Do transitive preferences always result in indifferent divisions?

TL;DR

This paper explores how intransitive preferences can be optimal in sequential decision games and uses quantum models to analyze how intransitivity influences strategic choices.

Contribution

It introduces a quantum game model showing that intransitive strategies can be optimal, challenging traditional views on preference transitivity and rationality.

Findings

Optimal strategies can be intransitive in certain game contexts.

Quantum models reveal increased importance of intransitive strategies.

Both players may find intransitive strategies optimal under quantum conditions.

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 \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.