This Month's Feature Column

Who Won!

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Introduction

Competitions are a staple of American life whether they take the form of an end of season basketball tournament between a collection of colleges, a little league baseball team playoff, or a group of individuals competing for the American chess championship. Many schemes have been devised to get at the best player or team.

One mathematical concern is how to schedule such tournaments. Here the situation for hockey, which requires a special kind of locale (ice rink), differs from chess or bridge where the space that can be used to conduct the tournament is more flexible. Such scheduling problems draw on ideas from combinatorics and finite geometry (balanced incomplete block designs).

After the tournament is played, an important aspect of tournaments is to decide how well the teams or players performed. Was there a clear winner? What might be a good ranking of the teams from strongest team to weakest team? Were some teams equally strong so that a "tie" should be declared? We shall see that some of the same tools that help with scheduling problems can also be brought to bear on these ranking questions.

To try to answer questions of this type we will begin by using a simple mathematical model as shown in Figure 1. Here the teams or players are represented by dots (small circles) and the number inside the dot is the name of the team or player. We consider tournament situations which cannot result in ties between the teams or players. Many sports, to make the matches more dramatic, have a way to avoid having ties occur. Thus, if player i has a match against player j, then either i beats j or j beats i. We show this in a diagram such as Figure 1 by putting a line segment with an arrow directed from team i to team j when in the match that they play team i beats team j. Now we can read off how many matches were won by a particular player. In Figure 1, for example, player 3 won 2 matches, those played against teams 4 and 5. Of course, if one knows how many matches a player wins in a round robin tournament without ties, one can figure out how many matches the player lost. Each team plays the same number of games as every other team and this number is one less than the number of teams in the tournament. As is often the case in constructing a mathematical model or representation for some situation in order to obtain insight, we have simplified (thrown away) some information in the process of drawing a graph such as the one in Figure 1. Thus, while i may beat j, we are not trying to record by "how much" i beat j. Sometimes this information is available as in basketball or baseball where each team has a score and we know by how much one team beat the other.

Who do you think should be declared the winner for the tournaments whose results are shown in Figure 1? Which team performed least well?

 

Joseph Malkevitch
York College (CUNY)
malkevitch at york.cuny.edu