We consider the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset's elements and maximal chains is satisfied? In this article, we present a combinatorial algorithm to positively resolve this question. We show that this result plays a crucial role in the equilibrium analysis of a generic security game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs, and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. The first (resp. second) player seeks to maximize the value of effective (resp. interdicted) flow net the total transportation (resp. interdiction) cost. Using our existence result on posets and strict complementary slackness in linear programming, we show that the equilibrium properties of this game can be described using primal and dual solutions of a minimum cost circulation problem. Our analysis provides a new characterization of the critical network components.

翻译：我们考虑以下问题:是否在有限部分定购的一组子集(装置)存在概率分布,从而满足了一组限制,涉及表面元素和最大链链的边缘概率?在本篇文章中,我们提出了一个组合算法,以积极解决这一问题。我们表明,这一结果在对功能化流量网络的一般安全游戏进行平衡分析方面发挥着关键作用。游戏涉及一个路由实体,该实体在面临路径运输成本时通过网络输送其流量,以及拦截器在面临边缘阻截成本时同时截取一个或多个边缘。第一个(重编)玩家寻求最大限度地提高有效(阻截)流动总运输(阻截)成本的价值。利用我们的存在在线性编程中的表面结果和严格的互补松懈状态,我们表明这一游戏的平衡特性可以用最低成本循环问题的原始和双重解决办法来描述。我们的分析为关键网络组成部分提供了新的特征。