Abstract
Epistemic game theory has shown the importance of informational contexts to understand strategic interaction. We propose a general framework to analyze how such contexts may arise. The idea is to view informational contexts as the fixed points of iterated, rational responses to incoming information about the agents’ possible choices. We discuss conditions under which such fixed points may exist. In the process, we generalize existing rules for information updates used in the dynamic epistemic logic literature. We then apply this framework to weak dominance. Our analysis provides a new perspective on a well known problem with the epistemic characterization of iterated removal of weakly dominated strategies.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
- 3.
See Pacuit (2015) for an extensive discussion of these different models of deliberation in games.
- 4.
We assume that the reader is familiar with the basic concepts of game theory (e.g., strategic games and various solution concepts such as iterated removal of strictly/weakly dominated strategies). Consult Leyton-Brown and Shoham (2008) for an introduction to game theory.
- 5.
Well-foundedness is only needed to ensure that for any set X, the set of minimal elements in X is nonempty. This is important only when W is infinite – and there are ways around this in current logics. Moreover, the condition of connectedness can also be lifted, but we use it here for convenience.
- 6.
A similar idea is found in standard models of differential information from the economics literature. In such models, it is assumed that there is a prior probability measure describing the players’ initial beliefs (often it is the same probability measure for all the players). The players’ posterior probabilities are defined by conditioning their prior probability measure on their private information (typically represented by some partition over the set of states).
- 7.
Consult any textbook on decision theory, such as Peterson (2009), for evidence of this fact.
- 8.
- 9.
Rabinovich takes this even further and argues that from the principle of indifference, players must assign equal probability to all choice-worthy options (Rabinowicz 1992).
- 10.
So, we assume that the models agree about which outcomes of the game have not been ruled out.
- 11.
An interesting extension would be to start with a multiagent belief model and allow players to incorporate information not only about which options are “choice-worthy”, but also about which beliefs their opponents may have. We leave this extension for future work and focus on setting up the basic framework here.
- 12.
Here, it is crucial that the language \(\mathcal{L}_{G}\) does not contain any modalities.
References
Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50, 510–530 (1985)
Anglberger, A.J.J., Gratzl, N., Roy, O.: Obligation, free choice and the logic of weakest permission. Rev. Symb. Log. 8(4), 807–827 (2015)
Apt, K., Zvesper, J.: Public announcements in strategic games with arbitrary strategy sets. In: Proceedings of LOFT 2010, Toulouse, France (2010a)
Apt, K., Zvesper, J.: The role of monotonicity in the epistemic analysis of strategic games. Games 1(4), 381–394 (2010b)
Arntzenius, F.: No regrest, or: edith piaf revamps decision theory. Erkenntnis 68, 277–297 (2008)
Baltag, A., Smets, S.: Group belief dynamics under iterated revision: fixed points and cycles of joint upgrades. In: Proceedings of Theoretical Aspects of Rationality and Knowledge, Stanford (2009)
Baltag, A., Smets, S., Zvesper, J.: Keep ‘hoping’ for rationality: a solution to the backwards induction paradox. Synthese 169, 301–333 (2009)
Board, O.: Dynamic interactive epistemology. Games Econ. Behav. 49, 49–80 (2004)
Boutilier, C.: Conditional logics for default reasoning and belief revision. PhD thesis, University of Toronto (1992)
Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in games. Econometrica 76, 307–352 (2008)
Cubitt, R., Sugden, R.: Common knowledge, salience and convention: a reconstruction of David Lewis’ game theory. Econ. Philos. 19(2), 175–210 (2003)
Cubitt, R., Sugden, R.: The reasoning-based expected utility procedure. Games Econ. Behav. 71(2), 328–338 (2011)
Cubitt, R., Sugden, R.: Common reasoning in games: a Lewisian analysis of common knowledge of rationality. Econ. Philos. 30(3), 285–329 (2014)
Dekel, E., Siniscalchi, M.: Epistemic game theory. In: Peyton Young, H., Shmuel, Z. (eds.) Handbook of Game Theory with Economic Applications, vol. 4, pp. 619–702. Elsevier, Amsterdam (2015)
Gerbrandy, J.: Bisimulations on planet Kripke. PhD thesis, University of Amsterdam (1999)
Halpern, J., Pass, R.: A logical characterization of iterated admissibility. In: Heifetz, A. (ed.) Proceedings of the Twelfth Conference on Theoretical Aspects of Rationality and Knoweldge, Stanford, pp. 146–155 (2009)
Harsanyi, J.: The tracing procedure: a Bayesian approach to defining a solution for n-person noncooperative games. Int. J. Game Theory 4, 61–94 (1975)
Holliday, W.: Trust and the dynamics of testimony. In: Logic and interaction rationality: Seminar’s yearbook 2009, pp. 147–178. ILLC technical reports (2009)
Lamarre, P., Shoham, Y.: Knowledge, certainty, belief and conditionalisation. In: Proceedings of the International Conference on Knoweldge Representation and Reasoning, pp. 415–424 (1994)
Levi, I.: Rationality, prediction, and autonomous choice. Can. J. Philos. 23(1), 339–363 (1993)
Lewis, D.: Convention. Harvard University Press, Cambridge (1969)
Lewis, D.: Counterfactuals. Blackwell, Oxford (1973)
Leyton-Brown, K., Shoham, Y.: Essentials of Game Theory: A Concise Multidisciplinary Introduction. Morgan & Claypool, San Rafael (2008)
Pacuit, E.: Dynamic epistemic logic II: logics of informaiton change. Philos. Compass 8(9), 815–833 (2013)
Pacuit, E.: Dynamic models of rational deliberation in games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds.) Models of Strategic Reasoning: Logics, Games and Communities. Springer (2015)
Pacuit, E., Roy, O.: Epistemic foundations of game theory. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2015)
Perea, A.: Epistemic Game Theory: Reasoning and Choice. Cambridge University Press, New York (2012)
Peterson, M.: An Introduction to Decision Theory. Cambridge University Press, Cambridge/New York (2009)
Plaza, J.: Logics of public communications. In: Emrich, M.L., Pfeifer, M.S., Hadzikadic, M., Ras, Z.W. (eds.) Proceedings, 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216 (1989)
Rabinowicz, W.: Tortous labyrinth: noncooperative normal-form games between hyperrational players. In: Bicchieri, C., Chiara, M.L.D. (eds.) Knowledge, Belief and Strategic Interaction. Cambridge University Press, Cambridge/New York, pp. 107–125 (1992)
Rabinowicz, W.: Does practical deliberation crowd out self-prediction. Erkenntnis 57, 91–122 (2002)
Rott, H.: Shifting priorities: simple representations for 27 itereated theory change operators. In: Lagerlund, H., Lindström, S., Sliwinski, R. (eds.) Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg. Uppsala Philosophical Studies, vol. 53, pp. 359–384 (2006)
Roy, O., Anglberger, A.J.J., Gratzl, N.: The logic of best action from a deontic perspective. In: Baltag, A., Smets, S. (eds.) Johan FAK van Benthem on Logical and Informational Dynamics. Springer (2014)
Samuelson, L.: Dominated strategies and common knowledge. Game Econ. Behav. 4, 284–313 (1992)
Schick, F.: Self-knowledge, uncertainty and choice. Br. J. Philos. Sci. 30(3), 235–252 (1979)
Skyrms, B.: The Dynamics of Rational Deliberation. Harvard University Press, Cambridge (1990)
Ullmann-Margalit, E., Morgenbesser, S.: Picking and choosing. Soc. Res. 44, 757–785 (1977)
van Benthem, J.: Dynamic logic for belief revision. J. Appl. Non-class. Log. 14(2), 129–155 (2004)
van Benthem, J.: Rational dynamics and epistemic logic in games. Int. Game Theory Rev. 9(1), 13–45 (2007)
van Benthem, J.: Logical Dynamics of Information and Interaction. Cambridge University Press (2010)
van Benthem, J.: Logic in Games. MIT, Cambridge/London (2014)
van Benthem, J., Gheerbrant, A.: Game solution, epistemic dynamics and fixed-point logics. Fund. Inf. 100, 1–23 (2010)
van Benthem, J., Pacuit, E., Roy, O.: Towards a theory of play: a logical perspective on games and interaction. Games 2(1), 52–86 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Pacuit, E., Roy, O. (2016). A Dynamic Analysis of Interactive Rationality. In: Redmond, J., Pombo Martins, O., Nepomuceno Fernández, Á. (eds) Epistemology, Knowledge and the Impact of Interaction. Logic, Epistemology, and the Unity of Science, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-26506-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-26506-3_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-26504-9
Online ISBN: 978-3-319-26506-3
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)