Abstract
Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving \(m+1\) rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality (\(R\infty AR\)), might not be satisfied by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy \(R\infty AR\). One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen’s technique of forcing.
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Notes
A strategy \(s_i\) strictly dominates another \(s^{\prime }_i\) if \(s_i\) yields player i a strictly better payoff than \(s^{\prime }_i\), independently of what all the other players decide. In turn, an action \(s_i\) weakly dominates \(s^{\prime }_i\) if i’s payoff with \(s_i\) is at least great as her payoff of choosing \(s^{\prime }_i\), no matter what the other players choose, while for some of the choices of the others it is strictly larger. These relations are defined for pure strategies, but weak dominance can be extended to cases in which a mixed strategy dominates a given pure strategy.
Continuity of the definition of types has no counterpart here since no topology is defined over the class of strategy-type pairs.
Abusing language, we will also allow the arguments of \(\pi _i\) to be mixed strategies, i.e. probability distributions over either \(S_i\) or \(\prod _{j \ne i}S_j\).
This is a weaker version of the concept of assumption in Lexicographic Probability Systems, namely that \(E_{-i}\) is believed only when \((S_{-i} \times T_{-i}) \setminus E_{-i}\) has been discarded. Another way of viewing this is noticing that all that player i knows, according to type \(t_i\) at the k-th level of attention is \(F^k\), belonging to the sequence \(P_i[t_i]\).
In what follows, \([i \ \text{ is } \text{ rational }]_{|S_i \times T_i}\) is the projection of the event \([i \ \text{ is } \text{ rational }]\) on \(S_i \times T_i\).
Notice the similarities between this notion and event-rationality (Barelli and Galanis 2013).
Recall that the range of each \(F^k\) in \(P_{i}[t_i]\) is the class of actions and types of the other players, \(\prod _{j \ne i} S_j \times T_j\). We denote with \(s_{-ij}\) (\(t_{-ij}\)) an element in \(\prod _{k \ne i, k \ne j} S_k\) (\(\prod _{k \ne i, k \ne j} T_k\)).
The definition of \(\curlyeqprec \) (not to be confused with the preference relation \(\preceq \)) depends on the intended application. Here, as in the problem of the convergence of the core in a market economy with infinite agents, we do not need a condition of incompatibility (Lewis 1990).
See Dirk Bergemann and Joan Feigenbaum’s course notes in https://zoo.cs.yale.edu/classes/cs455/fall08/2008/gt2.pdf.
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Acknowledgements
Tohmé was supported by the National Research Council of Argentina Grant PIP 112-200801-00804, Universidad Nacional del Sur’s PGI 24/E115 and by the EU H2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 690974 for the project MIREL: MIning and REasoning with Legal texts. Endicott College provided extra funding during the preparation of this paper. We are grateful to Adam Brandenburger, Andrés Perea, Esteban Peralta, Marciano Siniscalchi, Joel Sobel and an anonymous referee for their useful comments and criticisms. Previous versions of this paper can be found online (Tohmé et al. 2012).
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Tohmé, F., Caterina, G. & Gangle, J. Iterated Admissibility Through Forcing in Strategic Belief Models. J of Log Lang and Inf 29, 491–509 (2020). https://doi.org/10.1007/s10849-020-09317-4
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DOI: https://doi.org/10.1007/s10849-020-09317-4