Quantum Replicator Dynamics

Abstract

We propose quantization relationships which would let us describe and solution problems originated by conflicting or cooperative behaviors among the members of a system from the point of view of quantum mechanical interactions. The quantum analogue of the replicator dynamics is the equation of evolution of mixed states from quantum statistical mechanics. A system and all its members will cooperate and rearrange its states to improve their present condition. They strive to reach the best possible state for each of them which is also the best possible state for the whole system. This led us to propose a quantum equilibrium in which a system is stable only if it maximizes the welfare of the collective above the welfare of the individual. If it is maximized the welfare of the individual above the welfare of the collective the system gets unstable and eventually it collapses.

Learning in quantum games

“…The dynamics (QRD) and the coordinate-free expression (13) agree with the dynamics of Jain et al [23] (who derived an equivalent expression under the assumption that X and Ẋ commute), but not with the dynamics of Hidalgo [20] that follow a different, unrelated quantization paradigm. We provide the relevant calculations in Appendix B.…”

“…First, to achieve no-regret in a quantum setting, we introduce a flexible model for learning in general N -player quantum games based on the popular "follow the regularized leader" (FTRL) template for finite games [41,42]. The resulting model, which we call "follow the quantum regularized leader" (FTQL), contains as a special case the matrix multiplicative / exponential weights (MMW) dynamics that have been used extensively in quantum games and matrix learning [2,[22][23][24]46], and which give rise to the quantum replicator dynamics [20,23]. Importantly, as we show in Section 3, the mixed-state dynamics of FTQL decompose into a "classical" part (eigenvalues evolve as the FTRL dynamics in finite games), plus a "quantum" component capturing the evolution of the system's eigenfunctions (and which has no classical analogue).…”