Typical Margins of Victory with different voting systems in real life

Some have feared that margins of victory might typically be a lot larger or smaller with different voting systems. To investigate that, this page collects victory-margin data of the winner versus the 2nd place finisher, in various important elections using different voting systems.

There are some surprises. Range voting seems to produce larger victory-margins than approval voting (which can be regarded as range voting with only the endpoints of the score-range permitted as valid scores)?! – (happened in 6 our of 7 elections looked at so far) exactly the opposite of what one might naively expect from their Noise Levels arising from random votes.

[I contend noise levels are of little or no importance for predicting real life victory margins, and this surprise, assuming it is not just a statistical illusion/fluke, supports my contention.]

The Popular & Electoral College vote-margins in USA presidential elections 1908-2008

These elections, of course, were conducted using plurality voting.

The Closest US presidential elections

Gore=      50992335
G.W.Bush=  50455156   (wins in electoral college)
Difference=  537179 = 0.53% of the G+B votes = 58*noise
NoiseLevel=   10072

1960  (Election substantially fraudulent.)
Kennedy=   34226731   (wins)
Nixon=     34108157
Difference=  118574 = 0.174 of the K+N votes = 14.3*noise.
NoiseLevel=    8266

1876  (Election substantially fraudulent.)
Tilden    =4285992
Hayes=     4033768   (wins in electoral college)
Difference= 252224 = 3.03% of the T+H votes = 87*noise
NoiseLevel=   2884

In the above NoiseLevel=√N where N=total number of votes for both of the top 2 candidates combined. The "signal" (i.e. vote difference) far exceeded the "noise" in even the closest election ever, which was 14.3σ.

Biggest "Landslide" US presidential elections

Reagan=    58.8%
Mondale=   40.6%

Nixon=     60.7%
McGovern=  37.5%

Johnson=   61.1%
Goldwater= 38.5%

Landon=     36.5%
Difference= 24.3%

Harding=   60.3%
Cox=       34.1%

Parker=     37.6%
Difference= 18.8%

The Biggest Landslide French Presidential election

2002   (two-man runoff)
Jacques Chirac=   82.21%
Jean-Marie Le Pen=17.79%
Difference=       64.42%

The margin of victory here got within a factor of 1.55 of maximum possible (which would be 100%). In all 6 US landslides listed, it got within a factor ≤5.5.

Comparison of victory-margins in different voting systems

France 2007 comparison of victory-margins in different voting systems, given as a percentage of max theoretically possible margin:

 Voting system           victory margin
 Plurality:              31.2%-25.9% =    5.3%   (official results, first round)
 SimpleMajority:         53.1%-46.9% =    6.2%   (official runoff results)
 ApprovalVoting:         42.8%-41.6% =    1.2%
 (2,1,0)RangeVoting:     (1.08-0.96)/2  = 6.0%
 (5,4,3,2,1,0)RangeVote: (3.132-2.825)/5= 6.1% [or (3.041-2.774)/5=5.34%]
 (5,4,3,2,1,0)MedianScore:  (3-3)/5     = 0
 Condorcet:     52.0% to 48.0% pairwise = 4.0%  (also valid for Coombs final round)
 PseudoBorda:        (859.4-778.7)/1100 = 7.3%

Comparison of victory-margins in several voting systems (using poll data) from USA presidential elections 1988-2008 (expressed as percentage of max possible margin; median scores got from same poll/ballot data as range):

 voting system          year-1988
 Plurality(Official):    7.8%
 Approval:               2% (5canddts)
 Range (average-based):  6.33% (4pt allowed-score-range, 5canddts)
 MedianScore:            0

 voting system          year-1992
 Plurality(Official):    5.6%
 Approval:               1.2% (3canddts)
 Range:                  8.33% (7pt range, 3canddts)
 MedianScore:            0

 voting system          year-1996
 Plurality(Official):    8.5%
 Approval:               4% (4canddts)
 Range:                  6.66% (4pt range, 2canddts)
 MedianScore:            33.3%

 voting system          year-2000
 Plurality(Official):    0.51%
 Approval:               4% (4canddts)
 Range:                  4.17% or 6.64%  (101pt range, 4canddts)
 MedianScore:            0

 voting system          year-2004
 Plurality(Official):    2.47%
 Approval:               6% (3canddts)
 Range:                  9.3% (-5 to +5 range omitting 0 thus 10 possible scores, 2canddt)
 MedianScore:            10%

 voting system          year-2008
 Plurality(Official):    7.26%
 Approval:               21.3% (2canddts)
 Range:                  5.66%  (4pt range  among Obama & McCain only; ABC August poll)
 MedianScore:            0      (4pt range  among Obama & McCain only; ABC August poll)
 Range:                  4.4%  (101pt range among many canddts but we ignore everybody but 
                                         Obama & McCain; Gallup August poll)
 MedianScore:            3% ?    guess based on partial Gallup 101pt range-score data

Theoretical root-mean-square "noise levels" from random votes

For approval voting (each vote is 1=approve or 0=disapprove selected by a coin toss) N votes should lead to N/2±(N/4)1/2 approvals, i.e. the "noise level" for the average approval-level would be (4N)-1/2.

For continuum range voting with the real interval [0,1] as score range, but based on medians not averaging, assuming uniform random vote-scores, we'd expect noise level (8+4N)-1/2, i.e. essentially the same for large N.

For continuum average-based range voting with the real interval [0,1] as score range, we'd expect noise level (12N)-1/2, i.e. √3 times smaller.

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