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\begin{document}
In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
I've applied this to my model and extended the results.

% Marginal survival.
The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, 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.

Let $S_t = \sum^n_{j=1} s^j_t$.
Then the survival rates for a constellation and for society's fleet are respectively defined as:
\begin{align}
    R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
    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}
\end{align}
In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites-
in period $t+1$ that survived from period $t$.

The marginal survival rates when a given constellation $i$ changes size are:
\begin{align}
    \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)
        =  - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\
    \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N \left(  [1-l^i(s^i_t,S_t,D_t)]
            + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right)
            - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}
        {(S_t)^2} \notag{}\\
    =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
\end{align}
Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the weighted, average marginal survival
rate across constellation operators.
The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:SurvivalRates}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm will be compared to society.
In spite of this, conditions on the average effects can be developted as follows.

The marginal survival rate of the fleet is less than the weighted, arithmetic mean of marginal survival rates-
of the constellations when:
\begin{align}
    \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
        \leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
    \sum_{i=1}^N  R_i  - R \leq& 0\\
    \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\\
    \sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)]  \leq& 0 \label{EQ:ii}
\end{align}
which is true if every constellation has at least one satellite.
As any constellation of interest has at least one satellite
and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$,
we conclude that the marginal survival rate of the entire satellite fleet is lower
than the weighted arithmetic mean of marginal survival rates across constellations.
Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
but the survival rate for many operators must be higher than the societal rate.
Consequently, we would expect many operators to underestimate the impact of their behaviors on others
if they use their own observed or expected risk factors to estimate the risk they impose on others.
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