Why a tournament’s schedule changes its ceiling for unequal results

Equal numbers of matches do not mean equal possibilities for domination. In a preprint, Gergely Csáji and László Csató show how the structure of an incomplete round-robin schedule shapes the most unequal points distribution it can produce—a benchmark needed when comparing competitive balance across competitions.

A benchmark behind competitive balance

How unequal can a league table become? The answer depends not just on results, but on which teams are scheduled to meet. A preprint by Gergely Csáji and László Csató examines that dependence using mathematical proofs, rather than measurements of changing competitive balance in actual leagues.

Their focus is the Herfindahl–Hirschman Index, or HHI. Originally a measure of market concentration, it can also describe how tournament points are distributed: calculate each team’s share of all points awarded, square those shares, and add them together. Higher values indicate greater concentration.

To put that number on a comparable scale across seasons or competitions, researchers need its attainable minimum and maximum. If all teams play equally many matches, the minimum is straightforward: equal points produce an HHI of $1/N$, where $N$ is the number of teams. Finding the maximum is much harder.

The schedule becomes a network

The authors study incomplete round-robin tournaments, in which every team plays the same number of matches but faces only some of the other teams. Opponents are fixed in advance, unlike a Swiss-system competition whose pairings develop during play.

They represent the schedule as a network: teams are vertices and matches are edges. Every vertex has the same number of edges, reflecting equal match totals. The mathematical task is to assign wins, draws and losses to those scheduled matches so that HHI is as large as possible.

This is an extreme-outcome benchmark, not a prediction that any particular results will occur.

Two structures define the extremes

Under either two points or three points for a win, with one for a draw and zero for a loss, the authors prove that HHI cannot exceed $2/N$. A schedule reaches that ceiling exactly when its network is bipartite: teams can be split into two sets, with every match crossing between them.

The reason is intuitive. All teams on one side can win every match, while all teams on the other side lose every match. Because everyone plays equally often, half the teams then collect all the points.

The paper illustrates this possibility with a six-team group in the Romanian football cup. Its schedule permits the points distribution $(9,9,9,0,0,0)$ under three-point wins. This is a possible maximum-concentration configuration, not a reported set of actual results.

At the opposite extreme, independent, equally sized round-robin groups have the smallest maximum HHI among schedules with the same team and match totals, provided those groups can be formed exactly. If each team plays $d$ matches, that requires $N$ to be divisible by $d+1$.

These groups constrain concentration because teams must play every other member of their group. They cannot all win together. For win values of two through five points, this group structure uniquely achieves the smallest maximum. At higher win values, other structures can tie it. The paper also supplies a counterexample showing why the divisibility condition cannot simply be dropped.

Draws can increase concentration

With three-point wins, maximising inequality does not necessarily mean eliminating every draw. A decisive match awards three points overall; a draw awards only two. Draws can therefore reduce the total points against which a leading team’s share is measured.

In the paper’s four-team round-robin example, $(9,2,2,2)$ produces a higher HHI than the entirely decisive distribution $(9,6,3,0)$. The leader wins every match while the other teams draw with each other.

For independent round-robin groups, the authors derive an exact formula using configurations with a hierarchy of winners above a remaining tier of teams that draw among themselves. They also show that the universal upper bound changes when a win is worth at least four points: attaining it requires a particular mixture of decisive bipartite components and all-draw components.

Balanced pots do not remove computational difficulty

The preprint proves that calculating the exact maximum is NP-hard even for structured schedules with equal-sized seeding pots and equal numbers of opponents from each relevant pot. This includes constructions allowing matches within a team’s own pot and constructions excluding them.

Hardness persists for every fixed positive number of opponents per pot, under two-point wins, fixed integer win values of at least three, and settings prohibiting draws. The proofs encode logical satisfiability problems into tournament networks. These are worst-case complexity results, not demonstrations that every real schedule is impractical to analyse.

An appendix extends the fixed-total-points findings to a three-two-one-zero system with shootouts. Its maximum HHI equals that under two-one-zero scoring, and every maximising configuration contains only regular wins and losses.

The practical next step remains open: stronger approximation methods and tools for actual schedules. The authors also propose studying randomly generated balanced schedules, to distinguish exceptional possibilities for concentration from typical ones.