[EM] Lottery Methods

[Forest Simmons]

Recently Jobst introduced us to the idea of electing a "lottery" that, in 
turn, picks the winner.

Here's one way that idea could be applied to the obstreperous ballot set

49 C
24 B
27 A>B :

First we introduce the lotteries A', B', and C' :

   A' means a toss up between B and C.
   B' means a toss up between C and A.
   C' means a toss up between A and B.

Next we deduce the most likely relative rankings of the candidates and 
lotteries:

49 C>A'=B'>C'=A=B
24 B>A'=C'>B'=C=A
27 A>C'>B=B'>A'>C

Now notice that in every case X' is ranked above X on more ballots than 
not:

   A' beats A 73 to 27.
   B' beats B 49 to 24.
   C' beats C 51 to 49.

This suggests that we should throw out A, B, and C, and choose from among 
A', B', and C'.

We now have

49 A'=B'>C'
24 C'=A'>B'
27 C'>B'>A'

Here C' beats B' (51 to 49), B' beats A' (27 to 24), and A' beats C' (49 
to 27).

So the wv lottery winner is A' and the margins lottery winner is B'.

If we took the average of these (A'+ B')/2, the respective probabilities 
for A, B, and C would be 25%, 25%, and 50%, which is very close to the set 
of probabilities that random ballot would give: 27%, 24%, and 49%.

It is interesting to note that if we applied Rob's ballot-by-ballot DSV 
using Strategy A to the contest among A',B', and C', then B' would be the 
most likely winner (a sure winner if we multiplied this ballot set by 
100), even though it would make B the winner of the original ballot set.

This corresponds to the fact that the original ballot set and its reverse 
comprise an example showing that this version of DSV fails the Reverse 
Symmetry Criterion.


Forest