plot() method for the result of an experiment with varying numbers of ballots

The "adjusted rank" of a candidate is their ranking \(r\) plus their scaled "winning margin". The scaled margin is \(e^{-cx/\sqrt{n}}\), where \(x\) is the adjusted margin (i.e. the number of votes by which this candidate is ahead of the next-weaker candidate, adjusted for the number of ballots \(n\) and the number of seats \(s\)), and \(c>0\) is the margin-scaling parameter cMargin.

Usage

# S3 method for SafeRankExpt
plot(
  x,
  facetWrap = FALSE,
  nResults = NA,
  anBallots = 0,
  cMargin = 1,
  xlab = "Ballots",
  ylab = "Adjusted Rank",
  title = NULL,
  subtitle = "(default)",
  line = TRUE,
  boxPlot = FALSE,
  boxPlotCutInterval = 10,
  pointSize = 1,
  ...
)

Arguments

x: object containing experimental results
facetWrap: TRUE provides per-candidate scatterplots
nResults: number of candidates whose results are plotted (omitting the least-favoured candidates first)
anBallots, cMargin: parameters in the rank-adjustment formula
xlab, ylab: axis labels
title: overall title for the plot. Default: NULL
subtitle: subtitle for the plot. Default: value of nSeats and any non-zero rank-adjustment parameters
line: TRUE will connect points with lines, and will disable jitter
boxPlot: TRUE for a boxplot, rather than the default xy-scatter
boxPlotCutInterval: parameter of boxplot, default 10
pointSize: diameter of points
...: params for generic plot()

Value

graphics object, with side-effect in RStudio Plots pane

Details

The default value of cMargin=1.0 draws visual attention to candidates with a very small winning margin, as their adjusted rank is very near to \(r+1\). Candidates with anything more than a small winning margin have only a small rank adjustment, due to the exponential scaling.

A scaling linear in \(s/n\) is applied to margins when anBallots>0. Such a linear scaling may be a helpful way to visualise the winning margins in STV elections because the margin of victory for an elected candidate is typically not much larger than the quota of \(n/(s+1)\) (Droop) or \(n/s\) (Hare). The linear scaling factor is \(as/n\), where \(a\) is the value of anBallots, \(s\) is the number of seats, and \(n\) is the number of ballots. For plotting on the (inverted) adjusted rank scale, the linearly-scaled margin is added to the candidate's rank. Note that the linearly-scaled margins are zero when \(a=0\), and thus have no effect on the adjusted rank. You might want to increase the value of anBallots, starting from 1.0, until the winning candidate's adjusted rank is 1.0 when all ballots are counted, then confirm that the adjusted ranks of other candidates are still congruent with their ranking (i.e. that the rank-adjustment is less than 1 in all cases except perhaps on an initial transient with small numbers of ballots).

When both anBallots and cMargins are non-zero, the ranks are adjusted with both exponentially-scaled margins and linearly-scaled margins. The resulting plot would be difficult to interpret in a valid way.

Todo: Accept a list of SafeVoteExpt objects.

Todo: Multiple counts with the same number of ballots could be summarised with a box-and-whisker graphic, rather than a set of jittered points.

Todo: Consider developing a linear scaling that is appropriate for plotting stochastic experimental data derived from Condorcet elections.