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arXiv:1903.06454 · Lecture notes in computer science

A Quantum Annealing Algorithm for Finding Pure Nash Equilibria in Graphical Games

Christoph Roch, Thomy Phan, Sebastian Feld, Robert Müller, Thomas Gabor, Carsten Hahn, Claudia Linnhoff-Popien

Published Jan 1, 2020• 6 citations
arXivHTMLPDF
DOI
10.1007/978-3-030-50433-5_38

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The Grover algorithm and its underlying amplitude amplification process are one of the most essential and widely used building blocks of many other subsequent quantum methods. In this work, we adapt Grover’s search algorithm to find pure Nash equilibria in graphical games. In general, the main complexity lies in the construction of the algorithm’s oracle operator. We show how to set up the oracle from a given graphical game by translating it to a Boolean satisfiability problem, which can subsequently be logically synthesized to an oracle quantum circuit. We investigate the algorithm for random graphical game instances on IBM’s quantum simulator and give an outlook regarding the scaling of it.
The aim of the paper is to examine pure Nash equilibria in a quantum game that extends the classical bimatrix game of dimension 2. The strategies of quantum players are specific types of two-parameter unitary operations such that the resulting quantum game is invariant under isomorphic transformations of the input classical game. We formulate general statements for the existence and form of Nash equilibria and discuss their Pareto efficiency. We prove that, depending on the payoffs of a classical game, the corresponding quantum game may or may not have Nash equilibria in the set of unitary strategies under study. Some of the equilibria cease to be equilibria if the players’ strategy set is the three-parameter special unitary group.