Abstract
Reasoning about equilibria in normal form games involves the study of players’ incentives to deviate unilaterally from any profile. In the case of large anonymous games, the pattern of reasoning is different. Payoffs are determined by strategy distributions rather than strategy profiles. In such a game each player would strategise based on expectations of what fraction of the population makes some choice, rather than respond to individual choices by other players. A player may not even know how many players there are in the game. Logicising such strategisation involves many challenges as the set of players is potentially unbounded. This suggests a logic of quantification over player variables and modalities for player deviation, but such a logic is easily seen to be undecidable. Instead, we propose a propositional modal logic using player types as names and implicit quantification over players. With modalities for player deviation and transitive closure, the logic can be used to specify game equilibrium and interesting patterns of reasoning in large games. We show that the logic is decidable and present a complete axiomatisation of the valid formulas.
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References
Ågotnes, T., Harrenstein, P., Hoek, W., & Wooldridge, M. (2013). Boolean games with epistemic goals. In International workshop on logic, rationality and interaction (pp. 1–14). Springer.
Alur, R., Henzinger, T. A., & Kupferman, O. (2002). Alternating-time temporal logic. Journal of ACM, 49(5), 672–713.
Angluin, D., Aspnes, J., Eisenstat, D., & Ruppert, E. (2007). The computational power of population protocols. Distributed Computing, 20(4), 279–304.
Banerjee, A. V. (1992). A simple model of herd behaviour. The Quarterly Journal of Economics, 107(3), 797–817.
Blonski, M. (1999). Anonymous games with binary actions. Games and Economic Behaviour, 28, 171–180.
Blonski, M. (2000). Characterisation of pure strategy equilibria in finite anonymous games. Journal of Mathematical Economics, 34, 225–233.
Bonanno, G. (2001). Branching time logic, perfect information games and backward induction. Games and Economic Behaviour, 36(1), 57–73.
Bonzon, E., Lagasquie-Schiex, M. C., Lang, J., & Zanuttini, B. (2006). Boolean games revisited. In ECAI (Vol. 141, pp. 265–269).
Brandt, F., Fischer, F., & Holzer, M. (2009). Symmetries and the complexity of pure Nash equilibrium. Journal of Computer and System Sciences, 75(3), 163–177.
Christoff, Z., & Hansen, J. U. (2015). A logic for diffusion in social networks. Journal of Applied Logic, 13(1), 48–77.
Das, R., & Ramanujam, R. (2021). A logical description of strategizing in social network games. In Proceedings of the first international workshop on logics for new-generation artificial intelligence (pp. 111–123).
Daskalakis, C., & Papadimitriou, C. H. (2007). Computing equilibria in anonymous games. In Proceedings of the 48th symposium on Foundations of Computer Science (FOCS) (pp. 83–93). IEEE Computer Society Press.
Elkind, E., Faliszewski, P., Skowron, P., & Slinko, A. (2017). Properties of multiwinner voting rules. Social Choice and Welfare, 48(3), 599–632.
Ghosh, S. (2008). Strategies made explicit in dynamic game logic. In Proceedings of the workshop on logic and intelligent interaction, ESSLLI (pp. 74–81).
Ghosh, S., & Ramanujam, R. (2011). Strategies in games: a logic-automata study. In Lectures on logic and computation, pp. 110–159. Springer.
Grädel, E., & Ummels, M. (2008). Solution concepts and algorithms for infinite multiplayer games. New Perspectives on Games and Interaction, 4, 151–178.
Harrenstein, P., Hoek, W., Meyer, J., & Witteven, C. (2003). A modal characterisation of nash equilibrium. Fundamenta Informaticae, 57(2–4), 281–321.
Hoek, W., Jamroga, W., & Wooldridge, M. (2005). A logic for strategic reasoning. In Proceedings of the fourth international joint conference on autonomous agents and multi-agent systems (pp. 157–164).
Hoek, W., & Wooldridge, M. (2005). On the logic of cooperation and propositional control. Artificial Intelligence, 164(1–2), 81–119.
Julian, G., Harrenstein, P., & Wooldridge, M. (2015). Iterated boolean games. Information and Computation, 242, 53–79.
Kalai, E. (2005). Partially-specified large games. In International workshop on internet and network economics (pp. 3–13). Springer.
Kempe, D., Kleinberg, J., & Tardos, E. (2015). Maximizing the spread of influence through a social network. Theory of Computing, 11(4), 105–147.
Kozen, D., & Parikh, R. (1981). An elementary proof of the completeness of propositional dynamic logic. Theoretical Computer Science, 14(1), 113–118.
Mossel, E., Neeman, J., & Tamuz, O. (2013). Majority dynamics and aggregation of information in social networks. Autonomous Agents and Multi-Agent Systems, 28(3), 408–429.
Padmanabha, A. (2020). Propositional term modal logic. PhD thesis, Institute of Mathematical Sciences, Homi Bhabha National Institute, https://www.imsc.res.in/xmlui/handle/123456789/452
Parikh, R. (1985). The logic of games and its applications. Annals of Discrete Mathematics, 24, 111–140.
Paul, S., & Ramanujam, R. (2011). Neighbourhood structure in large games. In Proceedings of the 13th conference on theoretical aspects of rationality and knowledge (pp. 121–130).
Paul, S., & Ramanujam, R. (2013). Dynamics of choice restriction in large games. International Game Theory Review, 15(4), 1340031.
Paul, S., & Ramanujam, R. (2014). Subgames within large games and the heuristic of imitation. Studia Logica, 102(2), 361–388.
Padmanabha, A., & Ramanujam, R. (2019a). Propositional modal logic with implicit modal quantification. In Md. Aquil Khan & A. Manuel (Eds.), Logic and its applications—8th Indian conference, ICLA 2019, Delhi, India, March 1–5, 2019, Proceedings Lecture Notes in Computer Science, (Vol. 11600, pp. 6–17). Springer.
Padmanabha, A., & Ramanujam, R. (2019b). The monodic fragment of propositional term modal logic. Studia Logica,107(3), 533–557.
Ramanujam, R., & Simon, S. (2008). A logical structure for strategies. In Logic and the foundations of game and decision theory (LOFT 7), texts in logic and games (Vol 3, pp. 183–208). Amsterdam University Press.
Simon, S., & Apt, K. R. (2015). Social network games. Journal of Logic and Computation, 25(1), 207–242.
van Benthem, J. (2012). In praise of strategies. Games, Actions and Social Software, 7010, 96–116.
Weibull, J. W. (1997). Evolutionary game theory. MIT Press.
Acknowledgements
We thank the anonymous reviewers for their diligent reading of the manuscript, and immensely helpful suggestions. We thank Sujata Ghosh, Kamal Lodaya and S. P. Suresh, as well as the reviewers of LORI 2021 for their response to presentations on the paper.
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Das, R., Padmanabha, A. & Ramanujam, R. Implicit quantification for modal reasoning in large games. Synthese 201, 163 (2023). https://doi.org/10.1007/s11229-023-04156-9
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DOI: https://doi.org/10.1007/s11229-023-04156-9