Decision theory and discrete mathematics Original Research Article

This paper discusses recent developments that emphasize the role of discrete mathematics in decision theory. It focuses on three topics which illustrate this role. The first is the transitive simple majorities problem of determining the maximum number of linear orders on n candidates that prevent the occurrence of cyclic majorities when votersʹ preferences are confined to those orders. The second involves the varieties of unique solutions to simple systems of n — 1 linearly independent homogeneous linear equations in n variables of types that arise from qualitative equivalence comparisons in the measurement of subjective probabilities and utility differences. The third topic describes a computer-efficient hierarchy of stochastic-dominance relations for comparisons of risky alternatives whose outcomes lie in a unidimensional set of evenly spaced points. Although the paper is primarily expository, new results and new conjectures are included.