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 and . The value of this portfolio is

where and are the acceptance probabilities, and are the values and is the cost of hiring one or two candidates in total. This can be re-arranged to

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

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

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 accepts an offer, , increases you begin to drop others from the portfolio. Possibly even others who are better than .

For example, suppose that the department is slot-constrained and would incur the Dean’s wrath if it hired two people this year. If so that you prefer candidate , you will nevertheless make an offer only to if 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 and . If is a theorist and a macro-economist, the macro-economists will foresee that a high means no offer for .

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 is not too high, it is optimal to make offers to both and a hard offer constraint prevents that.)

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February 23, 2011 at 4:47 am

Robert WiblinIs this an instance of the winner’s curse?

http://en.wikipedia.org/wiki/Winner's_curse

February 23, 2011 at 8:58 am

jeffThe winner’s curse arises when the value of the candidate is also unknown. Then, conditional on the candidate not getting competing offers you infer that his value is low. That is a separate consideration.

Here I am talking about a tradeoff that exists even when the value of the candidate is known. I edited the post a bit to emphasize that.

February 23, 2011 at 4:55 pm

Lones SmithFor those youngsters out there eager to flaunt their chops, we found that the problem with a constant cost of filling the slot (making a college application), and an ability just to accept one college, one could use a greedy algorithm to recursively compute the optimal application portfolio. But stepping away from either assumption killed this intuitive way of applying, and in principle created an NP-hard problem (i.e. check all possible 2^N college portfolios, given N colleges). The dept problem is harder in two dimensions: non-constant cost of filling the slots, and a possible desire to accept more than one candidate. I think relaxing the second assumption is the smartest one to muse about.

Best thing about this problem: We could ignore all that yucky post high school math we learn in economics, and simply proceed by way of a double induction proof and elementary probability.