We see competition shows all the time, such as “Top Chef”, “Next Iron Chef”, “Chopped”, all of the baking competition shows, and even shows such as “Forged in Fire” and “Next Mythbuster”, where the loser of each round gets eliminated.

The goal of the show is to find “the best” out of a group of competitors, but this is not the likely outcome.

Let’s look at why, and how to fix it.

## Contents

2. “Rank and Yank” Elimination is Problematic

4.1. Linear Point Distribution

4.2. Non-Linear Point Distribution Schemes

5. Scoring With Elimination – Could This Be the Answer?

## 1. Show Format

All competition shows seem to follow a similar structure:

- Out of a starting group of “n” competitors, there are n-2 or n-3 shows where one competitor is eliminated on each show.
- This usually leads to a finale or showdown, where the last 2 or 3 competitors compete in one final round, and the winner of the season is chosen.

Each show generally follows this format:

**Recap of last week’s show****Warm-up or bonus challenge**: Contestants compete in a smaller challenge. Although the loser is not eliminated, the winner often receives a perk or bonus that they can use during the elimination round. Examples include extra time, a preferred ingredient, or the ability to tamper with other competitors.**Elimination challenge**: Contestants compete in a large challenge. The top 1-2 winners are proclaimed safe from elimination, and there might be an extra perk that the winner receives for next week’s show.**Elimination**: The bottom 1-2 losers go in front of the judges, and one of them is selected for elimination.

### 1.1. Variations

Some shows have some or all of these variations:

- Competitors are grouped in to teams (such as “Hell’s Kitchen”)
- Losers’ challenge: The bottom 1-2 losers compete to see who will be eliminated. The winner stays, the loser leaves.
- Competitors vote to see who should be eliminated.
- Point systems might be employed as a bridge to elimination, where the fewest points results in elimination.
- The first show is non-elimination. The winner gets perks, the losers get nothing, but also don’t face elimination.
- Overall performance: Elimination is based on the judges’ views of the loser’s overall performance. Without a scoring system, this is quite subjective.

## 2. “Rank and Yank” Elimination is Problematic

These shows generally use “rank and yank” elimination, where the competitors are ranked (or ordered) in terms of their performance, and the worst performance is eliminated.

This is attractive because, generally, for “n” competitors, you can have “n” shows, and there is some element of drama due to elimination on each show.

Although the intent is for each competitor to do their best, in reality, the incentive is for each competitor to NOT end up at the bottom of the list, which isn’t the same thing.

- Strong competitors are often sabotaged or voted off and eliminated early on, as no one wants to face them during the finale.
- Weak competitors often float around in the middle, in terms of rank, and last much longer than they should.
- The longer the weak competitors last, the greater the danger for the stronger competitors due to the prolonged risk of sabotage.

So on the one hand, you can have a strong competitor who gets eliminated due to a confluence of events – maybe they are unfamiliar with a key ingredient, on top of being at the short end of another competitor’s perk.

On the other hand, you often see a more or less average competitor who sits in the middle of the pack until the very end, and ends up winning.

There are much better elimination schemes that result in a significantly more fair outcome.

## 3. Double Elimination

Football often uses Double Elimination, where a competitor has to lose twice in order to be eliminated.

Among a group of “n” competitors, you would need 2 * n elimination rounds.

In the context of our competition shows:

- Either there must be 2 * n – 1 shows, or there must be 2 “ranking” rounds per show. Based on the average show format, it makes the most sense to have 2 ranking rounds per show.
- You could get half-way through the season before the first elimination. This happens in a worst-case scenario, where each competitor is ranked as the “loser” exactly once before the first elimination. Obviously, the lack of eliminations also eliminates most of the drama, and drama = ratings.
- Setting aside extreme scenarios, on average, there would be 1 elimination per every 2 ranking rounds.

Although this scheme overcomes the issue where a good competitor can be eliminated early, it doesn’t solve issues with sabotage, and it reduces the drama (and probably the ratings)

## 4. Point Tally / Total Score

In this scheme, there are no eliminations.

Instead, competitors are ranked after each round, and then receive points based on rank.

For example, if there are 10 competitors, perhaps first place receives 9 points, while the loser receives 0.

At the end of a specified number of rounds, the competitor with the highest total score is the winner.

Advantages:

- Stronger competitors will consistently score higher, while weaker competitors will consistently score lower.
- This scheme allows a strong competitor to stumble and recover, but also allows an average competitor to evolve over time, accruing enough points to end in a lead position.

The disadvantage of this scheme is that, without eliminations, there is no drama.

### 4.1. Linear Point Distribution

In the example above, we use a linear point distribution scheme, where there are 10 competitors, and each competitor gets more points the higher their position:

For “n” competitors, each competitor in rank “r” receives n-r points:

p = n-r

So for “n” competitors, the total points per round are:

n * (n-1) / 2

Assuming “n – 1” rounds, the total number of points for the entire competition are:

n^2 * (n-1) /2

For example, for 10 competitors:

- First place receives 9
- Last place receives 0
- Each round has a total of 45 points
- The entire competition has 405 points
- The highest score for the competition would be 81, while the lowest would be 0
- Top competitors would score 65 or above, average scores would range from 35 to 45, and low scores would be under 20.

A linear point distribution allows someone who has consistently performed at a high level – in the 7-9 range, to score a 4 without suffering too badly. With an average score of 8, and one low round, the competitor’s total score is still in the high 60 range.

Meanwhile, consistently poor performers will have consistently low scores.

As time progresses, the gap between high and low increases at a linear rate, but someone could still come back from behind to assume a leadership position. For example, if someone has an average score of 5 for the first 6 rounds, they could score 9 points (1st place) in the 3 remaining rounds for a total score of 57, which is a fairly competitive score.

The nice thing about a linear relationship is that we can change the minimum point score, increase the points per round, or increase the stepping between ranks, but as long as the relationship is linear, all of these systems equate to the one described above.

So, for example, we could aware double the points, or double plus 10 without changing the game dynamics.

### 4.2. Non-Linear Point Distribution Schemes

In a non-linear scheme, the points are raised to some exponent in order to create a greater or smaller gap between the highest and lowest rank.

Starting with our linear distribution, where a competitor out of a group of “n” ranks “r” and receives “p” points:

p = n-r

Using an exponent greater than 0 results in a much larger gap between first and last rank:

p = (n-r) ^ 1.5

Here, the top score out of a group of 10 is 27, while the lowest is 0. A middle rank of 5 would yield 11 points.

- A weaker competitor can gain points quickly by winning only a couple of rounds
- The gap between first and last place grows much more quickly after each round, making it much harder for last place to catch up
- Assuming 9 rounds, the top possible score is now 243 while the lowest possible score is still 0.
- A strong score for the season would be 200 points, while an average score would be 100, and a low score would be 30 or less. It’s easy to see that there is a much larger gap between average and high, vs. low and average.

Likewise, using an exponent less than 0 results in a much smaller gap:

p = 2 * (n-r) ^ (2/3)

In this scheme, winning a round doesn’t yield a tremendously high amount of points. The top score per round is only 8, with an average score being 5.

- There is very little differentiation between weak and strong competitors.
- A low-ranking competitor can quickly gain points by scoring average or better, and can therefore catch up quickly.
- High-ranking competitors can’t pull away from the pack.
- Assuming 10 rounds, the highest score would be 72, while the lowest is 0.
- A good score for the season would be 63, an average score 45, and a low score 24.

Comparison:

Rank | p=n-r | p^1.5 | 2p^(2/3) |
---|---|---|---|

1 | 9 | 27 | 8 |

2 | 8 | 22 | 7 |

3 | 7 | 18 | 7 |

4 | 6 | 14 | 6 |

5 | 5 | 11 | 5 |

6 | 4 | 8 | 4 |

7 | 3 | 5 | 4 |

8 | 2 | 2 | 3 |

9 | 1 | 1 | 2 |

10 | 0 | 0 | 0 |

Non-linear distribution is a way that we can reward either the winner or the loser disproportionately.

## 5. Scoring With Elimination – Could This Be the Answer?

Here is what I propose:

**Use a linear Point Tally / Scoring system combined with elimination.**

Structure:

- The first show is only for scoring.
- N-3 shows have an elimination round, resulting in the elimination of the competitor with the lowest total score.
- Once down to the final 3 competitors, they can trade their points for perks, and in the final round, winner takes all (by judges’ decree).

Show format:

- Small challenge – (n-r) points. Perhaps the winner also receives a perk, or some other variation.
- Large challenge – 2 * (n-r) points.
- Elimination: The competitor with the lowest score is eliminated.
- In the event of a tie, the two losers compete in a final elimination round for 1 point. The winner gets 1 point, the loser gets eliminated. By decree of the judges, there is one winner and one loser.
- In the event of an n-way tie, the losers compete and receive l-r points. For example, if the losing score is a 3-way tie, they compete for 3 positions, where the winner (of the losers) gets 3-1 = 2 points, and gets to stay. The middle score gets 3-2 = 1 point and gets to stay. The loser gets 3-3 = 0 points and gets eliminated.

**Advantages:**

- Consistently strong performers will have consistently high scores, and will be more resilient to elimination.
- Consistently poor performers will hover near the bottom, creating extra drama as they struggle not to be eliminated.
- Each show after the first show has an elimination (along with the drama it entails).
- During the finale / showdown, the points can be used as an extra game dynamic.

### 5.1. Point Distribution

Because the large challenge scores double that of the small challenge, the maximum points per show (small + large) = 3 * (n-1).

In our example of 10 competitors, this would be 3 * 9, or 27 points.

The maximum possible score is 243, and the minimum is still zero.

If a person scores consistently average, their cumulative score would be about 135.

Also, this introduces a new dynamic, where, even if a competitor consistently scores high in the small challenges but low in the large challenges, their score remains competitive for about 1/3 of the total number of rounds.

## 6. Conclusion

Contest / Elimination shows that use single-elimination (rank and yank) don’t necessarily end up with the superior contestant as the winner.

Other schemes, such as scoring with elimination provide a better ranking system using the same number of ranking rounds, and also introduces new game dynamics that potentially makes the contest itself more interesting.