2.7 What’s Wrong with IRV?

Example 1

Let’s return to our City Council Election

	342	214	298
$1^{\text {st }} \text { choice }$	Elle	Don	Key
$2^{\text {nd }} \text { choice }$	Don	Key	Don
$3^{\text {rd }} \text { choice }$	Key	Elle	Elle

Show solution

In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The 214 people who voted for Don have their votes transferred to their second choice, Key.

	342	512
$1^{\text {st }} \text { choice }$	Elle	Key
$2^{\text {nd }} \text { choice }$	Key	Elle

So Key is the winner under the IRV method.

We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don’s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.

Example 2

Consider the voting system below.

	37	22	12	29
$1^{\text {st }} \text { choice }$	Adams	Brown	Brown	Carter
$2^{\text {nd }} \text { choice }$	Brown	Carter	Adams	Adams
$3^{\text {rd }} \text { choice }$	Carter	Adams	Carter	Brown

Show solution

In this election, Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 for Brown.

Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to “jump on the bandwagon”, 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.

# Math in Society 2.7 -- What's Wrong with IRV? (the monotonicity failure)
# 100 voters.  In the re-vote, SWITCH of the 12 "Brown, Adams, Carter" voters
# jump on the bandwagon and move Adams up to first: "Adams, Brown, Carter".
# Nothing else changes, and not one vote is ever moved AWAY from Adams.
def election(switch):
    return [(37 + switch, "ABC"), (22, "BCA"), (12 - switch, "BAC"), (29, "CAB")]

def irv(schedule):
    alive = sorted({c for _, r in schedule for c in r})
    alive = set(alive)
    tied = False
    while True:
        tally = {c: 0 for c in alive}
        for n, r in schedule:
            for c in r:
                if c in alive:
                    tally[c] += n
                    break
        cont = sum(tally.values())
        lead = max(tally, key=lambda c: (tally[c], c))
        if tally[lead] * 2 > cont or len(alive) == 1:
            return lead, tally[lead], cont, tied
        low = min(tally.values())
        out = sorted(c for c in alive if tally[c] == low)
        tied = tied or len(out) > 1
        alive.discard(out[0])

names = {"A": "Adams", "B": "Brown", "C": "Carter"}
print("switch   first-place votes         IRV winner")
prev = None
for switch in range(0, 13):
    first = {}
    for n, r in election(switch):
        first[r[0]] = first.get(r[0], 0) + n
    w, got, cont, tied = irv(election(switch))
    flag = "   <-- the winner just changed" if prev is not None and w != prev else ""
    if tied:
        flag += "   (elimination tie: outcome depends on the tiebreak rule)"
    print(f"{switch:>4}     " + "  ".join(f"{c} {first.get(c, 0):>3}" for c in "ABC")
          + f"      {names[w]:<7}{got:>4}/{cont}{flag}")
    prev = w

print("\nAdams' column only ever goes UP, and yet partway down this table Adams")
print("stops winning. Gaining support cost Adams the election -- exactly what the")
print("Monotonicity Criterion forbids.")
print("\nTry it: change the 22 'BCA' voters to (22, 'BAC') and re-run. Now Brown's")
print("votes transfer to Adams instead of Carter and Adams wins every row, so IRV")
print("does not ALWAYS violate monotonicity -- it depends on where the eliminated")
print("candidate's second choices go.")

	47	22	2	29
$1^{\text {st }} \text { choice }$	Adams	Brown	Brown	Carter
$2^{\text {nd }} \text { choice }$	Brown	Carter	Adams	Adams
$3^{\text {rd }} \text { choice }$	Carter	Adams	Carter	Brown

In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with 51 votes to Adams’ 49 votes! Even though the only vote changes made favored Adams, the change ended up costing Adams the election. This doesn’t seem right, and introduces our second fairness criterion:

This criterion is violated by this election. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; just that IRV has the potential to violate the criterion in certain elections.

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.