Abstract
We extend significantly a result in Heifetz and Kets (Theor Econ 18:415–465, 2018) for Carlsson and van Damme’s (Econometrica 61:989–1018, 1993) global games by which even slight deviations from common belief in infinite depth of reasoning restore the robustness of rationalizable actions multiplicity, in contrast with the intriguing findings of Weinstein and Yildiz (Econometrica 75:365–400, 2007) under an idealized lack of such deviations. Here we show that multiplicity of rationalizable actions is a robust phenomenon even if finite depth of reasoning is an ‘extremely remote rumor’, where someone suspects that someone suspects (...) that somebody might have a finite depth of reasoning, and where the dots range over a transfinite range.
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Notes
Allowed perturbations are according to the topology generated by the minimal ‘vocabulary’ expressing mutual beliefs, namely the topology of weak convergence, generated by the open sets pertaining to states of nature and the events of the form ‘player i assigns probability larger than p to the open set ...’. The result does not obtain with topologies that restrict a priori the perturbations according to the behavior implied by them, e.g. the strategic topology or the uniform strategic topology (Dekel et al. 2006; Chen et al. 2010, 2017).
For example, Bosch Domenech et al. (2002, p. 1698) cite the following participants in a large-scale beauty-contest game:
E#3237: If everybody would choose 100, the maximum number that could be chosen is 66.6. Therefore, theoretically nobody will send a number over 66.6 and, if you multiply this by 2/3 we get 44.4. Therefore, in theory, nobody should be sending either a number over 44.4. Following this reasoning process the only number that should be sent is 1. However, I understand that many different people participate in this game and not everybody will apply the reasoning process explained above. Therefore, and taking into account that the majority of people would go all the way up to 1, I choose 6.8.
E#1811: I choose the number 15.93. The reasoning is the following:
I assume 10% do not have a clue and pick the mean 50
20% give a naive answer: 33 = \(50\times 2/3\)
50% go a second round: 22 = \(33\times 2/3\)
5% go a third round: 14 = \(22\times 2/3\)
5% are really devious and choose 10 = \(14\times 2/3\)
10% are crazy mathematicians who choose 1.
This is the well-foundedness property of sets, and of ordinals in particular.
A belief over a Hausdorff space X endowed with the Borel \(\sigma \)-field \({\mathcal {B}}\left( X\right) \) is a regular Borel probability measure, and the space of these beliefs \(\Delta \left( X\right) \) is endowed with the topology of weak convergence.
See Heifetz (1993) for the Mertens–Zamir construction using the Kolmogorov limit extension in case the base space is not necessarily compact.
On this choice of topology see Heifetz and Kets (2018), p. 418–419.
See e.g. Jech (2003), p. 25.
We employ the convention that for \(m=1\) the sequence \(\gamma _{1},\dots ,\gamma _{m-1}\) is empty, i.e. \(\omega ^{\gamma _{1}}+\cdots +\omega ^{\gamma _{m-1}}=0\).
See e.g. Takeuti (1987), definition 12.19, p. 120.
In Heifetz and Kets (2018) we defined a variant of such types to be types with almost common belief in infinite depth—types that are certain (i.e. with probability 1 and not just close to 1) that the other player is certain that [...n steps...] that the other player has finite depth of reasoning (though with no extra requirement that this finite depth is very likely to be large, as we require here).
However, since there are uncountably many countable limit ordinals, fixing a fundamental sequence increasing to each of them would have required an appeal to the axiom of choice; moreover, by definition increasing sequences to uncountable limit ordinals would be uncountable themselves. With these caveats, extensions of our results even further to these transfinite domains are conceivable, but we will not pursue them here.
For \(k=0,\) the range \(\ell =1,\dots k\) in the definitions below is to be understood as empty.
Notice that when restricted to the set of types in (2) , whose beliefs \(\mu _{i,\sigma }^{1}\left( x_{i}\right) \) on \(\Theta \) determine also their beliefs on the other player’s signal \(X_{-i} \), the \(\sigma \)-algebra \({\mathcal {N}}_{-i}\) is identical to the \(\sigma \)-algebra \({\mathcal {X}}_{-i}\) generated by player \(-i\)’s signals, i.e. by the events
$$\begin{aligned}&\left\{ h_{i}\in H_{i}:\psi _{i}\left( h_{i}\right) \left( \Theta \times \left\{ \left( x_{i},{\tilde{\mu }}_{-i}^{0},\left( \left( \mu _{-i,\sigma }^{\ell }\left( x_{-i}\right) \cdot \widetilde{\mathring{\mu } }_{-i}^{\ell }\right) \right. \right. \right. \right. \right. \\&\quad \times \left. \left. \left. \left. \left. \left( \mathrm {id}_{\Theta }\times {\tilde{g}}_{i,\sigma }^{\le \ell -1}\right) ^{-1}\right) _{\ell =1,2,\dots }\right) \in H_{-i}:x_{-i}\in E_{-i}\right\} \right) \ge p\right\} \end{aligned}$$for \(E_{-i}\in {\mathcal {B}}\left( X_{-i}\right) ={\mathcal {B}}\left( {\mathbb {R}} \right) \) and \(p\in \left[ 0,1\right] \).
That is, \(\delta _{i}\left( h_{i}^{\prime }\right) \in \mathring{H}_{i}^{\eta ,\alpha }.\)
More precisely: given \(\Theta =X_{1}=X_{2}={\mathbb {R}}\), for \(i=1,2\) one can define by induction the mapping
$$\begin{aligned} \rho _{i}=\left( \rho _{i}^{\infty },\rho _{i}^{0},\rho _{i}^{1},\ldots \right) :H_{i}=H_{i}^{\infty }\cup \bigcup _{k=0}^{\infty }H_{i}^{k}\rightarrow Z_{i} \cup \bigcup _{k=0}^{\infty }Z_{i}^{k} \end{aligned}$$from the HK universal space (Sect. 2.4) to the classical Mertens–Zamir universal space \(Z_{i}\) (Sect. 2.1) and its spaces of hierarchies \(Z_{i} ^{k}\) up to level k, such that \(\rho \) preserves mutual beliefs about \(\Theta \) and \(X_{1},X_{2}\) (but ‘abstracts’ from the information encoded in \(H_{i}\) on mutual beliefs about depth of reasoning). Then the types in \(H_{i} \) with belief hierarchies as in \(G_{i,\sigma }^{k}\) for \(k=0,1,\ldots ,\infty \), i.e. as in the \(\sigma \)-global game (Sect. 2.3), are
$$\begin{aligned} \rho _{i}^{-1}\left( G_{i,\sigma }\cup \bigcup _{k=0}^{\infty }G_{i,\sigma } ^{k}\right) \end{aligned}$$
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I am grateful and indebted to Willemien Kets for numerous discussions of this project, and to two anonymous referees.
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Heifetz, A. Robust multiplicity with (transfinitely) vanishing naiveté. Int J Game Theory 48, 1277–1296 (2019). https://doi.org/10.1007/s00182-019-00670-8
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DOI: https://doi.org/10.1007/s00182-019-00670-8