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Agenda Voting — when the order of the vote picks the winner

Same ranked ballots, different tabulation — taken to the extreme. In agenda voting the tabulation has a parameter: the order in which alternatives face each other. Change the agenda, change the winner — with the very same ballots. It is the cleanest illustration in the repo of why "the ballots" never decide anything by themselves; the procedure always co-decides.

→ Companions: Ranked Robin (the neutral way to run the same pairwise matchups) · Borda · Glossary: Condorcet

Source: notes digested from Sherif El-Helaly, The Mathematics of Voting and Apportionment (§1.3.5–1.3.7), collected in Adam's Tactical Voting doc. Paraphrased summary — see the book for the full text.

What it is

Agenda voting is the parliamentary/committee procedure: list the alternatives in a fixed order (the agenda), then run sequential pairwise majority votes — the first two alternatives face off, the survivor meets the third, and so on down the list. The survivor of the last vote wins. A tie between the i-th and j-th agenda entries (i < j) goes to the later entry, so the procedure always produces exactly one winner.

Two things follow immediately:

  • It is not neutral — alternatives are treated differently by construction (position in the agenda matters). That is why it is used for policies (motions vs. status quo), never for candidates.
  • It runs only a subset of the pairwise matchups, in sequence. Contrast Ranked Robin, which runs every matchup symmetrically — the neutral, order-free way to consume the same ranked ballots.

The agenda picks the winner (all four can win)

Nine voters, four alternatives:

4 voters 3 voters 2 voters
a c b
d b a
c d d
b a c

With these exact ballots, the winner is whatever the agenda-setter wants it to be:

Agenda Winner
c b a d a
c a b d b
d a b c c
a b c d d

(This profile has a Condorcet cycle, which is what gives the agenda its power — see below for when it doesn't.)

Properties — a mixed report card

  • Condorcet winner criterion: PASS. A Condorcet winner survives every matchup, so it wins under every agenda. Conversely, the left-most (first-listed) alternative wins if and only if it is the Condorcet winner — first position is the worst slot, having to survive every round.
  • Condorcet loser criterion: PASS. A Condorcet loser loses its first matchup under any agenda.
  • Monotone: PASS. Raising an alternative on some ballots never hurts it.
  • Weak Pareto efficiency: FAIL — the interesting one, next section.
  • An alternative that loses under every agenda need not be a Condorcet loser. Three voters, five alternatives:
Voter 1 Voter 2 Voter 3
a b c
b c e
d d a
e a d
c e b

No Condorcet winner exists and d is not a Condorcet loser — yet no agenda makes d win.

The Weak Pareto failure

Pareto dominated = some rival is ranked above you by every single voter. A method is Weakly Pareto efficient if it never elects a Pareto-dominated alternative — about the mildest quality bar imaginable. Agenda voting fails it. Three voters:

Voter 1 Voter 2 Voter 3
a b c
d c a
b a d
c d b

Every voter ranks a above d — yet agenda a c b d elects d. (The trick: a gets used up beating others early, and d, sitting last in the agenda, only has to win its one final matchup.)

Keep the two "loser" notions distinct:

  • Condorcet loser — loses by a majority to every opponent. Agenda voting protects against this.
  • Pareto dominated — loses unanimously to just one opponent. Agenda voting does not protect against this.

Neither implies the other, and agenda voting passing the first while failing the second is the proof.

The killer amendment (tactical nomination, not tactical voting)

In parliamentary practice the agenda is fixed by the bylaws: latest amendment first, status quo last (motion b vs. status quo a → agenda b a; amend the motion to c → agenda c b a). So you can't game the agenda order — but you can game what gets nominated onto it.

Setup: 11 voters, motion b vs. status quo a. Nine of the eleven prefer the motion — a straight vote kills the status quo 9–2. The two status-quo supporters rescue it by introducing a killer amendment (aka poison pill) c, engineered to split the pro-motion camp roughly in half — half love the amendment, half hate it worse than the status quo:

2 voters 4 voters 5 voters
a c b
c b a
b a c

Agenda c b a: c beats b 6–5, then a beats c 7–4. The status quo wins against the wishes of a 9-of-11 majority. The amendment manufactured a Condorcet cycle, and the bylaws' agenda walks straight into it. Crucially, no voter changed their a-vs-b preference — nobody was persuaded; the nomination did all the work. That makes this a tactical nomination, a different animal from tactical voting (the ballots stay sincere).

The textbook's memorable version: five students, a standing order of vegetarian pizza (a), a motion to add pepperoni (b) that 4 of 5 support. The lone vegetarian, knowing two of the four pepperoni fans love hot pepper and two hate it, amends the motion to "pepperoni and hot pepper" (c). Profile 1×(a c b), 2×(c b a), 2×(b a c); agenda c b a: c beats b 3–2, a beats c 3–2 — vegetarian pizza survives with 4 of 5 voters against it.

When does the poison pill work?

Generalize: m voters back the status quo (all ranking the amendment in the middle: a > c > b), and of the n pro-motion voters, n₁ rank the amendment on top (c > b > a) and n₂ rank it last (b > a > c), with m < n:

m n₁ n₂
a c b
c b a
b a c

The kill needs c to beat b first, then lose to a: m + n₁ > n₂ and m + n₂ > n₁. Both hold exactly when

|n₁ − n₂| < m — the killer amendment succeeds iff it splits the pro-motion camp more finely than the status-quo bloc's own size.

Fail modes (from the book's 23-voter drill: 17 pro-motion, 6 status quo): let too many rank the amendment last and the motion becomes a Condorcet winner (pill ignored); let too many rank it first and the amendment itself becomes the Condorcet winner — the killer turns "monster" and passes. Two of the four drill scenarios succeed (splits 6/11 and 11/6 against m = 6), two fail (5/12 and 12/5).

Why this earns a page in a STAR repo

  1. Procedure ≥ ballots. The nine-voter example is the sharpest possible "same ballots, four different winners" demo — stronger than any STAR-vs-IRV divergence, because here even the method is fixed and only a sequencing parameter moves.
  2. Neutrality is a feature you can lose. Ranked Robin = all pairwise matchups, order-free, neutral. Agenda voting = some pairwise matchups, in an order someone chose. Same ballot type, same majority-rule primitive, wildly different manipulability.
  3. Cycles are exploitable, not just curiosities. The killer amendment is a Condorcet cycle built on purpose. When we flag a cycle in an RR report, this is what a cycle is worth to a motivated actor under a sequential procedure.
  4. Tactical nomination ≠ tactical voting. Every ballot in the killer-amendment story is sincere. Manipulating the choice set is its own strategy family — worth keeping distinct in debate prep.