This paper studies random choice rules over finite sets that obey regularity but potentially fail to satisfy all of the Block-Marschak inequalities. Such random choice rules can be represented by non-additive random utility functions: that is, by capacities on the space of preferences. The higher-order Block-Marschak inequalities are shown to be related to the degree of monotonicity that can be achieved by a capacity representation. These results help to decipher the Block-Marschak inequalities, and are applied to study the relationship between random choice over finite sets and random choice over lotteries.