Christian List gave an interesting talk last week on a problem for group decision-making. It can be illustrated with the following example.
Suppose someone is accused of a crime, though there are questions both about whether the evidence against him is admissible, and whether that evidence suffices to prove his guilt. A panel of judges are to collectively decide the two questions, and hence whether the accused should be convicted. Suppose the individual judgments are as follows: One third of the judges answer affirmatively on all counts. Another third agree the evidence is admissible, but deny that it establishes guilt; hence they vote against conviction. The final third accept that the evidence implies guilt, but deny that the evidence is admissible, and hence likewise oppose conviction.
The majority conclusions are hence as follows: The evidence is admissible, the evidence establishes guilt, but the accused should (thereby?) not be convicted. This is clearly problematic! More generally, we can see that majorities of individually rational agents might collectively yield such irrational conclusions as holding each of A, B, and not A & B. (Philip Pettit uses this to argue that we should not always believe whatever the majority does.)
The obvious solution is to determine some fundamental or "privileged" set of base questions, get the individuals to vote only on them, and let their logical implications emerge indirectly. This is to give up on universal majoritarianism, for as we've seen, the implications might be disputed by the majority of members. Worse, the choice of base will influence the outcome, but there are no clear principled grounds against which to make the choice. To borrow Dave Chalmers' proposed terminology, there are any number of "reduction procedures" from logically interconnected question sets to safely non-redundant privileged bases. But if the choice of reduction procedure is both arbitrary and outcome-affecting, then we obtain the troubling conclusion that there may be no one true "democratic outcome" for any given set of individuals (with given beliefs and preferences).
I guess that may be one more argument for deliberative democracy, according to which individuals are to reason together to reach collective decisions, rather than treating individual opinions as raw inputs to be mechanically processed towards producing the so-called "group verdict". Not only is the latter undesirable; in many cases it is not even possible to implement (at least in a non-arbitrary fashion).