% Ver.2008/05/12
% Copyright (C) 2008- Mitsuru KIKKAWA, mitsurukikkawa@hotmail.co.jp
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\begin{document}
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{\small 2008�N�x���{�o�ϊw��t�G���(���k��w) 2008�N5/31(�y), 6/1(��)
��� A101 ${\big <}$ �����o�ϊw�ƃQ�[�����_ ${\big >}$}
\end{flushleft}
\vspace*{2mm}
\begin{center}
{\Huge \textbf{���v�͊w��p�����i���Q�[�����_}}
\\
\huge (Evolutionary Game Theory with Statistical Mechanics)
\\
{\Huge \textbf{�g�� �� (J)}}
\\
{\LARGE (Graduate School of Economics,
Kwansei Gakuin University, ��w�@������)
\\
{\tt E-mail:mitsurukikkawa@hotmail.co.jp,
URL: http://kikkawa.cyber-ninja.jp/index.htm}}
\end{center}
\begin{flushleft}
\Ovalbox{\parbox{79cm}
{\large \textbf{ABSTRACT}:
�i���Q�[�����_�𓝌v�͊w��p���Ē莮�����s����.
��̓I�ɂ͍ŋߐڂ̐l�Ƃ̂�, �����_���Ƀ}�b�`���O�����
�𓝌v�͊w�ōł��P����Ising���f��, SK���f�����Q�l��
�莮������. �����ŋύt(�헪�̈�v)�Ƃ����T�O�͑��]�ڂ𗘗p��
�Đ�������Ƃ���. ���̌��ʂ���p�����[�^�̑傫���ɂ����,
�`���I�Ȑi���Q�[�����_�ƈ�v����ꍇ�Ƃ��Ȃ��ꍇ�����邱��
����������. �܂������l�̃Q�[���ł͋ύt(�헪�̈�v)�͑��݂��Ȃ�
���Ƃ���������.
�܂��ύt���������Ă��Ȃ��p�����[�^�̈�ł�
Percolation��p����, ���ڂ����헪�̕��z�ɂ��čl�@����.
����ɊO����(���ӂ̃Q�[���ɉe��)���l���������̂𓱏o��,
Quenched�n�ɂ����đ��d�ύt�������Ă��邱�Ƃ�������.}}
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\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{MOTIVATION}
\\
�P�D��ԓI�ɕ��z���Ă����̂���ĂɃQ�[�����s��,
�ǂ̒��x�헪����v���Ă���̂��𒁏��p�����[�^��p����,
�Q�[���̋L�q.
\\
\textbf{RELATED LITERATURES}
\\
�P�DBlume [2](Ising Model),
Diederich and Opper [4] (SK model).}}
\\
\\
{\LARGE \textbf{MODEL}}
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{�ݒ�}
���ɑ����̎�̂�����, ���̊e��̂͐}1(��)�̂悤�Ɋi�q��ɂ���,
�ŋߐڂ̑���, �����_���}�b�`���O��, 2�̐헪�ŃQ�[�����s��.
}}
\\
\\
\hspace{5mm} \Ovalbox{\parbox{33.5cm}
{\Large
\textbf{���� 1.}
�e��͍̂��������E�K���x�𓾂邱�Ƃ�]��ł���.}}
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{���� 1.}
���� 1�̂��Ƃł̎��$x$��
����헪$\{S_i\},i=1,\cdots, N$�����, ���闘��$f$�𓾂�Ƃ���
�Q�[���̏󋵉��ɐ헪$\{S_i\}$�̊m�����z�͎��̂悤�ɂȂ�.
\\
(2.1) \hspace{3.5zw} $P(\{S_i\}) = Z^{-1}\exp(\gamma f)$.
\\
������ $\{S_i\}$�͎��$i$�̐헪, $\gamma$��
�ϐ�$f$ �͂���헪$\{S_i\}$��������Ƃ��̗����E�K�p�x, $Z$ �͋K
�i���萔��\���Ă���. �����
$\displaystyle \sum_{i=1}^N P(\{S_i\}) = 1$ �ƂȂ�.}}
\\
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{��` 1.}
�헪�����̒����������Ă��邩�ǂ�����
���f����ʂ�\textbf{�����p�����[�^}(order parameter)�Ƃ���
�T�O�����̂悤�ɓ�������.
\\
(2.2) \hspace{3.5zw} $m = {\Big <}S_i{\Big >} =
\dfrac{1}{N}\displaystyle \sum_{i=1}^N S_i \equiv
{\Big (}\displaystyle \sum_i S_i P(\{S_i\}){\Big )}$.
\\
������ ${\big <}\hspace{0.5zw} {\big >}$�͕��ς�\���Ă���.}}
\\
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{EXAMPLE}
\begin{center}
 \begin{tabular}{|c|c|c|}                     \hline
  \multicolumn{1}{|c|}{1�_2} & �헪1(+1)& �헪2(+2)\\ \hline
�헪1(+1) & $a$,$a$ & 0,0 \\ \hline
�헪2(+2) & 0,0 & $b$,$b$ \\ \hline
 \end{tabular}
\hspace{1zw}  \begin{tabular}{|c|c|c|}                     \hline
  \multicolumn{1}{|c|}{1�_2} & �헪1($-1$) & �헪2(+1) \\ \hline
�헪1($-1$) & $a$,$a$ & 0,0 \\ \hline
�헪2(+1) & 0,0 & $b$,$b$ \\ \hline
 \end{tabular}
\end{center}
\label{payoff1}
\hspace{8zw} �����\ 1
\hspace{12zw} �����\ 2
\\
�헪�̓Y������$\{1,2\}(\{-1,+1\})$�̏ꍇ:
�헪1���m��1�ō̗p����ꍇ,$m=1(-1)$,
�헪2���m��1�ō̗p����ꍇ,$m=2(+1)$,
�헪1�Ɛ헪2�������_���ɍ̗p����ꍇ,$m=\frac{3}{2}(0)$.}}

\begin{figure}[h]
 \begin{center}
  \scalebox{0.825}{\includegraphics{fig1.eps}\includegraphics{fig2a.eps}}\scalebox{1.25}{\includegraphics{fig2b-2.eps}}
  \caption{2���������i�q�ƍŋߐڊi�q�_�̗�.
�v�f$(x,y)$�͎�̂�\���Ă���. ��: Ising���f���ɂ����� (��) �ϐ�$\gamma$
�ƒ����p�����[�^$m$�Ƃ̊֌W, (�E) �����p�����[�^ $m$
�ƕϐ��Ɨ����̐� $\gamma f$ �Ƃ̊֌W.}
  \label{fig1}
 \end{center}
\end{figure}

\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{��` 2.}
$x\in \Delta$ ��\textbf{�i���I�Ɉ���Ȑ헪}
(Evolutionary Stable Strategy: ESS)�ł���Ƃ�,
�ǂ̂悤�Ȑ헪$y \ne x$ �ɑ΂��Ă�, ����$\bar{\epsilon}_y \in
(0,1)$ �����݂�, ���ׂĂ�$\varepsilon \in (0,\bar{\epsilon}_y)$
�ɂ��Ď��̕s���������藧���Ƃ�����.
\\
(2.3) \hspace{3.5zw} $u{\big [}x,\varepsilon y +
(1-\varepsilon)x{\big ]} > u{\big [}y,\varepsilon y +
(1-\varepsilon)x{\big ]}$.}}
\\
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{���� 3.}
���v�͊w��p�����i���Q�[�����_�ɂ�����
�i���I�Ɉ���Ȑ헪�Ƃ͎��̏����𖞂���.
\\
(2.4) \hspace{3.5zw} $u(y,x) \leq u(x,x)$,
\hspace{2zw} $\forall y$,
\hspace{2zw} (\textit{Equilibrium Condition})
\\
(2.6) \hspace{3.5zw} $|m-m^{\ast}| < \varepsilon$.
\hspace{7.5zw} (\textit{Stability Condition})
\\
������$m^{\ast}$�͐헪�̓Y�����������Ă���.}}
\\
\\
{\Large
\textbf{�헪����v���Ȃ��ꍇ}}
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{�藝 1.} (Coniglio, \textit{et al.} [3])
2����Ising���f���ɂ�����, $\mu^{s},s=\{+,-\}$�𕽍t�ɂ�����
($s$)�s�ϊm�����x�Ƃ���, ���̊֌W�����藧��.
\\
(i) $\gamma > \gamma_c$ �̂Ƃ�
\\
\hspace{5zw} $\mu^+_{\gamma,0}(\{|C^+_0|=\infty\}) > 0$,
\hspace{1zw} $\mu^-_{\gamma,0}(\{|C^-_0|=\infty\}) > 0$.
\\
(ii) Gibbs���z�̑S�� $\mathcal{G}(\gamma,h)$
�̔C�ӂ̒[�_ $\mu$ �ɑ΂���
\\
\hspace{5zw} $\mu(|C^+_0|=\infty)\mu(|C^-_0|=\infty) = 0$.}}
\\
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{�藝 2.} (��� [5])
$\gamma>0$ ���\������, $h$��
$\gamma' h' < \dfrac{1}{2}\log \dfrac{p_c}{1-p_c} - 4\gamma'$,
$\gamma h > \dfrac{1}{2}\log \dfrac{1-p_c}{p_c} + 4\gamma$
�𓯎��ɐ���������Ƃ�, $\mu_{\gamma,h}$ �Ɋւ��Ċm��1��
����$\ast$�N���X�^�[�̋������N����.}}
\\
\\
{\Large
\textbf{Random Matching}}
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{���� 4.}
�헪��2�����Ă����̂������_���}�b�`��,
�i�q������R�ɓ�����Ƃ���\underline{Annealed�n}�ɂ�����, ���җ�����
�ő�ɂ���悤�Ȓ����p�����[�^�͈ȉ��̓_�ŕ\�����.
��������̂���������Q�[���ł͒����p�����[�^��0�ƂȂ��
�������Ƃ���������.
\\
\hspace{5zw} $m = 0$\hspace{1zw} or \hspace{1zw}$\pm \sqrt{\dfrac{-J_0}
{\gamma J^2 N^2}}$.}}
\\
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{���� 5.}
�헪��2�����Ă����̂������_���}�b�`��,
�i�q������R�ɓ����Ȃ��Ƃ���\underline{Quenched�n}�ɂ�����,
���җ������ő�ɂ���悤�Ȓ����p�����[�^�͈ȉ��̊֐��ŕ\�����.
\\
\hspace{5zw}$m=\dfrac{1}{\sqrt{2\pi}}\displaystyle
\int_{-\infty}^{\infty}\exp{\Big (}-\dfrac{1}{2}z^2{\Big )}
\tanh (\gamma \tilde{J} \sqrt{q}z + \gamma \tilde{J}_0 n)dz$.}}
\\
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{���� 6.}
�e��͎̂�������邱�Ƃ��ł�, ����̉e��
�ɂ���Ă��������ω���(\underline{�O��̑���}), Quenched�n�̏ꍇ��
�����p�����[�^ $m$�̔�A���ȕω�, ���򂪋N��, ���d��������
(�}2(�E)).}}

\begin{figure}[h]
 \begin{center}
  \scalebox{0.825}{\includegraphics{fig3a.eps}\includegraphics{fig3b.eps}\includegraphics{fig4.eps}}
  \caption{(��) ���S�~�p�^�[��, (��) �`�F�X�Ճp�^�[��,
(�E) �����p�����[�^�̕���Ƒ��d��}
  \label{fig2}
 \end{center}
\end{figure}

\hspace{5mm} \Ovalbox{\parbox{38cm}
{\Large
\textbf{SUMMARY}
\\
\hspace{1cm} �P�D�ł��ȒP�ȍŋߐڂ̐l�Ƃ݂̂̏ꍇ�͋ύt��3����,
�헪�̓Y������$\{-1,1\}$�̏ꍇ(�����\ 2),
�����p�����[�^��$\tanh$�̊֐��ƂȂ���.
\\
\hspace{1cm} �Q�DCluster�����݂��Ă��Ȃ��ꍇ�͐헪�̕��z�͓��S�~�p�^�[����,
�`�F�X�Ճp�^�[�����l�����, ���� $\gamma$ ���\��������
�ꍇ����$\ast$Cluster�̋��������݂��Ă��邱�Ƃ���������
(�藝2).
\\
\hspace{1cm} �R�D�e��̂����R�Ɉړ����邱�Ƃ��ł��� Annealed �n��
�ꍇ��Replicator�n�Ɠ��l�ɋύt��3����, �����p�����[�^
�͓_�ŕ\����, �����l�o�ς̏ꍇ�͒����p�����[�^�̓����_����
�ꍇ�������݂��Ȃ����Ƃ���������(���� 4).
\\
\hspace{1cm} �S�D���ݍ�p���������猈�܂��Ă���, �e��̂����R�Ɉړ����邱
�Ƃ��ł��Ȃ���Ising type �̃��f���Ɠ��l��
�����p�����[�^�� $\tanh$ �̊֐��ƂȂ���(���� 5).
\\
\hspace{1cm} �T�D�O����(�W�ϐ�)�����݂���ꍇ,
Ising Type��Annealed �n�ł͐Êw�̏ꍇ�ƕς��Ȃ���,
Quenched�n�ɂ����đ��d�ύt�������Ă��邱�Ƃ���������(���� 6).
\\
\textbf{FUTURE WORKS}
\\
\hspace{1cm} �P�D���v�͊w�̎���Ԑ��ɒ��ڂ���. ��Ergodic���ɒ��ڂ�
Network�̋c�_��.
\\
\hspace{1cm} �Q�D$\gamma$�̓����� (�� �����v(superstatics)[1]).}}
\\
\\
\\
{\Large \textbf{REFERENCE}}
\\
\\
\hspace{5mm} \Ovalbox{\parbox{38cm}
{\normalsize
\hspace{5mm} [1] 
Beck C. and Cohen E.G.D.:
''Superstatistics,''
\textit{Physica A}, Vol. \textbf{322} (2003),
pp. 267-275.
\\
\hspace{5mm} [2]
Blume, Lawrence E.:
''The Statistical Mechanics of Strategic Interaction,''
\textit{Games and Economic Behavior}, Vol.\textbf{5} (1993),
pp. 387-424.
\\
\hspace{5mm} [3]
Coniglio,Antonio, Nappi, Chiara Rosanna, Peruggi, Fulvio
and Russo,  Lucio:
''Percolation and phase transitions in the Ising model,''
\textit{Communications in Mathematical Physics},
Vol. \textbf{51} (1976), pp. 315-323.
\\
\hspace{5mm} [4]
Diederich, S. and Opper, M.:
''Replicators with random interactions: A solvable model,''
\textit{Physical Review A}, Vol.\textbf{39} (1989),
pp. 4333-4336.
\\
\hspace{5mm} [5]
����ې�:
�u�C�W���O���f���̃p�[�R���[�V�����v
���w, Vol.\textbf{47} (1995), pp. 111-124.
\\
\hspace{5mm} [6]
Lanford III, O.E. and Ruelle, David:
''Observables at infinity and states with short range
correlations in statistical mechanics,''
\textit{Communications in Mathematical Physics},
Vol. \textbf{13} (1969), pp. 194-215.}}
\end{document}
