3 research outputs found
Randomness in Competitions
We study the effects of randomness on competitions based on an elementary
random process in which there is a finite probability that a weaker team upsets
a stronger team. We apply this model to sports leagues and sports tournaments,
and compare the theoretical results with empirical data. Our model shows that
single-elimination tournaments are efficient but unfair: the number of games is
proportional to the number of teams N, but the probability that the weakest
team wins decays only algebraically with N. In contrast, leagues, where every
team plays every other team, are fair but inefficient: the top of
teams remain in contention for the championship, while the probability that the
weakest team becomes champion is exponentially small. We also propose a gradual
elimination schedule that consists of a preliminary round and a championship
round. Initially, teams play a small number of preliminary games, and
subsequently, a few teams qualify for the championship round. This algorithm is
fair and efficient: the best team wins with a high probability and the number
of games scales as , whereas traditional leagues require N^3 games to
fairly determine a champion.Comment: 10 pages, 8 figures, reviews arXiv:physics/0512144,
arXiv:physics/0608007, arXiv:cond-mat/0607694, arXiv:physics/061221
Dynamics of Three Agent Games
We study the dynamics and resulting score distribution of three-agent games
where after each competition a single agent wins and scores a point. A single
competition is described by a triplet of numbers , and denoting the
probabilities that the team with the highest, middle or lowest accumulated
score wins. We study the full family of solutions in the regime, where the
number of agents and competitions is large, which can be regarded as a
hydrodynamic limit. Depending on the parameter values , we find six
qualitatively different asymptotic score distributions and we also provide a
qualitative understanding of these results. We checked our analytical results
against numerical simulations of the microscopic model and find these to be in
excellent agreement. The three agent game can be regarded as a social model
where a player can be favored or disfavored for advancement, based on his/her
accumulated score. It is also possible to decide the outcome of a three agent
game through a mini tournament of two-a gent competitions among the
participating players and it turns out that the resulting possible score
distributions are a subset of those obtained for the general three agent-games.
We discuss how one can add a steady and democratic decline rate to the model
and present a simple geometric construction that allows one to write down the
corresponding score evolution equations for -agent games
How to Choose a Champion
League competition is investigated using random processes and scaling
techniques. In our model, a weak team can upset a strong team with a fixed
probability. Teams play an equal number of head-to-head matches and the team
with the largest number of wins is declared to be the champion. The total
number of games needed for the best team to win the championship with high
certainty, T, grows as the cube of the number of teams, N, i.e., T ~ N^3. This
number can be substantially reduced using preliminary rounds where teams play a
small number of games and subsequently, only the top teams advance to the next
round. When there are k rounds, the total number of games needed for the best
team to emerge as champion, T_k, scales as follows, T_k ~N^(\gamma_k) with
gamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 for k=1,2,3.
These results suggest an algorithm for how to infer the best team using a
schedule that is linear in N. We conclude that league format is an ineffective
method of determining the best team, and that sequential elimination from the
bottom up is fair and efficient.Comment: 6 pages, 3 figure