Minimal Formal Note
0. Scope
This note defines a necessary condition for the decay of strategies in adaptive systems whose reproduction is mediated by a selection boundary. It introduces definitions and one primary theorem.
1. Preliminaries
Let:
S={s1,…,sn} denote a strategy set.
X denote outcome space.
Y:X→R≥0 denote a yield function.
B⊆X denote a selecting boundary (the locus of amplification/reproduction).
Each strategy sksk induces a distribution P(x∣sk)P(x∣sk) over outcomes.
Expected yield at the selecting boundary:Uk:=E[Y(x)∣x∈B,sk]Uk:=E[Y(x)∣x∈B,sk]
Replicator dynamics:p˙k=pk(Uk−Uˉ)p˙k=pk(Uk−Uˉ)
where Uˉ=∑jpjUj(system-wide mean yield)Uˉ=∑jpjUj(system-wide mean yield).
2. Externalized Loss
Let L(x)≥0L(x)≥0 denote loss imposed on outcomes.
Boundary-coupled: Loss is coupled iff Y′(x)=Y(x)−L(x)Y′(x)=Y(x)−L(x) for x∈Bx∈B.
Externalized: Loss is externalized iff it does not modify yield at the selecting boundary (i.e., Y(x)Y(x) remains invariant for x∈Bx∈B).
3. Theorem (Boundary-Coupled Decay Condition)
A strategy sksk decays asymptotically (pk→0pk→0) under replicator dynamics only if:Uk<Uˉ
Uk<Uˉ
In particular, loss L(x)L(x) produces decay if and only if it reduces expected yield at the selecting boundary:
E[Y(x)−L(x)∣x∈B,sk]<UˉE[Y(x)−L(x)∣x∈B,sk]<Uˉ
If loss is externalized (not coupled to BB), then UkUk remains unchanged and the strategy does not decay by inefficiency alone.
4. Corollary (Non-Inevitability)
Inefficiency, harm, or aggregate cost outside the selecting boundary does not imply decay. Selection responds only to boundary-coupled yield differences.
5. Converse (Expectation Detectability)
If a mechanism selectively suppresses a strategy class at scale, then ∃ΔUk≠0∃ΔUk=0. Such shifts are detectable in expectation under sufficient sampling.
6. Remarks
The model is domain-agnostic (biological, economic, informational).
Does not assume rational actors.
Specifies a necessary condition for decay, not a sufficient implementation.