[EM] are ranked pairs and river Schwartz-consistent?

James Green-Armytage jarmyta at antioch-college.edu
Wed Sep 15 20:52:14 PDT 2004


Dear election methods fans,

	I've often heard that ranked pairs is Schwartz consistent, but today I've
been wrestling with an example where it doesn't seem to be. Hopefully I'm
just making a mistake somewhere.
	First of all, by Schwartz, I mean in particular Schwartz's "union of
minimal undominated sets" set, or GOCHA set.... defined as follows:
1. An undominated set is a set such that no candidate within the set is
beaten by any candidate outside the set. 
2. A minimal undominated set does not contain any other undominated sets
besides itself.
3. The GOCHA set is the union of minimal undominated sets.

Here is my example.
Votes (with > symbol implied between each letter, e.g. 5 voters for
R>S>A>T)
5: RSAT
5: TARS
4: STAR
4: RAST
3: TRAS
3: SATR
2: ATRS
2: SRAT
1: ASRT
1: TRAS

Pairwise comparisons
A=R : 15-15
A>S : 16-14
A>T : 17-13
R>S : 20-10
S>T : 19-11
T>R : 18-12
	As far as I can tell, the GOCHA set in this example is only {A}. A is
surely undominated, and I cannot find any other sets of candidates other
than {A B C D} which are also undominated. So, a Schwartz-efficient method
should choose A with certainty.
	However, using ranked pairs, there is a tie between A and R, and if you
use a basic tie-breaking ranking of the options method, there is a 50%
chance that R will be elected.
	That is, using ranked pairs, the defeats are considered as follows
20-10		R>S		lock
19-11		S>T		lock
18-12		T>R		skip
17-13		A>T		lock
16-14		A>S		lock
	You are left with a tie, A=R>S>T. 
	The river method produces the same result.
20-10		R>S		lock
19-11		S>T		lock
18-12		T>R		skip
17-13		A>T		skip
16-14		A>S		skip
	A and R are both still undefeated.
	So, I don't understand what's going on. It doesn't seem like either
ranked pairs or river choose A with certainty. 
	Again, I'm hoping that I have made a mistake somewhere, and there is
someone on this list who can identify and correct it.

my best,
James







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