My game theory brings all the girls to the blog, and damn right, it > yours, damn right it > yours.

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.

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* 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.

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8 Responses to “My game theory brings all the girls to the blog, and damn right, it > yours, damn right it > yours.”

  1. Ed Says:

    Hey Paul-

    This is interesting, but the entire model hinges on the assumption that QX>C>RX, which is not very plausible. It’s not plausible because the signal of interest does not communicate any additional credible information on the part of the candidate independent of the what the department already knows about him.

    That is to say, if Z is a ordered set of candidates on the -1 to 1 interval for candidates with -1 being the worst candidate and 1 being the best candidate, with 0 being the average candidate, then a signal produces M=Q s.t. Q>R only if Z>0. This is because the model should as Z -> -1, C -> 0 because cor(Z,P)>0 which in turn implies that departments will know that quality is likely to be lower as P diminishes (which I assume you treat as a department’s reputation). This means that for weak candidates, the opportunity cost of sending a signal approaches zero because the there is no benefit to expending the signal elsewhere. As a candidate becomes better, the signal carries more weight because other departments know that the candidate is good and would respond to the signal. In essence, the value of the signal is tied to rank-ordering between the hiring and the producing department.

  2. Paul Gowder Says:

    Hmm… I think that depends on the shape of the market. Suppose that the market is such that top departments hire super-strong candidates, subtop departments hire merely above-average candidates, and nobody hires below-average candidates. Then the game I’ve specified is played only between candidates for whom Z>0 (that is, for “strong” read “super-strong” and for “weak” read “above-average”). In such a market, there is indeed a benefit for weak candidates, especially weak candidates who might otherwise be confused for strong candidates or (as the AEA’s advice suggests) have idiosyncratic preferences, to send the signal elsewhere (to signal their availability to subtop departments, per the AEA’s advice).

    Edit: actually, I don’t think I even need the market to be that shape — I just need there to be any low-end part of the market such that there are departments such that candidates who are weak per my model have a good chance of getting jobs at those departments and would have a better chance at getting jobs at those departments if they expressed interest in them. And that’s pretty obviously true of the real market. Think of a candidate who comes from a good school but produces crappy work because of imperfections in the admission process of good schools. The fact that he has MIT on his c.v. might keep him from getting looks from community colleges (who, let’s say, are the appropriate match), even though his crappy work will keep him getting hired at Harvard. Such a candidate would have a rather steep opportunity cost to sending a signal to Harvard, even though he’s toward the low end of the ordering.

    But perhaps I’m misunderstanding you? (Probably I’m misunderstanding you. Dear readers: Ed is better at math than I am. And I was afraid that the constant C assumption would come back and bite me in the ass. However, all is well even if Ed’s right and I’m wrong, because Ed is a political scientist, and so the important point — poli sci rulez –remains.)

  3. Ed Says:

    I think that the bad MIT student is a good counter-example to my assumptions, but think about what a weird example it is. I think that the model works really well, actually but it just needs to take into account the relative strengths of departments to one another. What I mean by this is that the market isn’t just divided into high and low, but rather is defined by the distances between departments.

    Here’s an example. Suppose there are 100 schools. Say Harvard is the best, so we set them to have a score of 100. The worst is Liberty with a score of 1. All the other schools are perfectly ordered between. The candidate from Liberty wastes his signal if she sends it to Harvard because she conveys little credible information with the signal. She does not waste it is she sends the signal to Liberty. She is a viable candidate for her department, but less and less so for 2,3,4… etc.
    Now, the Harvard candidate is ex ante assumed to be better so a signal to Liberty carries enormous weight because Liberty would not otherwise consider him. The value of this signal diminishes as it get closer to his rank.

    So, for a given department, X, and another one, Y, we have the interesting situation where as X – Y -> 100, the signal becomes more and more credible because of the opportunity cost of the signal goes higher and higher. As X – Y goes lower and even negative, the value of the signal diminishes.

    There are two points here. The first is that candidate quality should be taken relative to department quality. The best candidate will always be the best candidate by Z, but departments may have a range of good candidates for their ranking. The second is that candidates may have secondary preferences which break ties between places (e.g. if your family lives in CA, then you may prefer Berkeley to Penn). The model leaves that out, but if you incorporate that dynamic as a probability alongside individual candidate quality (with something like 0 < cor(Z, department ordering) < 1 or perhaps some noise so that Z’ = Z + N(0,0.1) or something), then you would probably also capture the MIT person who knows his quality but wants to ride off of the department name.

  4. Paul Gowder Says:

    Yeah, I think you’re basically right.

    The model as written probably works well enough where phd-granting school doesn’t carry enough information to distinguish candidates who are plausible for the top departments and candidates who are plausible only for middling departments. Suppose in the extreme case that the top departments produce 10 or 20 times the number of students as openings they have in each year, for example). Then there’s a need to distinguish between those that go to the top schools and those that go to middling schools, and department name doesn’t do really any work at all.

  5. david Says:

    Frankly, I spent some time on the econjobrumors board this year, and indeed this is a better contribution than anything I heard over there. Now, another question I have is, does the signaling improve the market in terms of reducing coordination costs? Is implementing the signaling mechanism welfare enhancing? I have no idea.

  6. Paul Gowder Says:

    Thanks! I’m flattered and gratified.

    No idea on the welfare-enhancing bit. I guess it reduces coordination costs to the extent adding signals of private information usually does?

  7. Art Says:

    This logic is right, but your facts are wrong. References for candidates are not inflated in this case. There are made by top professors to top professors with sealed letters within a fairly small community. The interview is relatively worthless relative to these sealed letters and the publication record of the candidate.

  8. Paul Gowder Says:

    Art, I’ll take your word for it: having not yet attained the level to see or write any of these letters, I can’t speak to the level of inflation…

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