Wooden-Spoon Theory

# Dr. Andrew BroadTennisWooden-Spoon Lists Explanation of the Theory

A wooden-spoon list is a compact representation of a tournament draw. It takes the form of a list of the players, ordered according to who beats whom in the tournament. Obviously the champion is first in the list. I call it a `wooden-spoon' list because any draw has exactly one `wooden-spoonist', defined as the player at the end of the longest losing chain, and that player has the `ignominy' of being identified as the last player on the list.

A simple example illustrates the theory. Suppose Monica Seles, Iva Majoli, Jelena Dokic and Karina Habšudová are the players in a four-player knockout draw. Suppose the results are as follows:

```Semi-finals:
Monica Seles beats Jelena Dokic
Iva Majoli beats Karina Habšudová

Final:
Monica Seles beats Iva Majoli
```
Obviously Monica is the number one player on the wooden-spoon list for this tournament (as well as number one in every other sense of the word). Iva is second, because she lost to Monica in the final. Jelena is third, because she lost to Monica in the semi-finals, and Monica went on to win the tournament. Karina is last on the list, because she lost to Iva, and Iva lost to Monica, so Karina is at the end of the longest losing chain - the `wooden-spoonist' of the tournament.

Study the following example of an eight-player draw to cement your understanding of wooden-spoon theory:

```Quarter-finals:
Monica Seles beats Iva Majoli
Daniela Hantuchová beats Iroda Tulyaganova
Jelena Dokic beats Vera Zvonarëva
Maria Sharapova beats Karina Habšudová

Semi-finals:
Monica Seles beats Daniela Hantuchová
Maria Sharapova beats Jelena Dokic

Final:
Monica Seles beats Maria Sharapova

Wooden-spoon list:
1. Monica Seles
2. Maria Sharapova
3. Daniela Hantuchová
4. Jelena Dokic
5. Iva Majoli
6. Karina Habšudová
7. Iroda Tulyaganova
8. Vera Zvonarëva
```

The wooden-spoon list has some neat properties:

• You can calculate whom a player beat or lost to in a particular round by respectively adding or subtracting a binary power (1 for final, 2 for semi-final, 4 for quarter-final, etc.). For example, given just the above wooden-spoon list, you can work out that the player whom Maria beat in the quarter-finals was Karina (because 2+4=6), or that the player Daniela lost to in the semi-finals was Monica (because 3-2=1).
• You can reverse-engineer the whole draw, given just the wooden-spoon list (strictly speaking, you can reverse-engineer a draw which is isomorphic to the actual draw, since the wooden-spoon list does not intrinsically preserve draw-order). The wooden-spoon list is thus a maximally compact representation of a draw.
• Entries in the wooden-spoon list could be tagged with extra information, such as the score by which each player (except the champion) lost the match that she lost, a Boolean flag for each entry to indicate draw-order, and of course extra information about each player such as her status in the draw (seeding, qualifier, wild card, etc.).
• You could even construct an anti-draw, by reversing the wooden-spoon list and then constructing a draw from the reverse wooden-spoon list. Effectively, it's as though the first-round losers went through to a second round to play each other, and so on - the losers playing in the next round instead of the winners. For what it's worth, the wooden-spoon list `predicts' who would lose to whom, culminating in the crowning of the anti-champion! ;-) The anti-draw for the above wooden-spoon list is as follows:
```Quarter-finals:
Iva Majoli loses to Monica Seles
Iroda Tulyaganova loses to Daniela Hantuchová
Vera Zvonarëva loses to Jelena Dokic
Karina Habšudová loses to Maria Sharapova

Anti-semi-finals:
Iroda Tulyaganova loses to Iva Majoli
Vera Zvonarëva loses to Karina Habšudová

Anti-final:
Vera Zvonarëva loses to Iroda Tulyaganova
```

The algorithm for constructing a wooden-spoon list from a given draw is as follows. This is how you assign players in a 128-player draw to positions in the wooden-spoon list:

• 1: the champion;
• 2: the runner-up;
• 3-4: the players who lost to players 1-2 in the semi-finals;
• 5-8: the players who lost to players 1-4 in the quarter-finals;
• 9-16: the players who lost to players 1-8 in the last sixteen;
• 17-32: the players who lost to players 1-16 in the last 32;
• 33-64: the players who lost to players 1-32 in the last 64;
• 65-128: the players who lost to players 1-64 in the first round.

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