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finalstate.aiRange Compression: Five Buckets, a Thousandfold Gap
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109ONE CELL, TWO WORLDS

Range Compression: Five Buckets, a Thousandfold Gap

Range compression discards differences within a category. Here, fictional annual risks with probabilities 1,000 times apart share one score under explicitly chosen bins.

Two fictional annual-risk records, A with probability 10 to the minus three and €100 million loss, and B with probability 10 to the minus six and €10 billion loss, send distinct markers into the bottom-right cell of a five-by-five matrix.Both markers retain their identities in probability row rank 1 and consequence column rank 5. Equal ordinal scores do not imply equal expected losses.A10⁻³€100mB10⁻⁶€10bnRow 1Column 51 × 5 = 5
Fictional annual fixed-loss cases. Selected bins: 0 ≤ p ≤ 10⁻³, loss ≥ €100m; ranks 1 and 5, score 5. Each event produces its stated loss or zero.

THE BINS THROW AWAY THE MAGNITUDE

The cutoff decides what remains distinguishable

Two logarithmic scale excerpts keep A and B fixed while brackets show that their probabilities are 1,000 times apart and their losses are 100 times apart.The scales show only the intervals between the two example values, inside the broader chosen rare and catastrophic bins. A remains paired with 10 to the minus three and €100m; B remains paired with 10 to the minus six and €10bn.Annual probability10⁻⁶10⁻⁵10⁻⁴10⁻³1,000×Loss per event€100m€1bn€10bn100×
Logarithmic excerpts between the fictional pair’s values. Brackets mark their differences, not the full widths of the broader bins.

These are chosen bins, not an industry standard. Remaining probability ranks 2–5 are (10⁻³, 10⁻²], (10⁻², 0.1], (0.1, 0.5], and (0.5, 1]. Loss ranks 1–4 are €0, €1m), [€1m, €10m), [€10m, €50m), and [€50m, €100m). The [frequency-impact matrix retains the category but discards each value's position within it.

  • Rare, rank 1: 0 ≤ annual probability ≤ 10⁻³
  • Catastrophic, rank 5: loss ≥ €100m
  • Both selected ranks produce 1 × 5 = 5

A SEPARATE INVERSION

Cox supplies a separate example of reversed ranking

Cox's normalized two-by-two matrix shows a medium point at consequence 0.65 and probability 0.10, and a low point at consequence 0.38 and probability 0.37, both on p equals 0.75 minus c.The fixed plotted inputs yield products 0.065 and 0.1406 respectively. The medium rating outranks low despite its smaller product. This mathematical example is distinct from the annual euro cases.p = 0.75 − cMediumHighLowMediumML10.5000.51pcM: 0.065 < L: 0.1406
Cox (2008), §2, pp. 499–501. Probability p and consequence c are normalized to 0–1; the points and products are mathematical examples, not euro data.

In Cox's 2008 paper, §2, pp. 499–501, the pair (p = 0.10, c = 0.65) is medium, with product 0.065. The pair (0.37, 0.38) is low, with product 0.1406. Here the rating reverses the product ranking. The fictional euro pair above merely ties.

  • A normalized 2 × 2 matrix splits each axis at 0.5
  • On p = 0.75 − c, a medium-rated point can have smaller p × c than a low-rated point

WHAT'S WRONG WITH RISK MATRICES?

What's Wrong with Risk Matrices?

Louis Anthony (Tony) Cox, Risk Analysis, 2008

Cox's Theorem 1 (§3.5, pp. 504–505) concerns coloring under weak consistency, betweenness, and consistent coloring. It places the left column and bottom row in the lowest-priority category and excludes the highest-priority category from the next column and row. That theorem is separate from §2's comparison with random choice.

A THOUSANDFOLD GAP, FOLDED FLAT

A score of five conceals a tenfold expected-loss difference

Two fixed calculation lines preserve case identity and show annual expected losses of €100k for A and €10k for B, while both score five.The mathematical values remain fixed. A shared rank product hides a difference in expected loss; the figure does not choose which risk to accept or mitigate.Expected loss per yearA10⁻³ × €100m = €100kB10⁻⁶ × €10bn = €10kOrdinal scoreAB1 × 5 = 51 × 5 = 5
Fictional common one-year horizon and fixed losses. Expected loss is p × loss; score is row rank × column rank. The tie does not reverse the quantitative ordering.

The probability gap is 1,000×; the consequence gap runs the other way at 100×. Their product leaves A's expected loss ten times B's. Expected loss need not settle a decision about catastrophic harm, affordability or constraints. This arithmetic shows why the cell is not a decision.

  • A: 10⁻³ × €100m = €100k per year
  • B: 10⁻⁶ × €10bn = €10k per year
  • Both ordinal scores: 1 × 5 = 5

The coin comparison requires its own decision problem

In Cox's §2 construction, choose one of two risks to eliminate, prefer the higher matrix category, and break category ties randomly. When the line p = 0.75 − c puts one risk in medium and another in low, the matrix selects the smaller product for elimination. A fair coin does better in expected loss removed. Negative correlation alone does not establish that result; the joint distribution, bins and decision rule need inspection. This model comparison does not demonstrate a particular avoided loss. Return to the decision behind the cell.

THE WORDS DON'T MEAN THE SAME THING

A shared word can hide different numerical cutoffs

Two explicitly hypothetical definitions of rare place the upper cutoff at annual probabilities of 10 to the minus three and 10 to the minus five, while A and B remain fixed.This is a sensitivity example, not a sample of study respondents. Moving the first probability boundary, and the adjacent rank-2 lower boundary with it, changes A's row from 1 to 2 while B stays in row 1. No observation dots converge.Illustrative cutoffsRare10⁻⁶10⁻³10⁻⁶10⁻³Rank 110⁻⁵Rank 1Rank 2Inputs unchanged
Illustrative cutoff change only: rare’s upper bound moves from 10⁻³ to 10⁻⁵, along with rank 2’s lower bound. Inputs and other boundaries stay fixed. No study observations are plotted.

Budescu, Broomell and Por's 2009 experiment found that interpretations of probability terms in 2007 IPCC sentences deviated from the guidelines even when readers could consult them. Ball and Watt report persistent assessor scatter for public-leisure hazards. These findings do not supply the illustration's cutoffs. Hubbard and Evans, pp. 2:4–2:5, distinguish ordinal ranks from ratio measurements.

  • Ask what probability interval each label denotes
  • Record the definition and the assessment horizon
  • Numerical definitions still need evidence

ANY BIN IS A DECISION

A boundary can hide distinctions without anyone intending it

A risk category gathers two distinct markers while a separate shortlist includes X, Y and Z and leaves documented candidate W outside its boundary.The two native constructions are separated. Combining numerical values into a category differs from excluding an option; neither construction establishes deliberate manipulation or superiority of the omitted option.CategoryShortlistXYZOmittedW
Analogy between categorization and exclusion. The omitted candidate is documented, not assumed superior.

The relationship to a three-option menu is an analogy. Categorization and exclusion are different operations, but both deserve visible rules. Choice framing can affect an available comparison; agenda control concerns what can enter it. A familiar format alone proves neither intent nor a hidden verdict.

  • A category merges values
  • A shortlist excludes alternatives
  • Inspect the rule that produced each summary

REOPEN THE INPUTS

Return to the inputs before drawing a finer picture

A return path leads from the shared score to the original A and B records; an assumption sheet joins them before an unsigned decision field.The closing figure reopens the inputs rather than morphing the grid into an invented loss curve. A and B keep their values, and the decision remains pending.5A10⁻³€100mB10⁻⁶€10bnAssumptionsDecision pending

A curve cannot recover distinctions already discarded by a score. Revisit the original evidence, use supported ranges, and leave gaps open where evidence is missing. A quantitative model still needs a criterion and an accountable decision. Continue to The Cell Is Not a Decision.

  • Recover source records and their assumptions
  • Check probability, consequence, horizon and dependence
  • Show what changes under plausible alternatives
Read the transcript

01 · ONE CELL, TWO WORLDS

Consider two fictional risks over the same year. If event A happens, the loss is one hundred million euros; its probability is one in a thousand. If event B happens, the loss is ten billion euros; its probability is one in a million. Otherwise, each produces no loss. Assign both records to the selected cell. The chosen rare row has rank one, and the catastrophic column has rank five. Both scores are five. The categories have preserved neither the probability gap nor the expected-loss ranking.

02 · THE BINS THROW AWAY THE MAGNITUDE

The boundaries explain the tie. Here, rare includes annual probabilities from zero through one in a thousand. Catastrophic includes losses of one hundred million euros or more. Keep the probability points fixed and compare their separation: one thousand times. Keep the loss points fixed and compare those: one hundred times. These are differences between the two cases inside broad bins, not measurements of the full widths of those bins.

03 · A SEPARATE INVERSION

Cox gives a separate example of inversion. His two-by-two matrix splits normalized probability and consequence at one half. Two points on the line probability equals zero point seven five minus consequence receive different ratings. The medium point has product zero point zero six five. The low point has product zero point one four zero six. The higher category therefore belongs to the smaller product. That reversal is a mathematical construction. The annual euro example produces a tie.

04 · WHAT'S WRONG WITH RISK MATRICES?

What's Wrong with Risk Matrices? Cox's paper studies several distinct problems. His first theorem constrains cell coloring under weak consistency, betweenness, and consistent coloring. Those are explicit axioms about how categories relate to quantitative risk. They are not the assumptions of his earlier comparison with random choice. Keep the two arguments separate.

05 · A THOUSANDFOLD GAP, FOLDED FLAT

Return to A and B. Multiply each annual probability by its fixed loss. A gives one hundred thousand euros per year. B gives ten thousand. The expected-loss difference is tenfold, while both ordinal scores remain five. That does not decide which event matters more. Catastrophic harm, available controls, constraints, and the consequences of being wrong may affect the action. It does show that the shared score has thrown away a distinction the decision may need.

06 · THE COIN COMPARATOR

The coin comparison asks which risk to eliminate. In Cox's construction, the matrix prefers medium to low and breaks category ties randomly. Along the specified line, whenever medium and low are distinguished, the matrix selects the smaller quantitative risk. Tossing a fair coin instead removes more expected loss on average in that setup. Negative correlation by itself is not enough to establish this result. The joint distribution, the bins, and the rule for acting on the categories all matter.

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08 · THE WORDS DON'T MEAN THE SAME THING

Words add another source of uncertainty. Budescu and colleagues found that numerical readings of probability phrases in IPCC sentences departed from the guidelines, even when readers had access to them. Ball and Watt reported persistent disagreement about matrix ratings for public-leisure hazards. In a separate illustrative test, change only the rare cutoff, from one in a thousand to one in a hundred thousand. A crosses into row two. B stays in row one. Both probabilities stay the same. Agreement on a label is not yet agreement on its boundary.

09 · ANY BIN IS A DECISION

Binning is a modeling choice. In the upper construction, two values share a category. In the lower construction, a shortlist leaves a documented alternative outside. These operations differ: one groups inputs, the other excludes a candidate. The analogy is a reason to inspect the rule and the information it removes. It does not establish that a designer intended to steer the decision, or that an omitted option deserves to win.

10 · REOPEN THE INPUTS

Reopen the original records. Follow the return path from the score to A and B. Bring their assumptions into the review: the time horizon, the loss model, the evidence behind each probability. A smoother line cannot restore what the score discarded. Use supported ranges and test whether a proposed action changes when assumptions change. If an input is unknown, keep that gap visible. The decision stays pending until someone judges the evidence and the criterion for acting.

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