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 Nash Equilibria
(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. GameTheory, Atomic Congestion Games, Coalitions, Convergence
to Equilibria, Price of Anarchy.

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 Nash Equilibria
(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. GameTheory, Atomic Congestion Games, Coalitions, Convergence
to Equilibria, Price of Anarchy.

Abstract: This chapter is an introduction to the basic concepts and advances of a new field, that of Computational (or Algorithmic) GameTheory. We study the computational complexity of Nash equilibria and review the related algorithms proposed in the literature. Then, given the apparent difficulty of computing exact Nash equilibria, we study the efficient computation of approximate notions of Nash equilibria. 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 Nash equilibria 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 Nash equilibria 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 Nash equilibria do exist for both these models.

Abstract: When one engineers distributed algorithms, some special characteristics
arise that are different from conventional (sequential or parallel)
computing paradigms. These characteristics include: the need for either a
scalable real network environment or a platform supporting a simulated
distributed environment; the need to incorporate asynchrony, where arbitrarya
synchrony is hard, if not impossible, to implement; and the generation
of “difficult” input instances which is a particular challenge. In this
work, we identifys ome of the methodological issues required to address
the above characteristics in distributed algorithm engineering and illustrate
certain approaches to tackle them via case studies. Our discussion
begins byad dressing the need of a simulation environment and how asynchronyis
incorporated when experimenting with distributed algorithms.
We then proceed bys uggesting two methods for generating difficult input
instances for distributed experiments, namelya game-theoretic one and another
based on simulations of adversarial arguments or lower bound proofs.
We give examples of the experimental analysis of a pursuit-evasion protocol
and of a shared memorypro blem in order to demonstrate these ideas.
We then address a particularlyi nteresting case of conducting experiments
with algorithms for mobile computing and tackle the important issue of
motion of processes in this context. We discuss the two-tier principle as
well as a concurrent random walks approach on an explicit representation
of motions in ad-hoc mobile networks, which allow at least for averagecase
analysis and measurements and may give worst-case inputs in some
cases. Finally, we discuss a useful interplay between theory and practice
that arise in modeling attack propagation in networks.

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, GameTheory and the concepts of Nash equilibria
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 Nash equilibria (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: Evolutionary GameTheory 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 Nash equilibria, 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 GameTheory, 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 GameTheory, 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 Nash Equilibria by showing that symmetric 2-person
games with random payoffs have on average exponentially less ESS than
Nash Equilibria. We conclude this article with some interesting areas of
future research concerning the synergy of Evolutionary GameTheory
and Algorithms.

Abstract: Evolutionary GameTheory 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 GameTheory, 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: The Internet and the Web have arguably surpassed the von Neumann Computer as the most complex computational artifacts of our times. Out of all the formidable characteristics of the Internet/Web, it seems that the most novel is its socio-economic complexity. The Internet and the Web are built, operated and used by a multitude of very diverse economic interests, which compete or collaborate in various degrees. In fact, this suggests that some very important insights about the Web may come from a fusion of ideas from Algorithms with concepts and techniques from Mathematical Economics and GameTheory. Since 2000, a new field has emerged (Algorithmic GameTheory and Computational Internet Economics), which examines exactly such ideas. We feel that this field belongs to Web Science. Some of the main topics of that field examined so far are Internet equilibria, the so called “Price of Anarchy” (a measure of loss of optimality due to selfishness [Koutsoupias,Papadimitriou (1999)], electronic markets and their equilibria, auctions, and algorithmic mechanisms design (inverse gametheory or how to design a game in the net in such a clever way that individual players, motivated solely by their selfish interests, actually end up meeting the goals of the game designer!). Our paper here surveys the main concepts and results of this very promising subfield.

Abstract: We consider applications of probabilistic techniques in the
framework of algorithmic gametheory. We focus on three distinct case
studies: (i) The exploitation of the probabilistic method to demonstrate
the existence of approximate Nash equilibria 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 address several recent developments in non-cooperative as well as
evolutionary gametheory, that give a new viewpoint to Complex Systems understanding.
In particular, we discuss notions like the anarchy cost, equilibria formation,
social costs and evolutionary stability. We indicate how such notions help
in understanding Complex Systems behaviour when the system includes selfish,
antagonistic entities of varying degrees of rationality. Our main motivation is the
Internet, perhaps the most complex artifact to date, as well as large-scale systems
such as the high-level P2P systems, where where the interaction is among
humans, programmes and machines and centralized approaches cannot apply.

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 Nash equilibria, 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 Nash equilibria 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 Nash equilibria and Fractional Perfect Matchings, we transfer a known bivaluedness result about Fractional Matchings to a certain class of Nash equilibria. So, the connection to Fractional Graph Theory may be the key to revealing the combinatorial structure of Nash equilibria 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].