Method note 01

Distance to flip

There is no cut line

Somewhere this quarter a committee will approve 35 initiatives worth $6.3m and decline 32 others. A line will be drawn across a ranked list. Above it, work that is going ahead; below it, work that is not. Everyone in the room will treat that line as a finding.

It is nothing of the sort. Move the envelope by $400,000 and initiatives cross it in both directions, their merits entirely unchanged. Nothing was learned about the work. Something was decided about the money.

Notice what happens when the budget does not bind at all. Every initiative worth doing is done. There is no cut, no in and out, no line — the question does not arise. The boundary appears only when capital is scarce, which means it is a property of the constraint rather than of the work. The line on the screen is not a finding. It is the shadow of a budget.

That would matter less if the presentation were harmless. It is not, and the cost falls in both directions.

What the ranking hides

Consider two funded initiatives. One scores 736 and costs $111,000. The other scores 507 and costs $1.33m. Ranked by score the first sits comfortably above the second, and both appear equally settled.

Now tighten the envelope by $400,000, or let the larger initiative's cost estimate rise by 15% — ordinary events in any planning cycle. The expensive initiative leaves the portfolio. The cheap one does not move, and would not move under almost any realistic change.

Both were in. One was never in question. The other was a single estimating error from being out. The ranked list conceals exactly that distinction, and conceals it symmetrically: an initiative excluded by $12,000 is presented identically to one excluded by $2m.

The reason is structural. A score is a property of an initiative. Membership in the selected set is a property of the solution — it depends on the envelope, on the constraints, and on every other initiative competing for the same capital. Using an initiative-level property to describe a solution-level outcome is a category error, and it is the standard practice of the field.

It follows that distance from the cut line, measured in score, proxies nothing useful. A high-scoring expensive initiative can be more fragile than a low-scoring cheap one. What is wanted is distance to flip, which is a different measurement entirely.

Figure 1. The same ten initiatives, ordered two ways. On the left the rows are identical: the ordering carries no information about how settled each decision is, and only the cut line distinguishes them. On the right, five of the ten sit near the boundary — three of them funded.

Distance to flip

Order the boundary by how far the world must move before an initiative's status changes, expressed in the units of whatever moved. If costs move, that is dollars: this leaves the portfolio if its cost rises $180,000, or 14%. For the envelope it is the change that would admit or eject it. For declared priorities it is the shift that would flip it.

The direction matters, and it is not symmetric. Cost estimates do not wander; they rise. Flyvbjerg's transport data puts the average overrun at 44.7% for rail, 33.8% for bridges and tunnels and 20.4% for roads, and finds no improvement across seventy years of records. So the measurement that earns its place is distance to flip in the direction estimates actually move, with the range taken from the organization's own history of overruns rather than from a figure somebody chose.

The output is not a ranked list but four bands: held, where no realistic change removes the initiative; borderline in, funded but close; borderline out, excluded but close; and out, excluded by a wide margin.

Readers from operations research will recognize the shape. This is a stability radius — the largest perturbation a solution tolerates before its structure changes — and readers from machine learning may find the margin of a classifier the nearer analogue. Neither idea is new and no claim is made that it is. What is proposed is that it belongs on the executive screen rather than in an appendix.

Two properties make it useful in a room where decisions are actually taken. It is symmetric: borderline-in and borderline-out sit adjacent and comparable, so a reader can see that a $67,000 initiative missed by very little while a $480,000 initiative that was funded is barely holding. And it is expressed in decision units rather than model units. This flips if the estimate is off by 14% is a sentence a finance director can weigh against their own experience. This scored 507 is not.

The instinct at this point is to reach for probability — a 0.97 chance of selection here, 0.4 there. The instinct is right and the instrument is wrong. A probability requires distributions over every cost and every declared weight, and organizations do not have them. Inventing them and reporting a figure to two decimal places would manufacture exactly the precision at issue. Distance avoids the problem: within a stated direction of change, the probability of flipping falls monotonically with distance, so distance establishes the ordering and the bands without asserting a number that cannot be defended. One cannot honestly say P = 0.97. One can honestly say these four sit in the fragile band.

What it costs

I ran this against a portfolio of 66 candidate initiatives. The solver committed 38 of them at $7.46m against a $7.78m envelope.

Ordered by score the committed set looks uniform. Ordered by distance to flip it resolves into something with structure. Twenty-three of the 38 hold past a doubling of their cost: no realistic overrun removes them, and they need no further discussion. One sits at +4%. The remaining fourteen spread between +15% and +145%.

Figure 2. The 38 committed initiatives by how far cost must rise before each leaves the portfolio.

Which turns a vague worry into arithmetic. Apply the mildest of Flyvbjerg's averages — 20.4% on roads — and two of the 38 leave. Apply the figure for bridges and tunnels, 33.8%, and four leave. Apply the rail figure of 44.7% and it is still four, because nothing else exits until +51%. The organization is exposed on four commitments and settled on the rest, and it can now say which four.

The excluded side is quieter, and that is a result too: nothing sits within 10% of entering. The nearest excluded initiative would need its cost to fall by a fifth. This boundary is not crowded — a fact worth knowing, and one no ranked list reports, because a ranked list cannot distinguish a decision that was close from one that was never in doubt.

Give a board 38 initiatives and an hour, and it will spread the hour evenly. Tell it that 23 are settled past any plausible overrun and four are not, and the hour goes where the decision actually is.

None of which changes the portfolio. The selected set is the same — the solver had already found it. What changes is what the organization knows about the set it selected.

The deeper cost is the belief the line creates. An initiative excluded by $12,000 is indistinguishable in quality from one funded by $12,000, but the presentation converts an artifact of the budget into a verdict. Good work below the line acquires the status of a loser and does not return next cycle, because organizational memory has recorded an arbitrary outcome as a judgment. Commitments that were never settled are staffed and announced as though they were, and when one reverses in month seven it is read as a failure of execution rather than as what it was.

Underneath is an error of precision. Declared weights carry a range; scores therefore carry one too, and near the boundary the ordering itself is uncertain. Drawing that boundary to the pixel is not a simplification but an assertion of certainty the model does not possess. And the consequence is behavioral, which is why it matters rather than merely being correct: a board that believes the line is meaningful will not discuss the marginal cases. The presentation suppresses precisely the deliberation the tool exists to support.

There are limits worth stating. The perturbation design — which parameters vary, over what ranges, how ties are resolved, where the band thresholds fall — is specific to the implementation and is not described here; those choices materially affect the result. More importantly, selection is not independent per initiative. One can flip because another's cost changed and consumed or released capacity. Distance measured one initiative at a time misses those couplings and therefore understates fragility when the constraint binds tightly, which is when it matters most. A perturbation ensemble catches the interactions; single-item distance does not. Where the two disagree, the disagreement is itself informative.

The narrow claim is this: ordering a portfolio boundary by fragility rather than by score is the correct modeling choice for an executive decision surface, and the standard practice of the field is a category error.

Reference

Flyvbjerg, B., Skamris Holm, M. and Buhl, S. (2002). Underestimating costs in public works projects: error or lie? Journal of the American Planning Association, 68(3), 279–295.

Optimization chooses a portfolio. Distance to flip tells executives which of its choices they can trust.

Cite as: Gruia, M. (2026). Distance to Flip: There Is No Cut Line. Frontnovo Method Notes.