[Election-Methods] [EM] Die Toss Version of FAWRB

[fsimmons at pcc.edu]

For those just joining in we're talking about designing lotteries that make
compromise C a sure winner in the following scenario: P: A>C>>B
Q: B>C>>A

Where P and Q are approximately 50%., and C is considered about 3/5 of the way between the lower and upper value in each faction from an expectation point of view; e.g. the A supporters would be indifferent between a sure election of C and a lottery that elects A vs. B with 3:2 odds.Well, we knew that the slope condition on  f at the right end point of the
interval [0,1] was only sufficient to make it a local equilibrium. However, your
original derivation of the formula f(x)=1/(5-4x) did not involve slope calculations.

The problem must have been in assuming that full B support for C would make C  the
approval winner, even when the A faction bullet voted. But in that case A, B, and C
would be tied for AW.I like your die throwing realization of FAWRB.The "bullet proportional probability property" (BPPP) is guaranteed by the
phrase at the end, "otherwise the favourite on the first drawn ballot is
elected," together with the fact if the first drawn ballot is a "bullet," then
this phrase will be invoked eventually.

It seems to me that if we are willing to give up the BPPP, then we can get C elected
with a higher likelihood (in other scenarios).Continuing with the present scenario for now ...If g is an increasing function on [0, 1], and g(1/2)=1/5, and g(1)=1, and C is
"60% good," then (if I am not mistaken) C is a globally attractive compromise in the following method applied to the above scenario:

1. Determine the approval winner X and the approval score x of X.

2.  Do a Bernoulli experiment with success rate g(x).  In case of success elect
X, else elect the favorite of a randomly drawn ballot.

If g(x) -> 0 as x -> 0, then this method has a kind of asymptotic BPPP.The only formula g(x) of the form (ax+b)/(cx+d) that satisfies the three conditions on g mentioned above is  g(x) = x/(4-3x).Let's check the sticky case where P = (50+) percent:If the A faction bullets, then A is the approval winner so their expectation isg(P) + (1-g(P))*Pwhich approaches 1/5 + (4/5)*(1/2)  or  3/5  as P approaches 50%.If, on the other hand, C has enough support to be the AW, the expectation for the A faction is(3/5)*g(x) +(1-g(x)*Pwhich increases in x (since P is near 50%, hence less than 3/5) to a maximum of 3/5.Bullet voting and full cooperation yield the same expectation.  Any uncertainty in whether P is greater than or less than exactly 50% shifts the advantage to the full cooperation equilibrium.Am I overlooking something?Note that in the case33 A1>A>A233 A2>A>A133 Bif the first two factions approve A, then A will be elected with probability g(2/3), which turns out  to be 1/3, compared with the value (2/3)*f(2/3) from the FAWRB method.This asymptotic BPPP seems reasonable to me: if the electorate is fragmented, then the BPPP holds approximately.  As the voters increasingly cooperate, the holdouts gradually lose proportional probability.Forest
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