Mathematics of Voting: Can Democracy Be Mathematically Fair?
If everyone votes honestly, surely mathematics should be able to find the fairest winner. Unfortunately, it isn’t that simple.
Ask a class to solve an equation and, usually, there is an answer that everyone can check.
Ask the same class to choose an end-of-term activity and things become rather more complicated.
Some want a film. Others prefer a quiz. A third group would choose a practical science challenge.
“Let’s vote,” somebody suggests.
That sounds straightforward. Count the votes, announce the winner and accept the democratic decision.
But there is a question we should ask first:
How are we going to count those votes?
We could count only first choices. We could eliminate the least popular option and vote again. We could award points for first, second and third places.
These methods can produce different winners, even when nobody changes their preferences.
For the final article in this mathematics-beyond-the-syllabus series, I want to explore something unexpected: how mathematics helps us question what we mean by a fair decision.
Three Sensible People Can Produce a Circular Decision
Let us begin with just three students choosing between activities A, B and C.
The symbol “>” means “is preferred to”.
Each student has a perfectly consistent ranking:
- Student 1: A > B > C
- Student 2: B > C > A
- Student 3: C > A > B
Now compare the activities two at a time.
A against B
Students 1 and 3 prefer A to B.
A wins by 2 votes to 1.
B against C
Students 1 and 2 prefer B to C.
B wins by 2 votes to 1.
C against A
Students 2 and 3 prefer C to A.
C wins by 2 votes to 1.
The group therefore produces this result:
A beats B. B beats C. C beats A.
We have gone round in a circle.
This is the Condorcet paradox: individually consistent preferences can produce a cycle when combined through pairwise majority voting. An option that beats every other option head-to-head is called a Condorcet winner. Here, there is no such winner.
Nobody has miscounted. Nobody has changed their mind.
The difficulty comes from combining their opinions.
The Mathematical Idea Hidden Inside: Transitivity
Students already use a related idea when comparing numbers.
If:
8 > 5
and:
5 > 2
then:
8 > 2
This property is called transitivity.
We often expect preferences to behave similarly. If I prefer A to B and B to C, we might expect me to prefer A to C.
Every student in our example has a transitive ranking.
The group’s majority comparisons do not.
That is what I find particularly interesting about this investigation. A property that holds for every individual does not necessarily survive when we combine their decisions.
“The majority prefers it” sounds decisive, but we need to ask:
The majority prefers it to what?
Could the Order of Voting Decide the Winner?
Return to those three students.
Suppose we organise a sequence of head-to-head votes, with the winner of the first contest facing the remaining activity.
We can make any of the three activities win simply by changing the order of the contests.
The preferences remain identical.
This provides a useful lesson for school councils, committees, clubs and businesses: the procedure used to reach a decision can influence the outcome.
It also suggests a sensible practical habit.
Agree the voting procedure before you know which option it will favour.
One Election, Three Different Winners
Now let us move to a larger classroom example.
Thirty students rank three activities:
- A: watch a film;
- B: take part in a quiz;
- C: complete a practical science challenge.
Their preferences are:
We will keep these rankings fixed throughout.
Everyone votes honestly. Everyone participates. Everyone ranks all three activities.
Only the counting method changes.
Method 1: First-Past-the-Post
Count first choices:
A wins.
A has more first-choice votes than either competitor.
However:
A’s share = 12 / 30 x 100 = 40%
This is a plurality, meaning the largest share. It is not an absolute majority, which requires more than half.
There is another striking detail: the other 18 students all put A last.
Should the largest group of first-choice supporters be enough to decide the result?
That is our first question about fairness.
Method 2: A Two-Round Run-Off
Suppose the rule says that, if nobody receives more than half the first-round votes, only the top two options enter a final vote.
A and B qualify. C is eliminated.
The eight students who preferred C have:
C > B > A
Assuming they vote according to those unchanged preferences, they choose B in the final.
B wins by 18 votes to 12.
The film won under first-past-the-post. The quiz wins under a run-off.
Nobody has changed their opinion.
Method 3: Ranked Voting Through Instant Run-Off
“Ranked voting” describes a family of methods using ordered preferences. It does not identify one particular counting rule.
One example is instant run-off voting.
For our complete ballots:
- Count each voter’s highest-ranked remaining option.
- If an option has more than half the votes, it wins.
- Otherwise, eliminate the option with the fewest votes.
- Transfer each affected ballot to its highest-ranked remaining option.
- Repeat as necessary.
Here, C is eliminated first. Its eight ballots transfer to B.
B wins again, with 18 votes.
With three options and these fixed, complete rankings, instant run-off produces the same result as our two-round run-off.
A separate second election can differ in practice because turnout or preferences may change. Our investigation deliberately holds those factors constant.
Method 4: The Borda Count
A Borda count uses positions throughout each ranking.
For three options, award:
- 2 points for first place;
- 1 point for second place;
- 0 points for third place.
Calculate the totals:
A’s score = 12 x 2 + 10 x 0 + 8 x 0 = 24
B’s score = 12 x 0 + 10 x 2 + 8 x 1 = 28
C’s score = 12 x 1 + 10 x 1 + 8 x 2 = 38
C wins.
C has the fewest first-choice votes, but everyone places it first or second.
Our results are now:
The same 30 students. The same preferences. Three different winners.
Does the Science Challenge Have Another Claim to Victory?
Compare C head-to-head with each rival.
C against A
The 10 students in the second group and eight in the third prefer C.
C beats A by 18 votes to 12.
C against B
The 12 students in the first group and eight in the third prefer C.
C beats B by 20 votes to 10.
In this example, C is the Condorcet winner.
Yet both first-past-the-post and instant run-off fail to elect it.
This does not establish that the Borda count always finds a Condorcet winner. It does not. Different methods reward different features of the rankings, and their outcomes can conflict.
The example gives us something more useful than a slogan about the “best” system: it gives us a precise disagreement to investigate.
What Do We Actually Mean by “Fair”?
Before choosing a voting system, we need to decide which properties matter.
These priorities can pull in different directions.
Even Borda points contain an assumption worth discussing. Awarding 2, 1 and 0 points gives the same numerical gap between first and second as between second and third.
But a ranking does not tell us how strongly somebody feels.
One student might slightly prefer a film to a quiz. Another might love the film and strongly dislike both alternatives.
Their rankings could look identical.
A ballot records particular information about preferences. It does not automatically capture everything that matters to the voter.
Arrow’s Impossibility Theorem: The Higher-Level Challenge
For A-level and Further Maths students, this leads towards Arrow’s impossibility theorem.
Consider a rule that combines individual rankings into a complete, transitive group ranking for three or more options.
We might require:
- Unrestricted preferences: every combination of consistent individual rankings is allowed.
- Unanimity: if everyone prefers A to B, the group ranks A above B.
- Independence of irrelevant alternatives: if nobody changes their relative preference between A and B, the group’s comparison of A and B stays unchanged, even if other comparisons change.
- No dictator: no single voter’s strict preferences always determine the group’s corresponding comparisons.
Arrow’s theorem says that no such aggregation rule can satisfy all these requirements simultaneously.
This is a precise mathematical limitation within a specified framework.
It does not prove that democracy is pointless, that all voting systems are equally good, or that every possible meaning of fairness is impossible.
It tells us to examine our assumptions and recognise the trade-offs.
Try It: Turn the Classroom Into a Voting Laboratory
What appeals to me as a teacher is how little equipment this investigation needs.
Paper, pencils and three options are enough to begin.
Start With a Real Choice
Choose three harmless alternatives: lesson activities, fictional school-trip destinations or possible names for a science club.
Ask each student to rank all three privately.
Count the same ballots using first-past-the-post, instant run-off and Borda points. Then compare every pair head-to-head.
Agree how ties will be resolved before counting.
The discussion afterwards matters as much as the arithmetic:
- Did the methods agree?
- Was there a Condorcet winner?
- Did the winning option appear near the bottom of many ballots?
- Which rule would students choose before knowing the result?
- Does their preferred rule change after they see who wins?
That last question can generate a particularly revealing conversation.
GCSE Challenge: Explain the Results
Use the 30-student example to calculate percentages, check totals and explain why a plurality differs from a majority.
For Borda scoring, every ballot distributes:
2 + 1 + 0 = 3 points
Therefore:
Total points = 30 x 3 = 90
Check:
24 + 28 + 38 = 90
This is a simple way to verify the arithmetic.
A-Level Challenge: Change the Scoring Rule
Keep the ballots fixed, but award 3 points for first place, 1 for second and 0 for third.
Which option wins now?
Then investigate a general rule:
First place = p points
Second place = q points
Third place = 0 points
Assume p > q > 0.
For our example:
A’s score = 12p
B’s score = 10p + 8q
C’s score = 8p + 22q
Students can use inequalities to find which relationships between p and q allow each option to win.
An apparently simple voting exercise has become an algebraic investigation.
Further Maths or Computing Challenge: Explore Many Elections
With three options, the number of strict rankings is:
3! = 6
With four options:
4! = 24
With five options:
5! = 120
A spreadsheet or short program could generate preference profiles and compare counting methods.
Students could investigate how often different methods disagree under their chosen model.
The phrase “under their chosen model” matters. Randomly generating rankings in one particular way does not automatically reproduce how real people vote.
Why This Belongs in Mathematics Teaching
Mathematics is sometimes presented as a subject in which every question has one correct answer and every problem ends when the calculation is complete.
Voting mathematics offers a richer picture.
The calculations can be correct while the decision remains open to debate.
Students must distinguish between:
- the information collected;
- the rule applied;
- the mathematical result;
- the judgement that the result is fair.
At Hemel Private Tuition, this is the kind of investigation I would use to connect mathematical reasoning with decisions students recognise.
It brings together arithmetic, percentages, logic, inequalities, algorithms and clear explanation.
It also rewards a valuable habit: asking what a calculation actually establishes.
Conclusion: Count the Votes—and Question the Rules
Democracy needs counting, but counting alone cannot decide what fairness should mean.
Our three-student example produced a circular majority. Our 30-student example produced three different winners. Arrow’s theorem revealed a deeper limit on combining rankings while preserving several attractive conditions.
These results give us reasons to design decision-making procedures carefully and explain them clearly.
For students, the most useful lesson may be this:
Before asking who won, ask how the winner was chosen.
And before declaring a voting system fair, ask:
Which kind of fairness are we trying to achieve?
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