CDTT

The Condorcet doubly-augmented 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.)

Uses

Limiting an election method's selection to the CDTT members can permit it to satisfy the Strong Defensive Strategy criterion (or Minimal Defense) and Majority, 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.

In order to maximize Later-no-harm compliance, the CDTT should be paired with a method that itself fully satisfies Later-no-harm. In order to ensure that Mono-raise is not failed, the paired method should be used to generate a ranking of the candidates which is not influenced by which candidates make it into the CDTT. Then the CDTT member who appears first in this ranking is 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.
• Descending Solid Coalitions: This is also somewhat complicated, but it's the only non-random option which satisfies Clone Independence and Mono-raise.
• 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.

Regardless of the method paired with the CDTT, it should be noted that the combined method necessarily fails the Plurality criterion and Condorcet criterion.