We consider the well-studied game-theoretic version of machine scheduling in which jobs correspond to self-interested users and machines correspond to resources. Here each user chooses a machine trying to minimize her own cost, and such selfish behavior typically results in some equilibrium which is not globally optimal: An equilibrium is an allocation where no user can reduce her own cost by moving to another machine, which in general need not minimize the makespan, i.e., the maximum load over the machines. We provide tight bounds on two well-studied notions in algorithmic game theory, namely, the price of anarchy and the strong price of anarchy on machine scheduling setting which lies in between the related and the unrelated machine case. Both notions study the social cost (makespan) of the worst equilibrium compared to the optimum, with the strong price of anarchy restricting to a stronger form of equilibria. Our results extend a prior study comparing the price of anarchy to the strong price of anarchy for two related machines (Epstein, Acta Informatica 2010), thus providing further insights on the relation between these concepts. Our exact bounds give a qualitative and quantitative comparison between the two models. The bounds also show that the setting is indeed easier than the two unrelated machines: In the latter, the strong price of anarchy is @math , while in ours it is strictly smaller.
The bad Nash equilibrium in Example appears in several works @cite_12 @cite_21 @cite_11 @cite_22 to show that even for two machines the price of anarchy is unbounded, thus suggesting that the notion should be refined. Among these, the , which considers NE, is studied in @cite_12 @cite_14 @cite_19 . The , which considers equilibria arising in extensive form games, is studied in @cite_21 @cite_3 @cite_7 @cite_22 . In @cite_11 the authors investigate equilibria and the resulting , while @cite_18 focuses on the equilibria produced by the . A further distinction is between (randomized) and (deterministic) equilibria: in the former, players choose a probability distribution over the strategies and regard their expected cost, in the latter they choose deterministically one strategy. In this work we focus on pure equilibria and in the remaining of this section we write @math to denote the bounds on the price of anarchy for mixed equilibria.
@cite_18 @cite_14 @cite_22 @cite_7 @cite_21 @cite_3 @cite_19 @cite_12 @cite_11
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The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA. 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 Nash equilibria. 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-Nash equilibria. In order to obtain the last two bounds, we adapt techniques of Lucier and Paes Leme [13] and significantly extend them with new arguments. The Generalized Second Price Auction has been the main mechanism used by search companies to auction positions for advertisements on search pages. In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full information setting, socially optimal Nash equilibria are known to exist (i.e., the Price of Stability is 1). This paper is the first to prove bounds on the price of anarchy, and to give any bounds in the Bayesian setting. Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated strategies. In the full information setting we prove a bound of 1.618 for the price of anarchy for pure Nash equilibria, and a bound of 4 for mixed Nash equilibria. We also prove a bound of 8 for the price of anarchy in the Bayesian setting, when valuations are drawn independently, and the valuation is known only to the bidder and only the distributions used are common knowledge. Our proof exhibits a combinatorial structure of Nash equilibria and uses this structure to bound the price of anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to the Bayesian setting requires the use of novel combinatorial techniques that can be of independent interest. The price of anarchy, defined as the ratio of the worst-case objective function value of a Nash equilibrium of a game and that of an optimal outcome, quantifies the inefficiency of selfish behavior. Remarkably good bounds on this measure are known for a wide range of application domains. However, such bounds are meaningful only if a game's participants successfully reach a Nash equilibrium. This drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash equilibria and correlated equilibria, and to sequences of outcomes generated by natural experimentation strategies, such as successive best responses and simultaneous regret-minimization. We establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives. First, we identify a “canonical sufficient condition” for an upper bound on the price of anarchy of pure Nash equilibria, which we call a smoothness argument. Second, we prove an “extension theorem”: every bound on the price of anarchy that is derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of every outcome sequence generated by no-regret learners. Smoothness arguments also have automatic implications for the inefficiency of approximate equilibria, for bicriteria bounds, and, under additional assumptions, for polynomial-length best-response sequences. Third, we prove that in congestion games, smoothness arguments are “complete” in a proof-theoretic sense: despite their automatic generality, they are guaranteed to produce optimal worst-case upper bounds on the price of anarchy. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. We study network design games where @math self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as @math . Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of @math for weighted games, where @math is the total weight of all the agents. This almost matches the known upper bound of @math . Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the well- known multiplicative-weights learning algorithm results in convergence to pure equilibria. Our results show that nat- ural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the load-balancing game introduced by Koutsoupias and Pa- padimitriou, which has super-constant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined game-theoretically, but we show that this property holds whenever a stability criterion from the theory of dy- namical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribu- tion). We further extend our results to show that players can use algorithms with different (sufficiently small) learn- ing rates, i.e. they can trade off convergence speed and long term average regret differently. We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a non-transient Nash equilibrium, for any α with 1 < α ≤ √n 2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for α ≥ 12n[log n], every Nash equilibrium forms a tree.Without relying on the tree conjecture, proved an upper bound on the price of anarchy of O(√α), where α ∈ [2, n2]. We improve this bound. Specifically, we derive a constant upper bound for α ∈ O(√n) and for α ≥ 12n[log n]. For the intermediate values we derive an improved bound of O(1 + (min α2 n, n2 α )1 3).Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing. We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upper-bound side, we define the framework of "local smoothness", which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is at most m, where m denotes the number of coalitions. For general networks, we present an instance showing that the price of anarchy is unbounded, even in the case of two coalitions. If latencies are restricted to polynomials with nonnegative coefficients and bounded degree, we prove upper bounds on the price of anarchy for general networks, which improve upon the current best ones except for affine latencies. In light of the negative results even for two coalitions, we analyze the effectiveness of Stackelberg strategies as a means to improve the quality of Nash equilibria. In this setting, an α fraction of the entire demand is first routed centrally by a Stackelberg leader according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by the coalitions (followers). For a single coalitional follower and parallel arcs, we develop an efficiently computable Stackelberg strategy that reduces the price of anarchy to one. For general networks and a single coalitional follower, we show that a simple strategy, called SCALE, reduces the price of anarchy to 1+α. Finally, we investigate SCALE for multiple coalitional followers, general networks, and affine latencies. We present the first known upper bound on the price of anarchy in this case. Our bound smoothly varies between 1.5 for α=0 and full efficiency for α=1. Many natural games have both high and low cost Nash equilibria: their Price of Anarchy is high and yet their Price of Stability is low. In such cases, one could hope to move behavior from a high cost equilibrium to a low cost one by a "public service advertising campaign" encouraging players to follow the low-cost equilibrium, and if every player follows the advice then we are done. However, the assumption that everyone follows instructions is unrealistic. A more natural assumption is that some players will follow them, while other players will not. In this paper we consider the question of to what extent can such an advertising campaign cause behavior to switch from a bad equilibrium to a good one even if only a fraction of people actually follow the given advice, and do so only temporarily. Unlike the "value of altruism" model, we assume everyone will ultimately act in their own interest. We analyze this question for several important and widely studied classes of games including network design with fair cost sharing, scheduling with unrelated machines, and party affiliation games (which include consensus and cut games). We show that for some of these games (such as fair cost sharing), a random α fraction of the population following the given advice is sufficient to get a guarantee within an O(1 α) factor of the price of stability for any α > 0. For other games (such as party affiliation games), there is a strict threshold (in this case, α 1 2 is enough to reach near-optimal behavior). Finally, for some games, such as scheduling, no value α < 1 is sufficient. We also consider a "viral marketing" model in which certain players are specifically targeted, and analyze the ability of such targeting to influence behavior using a much smaller number of targeted players. We consider strategic games in which each player seeks a mixed strategy to minimize her cost evaluated by a concave valuationV (mapping probability distributions to reals); such valuations are used to model risk. In contrast to games with expectation-optimizer players where mixed equilibria always exist (Nash 1950; Nash Ann. Math. 54, 286---295, 1951), a mixed equilibrium for such games, called a V-equilibrium, may fail to exist, even though pure equilibria (if any) transfer over. What is the exact impact of such valuations on the existence, structure and complexity of mixed equilibria? We address this fundamental question in the context of expectation plus variance, a particular concave valuation denoted as RA, which stands for risk-averse; so, variance enters as a measure of risk and it is used as an additive adjustment to expectation. We obtain the following results about RA-equilibria: A collection of general structural properties of RA-equilibria connecting to (i)E-equilibria and Var-equilibria, which correspond to the expectation and variance valuations E and Var, respectively, and to (ii) other weaker or incomparable properties such as Weak Equilibrium and Strong Equilibrium. Some of these structural properties imply quantitative constraints on the existence of mixed RA-equilibria.A second collection of (i) existence, (ii) equivalence and separation (with respect to E-equilibria), and (iii) characterization results for RA-equilibria in the new class of player-specific scheduling games. We provide suitable examples with a mixed RA-equilibrium that is not an E-equilibrium and vice versa.A purification technique to transform a player-specific scheduling game on two identical links into a player-specific scheduling game on two links so that all non-pureRA-equilibria are eliminated while no new pure equilibria are created; so, a particular player-specific scheduling game on two identical links with no pure equilibrium yields a player-specific scheduling game with noRA-equilibrium (whether mixed or pure). As a by-product, the first ?????-completeness result for the computation of RA-equilibria follows. We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ--approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5 2 + Oγ for arbitrarily small γ > 0; for latency functions with maximum degree d≥ 2, it is d2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1 γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ-games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications. We define smooth games of incomplete information. We prove an "extension theorem" for such games: price of anarchy bounds for pure Nash equilibria for all induced full-information games extend automatically, without quantitative degradation, to all mixed-strategy Bayes-Nash equilibria with respect to a product prior distribution over players' preferences. We also note that, for Bayes-Nash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games. We give several applications of our definition and extension theorem. First, we show that many games of incomplete information for which the price of anarchy has been studied are smooth in our sense. Thus our extension theorem unifies much of the known work on the price of anarchy in games of incomplete information. Second, we use our extension theorem to prove new bounds on the price of anarchy of Bayes-Nash equilibria in congestion games with incomplete information. We study network formation with n players and link cost α > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player ν incorporates the expected number of players to which ν will become disconnected. We focus on unilateral link formation and Nash equilibrium . We show existence of Nash equilibria and a price of stability of 1 + ο (1) under moderate assumptions on the adversary and n ≥ 9. We prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of player pairs to be separated. We show an Ο (1) bound on the price of anarchy for both adversaries, the constant being bounded by 15 + ο (1) and 9 + ο (1), respectively. We study bottleneck congestion games where the social cost is determined by the worst congestion on any resource. In the literature, bottleneck games assume player utility costs determined by the worst congested resource in their strategy. However, the Nash equilibria of such games are inefficient since the price of anarchy can be very high and proportional to the number of resources. In order to obtain smaller price of anarchy we introduce exponential bottleneck games, where the utility costs of the players are exponential functions of their congestions. In particular, the delay function for any resource r is MCr, where Cr denotes the number of players that use r, and M is an integer constant. We find that exponential bottleneck games are very efficient and give the following bound on the price of anarchy: O(log |R|), where R is the set of resources. This price of anarchy is tight, since we demonstrate a game with price of anarchy Ω(log |R|). We obtain our tight bounds by using two novel proof techniques: transformation, which we use to convert arbitrary games to simpler games, and expansion, which we use to bound the price of anarchy in a simpler game. We study the price of anarchy and the structure of equilibria in network creation games. A network creation game (first defined and studied by [4]) is played by n players 1, 2, . . . , n , each identified with a vertex of a graph (network), where the strategy of player i, i = 1, . . . , n, is to build some edges adjacent to i. The cost of building an edge is α > 0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame (the original variant of [4]) the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame (variant defined and studied by [3]) the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α > 273 · n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α > 129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this (almost) answers one of the basic open problems in the field – is price of anarchy of the network creation game constant for all values of α? – in an affirmative way, up to a tiny range of α. We consider the problem of a monopolist seller who wants to sell some items to a set of buyers. The buyers are strategic, unit-demand, and connected by a social network. Furthermore, the utility of a buyer is a decreasing function of the number of neighbors who do not own the item. In other words, they exhibit negative externalities, deriving utility from being unique in their purchases. In this model, any fixed setting of the price induces a sub-game on the buyers. We show that it is an exact potential game which admits multiple pure Nash Equilibria. A natural problem is to compute those pure Nash equilibria that raise the most and least revenue for the seller. These correspond respectively to the most optimistic and most pessimistic revenues that can be raised. We show that the revenues of both the best and worst equilibria are hard to approximate within sub-polynomial factors. Given this hardness, we consider a relaxed notion of pricing, where the price for the same item can vary within a constant factor for different buyers. We show a 4-approximation to the pessimistic revenue when the prices are relaxed by a factor of 4. The interesting aspect of this algorithm is that it uses a linear programming relaxation that only encodes part of the strategic behavior of the buyers in its constraints, and rounds this relaxation to obtain a starting configuration for performing relaxed Nash dynamics. Finally, for the maximum revenue Nash equilibrium, we show a 2-approximation for bipartite graphs (without price relaxation), and complement this result by showing that the problem is NP-Hard even on trees. We consider the problem of computing a Nash equilibrium in multiple-player games. It is known that there exist games, in which all the equilibria have irrational entries in their probability distributions [19]. This suggests that either we should look for symbolic representations of equilibria or we should focus on computing approximate equilibria. We show that every finite game has an equilibrium such that all the entries in the probability distributions are algebraic numbers and hence can be finitely represented. We also propose an algorithm which computes an approximate equilibrium in the following sense: the strategies output by the algorithm are close with respect to l ∞ -norm to those of an exact Nash equilibrium and also the players have only a negligible incentive to deviate to another strategy. The running time of the algorithm is exponential in the number of strategies and polynomial in the digits of accuracy. We obtain similar results for approximating market equilibria in the neoclassical exchange model under certain assumptions. 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.