Stable preference pairs
Match fictional labels so there is no unmatched pair that would both prefer one another to their assigned partners.
Logic & rule checking · Cooperation & communication
- Players
- 1–6 cooperative
- Time
- 15–25 minutes
- Materials and quantities
- Two groups of three fictional labels, printed preference lists, proposal log.
1–6 cooperative
Prepare
Proposers A,B,C rank X,Y,Z as follows: A: X,Y,Z; B: Y,X,Z; C: Y,Z,X. Receivers rank A,B,C: X: B,A,C; Y: A,C,B; Z: C,B,A. These are invented labels, not opinions about guests. Each label must get exactly one partner.
Play
An unmatched proposer offers to the highest-ranked receiver not yet tried. A receiver holds at most one proposal: keep the preferred of a new and held offer, and reject the other. A held proposal is tentative, not a final acceptance. Rejected proposers continue down their own lists.
Worked example
Process A→X, B→Y, C→Y. Y prefers C to B, so B is rejected. B then offers X; X prefers B to A, rejecting A. A offers Y; Y prefers A to C. C then offers Z. Final pairs A–Y, B–X, C–Z.
Finish and audit
Every label must be paired once. Then inspect each unmatched proposer/receiver pair: it blocks the matching only if both prefer each other to their current partners. For the worked pairs no such blocking pair exists. A matching is not required to give everyone their first choice; stability is this precise rule, not general fairness or happiness.
Adapt
Work as a group with six large cards and an adult reading ranks. Record every rejection so an offer is not repeated. Never ask participants to rank real peers, and do not present this puzzle as relationship advice.
Aim: Match fictional labels so there is no unmatched pair that would both prefer one another to their assigned partners.
Practice within this activity: Match fictional labels so there is no unmatched pair that would both prefer one another to their assigned partners.
Adult observation: Ask the participant to explain one legal choice. Accept passing and a partner writing; do not infer a lasting ability from a single round.
Mechanism and adaptation: A separately written PartyPageKit version of the mechanism described below, with original examples and materials. The earliest inventor is not established here.
Goals describe observable activity choices, not measured long-term gains or institutional endorsement.
Free alternatives: Write six labelled preference cards on scrap paper; never use real classmates’ personal rankings.
Buying recommendations reserved; no purchase required.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Fictional preference cards
Use invented labels only, not real classmates. A held proposal is tentative.
Proposal and blocking-pair audit
| Offer | Held by receiver | Rejected proposer |
|---|---|---|
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| Unmatched pair | Both prefer? |
|---|---|
| ________ | ________ |
| ________ | ________ |
| ________ | ________ |
Teacher instructions and checked example · separate
1–6 cooperative
Two groups of three fictional labels, printed preference lists, proposal log.
Write six labelled preference cards on scrap paper; never use real classmates’ personal rankings.
Print only the relevant player pages; keep this teacher page separate. Read the full web rules before leading. A helper may write or move pieces, and anyone may pause or pass. No classroom or physical-printer trial is claimed.
Offers A→X, B→Y, C→Y (reject B), B→X (reject A), A→Y (reject C), C→Z. Final A–Y, B–X, C–Z. A prefers X to Y, but X prefers its B to A; C prefers Y to Z, but Y prefers A to C. No blocking pair. Stable does not mean everyone gets first choice.