Information cascades

An in­for­ma­tion cas­cade is a prob­lem in group ra­tio­nal­ity. Wikipe­dia has ex­cel­lent in­tro­duc­tions and links about the phe­nomenon, but here is a meta-ish ex­am­ple us­ing like­li­hood ra­tios.

Sup­pose in some fu­ture ver­sion of this site, there are sev­eral well-known facts:

  • All posts come in two kinds, high qual­ity (in­sight­ful and rele­vant) and low qual­ity (old ideas re­hashed, long hy­po­thet­i­cals).

  • There is a well-known prior 60% chance of any­thing be­ing high qual­ity, rather than low qual­ity. (We’re do­ing well!)

  • Read­ers get a pri­vate sig­nal, ei­ther “high” or “low”, their per­sonal judge­ment of qual­ity, which is wrong 20% of the time.

  • The num­ber of up and down votes is dis­played next to each post. (Note the differ­ence from the pre­sent sys­tem, which only dis­plays up minus down. This hy­poth­e­sis makes the math eas­ier.)

  • Read­ers are com­pe­tent in Bayesian statis­tics and strive to vote the true qual­ity of the post.

Let’s talk about how the very first reader would vote. If they judged the post high qual­ity, then they would mul­ti­ply the prior like­li­hood ra­tio (6:4) times the bayes fac­tor for a high pri­vate sig­nal (4:1), get (6*4:4*1) = (6:1) and vote the post up. If they judged the post low qual­ity then they would in­stead mul­ti­ply by the bayes fac­tor for a low pri­vate sig­nal (1:4), get (6*1:4*4) = (3:8) and vote the post down.

There were two sce­nar­ios for the first reader (pri­vate in­for­ma­tion high or low). If we spec­u­late that the first reader did in fact vote up, then there are two sce­nar­ios for the sec­ond sce­nario: There are two sce­nar­ios for the sec­ond reader:

  1. Per­sonal judge­ment high: (6:4)*(4:1)*(4:1) = (24:1), vote up.

  2. Per­sonal judge­ment low: (6:4)*(1:4)*(4:1) = (6:4), vote up against per­sonal judge­ment.

Note that now there are two ex­pla­na­tions for end­ing up two votes up. It could be that the sec­ond reader ac­tu­ally agreed, or it could be that the sec­ond reader was fol­low­ing the first reader and the prior against their per­sonal judge­ment. That means that the third reader gets zero in­for­ma­tion from the sec­ond reader’s per­sonal judge­ment! The two sce­nar­ios for the third reader, and ev­ery fu­ture reader, are ex­actly analo­gous to the two sce­nar­ios for the sec­ond reader.

  1. Per­sonal judge­ment high: (6:4)*(4:1)*(4:1) = (24:1), vote up.

  2. Per­sonal judge­ment low: (6:4)*(1:4)*(4:1) = (6:4), vote up against per­sonal judge­ment.

This has been a night­mare sce­nario of group­think af­flict­ing even dili­gent bayesi­ans. Pos­si­ble con­clu­sions:

  • Don’t strive to vote the true qual­ity of the post, strive to vote your per­sonal judge­ment.

  • Try to avoid even notic­ing the score. (Maybe scores could even be oc­cluded, like spoiler-text?)

  • In­for­ma­tion cas­cades are dan­ger­ous and in­ter­est­ing. We should de­velop good cog­ni­tive cit­i­zen­ship tech­niques.

  • Broad­cast novel ev­i­dence, not con­clu­sions.

Note: Olle found an er­ror that ne­ces­si­tated a rewrite. I apol­o­gize.