Non-monotonicity — losing by gaining support

A voting method is monotone if ranking or scoring a candidate higher can never hurt them, and lower can never help. Non-monotonicity (Felsenthal: lack of monotonicity; a.k.a. the more-is-less or additional-support paradox) is the failure: a candidate loses because voters raised them, or wins because voters lowered them. (The model conditional paradox: one datum changes — a few voters raise a candidate — everything else held constant, and the outcome flips against the raise.)

→ Glossary: monotonicity · Felsenthal's taxonomy: README

The 17-voter demonstration (live pair)

BV2145 → BV2146 is Felsenthal's Example 2, run for real on BetterVoting. Under plurality-with-runoff (= IRV for three candidates), Ben wins the first election. Then two voters raise Ben from Cleo>Ben>Ada to Ben>Cleo>Ada — and Ben loses: his extra first-choice votes drain Cleo below Ada, the elimination flips from Ada to Cleo, and Ada beats Ben 9–8 in the final pair. More support, worse result.

The repo also has a standalone constructed pair: the monotonicity set (RCV-IRV flips; STAR, on the same story, doesn't).

Where it comes from

Non-monotonicity is a disease of elimination order. In RCV-IRV (and plurality-with-runoff) the candidate eliminated early hands their transfers to the finalists — so raising a candidate can demote a useful loser out of the elimination slot and reshuffle the final pair. This is IRV-specific, not a property of ranked ballots: Ranked Robin counts the same ballots pairwise and is fully monotone, and in the BV2145/46 pair both Ranked Robin and STAR elect Ada before and after the raise. (In BV2146's STAR race the raise does change the finalist pair — Ben displaces Cleo — but Ada still wins the runoff; score totals move with the voter's change, never against the raised candidate.)

Why it matters

A method that can punish a candidate for being liked more punishes voters for voting sincerely — it makes "should I really rank my favorite first?" a live strategic question, which is precisely what a good ballot is supposed to end. And unlike the Condorcet winner paradox, non-monotonicity is invisible in any single election result: it only shows when you compare the two worlds, which is why a frozen, re-runnable pair like BV2145/46 is the right teaching object.