Single Winner Systems

In this section, I will go through some voting systems that elect one person at a time, as in elections for governor, and discuss how well they stack up by the standards listed on the main page.

Plurality winner (AKA First-past-the-post, Our system)

See wikipedia for a complete description of this system.

This is the most common system for holding single-winner elections in the United States.

Instant runoff voting

See wikipedia for a complete description of this system.

This system is used in single-winner elections in Ireland and San Francisco. It is the most frequently advocated replacement for Plurality.

Approval voting

See wikipedia for a complete description of this system.

A variant of this system is used by the Harvest Coop Market.

Examples

In these examples, I will posit some grid of sincere preferences held by the electorate. I will then evaluate the expected outcomes of the elections for those sincere preferences under each system, with and without strategy. For Approval Voting, where there is always strategy, I have invented a suboptimal strategy to act as a baseline. The actual outcome will probably be in between the "with" and "without", depending on many factors. All strategies selected are generally accepted as the best strategies available under that particular system. They are listed here:

Example 1

This example is very similar to the situation many minor parties find themselves in today.

Sincere Preferences:

First Choice Aggregate Vote
 Votes Second Choice
Left
 6
 6
Center
0
 Right
Center
 43
 20
Left
20
 Right
3
 -
Right 46
 40
 Center
3
 Left
3
 -

Predicted Outcomes:

Candidate IRV (No strategy) IRV (Strategy) Approval (Top 2) Approval (Strategy) Plurality (No strategy) Plurality (Strategy)
Left:
 6     0
 6     0
 29
 6
 6
 0
Center:
 43  49
 43  49
 89
 49
 43
 49
Right:
 46  46
 46  46
 66
 46
 46
 46

In this example, IRV is superior to Approval and Plurality in that there is no advantage to using strategy.  Both Approval and IRV are superior to Plurality in that they give a more accurate reading of the voters' true intent.  

Example 2

This example is supposed to represent what would happen if support for "Left" grew, mainly at the expense of Center.

Sincere Preferences:

First Choice Aggregate Vote Votes
 Second Choice
Left: 31
25
 Center
3
 Right
3
 -
Center: 27
13
 Left
13
 Right
3
 -
Right: 39
 31
 Center
3
 Left
6
 -

Predicted Outcomes:

Candidate IRV (No strategy) IRV (Strategy) Approval (Top 2) Approval (Strategy) Plurality (No strategy) Plurality (Strategy)
Left:
 31  44
 6    0
 47
 31
 31
 3
Center:
 27  0
 52  52
83
 52
 27
 52
Right:
 39  52
 39  39
 56
 42
 39
 42

In this example, strategy plays a significant role in determining the outcome for both IRV and Plurality. If the strategy were to be taken to the maximum level, both systems would seriously underreport the true strength of Left's candidacy. Given the extremely strategic IRV or Plurality results, it is difficult to distinguish between the sincere preferences of Example 1 and Example 2.

Approval with strategy does a better job of describing Left's support than IRV with strategy. Approval also does a better job of degrading gracefully if some voters are unclear on what their best strategy is, by degrading toward the center.

Example 3

Support for Left has continued to grow until Left is the dominant player.

Sincere Preferences:
First Choice Aggregate Vote Votes
 Second Choice
Left: 45
 35
 Center
6
 Right
4
 -
Center: 25
 10
 Left
9
 Right
6
 -
Right: 21
 16
 Center
3
 Left
2
 -

Predicted Outcomes:

Candidate IRV (No strategy) IRV (Strategy) Approval (Top 2) Approval (Strategy) Plurality (No strategy) Plurality (Strategy)
Left:
 45  48
 45  48
 58
 48
 45
 48
Center:
 25  41
 41  41  
 76
 41
 25
 41
Right:
 21   0
 5     0
 36
 21
 21
 2

In this example, Strategic Voting only determines the outcome in Approval.  Under Approval, the voters in Left have reached the size where they no longer have to also support Center in order to defend against Right, so they abandon Center and win outright.

Example 4

A more complicated example.  Voters are almost evenly split between a "Left faction", represented by Left and Center-Left, and a "Right Faction", represented by Right and Center-Right. However, voters in the Right faction prefer their centrist, while voters in the Left faction prefer their extremist. (Some voters cross factions, though.)

Sincere Preferences:
First Choice Aggregate Vote Votes
 Second Choice Third Choice
Left: 30
 26
 Center-Left
Center-Right
1
 Center-Right
 Center-Left
1
 Right
 -
2
 -
Center-Left: 21
 6
 Left Center-Right
5
 Left
 -
7
 Center-Right
 Left
2
 Right
 Left
1
 -
Center-Right: 41
 13
 Center-Left Right
25
 Right
 -
3
 -
Right:
 7
 5
 Center-Right
 -
2
 Left
 Center-Left

Predicted Outcomes:

Candidate IRV (No strategy) IRV (Strategy) Approval (Top 2) Approval (Strategy) Plurality (No strategy) Plurality (Strategy)
Left:
 30   32   45
3    5     0
 43
 32
 30
 3
Center-Left
 21   21    0
 47  48   50
 60
 49
 21
 47  ?  TIE
Center-Right
 41   46   53
 47  48   48
 54
 47
 41
 47  ?  TIE
Right
 7    0       0
 2    0     0
 35
 8
 7
 2

Strategy comes into play in IRV and Plurality, and a bit less so in Approval.  Again, IRV and Plurality using Strategy fail to accurately report the strength of each party.

Conclusions

Revisit conclusions soon.

This is part of Dan Keshet's Voting Systems website.