To explain transitivity, let us look first at a totally different example that has a lot to do with transitivity but nothing to do with rationality. A combination of choices is rational if an observer can deduce someone else's preferences (that is, learn what that person likes best, medium, and least) just by observing his choices. Suppose I put your friends Adam and Ben side-by-side and I see that Adam is taller than Ben. Then I put Ben and Charlie side-by-side and I see that Ben is taller than Charlie. Can you tell me who is taller, Adam or Charlie? Adam, of course. How do we know? We simply use transitivity: if Adam is taller than Ben, and Ben is taller than Charlie, then Adam is taller than Charlie. This also means that I can rank your three friends by their height: from tallest (Adam) to shortest (Charlie).

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