Believe it or not that line of thinking does lie just below the surface in many recruiting discussions.  The recruiting committee wants to hire good people but because the market moves quickly it has to make many simultaneous offers and runs the risk of having too many acceptances.  There is very often a real feeling that it is safe to make offers to the top people who will come with low probability but that its a real risk to make an offer to someone for whom the competition is not as strong and who is therefore likely to accept.

This is not about adverse selection or the winner’s curse.  Slot-constraint considerations appear at the stage where it has already been decided which candidates we like and all that is left is to decide which ones we should offer.  Anybody who has been involved in recruiting decisions has had to grapple with this conundrum.

But it really is a phantom issue.  It’s just not possible to construct a plausible model under which your willingness to make an offer to a candidate is decreasing in the probability she will come.  Take any model in which there is a (possibly increasing) marginal cost of filling a slot and candidates are identified by their marginal value and the probability they would accept an offer.

Consider any portfolio of offers which involves making an offer to candidate F. The value of that portfolio is a linear function of the probability that F accepts the offer.  For example, consider making offers to two candidates F and O.  The value of this portfolio is

q_O [ q_F (v_F +v_O - C(2))+( 1 - q_F )(v_O - C(1) ) ]

+(1 -q_O)[q_F v_F-C(1)]

where q_O and q_F are the acceptance probabilities, v_O and v_F are the values and C(\cdot) is the cost of hiring one or two candidates in total.  This can be re-arranged to

q_F \left[q_O\left(v_F-MC(2)\right)+(1- q_O) \left(v_F - C(1)\right) \right] + const.

where MC(2) = C(2) - C(1) is the marginal cost of a second hire.  If the bracketed expression is positive then you want to include F in the portfolio and the value of doing so only gets larger as q_F increases. (note to self:  wordpress latex is whitespace-hating voodoo)

In particular, if F is in the optimal portfolio, then that remains true when you raise q_F.

It’s not to say that there aren’t interesting portfolio issues involved in this problem.  One issue is that worse candidates can crowd out better ones.  In the example, as the probability that F accepts an offer, q_F, increases you begin to drop others from the portfolio.  Possibly even others who are better than F.

For example, suppose that the department is slot-constrained and would incur the Dean’s wrath if it hired two people this year.  If v_O > v_F so that you prefer candidate O, you will nevertheless make an offer only to F if q_F is very high.

In general, I guess that the optimal portfolio is a hard problem to solve.  It reminds me of this paper by Hector Chade and Lones Smith.  They study the problem of how many schools to apply to, but the analysis is related.

What is probably really going on when the titular quotation arises is that factions within the department disagree about the relative values of F and O.  If F is a theorist and O a macro-economist, the macro-economists will foresee that a high q_F means no offer for O.

Another observation is that Deans should not use hard offer constraints but instead expose the department to the true marginal cost curve, understanding that the department will make these calculations and voluntarily ration offers on its own.  (When q_F is not too high, it is optimal to make offers to both and a hard offer constraint prevents that.)