Voting Rules for
 Accurate
Democracy

Different uses for voting need different types of voting.

Voting rules for setting policies; Condorcet rules

Voting Cycles

Setting policies with Condorcet rules, chapter contents

Condorcet's Rule and Voting Cycles

Condorcet's rule is not always decisive. There is no Condorcet winner if A beats B, B beats C, and C beats A. This is called a voting cycle. (It is also called a voting paradox because the collective ranking can be circular even if each voter has non-circular preferences.)
 A > B > C > A

Less than 1.5% of real elections lead to a chance voting cycle when there are 21 voters or more.[T] Less than 10% of simulated elections create a voting cycle when there are 4 options spread among 200 voters. But on a small council with 3 factions, inadvertent ties are more common. And manipulators of agenda voting or Condorcet's rule can sometimes cause a voting cycle — leading to a weak decision by a tie-breaking rule.

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 1 c
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
 B > D > C > B
 Table 2 c   Pairwise Tests of 4 Motions
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 cycle's arrows 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. 
These 7 ballots also show a Condorcet tally, an Instant Runoff, and suspended votes.

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, some 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.

|____________A_________B______________C____|

C > B > A

This simple tactic is sometimes called “raising turkeys;” you raise your rank of a “turkey” that you are sure will lose. It is also called “burying” the true winner, or “crossover votes” because the loser you raise must be across the center, on the far wing of the other side.

Conspirators risk enacting their least favorite policy. If, for example, many 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 one 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” of candidates.

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 to 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. Other manipulation-resistant rules break voting cycles using Instant Runoff Voting (IRV). The techncal name for these rules is “Condorcet-Hare Hybrids”. The LOR1 hybrid is explained next. Loring One-winner Rule

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