DO CLASSICAL (OR QUANTUM) TRANSITIVE PREFERENCES ALWAYS RESULT IN INDIFFERENT DIVISIONS? Marcin Makowski 1, *, Edward W. Piotrowski 1 ** and Jan Sladkowski 2 1. Institute of Mathematics, University of Bialystok, Akademicka 2, PL-15424, Bialystok, Poland * makowski.m@gmail.com, ** qmgames@gmail.com 2. Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland jan.sladkowski@us.edu.pl January 27, 2015 Abstract 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 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. 1 Introduction Games have long fascinated scholars, often contributing to the development of new theories [1]. In fact, the attempts to construct a systematic theory of rational behaviour have focused on games as simple examples of human rationality. The attractiveness of the new theory investigating the interactions between rational players (a problem reflected in many fields of science) has contributed to its various applications [2]. Game theory methods have been used in areas such as military science, biology, economics and other social sciences. Since the very beginning, the game theory has been closely connected with the information theory [3]. Therefore, during the development of the theory of quantum information [4], a quantum game theory has appeared naturally. In its 1 arXiv:1501.04063v1 [quant-ph] 30 Dec 2014