Experiment 16 Stable Matching
The Matchmaker’s Dilemma
Match applicants and teams, step through offers, and find pairs who would rather choose each other.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Predict, Then Make the Matches
Can Both Sides Get a Match They Will Keep?
You are Applicant A. Applicants A, B, and C each want one team; Oak, Pine, and Elm each have one place. A preference ranking is an ordered list from most wanted to least wanted. A stable matching has no applicant and team who are not matched to each other and who both prefer each other to their assigned partners. A pair like that is called a blocking pair.
A proposal is an offer to pair up. The receiving side holds its favorite offer so far and rejects the rest. Holding is temporary: a later, better offer can replace it. This procedure is called deferred acceptance. Deferred means delayed.
2. Reveal the Incentives
Follow Each Offer
Current Assignments
Proposal History
3. Change One Assumption
Change Who Goes First
Reverse which side proposes and replay with the same preferences. Then change one ranking. These lists have no ties and include every possible partner. Every match is acceptable; there are no empty places at the end.
Changing a scenario setting restarts the proposal history. The manual assignment checker does not change the proposal procedure.
Work It Out, Step by Step
- Suppose A ranks Oak first, Pine second, and Elm third. A’s choice numbers are 1, 2, and 3. Smaller means more preferred.
- If A has Pine but prefers Oak, that is only half of a blocking pair. Oak must also prefer A to its current applicant.
- If Oak prefers its current applicant, A cannot make a mutually welcome switch. Wanting a different assignment alone does not make a blocking pair.
- Each proposer tries its first choice, then its second if rejected, then its third. With three proposers and three possible partners, at most nine proposals are needed.
In words: three people can each make at most three offers. The multiplication sign means “three groups of three.” Stability is a check on pairs, not a promise of everyone’s favorite outcome.
For experts: formal model and assumptions
Complete, Strict, One-to-One Preferences
Strict means no tied ranks. One-to-one means each applicant receives one team and each team receives one applicant. The proposing side gets its best possible partners among stable matchings under these assumptions. Reversing the proposing side can change the stable result. “Best stable” does not mean best among all imaginable assignments.
- \(r_A(T)\)In words: A’s rank number for team T.
- The small A is a label naming the applicant. The parentheses name the team we are looking up. This is a function, which means a rule that gives an answer for an input. It is not multiplication.
Once a team holds an offer, its held applicant can only improve in its ranking. If an applicant preferred some team to its final match, it already proposed there and was rejected. That team ends with someone it prefers, so the pair cannot block. Swapping the side labels gives the same argument for team proposals.
The sum of ranks displayed here is a teaching score. Rankings alone do not tell us the size of people’s satisfaction, so this is not a claim about total well-being.
For more examples of the notation, use the math reading guide.
4. Transfer the Lesson
Where Would the Same Rule Help?
A school matches students to one-seat clubs. Would letting clubs make the proposals favor the same students as letting students propose? What changes if one club has ten seats or students are allowed to leave choices off their lists?
Compare your reasoning
Switching the proposing side can change whose preferred stable assignment is selected. Extra seats and incomplete lists require changes to this small model. Do not treat the three-by-three result as a complete school admissions policy.
Concept reference: the Nobel Prize committee’s explanation of stable matching and deferred acceptance.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.