Eye Ball Mathematics

[Steve Eppley]

Donald asked:
-snip-
>"If a full majority of all the voters indicate that they would
>rather have B than A, then if we choose A or B, it should be B."
>
>My question is: How do we know which candidate is the indicated
>majority candidate?
>             46ABC
>             20B
>             34CBA
>In Steve's example I can eye ball the example and I can tell in my
>mind that B has a majority of the choices within the first two
>selections - but with a larger example I would not be able to know
>by eye balling - I will need some mathematics. 

I'll trust that your question is serious.

Counting "selections" is unrelated to the principle at the top.
Assuming you want to adhere to that principle, what you should be
eye-balling is the relative position of pairs of candidates.  

For example, if some voter voted 
   1: IJKADFGMRBXZYCQ 
you can still eyeball that this voter prefers A more than B.  
Try it yourself: seek 'A' or 'B' in the ballot, scanning from left to
right.  Which do you encounter first?  (You'd do a similar scan for
each pair of candidates.)

In the 45/20/34 example, in 54 of those ballots B is ranked ahead of A, 
and in 46 of them A is ranked ahead of B.  You can easily eyeball the 
other two pairings (A vs C and B vs C) too.

B is the only candidate who doesn't violate the principle: in a 
choice between A and B, a majority would pick B.  In a choice between 
B and C, a majority would pick B.

Even in a circular tie such as
   46:A                            46:A
   10:BA              or           20:B
   10:BC                           34:CBA
   34:CBA
there may still be one or more candidates whose election wouldn't 
violate the principle.
   A<B  46<54                      A<B  46<54
   B<C  20<34         or           B<C  20<34
   C<A  44<56                      C<A  34<46
Here B's only pair-loss (to C) is 34 to 20.  Since 34 isn't a
majority, electing B doesn't violate the principle at all: in a 
choice between B and C, neither would be preferred by a majority.

The only case where it is necessary to violate the principle at all
is if *every* candidate loses some pairing with a majority against. 
In that unhappy (and hopefully rare, or the society is in trouble)
case, elect the candidate which minimizes the violation.  In other 
words, elect the one with the smallest largest pairloss.

Instant Runoff can grossly violate the principle, if you don't ignore 
the preferences of some of the voters:
   35:ABC
   16:BA
   16:BC
   33:CBA
Instant Runoff elects A, even though B would beat A in a landslide 
(65 to 35).  B would beat C in an even bigger landslide (67 to 33).
Feel free to tweak the second and third choices in these ballots a 
bit to make them appear more realistic, not lockstep.  It takes more
than tweaks to reduce those landslides significantly. 

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)