Almost finished last of suggested changes

CurrentWriting/Main.tex

@@ -23,7 +23,7 @@
 
 %Describe sections
 The paper is organized as follows.
-Section \ref{SEC:Model}, describes the laws of motion 
+Section \ref{SEC:Models}, describes the laws of motion 
 governing satellites and debris(\ref{SEC:Laws})%, 
 %places limits on various measures of risk (\ref{SEC:Survival}),
 and reviews various definitions of kessler syndrome in 
@@ -32,7 +32,8 @@ It then describes the dynamic problem faced by constellation operators
 (\ref{SEC:Operator}) and social planners (\ref{SEC:Planner}).
 Section \ref{SEC:Computation} describes the computational approach and the
 results are reported in \cref{SEC:Results}.
-Section \ref{SEC:Conclusion} concludes with a discussion of future changes and 
+Section \ref{SEC:Conclusion} concludes with a discussion of limitations, concerns, 
+and remaining policy questions.
 
 %\section{Modeling the Environment}\label{SEC:Environment}
 \section{Model}\label{SEC:Models}

CurrentWriting/readme.txt

@@ -1,13 +1,14 @@
 
 
 # TODO
- - Add section to computation that includes specifications of loss and cost functions. (Pg 9)
- - Explain utility better in the constellation operators program (pg 8) especially why I refer to it as utility. Maybe rename to "benefit function"?
-    Currently working on this. Decided to rename to benefit function.
- - In training section (4.0.1.1) clarify loss function. There is the satellite loss function and
-    the Approximation loss function.
  - Add a conclusion
- - Reorganize the sections.
-    Currently working on this, mostly done.
 
 
+
+ ## Done
+ - Explain utility better in the constellation operators program (pg 8) especially why I refer to it as utility. Maybe rename to "benefit function"?
+    Currently working on this. Decided to rename to benefit function.
+ - Add section to computation that includes specifications of loss and cost functions. (Pg 9)
+ - In training section (4.0.1.1) clarify loss function. There is the satellite loss function and
+    the Approximation loss function. This also just needs rewritten.
+        Decided to rename satellite loss to satellite destruction rate and ML loss to objective.

CurrentWriting/sections/00_Introduction.tex

@@ -132,19 +132,17 @@ us to examine complex scenarios similar to those encountered in actual policymak
 %   - Allows for non-firm participants
 %   - avoidance efficiencies
 
-Below I describe the similarities and differences to these previous models to the current one.
+This model inherits the laws of motion for debris and constellation stocks from 
-As far as similarities go, it directly inherits the general laws of motion for debris and constellation stocks, 
+\autocite{RaoRondina2020,Adilov2018} and follows the DSGE modelling approach chosen by Rao.
-and follows the DSGE modelling approach chosen by Rao.
+It is distinguished from both of the aformentioned models 
-
+by the way it allows for the following:
-It is distinguished from these most models by the way it accounts for the following factors:
 \begin{itemize}
-    \item Heterogeneous agent types (represented by utility functions), 
+    \item Heterogeneous agent types including commercial, scientific, and military.
-        including commercial, scientific, and military.
+    \item Asymetric constellations.
-    \item Neither constellations are not assumed to be symmetric.
+    \item Inter- and intra- constellation risk is not assumed to be equal.
-    \item Collision avoidance efficiencies, i.e. within-constellation collisions are highly unlikely.
-    \item Heterogeneous risk between various satellite constellations.
 \end{itemize}
-The heterogeneity that I permit is the distinguishing feature of the model.
+The heterogeneity that I permit is the distinguishing feature of the model and the major
+justification for this work, as orbits are used by many different types of operators.
 
 
 

CurrentWriting/sections/01_LawsOfMotion.tex

@@ -38,8 +38,7 @@ following general law of motion for each constellation $i$.
     %  Representing those might be:
     %  - \eta s^i_t - y^i_t
 \end{align}
-Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions, i.e.
+Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions.
-the satellite loss function.
 %Assumption: 
 
 \subsubsection{Collision Efficiencies}
@@ -67,7 +66,7 @@ but this may be due to the fact that evasive maneuvers are usually taken
 when collisions appear reasonably possible.
 
 
-These collision efficiencies can be represented in the satellite loss function $l^i$ when:
+These collision efficiencies can be represented in the satellite destruction rate $l^i(\cdot)$ when:
 \begin{align}
     \parder{l^i}{s^k_t}{} > 0 ~~\forall k \in \{1,\dots,N)\\
     \parder{l^i}{s^j_t}{} > \parder{l^i}{s^i_t}{}  ~~\forall j\neq i

CurrentWriting/sections/04_ConstellationOperator.tex

@@ -3,18 +3,17 @@
 
 \begin{document}
 With the laws of motion introduced in sections \cref{SEC:Laws}, we can now describe
-the optimization problem facing each constellation operator.
+the optimization problem facing each constellation operator in general terms. 
+Actual functional specifications are described in \cref{SEC:Computation} on computation.
 
-Each operator recieve benefits\footnote{
+Each operator recieve per-period benefits 
-These benefits may take many forms, including profits for firms 
+-- such as profits for firms and warfighting capability for militaries --
-and utility for militaries.
-}
 from their constellation
-according to $u^i(\vec s_t,D_t)$, which depends 
+according to $u^i(\{s^j_t\},D_t)$, which depends 
 on the current sizes of constellations and the level of debris.
 In addition, the operator pays for the launch of $x^i_t$ satellites
-according to the cost function $F(x)$.
+according to a general cost function $F(x)$.
-These satellites will become operational in the next period.
+These satellites will become operational in the subsequent period.
 
 Thus the $M$-period (possibly infinite), problem is:
 \begin{align}

CurrentWriting/sections/05_SocialPlanner.tex

@@ -13,19 +13,17 @@ The Social (Fleet) Planner's problem can be written in the bellman form as:
          + \gamma \sum^N_{i=1} l^i(\vec s_t, D_t)
          + \Gamma \sum^N_{i=1} x^i_t
 \end{align}
-Some particular features of the model include:
+%Some particular features of the model include:
-\begin{itemize}
+%The single period welfare function consists only of constellation operators.
-    \item The single period welfare function consists only of constellation operators.
+%        Although satellites do deorbit and occasionally pose a risk to humans living on the
-        Although satellites do deorbit and occasionally pose a risk to humans living on the
+%        earth's surface\footnote{Skylab fell in Australia, with some pieces landing near towns.}
-        earth's surface\footnote{Skylab fell in Australia, with some pieces landing near towns.}
+%        modeling this risk properly would require adding a deorbit decisions, 
-        modeling this risk properly would require adding a deorbit decisions, 
+%        including uncontrolled deorbits.
-        including uncontrolled deorbits.
+Although the social planner controls each constellation, note that they do not reap additional
-    \item Although the social planner controls each constellation, they do not reap additional
 collision avoidance efficiencies.
-        One justification is that no social planner could concieve of every use of orbit
+this is because no social planner could concieve of every use of orbit
-        at any single point in time, and thus constellations are added sequentially.
+at any single point in time, and thus constellations may be designed sequentially.
 This allows only the intra-constellation benefits to be achived.
-\end{itemize}
 
 \subsubsection{Euler Equation}
 In accordance with Appendix \cref{APX:Derivations:EulerEquations}, 

CurrentWriting/sections/07_ComputationalApproach.tex

@@ -9,8 +9,7 @@ neural network.
 The approach uses the fact that the euler equation implicitly defines the 
 optimal policy function, for example: 
 $[0] = f(x(\theta),\theta)$.
-This can easily be turned into a mean square loss function by squaring both
+This can easily be turned into a mean square objective function,
-sides,
 $0 = f^2(x(\theta),\theta)$,
 allowing one to find $x(\dot)$ as the solution to a minimization problem.
 
@@ -57,7 +56,7 @@ three general computational approaches exist:
         \end{itemize}
 \end{itemize}
 I have chosen to use the AD to generate a euler equation function, which will
-then be the basis of our loss function.
+then be the basis of our objective function.
 
 
 The first step is to construct the intertemporal transition functions 
@@ -73,7 +72,7 @@ and laws of motion functions, retuning a $k$-period transition function.
 
 The second step is to generate functions that represent the optimality conditions.
 By taking the appropriate derivatives with respect to the laws of motion and
-utility functions, this can be constructed explicitly.
+benefit functions, this can be constructed explicitly.
 Once these two functions are completed, they can be combined to create
 the euler equations, as described in appendix \ref{APX:Derivations:EulerEquations}.
 
@@ -92,7 +91,7 @@ the euler equations, as described in appendix \ref{APX:Derivations:EulerEquation
 
 \paragraph{Training}
 
-With the euler equation and resulting loss function in place, 
+With the euler equation and resulting objective function in place, 
 standard training approachs can be used to fit the function.
 I plan on using some variation on stochastic gradient descent.
 
@@ -114,28 +113,36 @@ simultaneously.
 I would like to verify this approach as I have not dived into 
 some of the mathemeatics that deeply.
 
+\subsection{Functional Forms}
+The simpleset functional forms for the model are similar to those in 
+\autocite{RaoRondina2020}, giving:
+\begin{itemize}
+    \item The per-period benefit function:
+        \begin{align}
+            u^i(\{s^j_t\}, D_t) = \pi s^i_t
+        \end{align}
+    \item The launch cost function:
+        \begin{align}
+            F(x^i_t) =  f \cdot x^i_t 
+        \end{align}
+    \item The satellite destruction rate function:
+        \begin{align}
+            l^i(\{s^j_t\}, D_t) = 1 - e^{- d\cdot D_t - \sum^N_{j=1} h^j s^j_t}
+        \end{align}
+    \item The debris autocatalysis function:
+        \begin{align}
+            g(D_t) = g\cdot D_t
+            \\
+            g > 1
+        \end{align}
+\end{itemize}
 
-
+\subsection{Existence concerns}
-\subsubsection{Existence concerns}
 %check matrix inverses etc.
 %
 I am currently working on a plan to guarantee existence of solutions.
-Some of what I want to do is check numerically crucial values as well as
+Some of what I want to do is check numerically crucial values and 
-examine the necessary Inada conditions.
+mathematically necessary conditions for existence and uniqueness.
-
+Unfortunately this is little more than just a plan right now.
-
-\subsection{Computational Results}
-Cases to consider
-\begin{itemize}
-    \item Reproduce Rao-Rondina single satellite model.
-    \item Reproduce Adilov, perfect competition, cornot-like market.
-    \item Add military operators to Adilov's model. 
-        This will involve some sort of competitive complementarity 
-        with diminishing marginal returns.
-    \item Competitive market where the number of satellites improves quality, i.e. allows 
-        for pricing differences (Orbital Internet, e.g. Starlink).
-    \item Interacting orbital shells, using a vector representation of heterogeneous risk
-        imposed by constellations and debris.
-\end{itemize}
 
 \end{document}

CurrentWriting/sections/08_Conclusion.tex

@@ -2,9 +2,19 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-Conclusion
 % Not done yet
 % - Results to follow soon
 % - reiterate points about how important space is
+% - Discuss needed analyses
+%   - Insurance as a method of inducing proper action.
+%   - Rights of way (reduce risk by establishing procedures to avoid collisions.)
 % - 
+
+Although the the tragedy of the commons has not yet occured in
+the orbits surrounding out planet, the persistance of orbital debris 
+and the trend of decreasing launch costs suggest that it may not be far away.
+If it does occur, the effects may last centuries, impeding both science and human spaceflight.
+Hopefully, by the time this project is finished, I'll be able to distinguish
+the types of policies that lead to optimal constellation sizes.
+
 \end{document}

CurrentWriting/sections/09_Results.tex

@@ -6,20 +6,22 @@ So far, I have not been able to actually analyze any models,
 but the following are cases of interest.
 \begin{itemize}
     \item Reproduce Rao-Rondina single satellite model.
-    \item Reproduce Adilov, perfect competition, cornot-like market.
+    \item Reproduce Adilov's cornot-like market. 
-    \item Add military operators to Adilov's model. 
+    \item Add military operators to Adilov or Rao's model. 
         This will involve some sort of competitive complementarity 
         with diminishing marginal returns.
     \item Competitive market where the number of satellites improves quality, i.e. allows 
         for pricing differences (Orbital Internet, e.g. Starlink).
     \item Interacting orbital shells, using a vector representation of heterogeneous risk
         imposed by constellations and debris.
+    \item Add a deorbit choice variable to the model.
 \end{itemize}
 
 Among these, policies that would be interesting to analyse include:
 \begin{itemize}
     \item Launch and Operation Taxes
-    \item Deorbit contingent bonds.
+    \item Deorbit-contingent bonds, similar to environmental cleanup bonds 
+        in mining operations.
 \end{itemize}