[EM] Consensus Lottery Method

[fsimmons at pcc.edu]

Here's a rather elaborate method that seeks to establish as much consensus as possible, and to elicit 
from among those voters that do not benefit by the greatest consensus alternative another partial 
consensus alternative on which to spend their probability.

1.  Individual voters, parties, and other interested institutions nominate as many consensus lotteries as 
they want to.  These nominations are collected into a set C (for consensus).

2.  Voters submit or endorse ballots with three parts:
(i) The first part is the name of their favorite alternative.
(ii) The second part is a range style ballot.
(iii) The third part is a degree one homogeneous function of the alternatives into the non-negative reals, 
which is non-decreasing in each argument.  [I will explain what this means and what purpose it 
accomplishes later.]

3.  The product of all of the homogeneous functions from the ballots is formed to get a homogeneous 
function H whose degree is the number of factors in the product.

4.  Each member L of C is evaluated by H.  This is done by substituting into each argument of H the 
corresponding probability from L.  By a very slight abuse of notation the result will be denoted H(L).

5.  The member L of C for which H(L) is the greatest is the lottery that goes up against the benchmark 
lottery, i.e. the random favorite lottery.

6.  If no submitted range ballot prefers the random favorite lottery to L, then use L to elect the winner.  
Else use the random favorite lottery to elect the winner.

A function (x,y,z,...)-->f(x,y,z,...) is homogeneous of degree one iff for every real number t, the equation

          f(t*x,t*y,t*z,...)=t*f(x,y,z,...) holds.

This condition (via Euler's Theorem on homogeneous functions) ensures that every voter has the same 
voting power.

Some examples of degree one homogeneous functions are

(x,y,z)--> max(x,y,z)
(x,y,z)---> a*x+b*y+c*z
(x,y,z)---> min(x,y,z)
(x,y,z)---> x^a*y^b*z^c, if a+b+c=1.
(x,y,z)---> (x^n+y^n+z^n)^(1/n),
etc.

Basically, via these (and similar) functions the voters can specify how they want their probability 
distributed contingent on the probabilities that would hold if their ballot were not in the mix.

I'm out of time.  More tomorrow.