[EM] Detailed stats for the ordinal methods
Kristofer Munsterhjelm
km_elmet at t-online.de
Wed May 8 17:56:58 PDT 2024
I said I was going to either put up a link to or just post the detailed
manipulability stats for the methods. Due to current simulator
limitations, the per-category breakdown can be considerably off for
cardinal and approval methods (as well as their hybrids), so I won't
post those. (Namely, it may register burial or compromising as "other".)
But for the ordinal methods, the per-category information should be
correct. So here goes!
There are a few points to note: each run for a single method starts with
the RNG set to a fixed seed. This means that two methods that behave
identically will have the exact same result.
In addition, Ranked Pairs and the elimination methods have non-neutral
tiebreaks: they break ties in favor of earlier numbered candidates.
However, since the spatial generator doesn't care about the order it
generates its candidates, this is in effect the same thing as a random
ordering tiebreak.
These effects mostly explain why Smith//IRV and Schwartz//IRV have the
same results, and why the elimination methods (as well as RP) have zero
ties. The thing they don't explain is how the Smith and Schwartz sets
seem to coincide (to get the identical behavior), and I'm not sure why
that happens.
There are four strategy types:
- Burial: every voter in the strategic coalition tries to get the
candidate X they all prefer to the current winner W, to win, by ranking
W uniquely at bottom.
- Compromising: every voter in the coalition tries to get X to win by
raising X to the top, above every other candidate.
- Two-way: Both burial and compromising at once.
- Other: everything else.
Taking the report for Schulze as an example:
> 11: Schulze
> Ties: 0.001724 (862)
> Of the non-ties:
>
> Burial, no compromise: 124456 0.249342
> Compromise, no burial: 751 0.00150459
> Burial and compromise: 1003 0.00200946
> Two-sided: 39359 0.0788539
> Other coalition strats: 695 0.0013924
> ===============================================
> Manipulable elections: 166264 0.333102
>
> Worked in 166264 (0.3324, 0.3338) out of 499138 for Schulze(wv) ties: 862
This says that there were 862 elections where the honest outcome was a
tie, which thus were skipped. Of those that did not have a honest tie,
24.93% of the elections were vulnerable to burial but not compromising,
0.15% of the elections were vulnerable to compromising but not burial,
0.20% of the elections were vulnerable to both compromising and burial,
7.89% of the elections were vulnerable to two-sided strategy but
neither burial nor compromising on their own,
and
0.14% of the elections were immune to the above strategies but
vulnerable to something else.
In total that gives 33.31% of the non-tie elections being vulnerable to
some kind of strategy: or 166264 out of 499138.
The detailed stats are, in no particular order, and with non-poll
methods marked with an asterisk as before:
Using ballot domain Gaussian, sigma = 1
0: *BTR-IRV
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 63778 0.127556
Compromise, no burial: 2397 0.004794
Burial and compromise: 198 0.000396
Two-sided: 104608 0.209216
Other coalition strats: 34603 0.069206
==========================================
Manipulable elections: 205584 0.411168
Worked in 205584 (0.4105, 0.4119) out of 500000 for
BTREliminate-[ER-Plurality]/fd ties: 0 (0)
3: *Borda
Ties: 0.007576 (3788)
Of the non-ties:
Burial, no compromise: 149385 0.301051
Compromise, no burial: 33338 0.067185
Burial and compromise: 77412 0.156006
Two-sided: 84390 0.170068
Other coalition strats: 1766 0.00355896
==========================================
Manipulable elections: 346291 0.697869
Worked in 346291 (0.6972, 0.6985) out of 496212 for ER-Borda ties: 3788
4: Smith//IRV (Woodall)
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 15443 0.030886
Compromise, no burial: 2427 0.004854
Burial and compromise: 114 0.000228
Two-sided: 3262 0.006524
Other coalition strats: 16050 0.0321
==========================================
Manipulable elections: 37296 0.074592
Worked in 37296 (0.0742, 0.075) out of 500000 for [Smith]//[ER-IRV/fd]
ties: 0
5: Schwartz//IRV (Schwartz-Woodall)
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 15443 0.030886
Compromise, no burial: 2427 0.004854
Burial and compromise: 114 0.000228
Two-sided: 3262 0.006524
Other coalition strats: 16050 0.0321
==========================================
Manipulable elections: 37296 0.074592
Worked in 37296 (0.0742, 0.075) out of 500000 for
[Schwartz]//[ER-IRV/fd] ties: 0
6: Benham
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 15627 0.031254
Compromise, no burial: 2572 0.005144
Burial and compromise: 70 0.00014
Two-sided: 3062 0.006124
Other coalition strats: 15724 0.031448
==========================================
Manipulable elections: 37055 0.07411
Worked in 37055 (0.0737, 0.0745) out of 500000 for
Benham-Meta[ER-IRV/fd] ties: 0
7a: Copeland//Borda (Ranked Robin)
Ties: 0.000498 (249)
Of the non-ties:
Burial, no compromise: 181882 0.363945
Compromise, no burial: 449 0.000898447
Burial and compromise: 1823 0.00364782
Two-sided: 53215 0.106483
Other coalition strats: 2460 0.00492245
==========================================
Manipulable elections: 239829 0.479897
Worked in 239829 (0.4792, 0.4806) out of 499751 for
[Copeland]//[ER-Borda] ties: 249
7b: RCIPE
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 16077 0.032154
Compromise, no burial: 2456 0.004912
Burial and compromise: 282 0.000564
Two-sided: 2765 0.00553
Other coalition strats: 15571 0.031142
==========================================
Manipulable elections: 37151 0.074302
Worked in 37151 (0.0739, 0.0747) out of 500000 for Eliminate-[[Condorcet
non-loser],[ER-Plurality]]/fd ties: 0 (0)
8: Minmax
Ties: 0.00165 (825)
Of the non-ties:
Burial, no compromise: 123185 0.246777
Compromise, no burial: 731 0.00146442
Burial and compromise: 1000 0.00200331
Two-sided: 38597 0.0773216
Other coalition strats: 1971 0.00394852
==========================================
Manipulable elections: 165484 0.331515
Worked in 165484 (0.3309, 0.3322) out of 499175 for Minmax(wv) ties: 825
9: Plurality
Ties: 0.020924 (10462)
Of the non-ties:
Burial, no compromise: 0 0
Compromise, no burial: 204344 0.417422
Burial and compromise: 0 0
Two-sided: 0 0
Other coalition strats: 0 0
==========================================
Manipulable elections: 204344 0.417422
Worked in 204344 (0.4167, 0.4181) out of 489538 for ER-Plurality ties: 10462
10: IRV
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 0 0
Compromise, no burial: 21348 0.042696
Burial and compromise: 0 0
Two-sided: 0 0
Other coalition strats: 15789 0.031578
==========================================
Manipulable elections: 37137 0.074274
Worked in 37137 (0.0739, 0.0747) out of 500000 for ER-IRV/fd ties: 0
11: Schulze
Ties: 0.001724 (862)
Of the non-ties:
Burial, no compromise: 124456 0.249342
Compromise, no burial: 751 0.00150459
Burial and compromise: 1003 0.00200946
Two-sided: 39359 0.0788539
Other coalition strats: 695 0.0013924
==========================================
Manipulable elections: 166264 0.333102
Worked in 166264 (0.3324, 0.3338) out of 499138 for Schulze(wv) ties: 862
12: Baldwin
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 106015 0.21203
Compromise, no burial: 72 0.000144
Burial and compromise: 2527 0.005054
Two-sided: 11253 0.022506
Other coalition strats: 54895 0.10979
==========================================
Manipulable elections: 174762 0.349524
Worked in 174762 (0.3489, 0.3502) out of 500000 for ER-Baldwin/fd ties: 0
13: Condorcet//Borda (Black)
Ties: 0.000266 (133)
Of the non-ties:
Burial, no compromise: 176214 0.352522
Compromise, no burial: 2215 0.00443118
Burial and compromise: 195 0.000390104
Two-sided: 79258 0.158558
Other coalition strats: 14555 0.0291177
==========================================
Manipulable elections: 272437 0.545019
Worked in 272437 (0.5443, 0.5457) out of 499867 for
[Condorcet]//[ER-Borda] ties: 133
14: Raynaud
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 107385 0.21477
Compromise, no burial: 131 0.000262
Burial and compromise: 2576 0.005152
Two-sided: 10936 0.021872
Other coalition strats: 45583 0.091166
==========================================
Manipulable elections: 166611 0.333222
Worked in 166611 (0.3326, 0.3339) out of 500000 for
Eliminate-[Minmax(wv)]/fd ties: 0
15: *Smith//DSC
Ties: 0.001368 (684)
Of the non-ties:
Burial, no compromise: 141718 0.283824
Compromise, no burial: 1246 0.00249541
Burial and compromise: 674 0.00134985
Two-sided: 62796 0.125764
Other coalition strats: 1565 0.00313429
==========================================
Manipulable elections: 207999 0.416568
Worked in 207999 (0.4159, 0.4173) out of 499316 for [Smith]//[DSC] ties: 684
16: Ranked Pairs
Ties: 0 (0)
Of the non-ties:
Burial, no compromise: 128541 0.257082
Compromise, no burial: 759 0.001518
Burial and compromise: 1901 0.003802
Two-sided: 27418 0.054836
Other coalition strats: 1846 0.003692
==========================================
Manipulable elections: 160465 0.32093
Worked in 160465 (0.3203, 0.3216) out of 500000 for Ranked Pairs(wv) ties: 0
That should be right.
-km
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