# Multi-class classification

I am studying the one-vs-one multi-class classification approach using these slides.

I understand the middle green region cannot be properly classified using this approach.

What don't I understand is the below line?

Two way preferences need not be transitive.


Can someone explain this in the context of multi-class classification?

Suppose a point lies in the green region, and look at the classifier pairs starting in the lower left, moving counter-clockwise.

The first classifier pair says 'prefer c2 over c1' the second classifier says 'prefer c3 over c2'. If the the two-way preferences were transitive you would expect the final classifier to say 'prefer c3 over c1', but it doesn't, that classifier prefers c1 over c3!

Although this question got a clear and accepted answer.

I would just like to share with readers that the word 'transitive' here is used with reference to a mathematical (logical) property 'transitivity'.

Transitivity means that a relation between three elements is such that if it holds between the first and second and it also holds between the second and third it must necessarily hold between the first and third.

For example; if A is larger then B and B is larger than C; then we can say Ais larger than C.

So, as @Max S. has described; "Two-way preferences need not be transitive." means that

"The first classifier pair says 'prefer c2 over c1' the second classifier says 'prefer c3 over c2'. If the two-way preferences were transitive you would expect the final classifier to say 'prefer c3 over c1', but it doesn't, that classifier prefers c1 over c3!"