Robust Draws in Balanced Knockout Tournaments

Balanced knockout tournaments are ubiquitous in sports competitions and are also used in decision-making and elections. The traditional computational question, that asks to compute a draw (optimal draw) that maximizes the winning probability for a distinguished player, has received a lot of attention. Previous works consider the problem where the pairwise winning probabilities are known precisely, while we study how robust is the winning probability with respect to small errors in the pairwise winning probabilities. First, we present several illuminating examples to establish: (a)~there exist deterministic tournaments (where the pairwise winning probabilities are~0 or~1) where one optimal draw is much more robust than the other; and (b)~in general, there exist tournaments with slightly suboptimal draws that are more robust than all the optimal draws. The above examples motivate the study of the computational problem of robust draws that guarantee a specified winning probability. Second, we present a polynomial-time algorithm for approximating the robustness of a draw for sufficiently small errors in pairwise winning probabilities, and obtain that the stated computational problem is NP-complete. We also show that two natural cases of deterministic tournaments where the optimal draw could be computed in polynomial time also admit polynomial-time algorithms to compute robust optimal draws

Who Can Win a Single-Elimination Tournament?

A single-elimination (SE) tournament is a popular way to select a winner in both sports competitions and in elections. A natural and well-studied question is the tournament fixing problem (TFP): given the set of all pairwise match outcomes, can a tournament organizer rig an SE tournament by adjusting the initial seeding so that their favorite player wins? We prove new sufficient conditions on the pairwise match outcome information and the favorite player, under which there is guaranteed to be a seeding where the player wins the tournament. Our results greatly generalize previous results. We also investigate the relationship between the set of players that can win an SE tournament under some seeding (so called SE winners) and other traditional tournament solutions. In addition, we generalize and strengthen prior work on probabilistic models for generating tournaments. For instance, we show that \emph{every} player in an $n$ player tournament generated by the Condorcet Random Model will be an SE winner even when the noise is as small as possible, $p=\Theta(\ln n/n)$; prior work only had such results for $p\geq \Omega(\sqrt{\ln n/n})$. We also establish new results for significantly more general generative models.Comment: A preliminary version appeared in Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI), 201

Single-Elimination Brackets Fail to Approximate Copeland Winner

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature

Weak transitivity and agenda control for extended stepladder tournaments

A tournament graph over n players is weakly transitive at player p if it contains a Hamiltonian path (p1,p2,…,pn) with p1=p such that for all odd integers i≤n−2 there is an arc from pi to pi+2. We show that weak transitivity at p suffices to make player p win any extended stepladder tournament of degree at most two

How to Design a Stable Serial Knockout Competition

We investigate a new tournament format that consists of a series of individual knockout tournaments; we call this new format a Serial Knockout Competition (SKC). This format has recently been adopted by the Professional Darts Corporation. Depending on the seedings of the players used for each of the knockout tournaments, players can meet in the various rounds (eg first round, second round, ..., semi-final, final) of the knockout tournaments. Following a fairness principle of treating all players equal, we identify an attractive property of an SKC: each pair of players should potentially meet equally often in each of the rounds of the SKC. If the seedings are such that this property is indeed present, we call the resulting SKC stable. In this note we formalize this notion, and we address the question: do there exist seedings for each of the knockout tournaments such that the resulting SKC is stable? We show, using a connection to the Fano plane, that the answer is yes for 8 players. We show how to generalize this to any number of players that is a power of 2, and we provide stable schedules for competitions on 16 and 32 player

A theory of knockout tournament seedings

This paper provides nested sets and vector representations of knockout tournaments. The paper introduces classification of probability domain assumptions and a new set of axioms. Two new seeding methods are proposed: equal gap seeding and increasing competitive intensity seeding. Under different probability domain assumptions, several axiomatic justifications are obtained for equal gap seeding. A discrete optimization approach is developed. It is applied to justify equal gap seeding and increasing competitive intensity seeding. Some justification for standard seeding is obtained. Combinatorial properties of the seedings are studied

Condorcet-Consistent and Approximately Strategyproof Tournament Rules

We consider the manipulability of tournament rules for round-robin tournaments of $n$ competitors. Specifically, $n$ competitors are competing for a prize, and a tournament rule $r$ maps the result of all $\binom{n}{2}$ pairwise matches (called a tournament, $T$) to a distribution over winners. Rule $r$ is Condorcet-consistent if whenever $i$ wins all $n-1$ of her matches, $r$ selects $i$ with probability $1$. We consider strategic manipulation of tournaments where player $j$ might throw their match to player $i$ in order to increase the likelihood that one of them wins the tournament. Regardless of the reason why $j$ chooses to do this, the potential for manipulation exists as long as $\Pr[r(T) = i]$ increases by more than $\Pr[r(T) = j]$ decreases. Unfortunately, it is known that every Condorcet-consistent rule is manipulable (Altman and Kleinberg). In this work, we address the question of how manipulable Condorcet-consistent rules must necessarily be - by trying to minimize the difference between the increase in $\Pr[r(T) = i]$ and decrease in $\Pr[r(T) = j]$ for any potential manipulating pair. We show that every Condorcet-consistent rule is in fact $1/3$-manipulable, and that selecting a winner according to a random single elimination bracket is not $\alpha$-manipulable for any $\alpha > 1/3$. We also show that many previously studied tournament formats are all $1/2$-manipulable, and the popular class of Copeland rules (any rule that selects a player with the most wins) are all in fact $1$-manipulable, the worst possible. Finally, we consider extensions to match-fixing among sets of more than two players.Comment: 20 page

A paradox of tournament seeding

A mathematical model of seeding is analysed for sports tournaments where the qualification is based on round-robin contests. The conditions of strategyproofness are found to be quite restrictive: if each team takes its own coefficient (a measure of its past performance), only one or all of them should qualify from every round-robin contest. Thus the standard draw system creates incentives for tanking in order to be assigned to a stronger pot as each team prefers to qualify with teams having a lower coefficient. Major soccer competitions are shown to suffer from this weakness. Strategyproofness can be guaranteed by giving to each team the highest coefficient of all teams that are ranked lower in its round-robin contest. The proposal is illustrated by the 2020/21 UEFA Champions League.Comment: 23 pages, 3 table