So in a previous post, I took note of the AEA’s signaling innovation in the economics faculty job market: candidates can send a signal of particular interest to no more than two schools. The AEA advises candidates to use that mainly to express idiosyncratic preferences or a willingness to take jobs from schools that might otherwise think the candidate is uninterested in them, and particularly advises candidates not to send signals to top schools. I suggested in the previous post that such advice is mistaken: there’s a candidate profile for which it’s utility-maximizing to send signals to the top schools. The strategic intuition is that a signal to a top school is a way of cheaply communicating private information about a candidate’s high quality by investing a scarce resource in a top department, and that such a signal will carry more information than things like inflated letters of recommendation or ambiguous resumes. Since it’s costly for top schools to get candidate quality information (by investing in an interview), such a signal would clue the school in that such an investment would be worthwhile in a given candidate’s case.
Ben, in the comments, questioned me, and, as we all know, the only appropriate response to being questioned about a strategic/economic interaction is to hurl models straight at the head of your questioner until they say “uncle.”
Also, in light of Larry Summers’s comments on the relative intelligence of political scientists and economists, I, in my official capacity as a (nominal) political scientist, can’t resist the chance to throw down on the AEA when they’ve made a mistake that’s right at the heart of economics, viz., how players in a market should allocate a scarce resource.
I also rather like the idea of a bunch of top-calibre economists and ranking econ departments making their market decisions based on advice from me.
For all these reasons, it’s time to put my math where my mouth is. And it’s actually a fairly simple problem! So… with no further ado:
For simplicity, let’s start by assuming that there is no cheap way (other than, potentially, the signal) for top departments to determine candidates’ relative quality prior to interviewing them — that is, reading their papers is costly (time-consuming), and CVs and letters of recommendation are too noisy. This is a possibly controversial assumption, and I think I’ll say some words about why it’s true toward the end and offer further elaboration should people have a problem with it, but for now, let’s just get it out of the way.
Right. Let’s call the following game the Interview Signaling Game. The solution concept will of course be perfect bayesian equilibrium. I proceed to set it up…
It’s a two-player game with a candidate player and a (top) department player. The candidate can be one of two types: a strong type or a weak type, and nature sets the probability of the candidate’s being either type as P, in (0,1), for strong type and (obviously) 1-P for weak type. The candidate has perfect knowledge of his type*, the department knows only the probability of nature’s producing a strong candidate. The department can invest I>0 to determine the candidate’s type, where that investment comes from some combination of taking the time to read a candidate’s papers and expending a scarce interview slot on that candidate. (For simplicity, let’s just say collapse all these costs into the cost of interviewing the candidate.) The candidate can send a scarce signal at a cost of C,** where that cost is the opportunity cost of not having sent the signal to some other department, and is some function of the increased probability of being hired by that other department and, of course, the utility of same. Let A stand for whether or not the candidate sends the signal or not (that is, if the candidate signals, A=C, and if not, A=0), and likewise let B stand for whether the department invests I or not.
Let the probability of the department’s making an offer to a given candidate be M. If the department invests I in a candidate, M=Q for a strong candidate and R for a weak candidate, where Q>R. If it does not invest I, of course, M=0. Let the payoff to a candidate for an offer from a top department be X. For simplicity, we’ll assume that the department gets utility directly from interviewing strong candidates***, of Y>I, and gets no utility from interviewing a weak candidate. Set Y at 1, because you can do that sort of shit with von Neumann-Morgenstern utility functions and it makes the algebra slightly easier in cases I won’t even consider (hah). We need one more probability term (oh god, I wish it were easier to do greek letters in html), so, screw it, let’s use N. Let N be the probability that a candidate is strong, conditional on the department’s having interviewed that candidate (which is not necessarily the same as P).
The candidate’s utility function =MX-A. The department’s utility function =NY-B.
Now here comes the move that makes it all work. Let’s say QX>C>RX. I propose that as the set of parameter values under which there’s a separating equilibrium where a candidate sends a signal with probability 1 if he’s strong and with probability 0 if he’s weak, and a department interviews with probability 1 if it sees a signal and with probability 0 if it does not.
(It’s also a set of parameter values that probably resembles reality — it expresses the fact that the chance of getting a job at a top department is worth trading off resources that would otherwise be devoted to a subtop department only for strong candidates, and not for weak candidates. It could also just be a tautology if we define a strong candidate as a candidate for whom MX>C.****)
SOOO… is the proposed equilibrium actually a separating equilibrium? Fairly obviously, yes.
The department has no incentive to deviate. In equilibrium, N=1, so the department gets Y-I in each interaction, which is its maximum possible utility: given the candidate’s strategy, which is such that the posterior probability of a candidate’s being strong is 1 given that he signaled, if the department declines to interview any signaling candidate, the department gets 0 utility from that candidate, and Y-I>0. Likewise, if the department interviews a non-signaler, it gets a payoff of -I, as compared to the payoff of 0 that it gets if it keeps to strategy and only interviews signalers.
The candidate also has no incentive to deviate. In equilibrium, strong candidates get QX-C and weak candidates get 0. A strong candidate who fails to signal gets no interview, and thus gets 0, and QX-C>0. A weak candidate who signals gets RX-C, and RX-C<0.
So there is an equilibrium where strong candidates send signals to top departments, and the point of the signaling mechanism is not only to express a candidate’s willingness to take a job in a department that might otherwise think him uninterested, but also to signal his quality to the top schools. Dare I say it? … QED.
Of course, this isn’t the only possible equilibrium. There might be a pooling equilibrium where nobody (or everybody!) signals, as well as various semi-separating equilibria. I’m not going to take the trouble to hunt them down. The point of this post is more limited — to prove that there are plausible cases where candidates would want to send signals to top departments in order to communicate private information about their own quality. The AEA is mistaken to think otherwise, and to so advise its candidates and departments.*****
And… Larry? Political scientists: 1. Economists: 0.
* This is a fairly strong assumption, but, for candidates with competent and honest advisors, it seems justifiable. Also, we could relax it without losing the model so long as the candidate has much better knowledge of his type than the department does.
** C is the same for strong and weak candidates. This is, perhaps, an oversimplification, but probably a harmless one.
*** In reality, of course, the department would get utility from the probability of actually having a strong candidate accept the offer, but since we’re talking about top departments here, we can safely assume that this probability is both uniform and high, and thus pretty much irrelevant for modeling purposes.
**** I’m just ignoring the case where MX=C. Who cares?
***** If there are such additional equilibria, one might defend the AEA’s advice as an attempt to get the players to coordinate on such other equilibria… but why would they do such a thing? The separating equilirbium I’ve identified is almost certainly the best equilibrium, even if it’s not the only equilibrium, in that it achieves a perfect match, at least at the high end, between candidate quality and interview selections.