The Skating System Algorithm for Scrutineering

Overview

The Skating System is the worldwide standard scoring method for ballroom dance competitions. It consists of 11 rules that determine how adjudicator marks are collected and processed during a competition.

Algorithm Structure

In qualifying rounds, adjudicators select couples to advance to the next round.

Rule 1: Voting in Qualifying Rounds

• Each adjudicator votes for exactly the number of couples specified by the Chairman.
• Votes are tallied across all adjudicators.
• Couples with the most votes advance.
• In case of ties at the cutoff point, all tied couples advance.

These rules govern how adjudicators must mark couples in final rounds.

Rule 2: Complete Placement

• Each adjudicator must place ALL couples in order of merit in each dance.

Rule 3: Sequential Placement

• Placements must be sequential: 1st place for the best couple, 2nd for the next best, etc.

Rule 4: No Ties Allowed

• Adjudicators cannot tie couples for any position.
• Every couple must receive a unique placement from each adjudicator.

These rules determine how to calculate final placements for a single dance based on all adjudicators’ marks.

Rule 5: Majority Winner

• Calculate absolute majority: (Number of adjudicators ÷ 2) + 1
• For each position (starting with 1st), count how many adjudicators placed each couple at that position or better.
• If exactly one couple has absolute majority, they win that position.
• Remove placed couples from further consideration.
• If multiple couples have majority, proceed to Rule 6.
• If no couples have majority, proceed to Rule 8.

Rule 6: Largest Majority

• Applied when multiple couples have majority for the same position but with different counts.
• The couple with the largest majority (most marks at or better than examining position) wins.
• This couple is removed from further consideration.
• Continue with remaining couples for subsequent positions.

Rule 7: Breaking Equal Majorities

• Applied when couples have equal majorities (same count of marks) for the same position.

7(a) Sum of Marks: - Sum all marks at or better than the examining position for each tied couple. - Couple with the lowest sum wins the better position.

7(b) Equal Sums: - When sums are also equal, examine the next place mark. - Recalculate majorities including the additional place. - Apply Rules 5-7 recursively until the tie breaks.

7(c) Unbreakable Ties: - When all marks are examined and couples remain tied. - Couples share the same fractional placement (e.g., two couples tied for 2nd both receive 2.5).

Rule 8: No Majority Found

• Applied when no couple has majority for the position under review.
• Move to the next examining position (e.g., from 1st to 2nd).
• Continue incrementing until at least one couple achieves majority.
• Once majority is found, apply Rules 5-7 as normal.
• This rule is handled implicitly in the implementation by the examining loop.

When an event consists of multiple dances, placements from each dance are combined to determine overall results.

Rule 9: Final Summary Compilation

• Sum each couple’s placement marks across all dances (e.g., 1st + 2nd + 1st + 3rd = 7 points).
• Sort couples by total sum (ascending).
• Lowest total wins overall.
• If ties exist, apply Rules 10 and 11.

When couples have identical total placements after all dances, the following tie-breaking procedures are applied:

Rule 10: Count of Better Placements

• For each tied couple, count the number of dances where they placed at or above each position (starting from 1st).
• The couple with the most dances at or better than the current position wins the tie.
• If still tied, increment the position and repeat until the tie is broken.
• If all positions are examined and couples remain tied, proceed to Rule 11.

Rule 11: Head-to-Head Comparison

• Only the tied couples’ marks are considered across all dances.
• The single dance placement algorithm (Rules 5-8) is re-applied to these couples.
• The couple ranked 1st in this head-to-head wins the tie.
• If still tied, couples share the same final position, and fractional placements may be assigned.

Key Data Structures:

• Scores: Stored with heat_id, judge_id, slot (dance number), and value (placement)
• Rankings: Hash mapping Entry objects to their final rank
• Explanations: Optional detailed trace of algorithm decisions for debugging

Algorithm Flow:

1. Single Dance Placement (Heat.rank_placement):
   • Iterates through positions 1 to max, examining each
   • Uses recursive runoff function for tie-breaking
   • Returns rankings hash and optional explanations
2. Multi-Dance Compilation (Heat.rank_summaries):
   • Sums placements across all dances (Rule 9)
   • Breaks ties using count of better placements (Rule 10)
   • Falls back to head-to-head comparison (Rule 11)

Special Cases:

• Fractional Placements: When couples tie after all rules, rank = base + (count-1)/2.0
• Set Return: Rule 11 may return a Set of backs when ties are unbreakable
• Focused Runoff: Rule 7(b) uses focused=true flag to limit examination to tied couples
• Empty Scores: Algorithm handles cases where no couples have been marked

Validation Rules

1. All couples must be placed by all adjudicators
2. No duplicate placements allowed from single adjudicator
3. Majority = (Number of judges ÷ 2) + 1
4. All ties must be resolved using rules in sequence 5→6→7→8

References:

• Tabulating DanceSport Competition Marks
• The Skating System Study Guide