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The Downward Monotonicity Paradox — when ranking a loser lower makes him win (San Francisco 2020)

The mirror of the upward paradox. A voting method should never reward a candidate for losing support. RCV-IRV can. In the real 2020 San Francisco Board of Supervisors, District 7 race, shifting the losing candidate Engardio down one rank on about 800 ballots — giving him less first-place support and changing nothing else — would have made Engardio the winner. This page reproduces it on the real full-scale counts, then shows that STAR and Ranked Robin cannot do it.**

→ Part of the monotonicity worked set. Companions: Upward paradox — Alaska 2022 (its mirror) · non-monotonicity (concept) · monotonicity topic hub.

Two flavours of the paradox — this is the second

Non-monotonicity comes in a matched pair (Graham-Squire & McCune, arXiv:2301.12075):

  • Upward monotonicity paradox ("more is less"): raise the winner and the winner loses. → Alaska 2022.
  • Downward monotonicity paradox ("less is more"): shift a losing candidate L down the rankings on some ballots — keeping every other candidate's relative order the same — and L wins. Giving a candidate less support elects them. ← this page.

The Alaska example is only an upward paradox — there, shifting Begich or Palin down leaves them losers. To see a real downward paradox, the paper turns to San Francisco.

The real example: San Francisco D7, 2020

The 2020 SF Board of Supervisors race in District 7 had seven candidates. Following the RCV-IRV eliminations down to the final three (the paper's Table 2), the profile is Engardio, Melgar, Nguyen. Counted by RCV-IRV:

BEFORE — Engardio loses; Melgar wins. (sf_d7_downward_before)

ROUND 1
Engardio  14119   Hopeful       ← leads on first-choices
Melgar    11652   Hopeful
Nguyen    10855   Rejected       ← fewest, eliminated
Melgar    18561   Elected        ← Nguyen's transfers carry Melgar past Engardio
Engardio  16370   Rejected       ← Engardio LOSES

AFTER — rank the loser lower, and he wins. Shift 800 Engardio > Nguyen > Melgar ballots down to Nguyen > Engardio > Melgar. Engardio simply loses 800 first-place votes; nothing else changes. (sf_d7_downward_after)

ROUND 1
Engardio  13319   Hopeful       ← Engardio LOST support (14119 → 13319)
Nguyen    11655   Hopeful       ← +800, now just 3 votes ahead of Melgar
Melgar    11652   Rejected       ← Melgar now has the fewest, eliminated instead of Nguyen
Engardio  15819   Elected        ← Melgar's transfers, and Engardio beats Nguyen
Nguyen    15155   Rejected

Engardio lost first-place support and turned from loser into winner.

Why less support wins

Exactly as in Alaska, the trick is the order of elimination. Those 800 ballots, moved off Engardio and onto Nguyen, lift Nguyen's first-round count just past Melgar's — by 3 votes — so Melgar is eliminated first instead of Nguyen. Melgar's ballots transfer, and in the resulting final Engardio defeats Nguyen. The candidate Engardio needed gone was Melgar, and the way to eliminate Melgar was to make Engardio look weaker. Sequential elimination made a candidate stronger by taking his votes away.

Knife-edge — why full scale. The flip hinges on a 3-vote Round-1 gap (Melgar 11652 vs Nguyen 11655). A reduced toy model can't hold a 3-in-36,000 margin, so this case uses the real full-scale counts. The Round-1 three-way (14119 / 11652 / 10855) and the BEFORE final (Melgar 18561, Engardio 16370) reproduce the paper's published figures exactly; Melgar's own second-preference split — not printed in the quoted table — is reconstructed to complete the three-way, consistent with the paper's stated result (Engardio wins the AFTER final). Reproduce: uv run python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/monotonicity/cases/sf_d7_downward_after.yaml.

STAR and Ranked Robin can't do this

Both methods are monotonic — lowering a candidate can never help them, and raising one can never hurt — so neither has a downward (or upward) monotonicity paradox.

  • Ranked Robin (Condorcet / Copeland) reads every head-to-head, so the elimination-order trick has no purchase. On the same ballots it elects Melgar both times — before and after — because Melgar is the Condorcet winner (he beats Engardio 18561–16370 and Nguyen 24971–11655). Note the sharp contrast: RCV-IRV's result moves off the Condorcet winner under the paradox; Ranked Robin stays on him. (Same ranked ballot, a monotonic count.)
  • STAR passes mono-raise and its downward form: a candidate losing points can only ever hurt their own total, never help. (See STAR & monotonicity and the runnable STAR before/after pair.)

Keep it in proportion — the fair reading

  • Real and documented — SF D7 2020 is a genuine recorded election, one of the cleanest downward-paradox cases in the literature. That's why it teaches well.
  • But rare. Like its mirror, the downward paradox needs a near-three-way race with a specific transfer structure; most IRV elections never trigger one. The durable point is structural: IRV's winner depends on elimination order, which can move the wrong way in both directions.
  • IRV-specific, not "ranked voting." The same ballots under Ranked Robin are monotonic. It's instant-runoff that fails here, not ranked ballots as such.
  • STAR's own limit stays on the table. STAR passes mono-raise but fails the stronger mono-raise-delete in lab constructions — conceded in the worked set. Naming both sides is the point.

The takeaway matches the upward page: a voter's honest ranking shouldn't be able to backfire — up or down — and STAR and Ranked Robin give that guarantee where instant-runoff can't.

Sources