[EM] SODA

[fsimmons at pcc.edu]

In our SODA development we came to something of an impasse for determining the "order of play" for 
the candidates casting their approval cutoffs.

Here's a suggestion:

Let the DSC winner go first, because the DSC winner is easily calculated, satisfies Later-No-Harm (so 
does not unduly encourage truncation), and can be thought of as the minimal acceptable modification of 
plurality, namely de-cloning it without destroying its montonicity.  In a way, DSC elegantly 
accomplishes what IRV attempts but botches. 

[In the context of SODA where there is only one faction for each of the n candidates, the DSC method 
has to score at most n*(n-1) subsets, and it takes no more than the order of n^2 steps to determine the 
DSC score of each of these subsets.  So the whole thing can be done in the order of n^4 steps at worst.]

>From then on the next player in the sequence is the candidate that ranked the previous player X the 
highest. If there is a tie, say Y1, Y2, and Y3 each ranks X equally high (and higher than anybody else 
does) then the member of {Y1, Y2, Y3} ranked highest by X is the next player.

This order is clone consistent, i.e. if Y is replaced by a clone set, then the entire clone set will be 
intercalated into the order in place of Y.

This order discourages burial, because if X is first in the order, and Y buries X,  then Y will not follow X, 
unless all of the other candidates bury X, too, in which case X could not have been first.

Note that we could reverse the roles of X and Y in determining the order and breaking ties:  The 
remaining candidate Y that X ranks the highest is next, and if X ranks no remaining candidate, then the 
candidate that ranks X the highest is next.  My intuition is that this order might not be quite as burial 
resistant, but it would be better at discouraging what we could call "fawning," namely ranking the 
presumed DSC winner artificially high for the sole purpose of getting into the order earlier.

Another option would be to use the DSC winner's rankings for all of the rest of the players, and passing 
to the second player's rankings to resolve any equal rankings made by the DSC winner, etc.

We need to experiment to see if any of these is adequate, and if so, which is best.

What are some good scenarios to test?