[EM] Elections methods performance criterion
Ken Johnson
kjinnovation at earthlink.net
Mon Mar 8 03:26:01 PST 2004
>Date: Sat, 06 Mar 2004 02:35:02 -0800
>From: Ken Johnson <kjinnovation at earthlink.net>
>
>
>... I think the cardinal
>rating concept provides a useful basis for defining the objective of
>election systems (i.e., maximize "social utility" or "social
>representation")...
>Cardinal ratings are, in my view, a more useful definitional standard
>than rank-based standards ...
>... I ran some simple numerical simulations to test out this idea.
>
Following are additional results showing the relative performance of
more methods (e.g. IRV, Top-2 Runoff). The methods are as follows:
(1) Sincere CR, "SiCR". This is not a practically realizable method
because voters will strategize, but it is the standard by which the
other methods are evaluated.
(2) Strategic CR, "StCR". This only represents the most trivial voting
strategy: Exaggerate. Individuals' CRs are linearly scaled so that the
lowest and highest values are at 0 and 1. (A more optimal strategy is to
polarize one's ratings, which is equivalent to #4, StAV.)
(3) Sincere Approval Voting, "SiAV". Sincere CRs are rounded to 0 or 1.
Voters can potentially approve all candidates or approve none.
(4) Strategic Approval Voting, "StAV". Same as SiAV, except applied to
exaggerated CRs. In other words, the approve/disapprove CR cutoff level
is the average of the highest and lowest sincere CRs.
(5) Majority. This is a Condorcet method with "Smith/MinMax" cycle
resolution. If there is no Condorcet winner, consideration is limited to
the candidates who win the most two-way contests, and the one who
suffers the least-worst defeat against any other candidate is selected.
(6) Borda. Voter's preference rankings are converted into ratings. Each
candidate's rating is the number of lower-ranked candidates plus
one-half the number of equally-ranked candidates.
(7) IRV. Equivalent to Plurality if the Plurality winner has a majority.
Otherwise, the Plurality loser is eliminated and the process is repeated
with remaining rankings. Voters may rank candidates equal. (If a voter
has multiple first choices, their vote is split and fractional votes are
accrued to the first choice candidates.) If there is a tie for the
Plurality loser, all losers are eliminated unless all candidates are tied.
(8) Top2Runoff. All candidates except the two Plurality front-runners
are eliminated, and the Plurality vote count is repeated (but just
once). As with IRV, voters may rank candidates equal. If there is an
initial tie for the Plurality winner, all but the first-place candidates
are eliminated. If there is one winner, but a tie for second place, all
second-place candidates are retained in the runoff.
(9) Plurality. The candidate who gets the most first-place rankings
wins. If a voter is indifferent between their top-rated candidates,
their vote is distributed fractionally as in IRV and Top2Runoff.
(10) RandV. "Random voter": Pick a single voter by random selection and
let them decide the election.
(11) RandC. "Random candidate": Pick the winner by random selection.
In all cases simulated ballots are generated from assumed sincere CR
profiles, which are randomly-generated. The performance statistics for
each test case are based on one million simulated elections.
Ties are NOT resolved by random choice. Tied front-runners are just
registered as having equal probability of winning, and the winning
candidate's CR is averaged over tied winners.
Trial runs are tabulated below. Some of the output information includes
matrices containing pairwise comparison data between all methods, but
for brevity I've only included the first column (comparison to SiCR).
A few observations on the results:
Ignoring the uninteresting 2-candidate case, it appears that IRV and
Top2Runoff generally have inferior performance, comparable to Plurality
in many cases.
Borda performed surprising well. I think this is because Borda, in
effect, tries to reconstruct a CR profile from preference rankings, and
thus tends to perform somewhat like CR. However, the simulations are
only for "sincere" Borda, and in practice voting strategy may degrade
performance in the same way that CR performance is degraded when people
cast polarized ballots (effectively equivalent to StAV). However, there
is a catch: With Borda, you can't adopt the same strategy without
incurring a loss of voting power. For example, suppose candidates are
rated on a scale of 0...5. There are 4 candidates, C1 ... C4 and your
corresponding sincere CRs are
C1(0), C2(1), C3(2), C4(3), C5(4), C6(5)
Your optimum CR strategy is polarize your ratings,
C1(0), C2(0), C3(0), C4(5), C5(5), C6(5)
With Borda, the sincere ranking translates to a Borda count identical to
the above sincere CR rating,
C1 < C2 < C3 < C4 < C5 < C6 --> C1(0), C2(1), C3(2), C4(3), C5(4),
C6(5)
But if you try to use rank equality to mimic CR strategy, it doesn't
work quite the same,
C1 = C2 = C3 < C4 = C5 = C6 --> C1(1), C2(1), C3(1), C4(4), C5(4),
C6(4)
The strategy reduces your rating range from 0-5 to 1-4. Does this mean
Borda is somewhat less susceptible to strategy than CR? (Or is there a
possible way to make CR more strategy-resistant by effectively
diminishing the voting power of polarized ballots?)
The methods' performance may be diminished by strategy, which I have
ignored except in the case of StCR and StAV - and these only represent
the most trivial strategy. Generally, voting strategy is probably only
beneficial when one faction adopts the strategy; when everyone does,
everyone is worse off (on average). However, AV appears to be an
exception to the rule. If there are more than two candidates, strategy
appears to marginally improve, or at least not degrade, the performance
of AV.
Ken Johnson
********** 2 CANDIDATES **********
num_voter=10
num_candidate=2
num_election=1000000
Proportion of voting population that is, on average, adequately
represented by winning candidate:
SiCR 0.55177
StCR 0.54116
SiAV 0.54421
StAV 0.54116
Majority 0.54116
Borda 0.54116
IRV 0.54116
Top2Runoff 0.54116
Plurality 0.54116
RandV 0.51687
RandC 0.50013
Proportion of elections in which 1st method (column) chooses a candidate
with higher CR than 2nd method (row):
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.33117
SiAV 0.2626
StAV 0.33117
Majority 0.33117
Borda 0.33117
IRV 0.33117
Top2Runoff 0.33117
Plurality 0.33117
RandV 0.39471
RandC 0.5
Mean difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.010604
SiAV 0.0075624
StAV 0.010604
Majority 0.010604
Borda 0.010604
IRV 0.010604
Top2Runoff 0.010604
Plurality 0.010604
RandV 0.034901
RandC 0.051642
Max difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.2502
SiAV 0.23856
StAV 0.2502
Majority 0.2502
Borda 0.2502
IRV 0.2502
Top2Runoff 0.2502
Plurality 0.2502
RandV 0.53138
RandC 0.61499
********** 3 CANDIDATES **********
num_voter=10
num_candidate=3
num_election=1000000
Proportion of voting population that is, on average, adequately
represented by winning candidate:
SiCR 0.57735
StCR 0.57085
SiAV 0.56592
StAV 0.56707
Majority 0.56533
Borda 0.5669
IRV 0.56045
Top2Runoff 0.56045
Plurality 0.55964
RandV 0.52529
RandC 0.50012
Proportion of elections in which 1st method (column) chooses a candidate
with higher CR than 2nd method (row):
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.19118
SiAV 0.36655
StAV 0.33917
Majority 0.34992
Borda 0.29609
IRV 0.41063
Top2Runoff 0.41063
Plurality 0.39371
RandV 0.54293
RandC 0.66652
Mean difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.0064936
SiAV 0.011427
StAV 0.010277
Majority 0.012019
Borda 0.010449
IRV 0.016893
Top2Runoff 0.016893
Plurality 0.017705
RandV 0.052054
RandC 0.077228
Max difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.23964
SiAV 0.28243
StAV 0.23987
Majority 0.25643
Borda 0.25706
IRV 0.33035
Top2Runoff 0.33035
Plurality 0.33035
RandV 0.53913
RandC 0.61339
********** 100 VOTERS **********
num_voter=100
num_candidate=3
num_election=1000000
Proportion of voting population that is, on average, adequately
represented by winning candidate:
SiCR 0.52438
StCR 0.52225
SiAV 0.52106
StAV 0.52129
Majority 0.52044
Borda 0.52109
IRV 0.51961
Top2Runoff 0.51961
Plurality 0.51844
RandV 0.5025
RandC 0.5
Proportion of elections in which 1st method (column) chooses a candidate
with higher CR than 2nd method (row):
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.19589
SiAV 0.28094
StAV 0.26823
Majority 0.30808
Borda 0.26127
IRV 0.33099
Top2Runoff 0.33099
Plurality 0.35381
RandV 0.62926
RandC 0.66652
Mean difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.0021298
SiAV 0.0033157
StAV 0.0030913
Majority 0.0039399
Borda 0.0032861
IRV 0.0047659
Top2Runoff 0.0047659
Plurality 0.0059363
RandV 0.021879
RandC 0.024385
Max difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.068893
SiAV 0.094898
StAV 0.08237
Majority 0.10163
Borda 0.085905
IRV 0.11299
Top2Runoff 0.11299
Plurality 0.11745
RandV 0.19549
RandC 0.19808
********** 10 CANDIDATES **********
num_voter=100
num_candidate=10
num_election=1000000
Proportion of voting population that is, on average, adequately
represented by winning candidate:
SiCR 0.54404
StCR 0.54362
SiAV 0.5379
StAV 0.53823
Majority 0.53994
Borda 0.54195
IRV 0.53369
Top2Runoff 0.53029
Plurality 0.52325
RandV 0.5041
RandC 0.5
Proportion of elections in which 1st method (column) chooses a candidate
with higher CR than 2nd method (row):
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.11569
SiAV 0.46372
StAV 0.4538
Majority 0.37123
Borda 0.26244
IRV 0.54667
Top2Runoff 0.60739
Plurality 0.73391
RandV 0.87391
RandC 0.9
Mean difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.00041211
SiAV 0.0061362
StAV 0.0058015
Majority 0.004095
Borda 0.0020863
IRV 0.010348
Top2Runoff 0.013742
Plurality 0.020791
RandV 0.039941
RandC 0.044033
Max difference between CRs of 1st and 2nd methods' chosen candidates:
SiCR ... (columns 2-11 truncated)
SiCR 0
StCR 0.024509
SiAV 0.09694
StAV 0.087157
Majority 0.073927
Borda 0.052216
IRV 0.12583
Top2Runoff 0.16532
Plurality 0.17002
RandV 0.20673
RandC 0.2083
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