Automated verification techniques for stochastic games allow formal reasoning about systems that feature competitive or collaborative behaviour among rational agents in uncertain or probabilistic settings. Existing tools and techniques focus on turn-based games, where each state of the game is controlled by a single player, and on zero-sum properties, where two players or coalitions have directly opposing objectives. In this paper, we present automated verification techniques for concurrent stochastic games (CSGs), which provide a more natural model of concurrent decision making and interaction. We also consider (social welfare) Nash equilibria, to formally identify scenarios where two players or coalitions with distinct goals can collaborate to optimise their joint performance. We propose an extension of the temporal logic rPATL for specifying quantitative properties in this setting and present corresponding algorithms for verification and strategy synthesis for a variant of stopping games. For finite-horizon properties the computation is exact, while for infinite-horizon it is approximate using value iteration. For zero-sum properties it requires solving matrix games via linear programming, and for equilibria-based properties we find social welfare or social cost Nash equilibria of bimatrix games via the method of labelled polytopes through an SMT encoding. We implement this approach in PRISM-games, which required extending the tool’s modelling language for CSGs, and apply it to case studies from domains including robotics, computer security and computer networks, explicitly demonstrating the benefits of both CSGs and equilibria-based properties.

随机游戏的自动验证技术可以对系统进行形式上的推理，这些系统具有不确定性或概率环境中理性主体之间竞争或协作行为的特征。现有的工具和技术专注于基于回合的游戏（游戏的每个状态都由一个玩家控制）和零和属性（两个玩家或联盟具有直接相反的目标）。在本文中，我们提出了用于并发随机游戏（CSG）的自动验证技术，该技术提供了更自然的并发决策和交互模型。我们还考虑（社会福利）纳什均衡，以正式确定方案，其中具有不同目标的两个参与者或联盟可以协作以优化其共同绩效。我们建议对时间逻辑rPATL进行扩展，以在这种情况下指定定量属性，并提出相应的算法，以用于验证和停止游戏变体的策略综合。对于有限水平的属性，计算是精确的，而对于无限水平的属性，则使用值迭代进行近似计算。对于零和属性，它需要通过线性规划来求解矩阵博弈；对于基于均衡的属性，我们通过SMT编码的标记多面体方法来找到双矩阵博弈的社会福利或社会成本纳什均衡。我们在PRISM游戏中实现了这种方法，该方法需要扩展CSG的工具建模语言，并将其应用于来自机器人，计算机安全和计算机网络等领域的案例研究，