Pairwise ballot cycle
Distinguish each person’s ranking from the group’s pairwise result.
Adult planning suggestion: about9–12 · Cooperation & communication
- Players
- 3–6 per group
- Time
- 15–20 minutes
- Materials
- Three fictional ranked ballots, A/B/C option cards and tally sheet.
Prepare
Use fictional options A,B,C, not classmates or personal traits. Ballot 1 ranks A>B>C, ballot 2 B>C>A, ballot 3 C>A>B. Each ballot is a complete strict order. Assign readers to the three ballots and a separate recorder when possible.
Play
Compare A with B only: each ballot contributes one vote to whichever appears earlier. Repeat B versus C and C versus A, preserving the same ballots. Do not ask voters to change preferences between comparisons. Write both counts rather than only the winning letter.
Worked example
A beats B by 2–1, B beats C by 2–1, and C beats A by 2–1. Each individual ballot is consistent even though the group relation cycles. These results do not give a single option beating every other option.
Finish and check
Count again after rearranging the order in which the pairs are discussed: the three tallies must stay the same. If a game agenda selects a winner through sequential elimination, explain that this adds a separate rule. Finish with one written statement distinguishing individual and group order.
Adaptation and limits
For thirty learners use ten groups of three and rotate the recorder role. Start with fictional ballots rather than a consequential real class vote. NRICH labels its original task 14–18; our supported upper-primary presentation is an editorial choice, not evidence of suitability for every learner.
Purpose, observation and provenance
Within this activity: Distinguish each person’s ranking from the group’s pairwise result.
Ask whether a majority cycle means any individual contradicted their own ballot.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
NRICH — Voting Paradox — Majority-cycle background, source age 14–18; our fixed ballots and supported lesson do not establish younger-age efficacy.
Age bands are planning suggestions. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
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.
Player materials · Pairwise ballot cycle
Ballots: A>B>C / B>C>A / C>A>B.
Tally A:B ___:___ ; B:C ___:___ ; C:A ___:___ .
| A:B | B:C | C:A |
|---|---|---|
| __:__ | __:__ | __:__ |
Player record · keep the host key separate
| Comparison | For left | For right | Winner |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Host key · separate from players
All listed pairwise tallies 2:1. No all-pairs winner.
Run the round
Use fictional options A,B,C, not classmates or personal traits. Ballot 1 ranks A>B>C, ballot 2 B>C>A, ballot 3 C>A>B. Each ballot is a complete strict order. Assign readers to the three ballots and a separate recorder when possible.
Compare A with B only: each ballot contributes one vote to whichever appears earlier. Repeat B versus C and C versus A, preserving the same ballots. Do not ask voters to change preferences between comparisons. Write both counts rather than only the winning letter.
A beats B by 2–1, B beats C by 2–1, and C beats A by 2–1. Each individual ballot is consistent even though the group relation cycles. These results do not give a single option beating every other option.
Count again after rearranging the order in which the pairs are discussed: the three tallies must stay the same. If a game agenda selects a winner through sequential elimination, explain that this adds a separate rule. Finish with one written statement distinguishing individual and group order.
For thirty learners use ten groups of three and rotate the recorder role. Start with fictional ballots rather than a consequential real class vote. NRICH labels its original task 14–18; our supported upper-primary presentation is an editorial choice, not evidence of suitability for every learner.