Friday, August 25, 2006

Group Inconsistency

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).


  1. The majority conclusions are hence as follows: The evidence is admissible, the evidence establishes guilt, but the accused should (thereby?) not be convicted.

    Surely not thereby, because that would change the result from three distinct conclusions each of which received a majority vote to one conclusion for which no one voted.

    I don't know enough to say anything in detail about relevant logics, but I do know that in relevant logics it is fairly common to restrict the conjunction introduction rule. Each possible conjunct is really indexed according to its grounds or bases; and the relevance conjunction introduction rule is that conjunction introduction is that A and B have to have the same index for conjunction to be introduced. It's possible that something at least vaguely analogous is going on here; the propositions have different grounds, and so don't mix.

    And that would make sense, I think. Consider how one would want such a result to function in one's judicial system. A and B may have been decided upon for this particular case, but given the way the result was obtained, so that A is a precedent and B is a precedent; but because of the way the result was obtained, it would be odd to accept (A&B) as a precedent rather than as a fluke. Or in the conviction case: the conclusion 'the evidence is admissible' has the force of precedent; the conclusion 'the vidence establishes guilt' has the force of precedent'; and the conclusion 'the evidence does not suffice to convict' has the force of precedent; but it would be absurd to take the conjunction of the three as having the force of precedent, because the grounds for the third are inconsistent with the grounds for the first. So each conclusion is relevant to further decisions, but their conjunction is not.

  2. Could deliberative democracy possibly solve the issue?

    I’m not sure your implied comparison is fair, the problem is that deliberative democracy cannot possibly consider anywhere near all of all issues at the same time.

    So fundamentally it will be exposed to the same problem where the issues are resolved separately and conclusions arise that are separate and contradictory.
    Representative systems would achieve consistency of policy, hopefully consistent in a way that the country wants. Any sort of direct democracy would probably result in inconsistency.

  3. Not sure if this undermines the wider point, but evidence will usually be inadmissible for one of two reasons:

    1) The judges don't trust the jury to attach proper weight to the evidence.

    2) There is some misconduct (e.g. illegal search) that is being punished by exclusion of otherwise relevant evidence.

    In case 1, provided you assume the judges themselves attach proper weight to the evidence, a guilty verdict is correct.

    In case 2, assuming the exclusion was correct in the first place, there is no problem with the not guilty result. Some otherwise guilty people must go free for the exclusion policy to have any deterrent effect.

  4. Yeah, I was thinking of a type-2 case. But note that only 1/3 of the judges in my scenario think that the evidence is inadmissable. The majority conclude that there are no grounds for excluding it. Another majority (with some but not total overlap, of course) conclude that the evidence, if admitted, entails guilt. And yet another majority nonetheless vote "not guilty" overall. So, if you ask for the majority verdicts, here's what we have: the evidence, if admitted, establishes guilt; the evidence is admissible; and the accused is not guilty.

    (The paradox arises because the "not guilty" votes are based on the sum of two very different reasons, each of which is individually in a minority, but which form a majority when summed in this way.)

    But that's just an example. To see the core point, consider it in the abstract: If 1/3 believe A, and a different 1/3 believe B, then we have a 2/3 majority in support of the proposition "A or B". But the majority think A is false, and another majority think B is false. So the majority verdicts are inconsistent: the group concludes that each of A and B, individually, is false, but also one of them (when considered together, i.e. "A or B") is true!


Visitors: check my comments policy first.
Non-Blogger users: If the comment form isn't working for you, email me your comment and I can post it on your behalf. (If your comment is too long, first try breaking it into two parts.)

Note: only a member of this blog may post a comment.