<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
<style type="text/css" style="display:none;"> P {margin-top:0;margin-bottom:0;} </style>
</head>
<body dir="ltr">
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
Not monotone?</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
44 A>B>C</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
28 C>A>B</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
28 B>C>A</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
An A>B>C>A cycle, but C has the least support, so C and, because C beats A pairwise, A as well, are eliminated so B wins. Yay, B!</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
Hold on, 17 A>B>C voters decide they like B better than A and become B>A>C voters.</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
27 A>B>C</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="font-size:12pt" class="ContentPasted0">28 C>A>B</span><br class="ContentPasted0">
<span style="font-size:12pt" class="ContentPasted0">28 B>C>A</span></div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
17 B>A>C</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
Now we have the same cycle, but A has the smallest MaxPairwiseSupport, so A and B are eliminated and C wins! The 17 voters made their new favorite lose.</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
Or I may be misunderstanding something.</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<br>
</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
Jim Faran<br>
</div>
<div id="appendonsend"></div>
<hr style="display:inline-block;width:98%" tabindex="-1">
<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Election-Methods <election-methods-bounces@lists.electorama.com> on behalf of Forest Simmons <forest.simmons21@gmail.com><br>
<b>Sent:</b> Sunday, July 30, 2023 8:36 PM<br>
<b>To:</b> Kristofer Munsterhjelm <km_elmet@t-online.de><br>
<b>Cc:</b> EM <election-methods@lists.electorama.com><br>
<b>Subject:</b> Re: [EM] One benefit to nonmonotone methods</font>
<div> </div>
</div>
<div>
<div dir="auto">Monotone Banks methods are not easy to come by ... but here is the simplest UD Monotone Banks method that I know of:
<div dir="auto"><br>
</div>
<div dir="auto">MaxPS Sorted Pairwise</div>
<div dir="auto"><br>
</div>
<div dir="auto">Initialize a list variable L as the list of candidates in order of their MaxPairwiseSupport.</div>
<div dir="auto"><br>
</div>
<div dir="auto">[Break ties by considering in turn 2nd, 3rd, ... levels of support.]</div>
<div dir="auto"><br>
</div>
<div dir="auto">While no member of L is pairwise undefeated, update L by removing its bottom member X as well as every candidate defeated by X.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Elect the remaining undefeated candidate highest on the list L.</div>
<div dir="auto"><br>
</div>
<div dir="auto">It seems to me that this procedure yields a very clean, monotone, clone-free, one pass, precinct summable, Banks efficient, UD, burial resistant method.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Are we over-looking anything?</div>
<div dir="auto"><br>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">fws</div>
</div>
<br>
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Sun, Jul 30, 2023, 6:21 AM Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de">km_elmet@t-online.de</a>> wrote:<br>
</div>
<blockquote class="x_gmail_quote" style="margin:0 0 0 .8ex; border-left:1px #ccc solid; padding-left:1ex">
Since it's pretty quiet at the moment, here's another observation. I've <br>
been testing some methods that pass DMT or DMTC, and I've found out two <br>
things:<br>
<br>
- The "monotonized" contingent vote, where if A is the Plurality winner <br>
and can give some of his first preferences to some other B to displace <br>
the other finalist C and get B into the top two instead, A's score <br>
becomes A>B instead of A>C, is not that much more strategy susceptible <br>
than the ordinary contingent vote (0.91 vs 0.87 for 97 candidates, 10k <br>
elections, 32k tries per election).<br>
<br>
- But it's much harder to get the true strategy resistance of <br>
nonmonotone methods, because coalitional strategy is much harder to find <br>
than two-sided "rank who you're compromising for top, the current winner <br>
bottom" strategy.<br>
<br>
So even if say, a method X and its monotone variant both have strategy <br>
resistance 0.8, it's often harder to execute strategy against the <br>
nonmonotone one in practice because you can overshoot.<br>
<br>
In a monotone method, if your honest vote is A>B>C>D>E, and the current <br>
winner is D, and you're compromising for C, then if A>C>B>D>E works, <br>
then C>A>B>D>E will also work and most likely C>A>B>E>D will also work. <br>
So you can slam your compromise to the top and your burial target (the <br>
current winner) to the bottom, and that's a pretty simple strategy.<br>
<br>
But in a nonmonotone method, it's possible that A>C>B>D>E will work but <br>
C>A>B>D>E won't. So even though the honest election is vulnerable to <br>
strategy with both methods, it's harder to find the correct strategy.<br>
<br>
Thus if you absolutely need all the strategy resistance you can get, <br>
nonmonotone is probably where it's at. I'll still try to find a good <br>
monotone burial-resistant method, though!<br>
<br>
Some stats to show this effect: impartial culture, 97 voters, 5 <br>
candidates, 50k elections, 32k coalitional tries per election:<br>
<br>
Smith,Contingent vote:<br>
Ties: 0.00612<br>
<br>
Burial without compromise: 0.11049<br>
Compromise without burial: 0.24123<br>
Burial and compromise: 0.00278<br>
Two-sided: 0.00254<br>
Other: 0.51588<br>
<br>
Total susceptibility: 0.87292<br>
<br>
Smith,Contingent vote with donation:<br>
Ties: 0.00904<br>
<br>
Burial without compromise: 0.13044<br>
Compromise without burial: 0.23402<br>
Burial and compromise: 0.01158<br>
Two-sided: 0.54180<br>
Other: 4e-05<br>
<br>
Total susceptibility: 0.91788<br>
<br>
The "Other" category (which contains pushover and pushover-like <br>
strategy) has been almost entirely emptied, and that strategy has become <br>
two-sided instead.<br>
<br>
(Two-sided is the fraction of elections where neither burial nor <br>
compromising works, but doing both at the same time works.)<br>
<br>
-km<br>
----<br>
Election-Methods mailing list - see <a href="https://electorama.com/em" originalsrc="https://electorama.com/em" shash="lcLwoPRQHYx8YxI9yD39ekDMB9uI5bSqt7TfAkDQc1nL9VhibMQ9NC4DI1NsSNeXWQUJ2qlqrfrfdC2B1x8LGxQswXg86XmQoETp6gMqEe1rsaCqUua0MDTipLnknXm/EnmTlgfE6x27BATRPKmFmwDH2O1n+v8PJloxasx5B64=" rel="noreferrer noreferrer" target="_blank">
https://electorama.com/em</a> for list info<br>
</blockquote>
</div>
</div>
</body>
</html>