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Why Build "Silly" Tie Elections? — and a Map of Every Tie Case

In praise of the contrived example. The tiny, symmetric elections in this repo — 5,5,5 / 4,4,4, or 4,0,0 / 0,4,0 / 0,0,4 — will never show up at a polling place. That's the point: they're probes, built on purpose to drive a tabulation algorithm into its corners so we can see exactly what it does there.

Part of the Ties & Tie-Breaking hub · companion to STAR Tie-Breaking — The Full Chain.

The apologia (why a "silly" case earns its keep)

A deliberately-degenerate election isn't pedantry — it's the same move as a unit test that feeds a function 0, -1, and MAX_INT. Five concrete payoffs:

  1. It isolates one behavior. A real election changes a dozen things at once. A two-ballot tie changes nothing but the tie, so whatever the engine does next is unambiguously the tie-break — nothing else can be causing it.
  2. It makes a real bug reproducible. The contrived jfk7pd ballots are what turned "STAR sometimes shows something weird on ties" into a one-click repro that surfaced actual defects — the NaN display, the "no ballots have been cast" message, the silent random tie-break. The ballots are fake; the bugs they flushed out are real.
  3. It pins down the spec. "What should happen when every rung ties?" is a real design question. A minimal case forces a definite, documented answer (here: the pre-published lot decides) instead of leaving it to whatever the code happens to do.
  4. It teaches the concept. The clearest way to explain the "dead rung," or why each round breaks its tie with the other round's yardstick, is the smallest election that shows it. Big elections bury the lesson in noise.
  5. It becomes a regression test. Once captured with an expected winner, the case guards the engine forever: if someone later reorders the tie-break cascade, these tiny files fail immediately.

What each of your two examples isolates:

  • 5,5,5 then 4,4,4 — every voter rates all candidates equally, so no ballot expresses any preference at all. Totals tie, every pairwise is "Equal Support," five-star ties: a fully flat dead heat. It probes "what happens when the ballots say literally nothing to separate anyone?" (See Flat scores, ties & tie-breaking.)
  • 4,0,0 / 0,4,0 / 0,0,4 — a perfect rotation: three equal, mutually symmetric camps, and (capped at 4) no 5s, so the five-star rung is a dead rung. It probes "a genuine k-way symmetric tie with no cardinal signal to break it" — the 3-candidate analog of jfk7pd. (See the three-way dead-rung tie.)

A sixth payoff, which outranks the other five: they aren't actually contrived. The rotation above is the exact profile the standard impossibility proof constructs. Moulin (1983) shows that anonymity + neutrality + Pareto are incompatible with always naming one winner whenever the voter count n has a divisor r with 1 < r ≤ m candidates — and the witness he builds is k voters per rotation of the top three, which at k = 1 is 4,0,0 / 0,4,0 / 0,0,4 and at k = 2 is the symmetric six-voter cycle this repo also runs. So these files weren't a hunt for a pathological corner; they were a reconstruction of the textbook example, arrived at independently. That also converts the arithmetic into a search rule for building more: want a forced tie at m candidates? Pick n with a prime factor ≤ m. See Ties Are Forced.

And the honest caveat, so it stays balanced: these exact symmetries are astronomically rare in a public election, and the tabulation is correct the whole way down — the interesting question is only who wins a genuine tie. But "rare" isn't "never": small electorates — clubs, boards, committees, local primaries, which are much of BetterVoting's actual use — tie far more often than statewide races do. The probe is how you make sure the rare case is handled before it decides a real seat.

A map of every tie case (single-winner STAR)

Where a STAR result can land, from a clean win down to the lot. Each round has its own ladder; the lot is the floor of both.

flowchart TD
    A["Ballots (each candidate scored 0–5)"] --> B["Scoring Round: sum every candidate's scores"]
    B --> C{"Clear top two?"}
    C -->|yes| E["Two finalists advance"]
    C -->|"tie for a finalist slot"| D1{"Pairwise separates the tied?"}
    D1 -->|yes| E
    D1 -->|no| D2{"Five-star (most 5s) separates?"}
    D2 -->|yes| E
    D2 -->|"no — dead rung / tied"| D3["LOT decides the finalist"]
    D3 --> E
    E --> F["Automatic Runoff: which finalist do more voters prefer?"]
    F --> G{"One finalist preferred?"}
    G -->|yes| W(["Winner — decided by the ballots"])
    G -->|"tie (equal preference)"| H1{"Higher total score separates?"}
    H1 -->|yes| W
    H1 -->|no| H2{"Five-star (most 5s) separates?"}
    H2 -->|yes| W
    H2 -->|"no — dead rung / tied"| H3["LOT decides the winner"]
    H3 --> WL(["Winner — decided by lot (rare, audit-worthy)"])

The left spine (all "yes") is the ordinary election: a clean top two and a decisive runoff, no tiebreak ever consulted. Every rightward branch is a tie rung — and each one has a probe.

Every branch has a test

Branch reached What it isolates Probe
Clean top two + decisive runoff the baseline (no tiebreak) Flat_scores_ties_01
Scoring tie → pairwise breaks it a finalist chosen by head-to-head dead-rung case 01
Scoring tie → five-star breaks it a finalist chosen by most 5s dead-rung case 05
Scoring tie → dead rung → lot no 5s; the lot picks the finalist dead-rung cap ladder
Runoff tie → score breaks it winner by higher total score dead-rung case 04
Runoff tie → five-star breaks it winner by most 5s dead-rung case 04/07
Runoff tie → dead rung → lot no 5s; the lot picks the winner BV jfk7pd
Runoff tie → five-star tied non-zero → lot rung runs, decides nothing dead-rung case 09
Fully flat (no preference anywhere) ties at both loci at once Flat_scores_ties_07
k-way symmetric (rotation) any of k wins by lot; divergence (k−1)/k three-way dead-rung

Generate your own along any branch with generate_dead_rung_scenarios.py.

This map is v1 — single-winner STAR only. Natural extensions later: abstentions / quorum interactions, multi-winner (Bloc / proportional) tie handling, and the RCV-IRV elimination-tie branch (see Tie-Breaking: STAR vs. RCV-IRV).

See also