Voting Cycles

Condorcet's Rule and Voting Cycles

Less than 10% of simulated elections lead to a chance voting cycle when there are 4 options spread among 200 voters. But on a small council with 3 factions, inadvertent ties are fairly common. And manipulators of Condorcet's rule try to create voting cycles.

Voters can often forge a cycle by ranking the central item below an opposition item. In our first example C was a Condorcet winner. But if Uri changes his ballot to rank C below D, we find a voting cycle in which D beats C, C beats B, and B beats D. Each wins by 4 ballots to 3.

 

Table 1c 7 Ballots 4 Ranks Uri Nic Mo Lil Kit Jo Gil 1st 2nd 3rd 4th A B D C B C A D B C D A C B D A D C B A D C A B D A C B

 

Table 2c,   Pairwise Tests of 4 Candidates
	Votes Against
	A	B	C	D
for A	-	3	2	2	2 prefer A over D.
for B	4	-	3	4	4 prefer B over D.
for C	5	4		3	3 now prefer C over D.
for D	5	3	4	-	4 now prefer D over C.

 

Click a number in table 2c to check which ballots contribute to it.
Or click a ballot to highlight the candidate totals it helps.
Click here to reset the ballots and the Condorcet pairwise table.

You can see that there are many different majorities even in this small group with 7 voters.  Proposals B, C, and D each win a majority; so it is not accurate to claim any of them win the majority.

Conspiring to create a cycle is hard and risky in a large, diverse electorate with many candidates. But the examples above and below show it can be easy in a council with a handful of factions.

If option B is the most central, supporters of moderate-left A may add their support to those who sincerely rank the right-wing C above B, helping C to beat B.

Conspirators risk enacting their least favorite policy. If, for example, supporters of C miscalculate and try to create a cycle by adding their support to A, she wins.

Thus strategic voters can manipulate Condorcet's rule to end indecisively in a voting cycle. But they cannot manipulate it to elect their preferred option nor to eliminate the 1 which would be the Condorcet winner if they cast sincere ballots. That option is always one of those tied in the top voting cycle or "Smith Set".

To resolve voting cycles, the by-laws may send a voting cycle to further discussion and vote trading; to tabling the motion or dividing it into parts; to a tie-breaking vote by the chairperson or a "Condorcet-completion rule", an alternative way to tally the preference ballots.

Blake Cretney's web site is the place to learn Tideman's Ranked Pairs and Schulze's Beat Path completion rules. These elect the Condorcet winner when there is one and break voting cycles without creating much opportunity for manipulation. Another manipulation-resistant rule breaks voting cycles using Instant Runoff Voting (IRV), LOR1 is detailed next.