Transitivity, contextuality and decision making

In recent years there has been interest in the relationship between intransitivity and the presence of true (Type II) contextuality (contextuality with context independent marginals). The latter has been considered to be a <i>sine qua non</i> of quantum mechanics, although it has been observed in experiments on human decision making. Transitivity has long been viewed as essential to rational decision making although intransitivity appears to be ubiquitous in the natural world. An analogue of a <i>preference</i> graph is introduced for non-deterministic dynamical systems (of which decision making is an example) and used to identify several conditions which appear necessary for true contextuality to be present. Comparing the preference graph for the system against a reference preference graph formed from the marginals of the observables, one sees losses of options and intransitivity as necessary conditions.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.