How to Play Macroscopic Quantum Game

Andrei A. Grib ; Georgy N. Parfionov

Contributions to game theory and management, Tome 2 (2009), pp. 137-146

Résumé

Quantum games are usually considered as games with strategies defined not by the standard Kolmogorovian probabilistic measure but by the probability amplitude used in quantum physics. The reason for the use of the probability amplitude or "quantum probabilistic measure" is the nondistributive lattice occurring in physical situations with quantum microparticles. In our paper we give examples of getting nondistributive orthomodular lattices in some special macroscopic situations without use of quantum microparticles. Mathematical structure of these examples is the same as that for the spin one half quantum microparticle with two non-commuting observables being measured. So we consider the so called Stern-Gerlach quantum games. In quantum physics it corresponds to the situation when two partners called Alice and Bob do experiments with two beams of particles independently measuring the spin projections of particles on two different directions In case of coincidences defined by the payoff matrix Bob pays Alice some sum of money. Alice and Bob can prepare particles in the beam in certain independent states defined by the probability amplitude so that probabilities of different outcomes are known. Nash equilibrium for such a game can be defined and it is called the quantum Nash equilibrium. The same lattice occurs in the example of the firefly flying in a box observed through two windows one at the bottom another at the right hand side of the box with a line in the middle of each window. This means that two such boxes with fireflies inside them imitate two beams in the Stern-Gerlach quantum game. However there is a difference due to the fact that in microscopic case Alice and Bob freely choose the representation of the lattice in terms of non-commuting projectors in some Hilbert space. In our macroscopic imitation there is a problem of the choice of this representation(of the angles between projections). The problem is solved by us for some special forms of the payoff matrix. We prove the theorem that quantum Nash equilibrium occurs only for the special representation of the lattice defined by the payoff matrix. This makes possible imitation of the microscopic quantum game in macroscopic situations. Other macroscopic situations based on the so called opportunistic behavior leading to the same lattice are considered.