Abstract: We propose a simple and intuitive cost mechanism which assigns
costs for the competitive usage of m resources by n selfish agents.
Each agent has an individual demand; demands are drawn according to
some probability distribution. The cost paid by an agent for a resource
she chooses is the total demand put on the resource divided by the number
of agents who chose that same resource. So, resources charge costs
in an equitable, fair way, while each resource makes no profit out of the
agents.We call our model the Fair Pricing model. Its fair cost mechanism
induces a non-cooperative game among the agents. To evaluate the Nashequilibria of this game, we introduce the Diffuse Price of Anarchy, as an
extension of the Price of Anarchy that takes into account the probability
distribution on the demands. We prove:
– Pure Nashequilibria may not exist, unless all chosen demands are
identical; in contrast, a fully mixed Nash equilibrium exists for all
possible choices of the demands. Further on, the fully mixed Nash
equilibrium is the unique Nash equilibrium in case there are only two
agents.
– In the worst-case choice of demands, the Price of Anarchy is {\`E}(n);
for the special case of two agents, the Price of Anarchy is less than
2 − 1
m.
– Assume now that demands are drawn from a bounded, independent
probability distribution, where all demands are identically distributed
and each is at most a (universal for the class) constant times its expectation.
Then, the Diffuse Price of Anarchy is at most that same
constant, which is just 2 when each demand is distributed symmetrically
around its expectation.

Abstract: We give an efficient local search
algorithm that computes a good vertex coloring of a graph $G$. In
order to better illustrate this local search method, we view local
moves as selfish moves in a suitably defined game. In particular,
given a graph $G=(V,E)$ of $n$ vertices and $m$ edges, we define
the \emph{graph coloring game} $\Gamma(G)$ as a strategic game
where the set of players is the set of vertices and the players
share the same action set, which is a set of $n$ colors. The
payoff that a vertex $v$ receives, given the actions chosen by all
vertices, equals the total number of vertices that have chosen the
same color as $v$, unless a neighbor of $v$ has also chosen the
same color, in which case the payoff of $v$ is 0. We show:
\begin{itemize}
\item The game $\Gamma(G)$ has always pure Nashequilibria. Each
pure equilibrium is a proper coloring of $G$. Furthermore, there
exists a pure equilibrium that corresponds to an optimum coloring.
\item We give a polynomial time algorithm $\mathcal{A}$ which
computes a pure Nash equilibrium of $\Gamma(G)$. \item The total
number, $k$, of colors used in \emph{any} pure Nash equilibrium
(and thus achieved by $\mathcal{A}$) is $k\leq\min\{\Delta_2+1,
\frac{n+\omega}{2}, \frac{1+\sqrt{1+8m}}{2}, n-\alpha+1\}$, where
$\omega, \alpha$ are the clique number and the independence number
of $G$ and $\Delta_2$ is the maximum degree that a vertex can have
subject to the condition that it is adjacent to at least one
vertex of equal or greater degree. ($\Delta_2$ is no more than the
maximum degree $\Delta$ of $G$.) \item Thus, in fact, we propose
here a \emph{new}, \emph{efficient} coloring method that achieves
a number of colors \emph{satisfying (together) the known general
upper bounds on the chromatic number $\chi$}. Our method is also
an alternative general way of \emph{proving},
\emph{constructively}, all these bounds. \item Finally, we show
how to strengthen our method (staying in polynomial time) so that
it avoids ``bad'' pure Nashequilibria (i.e. those admitting a
number of colors $k$ far away from $\chi$). In particular, we show
that our enhanced method colors \emph{optimally} dense random
$q$-partite graphs (of fixed $q$) with high probability.
\end{itemize}

Abstract: .We present a new methodology for computing approximate
Nashequilibria for two-person non-cooperative games based upon certain
extensions and specializations of an existing optimization approach pre-
viously used for the derivation of xed approximations for this problem.
In particular, the general two-person problem is reduced to an inde-
nite quadratic programming problem of special structure involving the
n x n adjacency matrix of an induced simple graph specied by the in-
put data of the game, where n is the number of players' strategies.

Abstract: Consider a network vulnerable to viral infection. The system security software can guarantee
safety only to a limited part of the network. Such limitations result from economy costs or processing
costs. The problem raised is to which part of the network the security software should
be installed, so that to secure as much as possible the network. We model this practical network
scenario as a non-cooperative multi-player game on a graph, with two kinds of players, a set
of attackers and a protector player, representing the viruses and the system security software,
respectively. Each attacker player chooses a node of the graph (or a set of them, via a probability
distribution) to infect. The protector player chooses independently, in a basic case of the
problem, a simple path or an edge of the graph (or a set of them, via a probability distribution)
and cleans this part of the network from attackers. Each attacker wishes to maximize the probability
of escaping its cleaning by the protector. In contrast, the protector aims at maximizing
the expected number of cleaned attackers. We call the two games obtained from the two basic
cases considered, as the Path and the Edge model, respectively. For these two games, we are
interested in the associated Nashequilibria, where no network entity can unilaterally improve
its local objective. We obtain the following results:
• The problem of existence of a pure Nash equilibrium is NP-complete for the Path model.
This opposed to that, no instance of the Edge model possesses a pure Nash equilibrium,
proved in [7].
• In [7] a characterization of mixed Nashequilibria for the Edge model is provided. However,
that characterization only implies an exponential time algorithm for the general case.
Here, combining it with clever exploration of properties of various practical families of
graphs, we compute, in polynomial time, mixed Nashequilibria on corresponding graph
instances. These graph families include, regular graphs, graphs that can be decomposed, in
polynomially time, into vertex disjoint r-regular subgraphs, graphs with perfect matchings
and trees.
• We utilize the notion of social cost [6] for measuring system performance on such scenario;
here is defined to be the utility of the protector. We prove that the corresponding Price of
Anarchy in any mixed Nashequilibria of the game is upper and lower bounded by a linear
function of the number of vertices of the graph.

Abstract: Consider a network vulnerable to viral infection, where the security software can guarantee safety only to a limited part of it. We model this practical network scenario as a non-cooperative multi-player game on a graph, with two kinds of players, a set of attackers and a protector player, representing the viruses and the system security software, respectively. We are interested in the associated Nashequilibria, where no network entity can unilaterally improve its local objective. We obtain the following results: for certain families of graphs, mixed Nashequilibria can be computed in polynomially time. These families include, among others, regular graphs, graphs with perfect matchings and trees. The corresponding price of anarchy for any mixed Nashequilibria of the game is upper and lower bounded by a linear function of the number of vertices of the graph. (We define the price of anarchy to reflect the utility of the protector). Finally, we introduce a generalised version of the game. We show that the existence problem of pure Nashequilibria here is NP complete.

Abstract: Known general proofs of Nash Theorem (about the existence of NashEquilibria (NEa) in finite strategic games) rely on the use of a fixed point theorem (e.g. Brouwer or Kakutani. While it seems that there is no general way of proving the existence of Nashequilibria without the use of a fixed point theorem, there do however exist some (not so common in the CS literature) proofs that seem to indicate alternative proof paths, for games of two players. This note discusses two such proofs.

Abstract: In this paper we propose a new methodology for determining
approximate Nashequilibria of non-cooperative bimatrix games and,
based on that, we provide an efficient algorithm that computes 0.3393-
approximate equilibria, the best approximation till now. The methodology
is based on the formulation of an appropriate function of pairs of
mixed strategies reflecting the maximum deviation of the players¢ payoffs
from the best payoff each player could achieve given the strategy
chosen by the other. We then seek to minimize such a function using
descent procedures. As it is unlikely to be able to find global minima
in polynomial time, given the recently proven intractability of the problem,
we concentrate on the computation of stationary points and prove
that they can be approximated arbitrarily close in polynomial time and
that they have the above mentioned approximation property. Our result
provides the best till now for polynomially computable -approximate
Nashequilibria of bimatrix games. Furthermore, our methodology for
computing approximate Nashequilibria has not been used by others.

Abstract: We focus on the problem of computing approximate Nashequilibria and well-supported approximate Nashequilibria in random bimatrix games, where each player's payoffs are bounded and independent random variables, not necessarily identically distributed, but with common expectations. We show that the completely mixed uniform strategy profile, i.e. the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is an almost Nash equilibrium for random bimatrix games, in the sense that it is, with high probability, an {\aa}-well-supported Nash equilibrium where {\aa} tends to zero as n tends to infinity.

Abstract: We consider selfish routing over a network consisting of m parallellinks through which $n$ selfish users route their traffic trying tominimize their own expected latency. We study the class of mixedstrategies in which the expected latency through each link is at mosta constant multiple of the optimum maximum latency had globalregulation been available. For the case of uniform links it is knownthat all Nashequilibria belong to this class of strategies. We areinterested in bounding the coordination ratio (or price of anarchy) ofthese strategies defined as the worst-case ratio of the maximum (overall links) expected latency over the optimum maximum latency. The loadbalancing aspect of the problem immediately implies a lower boundO(ln m ln ln m) of the coordinationratio. We give a tight (up to a multiplicative constant) upper bound.To show the upper bound, we analyze a variant of the classical ballsand bins problem, in which balls with arbitrary weights are placedinto bins according to arbitrary probability distributions. At theheart of our approach is a new probabilistic tool that we call ballfusion; this tool is used to reduce the variant of the problem whereballs bear weights to the classical version (with no weights). Ballfusion applies to more general settings such as links with arbitrarycapacities and other latency functions.

Abstract: We consider algorithmic questions concerning the existence,
tractability and quality of atomic congestion games, among users that
are considered to participate in (static) selfish coalitions. We carefully
define a coalitional congestion model among atomic players.
Our findings in this model are quite interesting, in the sense that we
demonstrate many similarities with the non–cooperative case. For example,
there exist potentials proving the existence of Pure NashEquilibria
(PNE) in the (even unrelated) parallel links setting; the Finite Improvement
Property collapses as soon as we depart from linear delays, but
there is an exact potential (and thus PNE) for the case of linear delays,
in the network setting; the Price of Anarchy on identical parallel
links demonstrates a quite surprising threshold behavior: it persists on
being asymptotically equal to that in the case of the non–cooperative
KP–model, unless we enforce a sublogarithmic number of coalitions.
We also show crucial differences, mainly concerning the hardness of algorithmic
problems that are solved efficiently in the non–cooperative case.
Although we demonstrate convergence to robust PNE, we also prove the
hardness of computing them. On the other hand, we can easily construct
a generalized fully mixed Nash Equilibrium. Finally, we propose a new
improvement policy that converges to PNE that are robust against (even
dynamically forming) coalitions of small size, in pseudo–polynomial time.
Keywords. Game Theory, Atomic Congestion Games, Coalitions, Convergence
to Equilibria, Price of Anarchy.

Abstract: We consider algorithmic questions concerning the existence, tractability and quality of Nash equi-
libria, in atomic congestion games among users participating in selsh coalitions.
We introduce a coalitional congestion model among atomic players and demonstrate many in-
teresting similarities with the non-cooperative case. For example, there exists a potential function
proving the existence of Pure NashEquilibria (PNE) in the unrelated parallel links setting; in
the network setting, the Finite Improvement Property collapses as soon as we depart from linear
delays, but there is an exact potential (and thus PNE) for linear delays; the Price of Anarchy on
identical parallel links demonstrates a quite surprising threshold behavior: it persists on being
asymptotically equal to that in the case of the non-cooperative KP-model, unless the number of
coalitions is sublogarithmic.
We also show crucial dierences, mainly concerning the hardness of algorithmic problems that
are solved eciently in the non{cooperative case. Although we demonstrate convergence to robust
PNE, we also prove the hardness of computing them. On the other hand, we propose a generalized
fully mixed Nash Equilibrium, that can be eciently constructed in most cases. Finally, we
propose a natural improvement policy and prove its convergence in pseudo{polynomial time to
PNE which are robust against (even dynamically forming) coalitions of small size.

Abstract: We consider algorithmic questions concerning the existence,
tractability and quality of atomic congestion games, among users that
are considered to participate in (static) selfish coalitions. We carefully
define a coalitional congestion model among atomic players.
Our findings in this model are quite interesting, in the sense that we
demonstrate many similarities with the non–cooperative case. For example,
there exist potentials proving the existence of Pure NashEquilibria
(PNE) in the (even unrelated) parallel links setting; the Finite Improvement
Property collapses as soon as we depart from linear delays, but
there is an exact potential (and thus PNE) for the case of linear delays,
in the network setting; the Price of Anarchy on identical parallel
links demonstrates a quite surprising threshold behavior: it persists on
being asymptotically equal to that in the case of the non–cooperative
KP–model, unless we enforce a sublogarithmic number of coalitions.
We also show crucial differences, mainly concerning the hardness of algorithmic
problems that are solved efficiently in the non–cooperative case.
Although we demonstrate convergence to robust PNE, we also prove the
hardness of computing them. On the other hand, we can easily construct
a generalized fully mixed Nash Equilibrium. Finally, we propose a new
improvement policy that converges to PNE that are robust against (even
dynamically forming) coalitions of small size, in pseudo–polynomial time.
Keywords. Game Theory, Atomic Congestion Games, Coalitions, Convergence
to Equilibria, Price of Anarchy.

Abstract: In this survey we present some recent advances in the literature
of atomic (mainly network) congestion games. The algorithmic
questions that we are interested in have to do with the existence of pure
Nashequilibria, the efficiency of their construction when they exist, as
well as the gap of the best/worst (mixed in general) Nashequilibria from
the social optima in such games, typically called the Price of Anarchy
and the Price of Stability respectively.

Abstract: This chapter is an introduction to the basic concepts and advances of a new field, that of Computational (or Algorithmic) Game Theory. We study the computational complexity of Nashequilibria and review the related algorithms proposed in the literature. Then, given the apparent difficulty of computing exact Nashequilibria, we study the efficient computation of approximate notions of Nashequilibria. Next we deal with several computational issues related to the class of congestion games, which model the selfish behavior of individuals when competing on the usage of a common set of resources. Finally, we study the price of anarchy (in the context of congestion games), which is defined as a measure of the performance degradation due to the the lack of coordination among the involved players.

Abstract: We propose a simple and intuitive cost mechanism which assigns costs for the competitive
usage of m resources by n selfish agents. Each agent has an individual demand; demands are
drawn according to some probability distribution. The cost paid by an agent for a resource
it chooses is the total demand put on the resource divided by the number of agents who
chose that same resource. So, resources charge costs in an equitable, fair way, while each
resource makes no profit out of the agents.
We call our model the Fair Pricing model. Its fair cost mechanism induces a noncooperative
game among the agents. To evaluate the Nashequilibria of this game, we
introduce the Diffuse Price of Anarchy, as an extension of the Price of Anarchy that takes
into account the probability distribution on the demands. We prove:
² Pure Nashequilibria may not exist, unless all chosen demands are identical.
² A fully mixed Nash equilibrium exists for all possible choices of the demands. Further
on, the fully mixed Nash equilibrium is the unique Nash equilibrium in case there are
only two agents.
² In the worst-case choice of demands, the Price of Anarchy is £(n); for the special case
of two agents, the Price of Anarchy is less than 2 ¡ 1
m .
² Assume now that demands are drawn from a bounded, independent probability distribution,
where all demands are identically distributed, and each demand may not exceed
some (universal for the class) constant times its expectation. It happens that the constant
is just 2 when each demand is distributed symmetrically around its expectation.
We prove that, for asymptotically large games where the number of agents tends to
infinity, the Diffuse Price of Anarchy is at most that universal constant. This implies
the first separation between Price of Anarchy and Diffuse Price of Anarchy.
Towards the end, we consider two closely related cost sharing models, namely the Average
Cost Pricing and the Serial Cost Sharing models, inspired by Economic Theory. In contrast
to the Fair Pricing model, we prove that pure Nashequilibria do exist for both these models.

Abstract: We investigate the existence of optimal tolls for atomic symmetric
network congestion games with unsplittable traffic and arbitrary non-negative and
non-decreasing latency functions.We focus on pure Nashequilibria and a natural
toll mechanism, which we call cost-balancing tolls. A set of cost-balancing tolls
turns every path with positive traffic on its edges into a minimum cost path. Hence
any given configuration is induced as a pure Nash equilibrium of the modified
game with the corresponding cost-balancing tolls. We show how to compute in
linear time a set of cost-balancing tolls for the optimal solution such that the total
amount of tolls paid by any player in any pure Nash equilibrium of the modified
game does not exceed the latency on the maximum latency path in the optimal
solution. Our main result is that for congestion games on series-parallel networks
with increasing latencies, the optimal solution is induced as the unique pure Nash
equilibrium of the game with the corresponding cost-balancing tolls. To the best
of our knowledge, only linear congestion games on parallel links were known to
admit optimal tolls prior to this work. To demonstrate the difficulty of computing
a better set of optimal tolls, we show that even for 2-player linear congestion
games on series-parallel networks, it is NP-hard to decide whether the optimal
solution is the unique pure Nash equilibrium or there is another equilibrium of
total cost at least 6/5 times the optimal cost.

Abstract: In this paper we study the notion of the Evolutionary Stable
Strategies (ESS) in evolutionary games and we demonstrate their qualitative
difference from the NashEquilibria, by showing that a random
evolutionary game has on average exponentially less number of ESS than
the number of NashEquilibria in the underlying symmetric 2-person
game with random payoffs.

Abstract: In this work we study the tractability of well supported approximate
NashEquilibria (SuppNE in short) in bimatrix games. In view
of the apparent intractability of constructing NashEquilibria (NE in
short) in polynomial time, even for bimatrix games, understanding the
limitations of the approximability of the problem is of great importance.
We initially prove that SuppNE are immune to the addition of arbitrary
real vectors to the rows (columns) of the row (column) player¢s
payoff matrix. Consequently we propose a polynomial time algorithm
(based on linear programming) that constructs a 0.5−SuppNE for arbitrary
win lose games.
We then parameterize our technique for win lose games, in order to
apply it to arbitrary (normalized) bimatrix games. Indeed, this new technique
leads to a weaker {\"o}−SuppNE for win lose games, where {\"o} = √5−1
2
is the golden ratio. Nevertheless, this parameterized technique extends
nicely to a technique for arbitrary [0, 1]−bimatrix games, which assures
a 0.658−SuppNE in polynomial time.
To our knowledge, these are the first polynomial time algorithms providing
{\aa}−SuppNE of normalized or win lose bimatrix games, for some
nontrivial constant {\aa} ∈ [0, 1), bounded away from 1.

Abstract: In large-scale or evolving networks, such as the Internet,
there is no authority possible to enforce a centralized traffic management.
In such situations, Game Theory and the concepts of Nashequilibria
and Congestion Games [8] are a suitable framework for analyzing
the equilibrium effects of selfish routes selection to network delays.
We focus here on layered networks where selfish users select paths to
route their loads (represented by arbitrary integer weights). We assume
that individual link delays are equal to the total load of the link. We
focus on the algorithm suggested in [2], i.e. a potential-based method
for finding pure Nashequilibria (PNE) in such networks. A superficial
analysis of this algorithm gives an upper bound on its time which is
polynomial in n (the number of users) and the sum of their weights. This
bound can be exponential in n when some weights are superpolynomial.
We provide strong experimental evidence that this algorithm actually
converges to a PNE in strong polynomial time in n (independent of the
weights values). In addition we propose an initial allocation of users
to paths that dramatically accelerates this algorithm, compared to an
arbitrary initial allocation. A by-product of our research is the discovery
of a weighted potential function when link delays are exponential to their
loads. This asserts the existence of PNE for these delay functions and
extends the result of

Abstract: We present three new coordination mechanisms for schedul-
ing n sel¯sh jobs on m unrelated machines. A coordination
mechanism aims to mitigate the impact of sel¯shness of jobs
on the e±ciency of schedules by de¯ning a local schedul-
ing policy on each machine. The scheduling policies induce
a game among the jobs and each job prefers to be sched-
uled on a machine so that its completion time is minimum
given the assignments of the other jobs. We consider the
maximum completion time among all jobs as the measure
of the e±ciency of schedules. The approximation ratio of
a coordination mechanism quanti¯es the e±ciency of pure
Nashequilibria (price of anarchy) of the induced game. Our
mechanisms are deterministic, local, and preemptive in the
sense that the scheduling policy does not necessarily process
the jobs in an uninterrupted way and may introduce some
idle time. Our ¯rst coordination mechanism has approxima-
tion ratio O(logm) and always guarantees that the induced
game has pure Nashequilibria to which the system con-
verges in at most n rounds. This result improves a recent
bound of O(log2 m) due to Azar, Jain, and Mirrokni and,
similarly to their mechanism, our mechanism uses a global
ordering of the jobs according to their distinct IDs. Next
we study the intriguing scenario where jobs are anonymous,
i.e., they have no IDs. In this case, coordination mechanisms
can only distinguish between jobs that have diffeerent load
characteristics. Our second mechanism handles anonymous
jobs and has approximation ratio O
¡ logm
log logm
¢
although the
game induced is not a potential game and, hence, the exis-
tence of pure Nashequilibria is not guaranteed by potential
function arguments. However, it provides evidence that the
known lower bounds for non-preemptive coordination mech-
anisms could be beaten using preemptive scheduling poli-
cies. Our third coordination mechanism also handles anony-
mous jobs and has a nice \cost-revealing" potential func-
tion. Besides in proving the existence of equilibria, we use
this potential function in order to upper-bound the price of stability of the induced game by O(logm), the price of an-
archy by O(log2 m), and the convergence time to O(log2 m)-
approximate assignments by a polynomial number of best-
response moves. Our third coordination mechanism is the
¯rst that handles anonymous jobs and simultaneously guar-
antees that the induced game is a potential game and has
bounded price of anarchy.

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 the well-known
game theoretic notion of Nashequilibria, as well as global guarantees of
convergence to equilibrium, are widely studied in the literature.
Our paper can be seen as a quick introduction to Evolutionary Game
Theory, together with a new research result and a discussion of many
algorithmic and complexity open problems in the area. In particular, we
discuss some algorithmic and complexity aspects of the theory, which
we prefer to view more as Game Theoretic Aspects of Evolution rather
than as 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. In particular, we concentrate on the notion of
the Evolutionary Stable Strategies (ESS). We demonstrate their qualitative
difference from NashEquilibria by showing that symmetric 2-person
games with random payoffs have on average exponentially less ESS than
NashEquilibria. We conclude this article with some interesting areas of
future research concerning the synergy of Evolutionary Game Theory
and Algorithms.

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 Nashequilibria 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 Nash equilibrium in bimatrix games.
We start by proposing a novel characterization of the set of Nashequilibria, 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 Nash equilibrium, 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 Nash equilibrium.
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 (Nash equilibrium 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 Nashequilibria, 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 Nash equilibrium, each user routes its traffic on links that minimize its expected latency cost.
Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nashequilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + epsi, of that of the fully mixed Nash equilibrium, 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 Nash equilibrium 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 Nashequilibria. 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: Consider an information network with harmful procedures called attackers (e.g., viruses); each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is the system protector scanning and cleaning from attackers some part of the network (e.g., an edge or a path), which it chooses independently using another probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the system protector; towards a conflicting objective, the system protector aims at maximizing the expected number of cleaned attackers.
We model this network scenario as a non-cooperative strategic game on graphs. We focus on the special case where the protector chooses a single edge. We are interested in the associated Nashequilibria, where no network entity can unilaterally improve its local objective. We obtain the following results:
{\^a}{\"i}¿½{\"i}¿½ No instance of the game possesses a pure Nash equilibrium.
{\^a}{\"i}¿½{\"i}¿½Every mixed Nash equilibrium enjoys a graph-theoretic structure, which enables a (typically exponential) algorithm to compute it.
{\^a}{\"i}¿½{\"i}¿½ We coin a natural subclass of mixed Nashequilibria, which we call matching Nashequilibria, for this game on graphs. Matching Nashequilibria are defined using structural parameters of graphs, such as independent sets and matchings.
{\^a}{\"i}¿½{\"i}¿½We derive a characterization of graphs possessing matching Nashequilibria. The characterization enables a linear time algorithm to compute a matching Nash equilibrium on any such graph with a given independent set and vertex cover.
{\^a}{\"i}¿½{\"i}¿½ Bipartite graphs are shown to satisfy the characterization. So, using a polynomial-time algorithm to compute a perfect matching in a bipartite graph, we obtain, as our main result, an efficient graph-theoretic algorithm to compute a matching Nash equilibrium on any instance of the game with a bipartite graph.

Abstract: We study network load games, a class of routing games in
networks which generalize sel{\^A}¯sh routing games on networks consisting
of parallel links. In these games, each user aims to route some tra{\^A}±c from
a source to a destination so that the maximum load she experiences in the
links of the network she occupies is minimum given the routing decisions
of other users. We present results related to the existence, complexity,
and price of anarchy of Pure NashEquilibria for several network load
games. As corollaries, we present interesting new statements related to
the complexity of computing equilibria for sel{\^A}¯sh routing games in net-
works of restricted parallel links.

Abstract: We study the fundamental problem 2NASH of computing a Nash equilibrium (NE) point in bimatrix games. We start by proposing a novel characterization of the NE set, via a bijective map to the solution set of a parameterized quadratic program (NEQP), whose feasible space is the highly structured set of correlated equilibria (CE). This is, to our knowledge, the first characterization of the subset of CE points that are in “1–1” correspondence with the NE set of the game, and contributes to the quite lively discussion on the relation between the spaces of CE and NE points in a bimatrix game (e.g., [15], [26] and [33]).
We proceed with studying a property of bimatrix games, which we call mutually concavity (MC), that assures polynomial-time tractability of 2NASH, due to the convexity of a proper parameterized quadratic program (either NEQP, or a parameterized variant of the Mangasarian & Stone formulation [23]) for a particular value of the parameter. We prove various characterizations of the MC-games, which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero-sum (SZS) games of Moulin & Vial [25]. This gives an alternative explanation of the polynomial-time tractability of 2NASH for these games, not depending on the solvability of zero-sum games. Moreover, the recognition of the MC-property for an arbitrary game is much faster than the recognition SZS-property. This, along with the comparable time-complexity of linear programs and convex quadratic programs, leads us to a much faster algorithm for 2NASH in MC-games.
We conclude our discussion with a comparison of MC-games (or, SZS-games) to kk-rank games, which are known to admit for 2NASH a FPTAS when kk is fixed [18], and a polynomial-time algorithm for k=1k=1 [2]. We finally explore some closeness properties under well-known NE set preserving transformations of bimatrix games.

Abstract: In sponsored search auctions, advertisers compete for a number
of available advertisement slots of different quality. The
auctioneer decides the allocation of advertisers to slots using
bids provided by them. Since the advertisers may act
strategically and submit their bids in order to maximize their
individual objectives, such an auction naturally defines a
strategic game among the advertisers. In order to quantify
the efficiency of outcomes in generalized second price auctions,
we study the corresponding games and present new
bounds on their price of anarchy, improving the recent results
of Paes Leme and Tardos [16] and Lucier and Paes
Leme [13]. For the full information setting, we prove a surprisingly
low upper bound of 1.282 on the price of anarchy
over pure Nashequilibria. Given the existing lower bounds,
this bound denotes that the number of advertisers has almost
no impact on the price of anarchy. The proof exploits
the equilibrium conditions developed in [16] and follows by
a detailed reasoning about the structure of equilibria and a
novel relation of the price of anarchy to the objective value
of a compact mathematical program. For more general equilibrium
classes (i.e., mixed Nash, correlated, and coarse correlated
equilibria), we present an upper bound of 2.310 on
the price of anarchy. We also consider the setting where
advertisers have incomplete information about their competitors
and prove a price of anarchy upper bound of 3.037
over Bayes-Nashequilibria. In order to obtain the last two
bounds, we adapt techniques of Lucier and Paes Leme [13]
and significantly extend them with new arguments

Abstract: We study the performance of approximate Nashequilibria for congestion
games with polynomial latency functions. We consider how much the price of anarchy
worsens and how much the price of stability improves as a function of the
approximation factor . We give tight bounds for the price of anarchy of atomic and
non-atomic congestion games and for the price of stability of non-atomic congestion
games. For the price of stability of atomic congestion games we give non-tight
bounds for linear latencies. Our results not only encompass and generalize the existing
results of exact equilibria to -Nashequilibria, but they also provide a unified
approach which reveals the common threads of the atomic and non-atomic price of
anarchy results. By expanding the spectrum, we also cast the existing results in a new
light.

Abstract: In this paper we study the support sizes of evolutionary stable strategies (ESS) in random
evolutionary games. We prove that, when the elements of the payo matrix behave either as uniform,
or normally distributed random variables, almost all ESS have support sizes o(n), where n is the
number of possible types for a player. Our arguments are based exclusively on a stability property
that the payo submatrix indicated by the support of an ESS must satisfy.
We then combine this result with a recent result of McLennan and Berg (2005), concerning the
expected number of NashEquilibria in normal{random bimatrix games, to show that the expected
number of ESS is signicantly smaller than the expected number of symmetric Nashequilibria of the
underlying symmetric bimatrix game.

Abstract: In this paper we present an implementation and performance evaluation of a descent algorithm that was proposed in \cite{tsaspi} for the computation of approximate Nashequilibria of non-cooperative bi-matrix games. This algorithm, which achieves the best polynomially computable \epsilon-approximate equilibria till now, is applied here to several problem instances designed so as to avoid the existence of easy solutions. Its performance is analyzed in terms of quality of approximation and speed of convergence. The results demonstrate significantly better performance than the theoretical worst case bounds, both for the quality of approximation and for the speed of convergence. This motivates further investigation into the intrinsic characteristics of descent algorithms applied to bi-matrix games. We discuss these issues and provide some insights about possible variations and extensions of the algorithmic concept that could lead to further understanding of the complexity of computing equilibria. We also prove here a new significantly better bound on the number of loops required for convergence of the descent algorithm.

Abstract: We focus on the problem of computing an epsilon (Porson)-Nash equilibrium of a bimatrix game, when epsilon (Porson) is an absolute constant. We present a simple algorithm for computing a View the MathML source-Nash equilibrium for any bimatrix game in strongly polynomial time and we next show how to extend this algorithm so as to obtain a (potentially stronger) parameterized approximation. Namely, we present an algorithm that computes a View the MathML source-Nash equilibrium, where {\"e} is the minimum, among all Nashequilibria, expected payoff of either player. The suggested algorithm runs in time polynomial in the number of strategies available to the players.

Abstract: We focus on the problem of computing an -Nash equilibrium
of a bimatrix game, when is an absolute constant. We present a simple
algorithm for computing a 3 -Nash equilibrium for any bimatrix game in
4
strongly polynomial time and we next show how to extend this algorithm
so as to obtain a (potentially stronger) parameterized approximation.
Namely, we present an algorithm that computes a 2+{\"e} -Nash equilibrium,
4
where {\"e} is the minimum, among all Nashequilibria, expected payoff of
either player. The suggested algorithm runs in time polynomial in the
number of strategies available to the players.

Abstract: We focus on the problem of computing an ²-Nash equilibrium
of a bimatrix game, when ² is an absolute constant. We present a simple
algorithm for computing a 3
4 -Nash equilibrium for any bimatrix game in
strongly polynomial time and we next show how to extend this algorithm
so as to obtain a (potentially stronger) parameterized approximation.
Namely, we present an algorithm that computes a 2+¸
4 -Nash equilibrium,
where ¸ is the minimum, among all Nashequilibria, expected payoff of
either player. The suggested algorithm runs in time polynomial in the
number of strategies available to the players.

Abstract: We consider applications of probabilistic techniques in the
framework of algorithmic game theory. We focus on three distinct case
studies: (i) The exploitation of the probabilistic method to demonstrate
the existence of approximate Nashequilibria of logarithmic support sizes
in bimatrix games; (ii) the analysis of the statistical conflict that mixed
strategies cause in network congestion games; (iii) the effect of coalitions
in the quality of congestion games on parallel links.

Abstract: We focus on the problem of computing approximate Nashequilibria and well-supported approximate Nashequilibria in random bimatrix games, where each player's payoffs are bounded and independent random variables, not necessarily identically distributed, but with almost common expectations. We show that the completely mixed uniform strategy profile, i.e., the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a TeX -Nash equilibrium and a TeX -well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ϵ-well-supported Nash equilibrium where ϵ tends to zero as n tends to infinity.

Abstract: We study a problem of scheduling client requests to servers. Each client has a particular latency requirement at each server and may choose either to be assigned to some server in order to get serviced provided that her latency requirement is met, or not to participate in the assignment at all. From a global perspective, in order to optimize the performance of such a system, one would aim to maximize the number of clients that participate in the assignment. However, clients may behave selfishly in the sense that, each of them simply aims to participate in an assignment and get serviced by some server where her latency requirement is met with no regard to overall system performance. We model this selfish behavior as a strategic game, show how to compute pure Nashequilibria efficiently, and assess the impact of selfishness on system performance. We also show that the problem of optimizing performance is computationally hard to solve, even in a coordinated way, and present efficient approximation and online algorithms.

Abstract: We study extreme Nashequilibria 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 identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.
We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nashequilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1+ɛ) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.
Considering pure Nashequilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is View the MathML source-hard to approximate the worst social cost within a multiplicative factor better than 2-2/(m+1).

Abstract: We study extreme Nashequilibria 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 identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nashequilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1 + {\aa}) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.Considering pure Nashequilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is N P-hard to approximate the worst social cost within a multiplicative factor better than 2 - 2/(m + 1).

Abstract: We study computational and coordination efficiency issues of
Nashequilibria in symmetric network congestion games. We first propose
a simple and natural greedy method that computes a pure Nash equilibrium
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 security problem on a distributed network.
We assume a network whose nodes are vulnerable to infection
by threats (e.g. viruses), the attackers. A system security
software, the defender, is available in the system. However,
due to the network¢s size, economic and performance reasons,
it is capable to provide safety, i.e. clean nodes from
the possible presence of attackers, only to a limited part of
it. The objective of the defender is to place itself in such a
way as to maximize the number of attackers caught, while
each attacker aims not to be caught.
In [7], a basic case of this problem was modeled as a
non-cooperative game, called the Edge model. There, the
defender could protect a single link of the network. Here,
we consider a more general case of the problem where the
defender is able to scan and protect a set of k links of the
network, which we call the Tuple model. It is natural to expect
that this increased power of the defender should result
in a better quality of protection for the network. Ideally,
this would be achieved at little expense on the existence and
complexity of Nashequilibria (profiles where no entity can
improve its local objective unilaterally by switching placements
on the network).
In this paper we study pure and mixed Nashequilibria
in the model. In particular, we propose algorithms for computing
such equilibria in polynomial time and we provide a
polynomial-time transformation of a special class of Nashequilibria, called matching equilibria, between the Edge
model and the Tuple model, and vice versa. Finally, we
establish that the increased power of the defender results in
higher-quality protection of the network.

Abstract: We consider non-cooperative unsplittable congestion games where players share resources, and each player's strategy is pure and consists of a subset of the resources on which it applies a fixed weight. Such games represent unsplittable routing flow games and also job allocation games. The congestion of a resource is the sum of the weights of the players that use it and the player's cost function is the sum of the utilities of the resources on its strategy. The social cost is the total weighted sum of the player's costs. The quality of Nashequilibria is determined by the price of anarchy (PoA) which expresses how much worse is the social outcome in the worst equilibrium versus the optimal coordinated solution. In the literature the predominant work has only been on games with polynomial utility costs, where it has been proven that the price of anarchy is bounded by the degree of the polynomial. However, no results exist on general bounds for non-polynomial utility functions.
Here, we consider general versions of these games in which the utility of each resource is an arbitrary non-decreasing function of the congestion. In particular, we consider a large family of superpolynomial utility functions which are asymptotically larger than any polynomial. We demonstrate that for every such function there exist games for which the price of anarchy is unbounded and increasing with the number of players (even if they have infinitesimal weights) while network resources remain fixed. We give tight lower and upper bounds which show this dependence on the number of players. Furthermore we provide an exact characterization of the PoA of all congestion games whose utility costs are bounded above by a polynomial function. Heretofore such results existed only for games with polynomial cost functions.

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 Nashequilibria, of the optimal gain of the defender over its gain at a Nash equilibrium.
We provide a comprehensive collection of trade-offs between the
Price of Defense and the computational efficiency of Nashequilibria.
– Through reduction to a Two-Players, Constant-Sum Game, we prove
that a Nash equilibrium 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 Nashequilibria:
• In a Matching Nash equilibrium, 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 Nash equilibrium, 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 Nash equilibrium, 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 NashEquilibria; however, it is NP-complete to decide
their existence.
• In an Attacker Symmetric and Uniform Nash equilibrium, 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: Consider a network vulnerable to security attacks and equipped with defense mechanisms. How much is the loss in the provided security guarantees due to the selfish nature of attacks and defenses? The Price of Defense was recently introduced in [7] as a worst-case measure, over all associated Nashequilibria, of this loss. In the particular strategic game considered in [7], there are two classes of confronting randomized players on a graph G(V,E): v attackers, each choosing vertices and wishing to minimize the probability of being caught, and a single defender, who chooses edges and gains the expected number of attackers it catches. In this work, we continue the study of the Price of Defense. We obtain the following results: - The Price of Defense is at least |V| 2; this implies that the Perfect Matching Nashequilibria considered in [7] are optimal with respect to the Price of Defense, so that the lower bound is tight. - We define Defense-Optimal graphs as those admitting a Nash equilibrium that attains the (tight) lower bound of |V| 2. We obtain: › A graph is Defense-Optimal if and only if it has a Fractional Perfect Matching. Since graphs with a Fractional Perfect Matching are recognizable in polynomial time, the same holds for Defense-Optimal graphs. › We identify a very simple graph that is Defense-Optimal but has no Perfect Matching Nash equilibrium. - Inspired by the established connection between Nashequilibria and Fractional Perfect Matchings, we transfer a known bivaluedness result about Fractional Matchings to a certain class of Nashequilibria. So, the connection to Fractional Graph Theory may be the key to revealing the combinatorial structure of Nashequilibria for our network security game.

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 Nash equilibrium 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: We study the problem of routing traffic through a congested network. We focus on the simplest case of a network consisting of m parallel links. We assume a collection of n network users; each user employs a mixed strategy, which is a probability distribution over links, to control the shipping of its own assigned traffic. Given a capacity for each link specifying the rate at which the link processes traffic, the objective is to route traffic so that the maximum (over all links) latency is minimized. We consider both uniform and arbitrary link capacities. How much decrease in global performace is necessary due to the absence of some central authority to regulate network traffic and implement an optimal assignment of traffic to links? We investigate this fundamental question in the context of Nashequilibria for such a system, where each network user selfishly routes its traffic only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the Coordination Ratio, originally defined by Koutsoupias and Papadimitriou, as a measure of the cost of lack of coordination among the users; roughly speaking, the Coordination Ratio is the ratio of the expectation of the maximum (over all links) latency in the worst possible Nash equilibrium, over the least possible maximum latency had global regulation been available. Our chief instrument is a set of combinatorial Minimum Expected Latency Cost Equations, one per user, that characterize the Nashequilibria of this system. These are linear equations in the minimum expected latency costs, involving the user traffics, the link capacities, and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed strategies, where each user assigns its traffic with a strictly positive probability to every link, to derive the first existence and uniqueness results for fully mixed Nashequilibria in this setting. Through a thorough analysis and characterization of fully mixed Nashequilibria, we obtain tight upper bounds of no worse than O(ln n/ln ln n) on the Coordination Ratio for (i) the case of uniform capacities and arbitrary traffics and (ii) the case of arbitrary capacities and identical traffics.

Abstract: In this work, we study the combinatorial structure and the
computational complexity of Nashequilibria 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 Nash equilibrium, 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 Nash equilibrium
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 Nashequilibria for the selfish routing game we consider.
In a nutshell, these problems relate to deciding the existence of a
Nash equilibrium, constructing a Nash equilibrium with given support
characteristics, constructing the worst Nash equilibrium (the one with
maximum social cost), constructing the best Nash equilibrium (the one
with minimum social cost), or computing the social cost of a (given) Nash
equilibrium. 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
Nashequilibria and on the parameters of the system.

Abstract: In view of the apparent intractability of constructing NashEquilibria (NE
in short) in polynomial time, even for bimatrix games, understanding the limitations
of the approximability of the problem is an important challenge.
In this work we study the tractability of a notion of approximate equilibria in bimatrix
games, called well supported approximate NashEquilibria (SuppNE in short).
Roughly speaking, while the typical notion of approximate NE demands that each
player gets a payoff at least an additive term less than the best possible payoff, in a
SuppNE each player is assumed to adopt with positive probability only approximate
pure best responses to the opponent¢s strategy.
As a first step, we demonstrate the existence of SuppNE with small supports and
at the same time good quality. This is a simple corollary of Alth{\"o}fer¢s Approximation
Lemma, and implies a subexponential time algorithm for constructing SuppNE of
arbitrary (constant) precision.
We then propose algorithms for constructing SuppNE in win lose and normalized
bimatrix games (i.e., whose payoff matrices take values from {0, 1} and [0, 1] respectively).
Our methodology for attacking the problem is based on the solvability of zero sum bimatrix games (via its connection to linear programming) and provides a
0.5-SuppNE for win lose games and a 0.667-SuppNE for normalized games.
To our knowledge, this paper provides the first polynomial time algorithms constructing
{\aa}-SuppNE for normalized or win lose bimatrix games, for any nontrivial
constant 0 ≤{\aa} <1, bounded away from 1.

Abstract: We study the existence and tractability of a notion of approximate
equilibria in bimatrix games, called well supported approximate
NashEquilibria (SuppNE in short).We prove existence of "−SuppNE for
any constant " 2 (0, 1), with only logarithmic support sizes for both players.
Also we propose a polynomial–time construction of SuppNE, both
for win lose and for arbitrary (normalized) bimatrix games. The quality
of these SuppNE depends on the girth of the Nash Dynamics graph in
the win lose game, or a (rounded–off) win lose image of the original normalized
game. Our constructions are very successful in sparse win lose
games (ie, having a constant number of (0, 1)−elements in the bimatrix)
with large girth in the Nash Dynamics graph. The same holds also for
normalized games whose win lose image is sparse with large girth.
Finally we prove the simplicity of constructing SuppNE both in random
normalized games and in random win lose games. In the former case we
prove that the uniform full mix is an o(1)−SuppNE, while in the case
of win lose games, we show that (with high probability) there is either a
PNE or a 0.5-SuppNE with support sizes only 2.

Abstract: A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium – the one that maximizes social cost. Worst-case Nashequilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9].
More specifically, we continue the study of the Conjecture of the Fully Mixed Nash Equilibrium, henceforth abbreviated as FMNE Conjecture, which asserts that the fully mixed Nash equilibrium, when existing, is the worst-case Nash equilibrium. (In the fully mixed Nash equilibrium, the mixed strategy of each user assigns (strictly) positive probability to every link.) We report substantial progress towards identifying the validity, methodologies to establish, and limitations of, the FMNE Conjecture.