Transitive tournament
The scores are (4, 3, 2, 1, 0). Every result agrees with the final ordering, so there are no strict upsets.
03 / MY RESEARCH
My Research
Predictability in Sports · Illinois Mathematics Lab, UIUC Mathematics
UNDERGRADUATE RESEARCHER
What can tournament
results tell us?
I investigate sports predictability using directed tournaments, score sequences, and strict upset frequencies. The work combines computational enumeration with mathematical proof.
In a tournament, each pair of teams plays once with one winner. Each team’s score is its total number of wins. A strict upset occurs when a team defeats an opponent that finishes with more wins; tied final scores do not count.
My Python work includes enumeration and isomorphism checks across all 1,024 labeled five-team tournaments, which fall into 12 isomorphism classes.
Reverse match results to update the graph, scores, and strict upsets. Then explore the exact distribution across all five-team tournaments and filter by score sequence.
This new HTML companion computes the five-team examples directly in your browser. Its enumerated results illustrate the definitions used in my work; it does not substitute for the project’s general proofs.
TWO USEFUL EXTREMES
Balanced tournament
Each team has two wins: (2, 2, 2, 2, 2). There are no strict upsets because all final scores are tied, even though the tournament has cycles.
Zero strict upsets does not imply a single consistent ranking. The score-based definition excludes all matches between teams with equal final scores.
Ongoing questions
I am studying constructions with many upsets and the structure of tournaments that maximize them. My work also includes proof arguments concerning a 75% asymptotic bound on strict upset frequency.
The five-team explorer is a finite example. It does not establish an asymptotic result, and this page presents the broader proof work as ongoing research.
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