[Election-Methods] Lotteries based on Approval Ballots with Favorites indicated

[fsimmons at pcc.edu]

Ballots indicate "approved" and "favorite" (of the approved).

A ballot is drawn at random.

Let X, Y, and Z , respectively, represent the ballot's "favorite" candidate, the
ballot's "approved" candidate with the most approval on the other ballots, and
the candidate (approved or not on this ballot) that is approved on the most
ballots, i.e. the "approval winner."

Let x, y, and z, be the respective approvals of the three candidates divided by
the total number of ballots.

Elect X, Y, or Z with probabilities in the proportion

F(x,y,z) : G(x,y,z) : H(x,y,z) .

[The respective candidates are treated as distinct for lottery purposes, even if
two or more of them are identical; e.g. it is likely that X = Y or Y = Z .]

Our design challenge is to find functions F, G, and H that will

(1) yield a monotone method
(2) encourage cooperation
(3) give reasonable results in test cases
(4) be relatively free from crowding or teaming effects, 

and perhaps

(5) give proportionality for factions that "bullet vote" for "favorite."

Here's a simple method that might serve as a benchmark:

F(x,y,z) = x,  G(x,y,z) = y, and H(x,y,z) = 0.

Then X and Y are elected with respective probabilities x/(x+y), and y/(x+y) .

In our test case

50 A>C>>B
50 B>C>>A,

as long as  C has utility greater than 62.5%, there is incentive to give C 100%
approval (if I'm not mistaken).

This results in a probability of 2/3 for the election of C, and the other two
candidates splitting the remaining probability.  In other words the winning
proportion for C, A, and B is  4 :1 : 1 .

This proportion is preserved relative to A, A1, and A2 in another test case:

33 A1>A>>A2=B
33 A2>A>>A1=B
33 B

The probabilities for A, A1, A2, and B are in the proportion

  4 : 1 : 1 : 3

In which of the five criteria is this method the most deficient?

What other suggestions do we have for F, G, and H ?

Several of our more recent methods fit into this framework, along with Approval
and Random Ballot.

Forest