Consider a 2-option election, with 2N voters, each of whom has probability p of choosing the first option. If p is a fixed number, then as N goes to infinity, (chances of an exact tie times N) go to 0 if N isn’t exactly .5, and to infinity if it is. Since the event of p is exactly .5 has measure 0, this model supports the paradox of voting (PoV).
But! If p itself is drawn from an ordinary continuous distribution with nonzero probability density d around .5, then (chances of an exact tie times N) go to … I think it’s just d/2. Maybe there’s some correction factor that comes into play for bizarre distributions of p, but if we make the conventional assumption that it’s beta-distributed, then d/2 is the answer.
I think that the PoV literature is relying on the “fixed p” model. I think the “uncertain p” model is more realistic, but it’s still worth engaging with “fixed p” and seeing the implications of those assumptions.
Consider a 2-option election, with 2N voters, each of whom has probability p of choosing the first option. If p is a fixed number, then as N goes to infinity, (chances of an exact tie times N) go to 0 if N isn’t exactly .5, and to infinity if it is. Since the event of p is exactly .5 has measure 0, this model supports the paradox of voting (PoV).
But! If p itself is drawn from an ordinary continuous distribution with nonzero probability density d around .5, then (chances of an exact tie times N) go to … I think it’s just d/2. Maybe there’s some correction factor that comes into play for bizarre distributions of p, but if we make the conventional assumption that it’s beta-distributed, then d/2 is the answer.
I think that the PoV literature is relying on the “fixed p” model. I think the “uncertain p” model is more realistic, but it’s still worth engaging with “fixed p” and seeing the implications of those assumptions.