Read on Twitter
Alright, time for a twitter summary of my former job market paper, coauthored with Xiaosheng Mu (who is not on twitter, which is unfortunate for us, though maybe not for him….). We spent SUCH a long time on the writing, so I would hope it's readable. But, these are also fun. https://twitter.com/RevEconStud/status/1287783337248141315
This is a bit more "history of the project" than straight summary, which is probably more interesting for everyone.
In grad school, Xiaosheng and I had been seeing a lot of papers and innovations around Bayesian persuasion (henceforth BP).
In grad school, Xiaosheng and I had been seeing a lot of papers and innovations around Bayesian persuasion (henceforth BP).
These suggested ways to model information acquisition/transmission without holding on dearly to particular functional form assumptions.
Before BP, theorists might have studied information arrival by saying “well, what if this signal has this distribution, e.g., normal?”
Before BP, theorists might have studied information arrival by saying “well, what if this signal has this distribution, e.g., normal?”
Lots of great insights to be had from taking such a parametric approach. On the other hand, it’s limiting in that there are only so many you can look at tractably. And even when it’s tractable, often there are other restrictions which are hard to interpret or know how to relax.
Ideally, you wouldn’t be tied to these. The BP framework provided a way of saying, “let’s just imagine there is *SOME* information structure. What happens?” Which may (arguably) be more satisfying, since where does normality or uniformity come from anyways?
We wanted to use these ideas in a mechanism design setting. Specifically, we thought we could put these tools to work toward other applications, to get some new insights.
To make it most seamless with the prior work we knew, we thought it made sense to think of the “agent” learning about some key determinant of their preferences, according to some information structure. The issue was finding a problem that was neither too easy nor too hard.
This is usually the rub in theory papers, no less so here. We thought we could make the problem “hard” by looking at a dynamic setting. There were some innovative dynamic BP papers we knew decently well. But we also thought any dynamic setting would be hard enough.
Among those, we thought durable goods pricing was the most natural first one. Why? Well, for one, this was already a big literature. But also, if you want to think about an agent making some dynamic decision, it helps if their decision is as simple as possible.
“When do I want to say 'now'” seemed as basic as it could be. It also (not coincidentally, as per the process of doing theory referenced above) inherited a bunch of properties of other dynamic BP settings, so we thought we could handle it.
You might be thinking, “why move to dynamics immediately?” In retrospect, at least in my opinion, it probably would have made more sense to start with just a static world. But it turned out to be a lucky mistake that we didn’t.
Unbeknownst to us, others were ahead of us there. After we finished our first draft, we read the beautiful paper by Anne-Katrin Roesler and Balazs Szentes, which solved a version of our single period problem. We now talk, at length, about the differences in the approach.
However, it turns out that just focusing on durable goods pricing wasn’t quite enough to whittle down to something tractable and interesting. We now understand why quite well, though we may or may not have fully appreciated it at the time.
The short answer is: it’s still very hard to describe what a buyer would do with an arbitrary information structure. It’s even harder to think how a seller would optimize given those constraints.
The missing piece was to focus on a particular objective. In principle, there are lots of different things the information structure might be doing, and so lots of ways one might think it could be endogenized.
Our idea was to use the robust mechanism design (henceforth RMD) literature as inspiration to single out one. Specifically, we imagined the information structure being chosen to make the seller’s life as bad as possible.
This might seem like a weird choice to some of you; e.g, why not make the seller’s life as good as possible? In RMD, the usual motivation is that, if the designer really has no idea about some aspect of the environment, they should try to do well no matter what it actually is.
This justifies trying to get the best PROFIT GUARANTEE, no matter what the information structure. This turned out to make things tractable enough to solve. We wrote down a version where we could work out a single-period example in minutes, using results we knew from BP.
We then thought about whether the answer changed with a longer selling horizon. Asking this question—does a seller’s optimal profit change with the time horizon?—is the standard benchmark in the intertemporal pricing literature: Stokey showed that the answer was “no”.
(Sidenote: Stokey actually showed that the answer is no, unless time provided some mechanism to do price discrimination. I think people usually cite her paper as saying “no” without the qualifier. But anyways...)
By itself, you would expect information arrival to introduce some dynamics into the seller's problem; if information arrives over time, then buyers may be more informed later in time, and so that should change the price the seller would want to charge.
Our main result in the paper is that despite this, you recover Stokey's result under the robust objective.
Constant price paths deliver the optimal profit guarantee when the seller does not know how buyers learn over time.
Constant price paths deliver the optimal profit guarantee when the seller does not know how buyers learn over time.
Early versions of the paper focused on highlighting how the way you model information arrival matters for the result. The extension that is now Section 6.1 (in a slightly rephrased format) was originally central. But expositionally, it was better to focus on ONE benchmark.
That led us to writing the paper around the constant price path result. This was much better, since it let us highlight several ways you could or could not break the constant price path result.
It also let us elevate the single period problem, which many people thought was cool.
It also let us elevate the single period problem, which many people thought was cool.
So that’s the story. I only mentioned one way we got “lucky”, that we didn’t work on the static first, since we would have been too late. There were many others. I'll mention the two big ones.
First, we had some PHENOMENAL referees. I’d be lying if I said I felt that way about all them :-p But it’s not lost on me that we got such incisive and detailed comments that made the paper as good as it is now. Authentically grateful, and something I think about a lot (really).
Second, we had faculty advisors (some of whom are on twitter) who immediately knew this had potential and recognized that even during rough patches. This was not a given, by any means, and very important.
This was enormously fun to work on. As an over-eager second year, I had thought it would be really cool to extend RMD to new applications. I was also maybe too infatuated with “dynamics.” This was about as close to something my second-year self would have loved to write. Wild.
Am hoping to have some follow-on stuff down the pike reasonably soon (so stay tuned). In any case, the area of “pricing with general information" has flourished since we started working, so I doubt this will be the first time you’re hearing about it if you're still reading.
Dynamic RMD I think is more difficult to build on quite honestly, but I do think there will be some good papers in this space in the future too. Lots still to work on and problems to figure out….
