Abstract: Intuitively, Braess's paradox states that destroying a part
of a network may improve the common latency of selsh
ows at Nashequilibrium. Such a paradox is a pervasive phenomenon in real-world
networks. Any administrator, who wants to improve equilibrium delays
in selsh networks, is facing some basic questions: (i) Is the network
paradox-ridden? (ii) How can we delete some edges to optimize equilibrium
ow delays? (iii) How can we modify edge latencies to optimize
equilibrium
ow delays?
Unfortunately, such questions lead to NP-hard problems in general. In
this work, we impose some natural restrictions on our networks, e.g.
we assume strictly increasing linear latencies. Our target is to formulate
ecient algorithms for the three questions above.We manage to provide:
{ A polynomial-time algorithm that decides if a network is paradoxridden,
when latencies are linear and strictly increasing.
{ A reduction of the problem of deciding if a network with arbitrary
linear latencies is paradox-ridden to the problem of generating all
optimal basic feasible solutions of a Linear Program that describes
the optimal trac allocations to the edges with constant latency.
{ An algorithm for nding a subnetwork that is almost optimal wrt
equilibrium latency. Our algorithm is subexponential when the number
of paths is polynomial and each path is of polylogarithmic length.
{ A polynomial-time algorithm for the problem of nding the best
subnetwork, which outperforms any known approximation algorithm
for the case of strictly increasing linear latencies.
{ A polynomial-time method that turns the optimal
ow into a Nash
ow by deleting the edges not used by the optimal
ow, and performing
minimal modications to the latencies of the remaining ones.
Our results provide a deeper understanding of the computational complexity
of recognizing the Braess's paradox most severe manifestations,
and our techniques show novel ways of using the probabilistic method
and of exploiting convex separable quadratic programs.

Abstract: Intuitively, Braess’s paradox states that destroying a part of a network may improve the common latency of selfish flows at Nashequilibrium. Such a paradox is a pervasive phenomenon in real-world networks. Any administrator who wants to improve equilibrium delays in selfish networks, is facing some basic questions:
– Is the network paradox-ridden?
– How can we delete some edges to optimize equilibrium flow delays?
– How can we modify edge latencies to optimize equilibrium flow delays?
Unfortunately, such questions lead to View the MathML sourceNP-hard problems in general. In this work, we impose some natural restrictions on our networks, e.g. we assume strictly increasing linear latencies. Our target is to formulate efficient algorithms for the three questions above. We manage to provide:
– A polynomial-time algorithm that decides if a network is paradox-ridden, when latencies are linear and strictly increasing.
– A reduction of the problem of deciding if a network with (arbitrary) linear latencies is paradox-ridden to the problem of generating all optimal basic feasible solutions of a Linear Program that describes the optimal traffic allocations to the edges with constant latency.
– An algorithm for finding a subnetwork that is almost optimal wrt equilibrium latency. Our algorithm is subexponential when the number of paths is polynomial and each path is of polylogarithmic length.
– A polynomial-time algorithm for the problem of finding the best subnetwork which outperforms any known approximation for the case of strictly increasing linear latencies.
– A polynomial-time method that turns the optimal flow into a Nash flow by deleting the edges not used by the optimal flow, and performing minimal modifications on the latencies of the remaining ones.
Our results provide a deeper understanding of the computational complexity of recognizing the most severe manifestations of Braess’s paradox, and our techniques show novel ways of using the probabilistic method and of exploiting convex separable quadratic programs.

Abstract: The presentwork considers the following computational problem:
Given any finite game in normal form G and the corresponding
infinitely repeated game G∞, determine in polynomial time (wrt1 the representation
ofG) a profile of strategies for the players inG∞ that is an equilibrium
point wrt the limit-of-means payoff. The problem has been solved
for two players [10], based mainly on the implementability of the threats
for this case. Nevertheless, [4] demonstrated that the traditional notion of
threats is a computationally hard problem for games with at least 3 players
(see also [8]). Our results are the following: (i) We propose an alternative
notion of correlated threats, which is polynomial time computable
(and therefore credible). Our correlated threats are also more severe than
the traditional notion of threats, but not overwhelming for any individual
player. (ii) When for the underlying game G there is a correlated strategy
with payoff vector strictly larger than the correlated threats vector,
we efficiently compute a polynomial–size (wrt the description of G) equilibrium
point for G∞, for any constant number of players. (iii) Otherwise,
we demonstrate the construction of an equilibrium point for an arbitrary
number of players and up to 2 concurrently positive payoff coordinates in
any payoff vector of G. This completely resolves the cases of 3 players, and
provides a direction towards handling the cases of more than 3 players. It
is mentioned that our construction is not a Nashequilibrium point, because
the correlated threats we use are implemented via, not only full synchrony
(as in [10]), but also coordination of the other players˘ actions. But
this seems to be a fair trade-off between efficiency of the construction and
players˘ coordination, in particular because it only affects the punishments
(which are anticipated never to be used).

Abstract: Evolutionary Game Theory is the study of strategic interactions among large populations of agents who base their decisions on simple, myopic rules. A major goal of the theory is to determine broad classes of decision procedures which both provide plausible descriptions of selfish behaviour and include appealing forms of aggregate behaviour. For example, properties such as the correlation between strategies' growth rates and payoffs, the connection between stationary states and Nash equilibria and global guarantees of convergence to equilibrium, are widely studied in the literature. In this paper we discuss some computational aspects of the theory, which we prefer to view more as Game Theoretic Aspects of Evolution than Evolutionary Game Theory, since the term "evolution" actually refers to strategic adaptation of individuals ' behaviour through a dynamic process and not the traditional evolution of populations. We consider this dynamic process as a self-organization procedure, which under certain conditions leads to some kind of stability and assures robustness against invasion.

Abstract: We study the fundamental problem of computing an arbitrary Nashequilibrium in bimatrix games.
We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria.
We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular, we introduce the notion of mutual (quasi-) concavity of a bimatrix game, which assures (quasi-) convexity of our quadratic program, for at least one value of the parameter. For mutually concave bimatrix games, we provide a polynomial-time computation of a Nashequilibrium, based on the polynomial tractability of convex quadratic programming. For the mutually quasiconcave games, we provide (to our knowledge) the first FPTAS for the construction of a Nashequilibrium.
Of course, for these new polynomially tractable subclasses of bimatrix games to be useful, polynomial-time certificates are also necessary that will allow us to efficiently identify them. Towards this direction, we provide various characterizations of mutual concavity, which allow us to construct such a certificate. Interestingly, these characterizations also shed light to some structural properties of the bimatrix games satisfying mutual concavity. This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constant-sum games (rank-0 games). It is though incomparable to the subclass of games with fixed rank [KT07]: Even rank-1 games may not be mutually concave (eg, Prisoner's dilemma), but on the other hand, there exist mutually concave games of arbitrary (even full) rank. Finally, we prove closeness of mutual concavity under (Nashequilibrium preserving) positive affine transformations of bimatrix games having the same scaling factor for both payoff matrices. For different scaling factors the property is not necessarily preserved.

Abstract: We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. In a Nashequilibrium, each user routes its traffic on links that minimize its expected latency cost.
Our structural results provide substantial evidence for the Fully Mixed NashEquilibrium Conjecture, which states that the worst Nashequilibrium is the fully mixed Nashequilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed NashEquilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nashequilibrium is within a factor of 6 + epsi, of that of the fully mixed Nashequilibrium, assuming that link capacities are identical.
Our complexity results include hardness, approximability and inapproximability ones. Here we show, that for identical link capacities and under a certain condition, there is a randomized, polynomial-time algorithm to approximate the worst social cost within a factor arbitrarily close to 6 + epsi. Furthermore, we prove that for any arbitrary integer k > 0, it is -hard to decide whether or not any given allocation of users to links can be transformed into a pure Nashequilibrium using at most k selfish steps. Assuming identical link capacities, we give a polynomial-time approximation scheme (PTAS) to approximate the best social cost over all pure Nash equilibria. Finally we prove, that it is -hard to approximate the worst social cost within a multiplicative factor . The quantity is the tight upper bound on the ratio of the worst social cost and the optimal cost in the model of identical capacities.

Abstract: In routing games, the network performance at equilibrium can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braess's paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve the equilibrium performance by edge removal. In this work, we investigate the computational complexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network. We first show that bottleneck routing games do not suffer from Braess's paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA is as large as \Omega(n^{0.121}) and cannot improve by edge removal. Thus, the network design problem for such games is NP-hard to approximate within a factor of O(n^{0.121-\eps}), for any constant \eps > 0. On the positive side, we show how to compute an almost optimal subnetwork w.r.t. the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. The running time is determined by the total number of paths, and is quasipolynomial when the number of paths is quasipolynomial.

Abstract: We study computational and coordination efficiency issues of
Nash equilibria in symmetric network congestion games. We first propose
a simple and natural greedy method that computes a pure Nashequilibrium
with respect to traffic congestion in a network. In this algorithm
each user plays only once and allocates her traffic to a path selected via
a shortest path computation. We then show that this algorithm works
for series-parallel networks when users are identical or when users are of
varying demands but have the same best response strategy for any initial
network traffic. We also give constructions where the algorithm fails if
either the above condition is violated (even for series-parallel networks)
or the network is not series-parallel (even for identical users). Thus, we
essentially indicate the limits of the applicability of this greedy approach.
We also study the price of anarchy for the objective of maximum
latency. We prove that for any network of m uniformly related links and
for identical users, the price of anarchy is {\`E}( logm
log logm).

Abstract: We consider a strategic game with two classes of confronting
randomized players on a graph G(V,E): {\'i} attackers, each choosing vertices
and wishing to minimize the probability of being caught, and a
defender, who chooses edges and gains the expected number of attackers
it catches. The Price of Defense is the worst-case ratio, over all Nash
equilibria, of the optimal gain of the defender over its gain at a Nashequilibrium.
We provide a comprehensive collection of trade-offs between the
Price of Defense and the computational efficiency of Nash equilibria.
– Through reduction to a Two-Players, Constant-Sum Game, we prove
that a Nashequilibrium can be computed in polynomial time. The
reduction does not provide any apparent guarantees on the Price of
Defense.
– To obtain such, we analyze several structured Nash equilibria:
• In a Matching Nashequilibrium, the support of the defender is
an Edge Cover. We prove that they can be computed in polynomial
time, and they incur a Price of Defense of {\'a}(G), the
Independence Number of G.
• In a Perfect Matching Nashequilibrium, the support of the defender
is a Perfect Matching. We prove that they can be computed
in polynomial time, and they incur a Price of Defense of
|V |
2 .
• In a Defender Uniform Nashequilibrium, the defender chooses
uniformly each edge in its support. We prove that they incur a
Price of Defense falling between those for Matching and Perfect
Matching Nash Equilibria; however, it is NP-complete to decide
their existence.
• In an Attacker Symmetric and Uniform Nashequilibrium, all
attackers have a common support on which each uses a uniform
distribution. We prove that they can be computed in polynomial
time and incur a Price of Defense of either
|V |
2 or {\'a}(G).

Abstract: Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nashequilibrium with unbounded Coordination Ratio [E. Koutsoupias, C. Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS, vol. 1563, 1999, pp. 404-413; T. Roughgarden, E. Tardos, How bad is selfish routing? in: 41st IEEE Annual Symposium of Foundations of Computer Science, FOCS, 2000, pp. 93-102]. A Leader can decrease the coordination ratio by assigning flow {\'a}r on M, and then all Followers assign selfishly the (1-{\'a})r remaining flow. This is a Stackelberg Scheduling Instance(M,r,{\'a}),0≤{\'a}≤1. It was shown [T. Roughgarden, Stackelberg scheduling strategies, in: 33rd Annual Symposium on Theory of Computing, STOC, 2001, pp. 104-113] that it is weakly NP-hard to compute the optimal Leader's strategy. For any such network M we efficiently compute the minimum portion @b"M of flow r>0 needed by a Leader to induce M's optimum cost, as well as her optimal strategy. This shows that the optimal Leader's strategy on instances (M,r,@a>=@b"M) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess's Paradox graph, such that no strategy controlling {\'a}r flow can induce ≤1/{\'a} times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess's graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [A. Kaporis, P. Spirakis, The price of optimum in stackelberg games on arbitrary single commodity networks and latency functions, in: 18th annual ACM symposium on Parallelism in Algorithms and Architectures, SPAA, 2006, pp. 19-28]. Some preliminary results have also appeared as technical report in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The price of optimum in stackelberg games, in: Electronic Colloquium on Computational Complexity, ECCC, (056), 2005].

Abstract: In this work, we study the combinatorial structure and the
computational complexity of Nash equilibria for a certain game that
models selfish routing over a network consisting of m parallel links. We
assume a collection of n users, each employing a mixed strategy, which
is a probability distribution over links, to control the routing of its own
assigned traffic. In a Nashequilibrium, each user selfishly routes its traffic
on those links that minimize its expected latency cost, given the network
congestion caused by the other users. The social cost of a Nashequilibrium
is the expectation, over all random choices of the users, of the
maximum, over all links, latency through a link.
We embark on a systematic study of several algorithmic problems related
to the computation of Nash equilibria for the selfish routing game we consider.
In a nutshell, these problems relate to deciding the existence of a
Nashequilibrium, constructing a Nashequilibrium with given support
characteristics, constructing the worst Nashequilibrium (the one with
maximum social cost), constructing the best Nashequilibrium (the one
with minimum social cost), or computing the social cost of a (given) Nashequilibrium. Our work provides a comprehensive collection of efficient algorithms,
hardness results (both as NP-hardness and #P-completeness
results), and structural results for these algorithmic problems. Our results
span and contrast a wide range of assumptions on the syntax of the
Nash equilibria and on the parameters of the system.