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.
Results
Table: Sponsor-bias β by final rank, small munis (< 200k registered voters)
n_self = 558.
| Final rank | β | SE | p |
|---|---|---|---|
| 1 (winner) | +7.67 | 2.27 | <0.001 |
| 2 | +10.14 | 2.05 | <0.001 |
| 3 | +6.10 | 6.22 | 0.327 |
| 4 | +6.73 | 6.27 | 0.283 |
(from source/analysis/heterogeneity.py → build/table/heterogeneity.csv)
Table: Sponsor-bias β by final rank, runoff-eligible munis (≥ 200k registered voters)
n_self = 83.
| Final rank | β | SE | p |
|---|---|---|---|
| 1 (winner) | +7.33 | 2.94 | 0.013 |
| 2 | +7.41 | 3.34 | 0.027 |
| 3 | +9.43 | 3.74 | 0.012 |
| 4 | +6.34 | 3.09 | 0.040 |
(from source/analysis/heterogeneity.py → build/table/heterogeneity.csv)
Peak β is at rank 2 in small munis and rank 3 in runoff munis — the position shifts at the institutional cutoff.
Interpretation
This is the cleanest design-level discriminator between coordination ([hyp:coordination-peak]) and bandwagon ([hyp:bandwagon-peak]) the project can run with current data. The theories make different predictions about where β should peak:
- Coordination (Cox's M+1 rule): peak at the marginal-viability rank. In small munis (plurality, M=1) the cutoff is between rank 2 and rank 3, so peak β at rank 2. In runoff munis (top-2 advance to round 2, effective M+1=3), the cutoff is between rank 3 and rank 4, so peak β at rank 3.
- Bandwagon: peak β just below the front-runner — i.e., at rank 2 regardless of muni size, because that's where slant has the most room to flip apparent leader identity.
- Observed pattern: peak at rank 2 in small → peak at rank 3 in runoff matches the coordination prediction and contradicts the bandwagon prediction.
Caveats on strength
- Runoff cell is thin. Wide SEs (n_self = 83 vs 558 in small munis). The rank-2-vs-rank-3 gap in runoff munis (β = +7.41 vs +9.43) is not statistically distinguishable on its own; the prediction passes on the position-of-peak test, not on the gap-between-ranks test.
- Rank-5+ candidates show β ≈ 0 (small or wrong-signed), consistent with both theories: sponsors of clearly-hopeless candidates don't slant because there's no return.
Confidence rationale (yellow). The position-of-peak shift lines
up with the coordination prediction across both subsamples, but the
runoff cell is small (n_self = 83) and the rank-2-vs-rank-3 gap is
not separately significant. Confidence stays at yellow rather than
green until the 2022 cycle extension lands and gives the runoff
cell more power.
Follow-ups
- 2022 governor races: same design transfers cleanly (state-level, single-winner, runoff threshold). Would expand the runoff-rank cell substantially.
- RD at the 200k cutoff: with ~190 munis near the threshold in the 2020 aptos data, a fuzzy-RD on the position-of-peak-β as the outcome is feasible if the running variable is sharp enough.
- Rank quintile interaction with race competitiveness (AN-005's margin tertiles): bandwagon predicts both rank-2 peak AND tighter-race amplification; coordination predicts shift in peak position with rank but no necessary interaction with margin. Can separate.