<div><div dir="auto"><span style="color:rgb(49,49,49);font-family:'-apple-system','helvetica neue';font-size:16px;font-style:normal;font-weight:400;letter-spacing:normal;text-indent:0px;text-transform:none;white-space:normal;word-spacing:1px;text-decoration:none;display:inline!important;float:none">> I'm sorry. This "proof" cannot be true. It is certainly possible *not* to elect the Condorcet winner using either Approval or Score. This "proof" can be successfully refuted with a counter example. I posted one a few days ago regarding STAR (but it also says the same about Score and wouldn't be hard to modify to show that for Approval). A single counter-example is sufficient to disprove that conclusion that Approval and Score will (always) elect the Condorcet winner.</span></div><div dir="auto"><span style="color:rgb(49,49,49);font-family:'-apple-system','helvetica neue';font-size:16px;font-style:normal;font-weight:400;letter-spacing:normal;text-indent:0px;text-transform:none;white-space:normal;word-spacing:1px;text-decoration:none;display:inline!important;float:none"><br></span></div><div dir="auto"><font color="#313131" face="-apple-system, helvetica neue"><span style="word-spacing:1px">Like I said, this holds true if voters are strategic. They pick an approval threshold between the frontrunners (more strictly, set it at the expected value of the election result).</span></font></div><div dir="auto"><font color="#313131" face="-apple-system, helvetica neue"><span style="word-spacing:1px"><br></span></font></div><div dir="auto"><font color="#313131" face="-apple-system, helvetica neue"><span style="word-spacing:1px">This is the minimal amount of strategy possible: just say you like a candidate if you think they’re above-average. I’m not aware of any system that requires less strategizing to get the same result, except *maybe* MDD//Score.</span></font></div><div dir="auto"><font color="#313131" face="-apple-system, helvetica neue"><span style="word-spacing:1px"><br></span></font></div><div dir="auto"><font color="#313131" face="-apple-system, helvetica neue"><span style="word-spacing:1px">I don’t know if the result extends to STAR, but I suspect no, and that any use of pairwise comparisons breaks this property. Monroe (2000) showed that every well-known Condorcet method (including Minimax and all of its variants, like Schulze and RP, as well as Nanson-Baldwin) will fail to elect the Condorcet winner if voters are strategic.</span></font></div></div><div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Tue, May 21, 2024 at 1:30 PM robert bristow-johnson <<a href="mailto:rbj@audioimagination.com">rbj@audioimagination.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><br>
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> On 05/21/2024 3:49 PM EDT Closed Limelike Curves <<a href="mailto:closed.limelike.curves@gmail.com" target="_blank">closed.limelike.curves@gmail.com</a>> wrote:<br>
> <br>
> <br>
> This list started in 1996. I was born 3 years later, in 1999.<br>
> <br>
> Six years before I was born, and 3 years before this list was started, Myerson and Weber proved that approval (and score, highest medians, or ) will elect the Condorcet winner.<br>
<br>
I'm sorry. This "proof" cannot be true. It is certainly possible *not* to elect the Condorcet winner using either Approval or Score. This "proof" can be successfully refuted with a counter example. I posted one a few days ago regarding STAR (but it also says the same about Score and wouldn't be hard to modify to show that for Approval). A single counter-example is sufficient to disprove that conclusion that Approval and Score will (always) elect the Condorcet winner.<br>
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> This is true as long as voters use a strategy even a brain-dead turnip could work out: set your approval threshold between the frontrunners.<br>
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But what are frontrunners? How do you know who these candidates are under different ballot conditions? Do you mean FPTP frontrunners?<br>
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Lastly, I consider it disadvantageous to require any tactical thinking or to incentivize any strategic thinking from voters. Any more than what Arrow of Gibbard or Satterthwaite say we cannot get away from. This is why I am for strictly an ordinal ballot, where voters are not required to rank any candidate they don't want to rank, and where possible equal rankings are allowed (this is one reason I have sorta soured regarding BTR-IRV), and the Condorcet winner should always be elected whenever such candidate exists. I am still not committed about which Condorcet method, but, besides disincentivizing strategic or tactical voting, I am also concerned about the quality and conciseness of the legislative language to describe the method (that was the sole reason I had previously considered BTR-IRV).<br>
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r b-j . _ . _ . _ . _ <a href="mailto:rbj@audioimagination.com" target="_blank">rbj@audioimagination.com</a><br>
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"Imagination is more important than knowledge."<br>
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