We consider a game on a graph \(G=\langle V, E\rangle \) with two confronting classes of randomized players: \(\nu \) attackers, who choose vertices and seek to minimize the probability of getting caught, and a single defender, who chooses edges and seeks to maximize the expected number of attackers it catches. In a Nash equilibrium, no player has an incentive to unilaterally deviate from her randomized strategy. The Price of Defense is the worst-case ratio, over all Nash equilibria, of \(\nu \) over the expected utility of the defender at a Nash equilibrium.

We orchestrate a strong interplay of arguments from Game Theory and Graph Theory to obtain both general and specific results in the considered setting:

(1) Via a reduction to a Two-Players, Constant-Sum game, we observe that an arbitrary Nash equilibrium is computable in polynomial time. Further, we prove a general lower bound of \(\frac{\textstyle |V|}{\textstyle 2}\) on the Price of Defense. We derive a characterization of graphs with a Nash equilibrium attaining this lower bound, which reveals a promising connection to Fractional Graph Theory; thereby, it implies an efficient recognition algorithm for such Defense-Optimal graphs.

(2) We study some specific classes of Nash equilibria, both for their computational complexity and for their incurred Price of Defense. The classes are defined by imposing structure on the players’ randomized strategies: either graph-theoretic structure on the supports, or symmetry and uniformity structure on the probabilities. We develop novel graph-theoretic techniques to derive trade-offs between computational complexity and the Price of Defense for these classes. Some of the techniques touch upon classical milestones of Graph Theory; for example, we derive the first game-theoretic characterization of König-Egerváry graphs as graphs admitting a Matching Nash equilibrium.

我们认为一个游戏上图\（G = \ langle V，E \ rangle \）与随机2面临类球员：\（\ NU \） 的攻击，谁选择的顶点，并寻求被抓到的可能性最小化，并一个单一的防御者，他选择优势并寻求最大化其捕获的攻击者的预期数量。在纳什均衡中，没有玩家有动机单方面偏离其随机策略。的国防部价格是最坏情况下的比率，在所有纳什均衡，的\（\ NU \）在中卫在纳什均衡的预期效用。

我们精心安排了博弈论和图论的论点之间的强大相互作用，以在考虑的背景下获得一般和具体结果：

（1）通过简化为两人，常数和博弈，我们观察到任意Nash均衡在多项式时间内都是可计算的。此外，我们证明了防御价格上\（\ frac {\ textstyle | V |} {\ textstyle 2} \）的一般下界。我们得出具有达到该下限的纳什均衡的图的特征，这揭示了与分数图理论的有希望的联系；因此，这意味着针对这种防御最优图的有效识别算法。

（2）我们研究了纳什均衡的某些特定类别，无论是它们的计算复杂度还是所产生的防御价格。这些类别是通过在参与者的随机策略上强加结构来定义的：支撑上的图论结构，或概率上的对称和均匀结构。我们开发了新颖的图论技术，以得出这些类在计算复杂度和防御价格之间的权衡。一些技术触及了图论的经典里程碑。例如，我们得出了第一的博弈论表征ķ ö NIG - Egerv á RY图作为图承认一个匹配的纳什均衡。