[EM] Monty Hall multiple doors problem as choice transfer conservation

[Richard Lung]

Multiple doors problem as Choice Transfer conservation

The Monty Hall problem deserves closer analysis as a model for elections.

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?

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.
Erdos was perhaps reconciled by seeing a computer simulation of Monty 
Hall results.

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.

/3-door problem/
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.
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.

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.

/1 and 2 door problems/
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.
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.
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.

/4-door problem/
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.

/Binomial count conservation law of election transfer/.
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.

/Election transfer proportional representation/.
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.
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.
Thus, a statistical demonstration of transferable voting for 
proportional representation, as personal representation, according to 
Carl Andrae, Thomas Hare and John Stuart Mill.

/Wright Hill live action STV/
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.
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.

/Probability Quota fluctuations for keep values/

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.
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).

Regards,

Richard Lung.


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