Quantum-like Representation of Extensive Form Games: Wine Testing Game

TL;DR

This paper explores applying quantum-like mathematical formalism to extensive form games, demonstrating how non-classical probabilities and Bell inequality violations can occur in multi-player game scenarios.

Contribution

It introduces a quantum-like framework for representing extensive form games, highlighting the role of noncommutative operators and contextual probabilities in modeling player behavior.

Findings

Quantum-like representation is applicable to extensive form games.

Bell's inequality can be violated in three-player quantum-like games.

Classical probability models cannot fully describe the game's probabilistic structure.

Abstract

We consider an application of the mathematical formalism of quantum mechanics (QM) outside physics, namely, to game theory. We present a simple game between macroscopic players, say Alice and Bob (or in a more complex form - Alice, Bob and Cecilia), which can be represented in the quantum-like (QL) way -- by using a complex probability amplitude (game's ``wave function'') and noncommutative operators. The crucial point is that games under consideration are so called extensive form games. Here the order of actions of players is important, such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information which is available to e.g. Bob about the previous action of Alice. In general one could not construct a classical probability space underlying a QL-game. This can happen even in a QL-game with two players. In a QL-game with three players Bell's inequality can be violated. The most natural probabilistic description is given by so called contextual probability theory completed by the frequency definition of probability.