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<font face="Helvetica, Arial, sans-serif">Forest - I read fpdk's
post as an implicit argument for cardinal voting (which was why it
was relevant to STAR). Each friend states the utility of each
topping to himself or herself: which topping do they choose
collectively? And the answer is the one whose sum of utilities is
greatest. I don't think there's a better answer.<br>
CJC<br>
</font><br>
<div class="moz-cite-prefix">On 18/08/2023 21:50, Forest Simmons
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CANUDvfrJJnzh67qG8s-By+D=MmhqiX0of23guuijpTRphqLcGQ@mail.gmail.com">
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<div dir="auto">He posed a pizza choice among friends pronlem.. a
problem of consensus as opposed to "tyranny of the majority" ...
how to find the best consensus decision when a simple majority
first place preference would not be ideal.</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Fri, Aug 18, 2023, 10:41 AM
Colin Champion <<a
href="mailto:colin.champion@routemaster.app"
moz-do-not-send="true">colin.champion@routemaster.app</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div> <font face="Helvetica, Arial, sans-serif">Forest – I
may be being slow, but... what problem are you trying to
solve? The problem which fpdk (quite plausibly, to my
mind) said was optimally solved by cardinal voting? Or the
problem which I claimed was optimally solved by decision
theory? Or something to do with tactical voting?<br>
CJC<br>
</font><br>
<div>On 18/08/2023 18:32, Forest Simmons wrote:<br>
</div>
<blockquote type="cite">
<div dir="auto">It's been a while since I thought about
this but here's something that somebody with some number
crunching resources should experiment with ... a lottery
method that I used to call "the ultimate lottery" back
before Jobst invented MaxParC, which arguably has at
least an equal claim to ultimateness:
<div dir="auto"><br>
</div>
<div dir="auto">Ballots are positive homogeneous
functions of the candidate probability variables. The
homogeneity degree doesn't matter as long as all of
the ballots are of the same degree.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The candidate probabilities are chosen
to maximize the product of the ballots.</div>
<div dir="auto"><br>
</div>
<div dir="auto">This candidate probability distribution
can be realized as a spinner. The spinner is spun to
determine the winner.</div>
<div dir="auto"><br>
</div>
<div dir="auto">How would this work for our pizza
example?</div>
<div dir="auto"><br>
</div>
<div dir="auto">For example, each voter's ballot could
be her pizza desirability [score] expectation as a
function of the lottery probabilities.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Then each A faction voter would submit
the same ballot ... namely the function given by the
expression</div>
<div dir="auto">100pA+80pC, while each B faction voter
would submit the expression</div>
<div dir="auto">100pB+80pC.</div>
<div dir="auto"><br>
</div>
<div dir="auto">When these ballots are multiplied
together, we get the product</div>
<div dir="auto">(<span style="font-family:sans-serif">100pA+80pC)^60×(</span><span
style="font-family:sans-serif">100pB+80pC)^40.</span></div>
<div dir="auto"><span style="font-family:sans-serif"><br>
</span></div>
<div dir="auto"><span style="font-family:sans-serif">The
p values that maximize this product (subject to the
constraint that they are non-negative and sum to 100
percent) are pA=pB=0, and pC=100%.</span></div>
<div dir="auto"><span style="font-family:sans-serif"><br>
</span></div>
<div dir="auto"><span style="font-family:sans-serif">The
lottery that maximizes the expectation product is
called the Nash lottery after John Nash who first
used this idea for efficient allocation of limited
resources.</span></div>
<div dir="auto"><span style="font-family:sans-serif"><br>
</span></div>
<div dir="auto"><font face="sans-serif">Since
expectations are linear combinations of the
probabilities, they are homogeneous of degree one
... one person, on vote. Their product is
homogeneous of degree n ... so n people, n votes.</font></div>
<div dir="auto"><font face="sans-serif"><br>
</font></div>
<div dir="auto"><font face="sans-serif">Instead of using
voter expectations for their ballots, the voters
could have used other homogeneous expressions ...
for example, by simply replacing </font><span
style="font-family:sans-serif">each sum of products
by a max of the same products.</span></div>
<div dir="auto"><font face="sans-serif"><br>
</font></div>
<div dir="auto"><font face="sans-serif">The product of
these modified ballots would be ...</font></div>
<div dir="auto"><font face="sans-serif"><br>
</font></div>
<div dir="auto"><span style="font-family:sans-serif">[max(100pA,80pC)]^60</span></div>
<div dir="auto"><span style="font-family:sans-serif">×[max(</span><span
style="font-family:sans-serif">100pB,80pC)]^40.</span><font
face="sans-serif"><br>
</font></div>
<div dir="auto"><br>
</div>
<div dir="auto">Maximization of this product with the
same constraints as before, yields the same consensus
distribution ... pC=100%.</div>
<div dir="auto"><br>
</div>
<div dir="auto">This information is new in the sense
that it has never been submitted for official
publication ... it's an exclusive bonus of Rob
Lanphier's EM list archive... first posted to this
list back in 2011 after Jobst and I published our 2010
paper on the use of mixed strategies for achieving
consensus.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Anyway, it turns out that using the Max
operator in place of the Sum operator yields a
distribution with less entropy whenever the two
distributions are not identical.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Less entropy means less randomness,
which means less chance, which in this context, means
more consensus.</div>
<div dir="auto"><br>
</div>
<div dir="auto">In our example, the candidate
distribution turned out to be 100 percent candidate C
... zero randomness ... zero entropy ... 100 percent
consensus.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Now you can see why I mentioned the need
for number crunching capability ... experimenting with
these ballot product maximizations requires some
serious number crunching.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The field is wide open. Is the Ultimate
Lottery Method strongly monotonic? For that matter,
how about even the Nash Lottery?</div>
<div dir="auto"><br>
</div>
<div dir="auto">Can MaxParC be formulated in terms of
the Ultimate Lottery?</div>
<div dir="auto"><br>
</div>
<div dir="auto">Somebody with some grad students should
get them going on this!</div>
<div dir="auto"><br>
</div>
<div dir="auto">fws</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Thu, Aug 17, 2023,
11:10 AM Forest Simmons <<a
href="mailto:forest.simmons21@gmail.com"
target="_blank" rel="noreferrer"
moz-do-not-send="true">forest.simmons21@gmail.com</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="auto">Suppose voter utilities for three
kinds of pizza are
<div dir="auto"><br>
</div>
<div dir="auto">60 A[100]>C[80]>>B[0]</div>
<div dir="auto">40 B[100]>C[80]>>A[0]</div>
<div dir="auto"><br>
</div>
<div dir="auto">Suppose the voters must choose by
majority choice between pizza C and the favorite
pizza of a voter to be determined by randomly
drawing a voter name from a hat.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The random drawing method would give
voter utility expectations of</div>
<div dir="auto"><br>
</div>
<div dir="auto">60%100+40%0 for each A groupie, and</div>
<div dir="auto">40%100+60%0 for each B groupie.</div>
<div dir="auto"><br>
</div>
<div dir="auto">The max utility expectation would be
60.</div>
<div dir="auto"><br>
</div>
<div dir="auto">On the other hand, if voters decide
to go with the sure deal C, the assured utility fo
every voter will be 80.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Every rational voter faced with this
choice will choose C.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Here we have an ostensibly random
method that is sure to yield a consensus decision
when voters vote ratkonally.</div>
<div dir="auto"><br>
</div>
<div dir="auto">More on this topic at</div>
<div dir="auto"><br>
</div>
<div dir="auto"><a
href="https://www.researchgate.net/figure/Properties-of-common-group-decision-methods-Nash-Lottery-and-MaxParC-Solid-and-dashed_fig3_342120971"
style="font-family:sans-serif" rel="noreferrer
noreferrer" target="_blank"
moz-do-not-send="true">https://www.researchgate.net/figure/Properties-of-common-group-decision-methods-Nash-Lottery-and-MaxParC-Solid-and-dashed_fig3_342120971</a><span
style="font-family:sans-serif"> </span><br>
</div>
<div dir="auto"><span style="font-family:sans-serif"><br>
</span></div>
<div dir="auto"><span style="font-family:sans-serif">fws</span></div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Thu, Aug 17,
2023, 1:18 AM Forest Simmons <<a
href="mailto:forest.simmons21@gmail.com"
rel="noreferrer noreferrer noreferrer"
target="_blank" moz-do-not-send="true">forest.simmons21@gmail.com</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="auto">The best methods that I know of
for the friends context are minimum entropy
lottery methods characterized by max possible
consensus (min entropy) consistent with a
proportional lottery method with higher entropy
fallback to disincentivize gratuitous
defection.
<div dir="auto"><br>
</div>
<div dir="auto">Jobst's MaxParC (Max Partial
Consensus) is the best example.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Too late to elaborate tonight.</div>
<div dir="auto"><br>
</div>
<div dir="auto">fws<br>
<div dir="auto"><br>
</div>
<div dir="auto">I'll </div>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Wed, Aug
16, 2023, 10:01 AM <<a
href="mailto:fdpk69p6uq@snkmail.com"
rel="noreferrer noreferrer noreferrer
noreferrer" target="_blank"
moz-do-not-send="true">fdpk69p6uq@snkmail.com</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0
0 0 .8ex;border-left:1px #ccc
solid;padding-left:1ex">
<div dir="ltr"><br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Mon,
Aug 14, 2023 at 12:09 AM C.Benham wrote:<br>
</div>
<blockquote class="gmail_quote"
style="margin:0px 0px 0px
0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex">
> I think this is an interesting
point. We can ask at a philosophical
level what makes a good voting method.
Is it just one that ticks the most
boxes, or is it one that most reliably
gets the "best" result?<br>
</blockquote>
<div><br>
</div>
<div>The one that most reliably gets the
best result in the real world. The
difficulty with this approach is
accurately modeling human voting
behavior and the consequent utility
experienced from the winner, but it's
still the better answer philosophically.</div>
<div><br>
</div>
<div>(Note that VSE predates Jameson Quinn
by decades, and has had several
different names: <a
href="https://en.wikipedia.org/wiki/Social_utility_efficiency"
rel="noreferrer noreferrer noreferrer
noreferrer noreferrer" target="_blank"
moz-do-not-send="true">https://en.wikipedia.org/wiki/Social_utility_efficiency</a>)</div>
<div><br>
</div>
<blockquote class="gmail_quote"
style="margin:0px 0px 0px
0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex"> >
And that's partly because the premise of
Condorcet is essentially built on a
logical fallacy - basically that if A is
preferred to B on more ballots that vice
versa then electing A must<br>
> be a better result than electing B.<br>
<br>
I'd be interested in reading your
explanation of why you think that is a <br>
"logical fallacy". What about if there
are only two candidates?<br>
</blockquote>
<div><br>
</div>
<div>Ranked ballots can't capture strength
of preference. It's possible for a
majority-preferred candidate to be very
polarizing (loved by 51% and hated by
49%), while the minority-preferred
candidate is broadly-liked and has a
much higher overall
approval/favorability rating. Which
candidate is the rightful winner?<br>
</div>
<div><br>
</div>
<div><a
href="https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html"
rel="noreferrer noreferrer noreferrer
noreferrer noreferrer" target="_blank"
moz-do-not-send="true">https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html</a></div>
<div><br>
</div>
<div>"Suppose you and a pair of friends
are looking to order a pizza. You, and
one friend, really like mushrooms, and
prefer them over all other vegetable
options, but you both also really, <i>really</i>
like pepperoni. Your other friend also
really likes mushrooms, and prefers them
over all other options, but they're also
vegetarian. What one topping should you
get?
<p>Clearly the answer is mushrooms, and
there is no group of friends worth
calling themselves such who would
conclude otherwise. It's so obvious
that it hardly seems worth calling
attention to. So why is it, that if we
put this decision up to a vote, do so
many election methods, which are
otherwise seen as perfectly reasonable
methods, fail? Plurality, <a
href="http://en.wikipedia.org/wiki/Two-round_system"
rel="noreferrer noreferrer
noreferrer noreferrer noreferrer"
target="_blank"
moz-do-not-send="true">top-two
runoffs</a>, <a
href="http://en.wikipedia.org/wiki/Instant-runoff_voting"
rel="noreferrer noreferrer
noreferrer noreferrer noreferrer"
target="_blank"
moz-do-not-send="true">instant
runoff voting</a>, all variations of
<a
href="http://en.wikipedia.org/wiki/Condorcet_method"
rel="noreferrer noreferrer
noreferrer noreferrer noreferrer"
target="_blank"
moz-do-not-send="true">Condorcet's
method</a>, even <a
href="http://en.wikipedia.org/wiki/Bucklin_voting"
rel="noreferrer noreferrer
noreferrer noreferrer noreferrer"
target="_blank"
moz-do-not-send="true">Bucklin
voting</a>; all of them,
incorrectly, choose pepperoni."</p>
</div>
<div>(And strength of preference is
clearly a real thing in our brains. If
you prefer A > B > C, and are
given the choice between Box 1, which
contains B, and Box 2, which has a 50/50
chance of containing A or C, which do
you choose? What if the probability
were 1 in a million of Box 2 containing
C? By varying the probability until
it's impossible to decide, you can
measure the relative strength of
preference for B > C vs A > C.)<br>
</div>
</div>
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