow with the School of Electrical
and Electronic Engineering, Nanyang Technological University, Singapore.
He is currently a Post-Doctoral Associate with the Singapore-MIT Alliance
for Research and Technology Centre, Singapore, where he is involved in the
Future Urban Mobility IRG Research Program. His current research interests include networked control systems, multi-agent systems, transportation
networks, and game theory.
2237
Lihua Xie (F’07) received the B.E. and M.E.
degrees from the Nanjing University of Science and
Technology, Nanjing, China, in 1983 and 1986,
respectively, and the Ph.D. degree from the
University of Newcastle, Callaghan, NSW, Australia,
in 1992, all in electrical engineering.
He has been with the School of Electrical
and Electronic Engineering, Nanyang Technological
University, Singapore, since 1992, where he is
currently a Professor. He served as the Head of
the Division of Control and Instrumentation from
2011 to 2014. His current research interests include robust control and
estimation, networked control systems, multiagent networks, and unmanned
systems.
Prof. Xie is a fellow of the International Federation of Automatic Control.
He has served as an Editor of IET Book Series in Control and an Associate
Editor of a number of journals, including the IEEE T RANSACTIONS
ON AUTOMATIC C ONTROL, Automatica, the IEEE T RANSACTIONS ON
C ONTROL S YSTEMS T ECHNOLOGY, and the IEEE T RANSACTIONS ON
C IRCUITS AND S YSTEMS -II.
Emilio Frazzoli (SM’07) received the Laurea
degree in aerospace engineering from the University
of Rome Sapienza, Rome, Italy, in 1994, and the
Ph.D. degree from the Department of Aeronautics
and Astronautics, Massachusetts Institute of
Technology, Cambridge, MA, USA, in 2001.
He is currently a Professor of Aeronautics and
Astronautics with the Laboratory for Information
and Decision Systems, and the Operations
Research Center with the Massachusetts Institute of
Technology. His current research interests include
autonomous vehicles, mobile robotics, and transportation systems, and the
area of planning and control for mobile cyber-physical systems.
Prof. Frazzoli is currently an Associate Fellow of the American Institute
of Aeronautics and Astronautics. He was a recipient of the NSF CAREER
Award in 2002.
Daniela Rus (F’10) received the Ph.D. degree in
computer science from Cornell University, Ithaca,
NY, USA.
She was a Professor with the Department of
Computer Science, Dartmouth College, Hanover,
NH, USA. She is the Andrew and Erna Viterbi
Professor of Electrical Engineering and Computer
Science and the Director of the Computer
Science and Artificial Intelligence Laboratory
with the Massachusetts Institute of Technology,
Cambridge, MA, USA. Her current research
interests include robotics, mobile computing, and data science.
Prof. Rus was a Class of 2002 MacArthur Fellow, a fellow of the
Association for Computing Machinery and the Association for the
Advancement of Artificial Intelligence, and a member of the National
Academy of Engineering.