How Mathematics Improves The World | Ranking Teams That Never All Play Each Other
Suppose Team A has won eight games and lost two.
Team B has won seven and lost three.
A is better?
Maybe.
Now discover that A played mostly weak opponents.
B played four of the strongest teams in the competition.
Still obvious?
Now make the schedules unequal.
Some teams played eleven games.
Some nine.
Some met twice.
Some never met at all.
One division is stronger than another.
Home advantage is unevenly distributed.
One star player missed half the season.
A league table that simply counts wins begins to confuse performance with schedule.
This is why ranking is a Mathematics problem.
The goal is not merely to count what happened.
It is to estimate relative strength from an incomplete network of comparisons.
Quick Read
Ranking is straightforward in a balanced round-robin tournament: every team plays every other team the same number of times, and total points or win percentage has a clear interpretation. Many real competitions are not balanced. Conferences, divisions, travel constraints, cancellations and incomplete seasons mean teams face different opponents and play different numbers of games.
Mathematics corrects this by modelling games as pairwise comparisons. Teams are nodes in a graph; games are edges. A rating method can estimate latent team strength so results against strong opponents count differently from results against weak ones. Elo-type systems update ratings after each game according to the difference between actual and expected outcome. Least-squares and generalized-row-sum systems use the whole comparison graph at once. Other methods incorporate margin of victory, home advantage, recency or uncertainty.
The International Chess Federation’s rating system is one prominent real-world pairwise model. FIDE converts rating differences into expected score probabilities, then updates each player from the difference between actual score and expected score, scaled by a development coefficient K. Sports teams can use related Elo-family ideas, although each sport requires choices about draws, score margin, home advantage and season structure.
Research on incomplete sports seasons shows why simple tables can become unfair when schedules are unequal. An Oxford Academic study of interrupted European football seasons analyses rankings that explicitly adjust for opponents’ strength; other research demonstrates that unbalanced schedules can distort league tables when teams do not face comparable opposition.
One-sentence answer: Mathematics improves the world by turning an incomplete schedule into a network of evidence, estimating how strong each team must be to explain its results against opponents of different strength rather than pretending every win was earned against the same difficulty.
A Win Is Not a Unit of Difficulty
Beat the weakest team.
Beat the champion.
Both enter the standings as one win in many sports.
They are not equally informative about team strength.
If a powerful team beats a very weak team, that result was expected.
If the weak team wins, the upset is highly informative.
A rating system should therefore care about surprise.
Actual result matters.
Expected result matters too.
The Comparison Graph
Represent each team as a node.
Draw an edge whenever two teams play.
A balanced round robin creates a complete or repeatedly complete graph.
An unbalanced league creates a sparse graph.
Some nodes have many edges.
Some fewer.
Some clusters play heavily within themselves.
Cross-cluster games are rare.
The ranking problem is now:
How can we infer one common strength scale from this incomplete graph?
Connectivity: Two Groups That Never Meet Cannot Be Ranked Reliably Against Each Other
Conference East plays only East.
Conference West plays only West.
No interconference games exist.
Within each conference we can rank teams.
Between conferences, the data contains no direct bridge.
The comparison graph is disconnected.
No mathematical cleverness can recover information the schedule never generated without adding prior assumptions.
One cross-conference match suddenly links the scales.
Schedule design creates identifiability.
Win Percentage Is a Good Statistic Under the Right Design
If everybody plays the same opponents under similar conditions, win percentage is powerful and transparent.
The problem begins when exposure differs.
A 70% record against the strongest schedule may indicate greater strength than an 80% record against the weakest.
Simple statistics are not primitive.
They are conditionally correct.
The question is whether the competition design satisfies those conditions.
Strength of Schedule: Count Who You Had to Beat
A first correction calculates opponents’ strength.
One crude approach averages opponents’ win percentages.
But that creates recursion.
Your opponents’ records depend on who they played.
The strength-of-schedule problem points naturally toward systems that solve all ratings simultaneously.
Every team’s strength depends partly on everyone else’s strength.
Elo’s Core Idea: Reward Surprise
Arpad Elo’s rating family became famous through chess.
The central idea is simple.
Given two current ratings, compute expected score.
Compare expectation with reality.
Update.
If a favourite wins, rating changes modestly.
If an underdog wins, both move more.
Information value is proportional to surprise.
Expected Score as a Probability
A common Elo-type logistic expression is:
EA = 1 / (1 + 10(RB−RA)/400)
If ratings are equal, expected score is 0.5.
A 200-point advantage corresponds to a substantially higher expected score under the conventional Elo scale.
FIDE currently implements the same general expected-score principle through official rating-difference probability tables rather than requiring users to apply this particular formula directly.
Probability converts rating difference into expectation.
Rating Update: Actual Minus Expected
The standard Elo-family update is:
R′ = R + K(S − E)
S is actual score.
E is expected score.
K controls update speed.
FIDE’s current regulations express the same structure explicitly: calculate scoring probability PD, compute ΔR = score − PD, sum across games and multiply by K.
The rating learns from prediction error.
K-Factor: How Quickly Should We Believe New Evidence?
A large K means ratings move quickly.
Useful for new teams whose true strength is uncertain.
Dangerous if one lucky game causes wild swings.
A small K creates stability.
Useful for established teams.
Slow if real strength changes quickly.
FIDE varies K by player experience, age and rating category.
In team sports, analogous systems may vary update rates by season stage or uncertainty.
Learning rate is part of the rating model.
Home Advantage: Same Teams, Different Conditions
A team often performs better at home.
Travel.
Crowd.
Familiarity.
Altitude or climate.
Officials and psychology may contribute depending on sport.
A ranking model can add a home parameter h:
effective rating difference = Rhome + h − Raway
If one team played eight home games and another four, raw records are not directly comparable without considering schedule location.
Draws: Half a Win Is Not Always the Best Probability Model
Chess Elo treats a draw as score 0.5.
Many sports award league points differently.
Football often awards 3 for a win and 1 for a draw.
Hockey may distinguish regulation and overtime results.
A rating system can model win/draw/loss probabilities separately using ordered-logit or Davidson-type extensions.
The scoring rule and predictive probability model need not be identical.
Margin of Victory: Is Winning 10–0 More Informative Than 1–0?
Binary ranking discards score margin.
A 10–0 win and 1–0 win both become W.
Margin can contain information about strength.
Using it creates incentives.
If rankings reward large margins aggressively, teams may continue attacking hopeless opponents merely to improve rating.
Sportsmanship and strategic behaviour enter the mathematics.
Many systems damp margin effects so the tenth goal matters far less than the first.
Least Squares: Fit Strengths to Score Differences
Suppose each game suggests:
team strength A − team strength B ≈ observed performance difference
Across many games, no set of ratings satisfies all equations exactly.
Least squares chooses ratings that minimise total squared residual.
In matrix form:
minimise ||Xr − y||²
X encodes who played whom.
r contains team ratings.
y contains observed outcomes or margins.
The whole season is solved simultaneously.
Why One Rating Must Be Anchored
If every rating increases by 100 points, predicted rating differences stay unchanged.
The data identifies relative strength, not absolute zero.
So least-squares systems need an identifying constraint:
- mean rating = 0;
- or one reference team fixed;
- or an external prior scale.
This is a general mathematical idea.
Some models identify only differences.
A coordinate origin is a convention.
Generalized Row Sum: Wins Plus Opponent Strength
In an incomplete tournament, one can adjust raw scores according to strength of opponents.
The generalized row-sum family does this recursively.
The Oxford study of incomplete European football seasons shows that such methods can satisfy desirable fairness axioms when teams have played different schedules.
The well-known least-squares ranking arises as an extreme case giving strong influence to opponent strength.
The mathematics makes schedule difficulty explicit instead of burying it inside the table.
Colley-Style Rankings: Use Wins and Losses but Regularise Early
Another family builds a linear system from wins, losses and opponents.
A prior pull toward average strength prevents teams with tiny samples from jumping to absurd extremes.
Beat one strong opponent in your only game.
You should rise.
You should not instantly become the unquestioned best team in the country.
Regularisation protects rankings from small-sample volatility.
Bradley–Terry Models: Pairwise Games as Logistic Regression
Let each team have positive strength parameter π.
A classic Bradley–Terry probability is:
P(A beats B)=πA/(πA+πB)
Write strengths on a log scale and this becomes logistic regression on rating difference.
Maximum likelihood estimates strengths from all game outcomes.
Home advantage and covariates can be added naturally.
Elo can be understood as an online relative of this pairwise probability approach.
Bayesian Ratings: A Team Has a Distribution, Not Just a Number
Team A rating: 1650.
Team B rating: 1630.
Is A definitely stronger?
No.
If A has played two games and B thirty, their uncertainties differ dramatically.
Bayesian systems represent strength as a posterior distribution.
A new team begins with a broad prior.
Results narrow it.
Rankings can then report:
- mean strength;
- credible interval;
- probability A is stronger than B.
Uncertainty belongs beside the ranking.
Glicko-Type Thinking: Rating Deviation Matters
Systems descended from Elo can add explicit rating uncertainty.
An inactive competitor’s uncertainty grows because current strength becomes less certain.
A highly active competitor’s rating deviation shrinks.
Then an upset against an uncertain new player should move established ratings differently from an upset against a precisely known elite player.
How certain we are about the opponent is part of the evidence.
Recency: Last Year’s Champion May Not Be Today’s Strongest Team
Teams change.
Players transfer.
Coaches change.
Injuries matter.
A ranking intended to predict tomorrow should discount old evidence eventually.
A historical championship table should not.
This creates an important distinction:
Are we ranking achievement, or estimating present strength?
Different jobs deserve different Mathematics.
Descriptive Ranking and Predictive Rating Are Not the Same
A league table may deliberately reward achievement.
Three points for a win.
Who accumulated the most?
A predictive rating asks:
Who would be favoured if these teams played tomorrow?
A team can lead the standings and have lower estimated present strength after injuries.
That is not necessarily contradiction.
The metrics answer different questions.
Incomplete Seasons: When the Table Stops Before the Schedule Does
A season is interrupted.
Some teams played difficult opponents early.
Others had their hardest games still ahead.
Simple points per game corrects unequal match count.
It does not correct unequal schedule strength.
The Oxford study on pandemic-interrupted football shows how axiomatic ranking methods can adjust incomplete schedules using opponent strength.
Fairness becomes a formal property rather than an argument on television.
Unbalanced Schedules Can Distort League Tables
North American professional leagues often schedule more games within divisions or conferences than across the entire league.
If one division is unusually strong, its teams may accumulate more losses simply because they play each other often.
Another division may inflate records internally.
Research on unbalanced sports schedules has documented distortions arising from this structure.
A schedule is not neutral measurement design.
It determines which comparisons exist.
Schedule Design Is Experimental Design
If the purpose is to rank teams accurately, schedule planners should create informative comparisons.
More cross-division games improve network connectivity.
Balanced home/away assignments reduce location bias.
Repeated games reduce random noise but consume schedule capacity.
Sports scheduling is therefore experimental design under travel, broadcast, stadium and fairness constraints.
Better comparison data can improve ranking before any algorithm changes.
Transitivity Can Fail
A beats B.
B beats C.
Does A beat C?
Not necessarily.
Styles create matchup effects.
One team handles pressing well and struggles against deep defence.
Another has the opposite pattern.
A single scalar rating assumes much of strength can be ordered on one dimension.
Real competition can contain rock-paper-scissors structure.
The ranking compresses a higher-dimensional matchup system into one number.
Team Strength Is Not Constant Within a Season
A team begins the year poorly.
Changes coach.
Improves dramatically.
A static season-wide model averages two different teams under one name.
Dynamic rating models allow strength to evolve as a random process:
Rt = Rt−1 + process noise
A Kalman-filter or state-space approach can estimate changing strength over time.
The team becomes a trajectory, not a constant.
Injuries and Lineups: The Team Name Hides Who Actually Played
Team A without its starting goalkeeper is not exactly Team A at full strength.
Player-level models estimate individual contributions and aggregate them to expected team performance.
This can improve predictions.
It can also overfit complex lineup combinations.
A simple team rating is robust and interpretable.
A detailed player model is richer and data hungry.
Model resolution should match data resolution.
Luck and Skill: Ten Games Is a Small Sample
Suppose two equal teams each have 50% win probability.
Across ten games, one can easily finish 7–3 by chance.
A ranking that treats 7–3 as definitive overstates evidence.
Confidence intervals or Bayesian shrinkage pull small samples toward league average.
As games accumulate, the data earns permission to move farther.
Regression to the Mean: Extreme Early Records Tend to Moderate
A team starts 5–0.
Some of that may be genuine strength.
Some may be favourable opponents and random close-game outcomes.
Future performance tends to move closer to underlying ability than the extreme observed start.
Statistical ranking separates latent strength from noisy record.
The ranking should not chase every streak.
Overtime and Penalties: One Game Can Contain More Than One Level of Evidence
A football match is level after 90 minutes and decided by penalties.
Should the rating treat that as a full-strength win?
A hockey game decided after overtime may award different standings points.
Sports-specific ratings decide how much evidence each outcome carries about ordinary-play strength.
Rules become statistical semantics.
Ranking and Seeding Create Feedback Loops
Rankings determine tournament seeds.
Seeds determine early opponents.
Early opponents affect future results and ranking.
A ranking system therefore changes the competition it later measures.
This feedback is usually intentional: strong teams are separated early.
It can also entrench historical advantage if ratings update too slowly.
Governance should understand the loop.
Promotion and Relegation: How Do We Compare Teams Across Divisions?
A second-division team wins 80% of its games.
A top-division team wins 40%.
Which is stronger?
Cross-division cup games, promoted-team performance and relegated-team performance create bridges.
Hierarchical models can estimate division-level strength offsets.
Without cross-level evidence, direct comparison is weak.
Again, the graph needs edges.
International Rankings: Geography Makes Schedules Sparse
National teams play infrequently.
Most matches happen within confederations.
Cross-confederation games are limited.
Home advantage, travel and competition importance differ.
International ranking is therefore especially dependent on modelling choices and bridge matches.
A sparse comparison network creates genuine uncertainty.
Ranking Fairness: Transparency Matters
A mathematically sophisticated ranking can be accurate and politically unusable if nobody understands it.
Sports rankings allocate:
- playoff places;
- seeds;
- qualification;
- prize money;
- reputation.
Stakeholders deserve to know:
- what results count;
- how recent games are weighted;
- whether margin matters;
- how opponent strength is handled;
- how ties are broken.
Transparency is part of fairness.
Goodhart’s Law: Teams Learn the Ranking Rules
If ranking rewards score margin heavily, teams run up scores.
If only wins matter, teams become risk-averse late in close games.
If strength of schedule matters, teams seek stronger non-league opponents.
Once a metric determines rewards, participants optimise against it.
A ranking rule therefore becomes part of game strategy.
Metric design should anticipate behaviour it creates.
Predictive Validation: Did the Rating Actually Predict Future Games?
A ranking can look plausible.
Test it.
Use ratings before each game to predict outcome probabilities.
Compare with results using:
- log loss;
- Brier score;
- calibration curves;
- accuracy;
- rank correlation.
If teams given 70% win probability actually win about 70% of such games, probabilities are calibrated.
Prediction supplies a reality receipt for rating quality.
Ranking Stability: Would One Result Rewrite the Whole Table?
Remove one close game.
Does Team A fall from first to eighth?
If yes, the ranking is fragile.
Bootstrap resampling can repeatedly resample games and recompute ranks.
Then report:
Team A ranks first in 52% of plausible resamples, second in 31%, third in 12%.
The single ordinal ranking hides uncertainty that the bootstrap reveals.
A Classroom Thought Experiment: The Three-Team Trap
Team A beats B.
B beats C.
C beats A.
Who is best?
There is no perfectly transitive ordering from those three results alone.
Add score margins.
Add repeated matches.
Add a fourth team that plays all three.
The ranking becomes better identified.
Students discover that more connected evidence improves inference.
A Second Thought Experiment: Same Record, Different Schedule
Team X goes 4–1 against opponents rated:
1000, 1000, 1000, 1000, 1000
Team Y goes 4–1 against:
1400, 1450, 1500, 1550, 1600
Same record.
Very different evidence.
Ask students to design a ranking that distinguishes them.
They have invented strength of schedule.
Primary Mathematics: Ranking Begins With Fractions and Tables
Primary students know:
- wins and losses;
- fractions;
- percentages;
- tables;
- averages;
- simple graphs.
Win percentage is a fraction.
Strength of schedule is an average.
A league table is ordered data.
The sophisticated rating systems grow from elementary comparison.
Secondary Mathematics: Ranking Becomes Probability
Secondary students add:
- probability;
- logistic functions;
- simultaneous equations;
- matrices;
- least squares;
- statistics.
Elo turns rating difference into probability.
Least squares fits an entire schedule graph.
Uncertainty becomes intervals.
Sports becomes applied statistics.
Advanced Mathematics: Rankings as Network Inference
Modern rating systems draw on:
- graph theory;
- maximum likelihood;
- Bayesian inference;
- linear algebra;
- optimisation;
- state-space models;
- bootstrap methods;
- decision theory.
Games are edges.
Strength is latent.
The schedule defines which differences can be learned.
The ranking is an inference from a graph.
Why This Improves the World
1. It makes unequal schedules more comparable
Opponent-adjusted systems distinguish easy and difficult records.
2. It converts games into calibrated expectations
Elo-type ratings say not only who is stronger, but how surprising a result should be.
3. It improves seeding and tournament design
Better ratings reduce the chance strongest competitors meet too early merely because schedules were uneven.
4. It makes uncertainty visible
Intervals and rating deviations distinguish “probably first” from “certainly first”.
5. It reveals weaknesses in competition design
Disconnected or heavily clustered schedules show where rankings cannot be strongly identified.
6. It teaches that fairness depends on the question
A championship table, predictive rating and seeding index may legitimately differ because they serve different purposes.
What Mathematics Does Not Do
A rating does not prove the higher-rated team will win the next game.
It does not remove injuries, styles or matchup effects.
It does not create cross-conference evidence if no cross-conference games were played.
It does not decide whether rankings should reward achievement or estimate present strength.
It does not make margin-of-victory incentives ethically neutral.
It does not guarantee a complicated method is fairer than a transparent simple one.
And one ranking number should never be mistaken for the full multidimensional reality of a team.
Frequently Asked Questions
What is an Elo rating?
An Elo-family rating estimates competitor strength from pairwise results. Expected outcome is calculated from rating difference, then ratings are updated according to actual outcome minus expected outcome.
Why does strength of schedule matter?
Teams facing stronger opponents should generally receive more credit for the same win-loss record because their observed results were generated under greater competitive difficulty.
Can two teams with the same record have very different ratings?
Yes. Their opponents, game locations, score margins and timing may differ. A rating model uses this additional evidence while a basic record does not.
Why can’t two disconnected leagues be ranked exactly?
Without cross-league games or external assumptions, there is no evidence fixing the relative strength scale between them. Each league can be ranked internally but their absolute offset remains unidentified.
Is the highest-rated team always the champion?
No. A championship rewards performance under the competition’s rules. A predictive rating estimates underlying strength. The two can differ, especially with short seasons, playoffs or changing team form.
Sources and Further Reading
- FIDE Handbook, FIDE Rating Regulations, including expected scoring probabilities and rating updates based on score minus expected score.
- Csató, Coronavirus and Sports Leagues: Obtaining a Fair Ranking When the Season Cannot Resume, IMA Journal of Management Mathematics, on incomplete schedules and opponent-strength adjustment.
- The Extent to Which Unbalanced Schedules Cause Distortions in Sports League Tables, on schedule imbalance in professional sports.
- FIDE, New FIDE Rating and Title Regulations Come Into Effect, background on the 2024 rating-system changes.
Continue Through eduKateSG
Continue with How Mathematics Works. This article connects to Finding Useful Pages in a Web Too Large to Read: both convert a network of relationships into an ordering. It also connects to Counting Wild Animals Without Seeing Every Animal, because both separate what was observed from the hidden quantity we actually want to estimate.
Final Thought: A League Table Is a Measurement Instrument
A team wins.
Another loses.
The scoreboard records the event perfectly.
The difficult part comes later.
How much should that result change what we believe about both teams?
Who did they play before?
How strong were those opponents?
Was the game home or away?
How uncertain were the ratings?
Did the schedules connect every part of the competition?
Mathematics improves the world here by refusing to pretend that every record was generated under the same test.
It reads the schedule as part of the evidence.
And that makes ranking not merely an act of ordering, but an act of measurement.