Abstract
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.
Highlights
Stochastic multi-player games are a versatile modelling framework for systems that exhibit cooperative or competitive behaviour in the presence of adversarial or uncertain environ- B Gethin Norman
We propose an extension of the rPATL logic which adds the ability to express quantitative nonzero-sum properties based on these notions of equilibria, for example “the two robots have navigation strategies which form a Nash equilibrium, and under which the combined expected energy usage until completing their tasks is below k”
In order to formalise properties of concurrent stochastic games (CSGs), we propose an extension of the logic rPATL, previously defined for zero-sum properties of turn-based stochastic multi-player games (TSGs) [19]
Summary
In order to investigate the performance, scalability and applicability of our techniques, we have developed a large selection of case studies taken from a diverse set of application domains including: finance, computer security, computer networks, communication systems, robot navigation and power control These illustrate examples of systems whose modelling and analysis requires stochasticity and competitive or collaborative behaviour between concurrent components or agents. PRISM-games 2.0 [51], which we have built upon in this work, provided modelling and verification for a wide range of properties of stochastic multi-player games, including those in the logic rPATL, and multiobjective extensions of it, but focusing purely on the turn-based variant of the model (TSGs) in the context of two-coalitional zero-sum properties. We mention the fact that the concept of equilibrium is used to analyze different applications such as cooperation among agents in stochastic games [39] and to design protocols based on quantum secret sharing [67]
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