Suppose you want to design a version of the propose-and-reject algorithm that avoids the property of guaranteed women-pessimality.

Suppose you want to design a version of the propose-and-reject algorithm that avoids the property of guaranteed women-pessimality. Identify valid strategies from the following list that would accomplish this goal without compromising the guarantee of a stable matching. Let X be the set of proposing entities and Y be the set of passive entities (entities receiving proposals).

i.) Swap the contents of X and Y.

ii.) Allow bilateral proposals from both X and Y.

iii.) Shuffle the contents of X and Y and partition the set at index ∣ X ∣.