Stochastic Equilibria under Imprecise Deviations in Terminal-Reward Concurrent Games

Bouyer, Patricia; Markey, Nicolas; Stan, Daniel

doi:10.4204/eptcs.226.5

Cited by 5 publications

(11 citation statements)

References 19 publications

(28 reference statements)

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“…when Assumption 1 does not hold, value iteration can fail to converge for expected reachability properties. Consider the CSG in Figure 7 (which again is an adaption of a TSG example from [9]) and the formula θ = R r1 [ F a ]+R r2 [ F a ] where a is the atomic proposition satisfied only by the states t 1 and t 2 . Clearly there are strategy profiles for which the targets are not reached with probability 1.…”

Section: Convergence Of Expected Reachability Propertiesmentioning

confidence: 99%

Equilibria-Based Probabilistic Model Checking for Concurrent Stochastic Games

Kwiatkowska

Norman

Parker

et al. 2019

Lecture Notes in Computer Science

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Probabilistic model checking for stochastic games enables formal verification of systems that comprise competing or collaborating entities operating in a stochastic environment. Despite good progress in the area, existing approaches focus on zero-sum goals and cannot reason about scenarios where entities are endowed with different objectives. In this paper, we propose probabilistic model checking techniques for concurrent stochastic games based on Nash equilibria. We extend the temporal logic rPATL (probabilistic alternating-time temporal logic with rewards) to allow reasoning about players with distinct quantitative goals, which capture either the probability of an event occurring or a reward measure. We present algorithms to synthesise strategies that are subgame perfect social welfare optimal Nash equilibria, i.e., where there is no incentive for any players to unilaterally change their strategy in any state of the game, whilst the combined probabilities or rewards are maximised. We implement our techniques in the PRISM-games tool and apply them to several case studies, including network protocols and robot navigation, showing the benefits compared to existing approaches. 4 1 4i.e. the values ( 3 4 , 1 4 ).

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Section: Convergence Of Expected Reachability Propertiesmentioning

confidence: 99%

Equilibria-Based Probabilistic Model Checking for Concurrent Stochastic Games

Kwiatkowska

Norman

Parker

et al. 2019

Lecture Notes in Computer Science

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“…Consider the CSG in Fig. 14 with players p 1 and p 2 (an adaptation of a TSG example from [8]) and the nonzero-sum state formula p 1 : p 2 max=? (θ ), where…”

Section: Appendix A: Convergence Of Zero-sum Reachability Reward Formulaementioning

confidence: 99%

“…Consider the CSG in Fig. 15 with players p 1 and p 2 (which again is an adaptation of a TSG example from [8]) and the nonzero-sum state formula p 1 : p 2 max=? (θ ), where θ = R r 1 [ F a ]+R r 2 [ F a ] and a is the atomic proposition satisfied only by the states t 1 and t 2 .…”

Section: Appendix A: Convergence Of Zero-sum Reachability Reward Formulaementioning

confidence: 99%

“…A notion of strong NE for a restricted class of CSGs is formalised in [27] and an approximation algorithm for checking the existence of such NE for discounted properties is introduced and evaluated. The existence of stochastic equilibria with imprecise deviations for CSGs and a PSPACE algorithm to compute such equilibria is considered in [8]. Finally, 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].…”

Section: Introductionmentioning

confidence: 99%

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Automatic verification of concurrent stochastic systems

Kwiatkowska

Norman

Parker

et al. 2021

Form Methods Syst Des

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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.

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“…Nevertheless, over game graphs, continuity of this best-response function is not ensured (hence the graph of BR is not closed). Let us consider for example game of Figure 2 (borrowed from [6]). It is assumed to be turn-based (vertex v i belongs to player A i ): from v i , player A i can easer continue or leave the game.…”

Section: Why Does the Standard Theory Not Apply?mentioning

confidence: 99%

A Note on Game Theory and Verification

Bouyer¹

2019

Automated Technology for Verification and Analysis

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We present some basics of game theory, focusing on matrix games. We then present the model of multiplayer stochastic concurrent games (with an underlying graph), which extends standard finite-state models used in verification in a multiplayer and concurrent setting; we explain why the basic theory cannot apply to that general model. We then focus on a very simple setting, and explain and give intuitions for the computation of Nash equilibria. We then give a number of undecidability results, giving limits to the approach. Finally we describe the suspect game construction, which (we believe) captures and explains well Nash equilibria and allow to compute them in many cases.

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Stochastic Equilibria under Imprecise Deviations in Terminal-Reward Concurrent Games

Cited by 5 publications

References 19 publications

Equilibria-Based Probabilistic Model Checking for Concurrent Stochastic Games

Equilibria-Based Probabilistic Model Checking for Concurrent Stochastic Games

Automatic verification of concurrent stochastic systems

A Note on Game Theory and Verification

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