# CDTT

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 Smith//Truncation set and beatpath GMC (where *GMC* means *Generalized Majority Criterion*) at the EM mailing list.

## Uses

Limiting an election method's selection to the CDTT members can permit it to satisfy the Minimal Defense criterion (and thus the Strong Defensive Strategy criterion) and the Majority criterion for solid coalitions, while coming close to satisfying the Later-no-harm criterion. Specifically, the CDTT completely satisfies Later-no-harm in the three-candidate case, and failures can only occur in the general case when there are majority-strength cycles.

(Please see the articles on the Minimal Defense criterion and Later-no-harm criterion for commentary on the significance of these criteria.)

The CDTT's Later-no-harm performance can be preserved by pairing the CDTT with a method which itself fully satisfies Later-no-harm. When the paired method is used to generate a ranking of the candidates which is *not* influenced by which candidates make it into the CDTT, then compliance with the Monotonicity criterion can be preserved when the paired method already satisfies this criterion. Then the CDTT member who appears first in this ranking would be elected.

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 Mono-raise and Clone Independence.**First-Preference Plurality**: This is decisive, simple, and monotone, but fails Clone Independence.**Instant Runoff Voting**: This is more complicated. It satisfies Clone Independence but not 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 Later-no-harm, clone independence, and Mono-raise.^{[citation needed]}^{[clarification needed]}**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 Mono-raise.

When the CDTT is paired with a method which satisfies Later-no-harm, the combined method fails the Plurality criterion and Condorcet criterion. Otherwise there need not be any incompatibility. For example, the Schulze method (using *winning votes*) always elects from the CDTT and also satisfies both of Plurality and Condorcet.

## Notes

The CDTT set is useful in situations where there is a majority "semi-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.)