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

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.