  Collective decision-making is vital for recent information and communications
technologies. In our previous research, we mathematically derived conflict-free
joint decision-making that optimally satisfies players' probabilistic
preference profiles. However, two problems exist regarding the optimal joint
decision-making method. First, as the number of choices increases, the
computational cost of calculating the optimal joint selection probability
matrix explodes. Second, to derive the optimal joint selection probability
matrix, all players must disclose their probabilistic preferences. Now, it is
noteworthy that explicit calculation of the joint probability distribution is
not necessarily needed; what is necessary for collective decisions is sampling.
This study examines several sampling methods that converge to heuristic joint
selection probability matrices that satisfy players' preferences. We show that
they can significantly reduce the above problems of computational cost and
confidentiality. We analyze the probability distribution each of the sampling
methods converges to, as well as the computational cost required and the
confidentiality secured. In particular, we introduce two conflict-free joint
sampling methods through quantum interference of photons. The first system
allows the players to hide their choices while satisfying the players'
preferences almost perfectly when they have the same preferences. The second
system, where the physical nature of light replaces the expensive computational
cost, also conceals their choices under the assumption that they have a trusted
third party. This paper has been published in Phys. Rev. Applied 18, 064018
(2022) (DOI: 10.1103/PhysRevApplied.18.064018).
