Synthesis of Controllable Nash Equilibria in Games with Quantitative Objectives

In Rational Synthesis, we consider a multi-agent system in which some of the agents are controllable and some are not. All agents have objectives, and the goal is to synthesize strategies for the controllable agents so that their objectives are satisfied, assuming rationality of the uncontrollable agents. Previous work on rational synthesis considers objectives in LTL, namely ones that describe on-going behaviors, and in Objective-LTL, which allows ranking of LTL formulas. In this paper, we extend rational synthesis to LTL[F ]– an extension of LTL by quality operators. The satisfaction value of an LTL[F ] formula is a real value in [0, 1], where the higher the value is, the higher is the quality in which the computation satisfies the specification. The extension significantly strengthens the framework of rational synthesis and enables a study its gameand socialchoice theoretic aspects. In particular, we study the price of stability and price of anarchy of the rationalsynthesis game and use them to explain the cooperative and non-cooperative settings of rational synthesis. Our algorithms make use of strategy logic and decision procedures for it. Thus, we are able to handle the richer quantitative setting using existing tools. In particular, we show that the cooperative and non-cooperative versions of quantitative rational synthesis are 2EXPTIMEcomplete and in 3EXPTIME, respectively – not harder than the complexity known for their Boolean analogues.

[1]  Axel Legay,et al.  Model Checking Quantitative Linear Time Logic , 2008, QAPL.

[2]  Krishnendu Chatterjee,et al.  Strategy logic , 2007, Inf. Comput..

[3]  Dana Fisman,et al.  Rational Synthesis , 2009, TACAS.

[4]  Jean-François Raskin,et al.  The Complexity of Rational Synthesis , 2016, ICALP.

[5]  P. Gács,et al.  Algorithms , 1992 .

[6]  Christos H. Papadimitriou,et al.  Algorithms, Games, and the Internet (Extended Abstract) , 2001 .

[7]  Johan van Benthem,et al.  Reasoning about Strategies , 2013, Computation, Logic, Games, and Quantum Foundations.

[8]  Fabio Mogavero.  Reasoning About Strategies , 2013, FSTTCS 2013.

[9]  Orna Kupferman,et al.  Repairing Multi-Player Games , 2015, CONCUR.

[10]  Michael Wooldridge,et al.  Iterated Games with LDL Goals over Finite Traces , 2017, AAMAS.

[11]  Tim Roughgarden,et al.  The price of stability for network design with fair cost allocation , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[12]  M. Wooldridge,et al.  Imperfect Information in Reactive Modules Games , 2016, KR.

[13]  Christos H. Papadimitriou,et al.  Worst-case equilibria , 1999 .

[14]  Noam Nisan,et al.  Algorithmic Mechanism Design , 2001, Games Econ. Behav..

[15]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Amir Pnueli,et al.  On the synthesis of a reactive module , 1989, POPL '89.

[17]  Michael Wooldridge,et al.  Iterated Boolean games , 2013, Inf. Comput..

[18]  Aniello Murano,et al.  What Makes Atl* Decidable? A Decidable Fragment of Strategy Logic , 2012, CONCUR.

[19]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[20]  Giuseppe Perelli,et al.  Synthesis with rational environments , 2016, Annals of Mathematics and Artificial Intelligence.

[21]  Aniello Murano,et al.  Reasoning about Strategies: on the Satisfiability Problem , 2016, Log. Methods Comput. Sci..

[22]  Michael Wooldridge,et al.  Rational Verification: From Model Checking to Equilibrium Checking , 2016, AAAI.

[23]  Orna Kupferman,et al.  Formalizing and Reasoning about Quality , 2013, ICALP.

[24]  Amir Pnueli,et al.  On the Development of Reactive Systems , 1989, Logics and Models of Concurrent Systems.

[25]  Michael Wooldridge,et al.  Nash Equilibria in Concurrent Games with Lexicographic Preferences , 2017, IJCAI.