I quoted: > >> In his classical contribution Condorcet (1785) described a committee > >> as a mechanism that effciently aggregates decentralized information. > >> In his famous jury theorem he argues that (i) increasing the number > >> of informed committee members raises the probability that an > >> appropriate decision is made and (ii) the probability of making the > >> appropriate decision will converge to one as the number of committee > >> members goes to infnity.< (from > >> http://www.ecb.int/pub/wp/ecbwp256.pdf.)
then I said: > > Despite this, my feeling is that the group is always right, except > > when it isn't. Just as the individual is always right, except when he > > or she isn't. But for any collective decision, only the democratic > > body of the collective has any legitimacy. > > > > (One problem with this theorem is that the larger the group, the > > harder it is to come to a decision.) Ted asks: > Is Condorcet's theorem true of a group of believers in the certainty > that they are soon to be carried off in Rapture? How about the group > composed of believers in neoclassical economics? I don't know, but I'd guess that it would apply. I'm not advocating C's theorem. Rather, I was throwing more weight behind the "collective wisdom" idea (and then, as Ted may have noticed, criticizing it). My feeling is that C was saying that a jury of 12 would be more accurate in its processing of the facts they were given -- to make a _binary decision_ (guilty/not guilty) -- than would be a jury of 1 or 6, assuming that one of the two verdicts is actually valid. It's like saying "two heads are better than one, while twelve are better than two." Obviously, the "facts" they were given will have been limited and biased by the attorneys and the judge, while their interpretation would be limited by any shared ideology or shared social position that limited and shaped their world-views. Further, discussions about complex theological beliefs such as the Rapture or neoclassical economics seem to go against the assumption of binary decision-making. Strictly speaking, the theorem assumes that the jurors vote independently, rather than discussing matters and voting collectively (which is the way juries I've been on have worked). (See "When Is Condorcet's Jury Theorem Valid?" by Daniel Berend and Jacob Paroush, _Social choice and welfare_, Volume 15, Number 4 (August 1998).) > In the case of the latter group, the basis for certainty of belief is > described by Roy Weintraub as follows: > > "mathematical (economic) models are rigorous (and 'true' in the only > useful scientific sense of the word) if they are built on a cogent > axiom base - like von Neumann and Morgenstern, and Debreu." > (Weintraub, How Economics Became a Mathematical Science, p. 100) > > "The Arrow-Debreu model was a major accomplishment; it presented an > economy composed of individual, self-interested agents - both > utility-maximizing households and profit-maximizing firms - pursuing > their own self-interest and whose actions produced an equilibrium in > which all choices were potentially reconciled. Put briefly, the > pursuit of individual self-interest could lead not to social > chaos but > to a coordinated social order. But how did a piece of work in > mathematical economics actually settle an economic question? How did > it come to pass that a particular paper, in a journal at that > time read > by very few economists, came to be accepted as having established a > foundational truth about market economics? These are not questions > economists typically ask. > ‘The theorem proves that ...' is enough information > to persuade > economists that knowledge associated with the theorem is secure > knowledge. Professional economists are confident about the > result and > the implications of the equilibrium proof, and no one needs to attend > to the means of its construction: the validity of the > equilibrium proof > in incontrovertible. Economists-in-training must learn that the > existence of a competitive equilibrium has been proved. All > economists > can make use of the proof of that result without subjecting it to > incessant challenge and reassessment. > "Scientists take some components of their research as given; > intellectual paralysis awaits the scientist who seeks to reopen every > foundational issue every day. For most economists the competitive > equilibrium proof is a tool to use with little regard to how the tool > was constructed. Those who study science use the idea of a > 'black box' > for settled results that are locked up and impenetrable, and thus > closed to current investigation. For every science, black boxes are > both healthy and necessary." (p. 184) the way I learned the Arrow/Debreu stuff in grad school (UC-Berkeley, then home of Debreu) was that it showed the assumptions one had to make for competitive equilibrium to exist, and that since the assumptions weren't realistic, the Arrow/Debreu conclusions didn't apply. Any deductive theorem cuts both ways. Jim D.
