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Characterizing Assumption of Rationality by Incomplete Information
Assumption of rationality is a concept in epistemic game theory introduced by Brandenburger et al. bfk08 and studied in Perea p12 by using lexicographic belief. A lexicographic belief is said to assume the opponents' rationality means that a \textquotedblleft good\textquotedblright\ choice always occurs in front of a \textquotedblleft bad\textquotedblright\ one. Here by good we mean a choice of the opponent can be supported by a cautious belief of him, that is, a belief that does not exclude any choice of the opponents; by bad we mean it cannot be supported by any such belief.
Like other concepts in epistemic game theory such as permissibility (Brandenburger b92) and proper rationalizability (Schuhmacher sc99, Ascheim a01), iterative admissibility is defined partly to alleviate the tension between caution and rationality (Blume et al. bbd91a, Brandenburger b92, B\"{o}rgers bo94, Samuelson s92, B\"{o}rgers and Samuelson bs94) by sacrificing rationality. Caution requires that every choice, be it rational or not, should appear in a belief; assumption of rationality only requires that those rational choices should occur in front of those irrational ones but cannot exclude the irrational ones. On the other hand, since rationality is a basic assumption on human behavior in game theory, it seems desirable to find an approach to have a \textquotedblleft complete\textquotedblright\ rationality while to keep the definition of iterative admissibility.
One approach is to use an incomplete information framework introduced by Perea and Roy ps17 which defined standard probabilistic epistemic model with incomplete information and used it to characterized $\varepsilon $ -proper rationalizability. Following their approach, Liu l17 defined lexicographic epistemic models with incomplete information, constructed a mapping between them and models with complete information, and characterized permissibility and proper rationalizability. In this paper, we still use the construction in Liu l17 and characterize assumption of rationality. We show that a choice is optimal for a belief hierarchy which expresses common assumption of rationality within a complete information framework if and only if it is optimal for a belief hierarchy within the corresponding incomplete information framework that expresses common full belief in caution, rationality, every good choice is supported, and prior belief in the original utility functions.
This paper is organized as follows. Section (ref) gives a survey of assumption of rationality in epistemic models with complete information and the lexicographic epistemic models with incomplete information. Section (ref) gives the characterization result and their proofs. Section (ref) gives some concluding remarks on the relationship between the result of this paper and characterization of permissbility in Section 4.6 of Liu l17.
In this subsection, we give a survey of lexicographic epistemic model with complete information and define iterative admissibility within it. We adopt the approach of Perea p12, Chapters 5 and 7. See Brandenburger et al. bfk08 for an alternative approach.
Consider a finite 2-person static game $\Gamma =(C_{i},u_{i})_{i\in I}$ where $I=\{1,2\}$ is the set of players, $C_{i}$ is the finite set of choices and $u_{i}:C_{1}\times C_{2}\rightarrow \mathbb{R}$ is the utility function for player $i\in I.$ In the following we sometimes denote $ C_{1}\times C_{2}$ by $C$. We assume that each player has a lexicographic belief on the opponent's choices, a lexicographic belief on the opponent's lexicographic belief on her, and so on. This belief hierarchy is described by a lexicographic epistemic model with types. \newline Definition (ref).1 (Epistemic model with complete information). Consider a finite 2-person static game $\Gamma =(C_{i},u_{i})_{i\in I}$. A finite lexicographic epistemic model for $ \Gamma $ is a tuple $M^{co}=(T_{i},b_{i})_{i\in I}$ where \newline (a) $T_{i}$ is a finite set of types, and \newline (b) $b_{i}$ is a mapping that assigns to each $t_{i}\in T_{i}$ a lexicographic belief over $\Delta (C_{j}\times T_{j}),$ i.e., $ b_{i}(t_{i})=(b_{i1},b_{i2},...,b_{iK})$ where $b_{ik}\in \Delta (C_{j}\times T_{j})$ for $k=1,...,K.$
Consider $t_{i}\in T_{i}$ with $b_{i}(t_{i})=(b_{i1},b_{i2},...,b_{iK}).$ Each $b_{ik}$ ($k=1,...,K$) is called $t_{i}$'s level-$k$\ belief. For each $(c_{j},t_{j})\in C_{j}\times T_{j},$ we say $t_{i}$ deems $(c_{j},t_{j})$ possible iff $b_{ik}(c_{j},t_{j})>0$ for some $ k\in \{1,...,K\}.$ We say $t_{i}$ deems $t_{j}\in T_{j}$ possible iff $t_{i}$ deems $(c_{j},t_{j})$ possible for some $c_{j}\in C_{j} $. For each $t_{i}\in T_{i},$ we denote by $T_{j}(t_{i})$ the set of types in $T_{j}$ deemed possible by $t_{i}$. A type $t_{i}\in T_{i}$ is \emph{cautious} iff for each $c_{j}\in C_{j}$ and each $t_{j}\in T_{j}(t_{i}),$ $t_{i}$ deems $(c_{j},t_{j})$ possible. That is, $t_{i}$ takes into account each choice of player $j$ for every belief hierarchy of $j $ deemed possible by $t_{i}.$
For each $c_{i}\in C_{i}$, let $ u_{i}(c_{i},t_{i})=(u_{i}(c_{i},b_{i1}).,..,u_{i}(c_{i},b_{iK}))$ where for each $k=1,...,K,$ $u_{i}(c_{i},b_{ik}):=\Sigma _{(c_{j},t_{j})\in C_{j}\times T_{j}}b_{ik}(c_{j},t_{j})u_{i}(c_{i},c_{j}),$ that is, each $ u_{i}(c_{i},b_{ik})$ is the expected utility for $c_{i}$ over $b_{ik}$ and $ u_{i}(c_{i},t_{i})$ is a vector of expected utilities. For each $ c_{i},c_{i}^{\prime }\in C_{i}$, we say that $t_{i}$ prefers $c_{i}$ to $c_{i}^{\prime }$, denoted by $u_{i}(c_{i},t_{i})>u_{i}(c_{i}^{ \prime },t_{i}),$ iff there is $k\in \{0,...,K-1\}$ such that the following two conditions are satisfied: \newline (a) $u_{i}(c_{i},b_{i\ell })=u_{i}(c_{i}^{\prime },b_{i\ell })$ for $\ell =0,...,k,$ and \newline (b) $u_{i}(c_{i},b_{i,k+1})>u_{i}(c_{i}^{\prime },b_{i,k+1})$. \newline We say that $t_{i}$ is indifferent between $c_{i}$ and $ c_{i}^{\prime },$ denoted by $u_{i}(c_{i},t_{i})=u_{i}(c_{i}^{\prime },t_{i}),$ iff $u_{i}(c_{i},b_{ik})=u_{i}(c_{i}^{\prime },b_{ik})$ for each $ k=1,...,K.$ It can be seen that the preference relation on $C_{i}$ under each type $t_{i}$ is a linear order. $c_{i}$ is rational (or optimal) for $t_{i}$ iff $t_{i}$ does not prefer any choice to $c_{i}$.
For $(c_{j},t_{j}),(c_{j}^{\prime },t_{j}^{\prime })\in C_{j}\times T_{j},$ we say that $t_{i}$ deems $(c_{j},t_{j})$ infinitely more likely than $(c_{j}^{\prime },t_{j}^{\prime })$ iff there is $k\in \{0,...,K-1\}$ such that the following two conditions are satisfied: \newline (a) $b_{i\ell }(c_{j},t_{j})=b_{i\ell }(c_{j}^{\prime },t_{j}^{\prime })=0$ for $\ell =1,...,k,$ and \newline (b) $b_{i,k+1}(c_{j},t_{j})>0$ and $b_{i,k+1}(c_{j}^{\prime },t_{j}^{\prime })=0$. \newline Definition (ref).2 (Assumption of rationality) A cautious type\ $t_{i}\in T_{i}$ assumes the $j$'s rationality iff the following two conditions are satisfied: \newline \textbf{(A1)} for all of player $j$'s choices $c_{j}$ that are optimal for some cautious belief, $t_{i}$ deems possible some type $t_{j}$ for which $ c_{j}$ is optimal; \newline \textbf{(A2)} $t_{i}$ deems all choice-type pairs $(c_{j},t_{j})$ where $ t_{j}$ is cautious and $c_{j}$ is optimal for $t_{j}$ infinitely more likely than any choice-type pairs $(c_{j}^{\prime },t_{j}^{\prime })$ that does not have this property.
Informally speaking, assumption of the opponent's rationality is that $t_{i}$ puts all \textquotedblleft good\textquotedblright\ choices in front of those \textquotedblleft bad\textquotedblright\ choices.The following definition extends assumption of rationality inductively into $n$-fold for any $n\in \mathbb{N}.$ \newline Definition (ref).3 ($n$-fold assumption of rationality) Consider a finite lexicographic epistemic model $ M^{co}=(T_{i},b_{i})_{i\in I}$ for a game $\Gamma =(C_{i},u_{i})_{i\in I}$. A cautious type $t_{i}\in T_{i}$ expresses $1$-fold assumption of rationality iff it assumes $j$'s rationality. For any $n\in \mathbb{N},$ we say that a cautious type $t_{i}\in T_{i}$ expresses $(n+1)$-fold assumption of rationality iff the following two conditions are satisfied: \newline \textbf{(nA1)} whenever a choice $c_{j}$ of player $j$ is optimal for some cautious type (not necessarily in $M^{co}$) that expresses up to $n$-fold assumption of rationality, $t_{i}$ deems possible some cautious type $t_{j}$ for player $j$ which expresses up to $n$-fold assumption of rationality for which $c_{j}$ is optimal; \newline \textbf{(nA2)} $t_{i}$ deems all choice-type pair $(c_{j},t_{j})$, where $ t_{j}$ is cautious and expresses up to $n$-fold assumption of rationality and $c_{j}$ is optimal for $t_{j},$ infinitely more likely than any choice-type pairs $(c_{j}^{\prime },t_{j}^{\prime })$ that does not satisfy this property.
We say that $t_{i}$ expresses common assumption of rationality iff it expresses $n$-fold assumption of rationality for every $n\in \mathbb{N}.$
In this subsection, we give a survey of lexicographic epistemic model with incomplete information defined in Liu l17 which is the counterpart of the probabilistic epistemic model with incomplete information introduced by Battigalli b03 and further developed in Battigalli and Siniscalchi bs03, bs07, and Dekel and Siniscalchi ds15. We also define some conditions on types in such a model. \newline Definition (ref).4 (Lexicographic epistemic model with incomplete information). Consider a finite 2-person static game form $ G=(C_{i})_{i\in I}.$ For each $i\in I,$ let $V_{i}$ be the set of utility functions $v_{i}:C_{1}\times C_{2}\rightarrow \mathbb{R}.$ A finite lexicographic epistemic model for $G$\ with incomplete information is a tuple $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ where \newline (a) $\Theta _{i}$ is a finite set of types, \newline (b) $w_{i}$ is a mapping that assigns to each $\theta _{i}\in \Theta _{i}$ a utility function $w_{i}(\theta _{i})\in V_{i},$ and \newline (c) $\beta _{i}$ is a mapping that assigns to each $\theta _{i}\in \Theta _{i}$ a lexicographic belief over $\Delta (C_{j}\times \Theta _{j}),$ i.e., $ \beta _{i}(\theta _{i})=(\beta _{i1},\beta _{i2},...,\beta _{iK})$ where $ \beta _{ik}\in \Delta (C_{j}\times \Theta _{j})$ for $k=1,...,K.$
Concepts such as \textquotedblleft $\theta _{i}$ deems $(c_{j},\theta _{j})$ possible\textquotedblright\ and \textquotedblleft $\theta _{i}$ deems $ (c_{j},\theta _{j})$ infinitely more likely than $(c_{j}^{\prime },\theta _{j}^{\prime })$\textquotedblright\ can be defined in a similar way as in Section (ref).1. For each $\theta _{i}\in \Theta _{i},$ we use $ \Theta _{j}(\theta _{i})$ to denote the set of types in $\Theta _{j}$ deemed possible by $\theta _{i}$. For each $\theta _{i}\in \Theta _{i}$ and \thinspace $v_{i}\in V_{i},$ $\theta _{i}^{v_{i}}$ is the auxiliary type satisfying that $\beta _{i}(\theta _{i}^{v_{i}})=\beta _{i}(\theta _{i})$ and $w_{i}(\theta _{i}^{v_{i}})=v_{i}$.
For each $c_{i}\in C_{i},v_{i}\in V_{i},$ and $\theta _{i}\in \Theta _{i}$ with $\beta _{i}(\theta _{i})=(\beta _{i1},\beta _{i2},...,\beta _{iK}),$ let $v_{i}(c_{i},\theta _{i})=(v_{i}(c_{i},\beta _{i1}),...,v_{i}(c_{i},\beta _{iK}))$ where for each $k=1,...,K,$ $ v_{i}(c_{i},\beta _{ik}):=\Sigma _{(c_{j},\theta _{j})\in C_{j}\times \Theta _{j}}\beta _{ik}(c_{j},\theta _{j})v_{i}(c_{i},c_{j}).$ For each $ c_{i},c_{i}^{\prime }\in C_{i}$ and $\theta _{i}\in \Theta _{i},$ we say that $\theta _{i}$ prefers $c_{i}$ to $c_{i}^{\prime }$ iff $ w_{i}(\theta _{i})(c_{i},\theta _{i})>w_{i}(\theta _{i})(c_{i}^{\prime },\theta _{i}).$ As in Section (ref).1, this is also the lexicographic comparison between two vectors. $c_{i}$ is rational (or optimal) for $\theta _{i}$ iff $\theta _{i}$ does not prefer any choice to $c_{i}.$ \newline Definition (ref).5 (Caution). $\theta _{i}\in \Theta _{i}$ is cautious iff for each $c_{j}\in C_{j}$ and each $\theta _{j}\in \Theta _{j}(\theta _{i})$, there is some utility function $v_{j}\in V_{j}$ such that $\theta _{i}$ deems $(c_{j},\theta _{j}^{v_{j}})$ possible.
This is a faithful translation of Perea and Roy ps17's definition of caution in probabilistic model (p.312) into lexicographic model. It is the counterpart of caution defined within the complete information framework in Section (ref).1; the only difference is that in incomplete information models we allow different utility functions since $c_{j}$ will be required to be rational for the paired type. \newline Definition (ref).6 (Belief in rationality). $\theta _{i}\in \Theta _{i}$ believes in $j$'s rationality iff $\theta _{i}$ deems $(c_{j},\theta _{j})$ possible only if $c_{j}$ is rational for $\theta _{j}.$
The following lemma shows that caution and a belief of full rationality can be satisfied simultaneously in an incomplete information model because each type is assigned with a belief on the opponent's choice-type pairs as well as a payoff function. The consistency of caution and full rationality is the essential difference between models with incomplete information and those with complete information. \newline Lemma (ref).1 (Belief in rationality can be satisfied). Consider a static game form $G=(C_{i})_{i\in I}$, $C_{i}^{\prime }\in C_{i}$ , and $\beta _{i}=(\beta _{i1},\beta _{i2},...,\beta _{iK})$ such that $ \beta _{ik}\in \Delta (C_{j})$ for each $k=1,...,K.$ Then there is $v_{i}\in V_{i}$ such that each $c_{i}\in C_{i}^{\prime }$ is optimal in $v_{i}$ for $ \beta _{i}.$ \newline Proof. There are various way to construct such a $v_{i}.$ Here we provide a simple one. For each $c\in C,$ let
It can be seen that each $c_{i}\in C_{i}^{\prime }$ is optimal in $v_{i}$ for $\beta _{i}.$ //
Caution and belief in rationality can be extended into $k$-fold for any $ k\in \mathbb{N}$ as follows. Let $P$ be an arbitrary property of lexicographic beliefs. We define that \newline (CP1) $\theta _{i}\in \Theta $ expresses $0$-fold full belief in $P$ iff $\theta _{i}$ satisfies $P;$ \newline (CP2) For each $n\in \mathbb{N}$ with $n\geq 2,$ $\theta _{i}\in T_{i}$ expresses $n$-fold full belief in $P$ iff $\theta _{i}$ only deems possible $j$'s types that express $n$-fold full belief in $P.$
$\theta _{i}$ expresses common full belief in $P$ iff it expresses $n$ -fold full belief in $P$ for each $n\in \mathbb{N}.$ By replacing $P$ with \textquotedblleft caution\textquotedblright\ or \textquotedblleft rationality\textquotedblright\ we can obtain common full belief in caution or in rationality.
The following two conditions are important in characterizing assumption of rationality. \newline Definition (ref).7 (Every good choice is supported). Consider a static game form $G=(C_{i})_{i\in I},$ a\ lexicographic epistemic model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ for\ $G$\ with incomplete information, and a pair $u=(u_{i})_{i\in I}$ of utility functions. A cautious type $\theta _{i}\in \Theta _{i}$ believes in that every good choice of $j$ is supported iff for each $ c_{j}$ that is optimal for some cautious type of $j$ (may not be in $M^{in}$ ) with $u_{j}$ as its assigned utility function, $\theta _{i}$ deems possible a cautious type $\theta _{j}\in \Theta _{j}$ such that $ w_{j}(\theta _{j})=u_{j}$ and $c_{j}$ is optimal for $\theta _{j}.$ \newline \textbf{Definition (ref).8 (Prior belief in }$u$\textbf{)}. Consider a static game form $G=(C_{i})_{i\in I},$ a\emph{\ }lexicographic epistemic model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ for\emph{\ }$G$\emph{ \ }with incomplete information, and a pair $u=(u_{i})_{i\in I}$ of utility functions. $\theta _{i}\in \Theta _{i}$ \emph{priorly believes in }$u$ iff for any $(c_{j},\theta _{j})$ with $\theta _{j}$ cautious deemed possible by $\theta _{i}$ satisfying that $w_{j}(\theta _{j})=u_{j},$ then $\theta _{i}$ deems $(c_{j},\theta _{j})$ infinitely more likely than any pair does not satisfy that property.
Common full belief in that every good choice is supported and prior belief in $u$ is different from that in caution or rationality. We have the following definition. \newline Definition (ref).9 ($n$-fold belief in that every good choice is supported and prior belief in $u$) Consider a static game form $G=(C_{i})_{i\in I},$ a\ lexicographic epistemic model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ for\ $G$\emph{\ }with incomplete information, and a pair $u=(u_{i})_{i\in I}$ of utility functions. $\theta _{i}\in \Theta _{i}$ \emph{express }$1$\emph{ -fold belief in that every good choice is supported and prior belief in }$u$ \emph{\ }iff it believes that every good choice of $j$ is supported and has prior belief in $u$. For any $n\in \mathbb{N},$ we say that a cautious type $ \theta _{i}\in \Theta _{i}$ expresses $(n+1)$\emph{-fold belief in prior belief in that every good choice is supported and prior belief in }$u$ iff the following two conditions are satisfied: \newline \textbf{(nP1)} whenever a choice $c_{j}$ of player $j$ is optimal for some cautious type (not necessarily in $M^{in}$) with $u_{j}$ as its assigned utility function that expresses up to $n$-fold belief in that every good choice is supported, $\theta _{i}$ deems possible some cautious type $\theta _{j}$ with $w_{j}(\theta _{j})=u_{j}$ for player $j$ which expresses up to $n $-fold belief in that every good choice is supported and prior belief in $u$ for which $c_{j}$ is optimal. \newline \textbf{(nP2)} $\theta _{i}$ deems all choice-type pair $(c_{j},\theta _{j})$ , where $\theta _{j}$ is cautious and expresses up to $n$-fold belief in that prior belief in $u$ and every good choice is supported and satisfies $ w_{j}(\theta _{j})=u_{j},$ infinitely more likely than any choice-type pairs $(c_{j}^{\prime },\theta _{j}^{\prime })$ that does not satisfy this property.
We say that $t_{i}$ expresses common full belief in that every good choice is supported and prior belief in $u$ iff it expresses $n$-fold belief in that every good choice is supported and prior belief in $u$ for every $n\in \mathbb{N}.$
So far we have introduced two different groups of concepts for static games: one includes assumption of rationality within a complete information framework, the other contains some conditions on types within an incomplete information framework. In this section we will show that there is correspondence between them. \newline Theorem (ref).1 (Characterization of iterative admissibility) . Consider a finite 2-person static game $\Gamma =(C_{i},u_{i})_{i\in I}$ and the corresponding game form $G=(C_{i}).$ $c_{i}^{\ast }\in C_{i}$ is optimal to some type expressing common full belief in caution and common assumption of rationality within some finite epistemic model with complete information if and only if there is some finite epistemic model $ M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ with incomplete information for $G$ and some $\theta _{i}^{\ast }\in \Theta _{i}$ with $w_{i}(\theta _{i}^{\ast })=u_{i}$ such that \newline (a) $c_{i}^{\ast }$ is rational for $\theta _{i}^{\ast }$, and \newline (b) $\theta _{i}^{\ast }$ expresses common full belief in caution, rationality, that every good choice is supported, and prior belief in $u$ .
To show Theorem (ref).1, we construct the mappings between finite lexicographic epistemic models with complete information and those with incomplete information. First, consider $\Gamma =(C_{i},u_{i})_{i\in I}$ and a finite lexicographic epistemic model $M^{co}=(T_{i},b_{i})_{i\in I}$ with complete information for $\Gamma .$ We first define types in a model with incomplete information in the following two steps: \newline Step 1. For each $i\in I$ and $t_{i}\in T_{i},$ let $\Pi _{i}(t_{i})=(C_{i1},...,C_{iL})$ be the partition of $C_{i}$ defined in Lemma (ref).1, that is, $\Pi _{i}(t_{i})$ is the sequence of equivalence classes of choices in $C_{i}$ arranged from the most preferred to the least preferred under $t_{i}.$ We define $v_{i\ell }(t_{i})\in V_{i}$ for each $\ell =1,...,L.$ We let $v_{i1}(t_{i})=u_{i}.$ By Lemma (ref).1, for each $C_{i\ell }$ with $\ell >1$ there is some $v_{i\ell }(t_{i})\in V_{i}$ such that each choice in $C_{i\ell }$ is rational at $ v_{i\ell }(t_{i})$ under $t_{i}.$ \newline Step 2. We define $\Theta _{i}(t_{i})=\{\theta _{i1}(t_{i}),...,\theta _{iL}(t_{i})\}$ where for each $\ell =1,...,L,$ the type $\theta _{i\ell }(t_{i})$ satisfies that (1) $w_{i}(\theta _{i\ell }(t_{i}))=v_{i\ell }(t_{i}),$ and (2) $\beta _{i}(\theta _{i\ell }(t_{i}))$ is obtained from $b_{i}(t_{i})$ by replacing every $(c_{j},t_{j})$ with $ c_{j}\in C_{jr}\in \Pi _{j}(t_{j})$ for some $r$ with $(c_{j},\theta _{j})$ where $\theta _{j}=\theta _{jr}(t_{j}),$ that is, $w_{j}(\theta _{j})$ is the utility function among those corresponding to $\Pi _{j}(t_{j})$ in which $c_{j}$ is the rational for $t_{i}.$
For each $i\in I,$ let $\Theta _{i}=\cup _{t_{i}\in T_{i}}\Theta _{i}(t_{i}). $ Here we have constructed a finite lexicographic epistemic model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ for the corresponding game form $G=(C_{i})_{i\in I}$ with incomplete information. In the following example we show how this construction goes. \newline Example (ref).2. Consider the following game $\Gamma $ (Perea p12, p.188):
and the lexicographic epistemic model $M^{co}=(T_{i},b_{i})_{i\in I}$ $ \Gamma $ where $T_{1}=\{t_{1}\},$ $T_{2}=\{t_{2}\}$, and
We show how to construct a corresponding model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$. First, by Step 1 it can be seen that $\Pi _{1}(t_{1})=(\{A\},\{B\})$ and $\Pi _{2}(t_{2})=(\{D\},\{C\}).$ We let $ v_{11}(t_{1})=u_{1}$ where $A$ is rational for $t_{1}$ and $v_{12}(t_{1})$ where $B$ is rational for $t_{1}$ as follows. Similarly, we let $ v_{21}(t_{2})=u_{2}$ where $D$ is rational under $t_{2}$ and $v_{22}(t_{2})$ where $C$ is rational under $t_{2}$ as follows:
Then we go to Step 2. It can be seen that $\Theta _{1}(t_{1})=\{\theta _{11}(t_{1}),\theta _{12}(t_{1})\},$ where
Also, $\Theta _{2}(t_{2})=\{\theta _{21}(t_{2}),\theta _{22}(t_{2})\},$ where
Let $M^{co}=(T_{i},b_{i})_{i\in I}$ and $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ be constructed from $M^{co}$ by the two steps above. We have the following observations. \newline Observation (ref).1 (Redundancy). For each $t_{i}\in T_{i}$ and each $\theta _{i},\theta _{i}^{\prime }\in \Theta _{i}(t_{i}),$ $\beta _{i}(\theta _{i})=\beta _{i}(\theta _{i}^{\prime }).$ \newline Observation (ref).2 (Rationality). Eeach $\theta _{i}\in \Theta _{i}(t_{i})$ believes in $j$'s rationality.
We omit their proofs since they hold by construction. Observation (ref).1 means that the difference between any two types in a $\Theta _{i}(t_{i})$ is in the utility functions assigned to them. Observation (ref).2 means that in an incomplete information model constructed from one with complete information, each type has (full) belief in the opponent's rationality. This is because in the construction, we requires that for each pair $(c_{j},t_{j})$ occurring in a belief, its counterpart in the incomplete information replaces $t_{j}$ by the type in $\Theta _{j}(t_{j})$ with the utility function in which $c_{j}$ is optimal for $b_{i}(t_{j})$. It follows from Observation (ref).2 that each $\theta _{i}\in \Theta _{i}(t_{i})$ expresses common full belief in rationality.
The following lemma shows that caution is preserved in this construction. \newline Lemma (ref).1 (Caution$^{co}\rightarrow $\ Caution$ ^{in}$). Let $M^{co}=(T_{i},b_{i})_{i\in I}$ and $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ be constructed from $M^{co}$ by the two steps above. If $t_{i}\in T_{i}$ expresses common full belief in caution, so does each $\theta _{i}\in \Theta _{i}(t_{i}).$ \newline Proof. We show this statement by induction. First we show that if $ t_{i}$ is cautious, then each $\theta _{i}\in \Theta _{i}(t_{i})$ is also cautious. Let $c_{j}\in C_{j}$ and $\theta _{j}\in \Theta _{j}(\theta _{i}).$ By construction, it can be seen that the type $t_{j}\in T_{j}$ satisfying the condition that $\theta _{j}\in \Theta _{j}(t_{j})$ is in $T_{j}(t_{i}).$ Since $t_{i}$ is cautious, $t_{i}$ deems $(c_{j},t_{j})$ possible. Consider the pair $(c_{j},\theta _{j}^{\prime })$ in $\beta _{i}(\theta _{i})$ corresponding to $(c_{j},t_{j}).$ Since both $\theta _{j}$ and $\theta _{j}^{\prime }$ are in $\Theta _{j}(t_{j}),$ it follows from Observation (ref).1 that $\beta _{j}(\theta _{j})=\beta _{j}(\theta _{j}^{\prime }).$ Hence $(c_{j},\theta _{j}^{w_{j}(\theta _{j}^{\prime })})$ is deemed possible by $\theta _{i}.$ Here we have shown that $\theta _{i}$ is cautious.
Suppose we have shown that, for each $i\in I,$ if $t_{i}$ expresses $n$-fold full belief in caution then so does each $\theta _{i}\in \Theta _{i}(t_{i})$ . Now suppose that $t_{i}$ expresses $(n+1)$-fold full belief in caution, i.e., each $t_{j}\in T_{j}(t_{i})$ expresses $n$-fold full belief in caution. By construction, for each $\theta _{i}\in \Theta _{i}(t_{i})$ and each $\theta _{j}\in \Theta _{j}(\theta _{i})$ there is some $t_{j}\in T_{j}(t_{i})$ such that $\theta _{j}\in \Theta _{j}(t_{i})$, and, by inductive assumption, each $\theta _{j}\in \Theta _{j}(\theta _{i})$ expresses $n$-fold full belief in caution. Therefore, each $\theta _{i}\in \Theta _{i}(t_{i})$ expresses $(n+1)$-fold full belief in caution. //
We also need a mapping from epistemic models with incomplete information to those with complete information. Consider a finite 2-person static game $ \Gamma =(C_{i},u_{i})_{i\in I},$ the corresponding game form $ G=(C_{i})_{i\in I},$ and a finite epistemic model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ for $G$ with incomplete information. We construct a model $M^{co}=(T_{i},b_{i})_{i\in I}$ for $\Gamma $ with complete information as follows. For each $\theta _{i}\in \Theta _{i},$ we define $E_{i}(\theta _{i})=\{\theta _{i}^{\prime }\in \Theta _{i}:\beta _{i}(\theta _{i}^{\prime })=\beta (\theta _{i})\}.$ In this way $\Theta _{i}$ is partitioned into some equivalence classes $\mathbb{E}_{i}= \{E_{i1},...,E_{iL}\}$ where for each $\ell =1,..,L,$ $E_{i\ell }=E_{i}(\theta _{i})$ for some $\theta _{i}\in \Theta _{i}.$ To each $ E_{i}\in \mathbb{E}_{i}$ we use $t_{i}(E_{i})$ to represent a type. We define $b_{i}(t_{i}(E_{i}))$ to be a lexicographic belief which is obtained from $\beta _{i}(\theta _{i})$ by replacing each occurrence of $ (c_{j},\theta _{j})$ by $(c_{j},t_{j}(E_{j}(\theta _{j})));$ in other words, $b_{i}(t_{i}(E_{i}))$ has the same distribution on choices at each level as $ \beta _{i}(\theta _{i})$ for each $\theta _{i}\in E_{i},$ while each $\theta _{j}\in \Theta _{j}(\theta _{i})$ is replaced by $t_{j}(E_{j}(\theta _{j})).$ For each $i\in I,$ let $T_{i}=\{t_{i}(E_{i})\}_{E_{i}\in \mathbb{E}_{i}}.$ We have constructed from $M^{in}$ a finite epistemic model $ M^{co}=(T_{i},b_{i})_{i\in I}$ with complete information for $\Gamma .$
It can be seen that this is the reversion of the previous construction. That is, let $M^{co}=(T_{i},b_{i})_{i\in I}$ satisfying that $b_{i}(t_{i})\neq b_{i}(t_{i}^{\prime })$ for each $t_{i},t_{i}^{\prime }\in T_{i}$ with $ t_{i}\neq t_{i}^{\prime }$, and $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ be constructed from $M^{co}$ by the previous two steps. Then $\mathbb{E} _{i}=\{\Theta _{i}(t_{i})\}_{t_{i}\in T_{i}}$ and $t_{i}(\Theta _{i}(t_{i}))=t_{i}$ for each $i\in I.$
In the following example we show how this construction goes. \newline Example (ref).3. Consider the game $\Gamma $ in Example (ref).2 and the model $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ for the corresponding game form where $\Theta _{1}=\{\theta _{11},\theta _{12}\},$ $\Theta _{2}=\{\theta _{21},\theta _{22}\},$ and
where $v_{1}=v_{12}(t_{1})$ and $v_{2}=v_{22}(t_{2})$ in Example (ref).2. It can be seen that $\mathbb{E}_{1}=\{\{\theta _{11},\theta _{12}\}\}$ since $\beta _{1}(\theta _{11})=\beta _{1}(\theta _{12})$ and $ \mathbb{E}_{2}=\{\{\theta _{21},\theta _{22}\}\}$ since $\beta _{2}(\theta _{21})=\beta _{2}(\theta _{22}).$ Corresponding to those equivalence classes we have $t_{1}(\{\theta _{11},\theta _{12}\})$ and $t_{2}(\{\theta _{21},\theta _{22}\}),$ and
We have the following lemmas. \newline Lemma (ref).2 (Caution$^{in}\rightarrow $ \ Caution$^{co}$). Let $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ and $M^{co}=(T_{i},b_{i})_{i\in I}$ be constructed from $ M^{in}$ by the above approach. If $\theta _{i}\in \Theta _{i}$ expresses common full belief in caution, so does $t_{i}(E_{i}(\theta _{i})).$ \newline Proof. We show this statement by induction. First we show that if $ \theta _{i}$ is cautious, then $t_{i}(E_{i}(\theta _{i}))$ is also cautious. Let $c_{j}\in C_{j}$ and $t_{j}\in T_{j}(t_{i}(E_{i}(\theta _{i}))).$ By construction, $t_{j}=t_{j}(E_{j})$ for some $E_{j}\in \mathbb{E}_{j},$ and there is some $\theta _{j}\in E_{j}$ which is deemed possible by $\theta _{i}.$ Since $\theta _{i}$ is cautious, there is some $\theta _{j}^{\prime }$ with $\beta _{j}(\theta _{j}^{\prime })=\beta _{j}(\theta _{j}),$ i.e., $ \theta _{j}^{\prime }\in E_{j},$ such that $(c_{j},\theta _{j}^{\prime })$ is deemed possible by $\theta _{i}.$ By construction, $(c_{j},t_{j})$ is deemed possible by $t_{i}(E_{i}(\theta _{i})).$
Suppose we have shown that, for each $i\in I,$ if $\theta _{i}$ expresses $n$ -fold full belief in caution then so does $t_{i}(E_{i}(\theta _{i}))$. Now suppose that $\theta _{i}$ expresses $(n+1)$-fold full belief in caution, i.e., each $\theta _{j}\in \Theta _{j}(\theta _{i})$ expresses $n$-fold full belief in caution. Since, by construction, for each $t_{j}\in T_{j}(t_{i}(E_{i}(\theta _{i})))$, there is some $\theta _{j}\in \Theta _{j}(\theta _{i})$ such that $t_{j}=t_{j}(E_{j}(\theta _{j})),$ by inductive assumption $t_{j}$ expresses $n$-fold full belief in caution. Therefore, $ t_{i}(E_{i}(\theta _{i}))$ expresses $(n+1)$-fold full belief in caution. // \newline Lemma (ref).3 (Assumption of rationality $ \longleftrightarrow $\ every good choice is supportedprior\ + belief in $u$ ). Let $M^{co}=(T_{i},b_{i})_{i\in I}$ and $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ be constructed from $M^{co}$. If $t_{i}\in T_{i}$ expresses common assumption of rationality, then each $\theta _{i}\in \Theta _{i}(t_{i})$ expresses common full belief in that every good choice is supported and prior belief in $u$.
On the other hand, let $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ and $ M^{co}=(T_{i},b_{i})_{i\in I}$ be constructed from $M^{in}$. If $\theta _{i}\in \Theta _{i}$ expresses common full belief in that every good choice is supported and prior belief in $u$, then $t_{i}(E_{i}(\theta _{i}))$ expresses common full assumption of rationality. \newline Proof. We show this statement by induction. Let $\theta _{i}\in \Theta _{i}(t_{i}).$ First we show that if $t_{i}$ assumes in $j$'s rationality, $\theta _{i}$ believes that every good choice is supported and prior belief in $u$. Let $c_{j}\in C_{j}$ be optimal for some cautious type of $j$ whose assigned utility function is $u_{j}$ within an epistemic model with incomplete information. It is easy to see that $c_{j}$ is optimal for its corresponding type, which is also cautious by Lemma (ref).2, in any complete information model constructed from the one with incomplete information by our approach above. Since $t_{i}$ assumes $j$'s rationality, $ t_{i}$ deems possible a cautious type $t_{j}$ for which $c_{j}$ is optimal. By construction, some $\theta _{j}\in \Theta _{j}(t_{j})$ is deemed possible by $\theta _{i}.$ Since $t_{i}$ is cautious, $(c_{j},t_{j})$ is deemed possible by $t_{i},$ and, by construction $(c_{j},\theta _{j1}(t_{j}))$ is deemed possible by $\theta _{i}$. Since $w_{j}(\theta _{j1}(t_{j}))=u_{j}$ and $c_{j}$ is optimal for $\theta _{j1}(t_{j}),$ it follows that $\theta _{i}$ believes in that every good choice is supported.
Let $(c_{j},\theta _{j})$ with $\theta _{j}$ cautious deemed possible by $ \theta _{i}$ satisfying $w_{j}(\theta _{j})=u_{j}$ and $(c_{j}^{\prime },\theta _{j}^{\prime })$ a pair which does not satisfy that condition. Let $ (c_{j},t_{j})$ and $(c_{j}^{\prime },t_{j}^{\prime })$ be the pairs occuring in the belief of $t_{i}$ corresponding to $(c_{j},\theta _{j})$ and $ (c_{j}^{\prime },\theta _{j}^{\prime }).$ Since $c_{j}$ is rational to $ \theta _{j}$ and $w_{j}(\theta _{j})=u_{j},$ it follows that $c_{j}$ is optimal for $t_{j}.$ On the other hand, $c_{j}^{\prime }$ is not optimal for $t_{j}^{\prime }.$ Since $t_{i}$ assumes $j$'s rationality, $t_{i}$ deems $ (c_{j},t_{j})$ infinitely more likely than $(c_{j}^{\prime },t_{j}^{\prime }).$ By construction, $\theta _{i}$ deems $(c_{j},\theta _{j})$ infinitely more likely than $(c_{j}^{\prime },\theta _{j}^{\prime }).$ Here we have shown that $\theta _{i}$ priorly believes in $u.$
Now we show the other direction: suppose that if $\theta _{i}\in \Theta _{i}$ believes in that every good choice is supported and priorly believes in $u$, we prove that $t_{i}(E_{i}(\theta _{i}))$ assumes $j$'s rationality. Suppose that $c_{j}$ is optimal for some cautious type within some epistemic model with complete information. It can be seen by construction that $c_{j}$ is optimal for some cautious type with $u_{i}$ as its assigned utility function within some epistemic model with incomplete information which corresponds to that complete information model. Since $\theta _{i}$ believes in that every good choice is supported, $\theta _{i}$ deems possible a cautious type $ \theta _{j}$ such that $w_{j}(\theta _{j})=u_{j}$ and $c_{j}$ is optimal for $\theta _{j}.$ By construction it follows that $t_{i}(E_{i}(\theta _{i}))$ deems $t_{j}(E_{j}(\theta _{j}))$ possible for which $c_{j}$ is optimal.
Let $(c_{j},t_{j})$ with $t_{j}$ cautious be a pair which is deemed possible by $t_{i}(E_{i}(\theta _{i}))$ satisfying that $c_{j}$ is optimal for $t_{j}, $ and $(c_{j}^{\prime },t_{j}^{\prime })$ be a pair deemed possible by $ t_{i}(E_{i}(\theta _{i}))$ which does not satisfy that condition. Let $ (c_{j},\theta _{j})$ and $(c_{j}^{\prime },\theta _{j}^{\prime })$ be the corresponding pairs occuring in the belief of $\theta _{i}.$ Since $\theta _{i}$ believes in rationality, by construction it follows that $u_{j}(\theta _{j})=u_{j}$ while $u_{j}(\theta _{j}^{\prime })\neq u_{j}.$ Since $\theta _{i}$ priorly believes in $u,$ $\theta _{i}$ deems $(c_{j},\theta _{j})$ infinitely more likely than $(c_{j}^{\prime },\theta _{j}^{\prime })$. It follows that $t_{i}(E_{i}(\theta _{i}))$ deems $(c_{j},t_{j})$ infinitely more likely than $(c_{j}^{\prime },t_{j}^{\prime }).$ Here we have shown that $t_{i}(E_{i}(\theta _{i}))$ assumes $j$'s rationality.
Suppose that, for some $n\in \mathbb{N},$ we have shown that for each $k\leq n,$ \newline (n1) if $t_{i}\in T_{i}$ expresses $k$-fold assumption of rationality, then each $\theta _{i}\in \Theta _{i}(t_{i})$ expresses $k$-fold full belief in that every good choice is supported and prior belief in $u$; \newline (n2) If $\theta _{i}\in \Theta _{i}$ expresses $k$-fold full belief in that every good choice is supported and prior belief in $u$, then $ t_{i}(E_{i}(\theta _{i}))$ expresses $k$-fold assumption of rationality.
Now we show that these two statements hold for $n+1.$ First, suppose that $ t_{i}\in T_{i}$ expresses $(n+1)$-fold assumption of rationality. Let $ c_{j}\in C_{j}$ be a choice of $j$ optimal for some cautious type whose assigned utility function is $u_{j}$ that expresses up to $n$-fold belief in that every good choice is supported. Then it is easy to see that (1) by inductive assumption, in the constructed complete information model the corresponding type expresses $n$-fold assumption of rationality, and (2) $ c_{j}$ is optimal for that type. Since $t_{i}$ expresses $(n+1)$-fold assumption of rationality, $t_{i}$ deems possible a cautious type $t_{j}$ that expresses up to $n$-fold assumption of rationality and for which $c_{j}$ is optimal. By construction, it follows that $\theta _{i}$ deems possible some $\theta _{j}\in \Theta _{j}(t_{j})$. By inductive assumption it follows that each $\theta _{j}\in \Theta _{j}(t_{j})$ expresses $n$-fold belief in that every good choice is supported. Since $\theta _{i}$ expresses common belief in caution and rationality it follows that $\theta _{i}$ deems $ (c_{j},\theta _{j1})$ for $\theta _{j1}\in \Theta _{j}(t_{j})$ (that is, $ w_{j}(\theta _{j1})=u_{j}$).
Let $(c_{j},\theta _{j})$ with $\theta _{j}$ cautious deemed possible by $ \theta _{i}$ satisfying that $\theta _{j}$ expresses up tp $n$-fold belief in prior belief in $u$ and that every good choice is supported and $ w_{j}(\theta _{j})=u_{j}$ and $(c_{j}^{\prime },\theta _{j}^{\prime })$ a pair which does not satisfy those conditions. Let $(c_{j},t_{j})$ and $ (c_{j}^{\prime },t_{j}^{\prime })$ be the pairs occuring in the belief of $ t_{i}$ corresponding to $(c_{j},\theta _{j})$ and $(c_{j}^{\prime },\theta _{j}^{\prime }).$ Since $c_{j}$ is rational for $\theta _{j}$ and $ w_{j}(\theta _{j})=u_{j},$ it follows that $c_{j}$ is optimal for $t_{j}.$ Also, by inductive assumption, it follows that $t_{j}$ expresses up to $n$ -fold assumption of rationality. On the other hand, it can be seen that $ (c_{j}^{\prime },t_{j}^{\prime })$ does not satisfy these conditions. Since $ t_{i}$ expresses $(n+1)$-fold of assumptions of rationality, $t_{i}$ deems $ (c_{j},t_{j})$ infinitely more likely than $(c_{j}^{\prime },t_{j}^{\prime }).$ By construction, $\theta _{i}$ deems $(c_{j},\theta _{j})$ infinitely more likely than $(c_{j}^{\prime },\theta _{j}^{\prime }).$ Here we have shown that $\theta _{i}$ expresses $(n+1)$-fold full belief in that every good choice is supported and prior belief in $u$.
Now suppose that $\theta _{i}\in \Theta _{i}$ expresses $(n+1)$-fold full belief in that every good choice is supported and prior belief in $u$. Let $ c_{j}\in C_{j}$ be a choice of $j$ optimal for some cautious type that expresses to $n$-fold assumption of rationality. By inductive assumption it follows that the corresponding type within some incomplete information model also expresses $n$-fold full belief in that every good choice is supported and prior belief in $u$. It can be seen that $c_{j}$ is optimal to the constructed type having $u_{j}$ as its utility functionand the type expresses up to $n$-fold full belief in that every good choice is supported and prior belief in $u$. Then $\theta _{i}$ deems possible a type $\theta _{j}$ with $w_{j}(\theta _{j})=u_{j}$ for player $j$ which expresses up to $n $-fold belief in that every good choice is supported for which $c_{j}$ is optimal. By inductive assumption it follows that $t_{i}(E_{i}(\theta _{i}))$ deems possible $t_{j}(E_{j}(\theta _{j}))$ which expresses $n$-fold assumption of rationality and for which $c_{j}$ is optimal.
Let $(c_{j},t_{j}$) be a pair with $t_{j}$ cautious deemed possible by $ t_{i}(E_{i}(\theta _{i}))$ where $t_{j}$ expresses up to $n$-fold assumption of rationality and $c_{j}$ is optimal for $t_{j}$, and let $(c_{j}^{\prime },t_{j}^{\prime })$ be a pair that does not satisfy this property. Let $ (c_{j},\theta _{j})$ and $(c_{j}^{\prime },\theta _{j}^{\prime })$ be the corresponding pairs occurring in the belief of $\theta _{i}.$ By inductive assumption and by construction, $\theta _{j}$ is cautious and expresses up to $n$-fold belief in that prior belief in $u$ and every good choice is supported and $w_{j}(\theta _{j})=u_{j},$ while $(c_{j}^{\prime },\theta _{j}^{\prime })$ does not satisfy this property. Therefore $\theta _{i}$ deems $(c_{j},\theta _{j})$ infinitely more likely than $(c_{j}^{\prime },\theta _{j}^{\prime }),$ which implies that $t_{i}(E_{i}(\theta _{i}))$ deems $(c_{j},t_{j})$ infinitely more likely than $(c_{j}^{\prime },t_{j}^{\prime }).$ Here we have shown that $t_{i}(E_{i}(\theta _{i}))$ expresses $(n+1)$-fold assumption of rationality. // \newline Proof of Theorem (ref).1. (Only-if) Let $ M^{co}=(T_{i},b_{i})_{i\in I}$, $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$ be constructed from $M^{co}$ by the two steps above, $c_{i}^{\ast }\in C_{i}$ be a permissible choice, and $t_{i}^{\ast }\in T_{i}$ be a type expressing common full belief in caution and common assumption of rationality such that $c_{i}^{\ast }$ is rational for $t_{i}^{\ast }.$ Let $ \theta _{i}^{\ast }=\theta _{i1}(t_{i}^{\ast }).$ By definition, $ w_{i}(\theta _{i}^{\ast })=u_{i}$ and $\beta _{i}(\theta _{i}^{\ast })$ has the same distribution on $j$'s choices at each level as $b_{i}(t_{i}^{\ast }) $. Hence $c_{i}^{\ast }$ is rational for $\theta _{i}^{\ast }.$ Also, it follows from Observation (ref).2, Lemmas (ref).1, and (ref).3 that $\theta _{i}^{\ast }$ expresses common full belief in caution, rationality,that a good choice is supported, and prior belief in $u$ .
(If). Let $M^{in}=(\Theta _{i},w_{i},\beta _{i})_{i\in I}$, $ M^{co}=(T_{i},b_{i})_{i\in I}$ be constructed from $M^{in}$ by the above approach, and $c_{i}^{\ast }\in C_{i}$ be rational for some $\theta _{i}^{\ast }$ with $w_{i}(\theta _{i}^{\ast })=u_{i}$ which expresses common full belief in caution, rationality, that a good choice is supported, and prior belief in $u$. Consider $t_{i}(E_{i}(\theta _{i}^{\ast })).$ Since $ w_{i}(\theta _{i}^{\ast })=u_{i}$ and $b_{i}(t_{i}(E_{i}(\theta _{i}^{\ast })))$ has the same distribution on $j$'s choices at each level as $\beta _{i}(\theta _{i}^{\ast })$, $c_{i}^{\ast }$ is rational for $ t_{i}(E_{i}(\theta _{i}^{\ast })).$ Also, by Lemmas (ref).2 and (ref).3, $t_{i}(E_{i}(\theta _{i}^{\ast }))$ expresses common full belief in caution and common assumption of rationality. //
Assumption of rationality is a refinement of permissibility (See Perea p12). This can also be seen within the framework of incomplete information. Comparing our characterization of the former of the characterization of the latter in Section 4.6 in Liu l17 it can be seen that there is correspondence between the conditions. Section 4.6 in Liu l17 characterizes permissibility by weak caution, rationality, and primary belief in $u$ within the incomplete information framework. The characteization of assumption of rationality shares rationality with it, while caution and prior belief are stronger than weak caution and primary belief in $u,$ respectively.
An interesting phenomenon is the role of rationality. Liu l17 provides two ways to characterize permissibility, one with rationality and one without it. The characterization of proper rationality there is a stronger version of the latter, while the characterization in this paper a stronger version of the former. So far, it seems that using or not using rationality in the characterization differentiate the two refinements of permissibility, that is, assumption of rationality and proper rationalizability, within the incomplete information framework. It would be interesting that any future research would confirm this statement or provide any counterexample, that is, show that proper rationalizability can be characterized with rationality while assumption of rationality can be done without it.
On the other hand, as shown in Liu l17 (and the construction here), it is always possible to construct epistemic models with incomplete information which satisfies rationality as well as all conditions for characterization of proper rationalizability. Further, prior belief in $u$ is voguely a condition between primary belief in $u$ and $u$-centered belief which is used in Theorem 3.2 of Liu l17 to characterize proper rationalizability. Those seem to correspond to the fact within the complete information framework that there is always possible to construct belief hierarchy which both assumes the opponent's rationality and respects the opponent's preferences.
The author would like to thank Andr\'{e}s Perea for his valuable discussion and encouragement. She thanks all teachers and students in the 4th Epicenter Spring Course on Epistemic Game Theory at Maastricht University to whom she owes inspiring teaching, stimulating discussions, and new ideas. She gratefully acknowledges the support of Grant-in-Aids for Young Scientists (B) of JSPS No.17K13707 and Grant for Special Research Project No. 2017K-016 of Waseda University.