# [EM] Discounting ties, how can MinMax differ from Ranked Pairs?

robert bristow-johnson rbj at audioimagination.com
Tue Jun 11 21:46:22 PDT 2019

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Let's assume Margins, but I think this works for Winning Votes also.
Begin with all candidates marked as "Plausible Winner" (which means they are not yet marked as "beaten") and order all N⋅(N-1) candidate Pairwise Defeats from strongest defeat strength to weakest defeat strength.  Call that ordered list the "Original List".
Now create another ordered list of Pairwise Defeats from the above Original List.

MinMax starts with
the entire Original List with N⋅(N-1) entries and successively eliminates entries from the bottom up.
Referring to a quote in this Wikipedia article: https://en.wikipedia.org/wiki/Minimax_Condorcet_method#Variants_of_the_pairwise_score , MinMax starts at the bottom and:
"Disregards the weakest Pairwise Defeat until one candidate is unbeaten."

Similarly Ranked Pairs starts with an empty list and successively includes entries from the Original List from the top down.
Cannot Ranked Pairs be concisely stated as:
"Include the strongest Pairwise Defeat until only one candidate is unbeaten."   ?

Is that not an accurate description of both
methods?
Now, a decreasing sequence of numbers (*not* strictly decreasing) of length N⋅(N-1)+1 that represents the number of "unbeaten" candidates (of all Pairwise Defeats above it) having
value of N at the top, that can be thought of as having each of the N⋅(N-1) Pairwise Defeats inserted between these numbers of decreasing integer value.
Assuming no ties, the value of this sequence at the bottom
must be either 0 or 1.  Also, assuming no ties, this decreasing sequence can only be decremented by 1 or 0.
If the value of this sequence is 1 at the bottom, there is a Condorcet Winner and all Pairwise Defeats are considered.  But regardless of whether MinMax or RP is used, this
ordered list is the same decreasing sequence of integer values.  If the value of this sequence of numbers is 0 at the bottom, there is no Condorcet Winner.  But, for all Pairwise Defeats having "1" below them, would not the undefeated candidate be the same
candidate?
This might look like:

Plausible Winners ----- Pairwise Defeat
N
A>D

N-1

A>C

N-2

B>D

N-2

B>A

N-3

B>C

N-3

C>D

N-3

Now if N=4 then the bottom N-3 is 1 and we have a Condorcet Winner.
Now suppose we have a cycle:
Plausible Winners ----- Pairwise Defeat
N

A>D

N-1

A>B

N-2

B>D

N-2

C>D

N-2

B>C

N-3

C>A

N-4

Now if N=4 then the bottom N-4 is 0 and we don't have a Condorcet Winner.  But the unbeaten candidate is A whether we include the B>C Pairwise Defeats or not.  Assuming no ties, how can the unbeaten candidate be different?
You guys may have discussed
this before when I wasn't paying attention, but it seems to me that if there are no ties and all of the Defeat strengths are unequal values, the ordering of this list must be the same and the number of Plausible Winners must decrease from N at the top to 1 at the bottom if there is a CW (or N to 0
if no CW).  How can MinMax and Ranked Pairs elect a different candidate?
Thank you for any attention and thought put to this.
--

r b-j                         rbj at audioimagination.com

"Imagination is more important than knowledge."

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