You walk to a berry bush and start stripping it. The first minutes are easy; after
that each one pays less than the last. Stay too long and you starve on bare twigs. Leave too soon
and you wasted the walk. There is an exact right moment, and you can draw it.
The walk, then the bush● food gathered● your rate for the whole trip● the best line
your rate, whole trip—this minute is paying—best moment to leave—bush stripped by then—
—
—
0.2 leave at once6.9 — the best40 strip it bare
0 no walk8 — default40 a long trek
1 empties fast5.0 — default15 deep bush
↳ drag You leave after and watch
the blue line pivot — your rate rises, peaks, then falls while you are still gathering food
What you're seeing
Left of the zero mark is the walk. Nothing is gathered during it, so the line is flat. Right of
zero is your time in the bush, and the amber curve is the food you have gathered so far.
The curve bends because the bush runs out. Easy berries first, then the awkward ones, then
almost nothing.
The blue line runs from the start of your walk to the moment you leave. Its steepness is what
you actually earn per minute across the whole trip — walking included. That is the number
to maximise, not the food in your hands.
Drag the leaving time. The blue line pivots. It gets steeper, reaches a best, then flattens
again while the amber curve is still climbing. You are still gathering food and already
losing.
The best line is the one that just grazes the curve instead of cutting it. At that instant the
bush is paying you exactly what the whole trip pays — so the next minute here is worth
exactly what a minute of walking-and-starting-fresh is worth, and after it, less.
Now lengthen the walk. The best moment moves later: a costly commute buys patience. And press
Double the bush's richness. The curve doubles, everything is worth twice as much, and
the best moment does not move at all. Charnov published this in 1976 for
animals. But it is the same shape as closing a book you have stopped enjoying, abandoning a
thinning seam, or leaving a conversation at a party. Every case where the cost of switching is
what makes staying tolerable.
The rule, exactly. Travel time T, then a stay of t in a
patch whose cumulative yield is
g(t) =G(1 − e−t/τ)
Over a long run of identical trips the intake rate is R(t) =
g(t)/(T+t). That is exactly the slope of the line
from (−T, 0) to (t, g(t)) — the line this
page draws. Setting its derivative to zero gives Charnov's marginal value theorem:
g′(t*) =g(t*) / (T + t*)
Leave when this minute's pay has fallen to the whole trip's average pay. Geometrically the best
line is the tangent from the start of the walk.
Verified in node (improve/verify/112-marginal-value.js, 16 checks):
The page's bisection, a 400,000-point brute-force search over R(t),
and a closed form written with the Lambert W function all return the same
t* at nine settings, agreeing to 1 part in 1011.
The Lambert W routine is validated on its own terms first, before any claim
rests on it.
The identity g′(t*) = R(t*) holds to
machine precision, and misses by at least 8.4% at 0.8× and 1.25× t*.
It is not true everywhere.
The chord this page actually draws has slope exactly g′(t*),
so the drawn construction is the real tangent and not an illustration of one.
t* rises monotonically with T, scales with τ, and is
exactly unchanged by G from 1 to 100,000.
Negative control. With T = 0 the solver returns
t* = 0 exactly. No commute, no reason to tolerate diminishing returns.
Negative control. Forget the walk and draw the tangent from the origin
instead, and the rule has no answer at all — not a wrong time, no time.
Negative control. "I walked ten minutes so I'll stay ten" never beats the
theorem and loses up to 38.6%. It is right for exactly one commute length.
Negative control. A linear yield curve has no best moment: the right
answer becomes "never leave", and the solver reports that instead of inventing a root.
Diminishing returns is the load-bearing assumption.
Negative control. A fixed "leave at 75% stripped" quota never wins and
loses up to 23.5%. The right fraction moves with the walk.
What the model leaves out: patches that differ from each other, a forager
who cannot tell how good this one is, competitors, and any risk of being eaten while walking.