Understanding the Stable Marriage Problem

Dr. Riehl introduces the stable marriage problem, a mathematical concept without numbers, focusing on arranging stable marriages to avoid instability. The problem is whether stable marriages can always be arranged, which is proven possible by the Gale-Shapley algorithm. The algorithm involves a process where women propose to men based on preferences, and men tentatively accept or reject proposals. The process repeats until stable engagements are achieved. An example with hypothetical characters illustrates the algorithm's steps and outcomes.

What makes the stable marriage problem unique compared to traditional mathematics?

• It is based on geometry.

• It involves complex calculations.

• It uses no numbers or calculations.

• It requires advanced algebra.

What is the main goal in arranging marriages in the stable marriage problem?

• To achieve stability by avoiding mutual preferences for others.

• To minimize the number of rejections.

• To maximize the number of marriages.

• To ensure everyone is happy with their partner.

What is the key question addressed in the stable marriage problem?

• Can we always find a stable marriage arrangement?

• How to calculate the number of possible marriages?

• What is the best way to make everyone happy?

• How to minimize the number of proposals?

Who developed the algorithm that solves the stable marriage problem?

• Alan Turing and John von Neumann

• Pythagoras and Euclid

• David Gale and Lloyd Shapley

• Albert Einstein and Isaac Newton

In the Gale-Shapley algorithm, what happens on the first day?

• Everyone gets married immediately.

• All proposals are rejected.

• Women propose to their top choice.

• Men propose to women.

What is the role of men in the Gale-Shapley algorithm?

• To propose to women.

• To reject all proposals.

• To accept the first proposal they receive.

• To reject all but their top suitor.

What happens if a woman is rejected in the Gale-Shapley algorithm?

• She is automatically engaged.

• She leaves the process.

• She waits for another proposal.

• She proposes to her next choice.

What is the outcome when the Gale-Shapley algorithm terminates?

• All engagements are stable.

• Some participants remain unengaged.

• All participants are unhappy.

• The process restarts.

In the example provided, who does Charlotte end up engaged to?

• Wickham

• Collins

• Darcy

• Bingley

What is demonstrated through the example with Charlotte, Elizabeth, Jane, and Lydia?

• The algorithm is too complex to understand.

• The process is random.

• The algorithm is flawed.

• The algorithm leads to stable engagements.