[Election-Methods] D(n)MAC

[fsimmons at pcc.edu]

Dear Jobst (and other open minded EM list participants),

Consider the case of two factions

P: A>C>B
Q: B>C>A,

where P>Q>0 and P+Q=100%.

Also suppose that there is a percentage R between 50% and 100%, such that
all voters in the first faction prefer C to the lottery
  R*A+(100%-R)*B, 
and all voters in the second faction prefer C to the lottery
  R*B+(100%-R)*A.

[Range voters can assume that sincere ratings for C are at R or above on all
ballots.]

It turns out that if the exponent "n" in the following formula is chosen so that

P+Q*P^(n-1) is less than or equal to R,

then the lottery method D(n)MAC that generalizes Jobst's  D2MAC method
has a stable equilibrium in which C is the sure winner.

Here's what I mean by D(n)MAC:

1. Ballots are approval style with favorites marked.

2. Draw n ballots at random (with replacement, if the ballot set is small).

3. If there is at least one candidate that is approved on all of the drawn
ballots, then (among those) elect the one that is approved on the most ballots
in the total collection of ballots.

4.  Otherwise, elect the favorite candidate on another randomly drawn ballot.

Example:

51%  A>C>B
49%  B>C>A

with R(C)=52%.

Since  .51+..49*51^7<.52, the method D7MAC has a stable equilibrium in which C
is the sure winner.

Note also that if P=Q=50%, then the relation simplifies to  1/2^n+1/2 < R .

So for example, if we cannot be certain which of the two factions is larger, 
then for R > 62.5%, candidate C is a stable D3MAC winner.

As Always,

Forest