Goal pursuit probability model

Goal pursuit probability at time $t$:

$$ P(A_{t}) = V \times (1 - e^{-s \times t}) \times \bigl(1 - d(1 - e^{-c \times t})\bigr) \times (1 - \frac{I_{t}}{I_0}) $$

Four forces shape persistence. Value alignment $V$ sets the ceiling. Skill growth $(1 - e^{-s \times t})$ drives early momentum. Depletion $\bigl(1 - d(1 - e^{-c \times t})\bigr)$ erodes willpower at rate $c$, but only down to a floor: $d$, between 0 and 1, is the most it ever takes. Interference $(1 - \frac{I_{t}}{I_0})$ compounds obstacles.

Without skill development ($s \rightarrow 0$), activation stalls regardless of motivation. As interference nears threshold ($I_{t} \rightarrow I_0$), probability crashes. Given time, skill saturates and depletion bottoms out, so $P(A_{t})$ settles toward a plateau:

$$ P_{\infty} = V \times (1 - d) \times (1 - \frac{I_{t}}{I_0}) $$

A goal survives when that plateau stays above the point where you quit. Value raises it; depletion and interference lower it.

The race between $s$ and $c$ sets the shape of the approach, not the destination. When skill grows faster than willpower drains ($s > c$), $P(A_{t})$ climbs to an early peak, then sags as depletion catches up, and levels off at the plateau: fast start, vulnerable middle, stable end. When depletion outpaces skill ($c > s$), there is no early peak—only a slow climb that never feels rewarding, which is its own way to lose a goal.

The formula holds $V$ and $d$ fixed, and its plateau moves only with $I_{t}$. Real goals move all of them: value grows as a practice pays off, depletion falls as habits automate, interference rises and falls with the rest of life. The examples below follow those shifts.

Learning Go from Python

Early weeks: High value ($V \approx 0.8$), fast skill gains, little depletion yet, minimal interference. $P(A_{t})$ rises through tutorials and first projects.

Months 2-3: Value holds. Skills slow (concurrency, errors) as depletion catches up. Work competes for attention. $P(A_{t})$ peaks, then sags toward its plateau.

Month 6+: Two paths. Persistence: patterns click, interference drops as Go integrates into the workflow, and the plateau sits above the quitting point—$P(A_{t})$ stabilizes, habit forms. Abandonment: $I_{t} \rightarrow I_0$ as priorities shift, the plateau falls below the quitting point, and the sag never ends.

Obsidian Knowledge System

Month 1: High value ($V \approx 0.9$), moderate skill growth, low depletion, low interference. $P(A_{t})$ rises through setup and note migration.

Months 2-4: Value tested before benefits materialize. Skill gains slow. Depletion rises from consistency demands. Novelty fades. $P(A_{t})$ sags—critical vulnerability.

Month 6+: Survivors see $V$ climb as note networks compound, and $d$ shrink as automation takes over the routine. Both raise the plateau, so $P(A_{t})$ crosses into habit with accelerating returns—the parameters changed, and the curve followed.

The middle phase kills most goals: $P(A_{t})$ falls from its early peak toward a plateau while $I_{t}$ grows, and if the plateau sits below the quitting point, nothing in the formula brings it back. Parameters vary by person—some deplete faster, others manage interference better. Interventions: strengthen $V$ through values alignment, accelerate $s$ through practice, lower $d$ through rest and routine, control $I_{t}$ through environment design.