Abstract
In this paper I scrutinize the “rational beliefs” in the concept of rationalizability in strategic games [Bernheim (Econometrica 52:1007–1028, 1984), Pearce (Econometrica 52:1029–1051, 1984)]. I illustrate through an example that a rationalizable strategy may not be supported by a “rational belief”, at least under one plausible interpretation of “rational belief”. I offer an alternative formulation of “rational belief” in the concept of rationalizability, which yields a novel epistemic interpretation of the notion of point-rationalizability.
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Notes
In an interesting paper, Kline (2013) demonstrated how to evaluate the famous Muddy Children Puzzle by using “rational” beliefs.
The notion of rationalizability captures the strategic implications of the assumption of “common knowledge of rationality” [see Tan and Werlang (1988)], which is different from the assumption of “commonality of beliefs” or “correct conjectures” in an equilibrium [see Aumann and Brandenburger (1995)].
The set of player i’s mixed strategies is a probability space on a finite set of pure strategies, which can be viewed as a simplex in a finite-dimensional Euclidean space.
Believing the truth is, certainly, intrinsically valuable. We intrinsically want to have true beliefs. Also, our original interest in the truth has an instrumental basis too. Generally speaking, believing the truth serves us better than falsehoods and better than no beliefs at all in coping with adverse uncertainties. To be sure, maximizing expected utility is not the epistemic goal for the rationality of belief. To achieve Bayesian rationality in a strategic context, we simply want the players’ beliefs to be epistemically rational. See Foley (1987), Moser et al. (1998), and Nozick (1993) for extensive discussions.
It is worth noting that, in these cases, this sort of “rational beliefs” also gives rise to some doubt about the appropriateness of the formalism of “rational belief”. According to BP-formula, never is there such a unique “rational belief” unless the state space is a singleton.
A strategy profile x is point-rationalizable if there is a subset Y of \(\ X\) such that (i) \(x\in Y\) and (ii) \(\forall i\in N\), \(\forall y_{i}\in Y_{i}\), \(\exists y_{-i}\in Y_{-i}\) such that \(y_{i}\in \mathop {\mathrm{argmax}}\nolimits _{z_{i}\in X_{i}}U_{i}\left( z_{i},y_{-i}\right) \). Let \(R^{*}\) be the set of all the point-rationalizable strategy profiles. Then, \(R^{*}\) is a largest (w.r.t. set inclusion) set satisfying (ii) and, moreover, \(R^{*}\) can be derived by the iterated elimination of never-best response with respect to “point-beliefs” [see Bernheim (1984, Proposition 3.1)].
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I am grateful to Yossi Greenberg for his encouragement and valuable comments. Financial support from National University of Singapore is gratefully acknowledged.
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Luo, X. Rational beliefs in rationalizability. Theory Decis 81, 189–198 (2016). https://doi.org/10.1007/s11238-015-9528-6
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DOI: https://doi.org/10.1007/s11238-015-9528-6