The Price of Anarchy in
Transportation Networks
Hyejin Youn1, Michael T. Gastner2, and Hawoong Jeong1
1
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon
305-701, Korea
2
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Keywords: Nash equilibrium, network optimization, efficiency of traffic flow, transportation
network
Correspondence:
Hawoong Jeong; Department of Physics, Korea Advanced Institute of Science and Technology,
373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Korea; E-mail: [email protected]; TEL:
+82-42-869-2543; FAX: +82-42-869-2510
1
Abstract
Uncoordinated individuals in human society pursuing their personally optimal
strategies do not always achieve the social optimum, the most beneficial state to the
society as a whole. Instead, strategies form Nash equilibria, which are, in general,
socially suboptimal. Society, therefore, has to pay a price of anarchy for the lack of
coordination among its members, which is often difficult to quantify in engineering,
economics and policymaking. Here we report on an assessment of this price of
anarchy by analyzing the road networks of Boston, London, and New York as well
as complex model networks, where one’s travel time serves as the relevant cost to
be minimized. Our simulation shows that uncoordinated drivers possibly spend up
to 30% more time than they would in socially optimal traffic, which leaves
substantial room for improvement. Counterintuitively, simply blocking certain
streets can partially improve the traffic condition to a measurable extent based on
our result.
Many real-world transportation systems in human societies are characterized by
networked structures and complex agents interacting on these networks (1, 2).
Understanding the agents’ behaviors has important consequences for the design and
control of, for example, the Internet, peer-to-peer, or vehicle networks (3, 4). It is
reasonable to assume that humans opt for the strategies that maximize their personal
utility (5). However, this does not mean that flows in transportation networks minimize
the cost for all users as is sometimes assumed (6, 7). On the contrary, we will
demonstrate that the flow can in reality still be far from optimal even if all individuals
search for the quickest paths and have complete information about the network and
other users’ behaviors.
2
In this paper, we study decentralized transportation networks where each directed
link from node i to j is associated with a delay lij, the time needed to travel along the
link. In most real networks, delays depend noticeably on the flow (8), that is, the
number of downloads, vehicles, etc. per unit time. For example, a single vehicle easily
moves at the permitted speed limit on an empty road, yet slows down if too many
vehicles share the same road. Suppose that there is a constant flow of travelers F in a
network starting at a source node s and seeking a path to a destination t. What is the
optimal traffic through the network? From the perspective of society as a whole, we
wish to minimize the total time spent by all users. On the other hand, every individual
wishes to follow the optimal route that minimizes his own travel time given the
decisions of all other users. Experimental tests indicate that human subjects approach
the problem of finding paths in a network from this latter self-interested perspective,
rather than from the former altruistic point of view (9, 10).
This observation is important since the two optimization strategies generally lead
to different outcomes (11). Consider, for instance, the simple network depicted in Fig.1a
(12). The nodes s and t are connected by two different types of links: a short but narrow
bridge A where the effective speed becomes slower as more cars travel on it, and a long
but broad multi-lane freeway B where delays due to congestion are negligible. Suppose
the delay on link A is proportional to the flow, lA ( fA ) = fA, while the delay on B is flowindependent, lB ( fB ) = 10, where fA(B) is the flow on link A(B). The total time spent by
all users is given by the “cost function” C ( fA ) = lA ( fA ) · fA + lB ( fB )· fB where the flow
on B is equal to fB = F − fA. It is easily verified that C attains its minimum for fA = 5
independently of the total flow F. If F = 10, for example, each link should be taken by
exactly half of the users, resulting in C = 75.
3
Fig. 1. Illustration of the price of anarchy. In the network shown in (a), suppose F = 10
users travel per unit time from s to t. If the flow along A is fA, the delay on this link is
lA=fA, whereas the delay along B is lB = 10 independently of the flow. The different
delays lA and lB reflect different physical features of the two links. For example, in a road
network, the links would be roads with different lengths, numbers of lanes, or driving
conditions. (b) The socially optimal flow sends five users along each link. Users on link
A spend a total time of l A ⋅ f A = 5 ⋅ 5 = 25 and users on link B pay a cost of
l B ⋅ f B = 10 ⋅ 5 = 50 , so the total cost is C = 75. This flow is optimal in the sense that all
other flows are more expensive. However, users traveling along B pay more than they
would on A, so there is an incentive for them to change paths. (c) In the Nash
equilibrium with fA =10 and fB =0, no user can reduce his cost unilaterally, i.e., there is
no incentive for any single user to change strategies. While the cost is now equal for
everybody – all ten users must pay an individual cost lA = 10 – the cost paid by all users
together, C = 100, is higher than before.
On the other hand, every user on link B could reduce his delay from 10 to 6 by
switching paths, which poses a social dilemma: as individuals, users would like to
reduce their own delays, but this reduction can cause unnecessary delay to the entire
group. In our example, as long as lA is not equal to lB, there will be an incentive for the
users experiencing longer delays to shift to another link. In contrast, if all users decide
4
to put their own interests first, the flow will be in a Nash equilibrium where no single
user can make any individual gain by changing his own strategy unilaterally (13). In our
case, all users take the link A, as shown in Fig. 1c, at the total cost of C=100.
Interestingly, not a single user experiences a shorter travel time in this Nash equilibrium
than in the social optimum in Fig. 1b, and still this suboptimal behavior is consistent
with experiments (9, 10). Furthermore, if all functions fij·lij(fij) are convex (as in most
realistic cases) and the flows fij are continuous, one can prove that there is always
exactly one Nash equilibrium (12).
Price of Anarchy in Boston
Although differences between Nash equilibria and social optima occur frequently
in other contexts (14), little is known about the situation in large-scale real
transportation networks. To shed light on this issue, we have analyzed Boston’s road
network shown in Fig. 2a. The 246 directed links in our network are segments of
principal roads, and their intersections form 88 nodes. Delays are assumed to follow the
“Bureau of Public Roads function” widely used in civil engineering,
β
 f ij  
d ij 
1 + α    .
lij =
 pij  
vij 
  

[1]
Here dij is the distance of the link between i and j, vij the speed limit (35 mph on all links,
for simplicity), fij the flow, and pij the capacity of the road segment. The parameters α
and β have been fitted to empirical data (15) as α=0.2 and β=10, so the delays increase
very steeply for large traffic volumes. Capacity is defined as the traffic volume at “level
5
Fig. 2. Networks of principal roads. (a) Boston-Cambridge area (both solid and dotted
lines; the thickness represents the number of lanes). We assume users travel from
Harvard Square in the northwest of the map to Boston Common in the southeast. For a
traffic volume of F = 10,000 vehicles per hour, drivers need 10 minutes from start to
end. The colour of each link indicates the additional travel time needed in the Nash
equilibrium if that link is cut (blue: no change, red: more than 60 seconds additional
delay). Black dotted lines represent links whose removal reduces the travel time, i.e.,
allowing drivers to use these links in fact creates more congestion than blocking these
streets. This counter-intuitive phenomenon is called "Braess’s paradox.” (b) London,
UK, and (c) New York City show similar results with 10,000 vehicles traveling from
Borough to Farringdon and 18,000 vehicles per hour traveling from Washington Market
Park to Queens Midtown Tunnel, respectively.
6
of service E” which is approximately 2000 vehicles per hour multiplied by the number
of lanes (16). We used Google Maps and Google StreetView to identify the principal
roads, measure the distances dij, and count the number of lanes for each direction.
Next we have calculated the flows fij for various total traffic volumes F from
Harvard Square to Boston Common. The socially optimal flows f ijSO are determined by
minimizing the cost to society per unit time which is the sum over everybody’s
individual delay
C=
∑l
ij
link ( i , j )
( f ij ) f ij .
[2]
This optimization problem, satisfying flow conservation at each intersection, can be
solved with standard convex minimum cost flow algorithms (17, 18). For the Nash
equilibrium, we can use the fact that the equilibrium flows f ijNE minimize the objective
function (12),
~
C=
∑ ∫
link ( i , j )
f ij
0
lij ( f ′) df ′ .
[3]
The price of anarchy (PoA) is defined as the ratio of the total cost of the Nash
equilibrium to the total cost of the social optimum (19); for example in Fig. 1,
PoA=100/75 = 1.33, or generally
∑l
PoA =
∑l
ij
( f ijNE ) ⋅ f ijNE
ij
( f ijSO ) ⋅ f ijSO
7
.
[4]
Figure 3a shows the PoA versus the total traffic volume F for Boston’s roads. For small
F, we find no difference between social optimum and Nash equilibrium cost; hence the
PoA is exactly 1. However, as F exceeds 1500 vehicles per hour, the PoA increases to
larger values and goes through several local maxima between PoA = 1.05 and 1.10. A
broad peak appears around the global maximum at 10,000 vehicles per hour with PoA ≈
1.30, that is, individuals waste 30% of their travel time for not being coordinated. For
larger F, the PoA then decreases monotonically and becomes practically
indistinguishable from 1 for F > 20,000. To interpret the relevance of these numbers, we
note that Storrow Drive, one of the busiest roads in downtown Boston, averages about
3800 vehicles per hour throughout the year (20). During rush hours, it is, therefore,
quite realistic to expect a flow of around 10,000 vehicles per hour, where the PoA
reaches its maximum of 30% above the optimum. Real traffic, of course, rarely flows
only between two ends. However, there are important situations in which traffic can be
described as having effectively a single source and destination, for example during an
evacuation or after a major public event (that is, a sports match or a concert). Our result
suggests that, under such circumstances, traffic volumes can easily reach values where a
high PoA can become a serious problem.
Further examples for the Price of Anarchy
To what extent are properties of the PoA observed in the Boston road network
characteristic of networks with flow-dependent costs? Among road networks, the results
appear to be typical as suggested by an analysis of the road networks of London and
New York (Fig.2). The road network in London, UK, consists of 82 intersections and
8
217 links marked as principal roads by Google Maps. Road junctions less than 50m are
merged maintaining the direction of the streets. Bus lanes are counted in the total
number of lanes in the street. Following the same procedure as in Boston, we find that
the PoA can increase up to 24% for trips between the Borough and the Farringdon
underground stations (Fig. 3a inset). Similar results also hold for New York, NY, USA.
The road network consists of 125 intersections and 319 streets - slightly more than in
Boston and London. Road junctions of less than 70m are merged maintaining the
direction of the streets. The inset of Fig. 3a shows that the price of anarchy can be as
high as 28% when 12,000 vehicles per hour travel from Washington Market Park to
Queens Midtown Tunnel in New York. The results remain qualitatively similar for
different sets of sources and destinations.
To gain further theoretical insight, we also constructed four ensembles of
bidirectional model networks with distinct underlying structures: a simple onedimensional lattice with connections up to the third-nearest neighbors and periodic
boundary conditions; Erdıs-Rényi random graphs (21) with links between randomly
drawn pairs of nodes; small-world networks (22) with a rewiring probability 0.1; and
Barabási-Albert networks (23) with broad degree distributions. All the networks contain
100 nodes and have an average degree of 6. Every link between nodes i and j has a
delay of the form
lij ( f ij ) = aij f ij + bij
[5]
where aij = aji is a random integer equal to 1, 2, or 3, and bij = bji between 1 and 100.
Equation [5] is a special case of the BPR function, Eq. [1], where β = 1, bij = dij/vij and
aij = α bij/pij. A cost function of this type captures essential properties of links in
9
important physical networks. In electric circuits, for example, the flow fij is an electric
current and the delay lij can be interpreted as the voltage difference between i and j.
Fig. 3. The price of anarchy (PoA), i.e., the ratio of Nash equilibrium to socially
optimal cost (Eq.[4]), as a function of the traffic volume F. (a) In Boston's road network
for journeys from Harvard Square to Boston Common with delays given by Eq. [1]. In
the inset, we plot the PoA in London’s road network for journeys from Borough to
Farringdon, and in New York’s road network for journeys from Washington Market
Park to Queens Midtown Tunnel. (b) The PoA in ensembles of model networks with
delays given by Eq. [5]. All networks have 100 nodes and 300 undirected links. The
first ensemble is a simple one-dimensional lattice with connections up to the thirdnearest neighbours and periodic boundary conditions. The second ensemble consists of
Erdıs-Rényi networks (21) where randomly drawn pairs of nodes are connected, but
networks which do not form one single connected component are discarded. As our
third ensemble, we have constructed small-world networks where the links of a onedimensional lattice are randomly rewired with probability 0.1 (22). The fourth ensemble
consists of Barabási-Albert networks with characteristically broad degree distributions
(23). The error bars represent one standard deviation in the PoA-distribution.
10
An affine current-voltage characteristic like Eq. [5] occurs in circuits with a
combination of Ohmic resistors (resistances aij) and Zener diodes (breakdown voltages
bij). Further examples with cost functions of the form of Eq. [5] include mechanical,
hydraulic, and thermal networks (24, 25).
For each network, we go through every pair of nodes to calculate the PoA for various
total flows F. Then the results are averaged over 50 networks to find the mean
〈 PoA(F) 〉 for each ensemble as plotted in Fig. 3b. After averaging over many pairs,
there are no longer multiple local maxima as in Fig. 3a. Instead, we find unimodal
functions for all ensembles with a steep increase for small F and a long tail for large
flows. The qualitative behavior can be understood as follows. From Eq. [2], [3] and [5],
the social optimum minimizes C = ∑ (aij f ij + bij f ij ) whereas the flow in the Nash
2
1
~
2
equilibrium minimizes C = ∑ ( aij f ij + bij fij ) . In the limit F → 0, both objective
2
functions become identical and, therefore, 〈 PoA 〉 → 1. For F → ∞, the quadratic
~
terms in the sums dominate, hence C / C → 2, i.e., both objective functions are
minimized by the same asymptotic flow pattern fij /F and 〈 PoA 〉 again approaches 1.
The maximum 〈 PoA 〉 occurs roughly where the quadratic and linear terms in the
objective functions are comparable, that is, aij fij ≈ bij for paths with positive flow.
Ignoring correlations between aij and fij, we have 〈 fij 〉 ≈
〈bij 〉
〈 aij 〉
. Since F = c 〈 fij 〉
where c is a factor bigger than but of the order of 1, we estimate the maximum 〈 PoA 〉
to be at Fmax ≈ c
〈bij 〉
〈 aij 〉
. In our example, 〈 aij 〉 = 2 and 〈 bij 〉 = 50.5, so we predict Fmax
to be bigger than but of the order of 25. Numerically, we find the maxima for our four
ensembles to be between 30 and 60 in good agreement with our estimate. Barabási-
11
Albert networks tend to have the lowest 〈 PoA(F) 〉 and small-world networks the
highest, but the statistical dependence between 〈 PoA 〉 and F is strikingly similar
among all ensembles.
Braess’s paradox
Knowing the PoA is important, but it is even more valuable to discover a proper method
to reduce it. In a road network, one could charge drivers toll fees to stimulate a more
cooperative behavior, but that strategy has problems of its own (26). For example, one
could charge a fee for using each link equal to the “marginal cost” fij · lij (fij) so that the
new Nash flow becomes equal to the social optimum, but it is difficult to convert the
marginal cost from time into money. And even if it was possible, we would have thrown
out the baby with the bath water: including the added fee, the total costs are just as far
away from the social optimum as before. However, as we have learned from Fig. 3b, we
can change the PoA by modifying the underlying network structure. For instance,
opening or closing roads to car traffic is relatively easy to implement and is, moreover,
equally effective for everybody. One might expect that opening new roads can improve
traffic, but closing them should increase congestion. However, contrary to common
intuition, Braess’s paradox suggests that road closures can sometimes reduce travel
delays (27). According to this counter-intuitive instance, opening all existing roads is
not always a straightforward solution to a traffic problem.
We investigated whether this apparent contradiction occurs in the road networks
of Boston, London and New York. In the case of Boston’s roads, the traffic volume is
12
set to F = 10,000 between Harvard Square and Boston Common (Fig. 2a). (At that point,
the PoA reaches its maximum, cf. Fig. 3a.) We then compare the costs of both the Nash
flows and optimal flows on the original network with those on networks where one of
the 246 streets is closed to traffic. In most cases, both costs increase when one street is
unavailable to through traffic, as intuitively expected. Nonetheless, we have identified
36 streets which, if deleted individually, decrease the PoA. Furthermore, there are six
connections which decrease the delay in the Nash equilibrium, shown as dotted lines in
Fig. 2. These are Staniford Street, four sections of Main Street, and one section of
Charles Street. All of them are convenient shortcuts between different routes from
Harvard Square to Boston Common, but create additional congestion if open to selfinterested users. If all drivers ideally cooperated to reach the social optimum, these
roads could be helpful; otherwise it is better to close these streets. Similar results are
also found in the road networks of London and New York (Fig.2). Figure 2b visualizes
seven links causing Braess’s paradox as dotted lines on a satellite image of London,
when 10,000 vehicles per hour travel from Borough to Farringdon. For the road network
of New York, we set the flow from Washington Market Park to Queens Midtown
Tunnel equal to 18,000 vehicles per hour. Here we identified twelve roads represented
as dotted lines which counter-intuitively decrease the travel time (Fig. 2c). The results
demonstrate that Braess’s paradox is more than an academic curiosity (27, 28, 29) or an
anecdote with only sketchy empirical evidence (11, 30, 31).
Braess’s paradox exists because the social optimum and the Nash equilibrium react in
different ways to changes in the network. After a link is cut, the socially optimal travel
time must be at least as long as before – otherwise the travel time was not the global
minimum when the link was present, contradicting the definition of the social optimum.
On the other hand, there is no a priori reason why severing a link could not improve the
Nash travel time (and therefore the PoA). By the same argument, adding new links can
13
potentially create more delay in the Nash equilibrium. Hence, a target for future policies
in transportation networks is to prevent unintended delays caused by, ironically, wellintentioned new constructions that form a disadvantageous Nash flow. Because convex
cost functions such as Eq. [1] occur in many networks – economic, biological, and
physical – Braess’s paradox is presumably also frequent outside vehicle transportation
networks. Further studies of the price of anarchy and Braess's paradox might therefore
lead to significantly improved flows in a number of important applications.
Acknowledgements
We thank Eric Smith, Yueyue Fan, D.-H. Kim, and H.-K. Lee for helpful discussion. HJ
acknowledges the motivation from the study group at the NECSI summer school. This
work was supported by KOSEF through the grant No. R17-2007-073-01001-0 and
is dedicated to the memory of Charles VanBoven.
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