homepage/content/blog/clrg/2022-10-09-CLRG-Scoring/index.md

---
title: CLRG Scoring Analyzed
date: 2022-10-09
tags:
    - clrg
stylesheets:
    - toys.css
scripts:
    - scorecard.mjs
---

Let's take a look how how CLRG does its scoring!
*With math!*


## How CLRG Scoring Works

As I am given to understand, the scoring works like so:

1. Adjudicators give you a "raw score": a real number between 0 and 100
2. The scoring system ranks each dancer per adjudicator, based on raw scores
3. These rankings are mapped into "award points"
4. All of a dancer's award points are summed
5. Final ranking is determined by comparing total award points

## Raw Scoring

The way raw scores translate into rankings and award points is a little
confusing, so I've made a little tool you can play with to get a feel for how it
works. Essentially, it's a way of normalizing places to an adjudicator: score
weights are only relative to the judge that assigns them.

Adjudicator A can assign scores between 80 and 100; 
adjudicator B can assign scores between 1 and 40; 
and they'll both have a first, second, third, fourth place, etc.
These places then get translated into award points.


## Award Points

Award points are handed out based on ranking against other dancers for that
adjudicator. I obtained these values from a FeisWorx results page for my kid:

<div class="awardPoints">
    <table>
        <thead>
            <tr>
                <th>Ranking</th><th>Award Points</th>
            </tr>
        </thead>
        <tbody></tbody>
    </table>
</div>

If there's a 2-way, 3-way, or n-way tie,
all tied dancers get the average of the next 2, 3, or n award points,
and the next 2, 3, or n rankings are skipped.


## What's with these values?

At first glance, the award points look like the output of an exponential function.

{{<figure src="chart.png" alt="Chart of scores vs. award points">}}

In an effort to figure out where these numbers came from,
I ran some curve fitting against the data.
Here's the best I could come up with:

| Ranking range | Award Points Function | Type of function |
| --: | --: | --- |
| 1 - 11 | 100 * x^-0.358 | Exponential |
| 12 -  50 | 51 - x | Linear |
| 51 - 60 | 14.2 - 0.46x + 0.00385x | Polynomial |
| 61 - 100 | 1 - x/100 | Linear |

If you, dear reader, are a mathematician,
I would love to hear your thoughts on why they went with this algorithm.

There are a few points to note here:

* 1st place is a *huge deal*. Disproportionately huge.
* Places 2-10 are similarly big deals compared to places 3-11.
* Places 12-50 operate the way most people probably assume ranking works: linearly.
* Places 51-60 fit best to a second degree polynomial, but it doesn't matter much for differences of hundreths of a point. This section is *really weird*, mathematically.
* Places 61-100 are all less than 1 point. If you're a judge trying to tank a top dancer, anywhere in this range is equivalent to anywhere else.


## Consequences of Exponential Award Points

Playing around with this,
I've found a few interesting consequences
of the exponential growth in the top 11 places.


### 1st place is super important

1st place is weighted so heavily that one judge could move a 5th place dancer into 2nd.

<table class="scorecard">
    <thead>
        <tr>
            <td></td>
            <th>Alice</th>
            <th>Bob</th>
            <th>Carol</th>
        </tr>
    </thead>
    <tbody>
        <tr>
            <th class="justify-left">Adj. 1</th>
            <td><input type="number" min=1 max=99 value=1 readonly></td>
            <td><input type="number" min=1 max=99 value=3></td>
            <td><input type="number" min=1 max=99 value=2></td>
        </tr>
        <tr>
            <th class="justify-left">Adj. 2</th>
            <td><input type="number" min=1 max=99 value=5></td>
            <td><input type="number" min=1 max=99 value=3></td>
            <td><input type="number" min=1 max=99 value=1></td>
        </tr>
        <tr>
            <th class="justify-left">Adj. 3</th>
            <td><input type="number" min=1 max=99 value=5></td>
            <td><input type="number" min=1 max=99 value=3></td>
            <td><input type="number" min=1 max=99 value=2></td>
        </tr>
    </tbody>
    <tfoot>
        <tr>
            <th class="justify-left">Award Points</th>
            <td class="justify-right"><output name="points"></td>
            <td class="justify-right"><output name="points"></td>
            <td class="justify-right"><output name="points"></td>
        </tr>
        <tr>
            <th class="justify-left">Ranking</th>
            <td class="justify-right"><output name="ranking"></td>
            <td class="justify-right"><output name="ranking"></td>
            <td class="justify-right"><output name="ranking"></td>
        </tr>
    </tfoot>
</table>

You can adjust these values to get a better feel for how scoring works.


### Tanking a high-ranked dancer is another way to cheat

Because of that exponential curve,
a low ranking from a single judge can carry a lot of weight.

<table class="scorecard">
    <thead>
        <tr>
            <td></td>
            <th>Alice</th>
            <th>Bob</th>
            <th>Carol</th>
        </tr>
    </thead>
    <tbody>
        <tr>
            <th class="justify-left">Adj. 1</th>
            <td><input type="number" min=1 max=99 value=3></td>
            <td><input type="number" min=1 max=99 value=11></td>
            <td><input type="number" min=1 max=99 value=2></td>
        </tr>
        <tr>
            <th class="justify-left">Adj. 2</th>
            <td><input type="number" min=1 max=99 value=3></td>
            <td><input type="number" min=1 max=99 value=1></td>
            <td><input type="number" min=1 max=99 value=2></td>
        </tr>
        <tr>
            <th class="justify-left">Adj. 3</th>
            <td><input type="number" min=1 max=99 value=3></td>
            <td><input type="number" min=1 max=99 value=2></td>
            <td><input type="number" min=1 max=99 value=1></td>
        </tr>
    </tbody>
    <tfoot>
        <tr>
            <th class="justify-left">Award Points</th>
            <td class="justify-right"><output name="points"></td>
            <td class="justify-right"><output name="points"></td>
            <td class="justify-right"><output name="points"></td>
        </tr>
        <tr>
            <th class="justify-left">Ranking</th>
            <td class="justify-right"><output name="ranking"></td>
            <td class="justify-right"><output name="ranking"></td>
            <td class="justify-right"><output name="ranking"></td>
        </tr>
    </tfoot>
</table>


### Being in 1st provides a nice buffer

Try playing around with Alice's rankings with Adjudicators 2 and 3 here.
She has to get ranked a lot lower before her overall ranking starts going down.

<div class="scrolly">
    <table class="scorecard">
        <thead>
            <tr>
                <td></td>
                <th>Alice</th>
                <th>Bob</th>
                <th>Carol</th>
                <th>Dave</th>
                <th>Erin</th>
            </tr>
        </thead>
        <tbody>
            <tr>
                <th class="justify-left">Adj. 1</th>
                <td><input type="number" min=1 max=99 value=1></td>
                <td><input type="number" min=1 max=99 value=3></td>
                <td><input type="number" min=1 max=99 value=2></td>
                <td><input type="number" min=1 max=99 value=4></td>
                <td><input type="number" min=1 max=99 value=5></td>
            </tr>
            <tr>
                <th class="justify-left">Adj. 2</th>
                <td><input type="number" min=1 max=99 value=7></td>
                <td><input type="number" min=1 max=99 value=1></td>
                <td><input type="number" min=1 max=99 value=2></td>
                <td><input type="number" min=1 max=99 value=3></td>
                <td><input type="number" min=1 max=99 value=4></td>
            </tr>
            <tr>
                <th class="justify-left">Adj. 3</th>
                <td><input type="number" min=1 max=99 value=5></td>
                <td><input type="number" min=1 max=99 value=2></td>
                <td><input type="number" min=1 max=99 value=1></td>
                <td><input type="number" min=1 max=99 value=3></td>
                <td><input type="number" min=1 max=99 value=4></td>
            </tr>
        </tbody>
        <tfoot>
            <tr>
                <th class="justify-left">Award Points</th>
                <td class="justify-right"><output name="points"></td>
                <td class="justify-right"><output name="points"></td>
                <td class="justify-right"><output name="points"></td>
                <td class="justify-right"><output name="points"></td>
                <td class="justify-right"><output name="points"></td>
            </tr>
            <tr>
                <th class="justify-left">Ranking</th>
                <td class="justify-right"><output name="ranking"></td>
                <td class="justify-right"><output name="ranking"></td>
                <td class="justify-right"><output name="ranking"></td>
                <td class="justify-right"><output name="ranking"></td>
                <td class="justify-right"><output name="ranking"></td>
            </tr>
        </tfoot>
    </table>
</div>