Consider three integers, a, b and c. If a is bigger than b and b is bigger than c, then we know a is also bigger than c. A relationship where this is true is called transitive.
More formally in mathematics, a relationship for examples a,b,c is transitive if : When a has that relationship to b, and b has that relationship to c, then it is implied that a also has that relationship to c.
Initially, it might be thought that all relationships must be transitive but this is not the case. For those familiar with Rock Paper Scissors[1]: Rock beats (blunts) scissors, scissors beat (cut) paper but paper beat (wraps) rock. Try ordering these by which one wins. For any pair, “better” is clear but across all three there is no “best”[2].
Intransitivity can occur in corporate decision making. Consider the following scenario where there are three people – Tom Dick and Harriet. They each have to express an opinion about three proposed options – A, B and C.
They agree that the group prefers one option over another if the majority of them prefer that option over the other.
Their opinions are as follows:
- Tom: “ I really like A. B is OK. I don’t like C”
- Dick: “I don’t like any. B is better than C. A is the worst”
- Harriet: “I really like A and C. I prefer C to A. B is OK.
Before getting into the analysis of the options – note that each set of opinions is clearly expressed and unambiguous. Each person’s preferences are known and able to be scrutinised or challenged and views are consistently held (over time and between comparisons). Each of these represent significant simplifications compared with real life situations.
So let’s look at the analysis – starting with comparing options A and B:
Tom prefers A to B. Harriet also prefers A to B. Dick prefers B to A. So in A vs B, A is preferred by 2 to 1.
Let’s look at B and C: Tom prefers B to C, Dick prefers B to C and only Harriet prefers C to B. So – A is preferred over B by 2:1 and B is preferred over C by 2:1. Things are not looking good for C.
But wait, let’s compare preferences for A and C directly:
Tom prefers A to C, Dick on the other hand prefers C to A. And Harriet? She prefers C to A. So C is preferred over A by 2:1.
So – the group (in each case by a majority of 2:1) says A is preferred over B, B is preferred over C and C is preferred over A. An intransitive result.[3]
If this were to happen in a real-world situation, what might be the consequences? As soon as a consensus began to occur about a particular option, one of the parties would be able to demonstrate that their preferred option would be a better choice – resulting in decision-making gridlock. Alternatively, if a decision were able to be achieved in the meeting, someone may feel justified in not abiding by the result, as they can demonstrate it wasn’t the preferred choice – leading to a collapse of joint decision making.
Heritage: I developed this example in the mid 1990s as a way to try to understand the sayings of Michael T Black at CSC Index who tried to explain why it was necessary to have a Value Metric. Part of the answer was to avoid intransitivity.
[1] Rock Paper Scissors – a playground game. See https://en.wikipedia.org/wiki/Rock_paper_scissors
[2] There are also intransitive dice where the expected value from each die is intransitive. See https://en.wikipedia.org/wiki/Intransitive_dice
[3] Earlier literature also refers to this as the Paradox of Voting, the Condorcet Effect and Cyclical Majorities.