Chinese and Mayan Dice: Circular Chains, Explicit Constructions, and Decision Theory

We study intransitive probabilistic preferences through dice whose face labels may be repeated. We distinguish restricted models, in which the set of permitted labels depends on the number of sides, from the unrestricted model, which allows arbitrary integer labels. We review existence results, including the exceptional impossibility of a Chinese chain of five four-sided dice, and give an explicit construction establishing the known existence of simple cycles of three-sided dice of every length m ≥3, withno ties and a net win margin of one. We examine mixed-integer programming, simulated annealing, and lexicographic optimization of the margins. We then extend pairwise comparison to probability measures by means of bounded, measurable, antisymmetric kernels. For complete comparison data with uniform weights, we derive the Hodge decomposition of the margin matrix into an additive score component and an orthogonal, divergence-free residual. We distinguish the failure of an exact additiverepresentation from the existence of strict preference cycles. The minimax theorem ensures that every finite family admits a mixed strategy with nonnegative expected margin against every mixture of that family. For the historical cycle with margins 4, 4, 10, we compute both the decomposition and the equilibrium mixture (2/9, 5/9, 2/9). Applications to portfolios, artistic evaluation, visual perception, and sports clarify the distinctions among independent comparisons, shared scenarios, and expected utility.