[EM] Hay Voting can emulate Approval Voting

[Peter de Blanc]

...or Condorcet voting, Plurality, Borda Count, IRV, or whatever else
you lke.

Or rather I will say that nearly any voting method can be Hayified.
Standard Hay Voting is actually Hayified Random Ballot.

I originally described Hay Voting by saying that we allocate voting mass
to various candidates; a random ballot is chosen, and then a random
point of voting mass is chosen from that ballot.

I now think a more revealing description of Hayified Random Ballot would
be to say that you allocate voting mass to all possible votes (in Random
Ballot, the set of possible votes happens to equal the set of
candidates). A random point of voting mass is selected from your ballot,
and whatever lies on that point becomes "your vote." Then all votes are
entered into a Random Ballot election.

But we can substitute any election method for Random Ballot.

What we discover about you, of course, is no longer your utility
function over candidates, but your utility function over votes (in RB
they happen to be equal). However, your utility function over votes is
determined by your utility function over candidates and your beliefs
about how the other voters will vote. As has been mentioned before,
these beliefs can be measured with a prediction game. Then we can solve
a system of linear equations to find your utility function over
candidates.

I conjecture that a Hayified voting method will asymptotically approach
the original voting method (ie the probability of producing the same
outcome approaches 1 as population size goes to infinity) if and only if
the original method possesses the property that adding one more of every
possible vote can never change the election outcome. Nearly all methods
have this property, but Random Ballot doesn't, which is why Hayified
Random Ballot is so much worse than ordinary Random Ballot.

- Peter de Blanc