On Cooperative Games and Quantum Algorithms

Tuesday, July 2, 2019, 3:00pm – 4:45pm
Presented by Demosthenes Ellinas
Located in 145Π58

Classical cooperative games are addressed from the point of view of quantum algorithms. The coalition formation (grand, non-overlapping, overlapping), problem and its possible quantization is formulated in terms of Hilbert space generalized bases constructed over the powerset of chosen index sets. Non-orthogonal and overcomplete vector bases (frames), and their projection operators labelled by coalition-subsets lead to a generalized resolution of unity. Vector redundancy, opercompletenes and coalition overlap form the first ingredient of quantization procedure. The two other ingredients of the classical game: the characteristic function and the Shapley value, are introduced respectively by means of the state density matrix of game’s quantum system and by the Moebius transform of the coalition projectors. Moebius transform quantifies the surplus value generated by the corresponding projector-coalition operators due to their non-commutativity. Based on this framework the question of novelty brought by the quantum features of a cooperative game is addressed. Two entry points are located, where alteration and outperformance of the quantum versus the classical game may be manifested: i) the characteristic function- state matrix/vector can be in superposition of states non having a classical analogue; ii) coalition vectors can be set in linear combinations, for which the quantum-mechanical probability interpretation applies. Some illustrating example of these possibilities will be presented. Towards establishing a theory of cooperative quantum games, questions of cooperation in the context of agents identified with quantum walk and quantum search algorithms will be addressed.