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Ordered Kripke Model, Permissibility, and Convergence of Probabilistic Kripke Model
In this section we give surveys on lexicographic belief and permissibility (Section (ref).1) and on probabilistic Kripke model for games (Section (ref).2). These will be preparation for the introduction of ordered Kripke model in Section (ref).
In this subsection we give a survey on lexicographic epistemic model (with complete information) and the definition of permissibility. For a details, see Perea p12, Chapter 5. Consider a finite 2-person strategic form game $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ where $I=1,2$. A finite lexicographic epistemic model for $G$ is a tuple $M^{lex}=(\Theta _{i},\beta _{i})_{i\in N}$ where \newline (a) $\Theta _{i}$ is a finite set of types, and \newline (b) $\beta _{i}$ is a mapping that assigns to every $\theta _{i}\in \Theta _{i}$ a lexicographic belief over $\Delta (S_{j}\times \Theta _{j}),$ i.e., $ \beta _{i}(\theta _{i})=(\beta _{i1},\beta _{i2},...,\beta _{iK})$ where $ \beta _{ik}\in \Delta (S_{j}\times \Theta _{j})$ for $k=1,...,K.$
Let $\theta _{i}\in \Theta _{i}$ with $\beta _{i}(\theta _{i})=(\beta _{i1},\beta _{i2},...,\beta _{iK}).$ Each $\beta _{ik}$ $(k=1,...,K)$ is called $\theta _{i}$'s level-$k$ belief. For $(s_{j},\theta _{j})\in S_{j}\times \Theta _{j},$ we say $\theta _{i}$ deems $(s_{j},\theta _{j})$ possible iff $\beta _{ik}(s_{j},\theta _{j})>0$ for some $k\in \{1,...,K\}.$ We say $\theta _{i}$ deems $\theta _{j}\in \Theta _{j}$ possible iff $\theta _{i}$ deems $(s_{j},\theta _{j})$ possible for some $s_{j}\in S_{j}$. For each $\theta _{i}\in \Theta _{i},$ we denote by $ \Theta _{j}(\theta _{i})$ the set of all $\theta _{j}\in \Theta _{j}$ deemed possible by $\theta _{i}$. \newline Definition (ref).1 (Caution) Type $\theta _{i}\in \Theta _{i}$ is \emph{cautious} iff for each $\theta _{j}\in \Theta _{j}(\theta _{i})$ and each $s_{j}\in S_{j},$ it deems $(s_{j},\theta _{j})$ possible.
For each $s_{i}\in S_{i}$, let $u_{i}(s_{i},\theta _{i})=(u_{i}(s_{i},\theta _{i1}).,..,u_{i}(s_{i},\theta _{iK}))$ where for each $k=1,...,K,$ $ u_{i}(s_{i},\theta _{ik}):=\Sigma _{(c_{j},t_{j})\in C_{j}\times T_{j}}\beta _{ik}(s_{j},\theta _{j})u_{i}(s_{i},s_{j}),$ that is, each $ u_{i}(s_{i},\theta _{ik})$ is the expected utility for $s_{i}$ over $\theta _{ik}$ and $u_{i}(s_{i},\theta _{i})$ is a vector of expected utilities. For each $s_{i},s_{i}^{\prime }\in S_{i}$, we say that $\theta _{i}$ prefers $s_{i}$ to $s_{i}^{\prime }$, denoted by $u_{i}(s_{i},\theta _{i})>u_{i}(s_{i}^{\prime },\theta _{i}),$ iff there is $k\in \{0,...,K-1\}$ such that the following two conditions are satisfied: \newline (a) $u_{i}(s_{i},\theta _{i\ell })=u_{i}(s_{i}^{\prime },\theta _{i\ell })$ for $\ell =0,...,k,$ and \newline (b) $u_{i}(s_{i},\theta _{i,k+1})>u_{i}(s_{i}^{\prime },\theta _{i,k+1})$ . \newline We say that $\theta _{i}$ is indifferent between $s_{i}$ and $ s_{i}^{\prime },$ denoted by $u_{i}(s_{i},\theta _{i})=u_{i}(s_{i}^{\prime },\theta _{i}),$ iff $u_{i}(s_{i},\theta _{ik})=u_{i}(s_{i}^{\prime },\theta _{ik})$ for each $k=1,...,K.$ It can be seen that the preference relation on $S_{i}$ under each type $\theta _{i}$ is a linear order. $s_{i}$ is rational (or optimal) for $\theta _{i}$ iff $\theta _{i}$ does not prefer any choice to $s_{i}$. \newline \textbf{Definition (ref).2 (Primary belief in the opponent's rationality) }Let $\theta _{i}\in \Theta _{i}$ with $\beta _{i}(\theta _{i})=(\beta _{i1},\beta _{i2},...,\beta _{iK}).$ $\theta _{i}$ \emph{ primarily believes in }$\emph{j}$\emph{'s rationality} iff $\theta _{i}$'s primary belief $\theta _{i1}$ only assigns positive probability to those $ (s_{j},\theta _{j})$ where $s_{j}$ is rational for $\theta _{j}.$ \newline \textbf{Definition (ref).3 (Common full belief in a property) }Let $P $ be an arbitrary property of lexicographic types. \newline (a) $\theta _{i}\in \Theta _{i}$ \emph{expresses }$0$\emph{-fold full belief in} $P$ iff $\theta _{i}$ satisfies $P;$ \newline (b) For each $n\in \mathbb{N},$ $\theta _{i}\in \Theta _{i}$ \emph{expresses }$(n+1)$\emph{-fold full belief in} $P$ iff $\theta _{i}$ only deems possible $j$'s types that express $n$-fold full belief in $P.$ \newline $\theta _{i}$ \emph{expresses common full belief in} $P$ iff it expresses $n$ -fold full belief in $P$ for each $n\in \mathbb{N}.$ \newline \textbf{Definition (ref).4 (Permissibility)}. Given a lexicographic epistemic model $M^{lex}=(\Theta _{i},\beta _{i})_{i\in N}$ for a game $ G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$, $s_{i}\in S_{i}$ is \emph{ permissible} iff it is optimal to some $\theta _{i}\in \Theta _{i}$ which expresses common full belief in caution and primary belief in rationality. \newline \textbf{Example (ref).1 }Consider the game $G$ as follows (Myerson m78):
and $M^{lex}=(\Theta _{i},\beta _{i})_{i\in N}$ for $G$ where $\Theta _{1}=\{\theta _{1}\},$ $\Theta _{2}=\{\theta _{2}\},$ and
It can be seen that $A$ is permissible since it is optimal to $t_{1}$ which expresses common full belief in caution and primary belief in rationality.
It is shown by Proposition 5.2 in Asheim and Dufwenberg ad03 that a strategy is permissible if and only if it survives an algorithm called Dekel-Fudenberg procedure (Dekel and Fudenberg df90). Given a game $ G,$ by Dekel-Fudenberg procedure we mean the process that (1) at first round we eliminate all weakly dominated strategies in $G,$ and (2) then iteratedly eliminate dominated strategies until no strategies can be eliminated.
In this subsection we give a survey of the probabilistic Kripke model for games which is a generalization of the standard Kripke model that is able to capture both pure and mixed strategies. For details, see Bonanno b08, b15. Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a 2-person strategic form game. A probabilistic Kripke model of $G$ is a tuple $ \mathcal{M}=(W,\{R_{i}\}_{i\in N},\{p_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N})$ where \newline (1) $W\neq \emptyset $ is the set of states (or possible worlds ), sometimes called the domain of $\mathcal{M}$ and is denoted by $ \mathcal{D}(\mathcal{M});$ \newline (2) For each $i\in N$, $R_{i}\subseteq S\times S$ is the accessibility relation for player $i.$ For each $w\in W,$ we use $R_{i}(w)$ to denote the set of all accessible states from $w,$ i.e., $R_{i}(w)=\{w^{\prime }\in W:wR_{i}w^{\prime }\};$ \newline (3) For each $i\in N,$ $p_{i}$ is a mapping from $W$ to $\Delta (W)$ satisfying (a) for each $w\in W,$ supp $p_{i}(w)\subseteq R_{i}(w),$ and (b) for each $w^{\prime }\in R_{i}(w),$ $p_{i}(w^{\prime })=p_{i}(w);$ \newline (4) For each $i\in N,$ $\sigma _{i}$ is a mapping from $W$ to $S_{i}$ such that for each $w^{\prime }\in R_{i}(w),$ $\sigma _{i}(w^{\prime })=\sigma _{i}(w)$.
$(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N})$ is a standard Kripke model of $G$. $\mathcal{M}^{o}=(W,\{R_{i}\}_{i\in N})$ is called the Kripke frame of $\mathcal{M}.$ Here we follow the literatures and assume that $\mathcal{M}^{o}$ is a KD45 frame, i.e., each $R_{i}$ is serial, transitive, and Euclidean. For each $i\in N,$ a semantic belief operator is a function $\mathbb{B}_{i}:2^{W}\rightarrow 2^{W}$ such that for each $E\subseteq W,$
A semantic common belief operator is a function $\mathbb{CB} :2^{W}\rightarrow 2^{W}$ such that for each $E\subseteq W,$
It can be seen that $\mathbb{B}_{i}$ and $\mathbb{CB}$ correspond to Aumann a76's standard concept \textquotedblleft knowledge\textquotedblright\ and \textquotedblleft common knowledge\textquotedblright .
At $w\in W$ the strategy $s_{i}\in S_{i}$ with $s_{i}$ is at least as prefered to $s_{i}^{\prime }$ iff $u_{i}(s_{i},\Sigma _{w^{\prime }\in R_{i}(w)}p_{i}(w)(w^{\prime })\sigma _{j}(w^{\prime }))\geq u_{i}(s_{i}^{\prime },\Sigma _{w^{\prime }\in R_{i}(w)}p_{i}(w)(w^{\prime })\sigma _{j}(w^{\prime })).$ $s_{i}$ is prefered to $s_{i}^{\prime }$ at $w$ iff the strict inequality holds. $s_{i}$ is optimal at $w$ iff there is no strategy preferred to $s_{i}$ at $w$. A state $w$ is rational for $i$ iff $\sigma _{i}(w)$ is optimal at $w$. We use $RAT_{i}$ to denote the set of all rational states for player $i,$ and define $ RAT=\cap _{i\in N}RAT_{i}$.
The following statement connects iterated elimination of pure dominated strategies (an algorithm) to rationality (an epistemic concept). Its proof can be found in Bonanno b15, p.452. \newline Theorem (ref).1 (Iterated elimination of dominated strategies and Kripke model). Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ and $S^{IEDS}$ be the set of strategy profiles surviving iterated elimination of dominated strategies. Then \newline (1) given an arbitrary probabilistic Kripke model of $G$, if $w\in \mathbb{CB }(RAT),$ then $\sigma (w)\in S^{IEDS};$ \newline (2) for each $s\in S^{IEDS}$, there is a probabilistic Kripke model of $G$ and a state $w$ such that $\sigma (w)=s$ and $w\in \mathbb{CB}(RAT)$.
In this section we define the ordered Kripke model as a modification of the standard one and show how it can be used to describe the lexicographic reasoning in game theory. \newline Definition (ref).1 (Ordered epistemic model) Let $ G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a 2-person strategic form game. An ordered Kripke model of $G$ is a tuple $\overline{\mathcal{M} }=(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N},\{\lambda _{i}\}_{i\in N})$ where \newline (1) $(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N})$ is a standard Kripke model of $G$, and \newline (2) For each $i\in N$, $\lambda _{i}$ assigns to each $w\in W$ an injection from a cut $\{1,...,K\}$ of natural numbers to the set of probability distributions (with finite supports) over $R_{i}(w)$, i.e., $\lambda _{i}(w):\{1,...,K\}\rightarrow \Delta (R_{i}(w)).$ $\lambda _{i}(w)$ can be interpreted as a linear order on a finite subset of $\Delta (R_{i}(w)).$ We use $\mathcal{D}(\lambda _{i}(w))$ and $\mathcal{R}(\lambda _{i}(w))$ to denote the domain and the range of $\lambda _{i}(w)$, i.e., $\mathcal{D} (\lambda _{i}(w))=\{1,...,K\}$ and $\mathcal{R}(\lambda _{i}(w))=\{\lambda _{i}(w)(1),...,\lambda _{i}(w)(K)\}$. \newline Definition (ref).2 (Caution). Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a strategic form game and $\overline{\mathcal{M}} =(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N},\{\lambda _{i}\}_{i\in N})$ an ordered Kripke model for $G.$ $R_{i}$ is cautious at $w\in W$ iff for any $s_{j}\in S_{j}$ ($j\neq i$), there exists $w^{\prime }$ which is assigned a possitive probability by some element in $\mathcal{R}(\lambda _{i}(w))$ such that $\sigma _{j}(w^{\prime })=s_{j}$. We say $\overline{ \mathcal{M}}$ is cautious iff for each $i\in N$, $R_{i}$ is cautious at every $w\in W$.
The difference between the ordered Kripke model and the standard one is that the former assigns a linear order $\lambda _{i}(w)$ on $R_{i}(w)$ for each state $w.$ This order is used to define the preferences in the model. We have the following defintion. \newline Definition (ref).3 (Lexicographic preferences) Let $ G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a strategic form game and $ \overline{\mathcal{M}}=(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N},\{\lambda _{i}\}_{i\in N})$ an ordered Kripke model for $G.$ At $w\in W$ the strategy $s_{i}\in S_{i}$ is at least as lexicographically prefered to $s_{i}^{\prime }$, denoted by $s_{i}\succeq _{w}s_{i}^{\prime }, $ iff $\exists k\in \{0,...,|\mathcal{D}(\lambda _{i}(w))|\}$ such that \newline (a) $u_{i}(s_{i},\sigma _{j}(\sigma _{j}(\lambda _{i}(w)(t))))=u_{i}(s_{i}^{\prime },\sigma _{i}(\lambda _{i}(w)(t)))$ for all $t\leq k$; \newline (b) $u_{i}(s_{i},\sigma _{j}(\lambda _{i}(w)(k+1)))=u_{i}(s_{i}^{\prime },\sigma _{i}(\lambda _{i}(w)(k+1))).$
Here by $\sigma _{j}(\lambda _{i}(w)(t))$ we mean the mixture of stategies in $\sigma _{j}(\lambda _{i}(w)(t)).$ Therefore
It can be seen that when $k=|\mathcal{D}(\lambda _{i}(w))|,$ $s_{i}$ and $ s_{i}^{\prime }$ generates the same payoff for player $i$ along $\lambda _{i}(w).$ This case is denoted by $s_{i}\simeq _{w}s_{i}^{\prime }.$ When $ k\neq |\mathcal{D}(\lambda _{i}(w))|,$ we say that $s_{i}$ is lexicographically prefered to $s_{i}^{\prime }$ at $w,$ denoted by $ s_{i}\succ _{w}s_{i}^{\prime }.$ $s_{i}$ is optimal at $w$ iff there is no $s_{i}^{\prime }\in S_{i}$ such that $s_{i}^{\prime }\succ _{w}s_{i}.$ We say a state $w$ is lexicographically rational for $i$ iff the choice $\sigma _{i}(w)$ is optimal for $i.$ For each $i\in N,$ let $LRAT_{i}$ be the set of rational states for player $i$ and $LRAT=\cap _{i\in N}LRAT_{i} $. \newline Example (ref).1. Consider the following game $G$ in Example (ref).1:
and an ordered Kripke model $\overline{\mathcal{M}}$ as follows:
It can be seen that $\overline{ \mathcal{M}}$ is cautious. It can be seen that $A$ and $C$ are optimal in each state, $w_{1}$ and $w_{2}$ are rational for player 1, and $w_{1}$ and $ w_{3}$ are rational for player 2. Therefore, $LRAT_{1}=\{w_{1},w_{2}\}$, $ LRAT_{2}=\{w_{1},w_{3}\},$ and $LRAT=\{w_{1}\}.$ On the other hand, as mentioned in Example (ref).1, since both $\sigma _{1}(w_{2})=A$ and $ \sigma _{2}(w_{2})=D$ are permissible strategies, lexicographic rationality in the ordered Kripke model here captures the concept of \textquotedblleft a strategy is rational under a lexicographic belief\textquotedblright\ in the first order. Now the problem is how to define belief hierarchy and common belief in this model. It can be seen that we cannot adopt $\mathbb{B}_{i}$ and $\mathbb{CB}$ in standard approach. Indeed, in Example (ref).1 $ \mathbb{B}_{i}(LART)=\mathbb{CB}(LART)=\emptyset ,$ which is contradictory to our intention to preserve $w_{2}$. Here we give one approach. For each $ i\in N$ and $w\in W,$ let $R_{i}^{1}(w)=\{w^{\prime }\in W:\lambda _{i}(w)(1)(w^{\prime })>0\}$ and $R^{1}=\cup _{i\in N}R_{i}^{1}.$ A semantic level-1 belief operator for player $i$ is a mapping $\mathbb{B} _{i}^{1}:2^{W}\rightarrow 2^{W}$ such that for each $E\subseteq W,$
Similarly, a semantic common level-1 belief operator is a mapping $ \mathbb{CB}^{1}:2^{W}\rightarrow 2^{W}$ such that for each $E\subseteq W,$
It can be seen that $\mathbb{B}_{i}^{1}(LRAT)=\mathbb{CB}^{1}(LRAT)=\{w_{1}\} $ in Example (ref).1. In general, we have the follwong result. \newline Theorem (ref).1 (Permissibility and semantic common level-1 belief). Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a strategic form game and $S^{PER}\subseteq S$ be the set of permissible strategy profiles. Then \newline (1) given an arbitrary cautious ordered Kripke model of $G,$ if $ w\in \mathbb{CB}^{1}(LRAT),$ then $\sigma (w)\in S^{PER}$, and \newline (2) for each $s\in S^{PER},$ there exists a cautious ordered Kripke model of $G$ such that $\sigma (w)=s$ and $w\in \mathbb{CB}^{1}(LRAT).$
To show Theorem (ref).1, we need the following lemma. \newline Lemma (ref).1. Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a strategic form game and $S^{DF}\subseteq S$ be the set of strategy profiles surviving Dekel-Fudenberg procedure. Then given an arbitrary cautious ordered epistemic model of $G,$ if $w\in \mathbb{CB}^{1}(LRAT),$ then $\sigma (w)\in S^{DF}$. \newline Proof. For each $n\in \mathbb{N},$ we use $S^{DFn}$ to denote the set of strategy profiles surviving the first $n$ rounds of Dekel-Fudenberg procedure. Let $\overline{M}=(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N},\{\lambda _{i}\}_{i\in N})$ be a cautious ordered epistemic model of $G$ and $w\in W.$ We show that if $w\in \mathbb{CB}^{1}(LRAT),$ then $\sigma (w)\in S^{DFn}$ for each $n\in \mathbb{N}.$ First, since $\overline{M}$ is cautious, it can be seen that $\sigma (w)\in S^{DF1}$. Indeed, if there is some $i\in N$ such that $\sigma _{i}(w)$ is eliminated in the first round of Dekel-Fudenberg procedure, then there is some $r_{i}\in \Delta (S_{i}).$ Then it follows from Theorem 5.8.3 in Perea p12 (p.215, 221-226) $ \sigma _{i}(w)$ cannot be optimal to any cauious belief, i.e., it cannot be optimal on $\lambda _{i}(w),$ which is contradictory since $w\in LRAT_{i}$.
Now we show that $\sigma (w)\in S^{DF2}$. Suppose for some $i\in N$, $\sigma _{i}(w)$ is eliminated in the second round of Dekel-Fudenberg procedure, i.e., there exists $r_{i}\in \Delta (S_{i}^{DF1})$ such that $ u_{i}(r_{i},s_{j})>u_{i}(s_{i},s_{j})$ for all $s_{j}\in S_{j}^{DF1}$. On the other hand, since $w\in \mathbb{CB}^{1}(LRAT)$, $\sigma _{i}(w)$ is optimal to $\lambda _{i}(w)(1).$ This implied that some strategies supporting $\lambda _{i}(w)(1)$ has been eliminated in the first round. However, since $w\in \mathbb{CB}^{1}(LRAT),$ it follows from the definition that supp $\lambda _{i}(w)(1)\subseteq LRAT,$ which, from the argument above, implies that all strategies $w^{\prime }\in $ supp $\lambda _{i}(w)(1) $ should have survived the first round and $\sigma _{j}(w^{\prime })$ stay in $S_{j}^{DF1},$ a contradiction.
Now suppose that $\sigma (w)\in S^{DF1}\cap ...\cap S^{DFn}$ but disappeared in $S^{DFn+1}.$ This could happen only if some strategies supporting $ \lambda _{i}(w)(1)$ had been eliminated in the $n$-th round, which is because some strategies supporting that strategy in $\lambda _{i}(w)(1)$ in ( $n-1$)-th round, etc. Finally this leads to the second and first rounds, which, by the argument above, is impossible. Therefore $\sigma (w)\in S^{DFn+1}$. // \newline Proof of Theorem (ref).1: (1) Since, by Proposition 5.2 in Asheim and Dufwenberg ad03, any strategy surviving Dekel-Fudenberg procedure is permissible and vice versa, i.e., $S^{PER}=S^{DF},$ (1) directly follows from Lemma (ref).1. \newline (2) Let $s\in S^{PER},$ that is, for each $i\in N,$ $s_{i}$ is optimal to some type expressing common full belief in caution and primary belief in rationality in a lexicographic epistemic model $ M^{lex}=(T_{j},b_{j})_{j\in N}.$ We construct an ordered Kripke model $ \overline{M}=(W,\{R_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N},\{\lambda _{i}\}_{i\in N})$ based on $M^{lex}$ as follows: \newline (1) Let $W=T\times S,$ here $T=\Pi _{i\in N}T_{i};$ \newline (2) for each $w=(t_{1},t_{2},s_{1},s_{2}),$ $\sigma _{i}(w)=s_{i};$ \newline (3) Connectiong each state in $W$ according to $M^{lex},$ i.e., for each $ w=(t_{1},t_{2},s_{1},s_{2}),w^{\prime }=(t_{1}^{\prime },t_{2}^{\prime },s_{1}^{\prime },s_{2}^{\prime })\in T\times S,$ $w^{\prime }=\lambda _{i}(w)(k)$ iff $t_{i}=t_{i}^{\prime },$ $s_{i}=s_{i}^{\prime },$ and $ (s_{j}^{\prime },t_{j}^{\prime })$ is the $k$-th entry in $b_{i}(t_{i});$ mixed strategy-type pairs are defined in a similar way. \newline Without loss of generality, we can assume that each type in $M^{lex}$ is cautious.\footnote{ For a state that is not cautious we can extend it into a cautious one. See Liu l18.} It can be seen that $\overline{\mathcal{M}}$ is also cautious, and there is $w\in W$ with $\sigma (w)=s$ and $w\in \mathbb{CB} ^{1}(LRAT).$ //
Though the ordered Kripke model is not the first framework combining standarad Kripke model with an order on (a subset of) each $R_{i}(w)$ (cf. Baltag and Smets bs06, bs07), here we are interested in how such a model can be connected to the probabilistic Kripke model for games introduced in Section (ref). In this section we will first introduce a probabilistic Kripke model with modified belief operators under which $ \varepsilon $-perfect rationalizability can be characterized. Then we will show that an ordered Kripke model can be seen as a \textquotedblleft limit\textquotedblright\ of a sequence of probablistic Kripke models.
In this subsection we give a survey on probabilitstic epistemic model (with complete information) and the definition of $\varepsilon $-perfect rationalizability. See Perea p12, Chapter 2 for the detail of the former. $\varepsilon $-permissible, which originates from Myerson m78 , is defined in a similar way as $\varepsilon $-proper rationalizability as in Schuhmacher s99 and Perea and Roy ps17. Consider a finite 2-person strategic form game $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$. A finite probabilistic epistemic model for $G$ is a tuple $ M^{pro}=(T_{i},b_{i})_{i\in N}$ where \newline (a) $T_{i}$ is a finite set of types, and \newline (b) $b_{i}$ is a mapping that assigns to every $t_{i}\in T_{i}$ a probability distribution over $\Delta (S_{j}\times T_{j}).$
For each $s_{i}\in S_{i}$ and $t_{i}\in T_{i},$ we define $ u_{i}(s_{i},t_{i})=\Sigma _{(s_{j},t_{j})\in S_{j}\times T_{j}}b_{i}(t_{i})(s_{j},t_{j})u_{i}(s_{i},s_{j}).$ $s_{i}$ is optimal (or rational) for $t_{i}$ iff $u_{i}(s_{i},t_{i})\geq u_{i}(s_{i}^{\prime },t_{i})$ for any $s_{i}^{\prime }\in S_{i}$. For each $ s_{i},s_{i}^{\prime }\in S_{i}$ and $t_{i}\in T_{i},$ $s_{i}$ is preferred to $s_{i}^{\prime }$ under $t_{i}$ iff $ u_{i}(s_{i},t_{i})>u_{i}(s_{i}^{\prime },t_{i}).$ Given $t_{i}\in T_{i},$ for each $(s_{j},t_{j})\in S_{j}\times T_{j},$ we say $t_{i}$ deems $ (s_{j},t_{j})$ possible iff $b_{i}(t_{i})(s_{j},t_{j})>0.$ We say $ t_{i}$ deems $t_{j}\in T_{j}$ \emph{possible} iff $t_{i}$ deems $ (s_{j},t_{j})$ possible for some $s_{j}\in S_{j}$. For each $t_{i}\in T_{i},$ we denote by $T_{j}(t_{i})$ the set of all $t_{j}$'s deemed possible by $ t_{i}$. \newline \textbf{Definition (ref).1 (Caution)} Type $t_{i}\in T_{i}$ is \emph{ cautious} iff for each $t_{j}\in T_{j}(t_{i})$ and each $s_{j}\in S_{j},$ $ t_{i}$ deems $(s_{j},t_{j})$ possible. \newline \textbf{Definition (ref).2 (}$\varepsilon $\textbf{-perfect trembling condition) }Type $t_{i}\in T_{i}$ satisfies $\varepsilon $\emph{ -perfect trembling condition} iff for any $s_{j}\in S_{j}$ and $t_{j}\in T_{j}(t_{i})$ such that $t_{i}$ deems $(s_{j},t_{j})$ possible, if $s_{j}$ is not optimal under $t_{j}$ then $b_{i}(t_{i})(s_{j},t_{j})\leq \varepsilon .$ \newline \textbf{Definition (ref).3 (Common full belief in a property) }Let $P $ be an arbitrary property of probabilistic types. \newline (a) $t_{i}\in T_{i}$ \emph{expresses }$0$\emph{-fold full belief in} $P$ iff $t_{i}$ satisfies $P;$ \newline (b) For each $n\in \mathbb{N},$ $t_{i}\in T_{i}$ \emph{expresses }$(n+1)$ \emph{-fold full belief in} $P$ iff $t_{i}$ only deems possible $j$'s types that express $n$-fold full belief in $P.$ \newline $t_{i}$ \emph{expresses common full belief in} $P$ iff it expresses $n$-fold full belief in $P$ for each $n\in \mathbb{N}.$ \newline \textbf{Definition (ref).4 (}$\varepsilon $\textbf{-Perfect rationalizability)}. Given a probabilistic epistemic model $ M^{pro}=(T_{i},b_{i})_{i\in N}$ for a game $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$, $s_{i}\in S_{i}$ is $\varepsilon $\textbf{-}\emph{ permissible} iff it is optimal to some $t_{i}\in T_{i}$ which expresses common full belief in caution and $\varepsilon $-perfect trembling condition. \newline \textbf{Example (ref).1}. Consider the following game $G$ (from Myerson m78):
and $M^{pro}=(T_{i},b_{i})_{i\in N}$ for $G$ where $T_{1}=\{t_{1}\},$ $ T_{2}=\{t_{2}\},$ and
where $\varepsilon \in (0,1).$ It can be seen that $A$ is $\varepsilon $ -permissible since it is optimal to $t_{1}$ which expresses common full belief in caution and $\varepsilon $-perfect trembling condition.
Originally, by the definition in Myerson m78, perfect equilibrium is the limit of $\varepsilon $-perfect equilibrium. Though permissibility is the concepts in epistemic game theory and is defined in a different manner, it still holds tha permissibility is the limit of $\varepsilon $ -permissibility. See Schuhmacher s99.
In this subsection, we show how to use probabilistic Kripke model to describe $\varepsilon $-permissibility. Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a 2-person strategic form game and $\mathcal{M} =(W,\{R_{i}\}_{i\in N},\{p_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N})$ a probabilistic Kripke model for $G.$ We give the following definitions \newline Definition (ref).5 (Caution). $\mathcal{M}$ is cautious at $w\in W$ for $i\in N$ iff for any $s_{j}\in S_{j}$ ($j\neq i$), there exists $w^{\prime }\in R_{i}(w)$ satisfying $p_{i}(w)(w^{\prime })>0$ and $\sigma _{j}(w^{\prime })=s_{j}$. We say $\mathcal{M}$ is cautious iff $\mathcal{M}$ is cautious at every $w\in W$ for each $i\in N$. \newline Definition (ref).6 ($\varepsilon $-perfect trembling condition). $\mathcal{M}$ satisfies $\varepsilon $-perfect trembling condition at $w\in W$ for $i\in N$ iff for each $w^{\prime }\in R_{i}(w),$ if $\sigma _{i}(w^{\prime })$ (i.e., $\sigma _{i}(w)$) is not optimal to $\sigma _{j}(w^{\prime }),$ then $p_{i}(w)(w^{\prime })\leq \varepsilon $.
The above two concepts are illustrated in the following example. \newline Example (ref).1. Consider the game $G$ in Example (ref).1:
and a probabilistic Kripke model depicted as in Figure 2.
It can be seen that $ RAT_{1}=\{w_{1},w_{2}\}$, $RAT_{2}=\{w_{1},w_{3}\},$ and $RAT=RAT_{1}\cap RAT_{2}=\{w_{1}\}.$ Since $\sigma (w_{1})=(A,C)$ is a pair of $\varepsilon $ -perfect rationalizable strategies, $RAT$ can still be used in this framework for the first-order. Now the problem is how to describe higher orders, i.e., interpersonal belief and common full belief in this framework. $\mathbb{B}_{i}$ and $\mathbb{CB}$ in standard probabilistic model do not work here since $\mathbb{B}_{i}(RAT)=\mathbb{CB(}RAT)=\emptyset ,$ while we want to keep $w_{1}.$ Here we provide an approach. Let $\mathcal{M} =(W,\{R_{i}\}_{i\in N},\{p_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N},)$ a probabilistic Kripke model for $G$ satisfying caution and $\varepsilon $ -perfect trembling condition. For each $i\in N,$ we define $ R_{i}^{>\varepsilon }=\{w^{\prime }\in R_{i}(w):p_{i}(w)(w^{\prime })>\varepsilon \}.$ An\ upper $\varepsilon $ semantic belief operator is a function $\mathbb{B}_{i}^{>\varepsilon }:2^{W}\rightarrow 2^{W}$ such that for each $E\subseteq W,$
An upper $\varepsilon $ semantic common belief operator is a function $\mathbb{CB}^{>\varepsilon }:2^{W}\rightarrow 2^{W}$ such that for each $E\subseteq W,$
When $\varepsilon <\frac{1}{2},$ it can be seen that in Example (ref) .1, $\mathbb{CB}^{>\varepsilon }(RAT)=\{w_{1}\}.$
In general, we have the following statement. \newline Theorem (ref).1 (Characterizing $\varepsilon $ -perfect rationalizability). Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a 2-person strategic form game, $\varepsilon <\frac{1}{2},\footnote{ It should be noted that $\varepsilon <\frac{1}{2}$ makes sure that $ R_{i}^{>\varepsilon }(w)\neq R_{i}(w),$ though from the viewpoint of convergence/limit this is just a technical requirement.}$ and $ S^{\varepsilon PER}\subseteq S$ be the set of $\varepsilon $-permissible strategy profiles. Then \newline (1) given an arbitrary probabilistic Kripke model of $G$ satisfying caution and $\varepsilon $-perfect trembling condition, if $w\in \mathbb{CB} ^{>\varepsilon }(RAT),$ then $\sigma (w)\in S^{\varepsilon PER}$, and \newline (2) for each $s\in S^{\varepsilon PER},$ there exists a probabilistic Kripke model of $G$ satisfying caution and $\varepsilon $ -perfect trembling condition such that $\sigma (w)=s$ and $w\in \mathbb{CB} ^{>\varepsilon }(RAT).$ \newline Proof. (2) can be proved in a similar way as Theorem (ref) .1. Here we only prove (1). Since there is no algorithm like Dekel-Fudenberg procedure that can screen out $\varepsilon $-permissibility, we show how to construct a type which expresses common full belief in caution and $ \varepsilon $-perfect trembling condition. Let $\mathcal{M} =(W,\{R_{i}\}_{i\in N},\{p_{i}\}_{i\in N},\{\sigma _{i}\}_{i\in N})$ be a probabilistic Kripke model for $G$ satisfying caution and $\varepsilon $ -perfect trembling condition and $w\in \mathbb{CB}^{>\varepsilon }(RAT)$. For $i\in N,$ we define a partition $\mathbb{E}_{i}=\{E_{i1},...,E_{i\ell _{i}}\}$ of $W$ and satisfies that for each $w^{\prime },w^{\prime \prime }\in W,$ $w^{\prime }$ and $w^{\prime \prime }$ belong to the same equivalent class $E_{i}$ if and only if $R_{i}(w^{\prime })=R_{i}(w^{\prime \prime })$ and $p_{i}(w^{\prime })=p_{i}(w^{\prime \prime }).$ For each $ E_{i}\in \mathbb{E}_{i}$ we assign a symble $t_{i}(E_{i}).$ Without loss of generality, we can assume that for each $s_{i}\in S_{i}$ and each $E_{ik},$ there is some $w^{\prime }\in E_{ik}$ such that $\sigma _{i}(w^{\prime })=s_{i}.\footnote{ This corresponds to caution for probabilistic epistemic model $M^{pro}.$ It should be noted that even this condition is not satisfied, we can construct \textquotedblleft dummies\textquotedblright\ to make this condition satisfied without hurt the model.}$ Let $T_{i}=\{t_{i}(E_{i})\}_{E_{i}\in \mathbb{E}_{i}}$, and define $b_{i}(t_{i}(E_{i}))$ with the same probability as $p_{i}(w^{\prime })$, where $w^{\prime }\in E_{i},$ and the corresponding $t_{j}(E_{j})$. It can be seen that $b_{i}(t_{i}(E_{i}))$ is well-defined since every state in one $E_{i}$ has identified distributions. It can be seen straightforwardly that $\sigma (w)\in S^{\varepsilon PER}$ since for each $i\in N,$ $\sigma _{i}(w)$ is optimal to the type corresponding to $w$ which expresses common full belief in caution and $\varepsilon $-perfect trembling condition since $\mathcal{M}$ satisfies caution and $\varepsilon $ -perfect trembling condition. //
In this subsection we show that any ordered Kripke model is the limit of a sequence of probabilistic Kripke models. This can be intuitively seen by comparing the two Kripke models in Example (ref).1 and (ref) .1: Indeed, Figure 2 can be obtained by replacing $ 1-\varepsilon $ with $1$ and $\varepsilon $ with $2$ in Figure 1. Also, this can be seen from that perfect rationalizability characterized by the former is the limit of a sequence of $\varepsilon $-perfect rationalizabilities characterized by the later. In this section we show how to formulate this idea.
Let $G=(N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})$ be a 2-person strategic form game and $\overline{\mathcal{M}}=(W,\{R_{i}\}_{i\in N},$ $\{\sigma _{i}\}_{i\in N},\{\lambda _{i}\}_{i\in N})$ be an ordered Kripke model of $G. $ Without loss of generality, we assume that $\overline{\mathcal{M}}$ satisfies the following two conditions: \newline (Disjoint supports) For each $i\in N$, $w\in W,$ and $k,k^{\prime }\in \mathcal{D}(\lambda _{i}(w)),$ supp $\lambda _{i}(w)(k)\cap $ supp $ \lambda _{i}(w)(k^{\prime })\neq \emptyset $ if and only if $k\neq k^{\prime };\footnote{ This condition is adopted in some papers such as Blume et al. \cite{bbd91a}, \cite{bbd91b} while is not required in some others such as the standard textbook of Perea \cite{p12}. Technically, this condition is not necessary in characterizing rationalizabilities. Here we use it out of simplification.} $ \newline (Surjection) For each $w\in W$ and each $w^{\prime }\in R_{i}(w),$ there is some $k\in \mathcal{D}(\lambda _{i}(w))$ such that $\lambda _{i}(w)(k)(w^{\prime })>0.\footnote{ Surjection is different from caution. Caution requires each strategy of the opponent should appear in the range. When there are multiple states in $ R_{i}(w)$ which are assigned the same strategy, to be cautious only means that at least one of those state should appear in the range, while surjection requires that each of these states should appear. On the other hand, it does not mean that surjection implies caution since surjection has nothing with strategies assigned to each state in $R_{i}(w)$. In finite models, when surjection is not satisfied, we can faithfully extend each model into one which satisfies surjection without hurting $LRAT$ and $ \mathbb{CB}(LRAT)$.}$
Let $\varepsilon \in (0,1)$. Consider a probabilistic Kripke model $\mathcal{ M}^{\varepsilon }=(W^{\varepsilon },\{R_{i}^{\varepsilon }\}_{i\in N},\{p_{i}^{\varepsilon }\}_{i\in N},\{\sigma _{i}^{\varepsilon }\}_{i\in N}) $ of $G$ satisfying \newline (a) $W^{\varepsilon }=W,$ $R_{i}^{\varepsilon }=R_{i}$ and $\sigma _{i}^{\varepsilon }=\sigma _{i}$ for each $i\in N;$ \newline (b) for each $i\in N,$ $w\in W$, and $w^{\prime },w^{\prime \prime }\in $ supp $\lambda _{i}(w)(k)$ for some $k\in \mathcal{D}(\lambda _{i}(w)),$ it is satisfied that $p_{i}^{\varepsilon }(w)(w^{\prime })/p_{i}^{\varepsilon }(w)(w^{\prime \prime })=\lambda _{i}(w)(k)(w^{\prime })/\lambda _{i}(w)(k)(w^{\prime \prime })$; \newline (c) for each $i\in N,$ $w\in W$, and $w^{\prime }\in R_{i}(w),$ $ 0<p_{i}^{\varepsilon }(w)(w^{\prime })\leq \varepsilon $ if $\lambda _{i}(w)(1)(w^{\prime })=0.$ \newline It can be seen that when $\varepsilon $ is small enough, such $\mathcal{M} ^{\varepsilon }$ exists (not unique). Let $\{\varepsilon _{n}\}_{n\in \mathbb{N}}$ be a sequence in $(0,1)$ that converges to $0$ such that for each $\varepsilon _{n},$ there is some probabilistic Kripke model satisfying (a) - (c). We choose an arbitrary $\mathcal{M}^{\varepsilon _{n}}$ satisfying (a) - (c) for each $\varepsilon _{n}.$ It can be seen that the sequence $\{\mathcal{M}^{\varepsilon _{n}}\}_{n\in \mathbb{N}}$ \textquotedblleft converges\textquotedblright\ to $\overline{\mathcal{M}}$ in the sense that \newline (1) for each $i\in N,$ $w\in W$, and $w^{\prime }\in R_{i}(w)$ such that $ p_{i}^{\varepsilon _{n}}(w)(w^{\prime })\rightarrow 0,$ $w^{\prime }$ does not appear in $\lambda _{i}(w)(1);$ \newline (2) for each $i\in N,$ $w\in W$, and $w^{\prime }\in R_{i}(w)$ such that $ p_{i}^{\varepsilon _{n}}(w)(w^{\prime })\nrightarrow 0,$ $w^{\prime }\in $ supp $\lambda _{i}(w)(1)$ and $p_{i}^{\varepsilon _{n}}(w)(w^{\prime })\rightarrow \lambda _{i}(w)(1)(w^{\prime });$ \newline (3) The convergence is propotional within each level of $\lambda _{i}(w). \footnote{ To dealing those \textquotedblleft irrational\textquotedblright\ choice which is assigned in probability $0$ in any rational belief is one of the motivation for the introduce of lexicographic belief and studies from conditional probability. See Blume et al. \cite{bbd91a}, \cite{bbd91b}, Brandenburger et al. \cite{bfk07}, Halpern \cite{h10}.}$ \newline More formally, this convergence can be seen from the rationalizabilities they characterize. We have the following statement. \newline Theorem (ref).2 (Probabilistic models converge to ordered model). Let $G$ be a 2-person strategic game, $\overline{\mathcal{M}}$ a cautious ordered Kripke model of $G$ satisfying disjoint supports and surjection$,$ $\{\varepsilon _{n}\}_{n\in \mathbb{N}}$ a sequence in $(0,1)$ converging to $0,$ and $\{\mathcal{M}^{\varepsilon _{n}}\}_{n\in \mathbb{N}}$ be a sequence of probabilistic model of $G$ satisfying condition (a) - (c) above for each $\varepsilon _{n}.$ Then each $w$ which is commonly believed to be lexicographically rational in $\overline{\mathcal{M}}$ is commonly believed to be $\varepsilon _{n}$-upper rational in $\mathcal{M} ^{\varepsilon _{n}}$ for each $\varepsilon _{n},$ i.e., $\mathbb{CB} ^{1}(LRAT)=\cup _{M\in \mathbb{N}}\cap _{n>M}\mathbb{CB}^{>\varepsilon _{n}}(RAT_{\varepsilon _{n}})$. \newline Proof. ($\subseteq $) Let $w\in \mathbb{CB}^{1}(LRAT).$ It follows from Theorem (ref).1 that $\sigma (w)\in S^{PER},$ that is, there is a lexicographic model $(\Theta _{i},\beta _{i})_{i\in N}$ such for each $ i\in N$, $\sigma _{i}(w)$ is optimal to some $\theta _{i}\in \Theta _{i}$ which expresses common full belief in caution and primary belief in rationality. Based on each $\mathcal{M}^{\varepsilon _{n}},$ $\theta _{i}$ can be accordingly translated into a state $t_{i}^{\varepsilon _{n}}$ \textquotedblleft starting\textquotedblright\ from $w$ in probabilistic model. Since $\sigma _{i}(w)$ is optimal to some $\theta _{i},$ when $ \varepsilon _{n}$ is small enough, $\sigma _{i}(w)$ is optimal to $ t_{i}^{\varepsilon _{n}},$ and $t_{i}^{\varepsilon _{n}}$ expresses common full belief on caution and $\varepsilon _{n}$-perfect trembling condition. This argument holds for each $i\in N.$ Then by Theorem (ref).1 $ \sigma (w)\in \mathbb{CB}^{>\varepsilon _{n}}(RAT_{\varepsilon _{n}}),$ and consequently $w\in \cup _{M\in \mathbb{N}}\cap _{n>M}\mathbb{CB} ^{>\varepsilon _{n}}(RAT_{\varepsilon _{n}})$.
($\supseteq $) Let $w\in \cup _{M\in \mathbb{N}}\cap _{n>M}\mathbb{CB} ^{>\varepsilon _{n}}(RAT_{\varepsilon _{n}})$, that is, for some $M\in \mathbb{N},$ $w\in \mathbb{CB}^{>\varepsilon _{n}}(RAT_{\varepsilon _{n}})$ for all $n\geq M.$ Since $\varepsilon _{n}\rightarrow 0,$ it follows that for each $i\in N,$ $\sigma _{i}(w)$ is optimal on $p_{i}^{\varepsilon _{n}}(w)$ for infinitely small $\varepsilon _{n}.$ Since each $\mathcal{M} ^{\varepsilon _{n}}$ keeps the propotion between states within each level of $\lambda _{i}(w),$ this implies that $\sigma _{i}(w)$ is optimal to $\lambda _{i}(w).$ Also, the state in the probabilistic model of $G$ corresponding to $\mathcal{M}^{\varepsilon _{n}}$ supporting $\sigma _{i}(w)$ expresses common full belief in caution and $\varepsilon $-perfect trembling condition. Since $\overline{\mathcal{M}}$ is surjective, it follows that the corresponding type in the lexicographic model for $\overline{\mathcal{M}}$ expresses common full belief in caution and primary belief in rationality. Therefore $\sigma _{i}(w)$ is perfect rationalizable in $\overline{\mathcal{M }}$. Since this argument holds for all $i\in N,$ it follows that $w\in \mathbb{CB}^{1}(LRAT).$ //
In Section (ref), we characterized permissibility by ordered Kripke model. Though it is desirable to characterize other rationalizability concepts, e.g., proper rationalizability (Asheim a01), in the ordered Kripke model, we think it is difficult, if not impossible. The reason is that in this framework, the difference between perfect and proper rationalizabilities is at what kind of order $\lambda _{i}(w)$ gives on $ R_{i}(w),$ which more relies on the interpretation than on the structure. In other words, by changing the order on accessable states we can characterize proper rationalizability; but this is attributed to the interpretation we give to each state, not to any structural properties of the Kripke frame $ (W,\{R_{i}\}_{i\in N})$ like seriality or transitivity.
On the other hand, using the approach introduced in Section (ref).3, proper rationalizability can be discussed as the limit of probabilistic Kripke models. Let $G$ be a 2-person strategic form game and a cautious ordered Kripke model $\overline{\mathcal{M}}$ satisfies a condition parallel to \textquotedblleft respecting the opponent's preferences\textquotedblright . For $\varepsilon >0,$ consider $\mathcal{M}^{\varepsilon }=(W^{\varepsilon },\{R_{i}^{\varepsilon }\}_{i\in N},\{p_{i}^{\varepsilon }\}_{i\in N},\{\sigma _{i}^{\varepsilon }\}_{i\in N})$ a probabilistic Kripke model of $G$ satisfying conditions (a), (b) in Section (ref).3 and (c$ ^{\prime }$) for each $i\in N,$ $w\in W$, and $w^{\prime },w^{\prime \prime }\in R_{i}(w),$ $0<p_{i}^{\varepsilon }(w)(w^{\prime })\leq \varepsilon p_{i}^{\varepsilon }(w)(w^{\prime \prime })$ if $\lambda _{i}(w)(k^{\prime })(w^{\prime })>0$, $\lambda _{i}(w)(k^{\prime \prime })(w^{\prime \prime })>0$, and $k^{\prime \prime }>k^{\prime }.$ It can be seen that (1) each probabilistic Kripke model satisfying (a), (b), and (c$^{\prime }$) characterizes some $\varepsilon $-perfect rationalizable strategies; (2) $ \overline{\mathcal{M}}$ is the limit of a sequence $\{\mathcal{M} ^{\varepsilon _{n}}\}_{n\in \mathbb{N}}$ satisfying (a), (b), and (c$ ^{\prime }$); and (3) the perfect rationalizable strategies characterized in $\overline{\mathcal{M}}$ is limits of $\varepsilon _{n}$-perfect rationalizable strategies characterized by $\{\mathcal{M}^{\varepsilon _{n}}\}_{n\in \mathbb{N}}$.
In this paper we have defined the ordered Kripke model to capture the concept of rationality under lexicographic belief hierarchy by a semantical approach. It is wondered that whether there exists a syntactic approach corresponding to that semantic framework, like the one developed in Bonanno b08 for the standard Kripke model for games. A critical property for that syntactic system, if exists, is that the change of the criterion for truth value from the first order to higher orders in the hierarchy, that is, in the first order we need (at most) to check every accessible state, while in the second order $\mathbb{B}_{i}^{1}$ we need only to check the first level states, etc. Works are expected in this direction.