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<p><font size="4"><br>
<br>
Multiple doors problem as Choice Transfer conservation<br>
<br>
The Monty Hall problem deserves closer analysis as a model for
elections.<br>
<br>
The generalised Monty Hall multiple doors choice problem
demonstrates by statistical probability, that a binomial count,
of election and exclusion, is required by a conservation law of
choice transfer, which promotes proportional representation as
personal representation. This does not just demonstrate the
statistics of election transfer PR, it demonstrates that
elections are a statistic. Whoever doubted it?<br>
<br>
It is tricky or counter-intuitive to a multitude of us,
including the second most productive mathematician in human
history (after Leonhard Euler). Paul Erdos was historys most
collaborative mathematician. Indeed he was mathematics greatest
ambassador, engaging with all people of any mathematical ability
whatsoever, in problem solving. <br>
Erdos was perhaps reconciled by seeing a computer simulation of
Monty Hall results.<br>
<br>
Different people have their own special ways of understanding
the problem. For me, the key is that more choice increases the
probability of (prize) success. This perspective is also a way
of generalising the Monty Hall problem. <br>
<br>
<i>3-door problem</i><br>
Looking at the arithmetic involved, you (the elector) get to
choose one of three doors, for a probability of success of one
third. This election is followed by (a compere like Monty Hall)
random opening a door, when turning-out not to be the prize,
effecting an exclusion of one of the doors not chosen.<br>
Election and exclusion are independent choices, which must not
agree, because a voters election vote cannot logically be the
same as his exclusion vote. Election and exclusion are
separately counted positive and negative choices, to balance the
books, for conservation of choice, like the conservation of
motion, or energy, in the cog-wheel train of a mechanical watch.<br>
<br>
The exclusion, of one door, leaves a further choice, between two
thirds of the original whole, of three doors. There is half a
chance of success, from said two thirds. One half of two thirds,
or, 1/2 x 2/3 = 1/3. But this is the probability of making a
second choice, by choosing the remaining unopened door. The
combined probability, of making a first choice and then a second
choice, is found by adding their respective probabilities, which
is one third plus one third equals two-thirds.<br>
<br>
<i>1 and 2 door problems</i><br>
The degenerate forms of the Monty Hall problem are the two door
and one door problems, respectively giving probabilities of a
half and zero or total improbability.<br>
In the one door case, there’s only a goat behind this door. Do
you want to open the door? There is zero election choice and
zero exclusion choice.<br>
In the two door case, the elector chooses one door, with one
half probability of prize success. The other door, if a failure
when opened, is excluded, leaving no other choice and no changed
probability of success.<br>
<br>
<i>4-door problem</i><br>
To generalise the Monty Hall problem, just add a door choice at
a time. The three door problem becomes a four-door problem. As
before, one (the elector) gets to choose a door, which is left
unopened. This time, the probability of success is one quarter.
One of the four doors, another one, is opened, and excluded, if
found a failed choice. Next, the elector gets to choose, to
stick with the original choice, or to choose a second of the
remaining three doors. Next, a second door is opened and
excluded, if found a failure. Finally, the elector has a third
choice to stick with the second choice or go back to the first
choice. w Three one-quarter probabilities add to a
three-quarters probability of success.<br>
<br>
<i>Binomial count conservation law of election transfer</i>.<br>
It is the change of choice, or choice transfer, which is
important. And each positive choice or election must be matched
by a negative choice or exclusion. So, what we appear to have
here is a conservation law of choice transfer, for a generalised
Monty Hall (multiple door) problem. This demonstrates that a
conservation law of choice transfer inter-depends on a binomial
count of election and exclusion, to promote proportional
representation, which is also personal representation.<br>
<br>
<i>Election transfer proportional representation</i>.<br>
The choices. of the contestant or elector and the compere or
excluder, could be reversed. As in Binomial STV, there is no
logical difference between preferences, as elections, and
reverse preferences, as exclusions.Thus, for one, two, three,
four, five, et cetera choices (positive or negative), there is a
corresponding series of probabilities: 0, 1/2, 2/3, 3/4, 4/5,
etc increasing probabilities of choice success, under a
conservation law of (positive or negative) choice transfer.
Increasing choice transfer creates probabilistic proportional
representation. <br>
Probabilistic PR amounts to one person running random trials of
door openings, or, many people, each running one of the
differing random trials. In either case, the sum of these trials
is a close enough approximation to exact proportional
representation, by the law of large numbers, in a sufficient
number of trials.<br>
Thus, a statistical demonstration of transferable voting for
proportional representation, as personal representation,
according to Carl Andrae, Thomas Hare and John Stuart Mill.<br>
</font></p>
<p><font size="4"><i>Wright Hill live action STV</i><br>
The multiple doors procedure could resemble the original live
action transferable vote, invented at the school of Thomas
Wright Hill two centuries ago, and videod of Irish school
children today. The children queue into quotas behind their
favorite candidates. Some candidates queues have more voters
than they need, and others have not enough voters, so surplus
and deficit votes transfer to a next prefered candidate, whose
queue still has chance of reaching the quota, and being elected,
too. <br>
This election transfer is like the Monty Hall dynamic: one
choice random trials to a probable quota, give or take a
fluctuation of support about the quota. There is usually a
fluctuation of the vote in surplus or deficit of the quota. The
fluctuation transfers quota surpluses and deficits, towards
electing a next prefered candidate.</font></p>
<p><font size="4"><i>Probability Quota fluctuations for keep values</i><br>
</font></p>
<p><font size="4">Knowing the probable proportional representation
for each number of door choices, also gives a (Monty Hall choice
success problem) quota equivalent, as the probability for voters
one, two, three choices etc, that is, respectively, one half,
one third, one quarter, etc. In the Monty Hall model, surplus
votes over the quota, and votes in deficit of the quota, are
related to random fluctuations about the probable quotas. <br>
STV measures these by Gregory method or keep values, which are
the quota, divided by each candidates total vote. The Monty Hall
problem equivalent, to the keep value, would be the probability
quota divided by the probability fluctuation (the probability
quota plus a surplus fluctuation or minus a deficit
fluctuation).<br>
</font></p>
<p><font size="4">Regards,</font></p>
<p><font size="4">Richard Lung.</font></p>
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