Derivatives of a Game Value Function in Connection with von Neumann Growth Models
Otto Moeschlin · Operations research proceedings · 1973
A possible way to show the existence of solutions to a generalized von Neumann growth model, due to Kemeny, Morgenstern and Thompson leads to a discussion of a game value function ∅: R+ → R, (1) $$\begin{gathered} \Phi \left( \alpha \right): = \mathop{{\max }}\limits_{{x \in \;{S^{{'m}}}}} \;\mathop{{\min }}\limits_{{y \in \;{S^{n}}}} \;x\;{M_{\alpha }}y = \mathop{{\min }}\limits_{{y\;{S^{n}}}} \;\mathop{{\max }}\limits_{{x\;{s^{m}}}} \;x\;{M_{\alpha }}y \hfill \\ : = v\left( {{M_{\alpha }}} \right) \hfill \\ \end{gathered} $$ where Mα: B -αA; B, A being nonnegative matrices of order m×n, α∈ R+; (2) $${S^{m}}: = \left\{ {x \in {R^{{m + }}}\left| {\sum\limits_{{i = 1}}^{m} {{x_{i}} = 1} } \right.} \right\} $$ (3) $${S^{n}}: = \left\{ {y \in {R^{{n + }}}\left| {\sum\limits_{{j = 1}}^{n} {{y_{i}} = 1} } \right.} \right\} $$