Say we're trying to choose r representatives out of n people, and let x_r, x_n be the fraction of representatives and people voting yes on a given issue, respectively. The code currently uses the metric
Pr((x_r < 1/2) == (x_n < 1/2))
which is the probability that representatives match the majority vote of the whole. This metric is highly nonlinear, but seems the best theoretical choice. In particular, the simpler quadratic metric
does not give reasonable proportional representation. However, there is an intermediate choice which hadn't occurred to me when first wrote the code:
namely to match the voting fraction as closely as possible. This is less nonlinear than the binary metric and far smoother, and thus may have much better convergence properties. Since it's smoother, it may also give more reasonable results earlier (with less data) than the binary metric.
Say we're trying to choose r representatives out of n people, and let x_r, x_n be the fraction of representatives and people voting yes on a given issue, respectively. The code currently uses the metric
which is the probability that representatives match the majority vote of the whole. This metric is highly nonlinear, but seems the best theoretical choice. In particular, the simpler quadratic metric
does not give reasonable proportional representation. However, there is an intermediate choice which hadn't occurred to me when first wrote the code:
namely to match the voting fraction as closely as possible. This is less nonlinear than the binary metric and far smoother, and thus may have much better convergence properties. Since it's smoother, it may also give more reasonable results earlier (with less data) than the binary metric.