DTM.4:5 - Archetypal Grounding

DTM.4:5.1 - Rare growth and persistence give different answers

Consider a constructed reversible uptake model for the share f using a practice. The modeled adoption flow is s f²(1−f): encounters must provide the reinforcement represented by f². The modeled abandonment flow is s θ f(1−f): access to alternative users contributes to departure. Thus:

f' = s f(1−f)(f−θ),    0≤f≤1,    s>0,    0<θ<1.

The two flows are nonnegative on the stated region. Their forms are hypotheses for this example, not universal properties of social learning. With θ=0.4 and s=1 per chosen time unit, the fixed points are 0, 0.4 and 1.

For 0<f<0.4 the derivative is negative; for 0.4<f<1 it is positive. The endpoint regimes attract interior states on their respective sides, while 0.4 separates them. At f=0.2 the rate is −0.032; at f=0.6 it is +0.048.

The rare-variant test reports decline near zero. It does not establish that an existing majority will disappear. The model instead gives persistence from the higher initial range. The first useful result is this distinction, before any choice of intervention.

DTM.4:5.2 - A temporary reduction can be followed by return

In the same model, suppose an admissible action temporarily changes f from 0.8 to 0.5. When the action ends and the old law resumes, f increases again because 0.5 remains above 0.4. A displacement to 0.3 would instead place the trajectory in the lower range.

These statements do not establish an available or legitimate way to change f. They tell the project what a claimed temporary intervention would need to achieve under the model. Changing ongoing support would require a revised law, not a fictitious one-time jump.

The endpoints in this model are invariant: departure is absent at f=1 and uptake absent at f=0. If the actual practice permits independent departure, outside introduction or invention, add those processes before using the endpoint predictions. A change in the underlying event account may remove the threshold or replace the two regimes.

DTM.4:5.3 - Compatibility changes a threshold but does not guarantee uptake

Suppose a switching group bears conversion costs while old-standard partners keep their process unchanged. Over one horizon, its gain relative to continuing is g(f)=2−0.5−3(1−f), where f is the adopter share. Assume equally weighted encounters in a homogeneous population, so f also gives the group’s relevant partner share on the new standard. The gain is positive only above f=0.5.

An available adapter costing 0.4 and reducing the conversion coefficient to 0.2 gives gₐ(f)=2−0.5−0.4−0.2(1−f). At f=0.2, g=−0.9 while gₐ=0.94.

This is a threshold of the group’s comparison, not yet a population stability result. Add the explicit event rule that a fraction e of the remaining groups receive a feasible, authorized offer and adopt when the gain is positive. Then one step gives:

f_next = f + (1−f)e × indicator(g>0).

For f=0.2 and e=0.25, the share remains 0.2 without the adapter and becomes 0.4 with it. For e=0, neither positive gain nor a favorable threshold produces uptake.

This constructed rule omits departure, unequal partners and changing expectations. To conclude persistence or reversibility, supply those mechanisms where they matter and repeat the corresponding question. ECO.8 retains the coordination and allocation of costs; the engineering method establishes compatibility.