Quantum Extensive Form Games

Abstract

We propose a concept of quantum extensive-form games, which is a quantum extension of classical extensive-form games. Extensive-form games is a general concept of games such as Go, Shogi, and chess, which have triggered the recent AI revolution, and is the basis for many important game theoretic models in economics. Quantum transitions allow for pairwise annihilation of paths in the quantum game tree, resulting in a probability distribution that is more likely to produce a particular outcome. This is similar in principle to the mechanism of speed-up by quantum computation represented by Grover's algorithm. A quantum extensive-form game is also a generalization of quantum learning, including Quantum Generative Adversarial Networks. Therefore it will become new theoretic basis of quantum machine learning, as well as a basis for a new game theoretic foundation for microeconomics. We propose the quantum angel problem as a new example of quantum extensive-form games. This is a quantum version of angel problem proposed by Conway in 1996. His original problem has already been solved, but by quantizing it, it becomes a non-trivial problem. In the quantum angel problem, Angel moves on a general graph as a quantum walker. By not only changing the dimensions and geometry of the graph, but also by adding/relaxing restrictions to the quantum resources available to Angel and Devil, the difficulty and complexity of the game is diversified in a way that is not possible in the traditional angel problem.

Keywords: Extensive-form game, quantum theory, quantum walk, quantum computation