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arXiv:1101.3543 · Journal of Physics A Mathematical and Theoretical

Decisions in elections—transitive or intransitive quantum preferences

Marcin Makowski, Edward W. Piotrowski

Published Apr 27, 2011• 32 citations
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Abstract

Our preferences depend on the circumstances in which we reveal them. We will introduce a dependence which allows us to illustrate the relation between the possibility of winning of a particular candidate in an election and the type of preference. It is generally observed that if voters start to clearly prefer one of the candidates, the significance of intransitive preferences in the quantum model decreases. This dynamic change cannot be observed in the case of the classical model.

DOI
10.1088/1751-8113/44/21/215303

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Our preferences depend on the circumstances in which we reveal them. We will introduce a dependence which allows us to illustrate the relation between the possibility of winning of a particular candidate in an election and the type of preference. It is generally observed that if voters start to clearly prefer one of the candidates, the significance of intransitive preferences in the quantum model decreases. This dynamic change cannot be observed in the case of the classical model.
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)’.
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.
Quantum communication protocols seek to leverage the unique properties of quantum systems for coordination or communication tasks, usually with guarantees of security or anonymity that exceed what is possible classically. One promising domain of application is elections, where strong such guarantees are essential to ensure legitimacy. We experimentally implement a recently proposed election protocol from Centrone et al. such that no one, including a potential central authority, can know the preferred candidate of any voter other than themself. We conduct a four-party election, generating and distributing four-partite GHZ states with ≈89%\approx 89\%≈89% fidelity and successfully recording voters' intentions ≈87%\approx 87\%≈87% of the time.