diff --git a/source/rst/von_neumann_model.rst b/source/rst/von_neumann_model.rst index 2005c7c..c6983f8 100644 --- a/source/rst/von_neumann_model.rst +++ b/source/rst/von_neumann_model.rst @@ -599,18 +599,18 @@ following arbitrage true x_0^T\left(B-\gamma^{* } A\right)p_0 &= 0 \end{aligned} -.. - - *Proof (Sketch):* Assumption I and II imply that there exist :math:`(\alpha_0, - x_0)` and :math:`(\beta_0, p_0)` that solve the TEP and EEP, respectively. If - :math:`\gamma^*>\alpha_0`, then by definition of :math:`\alpha_0`, there cannot - exist a semi-positive :math:`x` that satisfies :math:`x^T B \geq \gamma^{* } - x^T A`. Similarly, if :math:`\gamma^*<\beta_0`, there is no semi-positive - :math:`p` for which :math:`Bp \leq \gamma^{* } Ap`. Let :math:`\gamma^{* - }\in[\beta_0, \alpha_0]`, then :math:`x_0^T B \geq \alpha_0 x_0^T A \geq - \gamma^{* } x_0^T A`. Moreover, :math:`Bp_0\leq \beta_0 A p_0\leq \gamma^* A - p_0`. These two inequalities imply :math:`x_0\left(B - \gamma^{* } A\right)p_0 - = 0`. +.. note:: + + *Proof (Sketch):* Assumption I and II imply that there exist :math:`(\alpha_0, + x_0)` and :math:`(\beta_0, p_0)` that solve the TEP and EEP, respectively. If + :math:`\gamma^*>\alpha_0`, then by definition of :math:`\alpha_0`, there cannot + exist a semi-positive :math:`x` that satisfies :math:`x^T B \geq \gamma^{* } + x^T A`. Similarly, if :math:`\gamma^*<\beta_0`, there is no semi-positive + :math:`p` for which :math:`Bp \leq \gamma^{* } Ap`. Let :math:`\gamma^{* + }\in[\beta_0, \alpha_0]`, then :math:`x_0^T B \geq \alpha_0 x_0^T A \geq + \gamma^{* } x_0^T A`. Moreover, :math:`Bp_0\leq \beta_0 A p_0\leq \gamma^* A + p_0`. These two inequalities imply :math:`x_0\left(B - \gamma^{* } A\right)p_0 + = 0`. Here the constant :math:`\gamma^{*}` is both an expansion factor and an interest factor (not necessarily optimal). @@ -747,17 +747,17 @@ Using this interpretation, they restate Assumption I and II as follows V(-A) < 0\quad\quad \text{and}\quad\quad V(B)>0 -.. - - *Proof (Sketch)*: \* :math:`\Rightarrow` :math:`V(B)>0` implies - :math:`x_0^T B \gg \mathbf{0}`, where :math:`x_0` is a maximizing - vector. Since :math:`B` is non-negative, this requires that each - column of :math:`B` has at least one positive entry, which is - Assumption I. \* :math:`\Leftarrow` From Assumption I and the fact - that :math:`p>\mathbf{0}`, it follows that :math:`Bp > \mathbf{0}`. - This implies that the maximizing player can always choose :math:`x` - so that :math:`x^TBp>0` so that it must be the case - that :math:`V(B)>0`. +.. note:: + + *Proof (Sketch)*: \* :math:`\Rightarrow` :math:`V(B)>0` implies + :math:`x_0^T B \gg \mathbf{0}`, where :math:`x_0` is a maximizing + vector. Since :math:`B` is non-negative, this requires that each + column of :math:`B` has at least one positive entry, which is + Assumption I. \* :math:`\Leftarrow` From Assumption I and the fact + that :math:`p>\mathbf{0}`, it follows that :math:`Bp > \mathbf{0}`. + This implies that the maximizing player can always choose :math:`x` + so that :math:`x^TBp>0` so that it must be the case + that :math:`V(B)>0`. In order to (re)state Theorem I in terms of a particular two-player zero-sum game, we define a matrix for :math:`\gamma\in\mathbb{R}`