CDTT: Difference between revisions
Changed DSC: there's no theorem that proves this AFAIK, there's just a lack of methods that pass LNH, monotonicity, and clone independence.
imported>James Green-Armytage (→Uses: monotone?) |
(Changed DSC: there's no theorem that proves this AFAIK, there's just a lack of methods that pass LNH, monotonicity, and clone independence.) |
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The '''Condorcet (doubly-augmented gross) top tier''' or '''CDTT''' is defined by Douglas Woodall as the union of all minimal nonempty sets of candidates such that no candidate in each set has a majority-strength pairwise loss to any candidate outside of the set.
Equivalently it can be defined as the set containing each candidate ''A'' who has a majority-strength beatpath to every other candidate ''B'' who has a majority-strength beatpath to ''A''. That is, a candidate ''A'' is in the CDTT unless some candidate ''B'' has a majority-strength beatpath to ''A'' while ''A'' has no such beatpath to ''B''.
(The term ''doubly-augmented'' refers to Woodall's notion of the ''doubly-augmented gross score'' of one candidate against another. This score, for ''X'' against ''Y'', is defined as the number of voters ranking ''X'' above ''Y'', plus the full number of voters abstaining from this pairwise contest. Then the CDTT can be defined as the union of all minimal nonempty sets such that no candidate in each set has a doubly-augmented gross score of less than half the number of votes, against any candidate outside the set.)
Note that the CDTT is a subset of the [[Smith set]], because all candidates in the Smith set pairwise beat all other candidates, and therefore have no pairwise losses, let alone majority-strength pairwise losses, against those candidates.
== EM mailing list ==
In the pre-2005 years, CDTT was called [http://lists.electorama.com/pipermail/election-methods-electorama.com/1997-May/001484.html Smith//Truncation set] and [http://lists.electorama.com/pipermail/election-methods-electorama.com/1997-October/001544.html beatpath GMC] (where ''GMC'' means ''Generalized Majority Criterion'') at the EM mailing list.
== Uses ==
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Some methods which can be paired in this way with the CDTT:
*'''[[Random Ballot]]''': This can be very indecisive, but it is conceptually simple, and it satisfies [[Monotonicity criterion|Mono-raise]] and Clone Independence.
*'''[[Plurality voting|First-Preference Plurality]]''': This is decisive, simple, and [[Monotonicity criterion|
*'''[[Instant-runoff voting|Instant Runoff Voting]]''': This is more complicated. It satisfies Clone Independence but not [[Monotonicity criterion|monotonicity]]. The IRV ranking would be the reverse of the candidates' elimination order.
*'''[[Descending Solid Coalitions]]''': This is also somewhat complicated, but it's the only known non-random option which satisfies
*'''[[Minmax|MinMax (Pairwise Opposition)]]''': This has the advantage that it is calculated based on the pairwise matrix, just as the CDTT itself is. However, it is somewhat indecisive and fails Clone Independence. It satisfies [[Monotonicity criterion|Mono-raise]].
== Notes ==
The CDTT set is useful in situations where there is a majority "semi-[[Solid coalition|solid coalition]]":
26 A>B
25 B
49 C
Some [[Smith-efficient]] Condorcet methods, such as most Condorcet-IRV hybrids, elect C here. Yet, if A hadn't run, B would be the majority's 1st choice and would've won. The Smith set here is all candidates, but the CDTT set is A and B, because C suffers a 49 to 51 majority-strength loss to B. (Though note that this wouldn't be the case if, say, an irrelevant alternative D with 10 new voters bullet voting them had run, since a majority would then be 56 voters. So there is some limit to the usefulness or generality of CDTT.)
It may be possible to generalize the concept of a CDTT set for [[Condorcet PR]] so that with a 2-winner situation like:
18 A>B
16 B
18 C>D
16 D
32 E
One of (A, B) and one of (C, D) would be guaranteed to win, rather than there being a possible spoiler effect with A and C running helping to elect E. This could perhaps be done by defining it in terms of [[quota]]-strength losses.
== References ==
* Woodall, Douglas R. (2003). "Properties of single-winner preferential election rules II: examples and problems". (Draft.)
[[Category:Single-winner voting methods]]
[[Category:Voting theory]]
[[Category:Condorcet-related sets]]
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