Abstract
Rationalizability, originally proposed by Bernheim and Pearce, generalizes the notion of Nash equilibrium. Nash equilibrium requires common knowledge of strategies. Rationalizability only requires common knowledge of rationality. However, their original notion assumes that the payoffs are common knowledge.
I.e. agents do know what world they are in, but may be ignorant of what other agents are playing.
We generalize the original notion of rationalizability to consider situations where agents do not know what world they are in, or where some know but others do not know. Agents who know something about the world can take advantage of their superior knowledge. It may also happen that both Ann and Bob know about the world but Ann does not know that Bob knows. How might they act?
We will show how a notion of rationalizability in the context of partial knowledge, represented by a Kripke structure, can be developed.
Some of the ideas in this paper were included in [7], but the discussion on Dennett, and Theorem 4.1 are new.
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Notes
- 1.
She is separated from him by a big rock so he cannot see her but could hear her cries of pleasure.
- 2.
I am using the word belief in a weak sense in which we can use it for non-linguistic creatures.
- 3.
Something which puzzles me is how they passed the knowledge “it is a hoax” from one tiger to another. Tigers are solitary beasts and do not have cellphones.
- 4.
We have not yet defined ‘rational’ so we will temporarily rely on an intuitive meaning of the word. A precise definition will be provided in the next section.
- 5.
The \(R_i\) are often assumed to be equivalence relations and we shall follow this tradition.
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Acknowledgement
Dov Samet very kindly showed me some related work of his [2] which does talk about dominated strategies. This work was done independently of ours and has some elegant ideas. But he and his co-author do not make use of Kripke structures, or, for that matter, detectives and tigers! Thanks to David Makinson for comments. This research was supported by grants from the CUNY Faculty research assistance program.
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Parikh, R. (2017). An Epistemic Generalization of Rationalizability. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_21
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DOI: https://doi.org/10.1007/978-3-662-55386-2_21
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