Manipulating Tournaments in Cup and Round Robin Competitions

In sports competitions, teams can manipulate the result by, for instance, throwing games. We show that we can decide how to manipulate round robin and cup competitions, two of the most popular types of sporting competitions in polynomial time. In addition, we show that finding the minimal number of games that need to be thrown to manipulate the result can also be determined in polynomial time. Finally, we show that there are several different variations of standard cup competitions where manipulation remains polynomial.

💡 Research Summary

The paper investigates the computational problem of deliberately altering the outcomes of sports tournaments—specifically cup (single‑elimination) and round‑robin formats—by means of “throwing” games. The authors model a tournament as a complete directed graph (a tournament graph) where each vertex represents a team and each directed edge (vi, vj) indicates that vi defeats vj in a fair match. A coalition of teams (the manipulators) is allowed to reverse any edge that originates from a member of the coalition, reflecting the realistic constraint that a team can only throw a game it could have won. Two manipulation goals are considered: constructive (forcing a designated team vw to become the overall champion) and destructive (ensuring a designated team vl does not win).

Cup Competitions
For single‑elimination cups, the authors adapt the CSL algorithm originally developed by Conitzer, Sandholm, and Lang for the cup rule in elections. The algorithm proceeds recursively on the binary tree that defines the cup schedule. At each internal node, the set of “possible winners” from the left and right sub‑trees is computed. A team from the left can become a possible winner of the parent node if it either already defeats some team from the right in the original tournament graph or if the opponent belongs to the coalition and can be forced to lose by reversing the corresponding edge. The same condition applies symmetrically for teams from the right. This bottom‑up computation requires examining each pair of teams at most once, yielding a time complexity of O(m²), where m is the number of teams. Constructive manipulation is possible precisely when the target team appears in the possible‑winner set at the root. Destructive manipulation is handled by checking, for each opponent of the target, whether that opponent can be made a possible winner; if any such opponent exists, the target can be eliminated. Both tasks remain within the same polynomial bound.

Round‑Robin Competitions
Round‑robin tournaments are modeled with a scoring function S that assigns a pair of points (i, n‑i) to the two participants of a match, where n is a fixed total number of points per game. The authors show that when S has the normalized form S = {(i, n‑i) | 0 ≤ i ≤ n} (e.g., win‑loss, win‑draw‑loss, or any linear distribution of points), the manipulation problem can be reduced to a minimum‑cost flow problem. Fixed games (those not involving coalition members) have predetermined scores, while games involving coalition members are constrained: the coalition may only reduce its own points and increase the opponent’s points, which translates into a set of feasible edge capacities and costs in a flow network. Using results from Kern and Paulusma, they prove that determining whether a team can be made a winner under this normalized scoring is solvable in polynomial time. Conversely, for more general scoring schemes (such as the classic 3‑1‑0 win‑draw‑loss system), the manipulation decision becomes NP‑complete, aligning with prior work on election manipulation under arbitrary scoring rules. Destructive manipulation in round‑robin formats, however, remains polynomial: one simply maximizes the scores of all opponents while minimizing the target’s score under the allowed manipulations and checks whether any opponent can surpass the target.

Minimizing the Number of Manipulations
Beyond feasibility, the paper addresses the optimization problem of minimizing the number of games that must be thrown. For cups, the CSL recursion is augmented with a counter that tracks how many edge reversals are required to achieve a given possible winner; the algorithm selects, at each node, the option that yields the smallest cumulative count, still in O(m²) time. For round‑robin tournaments with normalized scoring, the flow network is extended with a unit cost on each “manipulation” edge, turning the feasibility flow into a min‑cost flow. Solving this yields the smallest set of games that need to be altered to guarantee the desired outcome, again in polynomial time.

Extensions to Variant Cup Formats
The authors further explore several variations of cup competitions: (1) reseeding after each round, (2) fixed versus unfixed seeding (where pairings can change dynamically), and (3) double‑elimination brackets (where a team must lose twice before exiting). They demonstrate that the same recursive approach can be applied after each reseeding step, and that a double‑elimination tournament can be treated as two sequential single‑elimination brackets, preserving polynomial‑time manipulability. Thus, a wide class of realistic tournament designs remains vulnerable to efficient manipulation.

Implications
The central insight is that when manipulation is limited to “throwing” games—i.e., reversing outcomes that the manipulators could have won—the structural simplicity of tournament graphs makes the problem tractable for the most common competition formats. Constructive and destructive manipulations can be detected quickly, and the minimal set of required manipulations can be computed efficiently. However, the choice of scoring rule critically affects complexity: normalized linear scoring keeps the problem easy, whereas richer scoring systems reintroduce NP‑hardness. This highlights a practical vulnerability: organizers seeking to safeguard fairness should consider tournament designs and scoring schemes that increase computational difficulty for potential manipulators, such as non‑linear point distributions or mechanisms that limit the ability to predict the impact of a single game on overall standings.

In summary, the paper provides a comprehensive algorithmic framework for analyzing and executing tournament manipulations in cup and round‑robin settings, establishes clear complexity boundaries based on scoring models, and extends the analysis to several realistic tournament variants, thereby offering both theoretical insight and practical guidance for the design of more manipulation‑resistant sporting competitions.