[EM] Smith-like sets

[Peter Gustafsson]

In elections where a ranked system is used and there are two dominant parties, there is a considerable risk of burying. I suspect that if an election system which mandates full ranking, or at least prohibits bullet voting, were to be introduced to USA shortly before an election, one would get voting results something like this:

45% GOP-LIB-nohoper1-nohoper2- ... -Random Crazy-GREEN-DEM
45% DEM-GREEN-nohoper2-nohoper1- ... -Random Crazy-LIB-GOP
4% LIB-GOP-nohoper1-nohoper2- ... -Random Crazy-GREEN-DEM
4% GREEN-DEM-nohoper2-nohoper1- ... -Random Crazy-LIB-GOP
2% nohoper2-nohoper1- ... -Random Crazy-DEM-GREEN-LIB-GOP

Obviously, those numbers are guesses, but the general idea is there: Voters favoring either big party will bury the main opponent below minor no-threat parties that they actually consider worse than the opposing main party.

When voters on both sides do this, we are liable to get big Smith sets, sets which are inflated (compared to what they would have been if voters would have voted non-strategically) due to this burying. If Borda or some other systems are used, we get some probably undeserving non-entity winning, which is not good for the country, and greatly increases the risk of plurality being reinstated. 

So, I wondered if there are sets that similar to the Smith set, but with less susceptibility to burying behavior. Here, "similar set" means any set such that there is some definable objective criteria set which divides the candidate set into two disjunct sets, and points out that the winner should be in one specific one of those sets. In most cases, the set containing the winner should be the smaller set, especially if the number of candidates is large.

A cursory googling did not reveal anything of interest, so I will hereby state a first suggestion on how such a criterion set could be laid out. If I have missed out on something during my googling, please point me in the right direction. If you have counterexamples showing how my suggestion would fail, please list them.

Without further ado, here is the rule set for how to construct an improved Smith-like set:
1. Calculate the % of vote bills that are headed by candidates of each party. If a party has several candidates and more than one of them placed first on any vote bill, add the percentages for all candidates of that party.
2. List all parties according to their total percentage of #1 spots on the voting bills
3. Take party in the top two spots of that list, and transfer them to Smith-like set.
4. Set aside all voting bills headed by any candidate from either of the two main parties, as decided in the prior step. None of those voting bills are used to calculate the remaining members of the Smith-like set. 
5. For all the remaining voting bills - those headed by candidates by non-major parties - temporarily redact all candidates from the major parties. We now have a set of voting bills from as of yet unfulfilled voters, where their preferences among the major party candidates are temporarily disregarded.
6. Calculate the Smith set of the remaining voting bills
7. Produce a Smith-like set by joining the Smith set of step 6 with the 2 major parties of step 3. 
8. For all voting bills, no matter whom the voter voted for as #1, delete all candidates from parties that are not in the Smith-like set of step 7. Reinstate preferences among major party candidates among voters preferring minor parties.
9. From those voting bills modified in step 8, perform preferred closure method (Schulze, MinMax, whatever)

I should probably illustrate this with the hypothetical voting results stated above. We have:
45% GOP-LIB-nohoper1-nohoper2- ... -Random Crazy-GREEN-DEM
45% DEM-GREEN-nohoper2-nohoper1- ... -Random Crazy-LIB-GOP
4% LIB-GOP-nohoper1-nohoper2- ... -Random Crazy-GREEN-DEM
4% GREEN-DEM-nohoper2-nohoper1- ... -Random Crazy-LIB-GOP
2% nohoper2-nohoper1- ... -Random Crazy-DEM-GREEN-LIB-GOP

Once step 2 is finished, we have the following list:
1. 45% GOP
2. 45% DEM
3. 4% LIB
4. 4% GREEN
5. 2% Nohope2

GOP and DEM are set aside to the Smith-like set, and the set of voting bills is culled to this:
40% LIB-nohoper1-nohoper2- ... -Random Crazy-GREEN
40% GREEN-nohoper2-nohoper1- ... -Random Crazy-LIB
20% nohoper2-nohoper1- ... -Random Crazy-GREEN-LIB

LIB-GREEN 40-60
LIB-NH1 40-60
LIB-NH2 40-60
GREEN-NH1 40-60
GREEN-NH2 40-60
NH1-NH2 40-60

Nohope2 emerges as the sole member of the Smith set in step 6. Obviously, this is strongly dependent on details of the minor party voter preferences.

The Smith-like set is then:
GOP
DEM
NH2

With all preferences reset, and culled for parties not in the Smith-like set, we get the following modified list of voting bills:
45% GOP-nohoper2-DEM
45% DEM-nohoper2-GOP
4% GOP-nohoper2-DEM
4% DEM-nohoper2-GOP
2% nohoper2-DEM-GOP

Adding together groups #1+#3 and #2+#4, we get:
49% GOP-nohoper2-DEM
49% DEM-nohoper2-GOP
2% nohoper2-DEM-GOP

DEM is the Condorcet Winner.

With this method of putting together a Smith-like set, voters favoring a major party have no motive to bury their main opponent, since they will get their candidates admitted into the Smith-like set in the beginning, and their votes will not play any part in figuring out which other candidates get into the set. Those voters already got their wish, they have no business having power over the rest. 

Party strategists of the major party will have one major and one minor goal. The major goal will be to get their voters out in sufficient numbers so as to make their party enter the Smith-like set in the first pass. That will probably be a trivially easy goal to fulfill, until minor parties start growing. The minor goal will be to influence minor party voters to vote strategically among the minor party candidates so as to create a total Smith-like set that is as good as possible for that major party. That minor goal will be much, much harder. For starters, it is unlikely that minor party voters will want to help the major parties. Secondly, the goal requires detailed data on the detailed preferences among a minority of voters. Thirdly, any changes in the message from a major party which is intended to mess with detailed preferences among minor party voters might well make truly undecided voters less interested in voting for the major party. 

It is my assumption that this Smith-like set strongly reduces the incentive to bury when election methods allowing fully ranked ballots are used, and that the tweak is relatively numerically simple. I do not as of now see any big drawback. I am looking forward to discussions.

Yours,

Peter Gustafsson