  The transitivity of preferences is one of the basic assumptions used in the
theory of games and decisions. It is often equated with rationality of choice
and is considered useful in building rankings. Intransitive preferences are
considered paradoxical and undesirable. This problem is discussed by many
social and natural sciences. The paper discusses a simple model of sequential
game in which two players in each iteration of the game choose one of the two
elements. They make their decisions in different contexts defined by the rules
of the game. It appears that the optimal strategy of one of the players can
only be intransitive! (the so-called \textsl{relevant intransitive
strategies}.) On the other hand, the optimal strategy for the second player can
be either transitive or intransitive. A quantum model of the game using pure
one-qubit strategies is considered. In this model, an increase in importance of
intransitive strategies is observed -- there is a certain course of the game
where intransitive strategies are the only optimal strategies for both players.
The study of decision-making models using quantum information theory tools may
shed some new light on the understanding of mechanisms that drive the formation
of types of preferences.
