Quantum Inf Process (2016) 15:3637–3650
Quantum approach to Bertrand duopoly
· Jan Sładkowski
Received: 14 December 2015 / Accepted: 24 May 2016 / Published online: 4 June 2016
© Springer Science+Business Media New York 2016
Abstract The aim of the paper is to study the Bertrand duopoly example in the
quantum domain. We use two ways to write the game in terms of quantum theory.
The ﬁrst one adapts the Li–Du–Massar scheme for the Cournot duopoly. The second
one is a simpliﬁed model that exploits a two qubit entangled state. In both cases, we
focus on ﬁnding Nash equilibria in the resulting games. Our analysis allows us to take
another look at the classic model of Bertrand.
Keywords Bertrand duopoly · Quantum game · Nash equilibrium
Quantum game theory is an interdisciplinary ﬁeld that combines game theory with
quantum theory [1–3]. The idea is to use the apparatus developed to describe quantum
phenomena to analyze macroscopic complex systems (including living systems) [4–8].
The ﬁrst attempt to describe a game in the quantum domain applied to ﬁnite nonco-
operative games in the normal form [1–3] but soon after that quantum game theory
has found applications in various ﬁelds including decision sciences [5,6,9], ﬁnance
theory [10–12] or mathematical psychology . Physical implementation of a quan-
This work was supported by the Ministry of Science and Higher Education in Poland under the project
Iuventus Plus IP2014 010973 in the years 2015–2017.
Institute of Mathematics, Pomeranian University, ul. Arciszewskiego 22d, 76-200 Słupsk, Poland
Institute of Physics, University of Silesia, ul Uniwersytecka 4, 40-007 Katowice, Poland