Derivatives of a Game Value Function in Connection with von Neumann Growth Models

Abstract

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,

$$\begin{gathered} \Phi \left( \alpha \right): = \mathop{{\max }}\limits_{{x \in \;{S^{{'m}}}}} \;\mathop{{\min }}\limits_{{y \in \;{S^{n}}}} \;x\;{M_{\alpha }}y = \mathop{{\min }}\limits_{{y\;{S^{n}}}} \;\mathop{{\max }}\limits_{{x\;{s^{m}}}} \;x\;{M_{\alpha }}y \hfill \\ : = v\left( {{M_{\alpha }}} \right) \hfill \\ \end{gathered} $$

((1))

where M_α: B -αA; B, A being nonnegative matrices of order m×n, α∈ R⁺;

$${S^{m}}: = \left\{ {x \in {R^{{m + }}}\left| {\sum\limits_{{i = 1}}^{m} {{x_{i}} = 1} } \right.} \right\} $$

((2))

$${S^{n}}: = \left\{ {y \in {R^{{n + }}}\left| {\sum\limits_{{j = 1}}^{n} {{y_{i}} = 1} } \right.} \right\} $$

((3))

Abstract of a paper presented to the Symposium on von Neumann Models, organized by the Institute of Mathematics of the Polish Academy of Sciences, held in Warsaw, July, 10-15, 1972, and subsequently to the Jahrestagung der Deutschen Gesellschaft für Operations Research, Hamburg, Sept., 6-8, 1972. The research described in this paper was carried out under grants from the Deutsche Forschungsgemeinschaft.