THE PRICE OF STABILITY FOR NETWORK DESIGN
WITH FAIR COST ALLOCATION
∗
ELLIOT ANSHELEVICH
†
, ANIRBAN DASGUPTA
‡
, JON KLEINBERG
§
,
´
EVA TARDOS
¶
,
TOM WEXLER
k
, AND TIM ROUGHGARDEN
∗∗
Abstract. Network design is a fundamental problem for which it is important to understand the
effects of strategic behavior. Given a collection of self-interested agents who want to form a network
connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite
different from the centrally enforced optimum. We study the quality of the best Nash equilibrium,
and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash
equilibrium solution has a natural meaning of stability in this context — it is the optimal solution
that can be proposed from which no user will defect.
We consider the price of stability for network design with respect to one of the most widely-studied
protocols for network cost allocation, in which the cost of each edge is divided equally between users
whose connections make use of it; this fair-division scheme can be derived from the Shapley value,
and has a number of basic economic motivations. We show that the price of stability for network
design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a
good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect
from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful
mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections
to the class of potential games defined by Monderer and Shapley, and extend our results to cases in
which users are seeking to balance network design costs with latencies in the constructed network,
with stronger results when the network has only delays and no construction costs. We also present
bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted
game.
Key words. network design, price of stability, Shapley cost-sharing
AMS subject classifications. 68Q99, 90B18, 91A43
1. Introduction. In many network settings, the system behavior arises from the
actions of a large number of independent agents, each motivated by self-interest and
optimizing an individual objective function. As a result, the global performance of the
system may not be as good as in a case where a central authority can simply dictate
a solution; rather, we need to understand the quality of solutions that are consistent
with self-interested behavior. Recent theoretical work has framed this type of question
in the following general form: how much worse is the solution quality of a Nash
∗
A preliminary version of this paper appeared in Proc. 45th Annual Symposium on Foundations
of Computer Science, 2004.
†
Rensselaer Polytechnic Institute, Department of Computer Science, Troy, NY. Research sup-
ported by ITR grant 0311333. Email: eanshel@cs.rpi.edu.
‡
Cornell University, Department of Computer Science, Upson Hall, Ithaca, NY 14853. Supported
by the Department of Computer Science. Email: adg@cs.cornell.edu.
§
Cornell University, Department of Computer Science, Upson Hall, Ithaca, NY 14853. Email:
kleinber@cs.cornell.edu. Supported in part by a David and Lucile Packard Foundation Fellowship
and NSF grants 0081334 and 0311333.
¶
Cornell University, Department of Computer Science, Upson Hall, Ithaca, NY 14853. Supported
in part by NSF grant CCR-032553, ITR grant 0311333, and ONR grant N00014-98-1-0589. Email:
eva@cs.cornell.edu.
k
Cornell University, Department of Computer Science, Upson Hall, Ithaca, NY 14853. Supported
by ITR grant 0311333. Email: wexler@cs.cornell.edu.
∗∗
Stanford University, Computer Science Department, Gates Building, Stanford, CA 94305. Sup-
ported in part by ONR grant N00014-04-1-0725, an NSF CAREER Award, and an Alfred P. Sloan
Fellowship. Email: tim@cs.stanford.edu.
1
2 ANSHELEVICH, DASGUPTA, KLEINBERG, TARDOS, WEXLER, ROUGHGARDEN
equilibrium
1
, compared to the quality of a centrally enforced optimum? Questions of
this genre have received considerable attention in recent years, for problems including
routing [37, 39, 13], load balancing [14, 15, 27, 38], and facility location [41]; see [34,
Chapters 17–21] for an overview of this literature.
An important issue to explore in this area is the middle ground between cen-
trally enforced solutions and completely unregulated anarchy. In most networking
applications, it is not the case that agents are completely unrestricted; rather, they
interact with an underlying protocol that essentially proposes a collective solution to
all participants, each of which can either accept it or defect from it. As a result, it
is in the interest of the protocol designer to seek the best Nash equilibrium; this can
naturally be viewed as the optimum subject to the constraint that the solution is sta-
ble, with no agent having an incentive to unilaterally defect from it once it is offered.
Hence, one can view the ratio of the solution quality at the best Nash equilibrium
relative to the global optimum as a price of stability, since it captures the problem
of optimization subject to this constraint. Some recent work [3, 13] has considered
this definition (termed the “optimistic price of anarchy” in [3]); it stands in contrast
to the larger line of work in algorithmic game theory on the price of anarchy [35] —
the ratio of the worst Nash equilibrium to the optimum — which is more suited to
worst-case analysis of situations with essentially no protocol mediating interactions
among the agents. Indeed, one can view the activity of a protocol designer seeking
a good Nash equilibrium as being aligned with the general goals of mechanism de-
sign [33] — producing a game that yields good outcomes when players act in their
own self-interest.
Network Design Games. Network design is a natural area in which to explore the
price of stability, given the large body of work in the networking literature on methods
for sharing the cost of a designed network — often a virtual overlay, multicast tree,
or other sub-network of the Internet — among a collection of participants. (See e.g.
[19, 22] for overviews of work in this area).
A cost-sharing mechanism can be viewed as the underlying protocol that deter-
mines how much a network serving several participants will cost to each of them.
Specifically, say that each user i has a pair of nodes (s
i
, t
i
) that it wishes to connect;
it chooses an s
i
-t
i
path S
i
; and the cost-sharing mechanism then charges user i a cost
of C
i
(S
1
, . . . , S
k
). (Note that this cost can depend on the choices of the other users
as well.) Although there are in principle many possible cost-sharing mechanisms,
research in this area has converged on a few mechanisms with good theoretical and
empirical behavior; here we focus on the following particularly natural one: the cost
of each edge is shared equally by the set of all users whose paths contain it, so that
C
i
(S
1
, S
2
, . . . , S
k
) =
X
e∈S
i
c
e
|{j : e ∈ S
j
}|
.
This equal-division mechanism has a number of basic economic motivations; it can be
derived from the Shapley value [32], and it can be shown to b e the unique cost-sharing
scheme satisfying a number of different sets of axioms [19, 22, 32]. For the former
reason, we will refer to it as the Shapley cost-sharing mechanism. Note that the total
edge cost of the designed network is equal to the sum of the costs in the union of all
S
i
, and the costs allocated to users in the Shapley mechanism completely pay for this
total edge cost:
P
k
i=1
C
i
(S
1
, S
2
, . . . , S
k
) =
P
e∈∪
i
S
i
c
e
.
1
Recall that a Nash equilibrium is a state of the system in which no agent has an interest in
unilaterally changing its own behavior.
THE PRICE OF STABILITY FOR NETWORK DESIGN WITH FAIR COST ALLOCATION 3
...
3
1
2
1
k-1
1
k
1
1
0 0 0 0 0
1+ε
s
t
k
t
k-1
t
1
t
2
t
3
Fig. 1.1. An instance in which the price of stability converges to H(k) = Θ(log k) as ε → 0.
Now, the general question is to determine how this basic cost-sharing mechanism
serves to influence the strategic behavior of the users, and what effect this has on
the structure and overall cost of the network one obtains. Given a solution to the
network design problem consisting of a vector of paths (S
1
, . . . , S
k
) for the k users,
user i would be interested in deviating from this solution if there were an alternate s
i
-
t
i
path S
0
i
such that changing to S
0
i
would lower its cost under the resulting allocation:
C
i
(S
1
, . . . , S
i−1
, S
0
i
, S
i+1
, . . . , S
k
) < C
i
(S
1
, . . . , S
i−1
, S
i
, S
i+1
, . . . , S
k
). We say that a
set of paths is a Nash equilibrium if no user has an interest in deviating. As we will see
below, there exists a set of paths in Nash equilibrium for every instance of this network
design game. (In this paper, we will only be concerned with pure Nash equilibria; i.e.,
with equilibria where each user deterministically chooses a single path.)
The goal of a network design protocol is to suggest for each user i a path S
i
so that
the resulting set of paths is in Nash equilibrium and its total cost exceeds that of an
optimal set of paths by as small factor as possible; this factor is the price of stability
of the instance. It is useful at this point to consider a simple example that illustrates
how the price of stability can grow to a super-constant value (with k). Suppose k
players wish to connect from the common source s to their resp ective terminals t
i
,
and assume player i has its own path of cost 1/i, and all players can share a common
path of cost 1 + ε for some small ε > 0 (see Figure 1.1). The optimal solution would
connect all agents through the common path for a total cost of 1 + ε. However, if
this solution were offered to the users, they would defect from it one by one to their
alternative paths. The unique Nash equilibrium has a cost of
P
k
i=1
1/i = H(k).
While the price of stability in this instance grows with k, it only does so loga-
rithmically. It is thus natural to ask how large the price of stability can be for this
network design problem. If we think about the example in Figure 1.1 further, it is
also interesting to note that a Nash equilibrium is reached by players taking turns up-
dating their paths (in other words, best-response dynamics) starting from an optimal
solution; it is natural to ask to what extent this holds in general.
Our Results. Our first main result is that in every instance of the network design
problem with Shapley cost-sharing, there always exists a Nash equilibrium of total
cost at most H(k) times optimal. In other words, the simple example in Figure 1.1 is
in fact the worst possible case.
We prove this result using a potential function method due to Rosenthal [36]
(based on [6]) and later generalized by Monderer and Shapley [30]: one defines a
potential function Φ on possible solutions and shows that every improving move of
4 ANSHELEVICH, DASGUPTA, KLEINBERG, TARDOS, WEXLER, ROUGHGARDEN
one of the users (to lower its own cost) reduces the value of Φ. Since the set of possible
solutions is finite, it follows that every sequence of improving moves leads to a Nash
equilibrium. The goal of Monderer’s and Shapley’s and Rosenthal’s work was to prove
existence statements of this sort; for our purposes, we make further use of the potential
function to prove a bound on the price of stability. Specifically, we give bounds relating
the value of the potential for a given solution to the overall cost of that solution; if
we then iterate using best-response dynamics starting from an optimal solution, the
potential does not increase, and hence we can bound the cost of any solution that we
reach. Thus, for this network design game, best-response dynamics starting from the
optimum do in fact always lead to a good Nash equilibrium.
We can extend our basic result to a number of more general settings. To begin
with, the H(k) bound on the price of stability extends directly to the case in which
users are selecting arbitrary subsets of a ground set (with elements’ costs shared
according to the Shapley value), rather than paths in a graph; it also extends to the
case in which the cost of each edge is a non-decreasing concave function of the number
of users on it. In addition, our results also hold if we introduce capacities into our
model; each edge e may be used by at most u
e
players, where u
e
is the capacity of e.
We arrive at a more technically involved set of extensions if we wish to add
latencies to the network design problem. Here each edge has a concave construction
cost c
e
(x) when there are x users on the edge, and a latency cost d
e
(x); the cost
experienced by a user is the full latency plus a fair share of the construction cost,
d
e
(x) + c
e
(x)/x. We give general conditions on the latency functions that allow us
to bound the price of stability in this case by d · H(k), where d depends on the
delay functions used. Moreover, we obtain stronger bounds in the case where users
experience only delays, not construction costs; this includes a result that relates the
cost of a best Nash equilibrium to that of an optimum with twice as many players,
and a result that improves the potential-based bound on the price of stability for the
single-source delay-only case.
Since a number of our proofs are obtained by following the results of best-resp onse
dynamics via a potential function, it is natural to investigate the speed of convergence
of best-response dynamics for this game. We show that with k players, it can run for
a time exponential in k. Whether there is a way to schedule players’ moves to make
best-response dynamics converge in a polynomial number of steps for this game in
general is an interesting open question.
Finally, we consider a natural generalization of the cost-sharing model that carries
us beyond the potential-function framework. Specifically, suppose each user has a
weight (perhaps corresponding to the amount of traffic it plans to send), and we
change the cost-allocation so that user i’s payment for edge e is equal to the ratio
of its weight to the total weight of all users on e. In addition to being intuitively
natural, this definition is analogous to certain natural generalizations of the Shapley
value [29]. The weighted model, however, is significantly more complicated: there is
no longer a potential function whose value tracks improvements in users’ costs when
they greedily update their solutions. We also show, using a construction involving
user weights that grow exponentially in k, that the price of stability can be as high as
Ω(k). We have obtained some initial positive results here, including the convergence
of best-response dynamics when all users seek to construct a path from a node s to
a node t (the price of stability here is 1), and in the general model of users selecting
sets from a ground set, where each element appears in the sets of at most two users.
THE PRICE OF STABILITY FOR NETWORK DESIGN WITH FAIR COST ALLOCATION 5
Related Work. Network design games under a different model were considered
by a subset of the authors in [3]; there, the setting was much more “unregulated”
in that users could offer to pay for an arbitrary fraction of any edge in the network.
This model resulted in instances where no pure Nash equilibrium existed; and in
many cases in [3] when pure Nash equilibria did exist, certain users were able to
act as “free riders,” paying very little or nothing at all. The present model, on the
other hand, ensures that there is always a pure Nash equilibrium within a logarithmic
factor of optimal, in which users pay for a fair p ortion of the resources they use.
Network creation games of a fairly different flavor — in which users correspond to
nodes, and can build subsets of the edges incident to them — have been considered in
[2, 12, 5, 16, 21, 31]. The model in this paper associates users instead with connection
requests, and allows them to contribute to the cost of any edge that helps them to
form a path that they need.
The bulk of the work on cost-sharing (see e.g. [19, 22] and the references there)
tends to assume a fixed underlying set of edges. Jain and Vazirani [23] and Kent and
Skorin-Kapov [26] consider cost-sharing for a single source network design game. Cost-
sharing games assume that there is a central authority that designs and maintains the
network, and decides appropriate cost-shares for each agent, depending on the graph
and all other agents, via a complex algorithm. The agents’ only role is to report their
utility for being included in the network.
Here, on the other hand, we consider a simple cost-sharing mechanism, the
Shapley-value, and ask what the strategic implications of a given cost-sharing mecha-
nism are for the way in which a network will be designed. This question explores the
feedback between the protocol that governs network construction and the behavior
of self-interested agents that interact with this protocol. An approach of a similar
style, though in a different setting, was pursued by Johari and Tsitsiklis [24]; there,
they assumed a network protocol that priced traffic according to a scheme due to
Kelly [25], and asked how this protocol would affect the strategic decisions of self-
interested agents routing connections in the network.
The special case of our game with only delays is closely related to the congestion
games of [39, 37]. They consider a game where the amount of flow carried by an
individual user is infinitesimally small (a non-atomic game), while in this paper we
assume that each user has a unit of flow, which it needs to route on a single path. In
the non-atomic game of [39, 37] the Nash equilibrium is essentially unique (hence there
is no distinction between the price of anarchy and stability), while in our atomic game
there can be many equilibria. Fabrikant, Papadimitriou, and Talwar [17] consider our
atomic game with delays only. They give a polynomial time algorithm to minimize the
potential function Φ in the case that all users share a common source, and show that
finding any equilibrium solution is PLS-complete for multiple source-sink pairs. Our
results extend the price of anarchy results of [39, 37] about non-atomic games to results
on the price of stability for the case of single source atomic games. Subsequent to
our work, further results on the price of anarchy and stability in atomic games with
delays were obtained in [4, 7, 11, 10, 40]. For games without delays, Agarwal and
Charikar [1] give improved bounds on the price of stability in single-source undirected
networks, and Fiat et al. [20] give bounds with the additional assumption that each
vertex is the destination of some player. Other aspects of these and closely related
games were recently explored in [9, 18].
A weighted game similar to ours is presented by Libman and Orda [28], with a
different mechanism for distributing costs among users. They do not consider the
