From de22ba1b4d4e067294ff71fdd1404e287174885b Mon Sep 17 00:00:00 2001 From: Miles Tiley Date: Tue, 26 Nov 2019 10:56:18 +1100 Subject: [PATCH] This commit aims to fix issue #698 - Removes redundant year in in-text citations - adds bibtex citations where missing - removes broken bibtex citations from docstrings - fixes typos --- source/rst/von_neumann_model.rst | 30 ++++++++++++++---------------- 1 file changed, 14 insertions(+), 16 deletions(-) diff --git a/source/rst/von_neumann_model.rst b/source/rst/von_neumann_model.rst index 76588910..2d25df42 100644 --- a/source/rst/von_neumann_model.rst +++ b/source/rst/von_neumann_model.rst @@ -14,8 +14,8 @@ Von Neumann Growth Model (and a Generalization) **Co-author:** Balint Szoke This notebook uses the class ``Neumann`` to calculate key objects of a -linear growth model of John von Neumann (1937) :cite:`von1937uber` that was generalized by -Kemeny, Moregenstern and Thompson (1956) :cite:`kemeny1956generalization`. +linear growth model of John von Neumann :cite:`von1937uber` that was generalized by +Kemeny, Morgenstern and Thompson :cite:`kemeny1956generalization`. Objects of interest are the maximal expansion rate (:math:`\alpha`), the interest factor (:math:`β`), and the optimal intensities (:math:`x`) and @@ -55,8 +55,7 @@ The code below provides the ``Neumann`` class """ This class describes the Generalized von Neumann growth model as it was - discussed in Kemeny et al. (1956, ECTA) :cite:`kemeny1956generalization` - and Gale (1960, Chapter 9.5) :cite:`gale1989theory`: + discussed in Kemeny et al. (1956, ECTA) and Gale (1960, Chapter 9.5): Let: n ... number of goods @@ -141,7 +140,6 @@ The code below provides the ``Neumann`` class """ Calculate the trivial upper and lower bounds for alpha (expansion rate) and beta (interest factor). See the proof of Theorem 9.8 in Gale (1960) - :cite:`gale1989theory` """ n, m = self.n, self.m @@ -409,7 +407,7 @@ for all :math:`j\in S`, :math:`\exists i\in T`, s.t. :math:`b_{i,j}>0`. The economy is **irreducible** if there are no proper independent subsets. -We study two examples, both coming from Chapter 9.6 of Gale (1960) :cite:`gale1989theory` +We study two examples, both coming from Chapter 9.6 of Gale :cite:`gale1989theory` .. code-block:: python3 @@ -540,7 +538,7 @@ and a number :math:`\alpha\in\mathbb{R}`, s.t. &\text{s.t. }\hspace{2mm}x^T B \geq \alpha x^T A \end{aligned} -Theorem 9.3 of David Gale’s book :cite:`gale1989theory` assets that if Assumptions I and II are +Theorem 9.3 of David Gale’s book :cite:`gale1989theory` asserts that if Assumptions I and II are both satisfied, then a maximum value of :math:`\alpha` exists and it is positive. @@ -578,7 +576,7 @@ that under Assumptions I and II, :math:`\beta_0\leq \alpha_0`. But in the other direction, that is :math:`\beta_0\geq \alpha_0`, Assumptions I and II are not sufficient. -Nevertheless, von Neumann (1937) :cite:`von1937uber` proved the following remarkable +Nevertheless, von Neumann :cite:`von1937uber` proved the following remarkable “duality-type” result connecting TEP and EEP. **Theorem 1 (von Neumann):** If the economy :math:`(A,B)` satisfies @@ -632,7 +630,7 @@ fact, it does not rule out (trivial) cases with :math:`x_0^TBp_0 = 0` so that nothing of value is produced. To exclude such uninteresting cases, -Kemeny, Morgenstern and Thomspson (1956) add an extra requirement +Kemeny, Morgenstern and Thomspson :cite:`kemeny1956generalization` add an extra requirement .. math:: x^T_0 B p_0 > 0 @@ -687,7 +685,7 @@ From the famous theorem of Nash (1951), it follows that there always exists a mixed strategy Nash equilibrium for any *finite* two-player zero-sum game. -Moreover, von Neumann’s Minmax Theorem (1928) :cite:`neumann1928theorie` implies that +Moreover, von Neumann’s Minmax Theorem :cite:`neumann1928theorie` implies that .. math:: V(C) = \max_x \min_p \hspace{2mm} x^T C p = \min_p \max_x \hspace{2mm} x^T C p = (x^*)^T C p^* @@ -730,7 +728,7 @@ of view) is the *dual* LP \end{aligned} -Hamburger, Thompson and Weil (1967) :cite:`hamburger1967computation` view the input-output pair of the +Hamburger, Thompson and Weil :cite:`hamburger1967computation` view the input-output pair of the economy as payoff matrices of two-player zero-sum games. Using this interpretation, they restate Assumption I and II as follows @@ -806,7 +804,7 @@ It is clear from the above argument that :math:`\beta_0`, :math:`\alpha_0` are the minimal and maximal :math:`\gamma` for which :math:`V(M(\gamma))=0`. -Moreover, Hamburger et al. (1967) :cite:`hamburger1967computation` show that the +Moreover, Hamburger et al. :cite:`hamburger1967computation` show that the function :math:`\gamma \mapsto V(M(\gamma))` is continuous and nonincreasing in :math:`\gamma`. @@ -817,7 +815,7 @@ input-output pair :math:`(A, B)`. Algorithm --------- -Hamburger, Thompson and Weil (1967) :cite:`hamburger1967computation` propose a simple bisection algorithm +Hamburger, Thompson and Weil :cite:`hamburger1967computation` propose a simple bisection algorithm to find the minimal and maximal roots (i.e. :math:`\beta_0` and :math:`\alpha_0`) of the function :math:`\gamma \mapsto V(M(\gamma))`. @@ -944,7 +942,7 @@ of the two methods we use. In particular, as will be shown below, in case of an irreducible :math:`(A,B)` (like in Example 1), the maximal and minimal roots of :math:`V(M(\gamma))` necessarily coincide implying -a ‘’full duality’’ result, i.e. :math:`\alpha_0 = \beta_0 = \gamma^*`, +a ‘‘full duality’’ result, i.e. :math:`\alpha_0 = \beta_0 = \gamma^*`, and that the expansion (and interest) rate :math:`\gamma^*` is unique. Uniqueness and Irreducibility @@ -983,7 +981,7 @@ is a self-sufficient part of the economy (a sub-economy) that in equilibrium can expand independently with the expansion coefficient :math:`\gamma^*_i`. -The following theorem (see Theorem 9.10. in Gale, 1960 :cite:`gale1989theory`) asserts that +The following theorem (see Theorem 9.10. in Gale :cite:`gale1989theory`) asserts that imposing irreducibility is sufficient for uniqueness of :math:`(\gamma^*, x_0, p_0)`. @@ -1005,7 +1003,7 @@ These assumptions imply that :math:`B=I_n`, i.e., that :math:`B` can be written as an identity matrix (possibly after reshuffling its rows and columns). -The simple model has the following special property (Theorem 9.11. in :cite:`gale1989theory`): if :math:`x_0` and :math:`\alpha_0>0` solve the TEP +The simple model has the following special property (Theorem 9.11. in Gale :cite:`gale1989theory`): if :math:`x_0` and :math:`\alpha_0>0` solve the TEP with :math:`(A,I_n)`, then .. math::