287 lines
10 KiB
Markdown
287 lines
10 KiB
Markdown
---
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title: CLRG Scoring Analyzed
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date: 2022-10-09
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stylesheets:
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- toys.css
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scripts:
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- speculator.mjs
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- scorecard.mjs
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---
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Let's take a look how how CLRG does its scoring!
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*With math!*
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## How CLRG Scoring Works
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As I am given to understand, the scoring works like so:
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1. Adjudicators give you a "raw score": a real number between 0 and 100
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2. The scoring system ranks each dancer per adjudicator, based on raw scores
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3. These rankings are mapped into "award points"
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4. All of a dancer's award points are summed
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5. Final ranking is determined by comparing total award points
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## Raw Scoring
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The way raw scores translate into rankings and award points is a little
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confusing, so I've made a little tool you can play with to get a feel for how it
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works. Essentially, it's a way of normalizing places to an adjudicator: score
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weights are only relative to the judge that assigns them.
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Adjudicator A can assign scores between 80 and 100;
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adjudicator B can assign scores between 1 and 40;
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and they'll both have a first, second, third, fourth place, etc.
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These places then get translated into award points.
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## Award Points
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Award points are handed out based on ranking against other dancers for that
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adjudicator. I obtained these values from a FeisWorx results page for my kid:
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<div class="awardPoints">
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<table>
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<thead>
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<tr>
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<th>Ranking</th><th>Award Points</th>
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</tr>
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</thead>
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<tbody></tbody>
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</table>
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</div>
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If there's a 2-way, 3-way, or n-way tie,
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all tied dancers get the average of the next 2, 3, or n award points,
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and the next 2, 3, or n rankings are skipped.
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### Award Points artifacts
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One quirk of awards points is that for any given overall
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score, there are only a handful of possible judge rankings that could have led
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to it. That means you can make some guesses about how each judge ranked an
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individual dancer, based on only their total award points.
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Here's a handy calculator!
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It (currently) doesn't consider the possibility of a tie.
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<div class="scrolly">
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<fieldset class="speculator">
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<legend>CLRG Award Points Speculator</legend>
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<div>
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Points: <input name="points" type="number" min=41 max=10000 value=188>
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<input name="adjudicators" type="hidden">
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</div>
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<table class="results">
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<caption>Possible Rankings</caption>
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<thead>
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<tr class="warning"><th>Computing: this could take a while!</th></tr>
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</thead>
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<tbody></tbody>
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</table>
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</fieldset>
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</div>
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## What's with these values?
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At first glance, the award points look like the output of an exponential function.
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{{<figure src="chart.png" alt="Chart of scores vs. award points">}}
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In an effort to figure out where these numbers came from,
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I ran some curve fitting against the data.
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Here's the best I could come up with:
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| Ranking range | Award Points Function | Type of function |
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| --: | --: | --- |
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| 1 - 11 | 100 * x^-0.358 | Exponential |
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| 12 - 50 | 51 - x | Linear |
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| 51 - 60 | 14.2 - 0.46x + 0.00385x | Polynomial |
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| 61 - 100 | 1 - x/100 | Linear |
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If you, dear reader, are a mathematician,
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I would love to hear your thoughts on why they went with this algorithm.
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There are a few points to note here:
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* 1st place is a *huge deal*. Disproportionately huge.
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* Places 2-10 are similarly big deals compared to places 3-11.
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* Places 12-50 operate the way most people probably assume ranking works: linearly.
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* 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.
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* 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.
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## Consequences of Exponential Award Points
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Playing around with this,
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I've found a few interesting consequences
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of the exponential growth in the top 11 places.
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### 1st place is super important
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1st place is weighted so heavily that one judge could move a 5th place dancer into 2nd.
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<table class="scorecard">
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<thead>
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<tr>
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<td></td>
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<th>Alice</th>
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<th>Bob</th>
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<th>Carol</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<th class="justify-left">Adj. 1</th>
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<td><input type="number" min=1 max=99 value=1 readonly></td>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=2></td>
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</tr>
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<tr>
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<th class="justify-left">Adj. 2</th>
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<td><input type="number" min=1 max=99 value=5></td>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=1></td>
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</tr>
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<tr>
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<th class="justify-left">Adj. 3</th>
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<td><input type="number" min=1 max=99 value=5></td>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=2></td>
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</tr>
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</tbody>
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<tfoot>
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<tr>
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<th class="justify-left">Award Points</th>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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</tr>
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<tr>
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<th class="justify-left">Ranking</th>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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</tr>
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</tfoot>
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</table>
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You can adjust these values to get a better feel for how scoring works.
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### Tanking a high-ranked dancer is another way to cheat
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Because of that exponential curve,
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a low ranking from a single judge can carry a lot of weight.
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<table class="scorecard">
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<thead>
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<tr>
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<td></td>
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<th>Alice</th>
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<th>Bob</th>
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<th>Carol</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<th class="justify-left">Adj. 1</th>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=11></td>
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<td><input type="number" min=1 max=99 value=2></td>
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</tr>
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<tr>
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<th class="justify-left">Adj. 2</th>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=1></td>
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<td><input type="number" min=1 max=99 value=2></td>
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</tr>
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<tr>
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<th class="justify-left">Adj. 3</th>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=2></td>
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<td><input type="number" min=1 max=99 value=1></td>
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</tr>
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</tbody>
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<tfoot>
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<tr>
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<th class="justify-left">Award Points</th>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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</tr>
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<tr>
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<th class="justify-left">Ranking</th>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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</tr>
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</tfoot>
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</table>
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### Being in 1st provides a nice buffer
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Try playing around with Alice's rankings with Adjudicators 2 and 3 here.
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She has to get ranked a lot lower before her overall ranking starts going down.
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<div class="scrolly">
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<table class="scorecard">
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<thead>
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<tr>
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<td></td>
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<th>Alice</th>
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<th>Bob</th>
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<th>Carol</th>
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<th>Dave</th>
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<th>Erin</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<th class="justify-left">Adj. 1</th>
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<td><input type="number" min=1 max=99 value=1></td>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=2></td>
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<td><input type="number" min=1 max=99 value=4></td>
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<td><input type="number" min=1 max=99 value=5></td>
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</tr>
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<tr>
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<th class="justify-left">Adj. 2</th>
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<td><input type="number" min=1 max=99 value=7></td>
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<td><input type="number" min=1 max=99 value=1></td>
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<td><input type="number" min=1 max=99 value=2></td>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=4></td>
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</tr>
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<tr>
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<th class="justify-left">Adj. 3</th>
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<td><input type="number" min=1 max=99 value=5></td>
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<td><input type="number" min=1 max=99 value=2></td>
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<td><input type="number" min=1 max=99 value=1></td>
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<td><input type="number" min=1 max=99 value=3></td>
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<td><input type="number" min=1 max=99 value=4></td>
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</tr>
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</tbody>
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<tfoot>
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<tr>
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<th class="justify-left">Award Points</th>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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<td class="justify-right"><output name="points"></td>
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</tr>
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<tr>
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<th class="justify-left">Ranking</th>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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<td class="justify-right"><output name="ranking"></td>
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</tr>
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</tfoot>
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</table>
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</div> |