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Methods: The mathematics of management (2)

by David Blakey

Some real examples of non-transitive sets.

[Monday 12 May 2003]

Since I wrote about the mathematics of management, two real examples of non-transitive sets have occurred to me. Both are about sport, rather than management, but they do illustrate how the mathematics works.

The first is in a game like football or cricket or baseball, with a number of teams in a league. During a season, each team will play each other team. Imagine it with cricket (or baseball).

Some teams will be stronger than others in batting. Some teams will be stronger than others in bowling (or pitching). Some teams will be stronger than others in the field.

There will be other characteristics, some related to psychology. For example, some teams will play more aggressively when they are losing. But we shall take just these three areas: batting, bowling and fielding.

Now imagine three teams, A, B and C, who each score high or medium or low in each of the three areas.

Team ATeam BTeam C

Now consider a match between Team A and Team B. Team A is stronger in batting and bowling; Team B is stronger in fielding. If these three areas were the main factors in winning, you would expect Team A to beat Team B.

For a match between Team B and Team C, Team B is stronger in batting and fielding, and Team C is stronger in bowling, so you would expect Team B to beat Team C.

If, because of these areas, Team A has indeed beaten Team B and Team B has beaten Team C, then a transitive set would be A > B > C. In a match between Team A and Team C, Team A would be expected to win.

In fact, Team C is stronger than Team A in bowling and fielding and weaker only in batting, so Team C would be expected to win. This would not surprise a student of management mathematics, but might surprise a sports commentator.

So much for theory. Here is a real example. It involves three boxers.

By October 1974, George Foreman had beaten Joe Frazier, and Joe Frazier had beaten Muhammad Ali. On October 30, Ali would fight Foreman for the world heavyweight boxing championship in Zaire - the ‘rumble in the jungle’.

If the relationships had been transitive, then Foreman > Frazier > Ali, so Foreman was expected to beat Ali. But Anthony Piel built the following table.


Foreman's power and technique had beaten Frazier, and Frazier's speed and power had beaten Ali. Ali's speed and technique might beat Foreman. In fact, Ali did win. (Mr Piel wrote a letter about it to the Scientific American, which was published in January 1975.)

Now this might be very interesting, and it might explain how one sports team or person can win over another ‘against the odds’, but it does have serious implications for management decisions. Imagine a selection process that eliminated options by comparing them on a one-by-one basis. If the selectors imagined a transitive relationship between the options, when there was in fact a non-transitive relationship, then the best option could actually be eliminated quite early in the process.

On major selection exercises, some clients do short-list options in this way. One team may be set the task of choosing between Option A and Option B. Another team may have to choose between Option C and Option D. They may short-list Option A and Option D. But if their tasks had been to choose between Option A and Option C in one team and between Option B and Option D in the other, they might have short-listed Option C and Option B.

As a consultant, you should always keep the realities of management mathematics in mind, and you should remind your clients of those realities when they need reminding.

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