Tournaments. Dale Zimmerman University of Iowa. February 3, 2017

Transcription

1 Tournaments Dale Zimmerman University of Iowa February 3, 2017

2 2 Possible tournament design objectives 1. Provide incentives for teams to maximize performance before and during the tournament 2. Maximize chances that the best teams meet in later rounds of tournament, as fan interest peaks 3. Maximize chance that the best team wins 4. Maximize profits (ticket sales, TV rights, etc.)

3 3 Two main categories of tournaments 1. Round robin tournament May be balanced (e.g. Big East basketball or English Premier League regular season) or unbalanced (Big 10 basketball or NFL regular season) 2. Knockout tournament May be single elimination (tennis, match play golf, NFL post-season) or double elimination (NCAA wrestling) May use a series of games to determine which team advances (NBA, MLB)

4 4 Screening and seeding Screening refers to determining which teams are eligible for competition Regular season results used for most major sports Confederation results used for World Cup Previous season s winnings/past performance used in professional golf After a balanced round robin regular season, tournament seeding is necessary only if there are ties. After unbalanced round robin seasons, similar tie-breakers are often used for seeding.

5 5 Screening and seeding, continued: NFL has an unusual approach for determining the next year s regular season schedule, which promotes competitive balance at the expense of the first design objective (drafts often achieve this alternative goal too) Some seeding schemes involve byes (NFL playoffs, Big East and Big Ten hoops playoffs, English FA Cup) Seeding versus re-seeding: World Cup, NBA, and NCAA basketball don t re-seed; NHL and NFL re-seed after each round The failure of NCAA hoops to re-seed after each round leads to some undesirable(?) results, as demonstrated by the following:

6 6 Number of Sweet 16 appearances, Seed Sweet 16 appearances #1 111 #2 81 #3 65 #4 59 #5 43 #6 42 #7 23 #8 12 #9 5 #10 23 #11 19 #12 20 #13 6 #14 2 #15 1 #16 0

7 7 Records per seed in first round of NCAA Tournament, Seeds Record % #1 vs. # % #2 vs. # % #3 vs. # % #4 vs. # % #5 vs. # % #6 vs. # % #7 vs. # % #8 vs. # %

8 8 Why so many Sweet 16 appearances for 10/11/12-seeds relative to 8/9-seeds? Probability arguments (shown in class):

9 9 Re-seeding after each round would reduce the magnitude of this disparity, but might not eliminate it (see first question below). Reseeding presents some practical difficulties, however. Additional questions of interest: Could the inclusion of conference champions from very weak conferences as automatic qualifiers (and 16-seeds) contribute significantly (by making it a virtual certainty that 1-seeds win their first-round game) to the relatively small chance that 8/9- seeds make it to the Sweet 16? Why do 12-seeds beat 5-seeds (in first round) as often as 11- seeds beat 6-seeds?

10 10 American Statistician article by Abdul-Chani and Frey Consider design of a knockout conference basketball tournament after a round robin season, with specific interest in the Big East tournament Current Big East tournament structure Teams seeded 1 to 10 by # of conference wins, with various tie-breakers Same format as a 16-team NCAA regional, except that seeds 1-6 get first-round byes Authors objective: determine if this is the best design for the truly best team to emerge as the conference champion

11 11 Model for team strength Let θ i represent the strength of team i Assume θ 1 θ 10 are iid N 0 σ 2 µ Assume P Team i beats Team jµ Φ θi θ j µ on neutral court Φ θ i H θ j µ at Team i s home σ 2 and H chosen to match data from Big East conference seasons

12 12 Comparison of tournament formats Using the model, 1 million Big East regular seasons were simulated, followed by a simulated conference tournament having each of several different formats: include fixed #, T, of teams include only those teams with no more than M fewer wins than the regular-season conference champion re-seeding versus not re-seeding bye formats among 10-team tournaments over 4 days Measure quality of format by how often the truly best team is the tournament champion

13 13 Summary of results Best fixed-number-of-teams format is T 1 (no tournament); generally the fewer teams in the tournament, the better. Best value of M is 0; generally the smaller that M is the better. M 0 is better than T 1. Re-seeding only improves things for tournaments with more than 5 teams, and then only very slightly. B 0 2µ B 3 1µ B 6 0µ, where B x yµ means that the top x teams get 1 bye and the top y teams get 2 byes. Current bye format is B 6 0µ (Big Ten tournament is B 10 4µ).

14 14 Regular-season tie-breaker for seeding If two or more Big East teams tie for first in the regular season, the first tie-breaker is the number of wins against the other teams tied for first. Makes sense, right? Not so fast, my friend. In simulations under the model, the use of this tie-breaker awards the highest seed to the weakest team in the tied cohort most often. In fact, under the assumed model, and in conferences with 4-12 teams at least, it is better to break ties by awarding the highest seed to the team that has the best record against teams not in the tied cohort. The Big East should change its tie-breaker to the exact opposite of what it is now.

15 15 Bradley-Terry model Bradley and Terry (1952, Biometrika) proposed a logit model for paired comparisons, including outcomes (win or loss) of games in a (possibly unbalanced) round robin tournament. Data: n i j # times Team i beats Team j Losing team Winning team Team 1 Team 2 Team m Team 1 n 12 n 1m Team 2 N 12 n 12 n 2m Team m N 1m n 1m

16 16 Model The Bradley-Terry model assumes that there are positive quantities π 1 π m, which can be assumed to sum to one, such that p i j P Team i beats Team jµ π i π i π j Alternatively, letting φ i logπ i, the model specifies that pi j log logπ i logπ j φ i φ j 1 p i j If all game outcomes are independent, then n i j independent bin N i j π i π i π j µ

17 17 Maximum likelihood estimation The log-likelihood of π π 1 π m is L πµ m m i 1 j i 1 m m i 1 j i 1 m i 1 n i j log p i j N i j n i j µlog 1 p i j µ n i j log πi N π i i j π j m n i j logπ i n i j log π i π j µ j i 1 n i j µlog π j π i π j L πµ must be maximized numerically. Can do this to obtain mles using the R package Ö Ð ÝÌ ÖÖݾ.

18 18 Maximum likelihood estimation, continued A home advantage can be incorporated into the model easily. Merely redefine pi j log logπ i logπ j λz 1 p i j where z 1 if Team i is at home 0 if Team i is away

19 19 Baseball example Results within Eastern Division of American League for 1987 season: Losing team Winning team Mil Det 2 Tor NY Bos Cle Bal Mil Det Tor NY Bos Cle Bal

20 20 Baseball example, continued Fit a Bradley-Terry model to these data, using Ö Ð ÝÌ ÖÖݾ (see handout).