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In his dissertation \cite{Rao:Dissertation} briefly examines the ''survival rates" of 
a satellite constellation.
Applying the same analysis to this formulation of the law of motion 
for satellite stocks (\cref{SEC:Laws}) clarifies some details on risk burdens
with highly asymmectical conditions.
%This approach allows us to construct a elasticity of survival and satellite additions, 
%i.e. an elasticity of risk.

%I should probably look up how to analyze changes in risk level and quantitative representations etc.

% Marginal survival.
The survival rate for a constellation $i$ is defined as $R^i = 1-l^i(\cdot)$, 
i.e. the proportion of satellites
that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
Thus the marginal survival rate represents the additional loss of
satellites due to a slightly larger constellation or fleet stock.

To extend this definition to all fleets, we can measure the total number of 
satellites that survive. 
This can be calculated as the weighted sum of survival rates.
\begin{align}
    R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t} \\
    %=& \frac{\text{Total Surviving Satellites}}\frac{\text{Total Starting Satellites}}
\end{align}

\subsubsection{Marginal Survival Rates}

We can find the marginal survival rate with respect to a given constellation $s^i_t$ as:

\begin{align}
    \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} \\
        =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)^2 
           \left[ 
                \left(\sum^n_{j=1}s^j_t\right) \left(\parder{}{s^i_t}{}\sum^n_{j=1} s^j_t R^j\right)
                - \sum^n_{j=1} s^j_t R^j
           \right] \\
        =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right) 
           \left[ 
                \left(\sum^n_{j \neq i} s^j_t \parder{R^j}{s^i_t}{}\right) 
                + \left( R^i + s^i_t \parder{R^i}{s^i_t}{}\right)
                - R 
           \right] 
        \\
        =&  
                \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) 
                + \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right)
        \\
        \parder{R}{s^i_t}{} 
        =&
        \sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}
        + \left(\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}\right) \label{EQ:MarginalSurvivalRelation}
\end{align}
This can also be written in differential form as
\begin{align}
        d{R} 
        =&
        \sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) d{R^j}
        + \left(\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}\right) d{s^i_t}\label{EQ:differentialSurvivalRelation}
\end{align}

From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation},
we can see that the fleetwide marginal survival rate
is made up of two components.
We'll call these the direct and relative survival effects, 
corresponding to the $dR^j$ and $ds^i_t$ terms respectively.
\begin{itemize}
    \item The direct survival effect,
        $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$,
        represents the effect of a new satellite on each constellation. 
        It is always negative because
        $\parder{R^j}{s^i_t}{} < 0$ by assumption.
        Thus each constellations' survival rate will decrease as satellites are added to 
        any constellation.
    \item The relative survival effect, found in
        $\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}$,
        represents the effect of averaging out marginal survival rates.
        Intuitively, when a constellation has a higher survival rate
        than the general fleet's survival rate, adding a satellite to 
        that constellation contributes less colision risk than if it were given 
        to another constellation.
        Thus when there are a large number of satellites in orbit, regardless of who 
        owns them, this effect is removed.
\end{itemize}

Consequently, we can see that in most cases, the marginal survival rate will be negative.
In most models, this is either not examined or is assumed, but now we have the opportunity
to examine incentives in the case that it is not true.
One particular case where this may be important is when there is low utilization,
low internal risk, and near-monopolistic use of an orbital shell.


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