423 lines
15 KiB
TeX
423 lines
15 KiB
TeX
\documentclass[12pt,a4paper]{article}
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\usepackage[margin=1in]{geometry}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{amsthm}
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\usepackage{graphicx}
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\usepackage{xcolor}
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\usepackage{booktabs}
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\usepackage{array}
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\usepackage{multirow}
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\usepackage{hyperref}
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\usepackage{tikz}
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\usepackage{pgfplots}
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% Theorem styles
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\theoremstyle{definition}
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\newtheorem{definition}{Definition}
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\newtheorem{example}{Example}
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\newtheorem*{tldr}{\textbf{TL;DR}}
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% Custom colors
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\definecolor{attention}{RGB}{200,0,0}
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\definecolor{success}{RGB}{0,100,0}
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\definecolor{info}{RGB}{0,0,150}
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% Title
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\title{\textbf{How Bad Am I, Actually?} \\[0.5em]
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{\Large Building a Pickleball Rating System That Doesn't Lie} \\[0.2em]
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{\normalsize (Now With 100\% Less Volatility and 100\% More Accountability)}}
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\author{Split (Implementation) \and Dane Sabo (System Design)}
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\date{February 2026}
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\begin{document}
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\maketitle
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% TL;DR BOX
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\begin{center}
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\fbox{%
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\parbox{0.9\textwidth}{%
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\vspace{0.3cm}
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\textbf{\Large TL;DR: What This System Does}
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\vspace{0.2cm}
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\begin{enumerate}
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\item \textbf{Single rating per player:} One number (usually 1500) instead of separate singles/doubles ratings.
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\item \textbf{Per-point scoring:} Your actual performance (points scored / total points) is compared to expected performance based on rating differences.
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\item \textbf{Smart doubles scoring:} When you play doubles, we calculate an ``effective opponent'' that accounts for partner strength using: \texttt{Effective Opp = Opp1 + Opp2 - Teammate}.
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\item \textbf{Simple math:} Rating changes are easy to understand and calculate. No volatility, no rating deviation—just you vs. opponents.
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\end{enumerate}
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\noindent\textbf{Result:} A fairer, simpler, easier-to-understand rating system.
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\vspace{0.3cm}
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}%
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}
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\end{center}
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\section{Introduction}
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Welcome to the simplified Pickleball ELO Rating System!
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After running our league with Glicko-2 for over a month, we realized the system had some problems. This document explains what was wrong with the old system, why we changed it, and how the new system works.
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\textit{(And yes, the system designer happens to be the biggest beneficiary of the new rating calculations. Coincidence? Probably. But we'll let you be the judge.)}
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\section{The Old System: What Went Wrong}
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\subsection{Glicko-2: A Brief Overview}
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The original system used \textbf{Glicko-2}, a rating system developed by Mark Glickman for chess. Unlike basic ELO (one number), Glicko-2 tracks \textit{three} values per player:
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\begin{enumerate}
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\item \textbf{Rating ($r$)}: Your skill estimate (like ELO, default 1500)
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\item \textbf{Rating Deviation ($RD$)}: How \textit{uncertain} the system is about your rating. High RD = ``we're not sure about this player yet.'' Low RD = ``we're confident.''
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\item \textbf{Volatility ($\sigma$)}: How \textit{consistent} you are. High volatility = your performance varies wildly.
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\end{enumerate}
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The math behind Glicko-2 involves converting ratings to a different scale, computing expected outcomes with a $g(\phi)$ function involving $\pi$, iteratively solving for new volatility using numerical methods, and... look, it's a lot. Most players had no idea why their rating changed the way it did.
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\subsection{The Arbitrary Margin Bonus (The ``Tanh Shit'')}
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Here's where things got sketchy. Standard Glicko-2 treats every win the same—whether you win 11-0 or 11-9. We wanted margin of victory to matter, so the old system added a \textit{margin bonus}:
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\begin{equation}
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\text{weighted\_score} = \text{base\_score} + \tanh\left(\frac{\text{margin}}{11} \times 0.3\right) \times (\text{base\_score} - 0.5)
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\end{equation}
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\textbf{Translation}: We took the hyperbolic tangent of a fraction involving the point margin, multiplied by an arbitrary constant (0.3), and added it to your win/loss.
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\textbf{Why 0.3?} No particular reason. It ``felt right.''
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\textbf{Why $\tanh$?} It squishes values between -1 and 1, which... seemed useful?
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This is what's known in the business as ``making stuff up.'' It worked, sort of, but it had no theoretical basis and was impossible to explain to players.
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\subsection{The Doubles Problem}
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The old system calculated team ratings by simply \textit{averaging} both partners:
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\begin{equation}
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R_{\text{team}} = \frac{R_{\text{player1}} + R_{\text{player2}}}{2}
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\end{equation}
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This seems reasonable until you think about it. If you (1400) play with a strong partner (1700) against two 1550s:
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\begin{itemize}
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\item Your team average: 1550
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\item Their team average: 1550
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\item The system thinks it's an even match!
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\end{itemize}
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But \textit{you} played against opponents rated 1550, while being ``carried'' by a 1700 partner. Winning this match shouldn't boost your rating as much as if you'd won with a weaker partner. The old system didn't account for this.
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\subsection{The RD Distribution Bug}
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When distributing rating changes between doubles partners, the old system gave \textit{more} change to players with \textit{lower} RD (more certain ratings). This is backwards.
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If we're uncertain about your rating (high RD), the system should update it \textit{more aggressively} to converge faster. Instead, we were doing the opposite—penalizing uncertain players by updating them slowly.
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\subsection{Summary: Why We Changed}
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\begin{enumerate}
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\item Glicko-2 was over-engineered for a recreational league
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\item The margin bonus was arbitrary (``the tanh shit'')
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\item Doubles averaging ignored partner strength effects
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\item The RD distribution was literally backwards
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\item Nobody understood why their rating changed
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\end{enumerate}
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Time for something simpler.
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\section{ELO System Basics}
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\subsection{The Core Idea}
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ELO is \emph{simple}:
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\begin{definition}[ELO Rating]
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Each player has one number: their \textbf{rating} (default: 1500). This represents their expected performance.
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\end{definition}
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When two players compete:
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\begin{enumerate}
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\item Calculate the expected probability that one player beats the other based on rating difference
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\item Compare expected to actual performance
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\item Adjust ratings based on the difference
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\end{enumerate}
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\subsection{Expected Winning Probability}
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The key formula is:
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\begin{equation}
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E = \frac{1}{1 + 10^{\frac{R_{\text{opponent}} - R_{\text{self}}}{400}}}
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\end{equation}
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\begin{definition}[Plain English]
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$E$ is the probability you ``should'' win against your opponent, based on rating difference alone.
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\end{definition}
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\textbf{What this means:}
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\begin{itemize}
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\item If you're rated 1500 and opponent is 1500: $E = 0.5$ (50-50 matchup)
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\item If you're rated 1600 and opponent is 1500: $E \approx 0.64$ (you should win about 64\% of the time)
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\item If you're rated 1400 and opponent is 1500: $E \approx 0.36$ (you should win about 36\% of the time)
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\end{itemize}
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The formula uses $10^x$ (powers of 10) because it's traditional in chess ELO. The 400 in the denominator is a scaling factor.
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\subsection{Rating Change Formula}
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After each match:
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\begin{equation}
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\Delta R = K \cdot (P_{\text{actual}} - E)
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\end{equation}
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\begin{definition}[Plain English]
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Your rating change ($\Delta R$) is:
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\begin{itemize}
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\item $K$ = How much weight each match has (32 for casual play)
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\item $P_{\text{actual}}$ = Your actual performance (0.0 to 1.0)
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\item $E$ = Expected performance
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\end{itemize}
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\end{definition}
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\textbf{Examples:}
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\begin{example}[Expected Win]
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You (1500) beat opponent (1500):
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\begin{align*}
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E &= 0.5 \\
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P_{\text{actual}} &= 1.0 \text{ (you won)} \\
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\Delta R &= 32 \cdot (1.0 - 0.5) = 16 \text{ points}
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\end{align*}
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\end{example}
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\begin{example}[Upset Win]
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You (1400) beat opponent (1500):
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\begin{align*}
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E &\approx 0.36 \\
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P_{\text{actual}} &= 1.0 \\
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\Delta R &= 32 \cdot (1.0 - 0.36) \approx 20.5 \text{ points}
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\end{align*}
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You gain more because you won an upset!
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\end{example}
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\begin{example}[Expected Loss]
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You (1600) lose to opponent (1500):
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\begin{align*}
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E &\approx 0.64 \\
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P_{\text{actual}} &= 0.0 \text{ (you lost)} \\
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\Delta R &= 32 \cdot (0.0 - 0.64) \approx -20.5 \text{ points}
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\end{align*}
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You lose more because it was an upset loss!
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\end{example}
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\section{Pickleball-Specific Innovations}
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\subsection{Per-Point Performance Scoring}
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In pickleball, matches are scored to 11 (win by 2). A 11-9 match is very different from an 11-2 match, even if both are wins.
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Instead of binary win/loss, we use:
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\begin{equation}
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P_{\text{actual}} = \frac{\text{Points Scored}}{\text{Total Points}}
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\end{equation}
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\begin{definition}[Plain English]
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Your actual performance is simply: how many points did you score out of total points played?
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\end{definition}
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\textbf{Examples:}
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\begin{itemize}
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\item 11-9 win: $P = 11/20 = 0.55$ (55\% of points)
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\item 11-2 win: $P = 11/13 = 0.846$ (84.6\% of points)
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\item 5-11 loss: $P = 5/16 = 0.3125$ (31.25\% of points)
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\end{itemize}
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This is more nuanced than binary outcomes and captures match quality.
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\subsection{The Effective Opponent Formula (Doubles)}
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In doubles, your partner's strength matters. If you have a strong partner, you're effectively facing a weaker opponent.
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We use:
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\begin{equation}
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R_{\text{effective opponent}} = R_{\text{opp1}} + R_{\text{opp2}} - R_{\text{teammate}}
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\end{equation}
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\begin{definition}[Plain English]
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Your effective opponent rating accounts for:
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\begin{itemize}
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\item How strong your actual opponents are
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\item How strong your teammate is (strong teammate = easier match for you)
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\end{itemize}
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\end{definition}
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\textbf{Examples:}
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\begin{example}[Balanced Teams]
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\begin{itemize}
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\item Opponents: 1500, 1500
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\item Your teammate: 1500
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\item Effective opponent: $1500 + 1500 - 1500 = 1500$
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\end{itemize}
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Neutral situation.
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\end{example}
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\begin{example}[Strong Partner]
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\begin{itemize}
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\item Opponents: 1500, 1500
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\item Your teammate: 1600
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\item Effective opponent: $1500 + 1500 - 1600 = 1400$
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\end{itemize}
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Your partner carried you! The system treats the match as easier (lower effective opponent).
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\end{example}
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\begin{example}[Weak Partner]
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\begin{itemize}
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\item Opponents: 1500, 1500
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\item Your teammate: 1400
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\item Effective opponent: $1500 + 1500 - 1400 = 1600$
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\end{itemize}
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You were undermanned. The system treats the match as harder (higher effective opponent).
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\end{example}
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This is fair: if you beat strong opponents with a weak partner, you gain more rating. If you barely beat weaker opponents with help, you gain less.
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\section{Before/After: System Migration}
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\subsection{What Changed}
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We migrated from Glicko-2 (complex, three parameters per player) to pure ELO (one parameter per player).
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Key differences:
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\begin{table}[h]
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\centering
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\begin{tabular}{|l|c|c|}
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\hline
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\textbf{Feature} & \textbf{Glicko-2} & \textbf{Pure ELO} \\
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\hline
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Parameters per player & 3 (rating, RD, volatility) & 1 (rating only) \\
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Complexity & High & Low \\
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Transparency & Medium & High \\
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Per-point scoring & Yes & Yes \\
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Effective opponent (doubles) & Weighted avg & Opp1+Opp2-Teammate \\
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\hline
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\end{tabular}
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\end{table}
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\subsection{Migration Data: Old vs New Ratings}
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We replayed all 29 historical matches through the new ELO system to see how ratings changed. Here's the comparison:
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\begin{table}[h]
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\centering
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\begin{tabular}{|l|r|r|r|r|}
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\hline
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\textbf{Player} & \textbf{Old Glicko Avg} & \textbf{New ELO} & \textbf{Change} & \textbf{Matches} \\
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\hline
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Andrew Stricklin & 1651 & 1538 & \textcolor{attention}{-113} & 19 \\
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David Pabst & 1562 & 1522 & \textcolor{attention}{-40} & 11 \\
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Jacklyn Wyszynski & 1557 & 1514 & \textcolor{attention}{-43} & 9 \\
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Eliana Crew & 1485 & 1497 & \textcolor{success}{+11} & 13 \\
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Krzysztof Radziszeski & 1473 & 1476 & \textcolor{success}{+3} & 25 \\
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Dane Sabo & 1290 & 1449 & \textcolor{success}{+159} & 25 \\
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\hline
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\end{tabular}
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\caption{Rating comparison after replaying all matches through the new system}
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\end{table}
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\textbf{Key observations:}
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\begin{itemize}
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\item \textbf{Rating spread compressed:} Old system had 361 points between top and bottom; new system has only 89 points. This makes sense—we're a recreational group, not pros.
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\item \textbf{Biggest winner:} Dane (+159 points). The old system was penalizing him for losses with weaker partners. The new effective opponent formula gives credit for ``carrying.''
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\item \textbf{Biggest loser:} Andrew (-113 points). Still ranked \#1, but the old system was over-crediting wins with strong partners.
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\item \textbf{Per-point scoring matters:} Close losses (11-9) now hurt less than blowout losses (11-2). This rewards competitive play even in defeat.
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\end{itemize}
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The new system rates players more fairly, especially in doubles where partner strength varies.
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\vspace{0.5em}
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\begin{center}
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\fbox{%
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\parbox{0.85\textwidth}{%
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\textbf{A Note on Conflicts of Interest} \\[0.3em]
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\small
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The astute reader may notice that the system designer (Dane) is also the biggest beneficiary of the new rating calculations, gaining a convenient 159 points.
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We want to assure you that this is \textit{purely coincidental} and the result of \textit{rigorous mathematical analysis}, not at all influenced by the fact that Dane was tired of being ranked last.
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The new formulas are based on \textit{sound theoretical principles} that just \textit{happen} to conclude that Dane was being unfairly penalized all along. Any resemblance to cooking the books is entirely accidental.
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\textit{Trust the math.}
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}%
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}
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\end{center}
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\section{Implementation Notes}
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\subsection{K-Factor}
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We use $K = 32$, which is standard for casual chess. This means:
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\begin{itemize}
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\item Each match typically changes your rating by 10--20 points
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\item It takes 5--10 matches to change rating by 100 points
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\item Reasonable for recreational play
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\end{itemize}
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Alternative: $K = 48$ (more volatile, faster changes) or $K = 16$ (slower, more stable).
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\subsection{Starting Rating}
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All new players start at 1500. This is arbitrary but standard in ELO systems.
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\subsection{Minimum Rating}
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Ratings never go below 1. This prevents the system from producing absurd values.
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\section{Frequently Asked Questions}
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\begin{enumerate}
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\item \textbf{Why not keep Glicko-2?}
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\begin{itemize}
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\item Glicko-2 is excellent for large, active chess communities.
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\item For a small pickleball league, it's over-engineered and hard to explain.
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\item Pure ELO is simpler and still fair.
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\end{itemize}
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\item \textbf{How do I know if my rating is accurate?}
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\begin{itemize}
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\item Your rating converges to your true skill over 10--20 matches.
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\item If you consistently beat players rated above you, your rating will rise.
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\item If you lose to players rated below you, your rating will drop.
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\end{itemize}
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\item \textbf{Why does my doubles rating matter in singles?}
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\begin{itemize}
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\item All matches (singles and doubles) update one unified rating.
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\item Your true skill is roughly the same in both formats.
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\item The effective opponent formula ensures partner strength doesn't artificially inflate/deflate your rating.
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\end{itemize}
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\item \textbf{Can I lose rating for a win?}
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\begin{itemize}
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\item No. If you have rating 1400 and opponent is 2000, you always gain rating for a win.
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\item The worst case: you have rating 1600, beat opponent at 1500, but played terribly (low point percentage). You gain less.
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\end{itemize}
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\end{enumerate}
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\section{Conclusion}
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The ELO system is transparent, fair, and easy to understand. It respects the nuances of pickleball (per-point play, partner strength) without the complexity of Glicko-2.
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Your rating now reflects your true skill more accurately than ever.
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\end{document}
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