If no one actually knows the plan other than the guy in charge, no one can leak the plan:
An example of compartmentalization was the Manhattan Project. Personnel at Oak Ridge constructed and operated centrifuges to isolate uranium-235 from naturally occurring uranium, but most did not know exactly what they were doing. Those that knew did not know why they were doing it. Parts of the weapon were separately designed by teams who did not know how the parts interacted.
True, and interesting since this can be used as a statistical lever to ignore the exponential scaling effect of conditional probability, with a minor catch.
Lemma:
Compartmentalization can reduce, even eliminate, chance of exposure introduced by conspirators.
Proof:
First, we fix a mean probability p of success (avoiding accidental/deliberate exposure) by any privy to the plot.
Next, we fix some frequency k1, k2, ... , kn of potential exposure events by each conspirators 1, ..., n over time t and express the mean frequency as k.
Then for n conspirators we can express the overall probability of success as
1 ⋅ ptk~1~ ⋅ ptk~2~ ⋅ ... ⋅ ptk~n~ = pntk
Full compartmentalization reduces n to 1, leaving us with a function of time only ptk. ∎
Theorem:
While it is possible that there exist past or present conspiracies w.h.p. of never being exposed:
they involve a fairly high mortality rate of 100%, and
they aren’t conspiracies in the first place.
Proof:
The lemma holds with the following catch.
(P1) ptk is still exponential over time tunless the sole conspirator, upon setting a plot in motion w.p. pt~1~k = pk, is eliminated from the function such that pk is the final (constant) probability.
(P2) For n = 1, this is really more a plot by an individual rather than a proper “conspiracy,” since no individual conspires with another. ∎
Compartmentalisation helps
If no one actually knows the plan other than the guy in charge, no one can leak the plan:
True, and interesting since this can be used as a statistical lever to ignore the exponential scaling effect of conditional probability, with a minor catch.
Lemma: Compartmentalization can reduce, even eliminate, chance of exposure introduced by conspirators.
Proof: First, we fix a mean probability p of success (avoiding accidental/deliberate exposure) by any privy to the plot.
Next, we fix some frequency k1, k2, ... , kn of potential exposure events by each conspirators 1, ..., n over time t and express the mean frequency as k.
Then for n conspirators we can express the overall probability of success as
1 ⋅ ptk~1~ ⋅ ptk~2~ ⋅ ... ⋅ ptk~n~ = pntk
Full compartmentalization reduces n to 1, leaving us with a function of time only ptk. ∎
Theorem: While it is possible that there exist past or present conspiracies w.h.p. of never being exposed:
Proof: The lemma holds with the following catch.
(P1) ptk is still exponential over time t unless the sole conspirator, upon setting a plot in motion w.p. pt~1~k = pk, is eliminated from the function such that pk is the final (constant) probability.
(P2) For n = 1, this is really more a plot by an individual rather than a proper “conspiracy,” since no individual conspires with another. ∎