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paper.qmd
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---
title: "Tournaments"
author: "Greg, Josie, Zach, Ryan"
format: html
editor: visual
---
@Glickman2025: Review of methods for estimating team and player strength. So this paper doesn't directly relate to our tournaments paper. We are interested in structure and this paper is interested in estimating strength. However, we are interested in knockout tournements (and other tournament structures) that require seeding, which is estimating player/team strength. So estimating team player/team could be considered part of the tournament structure.
Llull (1283)
@KendallSmith1940: Judges makes consistent preferences.
All about paired comparisons. Doesn't seem to be about tournaments.
@Slater1961 Slater's I (1961): Measure of inconsistency
Thurstone Mosteller model (1928): Bradley Terry Model with a normal CDF transformation.
Kendall-Wei (1952) Elo (1978) Glicko (1998) and Glicko2 (2006?): Essentially Bayesian Elo.\
Trueskill (2006): Used in halo 2 or 3.
Weak stochastic probablity. $P(A \rightarrow B) > 0.5$ and $P(B \rightarrow C) > 0.5$ then $P(A \rightarrow C) > 0.5$.
CRSP: Chain Rock Scissor Paper. Non transitive model.
Ratings vs rankings: Ratings are numeric estimates of strengths. Rankings are a function of ratings.
Let's start with what a BAD tournament would be.
Quarto manuscript template:https://github.com/mikemahoney218/quarto-arxiv
##Papers
https://www.sciencedirect.com/science/article/pii/S0378375803003215 "Maximum weight perfect match problem"
## Deterministic Binary Outcome Tournaments
- Knockout (KO) lKO
- 1KO
- 2KO
- True 3rd, college wrestling, repechage
- 3KO
- Seeding issues,.
- nRR
- 1RR
- 2RR
- partial RR
## Dynamic Binary Outcome Tournaments
- Swiss
- Dynamic Knockout (reseeding after every round)
## Tournaments
- Single elimination (NCAA tournament, NFL playoffs)
- Seeding issues (Tennis: https://www.liveabout.com/definition-of-seeding-3207821) Reseeding after each round. Straight knockout.\
- Second faces Third for silver medal. Guarantees than sivler has 1 loss and bronze has 2 losses.\
- With a 3rd place match (Badminton: https://www.nbcolympics.com/news/badminton-101-olympic-competition-format)
- Single elimination with series (NBA, MLB, NHL)
- Ladder tournaments (Korean baseball does best of 3 with one team starting up 1-0 in the opening round)
- Seeding issues (NFL re-seeds, NCAA tournament straight bracket)
- Group/Pool play followed by knockout stages (World Cup, Euro, Overwatch World CUp. )
- Double elimination (Overwatch Champions League)
- Repechage (Olympic Wrestling, Tae Kwon Do, Judo) (These also aware 2 bronze medals)
- True Double elimination
- Double elimination but a single loss before finals means you can onyl go for third place (College wrestling)
- Triple Elimination (Curling! https://thegrandslamofcurling.com/bracketology-101-triple-knockout-explained-2021/)
- Swiss Tournament (https://en.chessbase.com/post/125-years-swiss-system)
- Round Robin
- Single and Double.\
- Baseball is an imbalanced multiple round robin qualifying round.
- https://www.tandfonline.com/doi/abs/10.1080/01621459.1952.10501178
- Waterfall format
- Introduced in the Smash 64 community, [they claim](https://thesmashwriter.wordpress.com/2017/04/24/what-is-a-waterfall-bracket/)
- Recent example was [Super Smash Con 2023](https://www.start.gg/tournament/super-smash-con-2023/event/64-1v1-singles/overview)
- Players are seeded $1 \rightarrow n$ and grouped into $k$ divisions, 1 best, $n$ and $k$ worst.
- The worst $\frac{n + \epsilon}{k}$ play in [relatively small round robin pools](https://www.start.gg/tournament/super-smash-con-2023/event/64-1v1-singles/brackets/1184919/1833431). Some fraction advance. In this linked example, There are 16 pools in division 4, where each pool is roughly 13 players, with 4 from each division advancing.
- The advancing players are then entered into pools with the slightly better-seeded players, and again complete round robins. This iterates over and over again. In this example, to make the numbers work, there are also [wildcard rounds](https://www.start.gg/tournament/super-smash-con-2023/event/64-1v1-singles/brackets/1184927/1833460) to advance an extra few players to further divisions.
- In Smash 64, this ultimately leads to a top 16 or 32, which then becomes a double-elimination tournament (all prior losses are ignored). The [top 32](https://www.start.gg/tournament/super-smash-con-2023/event/64-1v1-singles/brackets/1184923/1833456) is seeded by the [top division pool results](https://www.start.gg/tournament/super-smash-con-2023/event/64-1v1-singles/brackets/1184922/1833451).
- Heat based events (Track 5000m and below, swimming)
- Ranking Round followed by single elimination (Olympic Archery https://www.nbcolympics.com/news/archery-101-olympic-competition-format-and-scoring)
- Elimination racing (Cycling, Bachelor/Bachelorette, Survivor)
Tournament: a series of contests between a number of competitors, who compete for an overall prize. Oxford Dictionary Wiki: "A tournament is a competition involving at least three competitors, all participating in a sport or game"
Tournament (graphy theory): [wikipedia definition](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) and in [McShane (2019)](https://scholar.smu.edu/hum_sci_statisticalscience_etds/13/) -- essentially a "round robin tournament" -- a graph where every node (competitor: A, B, C) is connected by a directed edge (win A -\> B or lose A \<- B), where there are no loops (self-directed edges -- competitors cannot play against themselves). AKA a complete directed graph with no loops; has $n$ nodes and $\choose{n}{2}$ edges. John W. Moon's [Topic on Tournaments](https://www.gutenberg.org/ebooks/42833) is the classic graph theoretical tournaments text. This definition should be used to disambiguate *tournament*, especially since graph theory may be used.
(https://www.espn.com/olympics/story/\_/id/40536088/2024-paris-summer-olympics-track-field-medaling-format-rules)
Curling and DOE! https://khazna.ku.ac.ae/en/publications/applying-design-of-experiments-tooptimize-the-performance-level-o
Poker: https://arxiv.org/abs/physics/0703122
Graph Theory Stuff: Chickens: https://link.springer.com/article/10.1007/BF02476378
Are elections tournaments?
Tournament of Tournaments.
- aggregation of rankings (golf fedex cup, formula one, tour de france)
https://liquipedia.net/overwatch/Overwatch_Champions_Series/2024/North_America/Stage_2 Swiss, Top Cut Swiss, Group play then a double elimiation bracket!
https://www.vgcguide.com/swiss-and-top-cut
```{r}
results <- data.frame(h = c("i","i","j","i","j","l"), a = c("j","l","l","k","k","k"), hpoints = c(3,3,3,1,1,1), apoints = c(0,0,0,1,1,1))
library(tidyverse)
h <-results %>% group_by(h) %>% summarize(sum = sum(hpoints))
a <- results %>% group_by(a) %>% summarize(sum = sum(apoints))
names(a) <- c("h","sum")
d <- rbind(h,a)
d %>% group_by(h) %>% summarize(sum = sum(sum)) %>% arrange(-sum)
#Now reverse the outcomes
results <- data.frame(h = c("i","i","j","i","j","l"), a = c("j","l","l","k","k","k"), hpoints = c(0,0,0,1,1,1), apoints = c(3,3,3,1,1,1))
library(tidyverse)
h <-results %>% group_by(h) %>% summarize(sum = sum(hpoints))
a <- results %>% group_by(a) %>% summarize(sum = sum(apoints))
names(a) <- c("h","sum")
d <- rbind(h,a)
d %>% group_by(h) %>% summarize(sum = sum(sum)) %>%arrange(-sum)
```
Note for Greg: Can we build a simulator that computes the probability of a team winning in two steps: 1) the team strength comes from a distribution itself and then the actual outcome come from flipping a coin based on the particular draw of the distribtuion.
Can we model a sport and how muhc variability it has? So like estimate the probability of rteam A beating team B as a function of relative strengths but also use that other distribution to model the variability of the sport. Does this exist?