2.2 Preference Schedules

To begin, we’re going to want more information than a traditional ballot normally provides. A traditional ballot usually asks you to pick your favorite from a list of choices. This ballot fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.

Example 1

A vacation club is trying to decide which destination to visit this year: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below:

# Math in Society 2.2 -- Preference Schedules
# The vacation club's ten individual preference ballots.
# H = Hawaii, O = Orlando, A = Anaheim.  EDIT any voter's ranking and re-run.
ballots = {
    "Bob": "AOH", "Ann": "AHO", "Marv": "OHA", "Alice": "HAO", "Eve": "AHO",
    "Omar": "OHA", "Lupe": "HAO", "Dave": "OHA", "Tish": "HAO", "Jim": "AHO",
}
place = ["1st choice", "2nd choice", "3rd choice"]
full = {"H": "Hawaii", "O": "Orlando", "A": "Anaheim"}

print("Individual preference ballots")
print(f"{'':>12}" + "".join(f"{v:>6}" for v in ballots))
for r in range(3):
    print(f"{place[r]:>12}" + "".join(f"{b[r]:>6}" for b in ballots.values()))

# Identical ballots collapse into a single column of the preference schedule.
counts = {}
for b in ballots.values():
    counts[b] = counts.get(b, 0) + 1
order = list(dict.fromkeys(ballots.values()))          # column order = first seen
total = sum(counts.values())

print("\nPreference schedule (identical ballots share one column)")
print(f"{'':>12}" + "".join(f"{counts[c]:>6}" for c in order))
for r in range(3):
    print(f"{place[r]:>12}" + "".join(f"{c[r]:>6}" for c in order))
print(f"{'':>12}  " + " + ".join(str(counts[c]) for c in order) + f" = {total} votes")

first = {}
for c in order:
    first[c[0]] = first.get(c[0], 0) + counts[c]
print("\nFirst-choice votes")
for k in sorted(first, key=lambda k: -first[k]):
    bar = "#" * first[k]
    print(f"   {full[k]:<8}{first[k]:>3}  {first[k]/total:>6.1%}  {bar}")
lead = max(first, key=lambda k: first[k])
print(f"\n{full[lead]} leads with {first[lead]} of {total} votes"
      f" -- a plurality, but {'a' if first[lead]*2 > total else 'NOT a'} majority.")
print("\nTry it: change Jim's ballot from 'AHO' to 'HAO' and re-run.")
print("Hawaii takes the lead without a single voter changing their mind about Hawaii")
print("versus Orlando -- only about which of the three they put on top.")

	Bob	Ann	Marv	Alice	Eve	Omar	Lupe	Dave	Tish	Jim
$1^{\text {st }} \text { choice }$	A	A	O	H	A	O	H	O	H	A
$2^{\mathrm{nd}} \text { choice }$	O	H	H	A	H	H	A	H	A	H
$3^{\mathrm{rd}} \text { choice }$	H	O	A	O	O	A	O	A	O	O

Show solution

These individual ballots are typically combined into one preference schedule, which shows the number of voters in the top row that voted for each option:

	1	3	3	3
$1^{\text {st }} \text { choice }$	A	A	O	H
$2^{\text {nd }} \text { choice }$	O	H	H	A
$3^{\text {rd }} \text { choice }$	H	O	A	O

Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: $1+3+3+3 = 10$ total votes.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.