Consider how the standard Gale-Shapley algorithm would behave if we ignore ties and incompleteness—what core property might break when preferences contain ties or partial lists?

How could you modify the preference lists to prioritize certain stable matchings over others when ties exist, and what data structure would help efficiently track these priorities?

Design a modified proposal mechanism where agents can strategically break ties based on additional criteria (e.g., randomness or external rankings)—how would this affect the algorithm’s correctness and termination guarantees?