He's best known for the Blackwell information ordering, the way to formalize when some signals give you more information than other signals.
A thread on Blackwell's lovely theorem and a simple proof you might not have seen.
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Blackwell was interested in how a rational decision-maker uses information to make decisions, in a very general sense. Here's a standard formalization of a single-agent decision and an information structure.
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One way to formalize that one info structure, φ, dominates another, φ', is that ANY decision-maker, no matter what their actions A and payoffs u, prefers to have the better information structure.
While φ seems clearly better, is it definitely MORE information?
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Blackwell found out a way to say that it is. That's what his theorem is about. Most of us, if we learned it, remember some possibly confusing stuff about matrices. This is a distraction: here I discuss a lovely proof due to de Oliveira that distills everything to its essence.
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We need a little notation and setup to describe Blackwell's discovery: that the worse info structure is always a *garbling* of a better one.
Let's start by defining some notation for the agent's strategy, which is an instance of a stochastic map -- an idea we'll be using a lot.
Stochastic maps are nice animals. You can take compositions of them and they behave as you would expect.
Here I just formalize the idea that you can naturally extend an 𝛼 to a map defined on all of Δ(X). And that makes it easy to compose it with other stochastic maps.
Okay! That was really all the groundwork we needed.
Now we can define Blackwell's OTHER notion of what it means for φ to dominate φ'.
It's simpler: it just says that if you have φ you can cook up φ' without getting any other information.
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Blackwell's theorem is that these two definitions (the "any decision-maker" and the "garbling" one) actually give you the same partial ordering of information structures.
"Everyone likes φ better" is equivalent to "you get φ' from φ by running it through a garbling machine 𝛾."
To state and prove the theorem, we need one more definition, which is the set of all things you can do with an info structure φ.
The set 𝓓(φ) just describes all distributions of behavior you could achieve (conditional on the state ω) by using some strategy.
Now we can state the theorem. We've discussed (1) and (3) already. Point (2) is an important device for linking them, and says that anything you can achieve with the information structure φ', you can achieve with φ.
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de Oliveira's insight is that, once you cast things in these terms, the proof is three trivialities and one application of a separation theorem.
(1) ⟹ (2). If φ' garbles φ and you HAVE φ, then just do the garbling yourself and get the same distribution.
(2) ⟹ (1). On the other hand, if φ can achieve whatever φ' can, it can achieve "drawing according to φ'(ω)," which makes you the garbling
(2) ⟹ (3) says if 𝓓(φ) contains 𝓓(φ') then you can do at least as well knowing φ': the easiest step.
Note that the agent's payoff depends only on the conditional distribution behavior given the state. Since all distributions in 𝓓(φ') are available w/ φ, agent can't do worse.
(3) ⟹ (2) is the step that's not unwrapping definitions.
Suppose (2) were false: then you could get some distribution 𝐝' with φ' that you can't get with φ. The set 𝓓(φ) of ones you can get with φ is convex and compact, so .... separation theorem! Separate 𝐝' from it.
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If we state what "separation" means in symbols, it gives us (*) below. But that tells us exactly how to cook up a utility function so that any distribution in 𝓓(φ), one of those achievable with φ, does worse than our 𝐝'. That's exactly what (3) rules out.
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That's it!
Happy birthday David Blackwell, and thanks Henrique de Oliveira. Though I am the world's biggest fan of Markov matrices, there's no need to use them for Blackwell orderings once you know this way of looking at things, which gets at the heart of the matter.
16/16
typo! that red arrow label should just be 𝛾
PS/ Tagging in @smorgasborb, who I didn't know was on Twitter and whose fault this all is.
A few typos above that I hope didn't interfere too much w/ exposition of his argument: In 6, the red label was wrong - fixed here. In 13, the first φ' should be φ.
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Our perceptions of some of the things we experience are deeply inaccurate. 🧵
Case 1: The vast majority of restaurants get few visits and go out of business quickly. This seems surprising because the typical restaurant you experience is busy and long-lived.
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The gap between reality and perception happens because few people experience any given unpopular, short-lived restaurant. Precisely because it is unpopular and short-lived!
The brilliant @CFCamerer, who gave this example, notes that it's not just curious but consequential.
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We, including aspiring restauranteurs, undersample unsuccessful restaurants so badly that it can make the restaurant business intuitively feel easy.
So too many people start restaurants who should have done other things instead.
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Excited to watch this talk by @renee_bowen_lyn : a model of echo chambers in social networks and how they take way less "behavioral error" than you might have thought to get started.
Behind the scenes there's a sort of puzzle based on a "naive martingale intuition": if there's abundant data and you understand the information process you're seeing, then a Bayesian should converge to accurate beliefs.
Here are some important statements that come up in economics:
"Nice estimators are consistent even in complicated models."
"Nice financial markets are informationally efficient."
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"Nice markets have price equilibria."
"Nice games have Nash equilibria."
The way these ideas are taught to Ph.D. economists in any field, even in core courses, involve very explicitly and extensively ideas extending ones in basic analysis.
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In particular, those ideas are: convergence (in fairly big spaces), integration and probability/martingales, continuity and fixed points.
Though you could get across aspects of these ideas at a high school level, econ grad school doesn't do them that way.
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One amusing feature of math classes at the master's level or above is that they almost take pride in not motivating the subject in external terms. For example, here's a page from a canonical textbook in algebraic geometry.
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This does a good job of reminding me of how Lecture 1 in such classes often felt, which is roughly, "The motivation for this class is fuck you. Let k be an arbitrary algebraically closed field. Now..."
Which was not a problem when I had my own motivations!
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At some point it stopped being enough. Incidentally, I don't think my economics courses were much better in the way of giving some great external motivation: I just found a cycle of self-reinforcing curiosity that kept me happily studying that subject.
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Was grateful to share at an #ASSA2021 session today a bit on what I've learned in teaching an undergraduate course on the Economics of Networks.
A short thread to serve as a focal point for any follow-up conversation.
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What networks is about (very rough and probably somewhat idiosyncratic description)
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I taught several variants of an undergraduate elective on this exciting and growing area. It was at the applied math/econ/CS intersection -- sometimes cross-listed, sometimes just economics but open to (and taken by) applied math, CS, other students.