Artists and scientists alike share a fascination for cyclic phenomena (also called strange loops), such as Escher’s drawings, Shepard’s musical scale, Condorcet’s voting paradox (inspiring Arrow’s impossibility theorem), the liar paradox, Gödel’s incompleteness theorem, the Rock-Paper-Scissors children game, to name but a few.
In this lecture, we study cyclic phenomena associated with the winning probably relation of a random vector. We briefly introduce the cycle-transitivity framework, ideally suited for characterizing the transitivity of reciprocal relations, a generalization of crisp complete relations encompassing winning probability relations. Focusing on winning probability relations, we lay bare the link with the underlying dependence structure, and complement elegant theoretical results with remarkable observations made through massive computation involving all 9.30 E+10 sets of 4 dice with 6 faces (independent random variables), and all 1 104 891 746 non-isomorphic posets of 12 elements (intricately dependent random variables). Throughout, we point out connections with species competition, environmetrics and chemometrics, economics and finance, and machine learning.
Since most attention so far was limited to cycles of length three, we initiate the study of cycles of length four, introducing the Rock-Paper-Scissors-Lizard metaphor. Although the picture is still far from complete, it already offers some interesting insights and challenging open problems.