[Question] Formalising continuous info cascades? [Info-cascade series]

This is a ques­tion in the info-cas­cade ques­tion se­ries. There is a prize pool of up to $800 for an­swers to these ques­tions. See the link above for full back­ground on the prob­lem (in­clud­ing a bibliog­ra­phy) as well as ex­am­ples of re­sponses we’d be es­pe­cially ex­cited to see.

Math­e­mat­i­cally for­mal­is­ing info-cas­cades would be great.

For­tu­nately, it’s already been done in the sim­ple case. See this ex­cel­lent LW post by Johni­cholas, where he uses up­votes/​down­votes as an ex­am­ple, and shows that af­ter the sec­ond per­son has voted, all fu­ture vot­ers are adding zero new in­for­ma­tion to the sys­tem. His ex­pla­na­tion us­ing like­li­hood ra­tios is the most in­tu­itive I’ve found.

The Wikipe­dia en­try on the sub­ject is also quite good.

How­ever, these two en­tries pri­mar­ily ex­plain how in­for­ma­tion cas­cades when peo­ple have to make a bi­nary choice—good or bad, left or right, etc. The ques­tion I want to un­der­stand is how to think of the prob­lem in a con­tin­u­ous case—do the prob­lems go away? Or more likely, what vari­ables de­ter­mine the speed at which peo­ple up­date to one ex­treme? And how far to­ward that ex­treme do peo­ple go be­fore they re­al­ise their er­ror?

Ex­am­ples of con­tin­u­ous vari­ables in­clude things like pro­ject time es­ti­mates, stocks, and prob­a­bil­is­tic fore­casts. I imag­ine it’s very likely that sig­nifi­cant quan­ti­ta­tive work has been done on the case of mar­ket bub­bles, and any­one can write an an­swer sum­maris­ing that work and ex­plain­ing how to ap­ply it to other do­mains like fore­cast­ing, that would be ex­cel­lent.

No answers.