Exercise 15 — Read the ballot, name the method

Every other exercise in this set hands you the method and asks who wins. This one runs backwards. You get two filled-in profiles and no method name — just the marks. Name the method from the shape of the ballot, then count. Then the part the textbooks stop before: one of these two profiles has a property that guarantees its winner is also the Condorcet winner, and the other has a winner so uncontested that no method in the library disagrees. Work out which is which, and why.

▶ Live on BetterVoting: profile (a) — vote · results ↗ (election d4v2dh, Test ID BV2258 — two races on the same 35 ballots: the Yes/No count, then the identical marks as 5/0 scores so the head-to-head view prints). Profile (b) — vote · results ↗ (election tfm64p, Test ID BV2259). BetterVoting's own count agrees with the engine on all three races — Blair, Blair, Clara. Two elections rather than one because the profiles have different electorates (35 voters and 4), and every BV voter votes every race.

You practice: reading a ballot as evidence — what a set of marks can and cannot record — plus the vocabulary that names each shape: dichotomous vs cardinal, approval shares vs vote shares, and the guarantee that comes free on a dichotomous profile.

Work each part on paper before opening its solution. All three YAMLs are runnable and their answer keys are regression-tested.

---

Profile (a) — 35 voters

The column headings are how many voters cast that ballot.

	×15	×8	×7	×5
Ada	No	Yes	Yes	No
Blair	Yes	Yes	No	Yes
Cosmo	Yes	No	Yes	No

Profile (b) — 4 voters

	Voter 1	Voter 2	Voter 3	Voter 4
Alice	0	3	0	4
Bruno	2	5	1	3
Clara	5	5	5	4
Diego	4	3	4	2

Your task

• (a) Which voting method produced profile (a)? Name two features of the ballot that rule out the alternatives.
• (b) Who wins profile (a), and with what share of the ballots? The three shares add up to far more than 100% — is something wrong?
• (c) Which method produced profile (b), and why is it called a cardinal method? (One sentence, and it should not be "because it uses numbers.")
• (d) Who wins profile (b)? Then the follow-up the prompt doesn't ask: can you tell from that table who each voter's favorite is?
• (e) Back to profile (a). Rank the three candidates by head-to-head wins — who beats whom. Compare that ranking to the approval totals from (b). Coincidence?
• (f) Does (e) mean Approval always elects the Condorcet winner? Answer carefully.

Solutions

(a) Approval voting — and here's how the ballot tells you

**Approval voting.** Two features settle it: 1. **Each candidate is marked independently, with only two states.** No ranks (nothing says which Yes is better than another Yes), no magnitudes. That rules out every ranked method — [RCV-IRV](../../06_Other/RCV_IRV/concepts/README.md), [Ranked Robin](../../05_Ranked_Robin/01_Learn/ranked_robin.md), Borda — and every score method. 2. **Voters mark different *numbers* of candidates.** The ×15 bloc approves two, the ×5 bloc approves one. That rules out [Choose-One](../../07_Concepts/topics/plurality.md) (exactly one mark) and k-approval (exactly k marks). Independent, binary, unlimited marks — that is the definition of an approval ballot. The theorists' name for a profile of them is a **[dichotomous profile](../../07_Concepts/GLOSSARY.md#the-wider-field-computational-social-choice)**: every voter's ballot is a ranking with exactly two levels and nothing said inside either one.

(b) Blair, 28 of 35 — and no, nothing is wrong with the percentages

Count the Yeses: Blair 15 + 8 + 5 = **28**, Cosmo 15 + 7 = **22**, Ada 8 + 7 = **15**.

--- Approval Voting (single winner) ---
 Tabulating 35 ballots (any non-zero score = approval).

   Blair -- 28 (80%) -- Elected
   Cosmo -- 22 (63%)
   Ada   -- 15 (43%)

[Approval Distribution] (how many candidates each ballot approved)
   65 approvals across 35 ballots — average 1.9 of 3 (range 1–2).
     approved 1: 5 ballots
     approved 2: 30 ballots

80 + 63 + 43 = 186%, and that is exactly right. **An approval percentage is a share of *ballots*, not a slice of one pie** — "80% of voters approve Blair" and "63% approve Cosmo" are both true of the same 35 people, because a voter can say yes to both. Reading approval shares as vote shares is the single most common mistake with these results. The **approval distribution** line is what tells you how much doubling-up there was: 65 approvals across 35 ballots, so the average voter approved 1.9 of the 3. Full report: [`ex15_approval_yes_no`](cases/cases_pages/ex15_approval_yes_no.md).

(c) Score (Range) voting — cardinal because each rating stands alone

**Score voting**, also called Range: rate every candidate on a fixed scale, add the columns, highest total wins. It's **cardinal** because each candidate is judged **against the scale, not against the other candidates** — the ballot records *how much* a voter likes each one, independently, so it can say "these two are exactly equal" and "this one is far better than that one." An ordinal ballot can only say *which comes first*; it has no way to distinguish a hair's-breadth preference from a chasm. (Voter 2 rating Bruno and Clara both 5 is a statement no ranking can make.) → [scores vs ranks](../../07_Concepts/scores_and_ranks/scores_vs_ranks.md) · [scoring methods vs ranked voting](../../07_Concepts/topics/scoring-methods-vs-ranked-voting.md) The giveaway that it isn't [STAR](../01_Learn/STAR_start_here.md), by the way, is that nothing in the prompt mentions a runoff — the ballots are identical. **A ballot alone never fully identifies the method**; Score and STAR share this one, which is the whole point of [exercise 3](ex03_five_verdicts.md).

(d) Clara, 19 — and no, you cannot always read off a favorite

Column sums: Clara 5+5+5+4 = **19**, Diego 4+3+4+2 = **13**, Bruno 2+5+1+3 = **11**, Alice 0+3+0+4 = **7**.

Scoring Round
   Clara         -- 19 -- First place
   Diego         -- 13 -- Second place
   Bruno         -- 11
   Alice         --  7

That scoring round **is** the Score-voting count — the file runs as STAR because the teaching CLI has no first-class `voting_method: Score`, and the totals it prints are the Score result. Here the runoff changes nothing:

Automatic Runoff Round
   Clara         -- 4 -- First place
   Diego         -- 0
   Equal Support -- 0

Clara wins **4–0**, and she's the Condorcet winner too. The engine prints **no `[Divergence from STAR]` block at all** — Choose-One, RCV-IRV, Approval and Score all land on Clara. She is scored top or joint-top by every single voter, so there is nothing for the methods to disagree about. Worth keeping one of these around: [most curated elections in this library diverge](../../method_comparisons/divergence_review/INDEX.md), and it would be dishonest to only ever show those. **Can you tell each voter's favorite?** Only for two of them. Voter 1 and voter 3 give Clara a clear 5. But voter 2 rates Bruno and Clara both 5, and voter 4 rates Alice and Clara both 4 — **tied at the top, with no tiebreaker on the ballot.** So you cannot reconstruct a Choose-One tally from these ballots without inventing information the voters didn't give you. (On profile (a) it's worse: an approval ballot has no top at all.) That asymmetry — score ballots can be read down to cruder ones, but never perfectly — is the [fidelity ladder](../../07_Concepts/scores_and_ranks/fidelity_ladder.md). Full report: [`ex15_score_profile`](cases/cases_pages/ex15_score_profile.md).

(e) Blair > Cosmo > Ada — the same order as the approval totals, and it is not a coincidence

Read the same 35 ballots pairwise ([`ex15_approval_pairwise`](cases/cases_pages/ex15_approval_pairwise.md) writes each approval as a 5 and each non-approval as a 0, which changes no head-to-head count — only the two-class order matters):

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 35 ballots.
Count × Ada,Blair,Cosmo
   15 ×   0,    5,    5
    8 ×   5,    5,    0
    7 ×   5,    0,    5
    5 ×   0,    5,    0

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Blair         -- 140 -- First place
   Cosmo         -- 110 -- Second place
   Ada           --  75
 Blair and Cosmo advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Blair         -- 13 -- First place
   Cosmo         --  7
   Equal Support -- 15
 Blair wins.
   Runoff math:
     35  ballots cast
   − 15  Equal Support (no preference between the two finalists)
     ──
     20  voters with a preference  (majority = 11)
           Blair 13 (65%)  ·  Cosmo 7 (35%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Blair

Blair beats Ada 20–7 and Cosmo 13–7; Cosmo beats Ada 15–8. **Blair > Cosmo > Ada — exactly the order of the approval totals 28 > 22 > 15.** Not luck. On a dichotomous profile, "more voters strictly prefer x to y" reduces to **"more voters approve x than approve y"** — so the head-to-head order *is* the approval order, a Condorcet winner is guaranteed to exist, and no cycle is possible. (Approval also coincides with [Borda](../../06_Other/other_ranked_methods/borda.md) on this domain, which is why the result is sometimes stated as *approval voting reconciles Borda and Condorcet*.) Notice where the preferences went, though: Blair vs Cosmo is 13–7 with **15 Equal Support** — the ×15 bloc approved both and has no say in that matchup. That column is the compression, and it's the hinge of (f).

(f) No — and the gap is worth being precise about

What's true is: **Approval elects the Condorcet winner of the approval ballots.** Always, guaranteed, as (e) shows. What does *not* follow: that it elects the Condorcet winner of what the voters actually think. Real approval ballots aren't cast by a dichotomous electorate — they're cast by people with richer opinions who compress them on the spot, and the compression can move the answer. The worked counterexample is five ballots long: in [Black Curtain #1](../../method_comparisons/black_curtain/condorcet_compression.md), Cal is the engine-confirmed Condorcet winner of the score ballots, Bob is the Condorcet winner of the same voters' approval ballots, and Bob wins — legitimately, because the three voters who prefer Cal approved both of them. So the honest phrasing of the guarantee names its domain: *on a dichotomous profile*, Approval is Condorcet-consistent. Drop those four words and you've promised something the method can't deliver. → [Approval in the theory literature](../../04_Approval/01_Learn/approval_in_the_literature.md) · [Approval's honest limits](../../04_Approval/01_Learn/approval_honest_limits.md)

The moral

Two profiles, two lessons about what marks can carry. A Yes/No ballot buys a guarantee — its winner beats everyone head-to-head, always — by throwing away every preference inside each group; the guarantee is real and it is about the ballots, not the voters. A 0–5 ballot keeps the intensity and the order, which is why it can be read down into an approval count, a ranking, or a runoff — and why the same marks support several different methods, so the ballot alone never names one.

Run them

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex15_approval_yes_no.yaml

Case	Method	Winner	Shows
ex15_approval_yes_no	Approval	Blair	The count, the shares, the approval distribution
ex15_approval_pairwise	STAR (5/0)	Blair	The same ballots pairwise — approval order = Condorcet order
ex15_score_profile	STAR	Clara	The scoring round as the Score count; every method agrees

See also

• Exercise 3 — one electorate, five verdicts — the same ballot, five methods, five answers
• Exercise 13 — where do you draw the line? — the other half of the approval story: one honest electorate, three thresholds, three winners
• Approval Voting · Approval in the theory literature
• Scores vs ranks · the fidelity ladder
• When compression moves the Condorcet winner — part (f), worked in full

file: ex15_read_the_ballot.md