Abstract
In this paper we develop a plausibility model by defining a new notion of rationality based on the assumption that a player believes that she doesn’t play a regret dominated strategy. Especially, we show that the interactive epistemic outcomes of this type of rationality are in line with the solutions of the Iterated Regret Minimization (IRM) algorithm. So, we state that one can achieve a characterization of the IRM algorithm by keeping upgrading the assumption of rationality, and we obtain common belief of rationality in the limit model. A benefit of our characterization is that it provides the epistemic foundation to the IRM algorithm and solve a dynamic information problem best expressed through the Traveler’s Dilemma. Meanwhile, we also link solutions of the IRM algorithm to modal \(\mu \)-calculus to deepen our understanding of the epistemic characterization.
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Notes
- 1.
In [6], the definition of the algorithm didn’t compute again the regrets of the strategies after the process of elimination of the regret dominated strategies, that’s why it couldn’t give the desired solution to the Traveler’s Dilemma. In this paper we correct this fact.
- 2.
An upgrade \(\Uparrow \phi \) is truthful in a plausibility model M if \(\phi \) is true at \(z^{*}\).
- 3.
By “hard” information we mean an information, typically referred to Knowledge, whose truth is beyond any doubt. By “soft” information we mean an information, typically referred to Belief, which can be subject to a change.
- 4.
This formula has the following syntactic interpretation: \(\left\| \mu p \cdot \phi (p)\right\| _{i} = \bigcap \{ T\subseteq S \mid T\subseteq \left\| \phi (p)\right\| _{i[\left\| p\right\| =T]} \}\) where i is the interpretation map and S is the domain of the Kripke model.
- 5.
We say that “player i prefers strategy \(s_{i}\) over strategy \(s_{i}'\) ”.
- 6.
In case \(\mathbf M _{G_{p}},w \vDash B_{i} s_i\), where \(s_i\in S_i\) is a (mixed or pure) strategy, we say that “player i believes she should play strategy \(s_i\)”.
- 7.
Corollary 6 says: Every repeated truthful radical upgrade definable in doxastic-epistemic logic (i.e. the language of simple belief and knowledge operators, without any conditional beliefs) stabilizes every model.
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Acknowledgements
For this work the first author benefited financial support from the University of Pisa, the paper is also supported by Kep Programm of National Social Science Foundation of China (No. 16AZX017) and by Kep Programm of National Social Science Foundation of China (No. 15AZX020). The authors would like to thank Alexandru Baltag, Alessandro Berarducci, Davide Grossi and Johan van Benthem for their precious suggestions and comments.
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Bobbio, F., Cui, J. (2018). A Plausibility Model for Regret Games. In: Belardinelli, F., Argente, E. (eds) Multi-Agent Systems and Agreement Technologies. EUMAS AT 2017 2017. Lecture Notes in Computer Science(), vol 10767. Springer, Cham. https://doi.org/10.1007/978-3-030-01713-2_14
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