H6: β peaks at the M+1 viability cutoff (coordination)
The sponsor-bias amplification β is heterogeneous in a candidate's ex-ante viability rank, and the heterogeneity peak should sit at the M+1 viability cutoff implied by Cox's rule. In small (plurality) mayoralties M+1 = 2, so the peak should appear at rank 2; in runoff-eligible mayoralties M+1 = 3, so the peak should appear at rank 3. The position of the peak is what shifts at the 200,000-voter institutional threshold — this is the cleanest design-level test the project can run on the coordination-vs-bandwagon discrimination.
Evidence strength: Supported by AN-004 (2026-06-02). Peak β sits at rank 2 in small munis (+10.14 pp, p < 0.001, n_self = 558) and at rank 3 in runoff-eligible munis (+9.43 pp, p = 0.012, n_self = 83). The position-of-peak shift matches Cox's M+1 rule uniquely; bandwagon would predict rank 2 in both cells. The magnitude gap within the runoff cell (rank 2 = +7.41 vs rank 3 = +9.43) is not statistically distinguishable on its own, so the qualitative position-of-peak test passes but the gap-magnitude test does not — confidence held to yellow at the analysis level until the 2022 governor cycle expands the runoff cell.
Theory
The framework is Polls as coordination devices (theory.md §"Polls as coordination devices (strategic voting)"). Voters in plurality systems avoid wasting their vote on candidates outside the seat-winning set; polls are the common-knowledge signal that coordinates them onto the M+1 viable candidates [cite:cox1997making]. In Brazilian mayoralties the institution itself shifts M+1: munis with fewer than 200,000 registered voters use plurality (M=1 → 2 viable), munis at or above the cutoff use two-round runoff with the top 2 advancing (effective first-round M+1 = 3). The 200k threshold is set by Constitution Art. 29 II + Lei 9.504/1997 Art. 3, computed by TSE each cycle, and is exogenous to candidate quality Andonie & Kuzmics (2012).
Prediction
In a heterogeneity regression of error ~ sponsored_by × rank_dummies
absorbing candidate FE, estimated separately for small and
runoff-eligible munis, the position-of-peak β should differ across
the cell. Specifically: peak at rank 2 in small munis, peak at rank 3
in runoff munis. The discriminating object is the cross-cell
pattern, not the magnitude of any single cell.
Competing predictions
Bandwagon (H7). Voters reward the apparent leader regardless of M+1 structure, so the sponsor incentive peaks just below the front-runner — at rank 2 in both muni-size classes. Bandwagon and coordination agree on small munis (both predict rank 2) but disagree on runoff munis: coordination predicts the peak slides to rank 3, bandwagon predicts it stays at rank 2.
Uniform-incentive null. Every candidate has the same strategic incentive to commission a slanted poll regardless of position, because a +7 pp shift could move voters across any viability threshold the voter is using. Under this null the rank gradient should be flat in both muni-size classes. AN-004 rejects this null for both classes.
Prior research
Granzier et al. (2023) is the closest institutional analog: in French two-round elections, first-round poll rankings shape second-round voting and candidate dropout — direct evidence that voters use the M+1 cutoff implied by the institution. Andonie & Kuzmics (2012) formalizes pre-election polls as game-theoretic coordination devices, giving the model that motivates the position-of-peak prediction. Lloyd et al. (2016) documents non-random Brazilian poll error operating through strategic voting specifically — the closest substantive predecessor for the claim that poll-induced coordination is consequential in Brazil. Bandwagon empirical evidence in multi-party plurality systems [cite:schmittbeck2008polls; cite:strijbis2015umfragen] supplies the discriminating alternative.
Evidence
| Analysis | Bearing | Key takeaway |
|---|---|---|
| AN-004 | Supports | Peak β shifts from rank 2 in small munis (+10.14 pp, p < 0.001, n_self = 558) to rank 3 in runoff munis (+9.43 pp, p = 0.012, n_self = 83). Position-of-peak matches Cox's M+1; bandwagon would predict rank 2 in both. Rank-5+ ≈ 0 in both cells, consistent with no return to slant for clearly-hopeless candidates. |
| AN-045 | Refines | Decomposing AN-004 by final_rank × tight_race, the largest tight-race amplification is at rank 1 (+4.80 → +12.20 pp), not rank 2 (+9.54 → +6.81). Rank-2 over-commissioning (AN-027) and rank-1 over-statement (AN-045) live in different cells of the rank × margin manifold — coordination at runner-up, bandwagon at leader. Yellow confidence: descriptive cell means with thin sponsored-side n (51–208 per cell); within-candidate FE underpowered for the 3-way. |
| AN-050 | Refines | Under rank-at-commission rather than final_rank, the AN-040 "rank-2 over-statement" vanishes (rac-2 sponsor effect: −0.46 pp non-tight, +4.20 pp tight, n = 22–23 sponsored polls per cell). The +11.5 pp ex-post rank-2 effect was a Simpson's-style aggregation artifact: candidates who ended rank 2 were polled at varied positions during the campaign. Coordination-demand at rac-2-in-tight remains real on the selection side (AN-027 confirmed) but not on the bias-amplification side at AN-040's level. |
Open tests
2022 governor cycle for runoff power
The runoff cell in AN-004 has n_self = 83, which is why the rank-2 vs rank-3 gap within runoff munis is not individually significant. The 2022 governor-race extension transfers the design cleanly (state-level, single-winner, identical runoff threshold) and should substantially expand the runoff cell, upgrading the position-of-peak finding from yellow to green.
Fuzzy-RD at the 200,000-voter threshold
With roughly 190 munis sitting near the 200k registered-voter
threshold in the 2020 aptos data, a fuzzy-RD on position-of-peak-β as
the outcome at the runoff-eligibility boundary is feasible if the
running variable is sharp enough. This is the design-level RD the
theory motivates; mass-near-the-cutoff is the open question
(thinking.md Q3).
Rank-quintile × margin separation
AN-045 and AN-050 already begin this work. Bandwagon predicts both rank-2 peak and tighter-race amplification at rank 1; coordination predicts the muni-size shift in peak position with no necessary margin interaction. Refining the partition with rank-at-commission (rather than final rank) is the highest-paper-value follow-up — the rank-at-commission swap turns the AN-040 ex-post artifact into a clean ex-ante test of where each voter-side mechanism actually lives.
Supporting analyses
Peak β shifts from rank 2 in small munis (+10.14) to rank 3 in runoff-eligible munis (+9.43). Direction matches Cox's M+1 rule (M+1=2 vs M+1=3); bandwagon would predict rank 2 in both. The cleanest design-level discriminator the project supports with current data.
Coordination-incentive prediction fails on selection (rank-1 winners *over*-commission by +28 pp; runners-up *under*-commission by −8 pp; χ²(4) = 169, p < 10⁻³⁵) but partially holds on the bias gradient (spec-3c: rank-1 β = +4.88 pp, p = 0.23, NOT significant; rank-2 β = +9.13 pp, p = 0.01; rank-3 β = +11.75 pp, p = 0.02; rank-4 β = +13.80, p = 0.02; rank-5+ β = +5.24, p = 0.26). The two halves point to a refined mechanism: winners commission polls in greater numbers (resourced campaigns can afford polling) but the slant on winner-sponsored polls is smaller and insignificant — the marginal value of a *slanted* poll, not the marginal value of *a* poll, varies by rank.
Rank-at-commissioning RECOVERS the coordination-incentive prediction in tight races. AN-026's selection finding ('winners over-commission by +28 pp on final-rank') was largely an artifact of (a) eventually-rank-2 candidates being labeled rank-2 even when leading at commissioning, and (b) safe-race rank-1 over-commissioning dominating the pooled count. When rank is measured at commissioning from the most recent prior neutral poll AND stratified by race competitiveness, the tight-margin tertile reverses: rank-1 share -9.7 pp (sensitivity -16.4 pp, p=0.007); rank-2 +19.2 pp (sensitivity +11.9 pp); rank-3 +14.4 pp. In wide-margin races rank-1 still over-commissions by +22 pp. Two distinct mechanisms: coordination-demand drives runner-up over-commissioning in competitive races; resourcing-without-strategic-need drives leader over-commissioning in safe races. The bias-gradient half is uninformative at current n — rank × tertile cells are too thin for clean inference.
AN-027's tight-margin coordination signal is ROBUST across three temporal anchors (date_start the commissioning-latest-bound; date_registered the regulatory-filing moment; date_end the AN-027 default). Rank-2 over-commissioning sits in [+17, +19] pp under primary spec across all anchors; rank-1 deviation in [−2, −9.7]. Sensitivity gives [+8, +12] / [−9, −16]. Pooled (all-races) rank-1 over-commissioning of +8 to +12 pp persists at all anchors. NOTABLE SHIFT from AN-027 in the bias gradient: under sharper temporal alignment, rank-1 β rises from +6.51 (date_end, AN-027) to +7.46–7.84 (date_start/date_registered) and crosses significance (p ≈ 0.013); the rank-3 β is stable at ~+12. The 'winner-commissioned polls show less slant' AN-027 framing is therefore shaded — leaders DO slant significantly under tight temporal alignment, but still less than mid-ranks. Empirical surprise: 26.6% of polls are registered AFTER fielding ends, so date_registered is the regulatory-filing moment rather than the commissioning moment per se.
Money does NOT drive the rank gradient in self-sponsoring. The AN-027 two-mechanism interpretation of safe-race rank-1 over-commissioning as 'resourcing without strategic need' is REFUTED on the literal money-control test. Baseline rank-2 vs rank-1 gap is −3.6 pp tight / −3.6 pp mid / −6.8 pp wide; under log(revenue) control it moves to −3.6 / −3.6 / −6.6 — essentially unchanged across all margin tertiles. log(revenue) main effect is +0.23 pp per log-unit (p=0.16 pooled); rank × log(revenue) interactions are all <0.012 pp / log-unit and insignificant (p>0.17). The cross-margin pattern (rank-1 dominance largest in safe races) is robust to money control. Substantive read: the safe-race rank-1 over-commissioning is real but is NOT explained by campaign-finance revenue. Alternative resourcing channels (campaign sophistication, party support, fixed-cost crossing of the polling-affordability threshold) remain candidates; pure-money resourcing as a stand-alone mechanism is not supported.
Joint test of the two-mechanism story SUPPORTS coordination + reveals money is orthogonal to both halves. Under rank-at-commission (date_start anchor) + race-FE LPM with log(receita) control: tight-race rank-2 vs rank-1 coefficient = **+1.81 pp (p=0.10) baseline, +1.81 pp (p=0.10) with money** — fully robust to money control. Wide-race rank-2 coefficient = −0.82 (p=0.24) baseline, −0.73 (p=0.30) with money — leader dominance preserved. log(receita) coefficient ≤ +0.14 pp / log-unit, all insignificant in race-FE specs. **The tight-vs-wide rank-2 contrast is 2.6 pp** and survives money control. Both mechanisms (tight-race coordination, safe-race rank-1 dominance) operate; money explains neither. Notable shift from AN-029 final-rank: rank-2 in tight races flips from −3.61 pp (final-rank) to +1.81 pp (rank-at-commission), a 5.4 pp swing showing the final-rank coding obscured the coordination signal that proper temporal alignment recovers. Marginal significance (p=0.10) at n_self=32 in tight cohort — direction is clean, statistical power is the binding constraint.
Rank-2 sponsor over-statement does NOT concentrate in tight races as the coordination story predicted. The descriptive cell means show the biggest tight-race amplification at RANK 1 instead: rank-1 sponsor effect jumps from +4.80pp (non-tight) to +12.20pp (tight); rank-2 sponsor effect runs +9.54 / +6.81 across the same cut (lower in tight). Pooled OLS confirms: `sp × tight` is +9.26pp (p<0.001), `sp × rank2 × tight` is −11.99pp (p=0.001) — the rank-2 amplification is OFFSET in tight races. Within-candidate FE attenuates these to n.s. levels, so the cross-candidate cell-mean reading is the cleanest takeaway. The AN-040 rank-2 over-statement spans both close and wider races; the close-race-specific bias is on rank-1 winners, consistent with bandwagon-effect demand rather than the coordination-demand story.