Support monotonicity — can more approvals cost you your seat?¶

Level: 301 · deep dive

One line: the obvious demand is that extra support never hurts a winner; the non-obvious finding is that it holds for one candidate far more often than for a group, and that gap is what the word cand marks in Table 3.1.

Two columns, two definitions¶

Support can grow in two different ways, and the rules do not treat them alike, so the table gives each its own column:

With additional voters — a new voter joins, approving exactly the set X. Written A+X.
Without additional voters — an existing voter widens her ballot to also approve X. Written Aᵢ+X.

Definition 3.3 (Sánchez-Fernández & Fisteus). R satisfies support monotonicity without additional voters if for every instance (A,k), voter i, and candidate set X ⊆ C: (1) if X ⊆ W for all winning committees W, then X ⊆ W′ for all W′ winning in Aᵢ+X; and (2) if X ⊆ W for some winning W, then X ⊆ W′ for some W′ winning in Aᵢ+X. With additional voters is the same with A+X in place of Aᵢ+X.

When X is a single candidate {c}, the axiom is called candidate monotonicity — and that is the whole content of the table's cand entries:

symbol	meaning
✓	support monotonicity holds — for any set `X`
`cand`	candidate monotonicity holds (single `X = {c}`) but support monotonicity fails for some larger group
✗	fails even for a single candidate

Read the "without additional voters" column and this is nearly the whole story: only AV and SAV are ✓, and everything else is cand. Not one of the eleven other rules can promise that a group already seated stays seated when a voter adds her approval of exactly that group.

Why a group is harder than a candidate¶

The intuition that makes candidate monotonicity feel obvious — "more support can only help" — is true of the candidate you approve and false of the arrangement around them. Approving a whole slate does two things at once: it raises each member's standing, and it changes how much your ballot is worth to the rules that reweight. Under any rule with diminishing returns (every Thiele method, every load-balancing rule), a voter who now approves three winners is a satisfied voter, and her remaining influence shrinks. She can push her slate in and simultaneously stop being the reason any of them stays.

That is not a pathology of one rule; it is the mechanism proportionality is built on. Which is why the column is a near-uniform cand.

Witness — seq-CC loses a candidate when a voter joins to support him¶

Support monotonicity with additional voters. Twelve voters, three seats:

3 × {a}    1 × {a,c,d}    1 × {b}    2 × {b,c}    1 × {b,d}    2 × {c}    2 × {d}

seq-CC elects {a,c,d}. Now one more voter arrives approving exactly {a,d} — a subset of the winning committee, so by the axiom a and d must both survive:

before          {a,c,d}
+1 × {a,d}      {a,b,c}      <- d is gone

The new voter supported a and d; d lost the seat. seq-CC is therefore only cand in that column.

Witness — seq-PAV, without a new voter¶

The other column. Nine voters, three seats:

1 × {c,d}   1 × {a,c}   1 × {a,d}   1 × {a,f}   1 × {b,c}   2 × {b,f}   1 × {c,e}

seq-PAV elects {a,c,f}. Voter 1 now widens her honest {c,d} to {a,c,d,f} — she has added approvals for a and f, both sitting winners:

before                    {a,c,f}
voter 1 adds {a,f}        {a,b,c}      <- f is gone

She strengthened a and f; f fell out. Note what this is not: it is not strategic. She reported more honest support and was punished for it, which is a different complaint from strategyproofness and lands on a rule (seq-PAV) that is otherwise well-behaved.

The three rules that fail even for one candidate¶

Monroe, Greedy Monroe and the Method of Equal Shares are ✗ in the "with additional voters" column — a single new voter approving a single sitting winner can remove him. For Equal Shares:

1×{b,d}  1×{a,b}  1×{b,d,e}  1×{a,e}  2×{c,d,e}  1×{c,e}  1×{a,c,e}  1×{b,c,d}

k=3        ->  {a,d,e}
+1 × {a}   ->  {b,c,e}        <- a is gone

The book is unusually blunt about what to make of this: the failure "can be seen as a serious argument against these rules" if you care about fair treatment of candidates — and a non-issue if the candidates are inanimate (items in a budget, features on a roadmap), which is exactly where Equal Shares is deployed. The axiom's importance is domain-dependent, and saying so is not hedging.

What this repo should and shouldn't claim¶

This is not the monotonicity that sinks IRV. Monotonicity in the single-winner sense — raising a candidate on your ballot must not defeat them — is a ranked-ballot pathology with a famous IRV witness. The axiom on this page is its committee-shaped relative, defined over approval sets and seat counts. They rhyme; they are not the same theorem, and citing one for the other is a real error.
AV and SAV pass both columns outright. For bloc Approval that is a clean, citable virtue: adding approvals for a sitting winner can never unseat them, individually or as a group.
Every proportional approval rule is cand at best. When this repo recommends proportional rules — and it does — that is the accompanying honest sentence.

Reproduce it¶

.venv/bin/python 06_Other/abcvoting_tabulation_engine/abc_axiom_check.py --verbose

Seven support-monotonicity witnesses replay: three for the with additional voters column (seq-CC, Greedy Monroe, Equal Shares) and four for without (seq-PAV, seq-CC, Equal Shares, Greedy Monroe).

Related: the table · committee monotonicity — the previous column · consistency — the next · single-winner monotonicity → the monotonicity hub.