%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%     A fuzzy perceptive value for multi-variate stopping     %
%     problem with a monotone rule                            %
%     by M.Kurano, M. Yasuda, J.Nakagami and Y.Yoshida        %
%                                                             %
%     2006/11/7                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bicfirstauthor[Faculty of Education, Chiba University, Chiba 263-8522 Japan,%
  %{\tt kurano@faculty.chiba-u.jp}
]{Masami}{Kurano}
\bicsecondauthor[Faculty of Science, Chiba University, %
  Chiba 263-8522 Japan,%
  %{\tt yasuda@math.s.chiba-u.ac.jp}
]{Masami}{Yasuda}
\bicthirdauthor[Faculty of Science, Chiba University, %
  Chiba 263-8522 Japan, 
  %{\tt nakagami@math.s.chiba-u.ac.jp}
]{Jun-ichi}{Nakagami}
\bicfourthauthor[Faculty of Economics and Business %
  Administration, The University of Kitakyushu, Kitakyushu 802-8577 Japan,
  %{\tt yoshida@kitakyu-u.ac.jp}
]{Yuji}{Yoshida}

\bictitle{A fuzzy perceptive value for multi-variate stopping
     problem with a monotone rule}
\bicvol{39}
%\bicno{99}
\bicyear{2007}

\bickeyword{
  Equilibrium point, Fuzzy perception equilibrium payoff, 
  Fuzzy perception stopping problem, Monotone logical rule, Multi-variate stopping problem.
}

%
%
%
%-------------------------the text-----------------------
\begin{document}
\maketitle   %-------------------------------------------

\begin{abstract}
   This paper discusses a perception-based theory for a multi-variate  
   stopping problem 
   with a monotone rule by \cite{YKN82}. 
   The problem is developed by using the perceptive analysis 
   of the previous work(\cite{KYNY04}). 
   We will show a recursive equation which determines the perceptive
   value of its equilibrium point for the model.
\end{abstract}

\addtocounter{page}{0}



\section{Introduction}
\label{Intro}% section 1

  In a social life or in business, a group decision-making cannot be
avoided and be carried up by taking each individual opinions into
the whole group decision. In our previous works(\cite{YKN82}), 
the multi-variate stopping problem by $p$-players($p\ge 2$) 
has been formulated. 
The key point of a group decision at each stage is to sum up 
each individual declaration(opinion) by cooperating a monotone logical
rule in order to make the group decision, that is, "stop" or "continue"
the multi-variate stochastic process.

In order to review the result and extend it to our fuzzy model, we firstly
consider the following processes: Let $X=(X_1,\cdots,X_N)$ be a sequence
of $p$-dimensional random vectors on a probability space 
$(\Omega, {\cal B}, P)$. The planning time horizon is $N$ and $p$ means
a number of players. That is,
$X_n =(X_n^1,\cdots,X_n^p), n=1,\cdots,N$ with $X_n^i \,  (i=1,\cdots,p)$ 
a payoff for player $i$ when the process have stopped at stage $n$.

To describe an individual declaration of each player, let us consider
$\{0,1\}$-valued, $\{X_n\}$-adapted process 
$d^i = (d_1^i, \cdots ,d_N^i), i=1,\cdots, p$.
Such a sequence $d^i$, called an individual strategy for player $i$,  
is interpreted as follows: On the basis of the observed value
$X_1, \cdots,X_n$ at stage $n$, the decision $d_n^i = 1$ means
that player $i$ declares to stop the process and $d_n^i = 0$ is
otherwise. The set of all individual strategies of player $i$ will be
denoted by $D^i(X)$ depending on $X = \{X_n^i, n=1,\cdots , N, i=1,\cdots ,p\}$ 

A \{0,1\}-valued $p$-variate logical function 
$\pi = \pi(x^1,\cdots, x^p ): \{0,1\}^p \rightarrow \{0,1\} $  
is said to be monotone if 
$ \pi(0,\cdots,0)=0, \pi(1,\cdots,1)=1 $
and
$ \pi(x^1,\cdots, x^p ) \leq  \pi(y^1,\cdots, y^p ) $
whenever $x^i \le y^i $ for each $i$ (\cite{F71, YKN82}).
%
A stopping time of the process is defined by the individual strategy 
and the monotone rule:
For any $d=(d^1,\cdots ,d^p) \in D(X):= \displaystyle\prod_{i=1}^p D^i(X)$
with $d^i = (d_1^i,\cdots ,d_N^i) \in D^i(X), i=1,\cdots ,p$,
\[ 
  t_{\pi}(d) \triangleq \{ \mbox{ smallest } \, n \, \mbox{ such that } 
  \pi (d_n^1,\cdots , d_n^p) = 1\} 
\] 
$ \mbox{ with } d=(d^i;i=1,\cdots ,p),
  d^i = (d_n^i; n=1,\cdots ,N).$ 
%
Thus, the expected payoff of player $i$ 
for $X = \{X_n^i, n=1,\cdots , N, i=1,\cdots ,p\}$ 
with the individual 
strategy $d$ under the monotone rule $\pi $
is defined by $ E[X_{t_{\pi }(d)}^i]$.

  Under the assumption that the value of random vectors is known to all
players, an equilibrium point 
${}^*\!d = ({}^*\!d^1,\cdots , {}^*\!d^p) \in D$  
is explicitly given and its corresponding equilibrium payoff is 
calculated by a recursive equation of Dynamic Programming
in \cite{YKN82}.
Hereafter the above problem will be denoted by PROB\{$X|\pi,N$\}
to distinguish it from a perceptive model given in the sequel section.
  In many practical situation, we are often forced with the case that 
the value of the sequence of random variable is partially observed 
by dimness of perception or measurement imprecision. In order to treat 
with such a situation, we use a fuzzy perception function(\cite{BT87})
by which the perception level of each random variable is represented.

  In this paper, we give a fuzzy perception model 
F-PROB\{$\widetilde{X}|\pi,N$\} which extends a probability model 
PROB\{$X|\pi,N$\} and characterize the perception value of each player's
equilibrium payoff. The payoff can be calculated by a fuzzy recursive 
equation. This problem is motivated by Zadeh's paper(\cite{Z02}) on
a perceptive-based probability theory in which the perception value of 
the objective function under possibility constraints is introduced and 
calculated using a generalized extension principle.

The perceptive analysis developed in this paper is related to our
previous works. 
A model of stopping problems is 
formulated in \cite{KYNY04} and  that of Markov decision processes is 
in \cite{KYNY05a} and \cite{KYNY05b}.
However the basic assumption implemented in the previous stopping
problem is different from this paper. The model which we want 
to consider here is restricted within two decision case, that is, 
$\{ 0, 1\}$. Its general formulation of a fuzzy decision case 
seems to be a quite difficult.



\section{A fuzzy perception function and a pseudo order}
\label{pseudo}%section 2
%-----------
In this section we will give some notion and 
the definition of fuzzy perception functions. Also, a pseudo order
on the set of fuzzy sets is described because the fuzzy perception
model for multi-variate stopping needs this notion. 

  For any set $A$, a fuzzy set on $A$ will be denoted by its membership
function $\widetilde{a} : A \rightarrow [0,1]$. The $\alpha$-cut of
$\widetilde{a}$ is given by 
$\widetilde{a}_\alpha := \{ x \in A \vert \widetilde{a}(x)\geq \alpha \} 
(\alpha \ne 0)$ and, for $\alpha =0$,
$\widetilde{a}_0 := \mbox{cl}\{ x \in A \vert \widetilde{a}(x) > 0 \} $ 
where cl means the closure of set. For other notions of fuzzy sets, 
refer to \cite{DP80,Z65}.
  Let $\mathbb{R}$ be the set of all real numbers and $\cal{F}(\mathbb{R})$
the set of all fuzzy sets on $\mathbb{R}$,  
and $\widetilde{\mathbb{R}} \subset \cal{F}(\mathbb{R})$ 
the set of all fuzzy numbers on $\mathbb{R}$,
i.e.,
$\widetilde{r} \in \widetilde{\mathbb{R}}$ means that 
$\widetilde{r} : \widetilde{\mathbb{R}} \rightarrow [0,1]$ is 
normal, upper semi-continuous and fuzzy convex and has a compact support. 
Let $\mathbb{C}$ be the set of all bounded and closed intervals of 
$\mathbb{R}$. Then, for any $\widetilde{r} \in \widetilde{\mathbb{R}}$, it
obviously holds that $\widetilde{r}_\alpha \in \mathbb{C} \,(\alpha \in [0,1])$.
So, we write $\widetilde{r}_\alpha = [{}^-\!r_\alpha , {}^+\!r_\alpha ]
\,(\alpha \in [0,1])$ as a closed interval.
  A map $\widetilde{X} : \Omega \rightarrow \widetilde{\mathbb{R}}$ is called
a fuzzy perception function(\cite{BT87}) or a fuzzy random variable
(\cite{PR86}) if, 
%for each $\alpha \in [0,1]$ and $\omega \in \Omega $, 
for each $\alpha$ and $\omega$,
the closed interval 
$\widetilde{X}_\alpha(\omega) = [{}^-\!\widetilde{X}_\alpha(\omega), 
{}^+\!\widetilde{X}_\alpha(\omega)]$ is endowed with each measurable end points.

  Let denote all of integrable random variable on $(\Omega , {\cal B}, P)$ 
by $\mathcal X$.
For any fuzzy random perception function $\widetilde{X}$, its expectation
$E\widetilde{X} \in \widetilde{\mathbb{R}}$ is defined by
\[
   E\widetilde{X}(x) \triangleq \sup\{ \widetilde{\mu}(\widetilde{X})(X); X \in {\mathcal X}, 
   E(X) = x \}   
\] 
provided that $\widetilde{\mu }(\widetilde{X})$ is a fuzzy set on $\mathcal X$ and
\( 
   \widetilde{\mu }(\widetilde{X})(X) = 
   \inf_{\omega \in \Omega }\widetilde{X}(\omega)(X(\omega ))  
\) 
for all $X \in \mathcal X$.
Obviously, we have
\[ 
  E(\widetilde{X}_\alpha ) =
  \left[ \int {}^-\!\widetilde{X}_\alpha(\omega ) \, dP(\omega ),
  \int {}^+\!\widetilde{X}_\alpha(\omega ) \, dP(\omega ) \right] 
\] 

  Next we define a binary relation $\preccurlyeq$ on ${\cal F}({\mathbb R})$
as follows.

\begin{df}
For any $\widetilde{s},\widetilde{r} \in {\cal F}({\mathbb R})$, 
a relation $\widetilde{s} \preccurlyeq \widetilde{r}$ holds 
if both of {\rm (i)} and {\rm (ii)} are satisfied simultaneously:
\begin{enumerate}
  \item[{\rm (i)}] For any $x \in {\mathbb R}$ and $\epsilon > 0$, there exists $y \in  
     {\mathbb R}$ such that $x \leq y$ and $\widetilde{s}(x) < \widetilde{r}(y) +
     \epsilon $.
  \item[{\rm (ii)}] For any $y \in {\mathbb R}$ and $\epsilon > 0$, there exists $x \in  
     {\mathbb R}$ such that $x \leq y$ and $\widetilde{s}(x) > \widetilde{r}(y) -
     \epsilon $.
\end{enumerate}
\end{df}

This binary relation is an extension of the well known partial order called 
the fuzzy max order(\cite{RR85}). We should prove its property of pseudo 
ordering.

\begin{lem}
  The binary relation $\preccurlyeq$ is a pseudo order on ${\cal F}(\mathbb R )$. 
\end{lem}
{\it proof.}
First it trivially holds that $ \widetilde{s} \preccurlyeq \widetilde{s}$
for $ \widetilde{s} \in {\cal F}(\mathbb R )$.
The next is to show the transitivity. 
Assume that $\widetilde{s} \preccurlyeq \widetilde{t}$ and 
$\widetilde{t} \preccurlyeq \widetilde{r}$
for $\widetilde{s},\widetilde{t},\widetilde{r} \in {\cal F}(\mathbb R )$.
From $\widetilde{s} \preccurlyeq \widetilde{t}$ it follows that for any
$x \in {\mathbb R}$ and $\epsilon_1 > 0$, there exists $y \in {\mathbb R}$
which satisfy $x \leq y$ and $\widetilde{s}(x) < \widetilde{t}(y) + \epsilon_1$.
For this $y$, we select $z \in {\mathbb R}$ and $\epsilon_2 >0$
with $\widetilde{t}(y) < \widetilde{r}(z) + \epsilon_2$ and $y \leq z $
since $\widetilde{t} \preccurlyeq \widetilde{r}$.
Thus, we have $\widetilde{s}(s) < \widetilde{r}(z) + \epsilon_1 + \epsilon_2$ 
with $x \leq z$.
Similarly, there exists $x \in {\mathbb R}$ such that 
$\widetilde{s}(x) > \widetilde{r}(z) - \epsilon$ and $x \leq z$ for any
$z \in {\mathbb R}$ and $\epsilon >0$.
These shows that $\widetilde{s} \preccurlyeq \widetilde{r}$, which completes the proof.
\hfill $\square$


%\section{Multi-variate stopping problem}
%\label{stop}% section 3



\section{Fuzzy perception model of multi-variate stopping}
\label{stopmodel}% section 2

In this section a fuzzy perception model for multi-variate stopping problem
is defined.
Before the definition, we need a next assumption on the given probability
space which is important to show the continuity of random vectors.
This cause the determination of each individual decision. 

\begin{asm}
\label{prb-meas}
The probability measure $P$ with underling space $(\Omega , {\cal B}, P)$
is assumed to be non atomic. 
\end{asm}

Let denote by ${\mathcal X}^{p\times N}$ the set of all $N$-dimensional row
random vector whose elements are mutually independent $p$-dimensional column
random vectors. That is, a random variable $X_n^i \in {\mathcal X}$ means 
a payoff for players $i=1,\cdots,p$ at time-parameter $n=1,\cdots,N$.
Composing each of time stage and whole of players payoff, it is 
constructed as
\[ 
  {\mathcal X}^{p\times N} \triangleq  
   \left\{ X=(X_1,\cdots,X_N) \Biggm|% 
    \begin{array}{l}
        X_n = (X_n^1, \cdots ,X_n^p)^T \in {\mathcal X}^p  
        \mbox{ and } X_n^i \in {\mathcal X} \\ 
        \mbox{ are mutually independent for } 
        n=1,\cdots, N, \\ i=1,\cdots, p
    \end{array}\right\}
\] 
where $(\cdots )^T$ is the transpose of a vector. 

\begin{df}
\label{indvstra}
  An individual stopping strategy function $\delta^i$ of player 
  $i(i=1,\cdots, p)$  are defined as
  the map $\delta^i=(\delta_n^i; n=1,\cdots, N) : 
  X \in {\mathcal X}^{p \times N} \rightarrow D^i(X)$. 
\end{df}

We denote by $\Delta^i$ the set of all individual stopping
strategy functions of player $i$ and
$\Delta = \Pi_{i=1}^p \Delta^i$. Simply call $\delta \in \Delta $
as a stopping strategy function.  
Let $\widetilde{X} =(\widetilde{X_n};n=1,\cdots,N)$ be a sequence of
$p$-dimensional fuzzy perception function with
$\widetilde{X_n} = (\widetilde{X}_n^i)$.
For any $\delta =(\delta^i;i=1,\cdots, p) \in \Delta$ and
a monotone rule $\pi$, the $\delta$-stopped fuzzy perception payoff
of player $i(i=1,\cdots, p)$ is a fuzzy set on ${\mathbb R}$ and 
defined by
\begin{equation}\label{vofi}
  \widetilde{v}^i(\delta)(x) \triangleq \sup_{X \in A} \mu_{\widetilde{X}}(X)  
\end{equation}  
where the supremum $X \in A$ is taken all over on the range
of set 
$$  A \triangleq \left\{ X \in {\mathcal X}^{p \times N} 
  \Bigm\vert E\left(X^i_{t_\pi (\delta (X))} \right) = x \right\} $$
and
\begin{equation}\label{mux}
  \mu_{\widetilde{X}}(X) \triangleq  \inf_{\omega \in \Omega } 
     \widetilde{X}_1(X_1(\omega)) \wedge \cdots \wedge \widetilde{X}_N(X_N(\omega))
\end{equation}
and
$
 \widetilde{X}_n(X_n(\omega )) \triangleq
 \widetilde{X}_n^1(X_n^1(\omega)) \wedge \cdots \wedge \widetilde{X}_n^p(X_n^p(\omega)).
$

The stopping strategy function ${}^*\!\delta = ({}^*\!\delta^i; i=1,\cdots,p )
\in \Delta $ is called an equilibrium point (cd. \cite{V77}) if for each 
$i (i=1,\cdots, p)$ it holds that
\begin{equation}\label{eqpoint}
  \widetilde{v}^i({}^*\!\delta ) \succcurlyeq 
  \widetilde{v}^i({}^*\!\delta^{-i} / \delta^i  )
\end{equation}
for all $\delta^i \in \Delta^i$, where 
${}^*\!\delta^{-i} / \delta^i \triangleq  
({}^*\!\delta^1, \cdots , {}^*\!\delta^{i-1}, 
\delta^{i} , {}^*\!\delta^{i+1},\cdots,{}^*\!\delta^{p})$.

The objective of our fuzzy perception problem is to find an equilibrium point 
${}^*\!\delta\in \Delta$ and to calculate
the equilibrium fuzzy perception payoff $\widetilde{v}^i({}^*\!\delta ) (i=1,\cdots,p)$.
This problem will be denoted by F-PROB$\{\widetilde{X} \vert \pi, N\}$ in the afterwards. 



\section{Equilibrium fuzzy perception payoffs}
% and recursive equation}
%model}
\label{fuzzy}% section 4

In order to discuss F-PROB$\{\widetilde{X} \vert \pi, N\}$, we need the following
results on PROB$\{X \vert \pi, N\}$ given in the previous work(\cite{YKN82}).

For any $p$-dimensional random variables $\{ X_n=(X_n^i; i=1,\cdots,p); n =1,\cdots,N\} $,
we define a sequence of vectors $\{ v_n=(v_n^i; i=1,\cdots,p); n =1,\cdots,N\} $
inductively by, for each $i=1,\cdots,p$,
\begin{equation} \label{EQ1}
    \left\{ \begin{split}
    v_1^i &= E(X_N^i) \\ 
%\end{equation}
%\begin{equation} \label{EQ2}
    v_{n+1}^i &= v_n^i + \alpha^i(v_n|X_{N-n}) E(X_{N-n}^i -v^i)^+ 
                    - \beta^i(v_n|X_{N-n}) E(X_{N-n}^i -v^i)^- 
    \end{split}
    \right. 
\end{equation}
where
\begin{equation} \label{EQ3}
    \alpha^i(v_n | X_{N-n}) = P[\pi(I_{C_n^1},\cdots,I_{C_n^{i-1}},
       1,I_{C_n^{i+1}},\cdots,I_{C_n^{p}}) = 1 ], 
\end{equation}
\begin{equation} \label{EQ4}
    \beta^i(v_n|X_{N-n}) 
     = P[\pi(I_{C_n^1},\cdots,I_{C_n^{i-1}},
      0 ,I_{C_n^{i+1}},\cdots,I_{C_n^{p}}) = 1] 
\end{equation}
and $C_n^j = \{ X_{N-n}^j \geq v_n^j \} (j=1,\cdots,p)$ and
$I_A$ is an indicator of set $A$. 

\begin{lem} {\rm (Corollary 2.2 in \cite{YKN82})} %\lemma 4.1
  For a sequence $\{v_n \}$ given in 
  {\rm (\ref{EQ1})},
  define a strategy
  ${}^*\!d = ({}^*\!d^i) $ with ${}^*\!d^i =({}^*\!d_n^i)
  \in D^{i}(X)$ by ${}^*\!d_n^i = I_{\{ X_n^i \geq v_{N-n}^i \} } \,\,  
  (n=1,\cdots,N; i=1,\cdots,p)$. Then
  ${}^*\! d$ is an equilibrium point for PROB($X \vert \pi, N $) and
  $ v_N^i = E\left(X_{t_{\pi}({}^*\!d)}\right)$.
\end{lem}

We note here that the equilibrium stopping strategy ${}^*\! d \in D^i(X)$ of player $i$
for PROB($X \vert \pi, N $)
is determined by $X \in {\mathcal X}^{p\times N}$. Therefore 
the equilibrium stopping strategy ${}^*\! d \in D^i(X)$ is recognizable or 
is reflected over $X \in {\mathcal X}^{p\times N}$.
Such an individual stopping function will be defined by
\begin{equation}\label{d*}
 {}^*\! \delta^i(X) = 
   \left({}^*\! \delta_1^i(X), \cdots, {}^*\! \delta_N^i(X)
   \right) 
\end{equation}
explicitly expressing of random variable $X \in {\mathcal X}^{p\times N}$.


\begin{thm}
  For a sequence of %$p$-dimensional 
  fuzzy perception functions 
  $\widetilde{X} = ( \widetilde{X}_1, \cdots, \widetilde{X}_N)$ 
  and a monotone rule $\pi$, a strategy function 
  ${}^*\! \delta^i $ defined in (\ref{d*}) 
  gives an equilibrium point for F-PROB$\{ \widetilde{X} \vert \pi, N \}$.
\end{thm}

\noindent
{\it Proof.}
Let select any $\delta^i \in \Delta^i$. For $x \in {\mathbb R}$ with 
$\widetilde{v}^i({}^*\! \delta^{-i} / \delta^i ) (x) > 0$ and 
$ \epsilon >0 $, there exists $X \in {\mathcal X}^{p\times N}$ such
that $E\left(X_{t_\pi({}^*\! \delta^{-i} / \delta^i ) }^i \right) = x$
and $\widetilde{v}^i({}^*\! \delta^{-i} / \delta^i ) (x) < 
\mu_{\widetilde{X}}(X) +  \epsilon $ from the definition of (\ref{vofi}). 
Since ${}^*\! \delta (X)$ is an equilibrium point for 
PROB$\{X \vert \pi, N \}$, it follows that
$ x \leq y \triangleq E\left(X_{t_{\pi} ({}^*\! \delta (X))}^i\right) $
implies $\widetilde{v}^i({}^*\! \delta^i)  \geq 
\mu_{\widetilde{X}}(X)$. And then
$ \widetilde{v}^i({}^*\! \delta^{-i} / \delta^i)(x) <  
\widetilde{v}^i({}^*\! \delta)(y) + \epsilon$. 

Similarly, for any $y \in {\mathbb R} $ with 
$\widetilde{v}^i({}^*\! \delta )(y) > 0 $ and $\epsilon > 0$,
we have 
$ \widetilde{v}^i({}^*\! \delta^{-i} / \delta^i)(x) + \epsilon >  
\widetilde{v}^i({}^*\! \delta)(x) $ provided that
$y \geq x \triangleq E\left(X_{t_{\pi ({}^*\! \delta (X))}}^i \right)$.
Thus, combining two order relations of (\ref{eqpoint}), we have
$\widetilde{v}^i({}^*\! \delta^{-i} / \delta^i)  \preccurlyeq   
\widetilde{v}^i({}^*\! \delta) $.
\hfill $\square$


\vspace{2ex}
The next lemma is to prepare an explicit form of the perception value 
corresponding the equilibrium point. This is a general case for 
the monotone rule.
\begin{lem}
  For $v = (v^1, \cdots,v^p) \in R^p$
  and $Y = (Y^1, \cdots,Y^p) \in {\mathcal X}^p$, let, for each
  $j=1,\cdots,p$, 
  \begin{equation}\label{Gop}
    G^i(Y,v) \triangleq v^i + \alpha^i(v \vert Y) E\left[Y^i - v^i\right]^+ 
                 - \beta^i(v \vert Y) E\left[Y^i - v^i\right]^-,
  \end{equation}
  where
  \begin{align}
    \alpha^i(v \vert Y)  & \triangleq P[\pi(I_{C^1},\cdots,I_{C^{i-1}},
      1, I_{C^{i+1}},\cdots,I_{C^p}) =1 ], \\  
    \beta^i(v \vert Y)  & \triangleq P[\pi(I_{C^1},\cdots,I_{C^{i-1}},
      0, I_{C^{i+1}},\cdots ,I_{C^p}) =1 ]
  \end{align}
and $ C^j \triangleq \{ Y^j \geq v^j\}$.  
Then, $ G^i(Y,v)$ $(i=1,\cdots,p)$ is continuous in $ (Y,v) \in {\mathcal X}^p\times {\mathbb R}^p$.
\end{lem}

\noindent
{\it Proof.}
Since the underlying $P$ is assumed to be non-atomic by (\ref{prb-meas}), 
the dominated convergence theorem implies that $ \alpha^i(v \vert Y)$, 
$\beta^i(v \vert Y)$ are continuous in $(Y,v)$. And both of 
$E\left[Y^i - v^i\right]^+$ and $E\left[Y^i - v^i\right]^-$ are also continuous. Hence
the assertion is obtained immediately.
\hfill $\square$

\vspace{2ex}
For a given sequence $\widetilde{X} = (\widetilde{X}_1, \cdots,\widetilde{X}_N)$,  
let define
$$
   \widetilde{X}(n) \triangleq (\widetilde{X}_{N-n+1}, \widetilde{X}_{N-n+2},
   \cdots,\widetilde{X}_N)
$$
for $n = 1, \cdots,N$.
By considering the F-PROB$\{ \widetilde{X}(n) \vert \pi, n\} $,
we denote $\widetilde{v}_n=(\widetilde{v}_n^i; i=1,\cdots,p) $
the fuzzy perception value of its equilibrium point.

\begin{thm}  \label{iteration}
It holds that, for $n=1,\cdots, N, \,\, i=1,\cdots , p $,
\begin{itemize}
   \item[{\rm (i)}] $\widetilde{v}_n^i \in \widetilde{{\mathbb R}}$.
   \item[{\rm (ii)}] The $\alpha $-cut is given as
        $\widetilde{v}_{n,\alpha }^i
       = [ {}^-\! \widetilde{v}_{n,\alpha }^i, 
       {}^+\! \widetilde{v}_{n,\alpha }^i]$
   where
   \begin{equation}\label{it-eq-a}
       {}^-\! \widetilde{v}_{n,\alpha }^i = E[{}^-\! \widetilde{X}_{n,\alpha }^i],
       \quad 
       {}^+\! \widetilde{v}_{n,\alpha }^i = E[{}^+\! \widetilde{X}_{n,\alpha }^i].
   \end{equation}
   \item[{\rm (iii)}]
       \begin{equation}\label{it-eq-b}
       \left\{
         \begin{array}{l}
          {}^-\! \widetilde{v}_{n,\alpha }^i 
           = \min_{Y,v} G^i (Y,v)            \\[1.5ex]
          {}^+\! \widetilde{v}_{n,\alpha }^i
           = \max_{Y,v} G^i (Y,v)
         \end{array}
       \right.
       \end{equation}
    where the $\min_{Y,v}$ and $\max_{Y,v}$ are taken over the set
    $\{ (Y,v) \in {\mathcal X}^p \times {\mathbb R}^p \, \vert \,
      {}^-\! \widetilde{X}_{N-n} \leq Y \leq {}^+\! \widetilde{X}_{N-n}
      {}^-\! \widetilde{v}_{n} \leq v \leq {}^+\! \widetilde{v}_{n} \} $
      with the order relation $\leq$ of component-wise in ${\mathbb R}^p$.
\end{itemize}
\end{thm}

\noindent
{\it Proof.}
The proof proceeds by induction on $n$.
From the definition of $\alpha$-cut and the expectation of fuzzy set,
$$
  \begin{array}{ll}
  \widetilde{v}_{1,\alpha }^i 
  & = \left\{ x \in {\mathbb R} \,\,  \vert \, \, 
    \widetilde{v}_{1}^i({}^*\!\delta )(x) \geq \alpha \right\} \\[2ex] 
  & = \left\{ E(X_N^i ) \in {} {\mathbb R} \, \, \bigm\vert \, \, 
    X_N^i \in \widetilde{X}_{N,\alpha }^i \right\}  \\[2ex] 
  & = \left[E({}^-\!\widetilde{X}_{N,\alpha }^i), \,\,  
     E({}^+\!\widetilde{X}_{N,\alpha }^i) \right],
  \end{array}
$$ 
which implies that $\widetilde{v}_1^i  \in \widetilde{{\mathbb R}}$
and (\ref{it-eq-a}) holds for $n=1$.
To prove it for $n$, suppose that (i) and (ii) are hold in case of
$n-1$.
So
$$
  \widetilde{v}_{n+1,\alpha }^i = 
  \left\{ v_{n+1}^i\{ X\} \, \bigm\vert \, 
    \mu_{\widetilde{X}}(X) \geq \alpha ,  X =(X_1,\cdots,X_N) \in 
    {\mathcal X}^{p\times N} \right\}
$$
where $\widetilde{v}_{n+1,\alpha }^i$ is given in (\ref{EQ1}) with respect
to $X \in {\mathcal X}^{p\times N}$ and $\mu_{\widetilde{X}}(X)$ is in 
(\ref{mux}).
Thus, by (\ref{EQ1}), (\ref{EQ3}) and (\ref{EQ4}), we can obtain that
$$
  \widetilde{v}_{n+1,\alpha }^i = 
  \left\{ G_i(Y,v) \, \bigm\vert \, 
    Y \in {\widetilde{X}_{N-n,\alpha }}(X) ,  v \in {\widetilde{v}_{n,\alpha }}
     \right\}  
  = \left[{}^-\!\widetilde{v}_{n+1,\alpha}, \,\,  
     {}^+\!\widetilde{v}_{n+1,\alpha} \right]   
$$ 
holds.
This completes the proof of theorem.
\hfill $\square$



%\section{Optimal fuzzy perceptive value}
%\label{optimal}% section 4




\section{An example of a simple majority rule}
\label{example}% section 5

In this section, we discuss the case of $p=2$ and
the monotone rule is $\pi (x^1, x^2) = \max \{ x^1, x^2\} $,
that is, it is a simple majority rule.
Applying this case to Theorem \ref{iteration},
the sequence of iteration: 
\begin{equation}\label{EX1}
       {}^-\! \widetilde{v}_{1,\alpha }^i = E[{}^-\! \widetilde{X}_{N,\alpha }^i],
       \quad 
       {}^+\! \widetilde{v}_{1,\alpha }^i = E[{}^+\! \widetilde{X}_{N,\alpha }^i].
\end{equation}
and
the recursive equation of (\ref{it-eq-b}), for $i=1,2$ and $n=1,\cdots,N$,
\begin{equation}\label{EX2}
   \left\{
    \begin{array}{l}
    {}^+\!v_{n+1}^i = {}^+\!v_n^i + E({}^+\!X_{N-n}^i - {}^-\!v^i)^+ 
                                  - P({}^-\!X_{N-n}^{3-i} \geq  {}^+\!v^{3-i}) 
                                      E({}^+\!X_{N-n}^i - {}^-\!v^i)^- 
    \\[1.5ex]
    {}^-\!v_{n+1}^i = {}^-\!v_n^i + E({}^-\!X_{N-n}^i - {}^+\!v^i)^+ 
                                  - P({}^+\!X_{N-n}^{3-i} \geq  {}^-\!v^{3-i}) 
                                      E({}^-\!X_{N-n}^{i} - {}^+\!v^{i})^- 
    \end{array}
   \right.
\end{equation}
are obtained. Thus the fuzzy perception value of equilibrium point 
is $\left( \widetilde{v}^1({}^*\! \delta), 
\widetilde{v}^2({}^*\! \delta) \right) 
= ( \widetilde{v}_N^1, \widetilde{v}_N^2)$ for
an individual stopping strategy function $ {}^*\!\delta = ({}^*\!\delta^1, 
{}^*\!\delta^2) $ with a usual result of fuzzy max order.
% since
%$ \alpha^i(v \vert Y)   = P(\max\{1 , \} =1 )  = 1$ and   
%$ \beta^i(v \vert Y)   = P(\max\{  , 0\} =1 ) = P(  )  $.

To show a concrete numerical calculation considered in \cite{KYNY04},
it is treated an independent, uniformly fuzzy random variables.
Let $ \widetilde{X} = ( \widetilde{X}_1, \cdots,\widetilde{X}_N)$
with two players $ \widetilde{X}_n = (\widetilde{X}_n^1, \widetilde{X}_n^2),
n=1,\cdots,N$ and a triangle-fuzzy random variable
\begin{equation}
  \widetilde{X}_n^i = \left(Y_n^i / X_n^i / Z_n^i \right)
  \quad  i =1,2
\end{equation}
i.e. the center is $X_n^i$ with a left and a right spread 
$X_n^i - Y_n^i, X_n^i + Z_n^i$ respectively.
In oder to simplify the calculation, we assume that
these are independent and uniformly distributed as
$$
   X_n^i \sim U[0,1], \quad Y_n^i, Z_n^i \sim U[0, 1/2]. 
$$

%=============================================================
\begin{table}
\caption{Triangular fuzzy number}
\vspace{-0.7cm}
\begin{center}
%WinTpicVersion3.08
\unitlength 1cm
\begin{picture}(  6.0000,  3.6750)(  0.5000, -3.7500)
% LINE 2 0 3 0
% 2 300 400 2500 400
% 
\special{pn 8}%
\special{pa 296 394}%
\special{pa 2461 394}%
\special{fp}%
% LINE 2 0 3 0
% 2 400 300 400 1500
% 
\special{pn 8}%
\special{pa 394 296}%
\special{pa 394 1477}%
\special{fp}%
% LINE 2 0 3 0
% 2 200 1400 2600 1400
% 
\special{pn 8}%
\special{pa 197 1378}%
\special{pa 2560 1378}%
\special{fp}%
% LINE 2 0 3 0
% 2 800 1400 1600 400
% 
\special{pn 8}%
\special{pa 788 1378}%
\special{pa 1575 394}%
\special{fp}%
% LINE 2 0 3 0
% 2 1600 400 2000 1400
% 
\special{pn 8}%
\special{pa 1575 394}%
\special{pa 1969 1378}%
\special{fp}%
% LINE 2 2 3 0
% 2 1600 400 1600 1400
% 
\special{pn 8}%
\special{pa 1575 394}%
\special{pa 1575 1378}%
\special{dt 0.045}%
% STR 2 0 3 0
% 3 1400 1500 1400 1600 2 0
% $X_n^i$
\put(3.5000,-4.0000){\makebox(0,0)[lb]{$X_n^i$}}%
% STR 2 0 3 0
% 3 1800 1500 1800 1600 2 0
% $X_n^i + Z_n^i$
\put(4.5000,-4.0000){\makebox(0,0)[lb]{$X_n^i + Z_n^i$}}%
% STR 2 0 3 0
% 3 600 1500 600 1600 2 0
% $X_n^i - Y_n^i$
\put(1.5000,-4.0000){\makebox(0,0)[lb]{$X_n^i - Y_n^i$}}%
% STR 2 0 3 0
% 3 2000 700 2000 800 2 0
% $\widetilde{X}_n^i = ( Y_n^i / X_n^i / Z_n^i)$
\put(5.0000,-2.0000){\makebox(0,0)[lb]{$\widetilde{X}_n^i = ( Y_n^i / X_n^i / Z_n^i)$}}%
% STR 2 0 3 0
% 3 600 100 600 200 2 0
% Trianglular Fuzzy Number 
%\put(1.5000,-0.5000){\makebox(0,0)[lb]{Trianglular Fuzzy Number }}%
% STR 2 0 3 0
% 3 230 450 230 550 2 0
% $1$
\put(0.5750,-1.3750){\makebox(0,0)[lb]{$1$}}%
% STR 2 0 3 0
% 3 230 1450 230 1550 2 0
% $0$
\put(0.5750,-3.8750){\makebox(0,0)[lb]{$0$}}%
\end{picture}%
\end{center}
\end{table}

%=============================================================

In this case the fuzzy perception value of the equilibrium point
$ \widetilde{v}_n = (\widetilde{v}_n^1, \widetilde{v}_n^2)$ could be 
explicitly expressed as follows.

Here we use a notation ${\mathbb T}$ and ${\mathbb F}$, 
a distribution function of $W$, cooperating with a parameter 
$a$ by
$$ {\mathbb T}(x) \triangleq E(W - x)^+  \quad
(- \infty < x < +\infty )$$
where random variables $W = X + Y$, $Z = X - Y$ with $X \sim U[0,1]$ and
$Y \sim U[0,a] \, (0< a < 1/2)$, then
$$
 {\mathbb T}(x) =
 \left\{ \begin{array}{l}
          1/2 + a/a - x, \, (x<0) \\
          1/2 + a/2 - x + x^3/(6a), \, (0 \leq  x < a) \\
          1/2 + a/2 + a^2/6 -(1+a/2)x + x^2/2 , \, (a \leq  x < 1) \\
          (1+a -x)^3/(6a), \, (1 \leq  x < 1+ a) \\
          0, \, ( 1+a \leq  x)
         \end{array}
 \right.
$$
$$
 {\mathbb F}(x) \triangleq P(W \leq x) =
 \left\{ \begin{array}{l}
          0, (x<0) \\
          x^2/(2a), (0 \leq  x < a) \\
          -a/2 + x, (a \leq  x < 1) \\
          -(a+1/a)/2 -(1+1/a)x - x^2/(2a), (1 \leq  x < 1+ a) \\
          1, ( 1+a \leq  x)
         \end{array}
 \right.
$$
in cases like this.
Since $E(W-x)^- = {\mathbb T}(x) - a/2 - 1/2 +x $, 
$E(Z-x)^+ = {\mathbb T}(x+a)$,
$E(Z-x)^- = {\mathbb T}(a + x) + a/2 - 1/2 +x$ and 
$ P(Z \leq x) = {\mathbb F}(a+x)$
are hold, 
we can express the above recursive iteration (\ref{EX1}) and (\ref{EX2})
as follows:
By the symmetry of players, $\widetilde{v}_n^1 = \widetilde{v}_n^2$ and, set
\begin{equation}
   \widetilde{u}_n \triangleq \widetilde{v}_n^1 = \widetilde{v}_n^2
\end{equation}
and we want to grasp a form of $\alpha$-cut 
$ \widetilde{u}_{n,\alpha } = 
    \left[  {}^-\!\widetilde{u}_{n,\alpha},
       {}^+\!\widetilde{u}_{n,\alpha } \right]$
for each $ \alpha \in [0, 1]$.
By substituting $a = (1 - \alpha )/2$,
\begin{equation}
  {}^-\!\widetilde{u}_{1,\alpha } = 1/2 - (1-\alpha )/4, \quad 
  {}^+\!\widetilde{u}_{1,\alpha } = 1/2 + (1-\alpha )/4,
\end{equation}
for $n= 1,\cdots, N-1$, we had calculated (\ref{EX2}) as 
\begin{equation}
  \left\{
   \begin{array}{ll}
     {}^-\!\widetilde{u}_{n+1,\alpha } 
     &=  {}^-\!\widetilde{u}_{n,\alpha } %
      + {\mathbb T}({}^+\!\widetilde{u}_{n,\alpha } + (1-\alpha )/2 )  
         - \left(1 - {\mathbb F}({}^+\!\widetilde{u}_{n,\alpha })\right) 
     \\ 
     & \hspace{5em} \times  \left\{ {\mathbb T}({}^+\!\widetilde{u}_{n,\alpha } + (1-\alpha )/2) 
           + (1-\alpha )/4 - 1/2 + {}^+\!\widetilde{u}_{n,\alpha } \right\}, 
     \\[1.5ex] 
     {}^+\!\widetilde{u}_{n+1,\alpha } 
     &=   {}^+\!\widetilde{u}_{n,\alpha } %
          + {\mathbb T}({}^-\!\widetilde{u}_{n,\alpha } + (1-\alpha )/2 )  
         - \left(1 - {\mathbb F}({}^-\!\widetilde{u}_{n,\alpha })\right) 
     \\ 
     & \hspace{5em} \times  \left\{ {\mathbb T}({}^-\!\widetilde{u}_{n,\alpha } + (1-\alpha )/2) 
           - (1-\alpha )/4 - 1/2 + {}^-\!\widetilde{u}_{n,\alpha } \right\}. 
   \end{array}
  \right.
\end{equation}


The numerical result is as follows:
We use the notation
$u[1] = 1/2,  u[n+1] := 1/2 + u[n]^{3/2}, n=1,2,\cdots $
with $a=(1-\alpha)/2$. The case of $\alpha=1, 
a=(1-\alpha)/2=0.0$ corresponds the non-fuzzy model 
PROB model with the equilibrium value $u[n]$ 
and, ^^ ^^ lower" ^^ ^^ upper"
means the extremal point of interval for each level 
$ \alpha$.

\begin{table}
 \caption{The $\alpha-$cuts of the fuzzy equilibrium 
 point}
\begin{center}
\begin{tabular}{|r|c|cc|cc|} \cline{2-6}
\multicolumn{1}{c|}{} & $\alpha=1$ & \multicolumn{2}{|c|}{$\alpha=0.7$} 
  & \multicolumn{2}{|c|}{$\alpha=0.5$}\\   
\multicolumn{1}{c|}{} & $(1-\alpha)/2=0.0$ 
  & \multicolumn{2}{|c|}{$(1-\alpha)/2=0.15$} 
  & \multicolumn{2}{|c|}{$(1-\alpha)/2=0.25$}\\ \hline \hline 
$n$ & $u[i]$ & lower  & upper  & lower  & upper  \\ \hline \hline 
  1 & 0.5000 & 0.4250 & 0.5750 & 0.3750 & 0.6250 \\
  2 & 0.5625 & 0.3811 & 0.7654 & 0.2669 & 0.9004 \\
  3 & 0.5890 & 0.2851 & 0.9995 & 0.1494 & 1.2709 \\ 
  4 & 0.6022 & 0.2418 & 1.3124 & 0.1494 & 1.7487 \\ 
  5 & 0.6092 & 0.2418 & 1.6604 & 0.1494 & 2.2265 \\ 
  6 & 0.6130 & 0.2418 & 2.0085 & 0.1494 & 2.7043 \\ 
  7 & 0.6152 & 0.2418 & 2.3566 & 0.1494 & 3.1821 \\ 
  8 & 0.6164 & 0.2418 & 2.7047 & 0.1494 & 3.6599 \\ 
  9 & 0.6171 & 0.2418 & 3.0528 & 0.1494 & 4.1377 \\ 
 10 & 0.6175 & 0.2418 & 3.4008 & 0.1494 & 4.6155 \\ \hline  
\end{tabular}
\end{center}
\end{table}

\vspace{2ex}

\begin{acknowledgement}
Authors should show thanks to anonymous referee 
who point out the typographical errors.
\end{acknowledgement}


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\end{thebibliography}

\vspace{5mm}

\begin{flushleft}
{\it Received April 24, 2006}

{\it Revised November 7, 2006}
\end{flushleft}

\end{document}

  Then,
  holds.



