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

• O. Moeschlin
Conference paper
Part of the Proceedings in Operations Research book series (ORP, volume 1972)

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 Mα: 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)

Preview

Literatur

1. [1]
J.G. Kemeny, O. Morgenstern and G.L. Thompson: A Generalization of the von Neumann-Model of an Expanding Economy, Econometrica 1956, pp. 115–135.Google Scholar
2. [2]
H.D. Mills: Marginal Values of Matrix Games and Linear Programs, in Linear Inequalities and Related Systems (Editors: H.W. Kuhn and A.W. Tucher), Princeton, New Jersey, 1956, pp. 183–198.Google Scholar
3. [3]
O. Morgenstern and G.L. Thompson: An Open Expanding Economy Model, Noval Research Logistics Quarterly 16, 1969, pp. 443–457.Google Scholar
4. [4]
G.L. Thompson: On the Solution of a Game Theoretic Problem, in Linear Inequalities and Related Systems (Editors: H.W. Kuhn and A.W. Tucher), Princeton, New Jersey, 1956, pp. 275–284.Google Scholar