Fair Play Worksheet
Math & Democracy Series
The Fair Play Dilemma
Exploring Arrow's 5 Conditions for Fair Elections
Name: ___________________
Date: _________
Class Period: ____________
In 1951, Kenneth Arrow mathematically proved that no rank-choice voting system with 3+ options can be perfectly fair. To be completely fair, an election must meet the following five common-sense conditions.
PART 1
The Rules of the Game (Matching)
Match each condition on the left to its explanation on the right. Write the correct letter in the box next to each condition.
1. Non-Dictatorship Match ___
2. Individual Sovereignty Match ___
3. Unanimity Match ___
4. Freedom from Irrelevant Alternatives Match ___
5. Uniqueness of Group Ranking Match ___
A. The voting system must always output a single, clear, logical ranking of options, with no circular ties or paradoxical loops.
B. No single voter should have a "magic ballot" that decides the entire election outcome on its own, completely ignoring what everyone else wants.
C. If every single voter ranks Candidate X above Candidate Y, then the final group decision must also rank Candidate X above Candidate Y.
D. Voters must be allowed to rank the candidates in absolutely any order they choose, with no personal ranking preferences banned or restricted.
E. If Option A is currently beating Option B, introducing or removing a third "irrelevant" option (Option C) shouldn't flip the head-to-head winner of A and B.
PART 2
Democracy Detours (Scenarios)
Determine which of Arrow's 5 Conditions is being violated and write your answer below each scenario.
Scenario 1: The Pizza Topping Ultimatum
Mr. Harris's class is voting on toppings (Pepperoni, Veggie, Cheese). Mr. Harris says, "You can vote for any order, but you are forbidden from ranking Pepperoni first and Cheese last. If you do, I will throw your ballot away."
Violated Condition:
Scenario 2: The Mascot Spoiler
A head-to-head vote shows 60% of students prefer the Bears and 40% prefer the Wolves. A third option, the Eagles, is added to the ballot. Nobody changes their opinion about Bears vs Wolves, but the Wolves are now declared the winner!
Violated Condition:
Scenario 3: The Board Game Coup
The Board Game Club is voting on games. Out of 15 members, 14 vote for Catan. The club president, Leo, is the only one who wants Monopoly. The bylaws state if the president has a preference, that choice wins instantly, ignoring the other 14 votes. Monopoly wins.
Violated Condition:
Scenario 4: The Infinite Loop Drama
A Drama Club votes on three plays. A majority prefers Play A over B, a majority prefers B over C, and a majority prefers C over A. Because of this circular tie, the voting algorithm crashes in an infinite loop without outputting a group ranking.
Violated Condition:
Scenario 5: The Ice Cream Conspiracy
In a school poll of 100 students, every single student ranks Chocolate higher than Vanilla. Yet, when the computerized algorithm calculates the final winner, it ranks Vanilla higher than Chocolate overall.
Violated Condition:
Course Elective: Democratic Mathematics Consolidated Activity Sheet
Fair Play Answer Key
Math & Democracy Series • TEACHER KEY
The Fair Play Dilemma (ANSWER KEY)
Exploring Arrow's 5 Conditions for Fair Elections
Name: TEACHER COPY
Date: ANSWER KEY
Class Period: All Sections
In 1951, Kenneth Arrow mathematically proved that no rank-choice voting system with 3+ options can be perfectly fair. To be completely fair, an election must meet the following five common-sense conditions.
PART 1 SOLUTIONS
The Rules of the Game (Matching)
Match each condition on the left to its explanation on the right. Write the correct letter in the box next to each condition.
1. Non-Dictatorship Match B
2. Individual Sovereignty Match D
3. Unanimity Match C
4. Freedom from Irrelevant Alternatives Match E
5. Uniqueness of Group Ranking Match A
A. The voting system must always output a single, clear, logical ranking of options, with no circular ties or paradoxical loops.
B. No single voter should have a "magic ballot" that decides the entire election outcome on its own, completely ignoring what everyone else wants.
C. If every single voter ranks Candidate X above Candidate Y, then the final group decision must also rank Candidate X above Candidate Y.
D. Voters must be allowed to rank the candidates in absolutely any order they choose, with no personal ranking preferences banned or restricted.
E. If Option A is currently beating Option B, introducing or removing a third "irrelevant" option (Option C) shouldn't flip the head-to-head winner of A and B.
PART 2 SOLUTIONS
Democracy Detours (Scenarios)
Determine which of Arrow's 5 Conditions is being violated and write your answer below each scenario.
Scenario 1: The Pizza Topping Ultimatum
Mr. Harris's class is voting on toppings (Pepperoni, Veggie, Cheese). Mr. Harris says, "You can vote for any order, but you are forbidden from ranking Pepperoni first and Cheese last. If you do, I will throw your ballot away."
Violated Condition: Individual Sovereignty
Scenario 2: The Mascot Spoiler
A head-to-head vote shows 60% of students prefer the Bears and 40% prefer the Wolves. A third option, the Eagles, is added to the ballot. Nobody changes their opinion about Bears vs Wolves, but the Wolves are now declared the winner!
Violated Condition: Freedom from Irrelevant Alternatives
