<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">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">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">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">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">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">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">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">top-two runoffs</a>,
<a href="http://en.wikipedia.org/wiki/Instant-runoff_voting" rel="noreferrer noreferrer noreferrer
noreferrer noreferrer" target="_blank">instant runoff voting</a>,
all variations of <a href="http://en.wikipedia.org/wiki/Condorcet_method" rel="noreferrer noreferrer noreferrer
noreferrer noreferrer" target="_blank">Condorcet's method</a>,
even <a href="http://en.wikipedia.org/wiki/Bucklin_voting" rel="noreferrer noreferrer noreferrer
noreferrer noreferrer" target="_blank">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>
</div>
----<br>
Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer
noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">https://electorama.com/em</a>
for list info<br>
</blockquote>
</div>
</blockquote>
</div>
</blockquote>
</div>
<br>
<fieldset></fieldset>
<pre>----
Election-Methods mailing list - see <a href="https://electorama.com/em" target="_blank" rel="noreferrer">https://electorama.com/em</a> for list info
</pre>
</blockquote>
<br>
</div>
----<br>
Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer noreferrer" target="_blank">https://electorama.com/em</a> for list info<br>
</blockquote></div>