Consistency — two electorates that agree should keep agreeing

Level: 301 · deep dive

One line: the shallowest-looking column in Table 3.1 is the deepest, because consistency doesn't merely describe a class of rules — it defines one, and the four rules that pass are exactly the ones you can write as a sum of per-voter scores.

The definition

Definition 3.4. An ABC rule R satisfies consistency if for every k ≥ 1 and disjoint profiles A, A′: if R(A,k) ∩ R(A′,k) ≠ ∅, then R(A + A′, k) = R(A,k) ∩ R(A′,k).

In words: split the electorate in two, count each half, and look at the committees that won in both. Consistency says the merged election must elect precisely those, and nothing else. It is the multi-winner form of the Smith–Young axiom that characterises single-winner scoring rules — the same property this repo demonstrates for STAR in Two districts, one office.

Monroe's failure, worked

Two profiles, two seats. Profile A (4 voters):

{a,y}   {a,y}   {b,y}   {b,y}

Profile A′ (12 voters):

{y}  {a}  {a,x} {a,x} {a,x} {a,x}   {y}  {b,y}  {b,x} {b,x} {b,x} {b,x}

Monroe's winners:

electorate	winning committees	Monroe-score
A	{a,b}, {a,y}, {b,y}	4
A′	{a,b}	10
shared	{a,b}	—
A + A′	{x,y} (score 15) — {a,b} scores only 14	—

Both halves could elect {a,b}; merged, Monroe elects {x,y} and {a,b} is not even winning. Replayed by the checker:

Consistency (Def. 3.4)
  [REPRODUCED] Monroe   Example 3.2   both elect {a,b}; merged elects {x,y}

The mechanism is the same one that costs Monroe Pareto efficiency: its score depends on partitioning voters into equal constituencies, and a partition of the merged electorate is not the union of partitions of the halves. Any rule whose score is not a plain sum over voters is exposed here.

The result that makes this column matter

Consistency is not just another row of ticks. It is one of five axioms that pin down a whole class of rules:

Theorem 3.2 (Lackner & Skowron). An ABC ranking rule is an ABC scoring rule if and only if it satisfies anonymity, neutrality, consistency, weak efficiency, and continuity.

An ABC scoring rule (Definition 3.5) gives each voter a score f(|A(i) ∩ W|, |A(i)|) — how many of her approved candidates are seated, and how many she approved in total — and elects the committee maximising the sum. Since anonymity, neutrality, weak efficiency and continuity are satisfied by every sensible rule, the theorem reads, practically: consistent = summable over voters.

That explains the entire column at a glance:

Rule	consistent?	why
AV, CC, PAV	✓	Thiele methods — a sum of per-voter satisfactions by construction
SAV	✓	an ABC scoring rule that is not a Thiele method — its f depends on |A(i)|, the ballot's own length
seq-PAV, seq-CC, rev-seq-PAV, seq-Phragmén, leximax-Phragmén	✗	sequential — the outcome depends on the order seats were filled, which is not a sum over voters
Monroe, Greedy Monroe	✗	scored by a partition of voters into constituencies
Method of Equal Shares	✗	scored by a budget process over rounds
MAV	✗	scored by the worst voter, and a max is not a sum

SAV is the interesting entry, and it is why Definition 3.5's f takes two arguments rather than one. A Thiele method may only look at how many of your approved candidates won. SAV also looks at how many you approved, dividing your single vote among your marks — which is outside the Thiele family but still a plain per-voter sum, so it stays consistent. The class of consistent rules is strictly larger than the Thiele class, and SAV is the witness.

The honest reading

• Consistency is a real virtue and a weak one. It is worth having, and it holds for rules with very different characters — utilitarian AV, egalitarian CC, proportional PAV. It cannot tell you which one to use, so it belongs in an argument about form, never in an argument about fairness.
• Failing it is not disqualifying. Every sequential rule fails, and sequential rules are what actually gets deployed, because the optimising versions are NP-hard. seq-PAV is PAV's practical stand-in and it is inconsistent; that is the price of computability, and it is a price the field pays knowingly.
• The characterisation is the transferable idea. "Which axioms force a rule into a known algebraic form" is the shape of the deepest results in social choice — May's theorem, Arrow, Smith–Young. Theorem 3.2 is that shape, for committees.

Reproduce it

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

The consistency section replays Example 3.2 — the three Monroe counts above, computed from the profiles rather than quoted.

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Related: the table · support monotonicity — the previous column · inclusion-strategyproofness — the next · consistency for a single-winner method → Two districts, one office · Thiele methods · SAV.