Title DEA ゲームの凸性 ( 数理最適化から見た 凸性の深み 非凸性の魅惑 ) Author(s) 中林 健 ; 刀根 薫 Citation 数理解析研究所講究録 (2004) 1349: 204-220 Issue Date 2004-01 URL http://hdl.handle.net/2433/24871 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
1349 2004 204-220 204 DEA (Ken Nakabayashi) (Kaoru Tone) National Graduate Institute for Policy Studies 1.. DEA (Data $\mathrm{c}^{\mathrm{r}}$nvelopment Analysis) DEA ( [4]) $\lceil \mathrm{d}\mathrm{e}\mathrm{a}$ 2 DEA } $\backslash$ (Egoist s Dilemma) 3 4 DEA $ $ DEA 4 4 - DEA. ( [8]) 5 DEA 6 2. 1I)
$[2\mathrm{l}\mathrm{p}\mathrm{p}$ $ \backslash$ $i$ $ \backslash$ 205 2.1. 1 $\mathrm{c}1$ $\mathrm{c}2$ $\mathrm{c}4$ 4 4 $\mathrm{t}^{\backslash }\wedge$ 1 \sim 2:3 : 2:1 1. $\mathrm{r}^{1}\mathrm{j}rightarrow$ DEA ( $[12\mathrm{l}\mathrm{p}\mathrm{p}.$ $-\acute{\mathrm{t}}$. 12.13 $10\cdot 11$ ) $\sum_{\iota=1}^{4}w_{l}^{k}x_{k}$ $c_{k}= \max-$ $w^{\mathrm{a}}$ $\sum_{=1}^{4}(\sum_{=1}^{4}w_{l}^{\mathrm{a}}.x_{/})$ (1) $s.l$. $w^{k}\geq 0$ $(i=1 \cdots4)$ (1) $x$ $w$ $j$ $i$ (1) $w^{k}=(w_{1}^{\lambda}. \cdots w_{4}^{k})$
$\mathrm{i}\mathrm{f}\mathrm{i}_{\backslash }^{-}\square \prime \mathrm{c}2\mathrm{i}_{\nearrow}\xi_{\hat{\backslash }}\mathrm{s}\mathrm{c}\mathfrak{z}----$ $\mathrm{c}1$ $\mathrm{c}\prime \mathit{2}$ $[2\mathrm{l}\mathrm{p}\mathrm{p}$ 206 DEA (.152-159 [12]pp.79-83) (1) 2 1 ( $[9]\mathrm{p}.23$ ) $ ^{\ulcorner}$ (Egoist s Dilemma) 2.2. 2 $\mathrm{c}\mathrm{s}$ 3 3 \sim 1:3:2 3: - 1 2 3 4 10 5 5 4 24 $23$ $43$ $-\underline{?3}$ $43$ $1212$ $1\cross O1+3\cross \mathrm{c}2+\underline{9}\cross\backslash \mathrm{c}3$ 22 22 18 22 84 22/84 22/84 18/84 22/84 1 $ \sum_{=1}^{4}w\int x_{k}$ $d_{k}= \min-$ $\mathrm{w}^{\lambda}$. $\sum_{=1}^{3}(\sum_{=1}^{4}w^{k}x_{j})$ (2) $s.l$. $w_{j}^{k}\geq 0$ $(i=1 \cdots4)$
$\mathrm{j}_{/}.$ $\Psi^{-}\mathit{1}_{\cap}^{\Delta}.\square \mathfrak{h}\backslash f_{\mathrm{c}c}\text{}\cdot\text{ }\not\in+\mathrm{a}\mathrm{a}\text{}\backslash$ 207 (1) $i$ (2) $i$ $w$ $j$ (2) 4 2.3. $\mathrm{b}_{-}$ $\lceil \mathrm{g}\mathrm{n}\mathrm{p}$ GNP ( [ll]p. $131_{\text{}}$ ) R\angle IJ- $\lceil^{-}\text{}..l_{\mathrm{u}}^{\angle\text{}}.\mathrm{h} \backslash \rfloor fx\not\leq$ $\mathrm{a}\rfloor\text{ _{}\mathrm{a}}$ } $ $ \sim 2.4. - $\mathrm{l} $ )]
$w_{l}^{\mathrm{a}}$ 208 A r E$ [ ] ($\lceil ( [3]p.l47. $216^{-}217$ [] ). pf $\dot{\mathrm{h}}^{1}\mathrm{j}\angle$ L [3] - - A ( ) b-- $\mathrm{t}\backslash$\acute. ( $[3]\mathrm{p}.130$ p. 158 p. 192 p.232) $\int\gamma$) 3. (Egoist s Dilemma) l^{-}}\mathrm{r}$ $\Pi^{\backslash.-. 1 $\triangleright^{\wedge}$ $\mathrm{u}$) 1: 3. 1. ( ) $\ovalbox{\tt\small REJECT}_{1} \downarrow^{-}$}$2.2\ulcorner$. 2 $n$ $m$ - (2) $\sum$ $\nu \mathrm{t} k$ x $d_{k}= \min_{\mathrm{w}^{\lambda}}\prime\prime\underline{\prime=1}$ $ \sum_{=1}(\sum_{=1}^{n\prime}wf_{x_{\mathit{1}/}})$ (3) $s.l$.. $\geq 0$ $(i=1 \cdots m)$.
$x_{1_{-}}.\cdot.$ $\propto\cdots\propto\{\begin{array}{l}x_{\mathrm{i}_{l7}}\vdots X \prime\end{array}).$ 208 (3) (4) [1] ( Charnes-Cooper ) $d_{\mathrm{a}}$. $= \min_{l1\mathfrak{l} }\sum_{=1}^{\prime\prime l}w^{k}\cdot x_{k}$ $s.l$. $\sum_{=1}^{1}$ ( $\sum_{=1} $ \sim $$ $\nu^{k}\cdot$ $(4\mathrm{I}$ x$ )=1$ $w^{k}\geq 0$ $(i=1\cdots m)$. (3) (4) $1\backslash$. R- 1( ) $\sum_{k=1}^{\prime 7}d_{k}\leq 1$. (5) $(\begin{array}{ll}x_{\mathrm{l}1} \vdots x \prime \mathrm{l}\end{array})\propto($ 2- \vdash $X_{\prime 2}$. $ k^{*}=(w_{1}^{\lambda^{*}k^{*}} \cdots \mathrm{v} )$ ( $ \sum_{=1}^{7}$ $\sum_{=1} w$ $x$ $)=1$ $w=(w \cdots w )$ $\sum_{=\mathrm{i}} w^{\mathrm{a}^{*}}\cdot xk\leq\sum_{=1}^{\omega?}w$ $X_{k}$ l $\sum_{\mathrm{a}\cdot=1}^{\prime 7}d_{k}=.\sum_{\mathrm{A}=1}^{J\mathfrak{l}}(\sum_{=1} w^{k^{*}}x_{\mathrm{a}}.)\leq\sum_{k=1}^{\prime 7}(\sum_{l=1}^{\prime?l}wx_{\Lambda})=1$ $\sum_{=1}^{n}$ ($\sum_{l=1} w$ i $x_{j})=1$ $\mathrm{m}^{\mathit{1}}$ $\sum_{l=1} wx_{lk}=\sum_{\iota=1} $ $w \int^{*} x_{k}=d_{k}$ (const.)
$\bigwedge_{\square }^{\text{}}-u$ $\text{}$ $(\begin{array}{l}x_{11}\vdots X_{nl1}\end{array}\}\propto(\begin{array}{l}x_{1\underline{7}}\vdots X7\underline{\gamma}\end{array})\propto\cdots\propto\{\begin{array}{l}x_{1\prime 7}\vdots X_{l\mathrm{l}n}\end{array}\}$ 210 ( $( \sum_{j^{=1}} x$ $)(= \sum_{j=1}^{l?}$ $\sum_{t=1}^{n1}w$ $\sum_{=1}^{ln}w$ $x$ $)=1)$ $ \sum_{=\mathrm{i}}^{\prime\prime?}w_{l}x_{k}=d_{k}$ $\sum_{j=1} w_{j}(\sum_{=1}^{n}x_{j} )$ $\sum_{t=1}^{n\prime}w$ $(x_{k}-d_{k} \sum_{=1}^{11}x)=0$. $w$ $x_{k}=d_{k}. \sum_{=/1}^{\prime 7}x_{/}$ $(i=1 \cdotsm)$ ( ) - $-A^{-}\Rightarrow k$ $-\mathrm{k}\hat{\mathrm{n}}$ 3.2. ( ) (4) $\min$ $\max$ (6) $\sum_{k=1} c_{k}\geq 1$ ( ) 1 $c_{k}= \max_{\lambdau)}\sum_{l=1}^{n1}w^{k}x_{k}$ $s.t$. $\sum_{j=1}^{n}(\sum_{=1}^{l \mathrm{j}}w^{k}x_{\dot{j}})=1$ (6) $w^{k}\geq 0$ $(i=1\cdotsm)$ 3.3. ) P4{\vee $\lceil$
211 5 5: 4. \Lambda --DEA 2 $\text{}$ 2: 4. 1. $f \mathrm{j}$ $\mathrm{l}$ R li ( ) $\text{}$ (4) $d_{k}$ $d_{k}$
$[^{\backslash }\mathrm{x}^{\overline{\backslash }}]1$ $.\sim\sim-.\cdot \mathrm{r}_{\tilde{1}\grave{\prime}} \backslash -\backslash. \backslash$ }}}_{)^{\backslash }}\wedge\backslash \backslash$ $\backslash \backslash --\prime\prime 2\neg \backslash \sim. \backslash \cdot$ $( \overline{3_{-}}\backslash \backslash \neg. -\vee \backslash \mathrm{v}_{1} $ $(_{\underline{ll}_{d})}^{\prime\backslash }--^{f.\prime}.\backslash -/$ $ -.\backslash _{\mathfrak{l}}j\prime^{-\tau_{1^{\prime^{-}}}}- ( \mathrm{t}^{1\mathrm{y}^{-\sim}\backslash }.)\backslash \prime \mathit{1}_{\mathit{1}^{\prime\backslash }}\{^{\prime/^{1/}/}\{--\backslash arrow_{\backslash } -arrow f/^{\prime\backslash }.\backslash \backslash /1^{\backslash }-\backslash \cdot-\cdot.-- --\neg\backslash.ff^{\backslash }\backslash \backslash \backslash \backslash -$ $\mathrm{c}$ $w^{\underline{\gamma}*}$ $\prime\prime\overline{)}1\backslash$ $/\cdot.$ 212 4.2. $.\wedge$ $ \Gamma_{\sim}$. ( ) 1\sim 3. $ \downarrow $ $\tilde{1^{\star}}\backslash$ ) ( $.\iota\overline{\nu^{\mathrm{y}^{\backslash $(.1l..) \overline 3^{\hat{\star}}$ $\langle 1\mathfrak{l}\nearrow $ ). /))k$\backslash$ ) \sim \sim -- $-^{-}$ $-.$ $ $ $(\backslash -\sim-\sim--$ $-^{12}-^{3^{\star}}----\sim 1\mathfrak{l} \grave{j}\sim_{\backslash }$ $\langle$ 1: 1 2 - - 1 2 (7) $d_{12}= \min_{1_{-}^{\urcorner}\mathrm{w})}(\sum_{=\mathrm{i}}^{l\mathrm{i}1}w^{1^{\underline{\gamma}}}x1+\sum_{=1}^{\prime \prime}w^{12}x\underline{7})$ $s.t$. $\sum_{=1} (\sum_{=1} w^{12}x_{/})=$ ] (7) $w^{1^{\underline{\gamma}}}\geq 0$ $(i=1 \cdots m)$. (4) (7) DEA 1 2 (7) {12} (4) (7) oi $w^{1^{*}}$ 1 2 $w^{12^{*}}$
$\ovalbox{\tt\small REJECT} \text{}\mathrm{s}^{\text{}}$ $\sigma\supset\text{}$. }$ 213 }\text{ $\pm \text{ }$ $\mathrm{h}^{1}\text{^{}\backslash. DEA DEA (7) $N=$ {l23 $\cdot$. $n\}$ ( $cl$) DEA rnin $(N d)$ $S$ $T$ $S\cup T$ $d\text{}\cup T$ ) (8) $d$ (\emptyset ) 0 $d(s \cup T)=\min_{\mathrm{W} }(\sum_{\mathrm{a}\in\backslash }.\cdot$ ( $x_{k})+ \sum_{\mathrm{a}\cdot\in \mathit{4}}(\sum_{=1} \uparrow 4 X_{k}))$ $\cdot\sum_{=1} w$ $s.t$. $\sum_{=1}^{j1}(\sum_{=1} \iota vx_{/})=1$ (8) $w$ $\geq 0$ $(i=1 \cdotsm)$. $(N d)$ $d$ (N) 1 \supset (8) $\min$ $\max$ $c(\cdot)$ DEA m $irl$ $(N d)$ $c(\emptyset)=0$ DEA mcll. $-arrow$ (N. $c$ ) DEA $\min$ $c(n)=1$ DEA $\wedge\neq$ 1 DEA DBA $Z$ 4.3. $\lceil 3.3$. $(N d)$ $(N c)$ DEA 2(Dual $G_{\ell l}me$ ) $S\subset N$ $d(n-s)=1-c.(s)$. (9)
$\gamma_{d}^{\text{}}$ $\text{}$ $\grave{\text{}}.\mathrm{j}\doteqdot\text{}$ }$ 214 $d(n-s)$ ( ) $d(n-s)= \min_{\mathrm{w}}.\sum_{k\in N-S}(\sum_{=1} wx_{k})=\min_{\mathrm{v}\iota}$. $( \sum_{k\in N}(\sum_{=1} wx_{k})-\sum_{\lambda\cdot\in_{\mathrm{l}}\backslash ^{\tau}}(\sum_{=1} w_{i}x_{k}))$ $= \min_{1\mathrm{i}^{j}}(1-\sum_{k\in S^{\urcorner}}(\sum_{=1}^{\Pi \mathrm{j}}w_{;}x_{jk}))=1-\max_{1\mathrm{w}}\sum_{k\in\delta}.(\sum_{=1}^{1}$ $w_{;}x_{k})=1-c(s)$. $n$ $S$ $\{N-S\}$ (9). $)$ $(N d)$ $d(\cdot)$ c $($ $(N c)$ $D\iota lal$ Game DEA Dual $d(\cdot)$ $($ c. $)$ $\{N-S\}$ $\text{ }$ $\text{ }\grave{\text{} }$ $S$ ) $\text{ $\overline{\mathfrak{n}}\phi-\text{}\mathrm{a}^{\text{ }}\mathit{0}$ ( 2 ) $S$ $\{N-S\}$ $\sigma$) $\text{ }$ $\Phi_{\mathit{4}\mathrm{u}\backslash }^{\mathrm{b}\grave{\mathrm{e}}\text{}\neq x}\xi\text{}.\text{ }$ $V^{\iota^{\sqrt}}\text{ - $\mathrm{s}\sigma$) }++_{\backslash }\grave{\xi}\lfloor\ovalbox{\tt\small REJECT}_{A}\}_{arrow(\kappa-}/\backslash$ - $[8]\mathrm{p}$ (. 14) 2 $S$ DEA $\{N-S\}$ $ $ \Phi --ffi
$\mathrm{f}_{-}\mathrm{e}\text{}\prime\not\equiv\text{}$ }(_{\backslash }\#\sqrt+_{\backslash -\text{}^{}\mathrm{a}}\text{ }$ $\lambda\iota\text{}$ 7\text{_{}\overline{\mathrm{J}}}$ $\text{^{}\mathrm{o}}\text{ }$ $- r\text{}$ $\zeta $\exists \mathrm{i}\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{l}\text{ $\text{ }+ \mathfrak{l}\mathrm{f}\mathrm{f}\mathrm{i}\text{}$ $rx_{\mathrm{p}}^{arrow-}\equiv$ $\mathrm{j}\mathrm{g}^{-}\not\in_{-\mathcal{j}\mathrm{j}}^{-}\text{ }$ lf ^-- \pm $\vdash_{-}$ $\grave{\mathrm{l}}\langle\langle\langle.\mathit{3}\text{}$ $\text{ }$ $\text{ }$ 215 $/\backslash \grave{\ovalbox{\tt\small REJECT}}\Delta\text{}$ $/\backslash \Delta \mathbb{r}\text{ }$ A\mbox{\boldmath $\theta$}l ( [8]pp. $15^{-}16$ p.17 pp. 18.19 ) g \sigma 74 \acute \supset ( $[8]\mathrm{p}.21_{\text{}}$ ) - 2 DEA $\text{}$ DEA $S$ $\{N-S\}$ $n$ Game $D_{Lt}al$ DEA $\ovalbox{\tt\small REJECT}_{\mathrm{B}}\ovalbox{\tt\small REJECT}_{i\epsilon_{-}^{\ovalbox{\tt\small REJECT}\mathrm{B}_{1}\text{}}^{}\mathrm{g}}\text{ $\text{}$ \gamma \rightarrow \leftarrow F. $\text{}$ l rf ^-- $\backslash \mathrm{r}^{\infty}\underline{\backslash }\prime_{-}\text{}$ \mbox{\boldmath $\tau$}x }- DEA }$r ) $\grave{l}(\mathrm{b}\propto\star \text{ }\cdot\text{ }$ $\nabla$] \gamma \leftarrow -\mbox{\boldmath $\tau$}.. ( ) $\mathrm{d}\mathrm{e}\mathrm{a}$ $\simarrow\sigma\supset$ $\text{}\grave{\text{}}[mathring]_{\text{}}$ $\grave{\text{}}$ $-\text{}v$ $7_{-\overline{\mathrm{f}}}^{\mathrm{f}} \text{}$.. DEA 4.4. ffl DEA $(N d)$ 3 DEA $(N d)$ $S$ $T$ $S\cup T$ (8) ( ) $d(s \cup T)=\min_{w}(\sum_{k\in_{\mathrm{L}}\backslash ^{\neg}}(\sum_{=1}^{n\mathfrak{l}}w\mathrm{x}_{k})+\sum_{k\in \mathit{1}}.(_{j}\sum_{=1} wx_{k}))$ $\geq\min_{w}.\sum_{k\in\backslash }.(\sum_{(=1}^{m}w_{l}x_{k})+$ min $\sum_{k\in/}.(\sum_{=1}^{\prime \prime}wx_{k})=d(s)+d(t)$. $(N d)$ $\mathrm{f}_{jb}^{\not\in}$ * (b \mbox{\boldmath $\tau$} ( )
216 4 MMy J}c \rightarrow 3 DEA $\max$ $(N c)$ $S$ $T$ $c(\cdot)$ $c(.s\mathrm{u}t)\leq c(s)+c(t)$. (10) DEA $\max$ $(N c)$. DEA -( $4^{-}\text{}$ $\iota_{-}\text{}$ \not\in ) 3 $\ovalbox{\tt\small REJECT}$ $\mathrm{h}$ $\vdash_{-}$ 4.5. $\mathrm{f}\mathrm{l}_{/}^{-\s\square }-\wedge\star\not\in l^{\mathrm{f}_{\backslash }}$] $\text{ }\neq x\text{}$ $\mathrm{i}\mathrm{j}\mathrm{d}\overline{\equiv}^{-}\mathrm{i}\underline{\prime\grave{ }_{\backslash }^{-}.}\mathrm{i}\backslash 0\text{ }$ ff DEA DEA. $)$. d$($ $)$ c $($ $ $ $ \backslash J$ ( $[10]\mathrm{p}.226$)
$\emptyset$ $= \sum_{\backslash ^{\tau}\llcorner\in\backslash \subset N} $ 217 ( ) DEA 3 5. DEA DEA 5.1. $(N v)$ $i$ $\emptyset$ $[6]\text{}$ $[\overline{\prime}]\mathrm{p}.343$ (v) ( ) $\frac{(s-1)!(n-s)!}{n!}\{.\dagger (S)-v(S-\int_{(}i\})\}$ $(v)=. \sum_{\backslash \llcorner\in\backslash \subset \mathrm{a}}.\cdot$ (s: ). $S$ (11) $n$ $N$ $i$ $v(s)-v(s-\{i\})$ Dllal Gurne DEA 4 DEA $\min$ $(Nd)$ DEA rnax $(N c)$ $r_{\overline{1}\wedge}$ $f1_{-}$ $D\iota\ell al$ Gclrfie $\phi$ $(c)= \sum_{\in\llcorner\backslash ^{\neg}s\subset\lambda }\frac{(s-1)!(n-_{1}s)!}{n!}\{c(s)-c(s-\{i\})\}$. $c(s)=1-d(n-s)$ $c\cdot(s-\{i\})=1-d(n-s+\{i\})$ $\frac{(s-1)!(n-s\cdot)!}{n!}$ [ $\{1-$ d(n-s)}-d-d(n-s $+$ {i})}] $= \sum_{s\prime\in.\backslash ^{\mathrm{t}}\subset N}\frac{(s-1)!(n-s)!}{n!}\{d(N-S+i\int_{(}\})-d(N-S)\}$. $S =N-S+\{i\}$ $S $ $s$ $=. \sum_{\backslash \mathrm{t}l\in_{1}\backslash \subset N}.\frac{(s -1)!(n-s )!}{n!}$ {$d(s )-d(s -\{$ i}$)$} $=\emptyset$ $(d)$. 4.
$\backslash \mathrm{h}\mathrm{c}3$ 4 A $\mathrm{c}$ $\mathrm{d}$ 218 4 $\lceil$ ( $[5]\mathrm{p}.57$ ) $\lceil 2.2$ 6 2. 2 $(N c)$ $(N d)$ 6: 1 2 34 $\mathrm{h}$ 3 4 1 2 10 4 4 3 1 12 1 3 31 8 2 1 3410 ( ) 0.333 0.4 0.375 0.4 1.508 ( ) 0.125 0.1 0.1 0.083 0.408 $(N c)$ $(N d)$ ( 0.229 0.271 0.260 0.240 1 ( 0.229 0.271 0.260 0.240 1 5.2. DEA $(N v)$ $S$ ( $T$ $[7]\mathrm{p}.313$ ) $v(s)+v(t)\leq v(s\mathrm{u}t)+v(s\cap T)$. (12) 3 DEA $(N d)$ 7 : DEA. 1 - $\sim $\mathrm{b}$ 0$ 4 1 3 3 1 1 2 1 1 7 $S=$ {AB} $T=\{AD\}$ $S\cup T=$ {ABD} $S\cap T=\{A\}$ $d(s)=0.75$ $d(t)=0.5.$ $d(s\cup T)=0.8$ : $d(s\cap T)=0.375$ $d(s)+d(t)=1.25>1.175=d(s\cup T)+d(S\cap T)$ ( DEA $\min$
219 5 $S\cup T=N$ $S$ $T$ $d(s)+d(t)\leq 1+d(S\cap T)$. (13) $(N c)$ $c(s-s\cap T)+c(T-S\cap T)\geq c(\{s-s\cap T\}+\{T-S\cap T\})$. (14) 2 $c(\{s-s\cap T\}+\{T-S\cap T\})=1-d(S\cap T)$ $c(s-s\cap T)=1-d$ (T) $c\cdot(t-s\cap T)=1-d($S) (14) $\{1-d(T)\}+\{1-d(S)\}\geq 1-d(S\cap T)$ $d(s)+d(t)\leq 1+d(_{\llcorner}\nabla\cap T)$. 5 6 3 DEA $\min$ $(N d)$ 5 $ii$ $d(\{ik\})+d(\{jk\})\leq d(\{ij k\})+d(\{k\})$. $\mathrm{c}15$ ) 5 6 3 DEA $\max$ $(N c)$ 6. $\mathrm{d}\mathrm{e}\mathrm{a}$ DEA $i^{\backslash }R$ (Egoist s Dilemma) 3 DEA DEA DEA if
.S..M. inear The 220 -^ $\underline{\mathrm{a}}^{[perp]}\overline{l}$ DEA 4. $\lceil 4.5$.. DEA DEA DEA AHP (Analytic Hierarchy Process) DEA NATO 2 $ ^{\ulcorner}2.3$ $\triangleright^{-}$ DEA Dual Gcnne $\mathrm{f}$ $\llcorner$ [1] Charnes A. and Cooper W.W. 1962 Programming with Fractional unctionals Naval Research Logistics Quarterly 15 $333\cdot 334$. $\llcorner$ [2] Cooper. W.W.. Seiford and Tone K. 1999 Data EnvelopmentAnalysis $-A$ Comprehensive $DE\mathrm{A}$ Text with Models Applications References and -Solver Software Klrwer Academic Publisher. [3] March $\mathrm{j}.\mathrm{g}$. and Olsen J.P. 1989 Rediscovering Institutions Organizational Basis of Politics The Free Press. () 1994 1. $\cdot$ [4] Nakabayashi K. and Tone K. 2003 Egoist s Dilemlna: A DEA Game GRIPS Research Report Series 1-9003$\sim$0002. [5] Schelling T.C. 1980 The Strategy ofconflict [1980 ed.] Cambridge MA: Harvard University Press. $\llcorner$ [6] Shapley 1953 A value for -person games in Kuhn and Tucker $n$ $(\mathrm{e}\mathrm{d}\mathrm{s}.)$ Contributions to the Theory of Games Vol. $\mathrm{u}_{\backslash }305-317$. 1996 $\ovalbox{\tt\small REJECT}$ [7]. [8] 1970 I ] $\ovalbox{\tt\small REJECT}$ [9] 2002 ]. $[searrow]- \text{}0$ [10] 1941 g-hp-i. $-\ddagger\backslash [11] 1996 \not\in \pi$. $\ovalbox{\tt\small REJECT}$ $+_{\backslash }^{c}g\prime xb\backslash F$ [12] 1993 \acute \dotplus + ^ r\not\subset DEA.