dorsal/arxiv
View SchemaHow to Choose a Champion
| Authors | E. Ben-Naim, N. W. Hengartner |
|---|---|
| Categories | |
| ArXiv ID | physics/0612217 |
| URL | https://arxiv.org/abs/physics/0612217 |
| DOI | 10.1103/PhysRevE.76.026106 |
| Journal | Phs. Rev. E 76, 026106 (2007) |
Abstract
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.
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"abstract": "League competition is investigated using random processes and scaling\ntechniques. In our model, a weak team can upset a strong team with a fixed\nprobability. Teams play an equal number of head-to-head matches and the team\nwith the largest number of wins is declared to be the champion. The total\nnumber of games needed for the best team to win the championship with high\ncertainty, T, grows as the cube of the number of teams, N, i.e., T ~ N^3. This\nnumber can be substantially reduced using preliminary rounds where teams play a\nsmall number of games and subsequently, only the top teams advance to the next\nround. When there are k rounds, the total number of games needed for the best\nteam to emerge as champion, T_k, scales as follows, T_k ~N^(\\gamma_k) with\ngamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 for k=1,2,3.\nThese results suggest an algorithm for how to infer the best team using a\nschedule that is linear in N. We conclude that league format is an ineffective\nmethod of determining the best team, and that sequential elimination from the\nbottom up is fair and efficient.",
"arxiv_id": "physics/0612217",
"authors": [
"E. Ben-Naim",
"N. W. Hengartner"
],
"categories": [
"physics.soc-ph",
"cond-mat.stat-mech",
"math.PR"
],
"doi": "10.1103/PhysRevE.76.026106",
"journal_ref": "Phs. Rev. E 76, 026106 (2007)",
"title": "How to Choose a Champion",
"url": "https://arxiv.org/abs/physics/0612217"
},
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