From 4a189109e6679184533c7dfd721606d6257978e6 Mon Sep 17 00:00:00 2001 From: youainti Date: Mon, 15 Nov 2021 20:44:10 -0800 Subject: [PATCH] Removed euler equation references from operator's problem --- .../sections/04_ConstellationOperator.tex | 96 +------------------ 1 file changed, 1 insertion(+), 95 deletions(-) diff --git a/CurrentWriting/sections/04_ConstellationOperator.tex b/CurrentWriting/sections/04_ConstellationOperator.tex index fa5e75d..3414fda 100644 --- a/CurrentWriting/sections/04_ConstellationOperator.tex +++ b/CurrentWriting/sections/04_ConstellationOperator.tex @@ -38,101 +38,7 @@ The inifinite period version of the problem above can be rewritten in the bellma \end{align} where $x^{\sim i}_t$ represents the launch decisions of all the other constellation operators. -This implies that the policy function is a best response function, allowing for +One important point is that the policy function is a best response function, allowing for a nash equilibrium interpretation of the result. -To solve for the policy function, we have a variety of methods available. -Due to the computational method chosen later, I'm going to examine the conditions -for the existence of an euler equation. - - -\subsubsection{Euler Equation} -Appendix \cref{APX:Derivations:EulerEquations} contains more details -on the math involved. -What follows is just a sketch of the method in matrix notation. - -As there is only one choice variable, we get a single optimality condition. -It can be written in various formats, with the latter matching the appendix the best. -\begin{align} -% 0 =& \parder{}{x^i_t}{} u^i(\vec s_t, D_t) -\parder{}{x^i_t}{}F(x) -% + \beta \left[ \parder{}{x^i_t}{} -% V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) -% \right] \\ - 0 =& -\der{F}{x^i_t}{} - + \beta \left[ - \nabla_{x^i_t} [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ] - \cdot - \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}} - V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) - \right] \label{EQ:OptimalityCondition}\\ - 0 =& -\der{F}{x^i_t}{} + \beta \vec a(\vec s_t,D_t) \cdot \nabla V^i_{t+1} - \label{EQ:SimplifiedOptimalityCondition}\\ - =& - f_{x_t} + \beta \vec a_t \cdot \nabla V^i_{t+1} -\end{align} - -As there are $N$ constellations we get $N$ satellite stocks, -$N-1$ decisions $x^{\sim i}$, -and $1$ debris state for a total of $2N$ state -variables\footnote{recall that $N$ is the number of constellations.}. -Thus there are $2N$ envelope conditions to be found: -\begin{align} -% \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} V^i(\vec s_t, \vec x^{\sim i}_t, D_t) -% =& \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} u^i(\vec s_t, D_t) \notag \\ -% &+ \beta \left[ -% \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} } -% V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) -% \cdot -% \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} -% [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ] -% \right] \label{EQ:EnvelopeConditions} -% \\ - \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} V^i(\vec s_t, \vec x^{\sim i}_t, D_t) - = - \nabla \vec V^i_t - = \vec u^i - + \beta B_t \cdot \nabla \vec V^i_{t+1} - \label{EQ:SimplifiedEnvelopeConditions} -\end{align} -%When interpreting this, note that -% $$ -% \nabla \vec V^i_{t+1} = \nabla_{[\vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1}] } -% V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) -% $$ -% is a $2N \times 1$ vector of first derivatives but -% $$ -% A = \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} -% [ \vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1} ] -% $$ -% is a $2N \times 2N$ matrix of first derivatives. - -% By solving for $\vec V^i_{t+1}$ as a function of $\vec V^i_{t}$ we get the -% intertemporal condition: -% \begin{align} -% \frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i_t \right) -% = \nabla \vec V^i_{t+1} -% \end{align} -% Thus one crucial condition for the existence of a solution is that $A^{-1}$ exists for -% all values the laws of motion and choice functions can take. - -% \subsection{Existence} -% I need to do some more diving into conditions for existence. -% Of particular concern is that the way I have specified the debris may lead to -% non-convergence. -% - -To finish constructing the euler equation, we would use the intertemporal -transition function \cref{EQ:SimplifiedEnvelopeConditions} and iterated -versions of \cref{EQ:OptimalityCondition,EQ:SimplifiedOptimalityCondition} -to construct the $2N+1$ euler equations.\footnote{Double check numbers} -Note that for even a small number of agents -- e.g. 3 -- this iterated substitution -becomes relatively complex, requiring caculating an iterated intertemporal tranisition -function and laws of motion 6 times. -To solve this symbolicly involves inverting a $6 \times 6$ matrix. -As matrix inversion has approximately an $O(n^3)$ computational complexity, -this becomes unsustainable very quickly. - -Section \cref{SEC:Computation} describes how to address this issue to generate -these euler equations using features of modern programming languages and linear algebra -libraries. - \end{document}