Math A-lympiad: Preliminary 1998-1999

The best one wins

Introduction

What do the Tour de France, a football competition, Wimbledon, the Eurovision song contest, political elections and a consumer study by the Consumers Association have in common? In all these cases a mutual comparison is made and the ultimate aim is for the best to be brought out. The way in which the best is brought out is quite different in each of the above cases. In a football competition for example, all teams play against each other twice and points are awarded according to the result. The team with the most points at the end is the champion. With political elections voters cast their votes and the party or politician with the most votes is then the winner. It is not always only the first, as it is sometimes desirable to construct a further order. We are going to look at different types of systems that lead to a certain ranking.

Introductory exercise
Set out a number of existing systems one by one. You can make use of the examples already mentioned, but you may also look at other existing systems. Give a description of each system and the application it is used in. Indicate the advantages and disadvantages of the system. Also explain why the system is used in this situation.

Meat products

The Dutch Association of Meat Preparers considers it important from a promotional point of view to set up an annual competition among themselves. It enables members to earn the title ‘Meat Preparer of the Year’ in a number of different categories. In each category there is a first prize (gold), second prize (silver) and third prize (bronze). The winners get a cup and a certificate to hang in their shop. The association has the following five categories:
A Liver sausages
B Smoked sausages
C Pâtés
D Salads
E Foreign meat specialities.

Alongside the winners of each category one meat preparer is also given the title of general meat preparer of the year. The association has an extra large challenge cup for this.

Participants

The association has vigorously encouraged participation in the competition. Many members have thus become enthusiastic about it. Because quite a bit of variation in preparation method is allowed in each category, each participant may make a maximum of three entries in each category. In so doing it is possible for a participant to enter with as many as fifteen products. A participant does not need to enter all categories. However, in order to come into consideration for the general meat preparer of the year title you must have at least one entry (and at most three) in each category. Because of the many possibilities for winning a prize and as a result of the enthusiasm of the members, the number of entries has so increased that the judging of all these entries has led to problems.

Judging

Up until now the judging has been done by a specialised panel who picked the winners in mutual consultation. This manner of consultation has never been revealed to the outside, but for a number of years participants have had their doubts about its fairness. It is simply not possible for every member of the panel to try every product entered. There are currently hundreds of entries.

The association now wants a new system of judging. The association has even thought that all participants should be required to judge the competition on one day per year. On this day the products have to be sampled in a central location and an assessment must be made. In addition, the members of the specialised panel will be available for one day of the year to sample and to judge.

Preconditions

The new system must be calculated against a maximum of 300 products entered per category. The total number of participants is estimated to be a maximum of 500, where a maximum of 100 will participate in all categories.

The participants who judge the products, and also the specialised panel members, cannot try an unlimited number of products on the judging day. The association knows from experience that saturation is reached at a certain point in time. The products that still have to be tried after that do not have a fair chance. That is why the association has decided that ten products per person would be the ideal situation. If that leads to practical problems, the number of products to be tried necessarily has to be increased. However, the association does not want to exceed a maximum figure of 20 products per person on one day under any circumstances. The association is happy to aim for a figure of 10 to 12 products per person. On the other hand the association would also like to set up a system that is as fair as possible.
The specialised panel consists of 10 people.

Main exercise
Devise a judging system that is as fair as possible. The system must in any case satisfy the requirements and preconditions described above. Remember that the above numbers of products entered and participants are not necessarily exact. They are maximum numbers: the precise amount will only be known on the closing day for entries.
Describe the system and explain the choices made. Also draft a letter that the association can send to the participants. The letter must clearly and concisely set out the system for the participants.