Check who loses a whole unit when the available pool grows

Specify what the allocation is allowed to decide

Start with the actual planning question. In this invented workshop, three workstreams share identical demonstration units for one event. Their supplied activity weights are fixed inputs, not estimates produced by the lesson. The exercise has no minimum guarantee, capacity ceiling, mixed pack composition or reservation already honored. Those conditions would change the decision problem and must not be hidden inside an apparently neutral calculation.

Name the owner of the weights and ask what they represent. A count of scheduled demonstrations, a forecast of requests and a stakeholder preference are different justifications. Preserve the input date and the reason the owner considers the basis suitable. Completing this worksheet does not validate the weights or prove that the resulting distribution best serves the event.

The Census Bureau describes Hamilton/Vinton allocation: compute each proportional quota, keep its integer part, then assign remaining units by descending fractional remainder. Here quota = pool × weight / total weight. Specify a tie rule before execution; this fixture has no ties.

Reconstruct the first plan without a hidden rounding step

Use weights A = 5, B = 3, C = 1. At pool 4, quotas are 20/9, 12/9, 4/9; floors are 2, 1, 0 and remainders 2/9, 3/9, 4/9. The remaining unit goes to C: allocation 2, 1, 1.

Keep five named columns in the working record: recipient, supplied weight, exact quota, initial integer allocation and awarded remainder unit. Show the total beside the pool input, and preserve the remainder ordering separately. A reviewer should be able to locate a discrepancy without guessing which spreadsheet column silently changed meaning or whether a displayed number was rounded before another formula used it.

Use exact fractions for this small fixture so a display format cannot determine the ranking. The executed Python check reconstructs each row with rational arithmetic. An independent integer-quotient and remainder calculation verifies the same ordering without using the fraction implementation. The check concerns these invented inputs only; no equipment system, procurement service or production spreadsheet was changed.

Retain an intentionally incomplete candidate that stops after the initial integer allocations. Ask the reviewer to identify its unassigned balance before looking at the completed record. This makes the reconciliation visible rather than treating a successful total as something the software must have handled somehow. Do not repair a mismatch by editing an output cell without recording the rule that justifies the change.

Recalculate after the pool changes

At pool 5, quotas become 25/9, 15/9, 5/9; floors remain 2, 1, 0. Remainders 7/9, 6/9, 5/9 award the two remaining units to A and B: 3, 2, 0.

Young’s Census symposium paper calls a loss under increased total quantity and fixed weights the Alabama paradox. This fixture exhibits that property; the overall increase is not a promise that every recipient keeps its previous allotment.

Place the two completed records side by side, matching recipients by identity rather than by their current rank. Preserve the prior plan as a separate version. The review question is not simply whether the new total reconciles, but whether the proposed change conflicts with a commitment already made to an affected workstream. Check the original communication before treating that commitment as either nonexistent or irreversible.

If the business requirement is to preserve prior allocations, state it explicitly and have the planning owner select a method appropriate to that requirement. Do not silently add a manual protection and continue labeling the output as the unchanged calculation. Conversely, do not defend an unwanted operational result merely because a named procedure generated it correctly. A calculation can expose a policy conflict without resolving the policy choice.

Deliver the comparison before making the promise

Prepare a compact decision packet with the supplied assumptions, both complete records, the independent check and the affected-recipient comparison. State which version is provisional. Ask the owner whether to accept the specified rule, change the allocation policy or revise the external commitment. Keep that decision distinct from the technical finding that the arithmetic was reproduced.

Use a boundary test when the worksheet is implemented elsewhere. Re-enter the same two input sets through the actual receiving route and request its intermediate values, not only a screenshot of the final plan. If the results differ, inspect the input basis, hidden constraints and tie policy before deciding that either the teaching fixture or the receiving tool is wrong.

Close with the scope of the evidence. The exercise checks a stipulated rule and a concrete change witness. It does not measure customer benefit, certify equitable distribution or recommend taking resources from an actual team. Any real redistribution, purchase or notice to recipients remains a separate authorized action. The useful outcome is an informed approval question before someone announces that the revised plan improves every workstream’s position.

Make the reasoning concrete.

A fictional event coordinator prepares a revised demonstration-unit plan and wants to tell every workstream that the larger pool has improved its position. The old plan was described as provisional, but the coordinator has not yet inspected the recipient-level comparison.

The coordinator requests the explicit owner decision before communicating or implementing a revised plan.

Take the questions into your next brief.

Use these prompts in your own notes. Nothing is submitted from this page.

1. Basis

   Who supplied the weights, and what do they mean?

2. Constraints

   Which guarantees or operational limits are absent from the teaching fixture?

3. Record

   Can another reviewer reconstruct every intermediate column?

4. Change

   Which recipient-level outcomes differ between versions?

5. Approval

   What commitment must be reconciled before the revised plan is communicated?

Watch for the wrong conclusion.

• Accepting a correct grand total without checking the recipient-level change.
• Adding an undocumented manual exception to the named procedure.
• Treating a supplied weight as evidence that the allocation is operationally appropriate.

Sources & scope

All business examples and numerical fixtures are original and fictional. Sources support the identified mathematical or implementation distinctions. Qualified analysts own model specification and inference. No causal result, model fit, actual customer data or operational change is claimed.

• Methods of Apportionment →Primary government methodological explanation · Supports: Supports the integer-and-remainder procedure.
• Fairness in Apportionment →Primary scholarly symposium paper, H. Peyton Young, 2004 · Supports: Supports the pool-change counterexample property.