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

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

### Award Points artifacts

One quirk of awards points is that for any given overall
score, there are only a handful of possible judge rankings that could have led
to it. That means you can make some guesses about how each judge ranked an
individual dancer, based on only their total award points.

Here's a handy calculator!
It (currently) doesn't consider the possibility of a tie.

<div class="scrolly">
    <fieldset class="speculator">
        <legend>CLRG Award Points Speculator</legend>
        <div>
            Points: <input name="points" type="number" min=41 max=10000 value=188>
            <input name="adjudicators" type="hidden">
        </div>
        <table class="results">
            <caption>Possible Rankings</caption>
            <thead>
                <tr class="warning"><th>Computing: this could take a while!</th></tr>
            </thead>
            <tbody></tbody>
        </table>
    </fieldset>
</div>


## 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 are a second degree polynomial, but it doesn't matter much for so few points.
* 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=2></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=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>