The transitivity axiom is common to nearly all descriptive and normative utility theories of choice under risk. Contrary to both intuition and common assumption, the little-known ’Steinhaus-Trybula paradox’ shows the relation ’stochastically greater than’ will not always be transitive, in contradiction of Weak Stochastic Transitivity. We bespoke-design pairs of lotteries inspired by the paradox, over which individual preferences might cycle. We run an experiment to look for evidence of cycles, and violations of expansion/contraction consistency between choice sets. Even after considering possible stochastic but transitive explanations, we show that cycles can be the modal preference pattern over these simple lotteries, and we find systematic violations of expansion/contraction consistency.

The voting paradox occurs when a democratic society seeking to aggregate individual preferences into a social preference reaches an intransitive ordering. However it is not widely known that the paradox may also manifest for an individual aggregating over attributes of risky objects to form a preference over those objects. When this occurs, the relation ‘stochastically greater than’ is not always transitive and so transitivity need not hold between those objects. We discuss the impact of other decision paradoxes to address a series of philosophical and economic arguments against intransitive (cyclical) choice, before concluding that intransitive choices can be justified.