A new set of axioms and new method (equal gap seeding) are designed. The equal gap seeding is the unique seeding that, under the deterministic domain assumption, satisfies the delayed confrontation, ...fairness, increasing competitive intensity and equal rank differences axioms. The equal gap seeding is the unique seeding that, under the linear domain assumption, maximizes the probability that the strongest participant is the winner, the strongest two participants are the finalists, the strongest four participants are the quarterfinalists, etc.
Tournaments can be used to model a variety of practical scenarios including sports competitions and elections. A natural notion of strength of alternatives in a tournament is a generalized king: an ...alternative is said to be a
k
-
king
if it can reach every other alternative in the tournament via a directed path of length at most
k
. In this paper, we provide an almost complete characterization of the probability threshold such that all, a large number, or a small number of alternatives are
k
-kings with high probability in two random models. We show that, perhaps surprisingly, all changes in the threshold occur in the range of constant
k
, with the biggest change being between
k
=
2
and
k
=
3
. In addition, we establish an asymptotically tight bound on the probability threshold for which all alternatives are likely able to win a single-elimination tournament under some bracket.
Tournaments are popularly used in sporting events to select a champion. They are also used in experiments where paired comparison procedures are needed. Knockout tournaments are most useful when the ...number of treatments is too large to use the more well-known round-robin tournament. The mathematical and statistical literature does not address questions such as how one should seed teams in a tournament or which tournament structures are appropriate. For double-elimination (DE) tournaments even basic questions such as how many structures there are remain unanswered. This article addresses some fundamental questions concerning DE tournaments, including the number of DE tournaments and the probabilities of teams winning a DE tournament. Edwards gives many results about single-elimination (SE) tournaments, such as the probability of winning an SE tournament and a notation for labeling and counting them. In this article I develop similar results for the DE tournaments. The results given apply to an arbitrary number of teams, and not just four or eight, as is popular in the literature.
I examine a game-theoretical model of two variants of double-elimination tournaments, and derive the equilibrium behavior of symmetric players and the optimal prize allocation assuming a designer ...aims to maximize total effort. I compare these theoretical properties to the well-known single-elimination tournament.
•I studied a game-theoretical model of double-elimination tournaments.•Compared to single-elimination tournaments, players have a second chance to compete.•The standard version produces higher total effort than single-elimination.•The variant version however may produce lower total effort than single-elimination.•Granting a second chance to symmetric players may create asymmetrical incentives.
Tournaments are used to select a single winner from a group of participants in a sporting event or a paired-comparison experiment. This study compares different draws, or pairings, of teams in ...single-elimination tournaments under a set of relatively nonrestricting assumptions about the participating teams' pairwise probabilities of winning. We analyze and compare draws for four-team tournaments using various criteria, then attempt to generalize the results to eight-team tournaments. For example, only one-four team draw maximizes the probability that the best team wins for all pairwise probabilities, whereas eight draws are possibly optimal for eight-team tournaments.
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