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A further thought, while the family decides which Thanksgiving movie to watch ... . I think I can see how to resolve the issue regarding nu that I raised in Point 3. In the situation of the HP theorem the expected value of the performance of both individual solvers and group of solvers is computed with respect to a distribution nu, which is fixed -- does not depend on solver or group of solvers. For each x in the solution space X, nu(x) gives the probability that the solver (or solver-team) starts her work on alternative x. Since nu is fixed it occurs to me that the most natural interpretation, in modelling applications, is that nu describes, not on a solver's dispositions about where best to start, but rather the "manager's" dispositions about where to start the solver or team. The manager presents a problem to the individual or team and says: "Here is x, an example of a solution. As you can see, it is worth V(x). Please improve upon it as much as you can." nu(x) is simply the probability that the manager presents x as the example. Inherent ability of solvers (relative, of course, to the valuation function V) can be expressed expressed entirely through their phi-functions. For example, the solver of maximal inherent ability works like this: letting x* denote the optimal solution and x' denote the second-best solution, the solver's phi-function is p(x) = x* if x != x', p(x') = x'. If you like you could define the inherent ability of a solver as the expected value of her solution, computed relative to a uniform distribution on the set X of starting points. But of course this is distinct from the "functional" ability defined in the HP theorem, namely: expected value of the solution relative to nu, which need not be uniform. nu just needs to have full support, i.e., assign positive probability to every element of X, in order for the theorem to hold. I don't see why a formal model should assume that solvers are presented with an initial example, and presumably Singer and colleagues have moved beyond the study of HP types of models anyway. My suspicion -- that in the simulations in the Singer paper performance will be more closely correlated to the number of guesses than to C-diversity -- is unaffected by the foregoing.
I should qualify the following the following (humorously intended) advice made at the end of point 3 above: "And the practical lesson for business firms is that if there is no ability in play (i.e., no solver exercises knowledge of any association between alternatives and their values) then management should ignore considerations of diversity and hire those workers who are able to make the most guesses in the time allotted to solve the problem (and, of course, to give each worker his/her own subset of alternatives to investigate, making sure that the assigned sets are mutually disjoint)." Nothing in the HP model warrants the assumption (implicit in the above "advice" to business firms) that anyone, managers or problem-solving agents, is aware in advance or eventually becomes aware of all of the elements in the solution space X. Hence managers might not be able to pre-assign solutions for workers to investigate, and anyway the heuristics of the available problem-solvers would probably limit the solutions they would be disposed to investigate in the first place. Still, the strategy for management would be to maximize the number of approaches investigated, and that strategy might not be tightly correlated to diversity.
Well, there is a difference here. When Hong and Page introduced the valuation function V, they did so in explicitly probabalistic terms: "We consider a random value function mapping the first n integers, {1, 2, . . ., n}, into real numbers. The value of each of the n points is independently drawn according to the uniform distribution on the interval [0, 100]. Agents try to find maximal values for this random function." the probability that such a function V:{1,2, ..., n} -> [0,100] is not one-to-on is 0. So Page and Hong can assert (and should have asserted in their paper) that the statement of the HP theorem holds with probability 1. In the case of the claim that uniformity of nu, along with assumptions 0, 1 and 2, implies Assumption 3, there is no probability involved and no way that I can see to cast it in probabilistic terms. It's just a false claim.