Can the game be quantum?

The game in which acts of participants do not have an adequate description in terms of Boolean logic and classical theory of probabilities is considered. The model of the game interaction is constructed on the basis of a non-distributive orthocomplemented lattice. Mixed strategies of the participants are calculated by the use of probability amplitudes according to the rules of quantum mechanics. A scheme of quantization of the payoff function is proposed and an algorithm for the search of Nash equilibrium is given. It is shown that differently from the classical case in the quantum situation a discrete set of equilibrium is possible.