[EM] Hay Voting

[Forest W Simmons]

Here's a slightly different approach to Hay Voting:

Suppose that a typical voter votes the range ballot v=[x1, x2, x3].

The ballot adds to the respective virtual accounts of the three 
candidates amounts of

    x1/r, x2/r, and x3/r ,

where r is the L_2 norm of the vector v.


A dart board is colored with three colors with areas in proportion to 
the candidate account totals.

A dart is thrown repeatedly  from a great distance until it lands in 
one of the three colors, etc.

Here's why I think this is close to the right thing to do:

For any given voter, the marginal utility expectation due to his own 
ballot is approximately

the dot product of his utility vector [u1, u2, u3] with the unit vector
       
                 [x1/r,x2/r,x3/r],

which product is maximal when this unit vector is a scalar multiple of 
the utility vector.

Here's an equivalent way of looking at it:

We have a drawing in which the likelihood of v being drawn is 
proportional to  w=(x1+x2+x3)/r .  Once the vector v has been chosen, 
the lottery [x1, x2, x3]/(x1+x2+x3) is used to pick one of the three 
candidates.

The expected utility given that the vector was drawn is

q=(x1u1+x2u2+x3u3)/(x1+x2+x3).

The product wq is the above mentioned dot product.

However this is not exactly proportional to the marginal expection, 
because the exact expected utility woud be

(WQ+wq)/(W+w)

where the uppercase letters represent the values contributed by the 
other ballots.

So the contribution of this one ballot is proportional to

wq/(W+w).

It's that pesky w in the denominator that makes wq not exactly 
proportional to the marginal expeced utility.

This is a big problem when W is small.

But there should be a way of fixing this problem.

Forest