How Mathematics Improves The World | Dividing Something Fairly When Everyone Values It Differently
Cut a cake exactly in half.
Fair?
Only if both people value the two halves equally.
Suppose one half is chocolate and one half vanilla.
One person loves chocolate and dislikes vanilla.
The other feels the opposite.
A physically unequal cut can now make both people happier than an equal one.
This is the beginning of fair division.
Mathematics does not assume fairness means identical quantities.
It asks what fairness property we actually want.
Everyone gets at least a fair share by their own values?
No one prefers somebody else’s bundle?
No alternative allocation can make someone better off without hurting another?
The worst-off person is protected?
The total value is maximised?
These are different mathematical jobs.
The word fair is too small to contain them all.
Quick Read
Fair division studies how to allocate resources when people value the same objects differently. A classic model is cake cutting, where a divisible resource can be cut into arbitrary pieces and each participant has a personal value measure over it. A division is proportional if each of n people receives at least 1/n of the total value according to their own valuation. It is envy-free if nobody prefers another person’s allocation to their own.
These properties are related but not identical. Envy-free allocations of divisible goods are especially strong because if nobody envies anyone else, each person must value their own portion at least as highly as the average of all portions and therefore receives a proportional share. Research on cake cutting proves that envy-free allocations exist under broad assumptions, including connected-piece results for certain settings.
Indivisible goods change the problem. One apartment, one bicycle or one rare painting cannot be split without destroying what makes it useful. Exact envy-freeness may be impossible. Mathematics therefore develops relaxations such as envy-freeness up to one good, or EF1, where any envy can be removed by taking away at most one item from the envied bundle. Other approaches maximise Nash social welfare, combining efficiency with a strong fairness effect.
The important lesson is that fairness is not one formula. It is a family of explicit criteria. Mathematics improves decisions by forcing us to say which criterion matters, when multiple criteria conflict, and when no allocation can satisfy all of them simultaneously.
One-sentence answer: Mathematics improves the world by turning vague arguments about “fair shares” into explicit properties such as proportionality, envy-freeness, efficiency and protection of the least advantaged, so competing values can be reasoned about instead of hidden inside intuition.
Equal Quantity Is Not Equal Value
Two children divide a pizza.
One loves mushrooms.
The other hates them.
If half the pizza has mushrooms and half does not, splitting by area can create two physically equal pieces and two unhappy children.
Give the mushroom half to the mushroom lover and the plain half to the other.
Each may value their own portion more than 50% of the whole.
Subjective valuation creates the possibility of mutual gain.
Difference is not always an obstacle.
Sometimes it is what makes agreement possible.
Valuation Functions: Put Preference Into Mathematics
Let vi(S) represent how much person i values bundle S.
For divisible cake, v can behave like a measure.
If two pieces do not overlap, values add:
v(A ∪ B) = v(A) + v(B)
after normalising the whole cake to value 1.
Different people have different v functions.
One values icing.
One values fruit.
One values only the chocolate centre.
Fair division respects those differences rather than pretending value is printed on the object.
Proportionality: Everyone Gets at Least 1/n by Their Own Measure
There are n people.
Each values the entire resource at 1.
An allocation is proportional if:
vi(Ai) ≥ 1/n for every i.
The physical size of Ai can vary.
The guarantee is subjective.
Each person believes they received at least their fair fractional share.
Proportionality is powerful because it creates a minimum guarantee.
It does not eliminate envy.
A Proportional Allocation Can Still Create Envy
Three people each get something they value at 1/3 or more.
Person A values their own piece at 0.35.
They value B’s piece at 0.55.
A has a proportional share.
A still envies B.
This distinction matters in real institutions.
“You received enough” is not the same as “you do not prefer someone else’s outcome”.
Envy-Freeness: Nobody Wants to Swap
An allocation is envy-free if for all people i and j:
vi(Ai) ≥ vi(Aj).
Each person evaluates everybody’s bundle using their own valuation.
No one sees another allocation they would prefer.
This is stronger than equal quantities because it asks directly about preference.
It also avoids requiring everyone to agree on a common value scale.
Why Envy-Free Implies Proportional for Divisible Goods
Suppose there are n pieces and person i does not envy any other piece.
Their own piece is worth at least as much as each of the other n−1 pieces according to them.
Therefore their piece is worth at least the average value of all pieces.
The total is 1.
Average is 1/n.
So:
envy-free ⇒ proportional
under the usual divisible-good assumptions.
One fairness property mathematically contains another.
Cut and Choose: Two People, One Elegant Protocol
Person A cuts the cake into two pieces they value equally.
Person B chooses first.
A cannot complain because they believed the pieces equal.
B cannot complain because they selected the one they preferred.
The protocol creates incentives aligned with fairness.
A is motivated to cut carefully because B has choice.
Mechanism design appears in a child’s birthday cake.
Three People Are Much Harder
Cut-and-choose does not generalise trivially.
With three people, someone can envy another even if each received what seemed like a third during an intermediate step.
Classic fair-division research developed procedures involving trimming, ranking and reallocation.
For larger numbers, exact envy-free algorithms can become surprisingly complicated.
Fairness complexity grows faster than the number of mouths.
Moving-Knife Procedures: Fairness Can Be Continuous
Some cake-cutting procedures imagine a knife moving continuously across the cake.
Participants call “stop” when a region reaches a threshold by their own value.
These procedures show existence of fair divisions elegantly.
They also expose a practical issue.
A mathematically continuous protocol may require exact timing or infinitely precise comparison that is difficult to implement digitally.
Existence and computability are different questions.
Efficiency: Fairness Can Waste Value
Suppose A values the left half at 90 and the right half at 10.
B values left at 10 and right at 90.
Give A the right half and B the left.
Each receives value 10.
Swap them.
Each receives 90.
The original allocation is Pareto dominated.
An allocation is Pareto efficient if no other feasible allocation can make someone better off without making someone else worse off.
Fair division often seeks fairness and efficiency simultaneously.
Utilitarian Welfare: Maximise the Sum
One objective maximises:
Σ vi(Ai).
This can create very high total value because resources go to whoever values them most.
It can also create severe inequality.
If one person values everything slightly more than everyone else, a pure utilitarian optimum can allocate almost everything to them.
Total welfare is not the same as distributive fairness.
Egalitarian Welfare: Protect the Worst-Off
Another objective maximises:
mini vi(Ai).
This raises the value of the least satisfied participant as much as possible.
It prioritises the floor rather than the total.
The policy choice resembles infrastructure design.
Do we maximise average performance?
Or protect the worst tail?
Mathematics reveals that the objectives can disagree.
Nash Social Welfare: Multiply Utilities
The Nash social welfare objective maximises the product:
∏ vi(Ai)
or equivalently the sum of logarithms where values are positive.
The product strongly penalises giving someone almost nothing because a near-zero utility collapses the product.
At the same time, it rewards efficiency.
This balance makes Nash welfare influential in modern fair-allocation theory.
Divisible and Indivisible Goods Are Different Worlds
A cake can be cut.
A bicycle cannot usefully be divided among three children.
A hospital appointment cannot be split into thirds.
A school place is discrete.
Exact fairness properties that always exist for divisible goods can become impossible for indivisible goods.
The mathematical category of the resource matters before the algorithm begins.
Why Exact Envy-Freeness Can Be Impossible
Two people want one indivisible painting.
Both value it positively.
Someone gets it.
The other receives nothing.
The person with nothing envies the owner.
No exact envy-free allocation exists if the painting cannot be sold or compensated with another divisible resource.
When perfection is impossible, fair division asks for principled approximations.
EF1: Envy-Free Up to One Good
An allocation is EF1 if, for any envious person, removing at most one good from the envied bundle eliminates that envy.
Example:
A receives three books.
B receives two.
B prefers A’s bundle.
But remove A’s most valuable book from B’s perspective and B no longer prefers A’s remainder.
The allocation is not perfectly envy-free.
It is close in a precisely defined way.
EFX: A Stronger Approximation
Envy-free up to any positively valued good, EFX, asks for an even stronger guarantee.
If person i envies j, remove any item from j’s bundle that i values positively.
The envy should disappear.
EFX existence for general indivisible goods remains a deep area of research.
This is an instructive fact.
Questions children can understand can sit at the frontier of mathematical research.
Money Can Turn Indivisible Goods Back Into a Divisible Problem
An inheritance contains one house.
Three heirs cannot each receive one third of the physical house independently.
Sell the house and divide money.
Or one heir keeps the house and compensates the others.
Transferable money expands the feasible set.
Fairness can become easier because utility can be balanced with side payments.
But people may value ownership sentimentally in ways money does not capture.
Assignment Problems: One Person, One Slot
Students choose projects.
Doctors choose shifts.
Families apply for school places.
Workers receive offices.
These are assignment problems.
Participants rank or value discrete options.
Capacity constraints prevent everyone receiving first choice.
Matching theory, priorities, lotteries and market design become relevant.
Fair division overlaps with but is not identical to matching.
Randomisation: Fairness Before the Outcome
One indivisible prize.
Ten equally eligible people.
No deterministic allocation can give everyone equal outcome.
A lottery gives everyone equal probability before the draw.
This is ex ante fairness.
After the draw, one person owns the prize and nine do not.
The realised outcome is unequal.
Fair process and equal outcome are different ideas.
Strategy-Proofness: Can People Benefit by Lying About Their Preferences?
A fair-looking allocation rule can encourage manipulation.
If exaggerating your value for one item improves your allocation, participants may misreport.
Mechanism design studies rules under strategic behaviour.
Strategy-proofness means truthful reporting is a dominant strategy.
Fairness, efficiency and strategy-proofness can conflict.
A theoretically fair rule can fail socially if it rewards deception.
Information Is Expensive
To calculate a perfectly optimised allocation, the system might need complete valuations for every person over every possible bundle.
That can be impossible.
With m indivisible items, one person has 2m possible bundles.
Even reporting preferences becomes exponential.
Practical fair division therefore uses structured valuation assumptions, rankings, queries or iterative procedures.
Computation and communication are both resources.
Fairness Can Be Incompatible With Maximum Efficiency
Suppose A values an object at 100.
B values it at 99.
A pure efficiency rule gives it to A.
Now suppose A already received nine valuable objects and B received none.
A fairness-aware rule may give this object to B.
Total reported value falls by 1.
Distribution becomes far more balanced.
This is not a mathematical error.
It is a policy trade-off.
The Price of Fairness
Researchers study how much efficiency can be lost when a fairness requirement is imposed.
This is sometimes called the price of fairness.
If envy-freeness reduces maximum possible welfare by only 2%, the fairness guarantee may be cheap.
If it reduces welfare by 60%, the policy trade-off is sharper.
Mathematics makes the cost visible rather than hiding it inside rhetoric.
Fairness Through Time
One allocation happens today.
Another tomorrow.
A person who receives less today may receive priority later.
Repeated allocation creates temporal fairness.
Cloud computing, shift scheduling, public housing queues and school access can all involve repeated opportunities.
Fairness can be measured per round or cumulatively.
The time horizon changes the judgement.
Fairness Under Uncertainty
Suppose future demand is unknown.
Allocating all capacity today may leave nothing for tomorrow’s higher-need group.
Reserve too much and current resources sit unused.
Stochastic fair allocation balances expected efficiency with future access.
The fair decision can depend on scenarios that have not happened yet.
Fairness and Need Are Different From Fairness and Preference
Cake cutting usually assumes value represents preference.
Public policy often cares about need.
A patient needing urgent treatment may receive priority even if another person values the slot more.
A disability accommodation may deliberately allocate more resources to one person.
Equal treatment can create unequal capability.
The model must therefore encode the correct normative variable.
Preference, entitlement, need and desert are not interchangeable.
Fair Division Does Not Decide What Society Owes People
Mathematics can prove:
this allocation is proportional under these valuations.
It cannot prove:
these valuations are the only morally relevant facts.
Property rights.
Historical injustice.
Legal obligations.
Need.
Consent.
All can sit outside a narrow utility model.
A fair division algorithm is powerful only when the right problem has been defined.
A Classroom Thought Experiment: Cut the Chocolate Cake
Draw a rectangle.
Colour the left half chocolate and the right half vanilla.
Student A values chocolate 90 and vanilla 10.
Student B values chocolate 20 and vanilla 80.
Ask students to create:
- equal-area allocation;
- proportional allocation;
- envy-free allocation;
- maximum-total-value allocation.
Then compare them.
The same cake can have several “fair” answers depending on the criterion.
A Second Thought Experiment: Three Objects, Two People
There are three indivisible objects:
- book;
- ball;
- headphones.
Give each student a different valuation table.
Ask for an envy-free allocation.
Sometimes none exists.
Ask for EF1.
Now a solution may appear.
The lesson is profound:
when perfection is impossible, mathematics can still define the smallest acceptable relaxation.
Primary Mathematics: Fairness Begins With Fractions
Primary students already know:
- fractions;
- ratios;
- ordering;
- comparison;
- tables;
- simple optimisation.
One half is geometry.
One half by value is preference.
Fair division teaches children that equal-looking and equal-valued are different concepts.
Secondary Mathematics: Fairness Becomes Inequalities and Optimisation
Secondary students can add:
- inequalities;
- functions;
- linear programming;
- graphs;
- probability;
- game theory.
Proportionality becomes a lower-bound inequality.
Efficiency becomes an optimisation objective.
Random allocation becomes probability.
Strategic manipulation becomes game theory.
Advanced Mathematics: Fair Division as Mechanism Design
Modern fair allocation draws on:
- measure theory;
- combinatorics;
- topology;
- convex optimisation;
- approximation algorithms;
- game theory;
- mechanism design;
- social choice;
- market design.
A cake becomes a measure space.
Objects become combinatorial bundles.
Fairness properties become constraints.
Human strategic behaviour becomes part of the theorem.
Why This Improves the World
1. It separates fairness criteria that ordinary language confuses
Proportionality, envy-freeness and efficiency can be tested separately instead of collapsed into one argument about “equal shares”.
2. It makes disagreement computable without requiring identical preferences
Different valuations can actually create mutually beneficial divisions.
3. It tells us when perfection is impossible
Indivisible goods can make exact envy-freeness unattainable, preventing institutions from promising a property no allocation can satisfy.
4. It creates principled approximations
Relaxations such as EF1 replace arbitrary compromise with explicit guarantees.
5. It exposes the cost of fairness
Efficiency losses created by fairness constraints can be quantified and debated openly.
6. It makes institutions explain their objective
A rule must say whether it values equality, need, welfare, absence of envy or something else.
What Mathematics Does Not Do
Mathematics does not define moral worth.
It does not decide whether preference should outweigh need.
It does not turn strategic self-reports automatically into truthful values.
It does not make exact envy-freeness possible for every indivisible allocation.
It does not erase legal rights or historical claims.
It does not guarantee the most efficient allocation will feel fair.
And a mathematically fair allocation can still be unjust if the model omitted something society rightly cares about.
Frequently Asked Questions
What is proportional fair division?
With n participants, a proportional allocation gives each person at least 1/n of the total value according to their own valuation.
What does envy-free mean?
An allocation is envy-free if no participant prefers another participant’s allocation to their own when evaluated using their own preferences.
Are equal shares always fair?
No. Equal physical quantities can have very different subjective values. Fair division often uses value-based rather than size-based guarantees.
Can indivisible goods always be divided envy-free?
No. Exact envy-free allocations may not exist when valuable objects cannot be split. Relaxations such as EF1 are therefore widely studied.
What is Pareto efficiency?
An allocation is Pareto efficient if no other feasible allocation can make at least one person better off without making somebody else worse off.
Sources and Further Reading
- Aumann and Dombb, The Efficiency of Fair Division with Connected Pieces, defining proportional, envy-free and equitable cake divisions and discussing existence and efficiency.
- Research literature in fair division and computational social choice on cake cutting, indivisible goods, EF1, EFX and Nash social welfare.
Continue Through eduKateSG
Continue with How Mathematics Works. This article connects to When the Right Match Can Save a Life, where allocation again combines mathematical efficiency with human values, and to Ranking Teams That Never All Play Each Other, where the meaning of fairness depends on what comparison the system is trying to make.
Final Thought: Fairness Begins When We Stop Pretending Everyone Wants the Same Thing
Two people look at the same resource.
They see different value.
That seems like the problem.
Often it is the solution.
You take what matters more to you.
I take what matters more to me.
Then Mathematics asks the harder questions.
Did everyone receive enough?
Does anyone want to swap?
Did we waste value?
Could somebody have been helped without hurting anybody?
And if no perfect answer exists, what guarantee can we still defend?
That is Mathematics improving the world not by making fairness easy, but by making it precise enough to argue about honestly.