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@ -4,58 +4,80 @@
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\begin{document}
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In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
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I've applied this to my model and extended the results.
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This approach allows us to construct a elasticity of survival and satellite additions, i.e. an elasticity
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of risk.
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%I should probably look up how to analyze changes in risk level and quantitative representations etc.
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% Marginal survival.
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The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites-
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The survival rate for a constellation $i$ is defined as $R^i = 1-l^i(\cdot)$, i.e. the proportion of satellites-
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that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
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Thus the marginal survival rate represents the additional loss of
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satellites due to a slightly larger constellation or fleet stock.
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Let $S_t = \sum^n_{j=1} s^j_t$.
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Then the survival rates for a constellation and for society's fleet are respectively defined as:
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To extend this definition to all fleets, we can measure the total number of
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satellites that survive.
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This can be calculated as the weighted sum of survival rates.
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\begin{align}
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R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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R =& \frac{S_{t+1}- X_t}{S_t} = \frac{\sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] }{S_t} \label{EQ:socsurv}
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R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t}
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\end{align}
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In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites-
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in period $t+1$ that survived from period $t$.
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The marginal survival rates when a given constellation $i$ changes size are:
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\subsubsection{Marginal Survival Rates}
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We can find the marginal survival rate with respect to a given constellation $s^i_t$ as:
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\begin{align}
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\parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right)
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= - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\
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\parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N \left( [1-l^i(s^i_t,S_t,D_t)]
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+ s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right)
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- \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}
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{(S_t)^2} \notag{}\\
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=& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
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\parder{R}{s^i_t}{} =& \parder{}{s^i_t}{}\frac{\sum_{j=1}^n s^j_t R^i}{\sum_{j=1}^n s^i_t} \\
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=& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)^2
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\left[
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\left(\sum^n_{j=1}s^j_t\right) \left(\parder{}{s^i_t}{}\sum^n_{j=1} s^j_t R^j\right)
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- \sum^n_{j=1} s^j_t R^j
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\right] \\
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=& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)
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\left[
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\left(\sum^n_{j \neq i} s^j_t \parder{R^j}{s^i_t}{}\right)
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+ \left( R^i + s^i_t \parder{R^i}{s^i_t}{}\right)
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- R
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\right]
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\\
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=&
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\left(\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}\right)
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+ \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right)
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\\
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\parder{R}{s^i_t}{}
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=&
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\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}
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+ \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) \label{EQ:MarginalSurvivalRelation}
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\end{align}
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Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the weighted, average marginal survival
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rate across constellation operators.
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The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:SurvivalRates}.
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Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
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cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
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and specifying which firm will be compared to society.
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In spite of this, conditions on the average effects can be developted as follows.
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The marginal survival rate of the fleet is less than the weighted, arithmetic mean of marginal survival rates-
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of the constellations when:
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This can also be written in differential form as
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\begin{align}
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\sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
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\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
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\sum_{i=1}^N R_i - R \leq& 0\\
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\sum_{i=1}^N [1- l^i(s^i_t,S_t,D_t)] - \sum_{i=1}^N s^i_t [1- l^i(s^i_t,S_t,D_t)] \leq& 0\\
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\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii}
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d{R}
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=&
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\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) d{R^j}
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+ \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) d{s^i_t}\label{EQ:differentialSurvivalRelation}
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\end{align}
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which is true if every constellation has at least one satellite.
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As any constellation of interest has at least one satellite
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and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$,
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we conclude that the marginal survival rate of the entire satellite fleet is lower
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than the weighted arithmetic mean of marginal survival rates across constellations.
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Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
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but the survival rate for many operators must be higher than the societal rate.
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Consequently, we would expect many operators to underestimate the impact of their behaviors on others
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if they use their own observed or expected risk factors to estimate the risk they impose on others.
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From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation},
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we can see that the fleetwide marginal survival rate
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is made up of two components.
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\begin{itemize}
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\item $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$
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represents the effect on each satellite constellation, and is always negative because
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$\parder{R^j}{s^i_t}{} < 0$ by assumption.
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Thus each constellations' survival rate will decrease as satellites are added to
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any constellation.
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\item $\frac{ R^i - R }{\sum_{j=1}^n s^j_t}$,
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represents the effect of averaging out marginal survival rates.
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Intuitively, when a constellation has a higher survival rate
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than the fleet's survival rate, adding a satellite to that fleet contributes
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less colision risk than if it were given to another
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Note that it is positive but only when $R^i > R$.
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Additionally, it disappears quickly as the total number of satellites increase.
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Thus when there are a large number of satellites in orbit, regardless of who
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owns them, it is almost certain that any increase in satellite stocks will
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lead to a reduction in the survival rate.
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\footnote{I believe Rao makes this an assumption, I show it is a result}
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\end{itemize}
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Consequently, we can see that in many cases, the marginal survival rate will be negative.
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\end{document}
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youainti
5 years ago