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A Logical Description of Priority Separable Games

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Logic, Rationality, and Interaction (LORI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14329))

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Abstract

When we reason about strategic games, implicitly we need to reason about arbitrary strategy profiles and how players can improve from each profile. This structure is exponential in the number of players. Hence it is natural to look for subclasses of succinct games for which we can reason directly by interpreting formulas on the (succinct) game description rather than on the associated improvement structure. Priority separable games are one of such subclasses: payoffs are specified for pairwise interactions, and from these, payoffs are computed for strategy profiles. We show that equilibria in such games can be described in Monadic Least Fixed Point Logic (MLFP). We then extend the description to games over arbitrarily many players, but using the monadic least fixed point extension of existential second order logic.

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References

  1. Abdulla, P.A., Delzanno, G.: Parameterized verification. Int. J. Softw. Tools Technol. Transfer 18(5), 469–473 (2016). https://doi.org/10.1007/s10009-016-0424-3

    Article  Google Scholar 

  2. Ågotnes, T., Harrenstein, P., van der Hoek, W., Wooldridge, M.: Boolean games with epistemic goals. In: Grossi, D., Roy, O., Huang, H. (eds.) LORI 2013. LNCS, vol. 8196, pp. 1–14. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40948-6_1

    Chapter  Google Scholar 

  3. Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49, 672–713 (2002). https://doi.org/10.1145/585265.585270

    Article  Google Scholar 

  4. Apt, K., Simon, S., Wojtczak, D.: Coordination games on directed graphs. In: Proceedings of the 15th International Conference on Theoretical Aspects of Rationality and Knowledge (2015). https://doi.org/10.4204/EPTCS.215.6

  5. Aziz, H., Savani, R.: Hedonic games. Handbook of Computational Social Choice, chap. 15, pp. 356–376. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  6. Benthem, J.: Games in dynamic epistemic logic. Bull. Econ. Res. 53(4), 219–248 (2001). https://doi.org/10.1111/1467-8586.00133

    Article  Google Scholar 

  7. Benthem, J.: Extensive games as process models. J. Logic Lang. Inform. 11, 289–313 (2002). https://doi.org/10.1023/A:1015534111901

    Article  Google Scholar 

  8. Bonanno, G.: Branching time logic, perfect information games and backward induction. Games Econom. Behav. 36(1), 57–73 (2001). https://doi.org/10.1006/game.1999.0812

    Article  Google Scholar 

  9. Bonzon, E., Lagasquie-Schiex, M., Lang, J., Zanuttini, B.: Boolean games revisited. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) Proceedings of the 17th ECAI. Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 265–269. IOS Press (2006)

    Google Scholar 

  10. Cai, Y., Candogan, O., Daskalakis, C., Papadimitriou, C.: Zero-sum polymatrix games: a generalization of minmax. Math. Oper. Res. 41(2), 648–655 (2016). https://doi.org/10.1287/moor.2015.0745

    Article  Google Scholar 

  11. Cai, Y., Daskalakis, C.: On minmax theorems for multiplayer games. In: Proceedings of the SODA 2011, pp. 217–234. SIAM (2011)

    Google Scholar 

  12. Chatterjee, K., Henzinger, T., Piterman, N.: Strategy logic. Inf. Comput. 208(6), 677–693 (2010). https://doi.org/10.1016/j.ic.2009.07.004

    Article  Google Scholar 

  13. Das, R., Padmanabha, A., Ramanujam, R.: Reasoning in large games with unboundedly many players. In: Ghosh, S., Icard, T. (eds.) LORI 2021. LNCS, vol. 13039, pp. 41–57. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-88708-7_4

    Chapter  Google Scholar 

  14. Das, R., Ramanujam, R., Simon, S.: Reasoning about social choice and games in monadic fixed-point logic. In: Moss, L.S. (ed.) Proceedings Seventeenth Conference on Theoretical Aspects of Rationality and Knowledge, TARK 2019, Toulouse, France, 17–19 July 2019. EPTCS, vol. 297, pp. 106–120 (2019). https://doi.org/10.4204/EPTCS.297.8

  15. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009). https://doi.org/10.1137/070699652

    Article  Google Scholar 

  16. Ebbinghaus, Heinz-Dieter., Flum, Jörg.: Finite Model Theory. SMM, Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-28788-4

    Book  Google Scholar 

  17. Ghodsi, M., Saleh, H., Seddighin, M.: Fair allocation of indivisible items with externalities. CoRR abs/1805.06191 (2018). http://arxiv.org/abs/1805.06191

  18. Goranko, V.: The basic algebra of game equivalences. Stud. Logica. 75(2), 221–238 (2003). https://doi.org/10.1023/A:1027311011342

    Article  Google Scholar 

  19. Harrenstein, B., van der Hoek, W., Meyer, J.J., Witteveen, C.: Boolean games. In: van Benthem, J. (ed.) Proceedings of the 8th TARK, pp. 287–298. Morgan Kaufmann, San Francisco (2001)

    Google Scholar 

  20. Harrenstein, P., Hoek, W., Meyer, J., Witteven, C.: A modal characterization of Nash equilibrium. Fund. Inform. 57(2–4), 281–321 (2003)

    Google Scholar 

  21. Herzig, A., Lorini, E., Maffre, F., Schwarzentruber, F.: Epistemic Boolean games based on a logic of visibility and control. In: Kambhampati, S. (ed.) Proceedings of the 25th IJCAI, pp. 1116–1122. IJCAI/AAAI Press (2016)

    Google Scholar 

  22. Hoek, W., Jamroga, W., Wooldridge, M.: A logic for strategic reasoning. In: Proceedings of the Fourth International Joint Conference on Autonomous Agents and Multi-Agent Systems, pp. 157–164 (2005). https://doi.org/10.1145/1082473.1082497

  23. Igarashi, A., Elkind, E.: Hedonic games with graph-restricted communication. In: Jonker, C.M., Marsella, S., Thangarajah, J., Tuyls, K. (eds.) Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems, Singapore, 9–13 May 2016, pp. 242–250. ACM (2016)

    Google Scholar 

  24. Immerman, N.: Descriptive Complexity. Springer, Berlin (2012)

    Google Scholar 

  25. Janovskaya, E.: Equilibrium points in polymatrix games. Litovskii Matematicheskii Sbornik 8, 381–384 (1968)

    Google Scholar 

  26. Kearns, M.J., Littman, M.L., Singh, S.: Graphical models for game theory. In: Breese, J.S., Koller, D. (eds.) UAI 2001: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence, University of Washington, Seattle, Washington, USA, 2–5 August 2001, pp. 253–260. Morgan Kaufmann (2001)

    Google Scholar 

  27. Libkin, L.: Elements of Finite Model Theory, vol. 41. Springer, Heidelberg (2004)

    Book  Google Scholar 

  28. Massand, S., Simon, S.: Graphical one-sided markets. In: Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19, pp. 492–498. International Joint Conferences on Artificial Intelligence Organization (2019). https://doi.org/10.24963/ijcai.2019/70

  29. Mogavero, F., Murano, A., Perelli, G., Vardi, M.: Reasoning about strategies: on the model-checking problem. ACM Trans. Comput. Logic 15(4), 1–47 (2014). https://doi.org/10.1145/2631917

    Article  Google Scholar 

  30. Parikh, R.: The logic of games and its applications. Ann. Discrete Math. 24, 111–140 (1985). https://doi.org/10.1016/S0304-0208(08)73078-0

    Article  Google Scholar 

  31. Ramanujam, R., Simon, S.: Dynamic logic on games with structured strategies. In: Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR-08), pp. 49–58. AAAI Press (2008)

    Google Scholar 

  32. Ramanujam, R., Simon, S.E.: Structured strategies in games on graphs. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives [in Honor of Wolfgang Thomas]. Texts in Logic and Games, vol. 2, pp. 553–574. Amsterdam University Press (2008)

    Google Scholar 

  33. Schweikardt, N.: On the expressive power of monadic least fixed point logic. Theoret. Comput. Sci. 350, 325–344 (2006). https://doi.org/10.1016/j.tcs.2005.10.025

    Article  Google Scholar 

  34. Shapley, L.S., Scarf, H.: On cores and indivisibility. J. Math. Econ. 1(1), 23–37 (1974). https://doi.org/10.1016/0304-4068(74)90033-0

    Article  Google Scholar 

  35. Simon, S., Wojtczak, D.: Constrained pure Nash equilibria in polymatrix games. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 31 (2017)

    Google Scholar 

  36. Walther, D., Hoek, W., Wooldridge, M.: Alternating-time temporal logic with explicit strategies. In: Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK-2007), pp. 269–278 (2007). https://doi.org/10.1145/1324249.1324285

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Acknowledgements

We thank the anonymous reviewers for their comments which were very helpful in improving the presentation. The first author was partially supported by the Research-I foundation, IIT Kanpur. The third author was partially supported by the grant CRG/2022/006140.

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Authors and Affiliations

  1. Institute of Mathematical Sciences and Homi Bhabha National Institute, Chennai, India

    Ramit Das

  2. Azim Premji University, Bengaluru, India

    R. Ramanujam

  3. Department of CSE, IIT Kanpur, Kanpur, India

    Sunil Simon

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  1. Ramit Das
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Correspondence to Sunil Simon .

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Editors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Natasha Alechina

  2. CNRS, IRIT, Toulouse, France

    Andreas Herzig

  3. Shandong University, Jinan, China

    Fei Liang

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Das, R., Ramanujam, R., Simon, S. (2023). A Logical Description of Priority Separable Games. In: Alechina, N., Herzig, A., Liang, F. (eds) Logic, Rationality, and Interaction. LORI 2023. Lecture Notes in Computer Science, vol 14329. Springer, Cham. https://doi.org/10.1007/978-3-031-45558-2_3

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  • DOI: https://doi.org/10.1007/978-3-031-45558-2_3

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-031-45558-2

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