Application 3: jurisdiction-level counts

Motivating example: at-large versus district elections (Online Appendix C.4)

Online Appendix C.4 of Atsusaka (2021) considers a jurisdiction with six seats and a jurisdiction-wide minority share of 47.5%. It compares an at-large arrangement with six single-member districts whose minority shares are c(50, 40, 60, 30, 50, 80) and whose adjusted racial margins are c(30, 40, 50, 65, 70, 30). Under this hypothetical comparison, the simulations concentrate on zero minority officeholders at large and two or three under district elections.

minorep() supplies the seat-level probabilities and n_minorep() converts them into a distribution of counts, using independent draws across seats. The district component can be reproduced directly from the appendix inputs:

appendix_M <- c(30, 40, 50, 65, 70, 30)
appendix_C <- c(50, 40, 60, 30, 50, 80)
appendix_prediction <- minorep(M = appendix_M, C = appendix_C)
appendix_counts <- n_minorep(
  model_pred = appendix_prediction, n_sim = 1000, seed = 2026
)
prop.table(table(appendix_counts))
#> appendix_counts
#>     2     3 
#> 0.836 0.164

The electoral-rule comparison is conditional on the voting assumptions and the independence of seat outcomes. It does not by itself establish legal vote dilution. The example below illustrates the same workflow with a different set of district margins.

Simulating jurisdiction totals

n_minorep() converts district-level probabilities into simulated counts of minority officeholders across a jurisdiction. It makes independent Bernoulli draws for each district in every simulation.

district_M <- c(50, 40, 40, 35, 70, 85)
district_C <- c(50, 40, 60, 30, 50, 80)
district_prediction <- minorep(M = district_M, C = district_C)

count_draws <- n_minorep(
  model_pred = district_prediction,
  n_sim = 1000,
  seed = 2026
)
summary(count_draws)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   2.000   2.000   3.000   2.659   3.000   4.000

n_sim controls the number of returned draws. Supplying seed makes the result reproducible without changing the caller’s random-number state.

Inspect the simulation distribution

hist(
  count_draws,
  xlab = "Simulated number of minority officeholders",
  main = ""
)

The distribution represents uncertainty implied by the supplied district-level probabilities. It should be interpreted alongside the assumptions used to construct those probabilities.