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Technical
To Seed or Not?

Samir illustrates that there are weaknesses in seeded blocks followed by a seeded knockout which can be exploited to give a player an easier ride to the final.

 

A number of major croquet events use a format based on some initial round (often a block stage) followed by a knockout and there are a number of opinions about how that knockout should be organised.

In this article, I set out a hypothetical scenario to indicate the problems that can arise from block-based seeding of the knockout draw, and the benefits of using ranking-based seeding instead.

Whilst this is a single scenario, rather than a robust analysis, it was also the first scenario that I analysed, so there is no reason to believe that it’s a particularly extreme case.


Once upon a time, at a croquet club not very far, far away…

There was a croquet tournament scheduled using a format of four blocks (W, X, Y, Z) which were seeded using standard striped seeding.  The top two players from each block progressed to a knockout and the knockout was drawn based both on block positions and in a way to minimise the chance of players from the same block meeting during the knockout, i.e. the first round knockout would be W1 vs Y2, X2 vs Z1, Y1 vs W2 and Z2 vs X1.  Block play was stripe-seeded such that the top two players played in the final round.

The top players were conveniently named A, B, C, D, E, F, G, H, with equally convenient grades of 2600, 2550, 2500, 2450, 2400, 2350, 2300, 2250.  Thus…

Block W

had players A (2600) and H (2250),

Block X

had players B (2550) and G (2300),

Block Y

had players C (2500) and F (2350) , and

Block Z

had players D (2450) and E (2400).

There were a number of other players in each block, but none of those players managed to beat any of the top eight players.

Remarkably, almost every player played to a consistent standard commensurate with their grade.  The two exceptions were:

Player D

who played to a standard of 2700 (and was the best player in the event)

Player G

who played to a standard of 2650 (and was the second-best player in the event)

Players A and B,

playing to their grades of 2600, 2550 respectively, were the third and fourth best players in the event.

Equally remarkably, almost every block game was won by the person playing to the higher standard.  The only exception was the game won by player F (playing to their grade of 2350) against player C (playing to their grade of 2500) - a result which the usual formula says was a 33.4% probability, so not that unexpected.

Thus, the final block finishing positions were therefore:

Block W:

1st A (grade 2600), 2nd H (grade 2250)

Block X:

1st G (grade 2300, but playing to a standard of 2650), 2nd B (grade 2550)

Block Y:

1st F (grade 2350), 2nd C (grade 2500)

Block Z:

1st D (grade 2450, but playing to a standard of 2700), 2nd E (grade 2400)

And therefore the knockout draw was A vs C, B vs D, F vs H, E vs G. Each player continued to play to the standard that they had played in the block stage.

Analysis of the knockout drawn revealed that the player who was playing the best in the event (Player D, playing to a grade of 2700) had a 29.9% probability of winning the event, whilst player G (playing to a grade of 2650) had a greater (33.9%) probability of winning the event.  (The pre-event favourite, Player A, had only a 15.8% chance of winning the knockout, but there were two players in the event playing better croquet.)

An investigation was launched into how the event resulted in the person who was playing the best in the event (Player D) did not have the highest probability of winning the event.  This uncovered that:

  • Had the knockout been drawn using pre-event grades, the four players playing the best in the event (players D, G, A, B, in that descending order) would have had the highest probabilities of winning the event (35.4%, 24.4%, 20.7% and 10.5%, again in that order).  It seemed to everyone that this would have been much more satisfactory.
     
  • Had player A "accidentally" lost to player H in block W, they would have increased their probability of winning the knockout from 15.8% to 19.8%.  Had player D "accidentally" lost to player E in block Z, they would have increased their probability of winning the knockout from 29.9% to 32.5%.  Neither Player A or Player D had noticed this strategic option at the time, but everyone was now wise to it and thus prepared for the next event to use this format.  Anyone who recalled the Olympic Badminton farce* felt that this was a situation best avoided.

Grade-based seeding was therefore deemed much better for future events, and everyone lived happily ever after…

Or not.

 

* The players had all qualified for the quarter-finals of the women’s doubles and were accused of trying to manipulate the draw by losing so they would be matched with easier opposition in the next round or avoid playing their teammates. See e.g. http://www.bbc.co.uk/news/world-asia-china-19074280

Author: Samir Patel
All rights reserved © 2014-2017


Updated 28.i.16
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