数理解析研究所講究録 1263 巻 2002 年 103-116

103
An interval matrix game andits extensions to fuzzy and stochastic games(区間値およびファジー値をもつ行列ゲーム)
千葉大学教育学部 蔵野正美 (Masa mi Kurano)
Faculty of Education, Chiba University
千葉大学理学部 安田正實 (Masami Yasuda)
千葉大学理学部 中神潤一 (Jun-ichi Nakagami)
Faculty of Science, Chiba University
北九州大学経済学部 吉田祐治 (Yuji Yoshida)
Faculty of Economics and Business Administration, Kitakyushu University

Abstract
In this paper, we consider an interval matrix game with interval valued payoffs, which is the generation of the traditional matrix game. The “saddle-points”
this interval matrix game are defined and characterized as equilibrium points ofcorresponding non-zero sum parametric games. Numerical examples are given toillustrate our idea. These results are extended to the fuzzy matrix games. Also, we
formulate two person zer0-sum stochastic interval games.
Keywords: Interval game, saddle point, interval payoffs, fuzzy payoffs, equilibrium point,parametric game.

1Introduction and notations
In usual matrix game theory(cf. [25, 26]), all the elements of the payoff matrix are assumed
to be exactly given. But in areal application, we often encounter the case where the information on the required data includes imprecision or ambiguity because of the uncertain
environment. In order to deal with such acase, it is more reasonable to estimate the elements of the payoff matrix by intervals. As for interval approaches to linear programming
problem and decision processes, refer, for example, to $[7, 21]$ and [9] respectively.
In this note, we consider the interval matrix game which is an interval generation of
the traditional matrix game. The saddle points of the interval matrix game are definedand characterized as equilibrium points of corresponding non-zero sum parametric games.
Also, these results are extended to the fuzzy matrix games. Recently, Kurano et al [10]have developed the theory of MDPs in which the immediate rewards are described by use
of fuzzy sets. So that, we consider the question whether these results can be extended tostochastic games with interval or fuzzy payoffs. We shall formulate two person zer0-sum
stochastic interval games in which one-step payoffs are estimated by intervals.
In the reminder of this section, we shall give some notation on interval arithmetics (cf.
[16] and some preliminaries related to the preference relation on intervals.
Let be the set of all real numbers and the set of all bounded and closed intervals
with $a=[a, a]\in \mathbb{C}$ . We will give apartial
by identifying
in R. Note thatby the following definition
order $4,$$)$$\mathbb{C}$$\mathbb{R}$$\mathbb{R}\subset \mathbb{C}$$\prec \mathrm{o}\mathrm{n}$$\mathbb{C}$$a\in \mathbb{R}$104
For [a, b], [c,and [a,, [a, b]d] if ad]. The Hausdorff metri c$d]\in \mathbb{C}$$\neg\prec[c,$$\leqq c$$b]\neq[c,$$\delta([a,$and b$[13])$ on$\leqq d$$\mathbb{C}$, and [a,$b]\prec[c,$d] if [a, b]
is defined by , i.e.,b], [c,$d]):=|a-c|\vee|b$ -d| for [a, b], [c,where $x \vee y=\max\{x, y\}$ . Obviously, the metric space
The following arithmetics are used in the sequel.
For $[a, b]$ ,
and
(A
),
$[c, d]\in \mathbb{C}$$\lambda\in \mathbb{R}$(1.1)$\neg\prec[c,$d]$\delta$$(\mathbb{C}, \delta)$$d]\in \mathbb{C}$,is complete.

$\geq 0$[a,$b]+[c, d]=[a+c, b+d]$ ,(1.2)$\lambda[a, b]=$[ a, Ab].

Then, we have the following.

Lemma 1.1 For any $[a, b]$ ,(i)$[\mathrm{a}1, \ ]$, $[c, d]$ ,$\delta(\lambda[a, b], \lambda[a’, b’])=\lambda\delta([a, b], [a’,b’])$and(scalar)$[d, \mathrm{d}’]\in \mathbb{C}$.

$\lambda\in \mathrm{R}$(A$\geq 0$).

(ii) $\delta([a, b]+[a’, b’], [c, d]+[d, d’])\leq\delta([a, b], [c, d])+\delta([a’, b’], [d, d’])$ .
(iii) $\delta([a, b]+[a’,b’], [a, b]+[d, d’])=\delta([a’, b’], [d, d’])$.

(triangle)(shift)

Let $\mathbb{C}_{+}:=\{a\in \mathbb{C}|a=[a,b]\succ, [0,0]\}$ be the set of nonnegative intervals. Letandbe the set of all $m$-dimensional column vectors andmatrices, called interval
vectors and interval matrices respectively, whose elements are in , i.e.,$\mathbb{C}^{m}$$\mathbb{C}^{m\mathrm{x}n}$$m\cross n$$\mathbb{C}$$\mathbb{C}^{m}:=\{a=(a_{1}, a_{2}, \ldots, a_{m})^{t}|a:\in \mathbb{C}(1\leqq i\leqq m)\}$$\mathbb{C}^{m\mathrm{x}n}:=${A$=(a_{\dot{l}j})|a_{j}\dot{.}\in \mathbb{C}(1\leqq i\leqq m,$$1\leqq j\leqq n)$,}.

We shall identify
interval matrices with interval vectors andinterval matrices
with intervals, so thatand
. Also, we denote byandthesubsets of componentwise non-negative elements inand. We equip
withcomponentwise relations $4,$ . Similarly, we can defineandas the set of realm-dimensional column vectors and real $m\cross n$ matrices. Note that.
For any
with, $A$ will be denoted by $A$ $=[A^{-}, A^{+}]$ ,
where
,
and
.
For
,
and
,
$m\cross 1$$1\cross 1$$\mathbb{C}=\mathbb{C}^{1\mathrm{x}1}$$\mathbb{C}^{m}=\mathbb{C}^{m\mathrm{x}1}$$\mathbb{C}_{+}^{m}$$\mathbb{C}^{m}$$\mathbb{C}_{+}^{m\mathrm{x}n}$$\mathbb{C}^{m\mathrm{x}n}$$\prec$$\mathrm{R}^{m}$$\mathbb{C}^{m\mathrm{x}n}$$\mathrm{R}^{m\mathrm{x}n}$$\mathrm{R}^{m\mathrm{x}n}\subset \mathbb{C}^{m\mathrm{x}n}$$A=(a_{\dot{|}j})\in \mathbb{C}^{m\mathrm{x}n}$$A^{-}=(a_{\dot{|}j}^{-})\in \mathrm{R}^{m\mathrm{x}n}$$a_{\dot{|}j}=[a_{j}^{-}\dot{.}, a_{\dot{|}j}^{+}]$$A^{+}=(a_{j}^{+}\dot{.})\in \mathrm{R}^{m\mathrm{x}n}$$[A^{-}, A^{+}]=\{A\in \mathrm{R}^{m\mathrm{x}1}’|A^{-}\neg\prec A\backslash \prec$$A^{+}\}$$A=(a_{j}\dot{.})$$B=(b_{\dot{|}j})\in \mathbb{C}^{m\mathrm{x}n}$(1.1’)(1.2’)$A+B=$ { $A+B$ |A $\in A$ and B $\in B$ }|A $\in A\}$ ,$\lambda A=\{\lambda A$

where for $C=(c_{ij})$ and$D=(d_{\dot{\iota}j})\in \mathrm{R}^{m\mathrm{x}n}$$[A^{-}+B^{-}, A^{+}+B^{+}]\in \mathbb{C}^{m\mathrm{x}n}$

For any given(1.3)$\lambda\in R_{+}$$\mathrm{D}$$\subset \mathbb{C}$

,

$c$

.

, $C+D=(\mathrm{q}_{j}.+d_{\dot{l}j})$ . Observing $A+B=$

is called a minimal (maximal) point of{d$\in \mathrm{D}$|d$\prec(\succ)c\}=\emptyset$$\mathrm{D}$if.

Th set of all minimal (maximal) point of will be defined bySince the partial order 4on is equivalent to the vector ordering on
the corresponding order cone, the following fact follows easily (cf. [2, 20])$\mathrm{e}$$\mathrm{D}$$\min \mathrm{D}(\max \mathrm{D})$$\mathbb{C}$(cf. [20, 24]).
withas$\mathrm{R}^{2}$$\mathrm{R}_{+}^{2}$

105
Lemma 1.2 Let D be acompact and convex subset of C. Tien [a,$b] \in\min \mathrm{D}(\max$ D)
1] such that $\gamma a+(1-\gamma)b\leqq(\geqq)\gamma a’+(1-\gamma)b’$ for all
if and only if there exists.
$\gamma\in[0,$$[a’, b’]\in \mathrm{D}$In Section 2, an interval matrix game is specified and its saddle points are characterized
as equilibrium points of the corresponding non-zero sum parametric game. Afuzzy matrix
game is investigated in Section 3. In Section 4, anumerical example is given to illustrate
our arguments. In order to formulate the interval stochastic game we need the conceptof aexpectation of interval-valued random variables.
adiscrete random quantity with
be aprobability space andLet. Then, we define the expectation of by
its range$r:\Omegaarrow \mathbb{C}$$(\Omega, \mathscr{B}, P)$$r$$\mathscr{B}(r)=\{c_{1}, c_{2}, \cdots, c_{l}\}\subset \mathbb{C}$(1.4)$E[r]= \sum_{i=1}^{l}c_{i}P(r =c_{i})$.

. The definition
Note that arithmetics in (1.4) is given in (1.1) and (1.2) and $E[r]$of (1.4) is corresponding to the discrete case of the expectation of general fuzzy randomvariables (cf. [18]).
In Section 5stochastic interval games are specified and their saddle points are defined,
which are characterized as equilibrium points of corresponding nonzer0-sum parametricstochastic games in Section 6. In Section 7stochastic interval game are extended to thecase of the multi-dimensional fuzzy payoffs.
$\in \mathbb{C}$2Interval matrix games
interval matrix $A=(a_{ij})\in$
The two person interval matrix game is defined by thehave $m$ pure strategies $\{i|$, where player
and player, $m$ } and pure strategies $\{j|j=1,2, \ldots, n\}$ and if player 1and 2select$i=1,2$ ,$i(1\leqq i\leqq m)$ and $j(1\leqq j\leqq n)$ respectively, the payoff for player 1or the loss for player.
2is estimated by the interval
Let $X$ and be the set of all mixed strategies for player 1and 2respectively, i.e.,$m\cross n$$\mathbb{C}^{m\mathrm{x}n}$$2(\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{r})$$1(\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{r})$$n$$\ldots$$a_{ij}\in \mathbb{C}$$\mathrm{Y}$,$X= \{x= (\mathrm{x}, x_{2}, \ldots,x_{m})^{t}\in \mathbb{R}_{+}^{m}|\sum_{j=1}^{m}x_{i}=1\}$$\mathrm{Y}=\{y=(y_{1}, y_{2}, \ldots, y_{n})^{t}\in \mathbb{R}_{+}^{n}|\sum_{j=1}^{n}y_{i}=1\}$.

Then, for any selected pair of strategies $(x, y)\in X\cross \mathrm{Y}$ the expected payoff for player 1
is estimated by(2.1)$f(x, y):=x^{t}Ay= \sum_{i,j}x_{i}yjaij$.

By arithmetics in (1.1’) and (1.2’) the following holds obviously.

Lemma 2.1. For any x $\in X$ and y(2.2)$\in \mathrm{Y}$, it holds that$f(x, y)=[x^{t}A^{-}y,x^{t}A^{+}y]\in \mathbb{C}$.

106
Definition 1. (cf. $[[14],$ $[24]]$ ) Let
and A. Thensaid to be asaddle point of the interval matrix game A if the following holds:$(x^{*}, y^{*})\in X\cross \mathrm{Y}$(2.3)$\in \mathbb{C}^{m\mathrm{x}n}$$f(x^{*}, y^{*}) \in\max f(X, y^{*})\cap\min f(x^{*},$

where for any (x,$y)\in X\cross \mathrm{Y}$Y}.

$(x^{*}, y^{*})$

is

Y),

, $f(X, y)=\{f(x’,$y) $|x’\in X\}$ and $f(x, \mathrm{Y})=\{f(x, y’)|y’\in$

We note that $f(X,$y) and $f(x,$Y) are compact and convex subset of C.
In order to characterize the saddle point of the interval matrix game $A$ , we introduce
aparametric matrix game
. For each $\gamma\in[0,1]$ and, let$A=[A^{-}, A^{+}]\in \mathbb{C}^{m\mathrm{x}n}$$A(\gamma)$$A(\gamma)=\gamma A^{+}+(1-\gamma)A^{-}$

Definition 2. For any , $\in[0,1]$ ,is said to be a$(A(\gamma),
A(\sqrt))$point for anon-zero sum parametric gameif the following$\gamma$(i)$(x^{*}, y^{*})\in X\cross \mathrm{Y}$$\sqrt$$x^{t}A(\gamma)y^{*}\leqq x^{*t}A(\gamma)y^{*}$$(\gamma, \sqrt)$-equilibrium$(\mathrm{i})-(\mathrm{i}\mathrm{i})$holds:

for all x $\in X$ ,(ii) $x^{*t}A(\sqrt)y\geqq x^{*t}A(\sqrt)y^{*}$ for all y$\in \mathrm{Y}$.

We note that the-equilibrium point
for anon-zero sum game $(A(\gamma),A(\gamma))$means thatis asaddle point for the zero sum matrix game, i.e.,$(x^{*}, y^{*})$$(\gamma, \gamma)$$(x^{*}, y^{*})$(2.4)$A(\gamma)$$x^{t}A(\gamma)y^{*}\leqq x^{*t}A(\gamma)y^{*}\leqq x^{*t}A(\gamma)y$for all x $\in X$ and y.

$\in \mathrm{Y}$Also, every non-zero sum finite game has an equilibrium point (cf. [15, 26]), so thatfor any 7, $\in[0,1]$ , a-equilibrium point exists.
Applying Lemma 1.1 and 2.1, we have the following useful theorem which tells us
the relation of the interval matrix game $A$ and the non-zero sum parametric game$(\gamma, \sqrt)$$\sqrt$$(A(\gamma), A(\sqrt))$.

Theorem 2.1. A point
is asaddle point for the interval matrix game
$A$ if and only if$\in[0,1]$
there exist 7,
such thatis a-equilibrium point for
the non-zero sum parametric game $(A(\gamma), A(\sqrt))$ .
$(x^{*}, y^{*})\in X\cross \mathrm{Y}$$\sqrt$Proof.
$\gamma’\in[0,$(2.5)$(x^{*}, y^{*})$$(\gamma,\sqrt)$

By Lemma 1.2 and 2.1, that $f(x^{*}, y^{*}) \in\min f(x^{*},$Y) means that there exists
1] satisfying$\gamma’x^{*t}A^{+}y^{*}+(1-\gamma’)x^{*t}A^{-}y^{*}\leqq\gamma’x^{*t}A^{+}y+(1-\gamma’)x^{*t}A^{-}y$for all y

Obviously, (2.5) is rewritten as follows.
(2.6)$x^{*t}A(\gamma’)y^{*}\leqq x^{*t}A(\gamma’)y$for all y(2.7)$x^{*t}A(\gamma)y^{*}\geqq x^{t}A(\gamma)y^{*}$.

$\in \mathrm{Y}$which is corresponding with (ii) of Definition 2.
Similarly, $f(x^{*}, y^{*}) \in\max f(X, y^{*})$ means that there exists$\gamma\in[0,$for all x $\in X$ .

.

$\in \mathrm{Y}$1] such that107
Thus, the proof is complete.

$\square$

The following results easily follow from Theorem 2.1.

is asaddle point for the matrix game
Corollary 2.1. If$\gamma\in[0,1])$ ,is asaddle point for the interval matrix game $A$ .
$(x^{*}, y^{*})\in X\cross \mathrm{Y}$$A(\gamma)$

for all$(x^{*}, y^{*})$

Corollary 2.2. For any$A=([a_{ij}^{-}, a_{ j}^{+}.])\in \mathbb{C}^{m\mathrm{x}n}$, the saddle point.
saddle point for the matrix game$j(1\leqq i\leqq m, 1\leqq j\leqq n)$with$(x^{*}, y^{*})$

independent of and$A$of is uniquely determined as$i$$a_{ij}^{+}-a_{ij}^{-}=c$$a$$A^{-}=(a_{ij}^{-})$

is rewritten as $A(\gamma)=A^{-}+\gamma(A^{+}-A^{-})$ . So that if $A^{+}-A^{-}=$
Proof. We note thatis essentially equivalent for all , $\gamma’\in[0,1]$ , where all the elements ofcE,
andare 1. Thus, the statement of Corollary 2.2 follows obviously.
$A(\gamma)$$A(\gamma’)$$A(\gamma)$$\gamma$$\square$$E\in \mathbb{R}^{m\mathrm{x}n}$The following is useful in finding the saddle point of the interval matrix game A by1].
for all ,solving the parametric matrix game$(A(\gamma), A(\gamma’))$Corollary 2.3 [cf. [22]]. The point
if and only if there exist
matrix gameof asolution to$\gamma$

is asaddle point for the interval
is apartsuch that$(x^{*}, y^{*})\in X\cross \mathrm{Y}$$A\in \mathbb{C}^{m\mathrm{x}n}$(2.8)$\gamma$,$\gamma’\in[0,$$\gamma’\in[0,1]$$(x^{*}, y^{*})$$\{\begin{array}{l}x^{t}A(\gamma)-\mu^{t}=x^{t}A(\gamma)y1_{n}^{t}A(\gamma’)y+\nu=x^{t}A(\gamma’)y1_{m}\nu^{t}x=0,\mu^{\mathrm{t}}y=0x^{\mathrm{t}}1_{m}=\mathrm{l},y^{t}1_{n}=1x,\nu\in \mathbb{R}_{+}^{m},y,\mu\in \mathbb{R}_{+}^{n}\end{array}$where $1_{m}=$ 1,$($$\ldots$,$1)’\in \mathbb{R}_{+}^{m}$and$1_{n}=(1, \ldots, 1)’\in \mathbb{R}_{+}^{n}$.

Remark. On the interval matrix game $A$ , if player 1(2) selects the strategy $i(j)$ player, where the actual value of
1(2) receives(loses) the interval valued payoff
the payoff is not known precisely for both players like avalue of abeautiful ancient urn or
thenof the future project. In general,
the interval matrix game $A$ is not a zero sum game in the strict sense of the word. The
player 1and 2may consider the interval game $A$ as anon-zero sum game $(A(\gamma), A(\gamma’))$for player 1
andfor some parameters and , where the parametric gameand 2may be their subjective values for the interval game $A$ . Consider the extreme case$(\gamma, \gamma’)=(0,1)$ , a $(0, 1)$ -equilibrium pointmeans that$a_{ij}=[a_{ij}^{-}, a_{ij}^{+}]\in \mathbb{C}$$[a_{\dot{\iota}j}^{-}, a_{ij}^{+}]+[-a_{ij}^{+}, -a_{ij}^{-}]=[a_{ij}^{-}-a_{ij}^{+}, a_{ij}^{+}-a_{ij}^{-}]\neq 0(\ni 0)$$A(\gamma)$$\gamma’$$\gamma$$A(\gamma’)$$(x^{*}, y^{*})\in X\cross \mathrm{Y}$(i)for all $A\in A$ and $x\in X$ ,$x^{t}A^{-}y^{*}\leqq x^{*t}A^{-}y^{*}$(ii) $x^{*t}A^{+}y\geqq x^{*t}A^{+}y^{*}$ for all $A\in A$ and$y\in \mathrm{Y}$.

guarantees the best in the worst case for both players. Thus,
This shows that$(0, 1)$ -equilibrium point
will be called apessimistic-pessimistic pair. By the same$(1,
0)$
will be called an optimistic-equilibrium point
discussion as the above, theoptimistic pair. Then the parameter 7 $(0\leq\gamma\leq 1)$ is agrade of optimism for player 1oragrade of pessimism for player 2.
$(x^{*}, y^{*})$$(x^{*}, y^{*})$$(x^{*}, y^{*})$

108
3Extensions to fuzzy games
In this section, the results in the preceding section will be extended to the multi-dimensiona
fuzzy payoff games.
We write afuzzy set on
by its membership function(see Novak [17]and Zadeh [27] . The -cut $(\alpha\in[0, 1])$ of the fuzzy set on
is defined as$\mathrm{R}^{p}$$\tilde{s}:\mathrm{R}^{p}arrow[0,1]$$)$$\alpha$$\mathrm{R}^{p}$$\overline{s}$$\overline{s}_{\alpha}:=\{x\in \mathrm{R}^{p}|\overline{s}(x)\geq\alpha\}(\alpha>0)$

and$\overline{s}_{0}:=\mathrm{c}1\{x\in \mathrm{R}^{p}|\tilde{s}(x)>0\}$where cl denotes the closure of the set. Afuzzy set$\tilde{s}(\lambda x+(1-\lambda)y)\geq\tilde{s}(x)\Lambda\tilde{s}(y)$$\tilde{s}$

$x$

,

,

is called convex if$y\in \mathrm{R}^{p}$,$\lambda\in[0,1]$,

where a $\Lambda b=\min\{a, b\}$ . Note that is convex if and only if the -cut
is aconvex
set for all $\alpha\in[0,1]$ . Let
be the set of all convex fuzzy sets whose membershipfunctions :are upper-semicontinuous and normal
andhave acompact support. In the one-dimensional case $n=1$ ,denotes the set of all
fuzzy numbers. Let
be the set of all compact convex subsets of .
The definitions of addition and scalar multiplication on
are as follows: For
,
and A
,
$\tilde{s}$$\cdot$$\tilde{s}_{\alpha}$$\alpha$$\mathcal{F}(\mathrm{R}^{p})$$\mathbb{R}^{p}arrow[0,1]$$\tilde{s}$$( \sup_{x\in \mathrm{R}^{p}}\tilde{s}(x)=1)$$\mathcal{F}(\mathrm{R})$$\mathbb{C}(\mathrm{R}^{p})$$\mathrm{R}^{p}$$\mathcal{F}(\mathrm{R}^{p})$$\overline{s}$$\geq 0$$\tilde{r}\in \mathcal{F}(\mathrm{R}^{p})$(3.1)$( \tilde{s}+\tilde{r})(x):=\sup_{l\sim_{1}+=*}\{\tilde{s}(x_{1})\Lambda\tilde{r}(x_{2})\}x_{1}\rho_{2}\epsilon_{2}\mathrm{n}^{\mathrm{p}}$(3.2)$\tilde{s}(x/\lambda)$$(\lambda\tilde{s})(x):=\{$$1\{0\}(x)$ifif A $=0$’$\lambda>0$$(x\in \mathrm{R}^{p})$

,

where
is an indicator.
By using set operations $A+B:=\{x+y|x\in A, y\in B\}$ and $\lambda A:=\{\lambda x|x\in A\}$ for
any non-empty sets ,, the following holds immediately.
$1_{\{\cdot\}}(\cdot)$

$A$

(3.3)

$B\subset \mathrm{R}^{p}$$(\tilde{s}+\tilde{r})_{\alpha}=\tilde{s}_{\alpha}+\tilde{r}_{\alpha}$and$(\lambda\tilde{s})_{\alpha}=\lambda\tilde{s}_{\alpha}$$(\alpha\in[0, 1])$.

Let $K$ be anon-empty cone of . Using this $K$ , we can define apseudo order relationonby
if and only if $y-x\in K$ . We introduce apseudo orderon(cf. [8]). Let ,. The relationmeans the following (i) and (ii):$\mathrm{R}^{p}$$\mathrm{R}^{p}$$\neg K\prec$$x\prec_{K}y\neg$$\tilde{s}$(i) For any x(ii) For any y

For any a$\neg\prec K$$\tilde{r}\in \mathcal{F}(\mathrm{R}^{p})$$\in \mathrm{R}\mathrm{p}$, there exists y$\in \mathrm{R}^{p}$such that$\in \mathrm{R}^{p}$, there exists x$\in \mathrm{R}^{p}$such that x$\in \mathrm{R}^{p}$and d$\in \mathbb{C}(\mathrm{R}^{p})$(3.4)$\in \mathbb{C}$$\neg\prec_{K}y$and$\mathrm{s}(\mathrm{x})\leq\tilde{r}(y)$and$\mathrm{s}(\mathrm{x})\geq\tilde{r}(y)$.

.

, the product of a and d is defined as.

Lemma 3.1 [8]. For any
and$x\backslash \prec_{K}y$ad $=\{a^{t}d|d\in d\}$ .

We note that ad$\alpha\in[0,$$\mathcal{F}(\mathrm{R}^{p})$$\tilde{s}\prec_{\backslash K}\tilde{r}$1].

$\tilde{s},\tilde{r}\in \mathcal{F}(\mathrm{R}^{p})$,$\tilde{s}\prec_{\neg K}\tilde{r}$if and only if$a\tilde{s}_{\alpha}\leq a\tilde{r}_{\alpha}$

for all a$\in K^{+}$

109
fuzzy matrix
Here, we consider the two person fuzzy matrix game defined by the,. For any x $=(x_{1}, x_{2},$ $x_{m})^{t}\in X$ and y $=(y_{1}, y_{2},$the expected payoff for player 1is estimated (cf. [18]) by$m\cross n$$y_{n})^{t}\in \mathrm{Y}$$\overline{A}=(\overline{a}_{ij})\in \mathcal{F}(\mathbb{R}^{p})^{m\mathrm{x}n}$\ldots ,\ldots ,(3.5)$f(x, y):=x^{t} \overline{A}y=\sum x_{i}y_{j}\overline{a}_{ij}$We note that$f(x, y)\in \mathcal{F}(\mathbb{R}^{p})$

(3.6)

.

and its -cut is given by$\alpha$$f(x, y)_{\alpha}= \sum x_{i}y_{j}\overline{a}_{ij,\alpha}\in \mathbb{C}(\mathbb{R}^{p})$,is the -cut of .
where
The saddle point of the fuzzy matrix game is defined similarly as that of the interval
matrix game (see Definition 1in Section 2).
and setby, we denote, noting
For any$\alpha\in[0,1],\gamma\in[0,1]$. Here, forand, we putand$\alpha$$\tilde{a}_{ij,\alpha}$$\overline{a}_{\dot{\iota}j}$$\overline{A}$$[\tilde{a}_{ij,\alpha}^{-}(a),\tilde{a}_{ij,\alpha}^{+}(a)]$$a\overline{a}_{ij,\alpha}$$a\overline{a}_{ij,\alpha}\in \mathbb{C}$$a\in \mathbb{R}\mathrm{p}$$A_{\alpha}^{+}(a):=(\overline{a}_{ij,\alpha}^{+}(a))\in \mathbb{R}^{m\mathrm{x}n}$$A_{\alpha}^{-}(a):=(\overline{a}_{ij,\alpha}^{-}(a))\in \mathbb{R}^{m\mathrm{x}n}$$a\in \mathbb{R}^{p}$(3.7)$A_{\alpha,a}(\gamma)=\gamma A_{\alpha}^{+}(a)+(1-\gamma)A_{\alpha}^{-}(a)$.

Then, the saddle points of the fuzzy matrix game will be characterized in the following
theorem, whose proof is done by applying Lemma 3.1 and the ideas used in Section 2.
$\overline{A}$is asaddle point of the fuzzy matrix game
Theorem 3.1. A point
if and only if there exist two functions 7, 7’ : $[0, 1]$ $\cross K^{+}arrow[0,1]$ such that$\overline{A}$$(x^{*}, y^{*})\in X\cross \mathrm{Y}$(3.8)$x^{t}A_{\alpha,a}(\gamma(\alpha, a))y^{*}\leqq x^{*t}A_{\alpha,a}(\gamma(\alpha, a))y^{*}$$x^{*t}A_{\alpha,a}(\gamma’(\alpha, a))y\geqq x^{*t}A_{\alpha,a}(\gamma’(\alpha, a))y^{*}$

for all $\alpha\in[0,1]$ and $a\in K^{+}$ .

4Numerical Example
Here, we give numerical examples.
Example 1.

Let $A=$$A^{+}=$

and$(\begin{array}{ll}4 02 3\end{array})$$(\begin{array}{ll}[2,4] [-2,0][0,2] [\mathrm{l},3]\end{array})$$A^{+}-A^{-}=(\begin{array}{ll}2 22 2\end{array})$$\in \mathbb{C}^{2\cross 2}$$x^{*}=(\begin{array}{l}14\overline{5}’\overline{5}\end{array})$

Example 2.

, $y^{*}=$

Let A$(\begin{array}{l}32\overline{5}’\overline{5}\end{array})$

$=$

(

and

$A^{-}$]’$32]$$[- \frac{3}{2’}\frac{1}{]2}][1,2)$’

and$(x^{*}, y^{*})$. After asimple calculation, we find$f(x^{*}, y^{*})=[ \frac{2}{5},$$,’-4$$(\begin{array}{ll}2 -20 1\end{array})$. Thus, by Corollary 2.2, asaddle pointof $A$ is unique and given by asaddle point forthat. Noting that $A^{-}=$$\frac{12}{5}]$

with

.

$A^{-}--(_{\frac{31}{2’}},$$- \frac{3}{2})1$

and $A^{+}=$

110

(’, )
$\frac{1}{22}$. Noting$A(\gamma)=(_{\gamma+\frac{31}{2’}}^{\gamma+},$$2 \gamma-\frac{3}{2})\gamma+1$

metric equation (2.8) and find that the$(\gamma, \sqrt)$$x^{*}=( \frac{1}{10-2\gamma}\frac{9-2\gamma}{10-2\gamma})$,’for each $\gamma\in[0,1]$ , we solve the para-equilibrium point$y^{*}=( \frac{5-2\sqrt}{1-2\sqrt}, \frac{5}{1-2\gamma’})$$(x^{*}, y^{*})$

is given byw.th$f(x^{*}, y^{*})=[ \frac{2\gamma\sqrt-15\sqrt-15\gamma+75}{(10-2\gamma)(10-2\sqrt)}, \frac{6\gamma\sqrt-35\gamma-35\sqrt+75}{(10-2\gamma)(10-2\gamma)},]$.

By Theorem 2.1, the set of all saddle points is specified by the set of all
equilibrium points. Some saddle points and their values are given in Table 1.

$(\gamma, ’\sqrt)-$

Table 1. Saddle points and their values.

5Interval stochastic gameIn this section, we formulate two person zer0-sum stochastic games with interval payoffs, called interrval stochastic games, and define the saddle points under acriterion of
discounted interval gains.
Atwo person interval stochastic game is determined by five objects:$\{S, A, B, r, q\}$.

Where $S=\{1,2, \ldots, N\}$ denotes the state space, $A=\{1,2, \ldots, m\}$ and $B=\{1,2, \ldots, n\}$denote the set of actions available to player 1(maximizer) and player 2(minimizer) respectively. An interval-valued map:
denotes interval estimate of
one-step payoff function and $q=\{q_{ss’}(i,j)|s, s’\in S, i\in A,j\in B\}$ is atransition law,
, $q_{ss’}(i,j)\geq 0$ and $\sum_{s\in S},q_{ss’}(i,j)=1$ for , $s’\in S$, $i\in A,j\in B$ .
$r$

$S\cross A\cross Barrow \mathbb{C}$$\mathrm{i}.\mathrm{e}.$

$s$

111
Agame is played as follows: At each time of epoch, two players observe the current
state $s\in S$ of the system and players 1and 2independently choose actions $i\in A$ and$j\in B$ respectively. Then two events happen; (i) player 1receives an immediate payoff
estimated by the interval $r(s, i,j)\in \mathbb{C}$ and (ii) the system moves to anew state $s’\in S$
selected according to the distribution $q_{s}.(i,j)$ . This process is then repeated from the newstate $s’\in S$ .
The sample space is the product space $\Omega=(S\cross A\cross B)^{\infty}$ such that the projection
on the -th factor , and describe the state and the actions chooses,
andrespectively by players 1and 2at the -th time of the process $(t=1,2, \ldots)$ . Let $P(A)$and $P(B)$ be the sets of all probability distributions on and respectively, i.e.,$X_{t}$$\Delta_{t}^{B}$$\Delta_{t}^{A}$

$S$

$t$

$B$

$A$

$t$

$B$

$A$

$P(A)= \{x=(x_{1}, x_{2}, \ldots, x_{m})|x_{i}\geq 0,.\cdot\sum_{=1}^{m}=1\}$

and$P(B)= \{y=(y_{1}, y_{2}, \ldots, y_{n})|y_{j}\geq 0, \sum_{j=1}^{n}=1\}$.

and for player 1and 2are sets of probability distributions$\{\pi(\cdot|s)|s\in S\}\subset P(A)$ and $\{\sigma(\cdot|s)|s\in S\}\subset \mathcal{P}(B)$ respectively. The sets of all
stationary strategies for player 1and 2will be denoted by and . We assume that forwith , $s’\in S$, $i\in A,j\in B$ and $t\geq 1$ ,each pairA(stationary) strategy$\pi$$\sigma$$\Pi$$(\pi, \sigma)\in\Pi\cross\Sigma$$\Sigma$$s$,$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\{X_{t+1}=s’|X_{1}, \Delta_{1}^{A}, \Delta_{1}^{B}, \cdots, X_{t}^{\cdot}=s, \Delta_{t}^{A}=i, \Delta_{t}^{B}=j\}=q_{ss’}(i,j)$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\{\Delta_{t}^{A}=i|X_{1}, \Delta_{1}^{A}, \Delta_{1}^{B}, \cdots, X_{t}=s\}=\pi(i|s)$and$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\{\Delta_{t}^{B}=j|X_{1}, \Delta_{1}^{A}, \Delta_{1}^{B}, \cdots, X_{t}=s\}=\sigma(j|s)$.

determine theThen, the initial state $s\in S$ and the pair of strategies
on by the usual way.
probability measure
Here, we consider the total expected payoff for player 1in which the future payoff isand any starting
discounted with afactor $\beta(0<\beta<1)$ . For any pair$(\pi, \sigma)\in \mathrm{I}\mathrm{I}$$\cross\Sigma$$\Omega$$P_{\pi,\sigma}^{s}$$(\pi, \sigma)\in\Pi\cross\Sigma$

state $s\in S$ , let
(5.1)

where

$\mathcal{J}_{T}(s, \pi, \sigma)=\sum_{t=1}^{T}\beta^{t-1}E_{\pi,\sigma}^{s}[r(X_{t}, \Delta_{t}^{A}, \Delta_{t}^{B})]$$E_{\pi,\sigma}^{s}$is the expectation with respect to$P_{\pi,\sigma}^{s}$. Obviously,,$\mathscr{I}_{T}(s, \pi, \sigma)\in \mathbb{C}$and any starting state $s\in S$ ,Lemma 5.1 For any pair.
is aCauchy sequence with respect to the Hausdorffmetric$(\pi, \sigma)\in\Pi\cross\Sigma$$\delta\in \mathbb{C}$.

$\{J_{T}(s, \pi, \sigma)\}_{T=1}^{\infty}$

112
Proof. For any $T>H$ , it holds from Lemma 1.1 (iii) that$\delta(J_{T}(s,\pi, \sigma), J_{H}(s,\pi,\sigma))$$= \delta(0,\sum_{t=H+1}^{T}\beta^{t-1}E_{\pi,\sigma}^{s}[r(X_{t},\Delta_{t}^{A}, \Delta_{t}^{B})])$$– \beta^{H}\delta(0,\sum_{t=H+1}^{T}\beta^{t-H-1}E_{\pi,\sigma}^{s}[r(X_{t}, \Delta_{t}^{A}, \Delta_{t}^{B})])$$\leq\beta^{H}.\max_{\in s\in S,.Aj\in B}\delta(0, r(s, i,j))/(1-\beta)$

This completes the proof.
by.

$\square$Prom Lemma 5.1, the infinite horizon total expected payoff for player 1can be defined

(5.2)

Since

$J(s, \pi, \sigma)=\lim_{Tarrow\infty}J_{T}(s, \pi, \sigma)$$\mathrm{J}(\mathrm{s}, \pi, \sigma)\in \mathbb{C}$

(5.3)

.

, it will be written as$J(s, \pi, \sigma)=[J^{-}(s, \pi, \sigma), J^{+}(s, \pi, \sigma)]$

For any pair$(\pi,\sigma)\in\Pi\cross\Sigma$.

and $s\in S$ , let$J(s, \Pi,\sigma)=\{J(s,\pi,\sigma)|\pi\in\Pi\}$and$J(s,\pi, \Sigma)=\{J(s,\pi,\sigma)|\sigma\in\Sigma\}$.

The following is easily shown by applying the idea of Borkar’s discounted occupation

measure (cf. Theorem 1.2 [3]).
Lemma 5.2 For any paircompact and convex subsets$(\pi, \sigma)\in\Pi\cross\Sigma$$of\mathbb{C}$.

and s $\in S$ , $J(s, \Pi, \sigma)$ and $J(s, \pi, \Sigma)$ are

Definition 1’ (cf. [20, 24]) Let
and $s\in S$ . Then, the pairissaid to be asaddle point at $s\in S$ for the interval stochastic game if the folowing holds.
$(\pi^{*}, \sigma^{*})\in\Pi\cross\Sigma$$(\pi^{*}, \sigma^{*})$$J(s, \pi^{*}, \sigma^{*})\in\max J(s, \Pi, \sigma^{*})\cap\min J(s, \pi^{*}, \Sigma)$.

6Characterization of saddle points
In order to characterize the saddle point we introduce aparametric stochastic game.
For any $\in[0,$ 1], we put$\gamma$(6.1)$r^{\gamma}(s,i,j)=\gamma r^{+}(s, i,j)+(1-\gamma)r^{-}(s,i,j)\in \mathrm{R}$(s $\in S,$ i $\in A,j\in B)$ ,

where
and
are extreme points of the interval r and r $=[r^{-}(s,$i,j),$r^{+}(s, i,j)]$ .
For any pair
, let
and s
$r^{-}$$r^{+}$$(\pi,\sigma)\in\Pi\cross\Sigma$(6.2)$\in S$$I^{\gamma}(s, \pi,\sigma)=\lim_{Tarrow\infty}F_{T}(s,\pi,\sigma)$

,

113

$I_{T}^{\gamma}(s, \pi, \sigma)=\sum_{t=1}^{\alpha}\beta^{t-1}E_{\pi,\sigma}^{s}[r^{\gamma}(X_{t}, \Delta_{t}^{A}, \Delta_{t}^{B})]$$(T\geq 1)$.

and $s\in S$ . Then, the pairDefinition 2’ Let ,-equilibrium point at state $s\in S$ if the following (i) and (ii) hold:a.
for all(i).
for all(ii)$\gamma$$(\pi^{*}, \sigma^{*})\in\Pi\cross\Sigma$$\gamma’\in[0,1]$

is said to be$(\gamma, \gamma’)$$\pi\in\Pi$$I_{T}^{\gamma}(s, \pi, \sigma^{*})\leq I_{T}^{\gamma}(s, \pi^{*}, \sigma^{*})$$\sigma\in\Sigma$$I_{T}^{\gamma’}(s, \pi^{*}, \sigma)\geq I_{T}^{\gamma’}(s, \pi^{*}, \sigma^{*})$

Every finite noncooperative stochastic game has an equilibrium point (cf. [1]), so-equilibrium point exists. The following lemma follows
that for any 7, $\gamma’\in[0,1]$ , aobviously from (2.2), (2.3) and (6.1).
$(\gamma, \gamma’)$Lemma 6.1 For any pair$(\pi, \sigma)\in\Pi\cross\Sigma$

and s$\in S$,$I^{\gamma}(s, \pi, \sigma)=\gamma J^{+}(s, \pi, \sigma)+(1-\gamma)\mathscr{I}^{-}(s, \pi, \sigma)$

Theorem 6.1 A pairexist , $\gamma’\in[0,1]$ such that$(\pi^{*}, \sigma^{*})\in\Pi\cross\Sigma$

is a$(\pi^{*}, \sigma^{*})$$\gamma$.

is asaddle point at $s\in S$ if and only if there-equilibrium point at state $s\in S$ .

$(\gamma, \gamma’)$Proof. By Lemmas 1.2 and 6.1, that $J(s, \pi^{*}, \sigma^{*})\in\min J(s, \pi^{*}, \Sigma)$ means that there
1] satisfying
exists$\gamma’\in[0,$(6.3)$\gamma’J^{+}(s, \pi^{*}, \sigma^{*})+(1-\gamma’)\mathscr{I}^{-}(s, \pi^{*}, \sigma^{*})$$\leq\gamma’J^{+}(s, \pi^{*}, \sigma)+(1-\gamma’)J^{-}(s,$$\pi^{*}$,$\sigma$} for all$\sigma\in\Sigma$.

By Lemma 6.1, (6.3) is rewritten to (ii) of Definition 2’. Similarly,
holds.
$\max J(s, \Pi, \sigma^{*})$ means that there exists $\gamma\in[0,1]$ for which (i) of Definition 2’
Thus, the proof is complete.
$J(s, \pi^{*}, \sigma^{*})\in$$\square$

The following results easily follow from Theorem 6.1.
Corollary 6.1 If pair$\pi\in\Pi$ ,, $\gamma\in[0,1]$ , the pair$(\pi^{*}, \sigma^{*})\in\Pi\cross\Sigma$

is a saddle point for a zero-sum gameis asaddle point at state $s\in S$ for the interval$\{I^{\gamma}(s, \pi, \sigma)|$$(\pi^{*}, \sigma^{*})$$\sigma\in\Sigma\}$stochastic game.
Proof. The saddle point
is awhich implies that
6.1 follows from Theorem 6.1.

$(\pi^{*}, \sigma^{*})$$(\pi^{*}, \sigma^{*})$

satisfies that$(\gamma, \gamma)$$I^{\gamma}(s, \pi^{*}, \sigma)\geq I^{\gamma}(s,\pi^{*}, \sigma^{*})\geq I^{\gamma}(s,\pi, \sigma^{*})$,-equilibrium point. Thus, the statement of Corollary$\square$Corollary 6.2 If $r^{+}(s, i,j)-r^{-}(s, i,j)(=const.)$ is independent of $s\in S$ , $i\in A,j\in B$
for the interval stochastic game is uniquely determined as a
the saddle point
saddle point for azerO-sum game $\{I^{0}(s,\pi, \sigma)|\pi\in\Pi, \sigma\in\Sigma\}$ .
$(\pi^{*}, \sigma^{*})$

Proof. We note from (6.1) that $r^{\gamma}(s, i,j)=r^{-}(s, i,j)+\gamma(r^{+}(s, i,j)-r^{-}(s, i,j))$ . So
is independent of $s\in S$ , $i\in A,j\in B$ ,that if. The proof is completed byare essentially equivalent for any
andobserving Theorem 6.1.
$I^{\gamma}(s, \pi,\sigma)$$r^{+}(s, i,j)-r^{-}(s, i,j)(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)$$\gamma,\gamma’\in[0,1]$$I^{\gamma’}(s, \pi, \sigma)$$\square$114
The following is useful in finding the saddle points for interval stochastic games.
Corollary 6.3 (cf. [22]) Tie pairis asaddle point at $s\in S$ if
and only if there exist , $\in[0,1]$ such that $\pi^{*}(\cdot|s)=(x_{s1}, x_{s2}, \ldots, x_{sm})\in \mathcal{P}(A)$ and, ) $\in \mathcal{P}(B)(s\in S)$ is apart of asolution to$(\pi^{*}, \sigma^{*})\in \mathrm{I}\mathrm{I}$$\gamma$$\sigma^{*}(\cdot|s)=(y_{s1}, \mathrm{y}\mathrm{s}\mathrm{n})\ldots$(6.4)$\cross\Sigma$$\sqrt$$y_{sn}$$\{\begin{array}{l}v_{s}=\nu_{sj}+\Sigma(r^{\gamma}(s,i,j)+\sqrt\Sigma q_{ss’}(i,j)v_{s’})x_{s}..(s\in S)v_{s},=\mu_{s}..+..\sum_{j\in B}^{\in A}(r^{\sqrt}(s,i,j)+\sqrt\sum_{s’\in S}^{s’\in S}q_{ss’}(i,j)v_{s}’,)y_{sj}\Sigma\Sigma\nu_{s}.x_{s}..=0,\Sigma\Sigma\mu_{sj}y_{sj}=0s\in S\cdot.\in As\in Sj\in B\Sigma x_{\epsilon}..=1,\Sigma y_{sj}=1,(s\in S)i_{\dot{n}}^{\epsilon A}\geq 0,y_{sj}\geq 0j\in B(s\in S,i\in A,j\in B)\end{array}$

7Extensions to fuzzy payoff cases
In this section, we consider the stochastic game similar to that specified in Section 5exceptthat for each $s\in S$, $i\in A$ and $j\in B$ the multi-dimensional fuzzy payoff
is assigned.
Then, for apair
and $s\in S$ , we let$\mathrm{r}(\mathrm{s}, \mathrm{i},\mathrm{j})\in \mathcal{F}(\mathrm{R}^{p})$$(\pi, \sigma)\in\Pi\cross\Sigma$(7.1)$\overline{J(}s$,,$\pi,\sigma)=\sum_{t=1}^{\infty}\beta^{t-1}E_{\pi,\sigma}^{s}[\tilde{r}(X_{t}, \Delta_{t}^{A}, \Delta_{t}^{B})]$

where the expectation of afuzzy random variable is defined similarly as (1.3) by use of(3.1) and (3.2), and the convergence in (7.1) is taken with respect to the usual Hausdorff
metric on.
We note thatand its a-cut,
is given by$\mathcal{F}(\mathrm{R}^{p})(\underline{\mathrm{c}\mathrm{f}.}[4])$$J\overline{(}s$$J(s, \pi,\sigma)\in \mathcal{F}(\mathrm{R}^{p})$(7.2)$\overline{J(}s,\pi$,$\pi,\sigma)_{\alpha}$$\sigma)_{\alpha}=\sum_{t=1}^{\infty}\beta^{t-1}E_{\pi,\sigma}^{s}[\tilde{r}(X_{t}, \Delta_{t}^{A}, \Delta_{t}^{B})_{\alpha}]$

,

where
is an -cut of.
The saddle point of the stochastic game with fuzzy payoff is defined similarly as thatof the interval stochastic game (see Definition 1’ in Section 5).
For any, since the product, it will be written as$\overline{r}(s, i,j)_{\alpha}$$\mathrm{r}(\mathrm{s}, i, j)\in \mathcal{F}(\mathrm{R}^{p})$$\alpha$$a\in \mathrm{R}^{p}$$a\overline{r}(s, i,j)_{\alpha}\in \mathbb{C}$$a\tilde{r}(s,i,j)_{\alpha}=[(a\tilde{r}(s, i,j)_{\alpha})^{-}, (a\tilde{r}(s,i,j)_{\alpha})^{+}]$For $\alpha\in[0,1]$ , $d\in[0, 1]$ and(7.3)

For each pair
(4.8)

$a\in \mathrm{R}^{p}$.

, we put$r(s, i,j|\alpha,\gamma, a)=\gamma(a\tilde{r}(s, i,j)_{\alpha})^{+}+(1-\gamma)(a\overline{r}(s, i,j)_{\alpha})^{-}$$(\pi, \sigma)\in\Pi\cross\Sigma$

and $s\in S$ , we define.

$I(s, \pi, \sigma|\alpha, \gamma, a)=\sum_{t=1}^{\infty}\beta^{t-1}E_{\pi,\sigma}^{s}[r(X_{t}, \Delta_{t}^{A}, \Delta_{t}^{B}|\alpha, \gamma, a)]$

115
Then, the saddle point for the stochastic game with fuzzy payoff can be characterized in5.
the following, whose proof is done by applying Lemma 3.1 and the idea used in Section

is asaddle point for the stochastic game with
Theorem 7.1 A pair$\cross Karrow[0,1]$ such thatfuzzy payoffs if and only if there exist two function 7, : $[0, 1]$$(\pi^{*}, \sigma^{*})\in\Pi\cross\Sigma$$\gamma’$(4.9)$I(s, \pi, \sigma^{*}|\alpha, \gamma(\alpha, a), a)\leq I(s, \pi^{*}, \sigma^{*}|\alpha, \gamma(\alpha, a), a)$,$I(s, \pi^{*}, \sigma|\alpha, \gamma’(\alpha, a), a)\geq I(s, \pi^{*}, \sigma^{*}|\alpha, \gamma’(\alpha, a), a)$for all $\pi\in\Pi$ ,$\sigma\in\Sigma$, $\alpha\in[0,1]$ and $a\in K^{+}$ .

References[1] Altman, E. and Schwartz, A., Constrained Markov Games: Nash Equilibria,
J.A.Filar, V.Gartsgory and K.Mizukami(eds): Advances in Dynamic Games andApplications (Annals of the International Society of Dynamic Games, Volume 5),257-266 (1999).
[2] Benson, H.P., An improved definition of proper efficiency for vector maximization
with respect to cones, J. Math. Anal. Appl. 71, 232-241 (1979).
[3] Borkar, V.S., Topics in Controlled Markov-Chains, Pitman Research Notes in Mathematics 240, Longman Scientific-Wiley, New York, (1991).

[4] Diamond, P. and Kloeden, P., Metric Spaces of Fuzzy Sets, Theory and Applications,
World Scientic, (1994).
[5] Hartfiel, D.J., Markov Set-chains, Springer-Verlag, Berlin, (1998).
[6] Howard, R., Dynamic Programming and Markov Processes, MIT Press, Cambridge
MA, (1960).
[7] Ishibuchi, H. and Tanaka, H., Multiobjective programming in optimization of the interval objective formulation, European J. of Operational Research 48, 219-225 (1990).
[8] Kurano, M., Yasuda, M., Nakagami, J. and Yoshida, Y., Ordering of fuzzy sets -A
brief survey and new results, J. Operaiions Research Society of Japan 43, 138-148(2000).

[9] Kurano, M., Yasuda, M. and Nakagami, J., Interval methods for uncertain Markov
decision processes, In:Markov Processes and Contracted Markov Chains, edited by H.
Zhenting, J. A. Filer and A. Chen, Kluwer, Dordrecht, The Netherlands, (2001 to
appear).

[10] Kurano, M., Yasuda, M., Nakagami, J. and Yoshida, Y., Markov decision processes
with fuzzy rewards, The second international conference on NACA, Hirosaki, Japan,
July 30-August 2, (2001).
[1] Kurano, M., Yasuda, M., Nakagami, J. and Yoshida, Y., A note on interval games
and their saddle points, 京都大学数理解析研究所・研究集会 「数理最適イヒの理論とアノレゴリズム」 July 17-19, (2001).

116[12] Kurano, M., Yasuda, M., Nakagami, J. and Yoshida, Y., Stochastic games withinterval payoffs, 科学研究費研究集会 「不確実性の下での数理的意思決定の研究」 Oct.
18-19, (2001).

[13] Kuratowski, K., Topology. Academics Press, New York, (1966).
[14] Luc, D. T., Theory of vector optimization, Springer-Verlag, (1989).
[15] Nash, J., Noncooperative Games, Ann. of Math. 54, 286-295 (1951).
[16] Nenmaier, A., New techniques for the analysis of linear interval equations, Linear
Algebra and Applications, 58, 273-325 (1984).
[17] Novak, V., Fuzzy Sets and Their Applications, Adam Hilder, Bristol-Boston, (1989).

[18] Puri, M.L. and Ralescu, D.A., Fuzzy random variables, J. of Math. Anal, and Appli.,
114, 409-422 (1986).
[19] Puterman, M.L., Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley&Sons, INC, (1994).
[20] Sawaragi, Y., Nakayama, H. and Tanino, T., Theory of multiobjective optimization,
Academics Press, Inc. (1985).
[21] Shaochang, T., Interval number and fuzzy number linear programmings, Fuzzy Sets
and Systems, 66, 301-306 (1994).

[22] Sobel, M., Noncooperative stochastic games, Ann. Math. Statist. 42, 1930-1935(1971).
[23] Stowinski, R. (ed.), fibzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, (1998).
[24] Tanaka, T., Generalized quasiconvexities, cone saddle points and minimax theorem
for vector-valued functions. J. Optim. Theory Appl. 81, 355-377 (1994).
[25] von Neumann, J. and Morgenstein, Theory of Games and Economics Behavior
Princeton:Pi.ceton University Press, (1944).

[26] Wang, J., The Theory of Games, Oxford Science Publications, (1988).

[27] Zadeh, L.A., Fuzzy sets, Inform, and Control, 8, 338-353, (1965)