  In this paper, a novel quantization scheme for cooperative games is proposed.
The considered circuit is inspired by the Eisert-Wilkens-Lewenstein protocol
modified to represent cooperation between players and extended to $3$-qubit
states. The framework of Clifford algebra is used to perform necessary
computations. In particular, we use a direct analogy between Dirac formalism
and Quantum Register Algebra to represent circuits. This analogy enables us to
perform automated proofs of the circuit equivalence in a simple fashion. To
distribute players' payoffs after the measurement, the expected value of the
Shapley value with respect to quantum probabilities is employed. We study how
entanglement, representing the level of pre-agreement between players, affects
the final distribution of utility. The paper also demonstrates how all
necessary calculations can be automatized using the Quantum Register Algebra
and GAALOP software.
