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arXiv:1101.3568 · Journal of Physics A Mathematical and Theoretical

Transitivity of an entangled choice

Marcin Makowski, Edward W Piotrowski

Published Jan 24, 2011• 8 citations
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Abstract

We describe a quantum model of a simple choice game (constructed upon the entangled state of two qubits), which involves the fundamental problem of transitive–intransitive preferences. We compare attainability of optimal intransitive strategies in both classical and quantum models with the use of geometrical interpretation.
DOI
10.1088/1751-8113/44/7/075301

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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.
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 sine qua non 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 preference 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)’.
We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy at least as good as in the classical case, while Q's best strategy results in a game having the same value as the classical game. We investigate the consequences of Q storing his information in classical or quantum ways. It turns out that Q's optimal strategy is to use a completely entangled quantum notepad, on which to encode his prior information.