The Reinforcement paradox — winning everywhere, losing overall

The Reinforcement postulate: if two disjoint electorates each elect X, their union must elect X. The Reinforcement paradox (a.k.a. the multiple-districts or inconsistency paradox) is the failure: a candidate wins every district separately, yet loses when the districts are counted together. (A conditional paradox in Felsenthal's taxonomy: the one changed datum is the partition — amalgamate the districts, hold every ballot constant, and the winner changes.)

→ Glossary: reinforcement · Felsenthal's taxonomy: README · related counting property: the summability demo

The live demonstration (a trio of elections)

Felsenthal's Example 3, run for real: BV2147 — District I (17 voters, runoff → Bruno), BV2148 — District II (15 voters, Bruno wins round one with 8 of 15 — a majority favorite and the district's Condorcet winner), then BV2149 — Combined (the same 32 voters as one electorate → Alma). Two districts say Bruno; their union says Alma.

Where it comes from

Like non-monotonicity, this is a disease of elimination order. Plurality-with-runoff (and RCV-IRV generally) decides everything by who gets deleted: District I deletes Alma; the combined count deletes Cora; and the deletion determines the final pair. Amalgamating electorates reshuffles the first-choice tallies that drive deletion, so two agreeing districts can produce a disagreeing union. No appeal to a "truer" winner is needed — the procedure contradicts its own two verdicts. (In this construction the combined electorate is a Condorcet cycle, so the inconsistency really is the whole story.)

Methods that sum a single number per candidate from each ballot — Score, Approval, Borda — satisfy reinforcement outright: sums over a union are sums of sums. STAR's runoff stage costs it the formal guarantee, but in this trio STAR elects Bruno in District I, District II, and combined (102/98/88 in the union; runoff 18–14). The STAR-side failure is real all the same: the Two districts, one mayor exercise constructs it and runs it live as a second trio — West / East / Combined (BV2188–90): Avery wins both districts, Carmen wins their union. RCV-IRV also fails the related summability property — district subtotals can't even be added meaningfully, which is the operational face of the same problem: see the summability demo.

Why it matters

Real elections are counted in precincts, districts, and states, and results are audited by comparing parts to wholes. A method where "won every district" and "won the election" can point at different candidates makes district-level results uninterpretable as evidence — and hands every close loss a ready-made grievance. It takes 32 voters and three BetterVoting elections to show the whole thing end to end.