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The Price of Defense
 Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science
, 2006
"... Abstract. We consider a strategic game with two classes of confronting randomized players on a graph G(V, E): ν 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 Pric ..."
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Cited by 10 (4 self)
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Abstract. We consider a strategic game with two classes of confronting randomized players on a graph G(V, E): ν 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 worstcase ratio, over all Nash equilibria, of the optimal gain of the defender over its gain at a Nash equilibrium. We provide a comprehensive collection of tradeoffs between the Price of Defense and the computational efficiency of Nash equilibria. – Through reduction to a TwoPlayers, ConstantSum 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 Nash equilibria: • In a Matching Nash equilibrium, the support of the defender is an Edge Cover. We prove that they can be computed in polynomial
The Power of the Defender
, 2005
"... 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, ..."
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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 noncooperative 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 Nash equilibria (profiles where no entity can improve its local objective unilaterally by switching placements on the network). In this paper we study pure and mixed Nash equilibria in the model. In particular, we propose algorithms for computing such equilibria in polynomial time and we provide a polynomialtime transformation of a special class of Nash equilibria, 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 higherquality protection of the network.
A network game with attackers and a defender
 Algorithmica
"... Consider an information network with threats called attackers; each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is a protector entity called defender; the defender scans and cleans from attacks some part of the network (in particular, ..."
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Consider an information network with threats called attackers; each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is a protector entity called defender; the defender scans and cleans from attacks some part of the network (in particular, a link), which it chooses independently using its own probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the defender; towards a conflicting objective, the defender aims at maximizing the expected number of attackers it catches. We model this network security scenario as a noncooperative strategic game on graphs. We are interested in its associated Nash equilibria, where no network entity can unilaterally increase its local objective. We obtain the following results: • We obtain an algebraic characterization of (mixed) Nash equilibria. • No (nontrivial) instance of the graphtheoretic game has a pure Nash equilibrium. This is an immediate consequence of some covering properties we prove for the supports of the players in all (mixed) Nash equilibria.
How many attackers can selfish defenders catch?
, 2008
"... In a distributed system with attacks and defenses, an economic investment in defense mechanisms aims at increasing the degree of system protection against the attacks. We study such investments in the selfish setting, where both attackers and defenders are selfinterested entities. In particular, we ..."
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In a distributed system with attacks and defenses, an economic investment in defense mechanisms aims at increasing the degree of system protection against the attacks. We study such investments in the selfish setting, where both attackers and defenders are selfinterested entities. In particular, we assume a rewardsharing scheme among interdependent defenders; each defender wishes to maximize its own fair share of the attackers caught due to him (and possibly due to the involvement of others). Addressed in this work is the fundamental question of determining the maximum amount of protection achievable by a number of such defenders against a number of attackers if the system is in a Nash equilibrium. As a measure of system protection, we adapt the DefenseRatio [12], which describes the expected proportion of attackers caught by defenders. In a DefenseOptimal Nash equilibrium, the DefenseRatio is optimized. We discover that the answer to this question depends in a quantitatively subtle way on the invested number of defenders. We identify graphtheoretic thresholds for the number of defenders that determine the possibility of optimizing a DefenseRatio. In this vein, we obtain, through an extensive combinatorial analysis of Nash equilibria, a comprehensive collection of tradeoff results.
A Game on a Distributed Network∗
"... Consider a distributed 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 networ ..."
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Consider a distributed 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 simple 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. In [8, 9], we model this network scenario as a noncooperative strategic game on graphs. We focus on two basic cases for the protector; where it may choose a single edge or a simple path of the network. The two games obtained are called as the Path and the Edge model, respectively. For these games, we are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. For the Edge model we obtain the following results: ∗This work was partially supported by the IST Programs of the European Union under contract
A Network Game with Attackers and a Defender: A Survey
"... We survey a research line recently initiated by Mavronicolas et al. [14, 15, 16], concerning a strategic game on a graph G(V,E) with two confronting classes of randomized players: ν attackers who choose vertices and wish to minimize the probability of being caught by the defender, who chooses edges ..."
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We survey a research line recently initiated by Mavronicolas et al. [14, 15, 16], concerning a strategic game on a graph G(V,E) with two confronting classes of randomized players: ν attackers who choose vertices and wish to minimize the probability of being caught by the defender, who chooses edges and gains the expected number of attackers it catches. So, the defender captures system rationality. In a Nash equilibrium, no single player has an incentive to unilaterally deviate from its randomized strategy. The Price of Defense is the worstcase ratio, over all Nash equilibria, of the optimal gain of the defender (which is ν) over the gain of the defender at a Nash equilibrium. We present a comprehensive collection of tradeoffs between the Price of Defense and the computational efficiency of Nash equilibria proved in [14, 15, 16]. • We present an algebraic characterization of (mixed) Nash equilibria. • No (nontrivial) instance of the graphtheoretic game has a pure Nash equilibrium. This is an immediate consequence of some covering properties proved for the supports of the players in all (mixed) Nash equilibria. • We present a reduction of the game to a ZeroSum TwoPlayers Game that proves that
IPFP6015964 AEOLUS Algorithmic Principles for Building Efficient Overlay Computers Deliverable D1.4.1 Stability and FaultTolerance:
, 2006
"... 2 Adversarial Queueing Theory 2 2.1 Windowed and leakybucket adversaries.................... 3 2.2 Greedy scheduling protocols........................... 5 ..."
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2 Adversarial Queueing Theory 2 2.1 Windowed and leakybucket adversaries.................... 3 2.2 Greedy scheduling protocols........................... 5
The Price of Defense and Fractional Matchings ∗
"... Consider a network vulnerable to security attacks andequippedwithdefense 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 worstcase measure, over all associated Nash equil ..."
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Consider a network vulnerable to security attacks andequippedwithdefense 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 worstcase 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): ν 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: V  • ThePriceofDefenseisatleast 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 DefenseOptimal graphs as those admitting a Nash equilibrium that attains the
EXTENSIONS AND REFINEMENTS OF STABILIZATION
, 2009
"... Selfstabilizing system is a concept of faulttolerance in distributed computing. A distributed algorithm is selfstabilizing if, starting from an arbitrary state, it is guaranteed to converge to a legal state in a finite number of states and remains in a legal set of states thereafter. The property ..."
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Selfstabilizing system is a concept of faulttolerance in distributed computing. A distributed algorithm is selfstabilizing if, starting from an arbitrary state, it is guaranteed to converge to a legal state in a finite number of states and remains in a legal set of states thereafter. The property of selfstabilization enables a distributed algorithm to recover from a transient fault regardless of its objective. Moreover, a selfstabilizing algorithm does not have to be initialized as it eventually starts to behave correctly. In this thesis, we focus on extensions and refinements of selfstabilization by studying two nontraditional aspects of selfstabilization. In traditional selfstabilizing distributed systems [15], the inherent assumption is that all processes run predefined programs mandated by an external agency which is the owner or the administrator of the entire system. The model works fine for solving problems when processes cooperate with one another, with a global goal. In modern times it is quite common to have a distributed system spanning over multiple administrative domains, and processes have selfish motives to optimize their own pay