Suppose two people each have some private information about, say, a stock. The first says her posterior expectation of its price, then the second learns from her statement and says his, etc.
Will they reach a consensus? Under common priors, yes. (See below.)
People often learn this from GP (below), but they assume finite probability spaces & really use that.
Tweet proof shows that convergence to consensus doesn't rely on finiteness, and holds much more generally. (When was the martingale proof first written down?)
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If we do assume each of us has a finite partition of the states, the tweet proof immediately implies convergence in finitely many steps:
our partitions can only change finitely many times, given that convergence happens, it must happen in finite time.
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A thread on what we can learn about polarization of opinions from a simple behavioral network model.
Remember the DeGroot model? It says you decide what to think tomorrow by taking an average of what you and your friends think today.
Examples:
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Here is a condensed way of writing it using vector notation, which turns out to be very useful!
Here W has no negative entries, and each row sums to 1 (that's called "row-stochastic"). That makes sense, since each person is averaging others' views.
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We often want to think of the weights as coming from a social network, maybe something like this. The network tells you who are the friends that you listen to.
So we'd better establish a way of thinking of the updating weights as coming from a network.
An application of Aumann's agreeing-to-disagree result: certain kinds of war between rational countries are puzzling.
Since war involves destruction, better for one to surrender and bargain. But maybe each believes it'll get more by fighting? Suppose a country fights iff
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it thinks it will win with prob > 0.5. In the notation below, let X = indicator that I win. Then war means it's common knowledge that Y>.5>Z.
That's impossible, but the proof below doesn't quite show it. Exercise: show there's no CK event E on which Y>Z.
War here is like speculative trade (cf. no-trade theorem): it can't happen due to different information ALONE, because by Aumann both of us can't rationally expect to win.
Thus, if war is zero-sum or worse, it entails irrationality or different priors.
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One of my favorite things about Bob Wilson, co-winner of today's prize, is how gentle a giant he is, how modest yet understatedly charismatic and funny. This rare recording gives a sense.
"I'm here to today to argue that sequential equilibrium [his own invention with Kreps] -- which you said ... in 1982 completed the answer to the questions that Luce and Raiffa raised in 1957 -- well, MY stance was that that was a mild disaster!
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"Sequential equilibrium turned out to have enormous flaws. And the revelation of those flaws has, I think, been actually opening up what the real challenge is for game theory in terms of establishing what its foundations are.
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We look at a society where people update their opinions according to the _DeGroot model of updating_. It says you decide what to think tomorrow by taking a weighted average of what you and your friends think today.
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Despite its simplicity and strong assumptions, DeGroot's model has been a surprisingly helpful workhorse in networks.
We ask: suppose initial estimates are centered at the truth θ and conditionally independent.
Do we get a "wisdom of crowds" in the long run? More precisely...
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The average course size that students experience is bigger than the average course, because by definition the big courses have more students experiencing them ("the class size paradox.")
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When you go to the gym and look around, you feel relatively bad because the very frequent gym-goers are oversampled in your looking around, whereas the never-gym-goers are not sampled at all and don't make you feel (relatively) better.
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This is the assertion that "your friends are more popular than you are."
Why? Simplest way to see it: some people have no friends. But because they appear in nobody's friendship circles, they're not making anyone else feel unpopular.
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The selection effect that applies to these friendless (a.k.a. degree zero) people also applies to other people: the more friends you have, the likelier you are to be represented in people's friendship circles. So popular people are oversampled as friends. Hence the paradox.
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Still, what is the paradox exactly, as a quantitative statement?
Scott Feld, who coined the term and made the paradox famous, had one way of formalizing it. It isn't my favorite way, but it's a classic, and worth meeting first.
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