Document Open Access Logo

The Adversarial Stackelberg Value in Quantitative Games

Authors Emmanuel Filiot, Raffaella Gentilini, Jean-François Raskin

Thumbnail PDF


  • Filesize: 0.61 MB
  • 18 pages

Document Identifiers

Author Details

Emmanuel Filiot
  • Université libre de Bruxelles (ULB), Belgium
Raffaella Gentilini
  • University of Perugia, Italy
Jean-François Raskin
  • Université libre de Bruxelles (ULB), Belgium


We thank anonymous reviewers, Dr. Shibashis Guha and Ms. Mrudula Balachander for useful comments on a preliminary version of this paper.

Cite AsGet BibTex

Emmanuel Filiot, Raffaella Gentilini, and Jean-François Raskin. The Adversarial Stackelberg Value in Quantitative Games. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 127:1-127:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


In this paper, we study the notion of adversarial Stackelberg value for two-player non-zero sum games played on bi-weighted graphs with the mean-payoff and the discounted sum functions. The adversarial Stackelberg value of Player 0 is the largest value that Player 0 can obtain when announcing her strategy to Player 1 which in turn responds with any of his best response. For the mean-payoff function, we show that the adversarial Stackelberg value is not always achievable but ε-optimal strategies exist. We show how to compute this value and prove that the associated threshold problem is in NP. For the discounted sum payoff function, we draw a link with the target discounted sum problem which explains why the problem is difficult to solve for this payoff function. We also provide solutions to related gap problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Solution concepts in game theory
  • Theory of computation → Logic and verification
  • Non-zero sum games
  • reactive synthesis
  • adversarial Stackelberg


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Udi Boker, Thomas A. Henzinger, and Jan Otop. The target discounted-sum problem. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, pages 750-761, 2015. URL:
  2. Romain Brenguier, Lorenzo Clemente, Paul Hunter, Guillermo A. Pérez, Mickael Randour, Jean-François Raskin, Ocan Sankur, and Mathieu Sassolas. Non-zero sum games for reactive synthesis. In Language and Automata Theory and Applications - 10th International Conference, LATA 2016, Prague, Czech Republic, March 14-18, 2016, Proceedings, volume 9618 of Lecture Notes in Computer Science, pages 3-23. Springer, 2016. URL:
  3. Romain Brenguier and Jean-François Raskin. Pareto curves of multidimensional mean-payoff games. In Computer Aided Verification - 27th International Conference, CAV 2015, San Francisco, CA, USA, July 18-24, 2015, Proceedings, Part II, volume 9207 of Lecture Notes in Computer Science, pages 251-267. Springer, 2015. URL:
  4. Thomas Brihaye, Julie De Pril, and Sven Schewe. Multiplayer cost games with simple nash equilibria. In Logical Foundations of Computer Science, International Symposium, LFCS 2013, San Diego, CA, USA, January 6-8, 2013. Proceedings, volume 7734 of Lecture Notes in Computer Science, pages 59-73. Springer, 2013. URL:
  5. Krishnendu Chatterjee, Laurent Doyen, Herbert Edelsbrunner, Thomas A. Henzinger, and Philippe Rannou. Mean-payoff automaton expressions. In CONCUR 2010 - Concurrency Theory, 21th International Conference, CONCUR 2010, Paris, France, August 31-September 3, 2010. Proceedings, volume 6269, pages 269-283. Springer, 2010. URL:
  6. Krishnendu Chatterjee, Vojtěch Forejt, and Dominik Wojtczak. Multi-objective discounted reward verification in graphs and mdps. In Ken McMillan, Aart Middeldorp, and Andrei Voronkov, editors, Logic for Programming, Artificial Intelligence, and Reasoning, pages 228-242, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. Google Scholar
  7. Rodica Condurache, Emmanuel Filiot, Raffaella Gentilini, and Jean-François Raskin. The complexity of rational synthesis. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 121:1-121:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL:
  8. Jeanne Ferrante and Charles Rackoff. A decision procedure for the first order theory of real addition with order. SIAM J. Comput., 4(1):69-76, 1975. URL:
  9. Emmanuel Filiot, Raffaella Gentilini, and Jean-François Raskin. The adversarial stackelberg value in quantitative games, 2020. URL:
  10. Emmanuel Filiot, Raffaella Gentilini, and Jean-François Raskin. Rational synthesis under imperfect information. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 422-431. ACM, 2018. URL:
  11. Dana Fisman, Orna Kupferman, and Yoad Lustig. Rational synthesis. In Tools and Algorithms for the Construction and Analysis of Systems, 16th International Conference, TACAS, volume 6015 of Lecture Notes in Computer Science, pages 190-204. Springer, 2010. URL:
  12. Oded Goldreich. On promise problems: A survey. In Theoretical Computer Science, Essays in Memory of Shimon Even, volume 3895 of Lecture Notes in Computer Science, pages 254-290. Springer, 2006. URL:
  13. Anshul Gupta and Sven Schewe. Quantitative verification in rational environments. In 21st International Symposium on Temporal Representation and Reasoning, TIME 2014, Verona, Italy, September 8-10, 2014, pages 123-131. IEEE Computer Society, 2014. URL:
  14. Anshul Gupta, Sven Schewe, Ashutosh Trivedi, Maram Sai Krishna Deepak, and Bharath Kumar Padarthi. Incentive stackelberg mean-payoff games. In Software Engineering and Formal Methods - 14th International Conference, SEFM 2016, Held as Part of STAF 2016, Vienna, Austria, July 4-8, 2016, Proceedings, volume 9763 of Lecture Notes in Computer Science, pages 304-320. Springer, 2016. Google Scholar
  15. Anshul Gupta, Sven Schewe, and Dominik Wojtczak. Making the best of limited memory in multi-player discounted sum games. In Proceedings Sixth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2015, Genoa, Italy, 21-22nd September 2015, volume 193 of EPTCS, pages 16-30, 2015. URL:
  16. Richard M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete Mathematics, 23(3):309-311, 1978. URL:
  17. Orna Kupferman, Giuseppe Perelli, and Moshe Y. Vardi. Synthesis with rational environments. Ann. Math. Artif. Intell., 78(1):3-20, 2016. URL:
  18. J. F. Nash. Equilibrium points in n-person games. In PNAS, volume 36, pages 48-49. National Academy of Sciences, 1950. Google Scholar
  19. Martin J. Osborne. An introduction to game theory. Oxford Univ. Press, 2004. Google Scholar
  20. Amir Pnueli and Roni Rosner. On the synthesis of a reactive module. In Conference Record of the Sixteenth Annual ACM Symposium on Principles of Programming Languages, Austin, Texas, USA, January 11-13, 1989, pages 179-190, 1989. URL:
  21. Yaron Velner, Krishnendu Chatterjee, Laurent Doyen, Thomas A. Henzinger, Alexander Moshe Rabinovich, and Jean-François Raskin. The complexity of multi-mean-payoff and multi-energy games. Inf. Comput., 241:177-196, 2015. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail