In 1951, K. J. Arrow proved that, under certain assumptions, it is impossible to have group decision making rules which satisfy reasonable conditions like symmetry. This Impossibility Theorem is often cited as a proof that reasonable group decision making is impossible.
We start our paper by remarking that Arrow's result only covers the situations when the only information we have about individual preferences is their binary preferences between the alternatives. If we follow the main ideas of modern decision making and game theory and also collect information about the preferences between lotteries (i.e., collect the utility values of different alternatives), then reasonable decision making rules are possible: e.g., Nash's rule in which we select an alternative for which the product of utilities is the largest possible.
We also deal with two related issues: how we can detect individual preferences if all we have is preferences of a subgroup, and how we take into account mutual attraction between participants.