Comparative Cheap TalkArchishman Chakrabortyand forthcoming - PDF document

1 Comparative Cheap TalkArchishman Chakrab
Comparative Cheap TalkArchishman Chakrabortyand, forthcoming •Biased Sender S knows realization of r.v. iased Sender

2 S knows realization of r.v. •Receiver R
S knows realization of r.v. •Receiver R takes action a •Informed R chooses a = •But S ideal is a = •For any realizati

3 on that S reports, R will believe true •
on that S reports, R will believe true •So S must exaggerate even more and R compensates even more•So communication b

4 reaks down and R just chooses a=E[•S rec
reaks down and R just chooses a=E[•S receives expected payoff of only receives expected payoff of only S]= -Var[•Whi

5 ch is less than if S told truth! hich is
ch is less than if S told truth! hich is less than if S told truth! S]= -b •Uncertainty over quality of a good -selle

6 r tells buyer it’s •Uncertainty over pro
r tells buyer it’s •Uncertainty over prospects for a stock –analyst says it’s •Uncertainty over whether a spending pr

7 ogram is worthwhile –administrator says
ogram is worthwhile –administrator says it’s essential•Uncertainty over quality of a job applicant – •Good A is bette

8 r than B•Stock A is better than B•Propos
r than B•Stock A is better than B•Proposal A is better than B•Student A is better than B•Comparative statements are p

9 ositive along one •Can’t exaggerate!•But
ositive along one •Can’t exaggerate!•But still might have an incentive to invert the ordering •Professor (S) has N st

10 udents to recommend to employer •Employe
udents to recommend to employer •Employer hires a student if expected quality •Professor wants a student to be hired

11 if •“CS”equilibrium in one dimension if
if •“CS”equilibrium in one dimension if (“sorting condition”), so any ordering is an equilibrium. Rankings become mo

12 re informative as number of issues incre
re informative as number of issues increases 1 0.75 0.5 0.25 0 10 7.5 5 2.5 0 N=3N=2Distribution of top-ranked issue

13 For given N, partition cheap talk outper
For given N, partition cheap talk outperforms ordinal cheap talkfor small For given b, ordinal cheap talk outperforms

14 partition cheap talkfor large As befor
partition cheap talkfor large As before let aj:Nbe R’s best action for issue ranked jfrom the bottom. And let a(kki

15 s known. For each q(0,1), by the Gliven
s known. For each q(0,1), by the Glivenko-CantelliTheorem= F¹(q) a.s.where F(k) is distribution of each kTherefore f

16 or i=S,R,limjkkSo in the limit as the nu
or i=S,R,limjkkSo in the limit as the number of issues increases, the sender reveals allinformation and sender and re

17 ceiver payoffs are equivalent •E[•N=2: E
ceiver payoffs are equivalent •E[•N=2: E[[ș2:2]=2/3•N=3: E[[ș2:3]=2/4, E[ș3:3]=3/4•N=4: E[[ș2:4]=2/5, E[ș3:4]=3/

18 5, E[ș4:4]=4/5•N=5: E[[ș2:5]=2/6, E[ș
5, E[ș4:4]=4/5•N=5: E[[ș2:5]=2/6, E[ș3:5]=3/6, E[ș4:5]=4/6, E[ș5:5]=5/6Partial ordering:•N=3: E[[ș{2,3}:3•N=4:

19 E[{1,2}:44ș{3,4}:4]=7/10•N=5: E[{1,2}:5
E[{1,2}:44ș{3,4}:4]=7/10•N=5: E[{1,2}:55ș{3,4,5}:5 •Interdependent actions–Can only hire one person–Must buy goods

20 in bundle•Sender and receiver take actio
in bundle•Sender and receiver take actions, e.g. •Private receiver information, e.g. auctions•Non-additive payoffs•Re