<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">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" target="_blank" rel="noreferrer">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" 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" 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"><div dir="ltr"></div><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" 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" 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" target="_blank">top-two runoffs</a>, <a href="http://en.wikipedia.org/wiki/Instant-runoff_voting" rel="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" target="_blank">Condorcet's method</a>, even <a href="http://en.wikipedia.org/wiki/Bucklin_voting" rel="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>
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