Strategic information selection

P eke , Ju ek; Ka os, Dominik
Wo king Pape
S a egic in o ma ion selec ion
Cen e o Ma hema ical Economics Wo king Pape s, No. 689
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: P eke , Ju ek; Ka os, Dominik (2024) : S a egic in o ma ion selec ion, Cen e o
Ma hema ical Economics Wo king Pape s, No. 689, Biele eld Uni e si y, Cen e o Ma hema ical
Economics (IMW), Biele eld,
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689
Ma ch 2024
S a egic In o ma ion Selec ion
Ju ek P eke and Dominik Ka os
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
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S a egic In o ma ion Selec ion∗
Ju ek P eke †Dominik Ka os‡
Ma ch 28, 2024
Abs ac
Be o e choosing he ac ion o ma ch he s a e o he wo ld, an agen obse es a
s eam o messages gene a ed by some unknown bina y signal. The agen can ei he
lea n he unde lying signal o ee and upda e he belie acco dingly o igno e he
obse ed message and keep he p io belie . A e each pe iod he s eam s ops wi h
posi i e p obabili y and he inal choice is made.
We show ha a Ma ko ian agen wi h Gilboa-Schmeidle p e e ences lea ns and
upda es a e con i ming messages, bu she igno es con adic ing messages i he
belie is su icien ly s ong. He h eshold solely depends on he leas p ecise signal.
The agen has s ic ly highe an icipa o y u ili y han an agen who uses e e y
message o upda e. Howe e , he la e has a highe chance o choose he co ec
ou come in he end. In a popula ion o s a egic agen s, who only di e in hei
ini ial belie s, pola iza ion is ine i able.
Keywo ds: Dynamic Decision P oblem, Ambigui y, Gilboa-Schmeidle P e e ences,
Con i ma ion Bias, Pola iza ion
JEL Classi ica ion: D81, D83, D91
∗The au ho s hank Ch is Woolnough, Ge i Bauch, Gio gio Fe a i, Ehud Leh e , Fynn Louis
N¨a mann, and F ank Riedel as well as many semina pa icipan s a he uni e si ies o Biele eld, Du ham,
and Tel A i o hei commen s and sugges ions.
†Cen e o Ma hema ical Economics, Biele eld Uni e si y, Pos ach 100131, 33501 Biele eld, Ge many.
ju ek.p ek[email p o ec ed], co esponding au ho
‡Cen e o Ma hema ical Economics, Biele eld Uni e si y, Pos ach 100131, 33501 Biele eld, Ge many.
dominik.k[email p o ec ed]
1
1 In oduc ion
In an a e age minu e o Decembe 2023, social media use s sen ou 360,000 wee s on
X, sen 694,000 eels on Ins ag am ia di ec messages, and liked ou million Facebook
pos s.1A use o any o hese social media pla o ms canno ead all he a icles whose
headlines appea in he eed, bu has o choose wha headlines o ollow up on and wha
o igno e. This makes in o ma ion acqui emen on social media undamen ally di e en
om using p in media. While a p in ed newspape , e en i ead om co e o co e , will
p o ide he eade only wi h a ixed numbe o a icles, incoming news in social media o m
an endless s eam. Thus, he in o ma ion acqui emen p ocess in social media equi es a
p e-s ep: be o e eading and p ocessing an a icle, he use needs o selec wha o ead
om a huge supply.
I has been well es ablished ha his causes people o lea n di e en ly: o ins ance,
hey lea n less om online han om p in sou ces (E eland and Dunwoody, 2002; Yang
and G abe, 2011); bu a he same ime hey a e mo e ocused on speci ic opics se-
lec ed by hemsel es a he han he news edi o s (K uikemeie , Lechele , and Boye ,
2017). Empi ical s udies sugges he main d i e s behind in o ma ion selec ion be a i-
ude consis ency and sou ce c edibili y. S¨ul low, Sch¨a e , and Win e (2019) in es iga ed
he impac o bo h and showed ha a i ude consis ency has no impac on he ime people
spend looking a a headline in hei news eed bu inc eases he p obabili y hey ollow
up on i and ac ually ead he unde lying a icle. Mo eo e , hey ound ha he assumed
sou ce c edibili y has a posi i e e ec on bo h he ime spen wi h a headline as well as
on he ollow-up p obabili y.2
As social media ha e become inc easingly impo an as a sou ce o news,3unde -
s anding he mechanics behind in o ma ion selec ion as well as i s indi idual and social
consequences, such as pola iza ion4and collec i e decision making, a e c ucial. This a i-
1See S a is a (2024).
2See also Chiang and Knigh (2011) o he impo ance o c edibili y.
3The sha e o people in Ge many who weekly use social media as a sou ce o news has inc eased om
18 % in 2013 o 29 % in 2023, ha ing o e aken classical p in media in 2019. In he US, his numbe has
isen om 27 % o 48 % in he same ime. The e ec is pa icula ly s ong in he g oup o young adul s
(Reu e s Ins i u e o he S udy o Jou nalism, 2023). Acco ding o a s udy by he Pew Resea ch Cen e ,
a majo i y ac oss 19 coun ies ag ees ha social media is an e ec i e way o aise public awa eness
abou poli ical o social issues (77 % ag ee), o change people’s minds (65 %), ge elec ed o icials o pay
a en ion o issues (64 %), and o in luence policy decisions in hei own coun y (61 %) (Pew Resea ch
Cen e , 2022).
4In he abo e men ioned epo by he Pew Resea ch Cen e , 65 % o he in es iga ed subjec s say
2
cle p o ides a dynamic model o s a egic in o ma ion selec ion ha a ionalizes obse ed
beha io and in es iga es i s implica ions.
We conside a decision make who has o ma ch an unknown bina y s a e o he wo ld.
Be o e making he choice, she ecei es a disc e e s eam o bina y and symme ic messages,
o be hough o as a icles on social media ha endo se one o he wo s a es, ha a e
co ec wi h some unknown p obabili y o a leas 50%. She has he op ion o lea n his
p obabili y; in he news media example, a e ob aining a headline ( he message) she may
click on he link, ead he a icle, lea n how well he a icle is esea ched, and in e he
p obabili y ha he message o he headline was co ec . I she chooses o igno e he
message, he belie does no change; i she chooses o lea n he unde lying signal, she has
o pe o m a Bayesian upda e. The unde lying assump ion is ha people can easily igno e
a headline hey only see b ie ly, bu once hey ha e engaged wi h i , i canno be unseen
and will a ec hei belie . Fo he belie upda e she uses he p e ious belie , he message,
and he signal. Tha is, we assume ha he decision make is Ma ko ian, meaning ha
she does no use he en i e his o y o p e ious messages and signals.5
In each pe iod he s eam o messages s ops wi h some exogenous p obabili y and he
decision make has o make he choice abou he s a e; o he wise a new signal and a
new message will be d awn. Maximizing he u ili y hus becomes a dynamic op imiza ion
p oblem, whe e wi h e e y decision she has o ake in o accoun how he belie will de elop
in he u u e. The Ma ko ian s uc u e gi es ise o a Bellman equa ion whose solu ion
is he decision make ’s an icipa o y u ili y unc ion.6We show ha op imal s a egies
exis and desc ibe hei gene al s uc u e. A e wa ds, we ocus on a decision make
wi h max-min p e e ences (c . Gilboa and Schmeidle , 1989) and show ha any op imal
s a egy equi es he o igno e messages ha oppose he belie whene e he belie is
su icien ly s ong. Hence, she exhibi s wha S one and Wood (2018) call “non-ego-based
ha access o he in e ne and social media has made people mo e di ided in hei poli ical opinions.
Allco , B aghie i, Eichmeye , and Gen zkow (2020) showed ha egula Facebook use s who u ned o
he ne wo k o a mon h we e less pola ized in hei a i udes a e wa ds. See P io (2013) and Kubin
and on Siko ski (2021) o poli ical e iews on how (social) media d i e pola iza ion.
5The e a e wo majo easons o his assump ion: On he one hand, people see and ead so many
headlines and a icles on social media ha e en ually, hey canno keep ack o he en i e his o y o hei
messages, signals, and belie s. On he o he hand, i allows us o condi ion he agen ’s beha io only on
he belie and he signal, independen o ime and signal his o y, so ha he analysis emains ac able.
6We call his u ili y an icipa o y as i akes in o accoun all possible u he pa hs on which belie s
migh change. This ela es o he concep o u ili y om an icipa ion, in oduced by Loewens ein (1987)
and la e in es iga ed, o ins ance, by Caplin and Leahy (2001) and K˝oszegi (2017).
3

cogni i e dissonance”7: media consume s a oid con adic ing in o ma ion “no because
his h ea ens hei egos, bu simply because unce ain y is unse ling”.
Igno ing ee in o ma ion is ob iously no op imal in he amewo k o Blackwell (1951,
1953). Howe e , we show ha i is op imal in ou se ing, a ionalizing human beha io in
social media.8In pa icula , when compa ing ou s a egic decision make wi h a Bayesian
decision make , who upda es his belie a e e e y ound, we ind ha she ob ains a s ic ly
highe an icipa o y u ili y, while he p obabili y o ma ch he ue s a e o he wo ld is
s ic ly lowe .
Su ely, his model desc ibes social media only in a e y simpli ied way. Fo ins ance,
we assume ha he obse ed messages a e independen o he use ’s belie , and we igno e
algo i hms, echo chambe s, as well as he a ge ed use o ake news by poli ical agen s.
Ye , e en wi hou all hese o any o he o m o homophily in social media ne wo ks9we
ob ain qui e ex eme pola iza ion esul s whe e socie y only becomes mo e di ided o e
ime.
The es o he pape is o ganized as ollows. In Sec ion 2, we p esen u he ela ed
li e a u e. Sec ion 3de elops he model, discusses i s assump ions, and de i es gene al
exis ence and uniqueness esul s. In Sec ion 4, we de i e op imal s a egies in he absence
o ambigui y, while Sec ion 5co e s agen s wi h Gilboa-Schmeidle p e e ences unde am-
bigui y. In Sec ion 6, we compa e ou s a egic decision make wi h one who upda es a e
e e y message he obse es. Social consequences in e ms o pola iza ion and collec i e
decision making a e co e ed in Sec ion 7. Sec ion 8p o ides se e al ways o ex end he
model and Sec ion 9concludes he pape . P oo s a e elega ed o he appendix.
2 Fu he Rela ed Li e a u e
A la ge class o models in which agen s do no ully exploi he a ailable in o ma ion
ocuses on limi ed cogni i e abili ies. Fo ins ance, he obse ed endency o a o and
sea ch o in o ma ion ha ein o ces p io belie s while igno ing opposing in o ma ion
7The psychological concep o cogni i e dissonance has been popula ized by Fes inge (1957): Unce -
ain y abou wha o belie e and acing a ade-o be ween eeling alida ed and wan ing o know he
u h makes people eel uncom o able.
8Social media a e no he only place whe e in o ma ion migh be igno ed: e en some eade o his
pape migh ejec ou model and pu i aside wi hou le ing i a ec hei belie abou human beha io .
9See Aiello e al. (2012) o a s a is ical analysis o homphily in social media.
4
is known as selec i e exposu e.10 Lo d, Ross, and Leppe (1979) demons a e ha when
people wi h s ong opinions on con o e sial issues—in ha case, he dea h penal y in
he USA—a e gi en equally con incing e idence o bo h posi ions, hey become e en
s onge in hei opinions. So, di e en pieces o in o ma ion a o ing con a y s a es
o he wo ld do no neu alize, bu people pick wha hey wan o hea , in e p e i as
con i ming e idence, and igno e he es . This con i ma ion bias11 is no wha d i es
ou esul s, as ou decision make co ec ly p ocesses all in o ma ion ha she uses, bu
delibe a ely chooses o igno e some o i . This dis inc ion ela es o he psychological
dis inc ion be ween au oma ic and con olled p ocesses (Schneide and Shi in, 1977).
Among he models o s a egic in o ma ion selec ion, an impo an s and o he li -
e a u e, ollowing Sims (2003), ocuses on a ional ina en ion whe e agen s only ob ain
pa ial in o ma ion due o cos . (See Ma´ckowiak, Ma ˇejka, and Wiede hol (2023) o a
e iew.) In con as , in o ma ion in ou model comes a no cos and may s ill be ejec ed.
In many models agen s choose ex an e how o ga he in o ma ion, ha is, hey only
ob ain he ac ual message a e hei decision. Machina (1989) gi es a s ylized example
o such beha io based on he Allais Pa adox, whe e non-expec ed u ili y maximize s
choose no o ob ain in o ma ion. Ca illo and Ma io i (2000) p esen a decision make
wi h ime-inconsis en p e e ences who decides o acqui e only incomple e in o ma ion
abou a u u e ex e nali y. In Suen (2004), agen s obse e coa sened signals and decide
hemsel es on a coa sening ule. Simila ly, decision make s in Che and Mie endo (2019)
ha e access o di e en biased sou ces and choose he op imal one.
In con as , ou decision make i s ob ains a message and hen chooses wha o
do wi h i . This b ings ou model close o Chen (2022), who conside s agen s wi h
di e en models a hand who upda e hei belie s acco ding o he one ha bes suppo s
hei bias, and o F ye , Ha ms, and Jackson (2019) whe e agen s deal wi h “ambiguous”
signals ha can be used in di e en ways.12 Ye , ou model is dis inc in se e al key
ea u es. Fi s , agen s can decide o en i ely disca d and igno e a message. This b ings
us close o Allah e dyan and Gals yan (2014), whe e an agen does no eac o ad ice
(sen by ano he agen who wan s o pe suade he o me one) ha con lic s wi h he
p io con ic ion. Second, a decision make wi h Gilboa-Schmeidle p e e ences will no
10Ha e al. (2009) p o ide a psychological me a-s udy on ha subjec .
11See Nicke son (1998) o a psychological o e iew as well as Rabin and Sch ag (1999).
12No e ha hei use o “ambiguous” di e s om ou e minology as we e e o ambigui y as missing
knowledge abou p obabili ies.
5
ope a e unde he assump ion ha he message has been gene a ed by he signal ha
i s bes , bu a he by he signal ha p o ides he wi h he lowes u ili y. Thi d, e en i
he e is no ambigui y a all, ha is, i he signal is known, some o he messages will be
igno ed.
The concep o mo i a ed belie s migh se e as a b idge be ween ou model and he
li e a u e on ex-an e in o ma ion design. Acco ding o Epley and Gilo ich (2016), agen s
do no di ec ly choose hei belie s, bu “ hei mo i a ions guide wha in o ma ion hey
conside , esul ing in a o able conclusions ha seem manda ed by he a ailable e idence”.
E en hough ou agen ac s ex pos , she ollows p ecisely his app oach: i she expec s a
mo e a o able conclusion om lea ning he signal and in e p e ing he message, she does
so; o he wise she disca ds he message en i ely.
The good news-bad news e ec o op imism bias—expe imen ally con i med by Eil
and Rao (2011) and axioma ically cha ac e ized by B acha and B own (2012)—p esc ibes
people o espec he s eng h o a signal when he news is a o able. In B unne meie and
Pa ke (2005), agen s choose subjec i e p obabili ies such ha hei cu en well-being is
maximized, a he cos o possibly aking a w ong decision a e wa ds. This e ec di e s
om ou model as ou s a es a e neu al. Howe e , agen s in ou se up a e happie , he
mo e con inced hey a e o any s a e; being unsu e abou he ue s a e o he wo ld
makes hem uncom o able.
Signal ambigui y as a eason o in o ma ion selec ion has been desc ibed by Gen zkow
and Shapi o (2006). The main a gumen is ha a decision make who does no like wha
she eads can con enien ly ell he sel ha he a icle mus be poo ly esea ched and can
be sa ely igno ed. Ambigui y as a d i e o pola iza ion has been modeled by Glaese and
Suns ein (2013). In hei model, people belie e di e en pa s o he in o ma ion hey
ge . In Baliga, Hanany, and Klibano (2013), pola iza ion occu s as a esul o ambigui y
a e sion. This is no he case he e: e en hough we conside ex emely ambigui y a e se
decision make s, ou esul s abou pola iza ion would emain ue e en i he e we e no
ambigui y a all.
3 Op imal In o ma ion Selec ion
An agen has o ma ch he co ec s a e o he wo ld om he s a e space Ω = {0,1}.
Tha is, he choice space is C= Ω and he on Neumann-Mo gens e n u ili y unc ion
6
:C×Ω→ {0,1}is gi en by (c, ω) = 1 i c=ωand (c, ω) = 0 o he wise.
To keep no a ion simple, we iden i y any belie Λ ∈∆ (Ω) she migh en e ain wi h
λ= Λ(1) ∈[0,1]. Wi h such a belie he expec ed u ili ies om choosing 0 o 1 a e
E[ (0,·)] = 1 ·(1 −λ)+0·λ= 1 −λ, and
E[ (1,·)] = 1 ·λ+ 0 ·(1 −λ) = λ,
espec i ely. Thus, he indi ec expec ed u ili y a belie λis u(λ):= max {λ, 1−λ}.
P io o he choice he agen ob ains in o ma ion, which is modeled as a disc e e
s eam o symme ic bina y signals, ha is, maps ζ: Ω →∆ (Ω) wi h ζ(1|1) = ζ(0|0),
and ealized messages m∈ {0,1}. We assume wi hou loss o gene ali y ha ζ(1|1) ≥1
2,
so ha any signal ζcan be iden i ied wi h z=ζ(1|1) ∈1
2,1.13
Le Z⊆1
2,1be a compac se o signals. A each pe iod ∈N, he signal s eam
ends wi h some (exogenous) p obabili y 1 −pand he agen has o make he decision.
Wi h p obabili y p, he s eam con inues. In his case a signal z ∈Zis gene a ed, in a
po en ially unknown way, and a e wa ds a message m ∈ {0,1} ealizes acco ding o z ,
ha is, Pz (m = 0|ω= 0) = Pz (m = 1|ω= 1) = z .14 The decision make only obse es
m .
Fo any signal z he expec ed p obabili ies ha 0 and 1 ealize a e
q0
z(λ):=λ(1 −z) + (1 −λ)zand q1
z(λ):=λz + (1 −λ) (1 −z),
espec i ely. De ine o each z∈1
2,1 wo unc ions gz, g−1
z: [0,1] →[0,1] by
g−1
z(λ):=λ(1 −z)
λ(1 −z) + (1 −λ)zand gz(λ):=λz
λz + (1 −λ) (1 −z)(1)
o z∈1
2,1, as well as g−1
1≡0 and g1≡1.15 Su ely, i he agen knew z and m , hen
he pos e io belie s in case o a message o 0 o 1 we e λ =g−1
z (λ −1) o λ =gz (λ −1),
13The iple (Ω, ζ (·|0) , ζ (·|1)) is a symme ic bina y Blackwell expe imen , see Blackwell (1951,1953).
14The signals (z ) ≥1can be co ela ed. Ye , gi en he signals, he p obabili y measu es (Pz ) ≥1a e
independen .
15The la e de ini ions coincide wi h (1) o λ∈[0,1) o λ∈(0,1], espec i ely. The alues o g−1
1(1)
and g1(0) a e i ele an o he agen ’s op imal decision: i she is ce ain ha he ue s a e is 0 (o 1,
espec i ely) and ecei es a message s a ing he opposi e, she will conside i impossible ha he signal
is pe ec ly accu a e.
7
0z
0.5
λ
0.5
1
Figu e 2: The g aphs o z7→ λ∗(z) (solid line) and z7→ 1−z(dashed line)
Co olla y 4.2. The unc ion U∗
{z}is con inuous, sa is ies U∗
{z}(1 −λ) = U∗
{z}(λ) o
all λ∈[0,1] as well as U∗
{z}(0) = U∗
{z}(1) = 1, and is s ic ly dec easing on [0, λ∗(z)].
Mo eo e ,
U∗
{z}(gz(λ)) < U∗
{z}g−1
z(λ) o λ∈0,1
2and
U∗
{z}(gz(λ)) > U∗
{z}g−1
z(λ) o λ∈1
2,1.
Theo em 4.1 shows ha e en wi hou ambigui y he e a e belie s whe e i is op imal
o a s a egic decision make o igno e con adic ing messages. This egion, deno ed by
I(z) = [0, λ∗(z)] ∪[1 −λ∗(z),1] inc eases as zdec eases.
Co olla y 4.3. Le z, z0∈1
2,1. Then I(z)⊆I(z0)i and only i z≥z0.
The ela ion be ween zand λ∗(z) is depic ed in Figu e 2. As can be seen he e (and
analy ically e i ied), λ∗(z)>1−z o all z∈1
2,1.
5 In o ma ion Selec ion unde Ambigui y
In his sec ion we shall e u n o he decision p oblem when zis no ixed, bu s ems om
some compac se Z⊆1
2,1.23 We i s de i e he op imal s a egy o a decision make
23We exclude 1
2∈Z o a oid some edious case dis inc ions.
14

wi h Gilboa-Schmeide p e e ences. A e wa ds we conside an al e na i e app oach in
which he decision make is unawa e ha she migh eassess he iew on possible signals
a e ha ing ecei ed a message.
5.1 Op imal S a egies
Ou main inding is ha a Gilboa-Schmeidle decision make who aces a compac se
Zo signals will beha e exac ly as i she knew ha e e y message we e gene a ed by
z= min Z.
Theo em 5.1. Le Z⊆1
2,1be compac and and le H:F × P → Rbe he Gilboa-
Schmeidle agg ega o in (2). Le z= min Z. Then a s a egy is op imal gi en Zi and
only i i is op imal gi en {z}. In pa icula , U∗
Z=U∗
{z}.
A Gilboa-Schmeidle decision make acing signal se Zis beha io ally indis inguishable
om a decision make who knows ha zis he only possible signal. Mo eo e , such an
agen always ollows up on messages ha con i m he p io belie bu igno es con adic o y
in o ma ion i he p io is su icien ly s ong. The espec i e h eshold λ∗only depends
on he leas p ecise signal ha is deemed possible.
One migh be inclined o a gue ha Theo em 5.1 is no su p ising: a e all a Gilboa-
Schmeidle decision make is pessimis ic and should, hence, p esume he message s em
om he “wo s ” possible signal. We shall b ie ly illus a e ha his line o hough is
inco ec hough as he e is a di e ence be ween he “wo s ” and he “leas p ecise” signal.
Example 5.2. Le he signal se be Z=8
10,9
10such ha U∗
Z=U∗
{8
10 }and λ∗8
10=1
3.
Le he agen ’s belie be λ=1
20 and suppose ha message 1 ealizes. Upda ing wi h
espec o signal z=8
10 leads o an upda ed belie o g8
10 1
20=4
23, while upda ing wi h
espec o z=9
10 would lead o g9
10 2
10=9
28. As 0 <4
23 <9
28 <1
3and U∗
{8
10 }is s ic ly
dec easing in his a ea by Co olla y 4.2, he mo e p ecise signal would lead o a lowe
u ili y. 
The eason why he cons ella ion in he p e ious example does no a ec ou inding in
Theo em 5.1 is ha all belie s whe e he highe signal would lead o a lowe u ili y lie in
he a ea I(z) whe e he s a egic decision make does no upda e a all.
15
5.2 Nai e Decision Make s
Recall ha a he beginning o any pe iod he decision make aces wo sou ces o unce -
ain y: unce ain y abou wha message she will obse e and unce ain y abou he signal
ha will ha e gene a ed ha message. The de ini ion o he an icipa o y u ili y unc ion
in (3) made he implici assump ion ha she e alua es hese wo sou ces independen ly
o one ano he and ha she is awa e o his independence a he beginning o he pe iod.
Al e na i ely, she migh be unawa e ha he p e e ences could equi e he o eassess
wha cons i u es a wo s case scena io. The ollowing example illus a es he issue.
Example 5.3. Recall Example 5.2 wi h Z=8
10,9
10. A he p io o λ=1
20 <1
3=
λ∗8
10, he wo s ha could happen o he decision make is obse ing message 1. Thus,
ex an e, she will e alua e he si ua ion unde he p esump ion ha z=8
10 as o his z
he p obabili y o obse ing message 1 is maximal. Upon obse ing message 1, howe e ,
he wo s case o he is ha he message was gene a ed by signal z=9
10 as seen abo e.
The s a egic decision make in (3) akes his in o accoun and uses wo di e en alues
o za he wo di e en s ages. 
The decision make we conside ed so a was awa e ha she migh ha e a change o hea
along he way, e lec ed by he use o Bm
Z,σm(λ)(λ) as he se o possible belie s o he inne
agg ega o in (3). A decision make who is no awa e o his will ha e an an icipa o y
u ili y unc ion ha sa is ies
ˆ
U(λ) = (1 −p)u(λ) + pH hq0
·(λ)Hhˆ
U, B0
{·},σ0(λ)(λ)i+q1
·(λ)Hhˆ
U, B1
{·},σ1(λ)(λ)i, Zi.
(9)
We call a decision make wi h such an an icipa o y u ili y unc ion nai e, as opposed o
sophis ica ed, which would e e o he decision make in (3), ollowing a simila dis inc ion
made by O’Donoghue and Rabin (1999) in he con ex o ime-inconsis en p e e ences.
Showing ha P oposi ion 3.5 ca ies o e o he nai e decision make , i.e., ha he e
exis s a unique nai e an icipa o y u ili y unc ion o each σ∈ S, hence o h deno ed by
ˆ
Uσ
Z, can be done analogously o he p e ious analysis and is omi ed.
Example 5.4. Recall Example 3.4 wi h s a egy ρde ined by ρ(λ) = (l, l) o all λ∈
[0,1]. The an icipa o y u ili y unc ion o a nai e Gilboa-Schmeidle decision is he unique
16
solu ion o
ˆ
Uρ
Z(λ) = (1 −p)u(λ) + pmin
z∈Znq0
z(λ)ˆ
Uρ
Zg−1
z(λ)+q1
z(λ)ˆ
Uρ
Z(gz(λ))o
o all λ∈[0,1]. 
Since Bm
{z},σm(λ)(λ) is a single on se o all z∈Zand all σ∈ S, he nai e Gilboa-
Schmeidle decision make in (9) p esumes only one exp ession ake i s minimum, while
hei sophis ica ed coun e pa in (3) expec s he minimum o e all possible signals
wice—she is being pessimis ic “once mo e”. Hence, i is no su p ising ha he an-
icipa o y u ili y o a nai e decision make is highe .24
Lemma 5.5. Le H:F ×P → Rbe he Gilboa-Schmeidle agg ega o in (2). Fo any
compac se Z⊆1
2,1and any σ∈ S, i holds ha Uσ
Z(λ)≤ˆ
Uσ
Z(λ) o all λ∈[0,1].
Fo he nai e decision make , he op imal u ili y a p io λis
ˆ
U∗
Z(λ) = sup
σ∈S
ˆ
Uσ
Z(λ).(10)
We show ha in e ms o op imal s a egies and op imal an icipa o y u ili ies he e is no
di e ence be ween a nai e and a sophis ica ed decision make .
Theo em 5.6. Le Z⊆1
2,1be compac and and le H:F × P → Rbe he Gilboa-
Schmeidle agg ega o in (2). Le z= min Z. Then ˆ
U∗
Z=U∗
Z, and ˆ
U∗
Z=ˆ
Uσ
Zi and only
i σsa is ies (8) o z=z.
While Example 5.3 demons a es ha nai e and sophis ica ed Gilboa-Schmeidle deci-
sion make s migh e alua e some si ua ions di e en ly, Theo em 5.6 shows ha hese
di e ences ha e no e ec on he agen s’ beha io o he an icipa o y u ili y unc ion. In
pa icula , ˆ
U∗
Z=U∗
Z=U∗
{z}=ˆ
U∗
{z}, so ha no ma e whe he she is sophis ica ed o
nai e, a Gilboa-Schmeidle decision make will always ac and eel as i she knew ha z
we e he only possible signal.
24We a e no he i s o highligh his con lic be ween being happy and o e hinking unce ain y. I
has been beau i ully illus a ed, o ins ance, in Woody Allen’s “Annie Hall”. Bu he e is also empi ical
e idence suppo ing he o he di ec ion: ha happy people appea less cle e (c . Ba asch, Le ine, and
Schwei ze , 2016).
17
6 Benchma k Compa ison
We ha e de i ed he op imal beha io o a decision make who can s a egically igno e
in o ma ion. We shall call such a decision make s a egic o dis inguish he om a
Bayesian decision make , who mechanically upda es his belie a e e e y signal. In his
sec ion, we shall con as he wo wi h espec o hei an icipa o y u ili y as well as hei
success p obabili ies, i.e., he p obabili ies wi h which hey make he co ec choice a e
he s eam o messages s ops.
6.1 Igno ance is a Bliss
Recall ha a Bayesian decision make uses s a egy ρwi h ρ(λ) = (l, l) o all λ∈
[0,1]. He will lea n za e e e y message and upda e his belie acco dingly. Bo h o
he sophis ica ed and he nai e decision make he e a e, by P oposi ion 3.5, unique
an icipa o y u ili y unc ions Uρ
Zand ˆ
Uρ
Z, espec i ely. Mo eo e , U∗
Z(λ)≥Uρ
Z(λ) and
ˆ
U∗
Z(λ)≥ˆ
Uρ
Z(λ) o all λ∈[0,1] by de ini ion. We show ha hese inequali ies a e s ic
o all λ∈(0,1).
Theo em 6.1. Le Z⊆1
2,1be compac . Then U∗
Z(λ)> Uρ
Z(λ)and ˆ
U∗
Z(λ)>ˆ
Uρ
Z(λ)
o all λ∈(0,1).
Fo any non-degene a e belie , he an icipa o y u ili y o a s a egic decision make is
highe han ha o a Bayesian decision make . Tha is, s a egic igno ance makes people
happie , as long as he s eam o signals con inues.
6.2 Final Ou come
We shall u n o he p obabili ies wi h which he s a egic and he Bayesian decision
make choose co ec ly a he end o he message s eam. Th oughou he subsec ion, we
assume ha he agen s’ p io belie λis co ec and ha hey a e acing a known signal
z∈1
2,1. The ex-an e p obabili y ha a decision make wi h ini ial belie λand s a egy
σmakes he co ec choice is deno ed by σ
z(λ).25
25A his poin , we a e a bi inaccu a e as we do no o mally de ine he conce ning p obabili y space.
This space would ha e o con ain he co ec s a e o he wo ld ω, he numbe o signals T, and o each
T, he sequence o messages (m )T
=1. De ining such a space is no e y di icul ; howe e , i equi es some
cumbe some no a ion and does no p o ide addi ional insigh .
18
As he op imal s a egy o he s a egic decision make is no unique, we will ocus
on σ∗∗
z, which is de ined by
σ∗∗
z(λ) = 








(l, i) o λ∈[0, λ∗(z)] ,
(l, l) o λ∈(λ∗(z),1−λ∗(z)) ,
(i, l) o λ∈[1 −λ∗(z),1] .
(11)
By Theo em 4.1, his s a egy is op imal. Obse e ha i T≥1, i.e., i he e is a leas
one message, hen λ1∈I(z), so ha o ≥2 he s a egic decision make will no lea n
upon a con adic ing message. Thus, he agen exhibi s an ex eme p imacy e ec and
he inal choice is de e mined a la es a e he i s message. This allows us o easily
de i e he p obabili y ha he choice is co ec .
P oposi ion 6.2. Fo ini ial belie λ∈[0,1], he p obabili y o a co ec decision unde
σ∗∗
zis
σ∗∗
z
z(λ) = 


u(λ)i λ∈I(z),
pz + (1 −p)u(λ)o he wise.
An immedia e consequence o his p oposi ion is ha he ex-an e p obabili y o a co ec
choice is inc easing in z. While i does no change o “ex eme” ini ial belie s, i s ic ly
inc eases o “mode a e” belie s.
Co olla y 6.3. Le 1≥z > z0≥1
2. Then σ∗∗
z
z(λ) = σ∗∗
z0
z0(λ) o all λ∈I(z)and
σ∗∗
z
z(λ)> σ∗∗
z0
z0(λ) o all λ∈(λ∗(z),1−λ∗(z)).
Finding he ex-an e p obabili y ha a Bayesian decision make chooses co ec ly is mo e
complex, as i does no solely depend on he i s message. I s p ecise o mula o each
λ∈(0,1) is de i ed wi hin he p oo o he nex heo em.
Theo em 6.4. Fo any λ∈(0,1) and z∈1
2,1i holds ha ρ
z(λ)> σ∗∗
z
z(λ).
Theo em 6.4 shows ha s a egic news selec ion lowe s he decision make ’s long- un wel-
a e as i dec eases he p obabili y o choose co ec ly in he end. This con as s wi h
Theo em 6.1 which s a es ha as long as he s eam o signals con inues, he s a egic
19

decision make is happie han he Bayesian decision make . Thus, while s a egic igno-
ance inc eases ins an aneous well-being, i is de imen al in he long un i he ini ial
belie is co ec .
7 Social Ou come
Conside a popula ion No decision make s wi h ini ial belie s (λi)i∈Nwho ace a s eam
o signals d awn om some compac se Z⊆1
2,1and who selec hei in o ma ion
acco ding o σ∗∗
zas in (11). By Theo ems 4.1 and 5.6 i holds ha λi
∈I(z) o all
≥1 and all i∈N. Thus, al eady a e he i s signal, pola iza ion is ine i able and
ex eme: The e will only be “ex emis s”, who ne e look a messages con adic ing hei
wo ld iew. The “poli ical cen e ” (λ∗(z),1−λ∗(z)) is emp y and will s ay so o e e .
Pola iza ion e en inc eases o e ime. A e he i s message, all belie s will be in
[0, λ∗(z)] (“le -ex emis s”) o in [1 −λ∗(z),1] (“ igh -ex emis s”). Subsequen ly, he
belie s o all le -ex emis s will ( o z6= 1) mono onically dec ease and app oach 0, while
he belie s o all igh -wing ex emis s will mono onically inc ease owa ds 1. In pa icula ,
o la ge T, all belie s will be close o 0 o 1.
The se o decision make s who choose he co ec ou come depends on he ini ial
dis ibu ion o p io s and Z. Co olla ies 4.3 and 6.3 imply ha his se inc eases as z
inc eases. Hence, mo e people will make a co ec choice i he leas p ecise in o ma ion
sou ce becomes mo e p ecise.
8 Ex ensions
Two ex ensions o ou model seem s aigh o wa d: using agg ega o s o he han Gilboa-
Schmeide , and using a non-bina y s a e space. Nei he o hem is i ial and gene al
esul s abou he s uc u e o op imal s a egies emain unclea .
8.1 Agg ega ion o Second O de Belie s
I he decision make knows he dis ibu ion o signals, she can exploi his addi ional
in o ma ion. Modi ying De ini ion 3.2, one can hen de ine an agg ega o as a map
H:F(D)×∆(D)→R ha sa is ies:
20
Consis ency. Fo all x∈Dand he poin measu e δxon xi holds ha H(U, δx) = U(x).
Mono onici y. Fo all φ∈∆(D) and U, V :D→Rwi h U|supp(φ)≥V|supp(φ)i holds
ha H(U, φ)≥H(V, φ). I U|supp(φ)≥V|supp(φ)+ε o some ε > 0 i holds ha
H(U, φ)> H(V, φ).
Fo ins ance, an ambigui y neu al decision make wi h smoo h p e e ences in he sense
o Klibano , Ma inacci, and Muke ji (2005) wi h expec ed u ili y unc ion Uand second
o de belie F∈∆(D) uses he agg ega o
H[U, F] = EF[U].(12)
Le F∈∆1
2,1be a p obabili y measu e o e he se o signals and le Fm
λdeno e he
p obabili y measu e on Zcondi ional on he p io belie being λ∈(0,1) and he message
being m. Tha is, o any measu able subse Z0⊆Z,
Fm
λ(Z0) = RZ0qm
z(λ)F(dz)
RZqm
z(λ)F(dz).
Mo eo e , de ine ou maps
ψλ,l :Z→[0,1], z 7→ g−1
z(λ), ψλ,i :Z→[0,1], z 7→ λ,
ϕλ,l :Z→[0,1], z 7→ gz(λ), ϕλ,i :Z→[0,1], z 7→ λ
and push- o wa d measu es G0
F,a(λ):=F0
λ◦ψ−1
λ,a and G1
F,a(λ):=F1
λ◦ϕ−1
λ,a. Then G0
F,l(λ)
is he p obabili y measu e o e pos e io belie s gi en p io λand obse ed message 0
be o e he gene a ed signal is lea ned. The measu e G1
F,l(λ) is de ined acco dingly o
obse ed message 1. The an icipa o y u ili y unc ion o a sophis ica ed decision make
wi h agg ega o His hen de ined by
Uσ
F(λ) = (1 −p)u(λ) + pH q0
·(λ)HUσ
F, G0
F,σ0(λ)(λ)+q1
·(λ)HUσ
F, G1
F,σ1(λ)(λ), F
and he an icipa o y u ili y unc ion o a nai e decision make is de ined by
ˆ
Uσ
F(λ) = (1 −p)u(λ) + pH hq0
·(λ)Hhˆ
Uσ
F, G0
δ·,σ0(λ)(λ)i+q1
·(λ)Hhˆ
Uσ
F, G1
δ·,σ1(λ)(λ)i, Fi.
I is easy o see ha P oposi ion 3.5 ca ies o e and ha he agg ega o in (12) sa is ies
21
i s p emise. Hence, bo h he sophis ica ed and he nai e ambigui y neu al decision make
wi h smoo h p e e ences ha e a unique an icipa o y u ili y unc ion which a e ixed poin s
o he unc ionals Tσ
F,ˆ
Tσ
F:F([0,1]) → F ([0,1]), de ined by
Tσ
FU(λ) = (1 −p)u(λ) + pH q0
·(λ)HU, G0
F,σ0(λ)(λ)+q1
·(λ)HU, G1
F,σ1(λ)(λ), F,
ˆ
Tσ
FU(λ) = (1 −p)u(λ) + pH q0
·(λ)HU, G0
δ·,σ0(λ)(λ)+q1
·(λ)HU, G1
δ·,σ1(λ)(λ), F.
We show ha hese unc ionals coincide. This con as s wi h Example 5.3 which shows
ha such a s a emen ails o hold in he case o Gilboa-Schmeidle p e e ences.
Lemma 8.1. Le Hbe de ined as in (12). Then Tσ
FU(λ) = ˆ
Tσ
FU(λ) o any F∈
∆1
2,1,σ∈ S,U∈ F, and λ∈[0,1].
The p e ious lemma migh no be en i ely su p ising as he sophis ica ed decision make
essen ially akes an expec ed alue o a condi ional expec ed alue, while he nai e decision
make ake he expec ed alue only once. In con as , i migh no be ue o a decision
make who is no ambigui y neu al. An immedia e consequence is ha he an icipa o y
u ili y unc ions coincide o he sophis ica ed and he nai e decision make .
Co olla y 8.2. Le Hbe de ined as in (12). Fo all F∈∆1
2,1and all σ∈ S, i holds
ha Uσ
F=ˆ
Uσ
F.
Hence, in case o ambigui y neu al p e e ences, we do no ha e o dis inguish be ween
sophis ica ed and nai e decision make s. The exis ence o op imal s a egies can be de-
i ed as in P oposi ion 3.6, and hey a e iden ical o bo h ypes o decision make s, so
ha U∗
F(λ) = supσ∈S Uσ
F(λ) = supσ∈S ˆ
Uσ
F(λ) = ˆ
U∗
F(λ) o all λ. Wha emains unclea ,
howe e , is whe he op imal s a egies ha e a simila h eshold s uc u e as hose o he
Gilboa-Schmeidle decision make . In he ollowing example, a nume ical analysis sugges s
ha hey migh .
Example 8.3. Recall Example 5.2 whe e Z=8
10,9
10and le Fbe he uni o m dis ibu-
ion on Z. Suppose he decision make has smoo h p e e ences and is ambigui y neu al.
The an icipa o y u ili y unc ion U∗
Fis depic ed by he solid line in Figu e 3. The unc-
ion λ7→ EF1
λ[U∗
Z(g·(λ))] is depic ed by he do ed line. The wo unc ions ha e a unique
in e sec ion, deno ed λ∗(F). The op imal s a egies equi e ha he agen igno es any
1-message o λ<λ∗(F) and upda es a e any 1-message o λ>λ∗(F). Nume ically,
we ob ain ha λ∗(F)≈0.2823 ∈1
4,1
3=λ∗9
10, λ∗8
10.
22
0z
0.5

0.5
1
⇤(z)
Figu e 2: The g aph o z7! ⇤(z)
U⇤
z(gz()) >U
⇤
zg1
z() o 21
2,1.473
474
Theo em 4.1 p o es ha e en wi hou ambigui y he e will be belie s a which a decision475
make will igno e con adic ing messages. This egion, deno ed by I(z)=[0,⇤(z)] [476
[1 ⇤(z),1] inc eases as zdec eases.477
Co olla y 4.3. Le z,z021
2,1⇤. Then I(z)✓I(z0)i and only i zz0.478
P oo . I is su icien o show ha ⇤(z)⇤(z0)i andonlyi zz0. Bu his ollows479
om he obse a ion ha ⇤(z)=z1+p(1z)z
2z1, which is s ic ly dec easing on 1
2,1⇤.⌅480
The ela ion be ween zand ⇤(z) is depic ed in Figu e 2. No e ha ⇤(z)>1z o all481
z21
2,1,482
5 In o ma ion Selec ion unde Ambigui y483
In his sec ion we shall e u n o he decision p oblem when zis no ixed, bu s ems om484
some compac se Z✓1
2,1⇤.28 We i s de i e he op imal s a egy o a decision make 485
wi h Gilboa-Schmeide p e e ences. A e wa ds we conside an al e na i e app oach in486
28We exclude 1
22Z o a oid some edious case dis inc ions.
19
0z
0.5

0.5
1
⇤(z)
Figu e 2: The g aph o z7! ⇤(z)
U⇤
z(gz()) >U
⇤
zg1
z() o 21
2,1.473
474
Theo em 4.1 p o es ha e en wi hou ambigui y he e will be belie s a which a decision475
make will igno e con adic ing messages. This egion, deno ed by I(z)=[0,⇤(z)] [476
[1 ⇤(z),1] inc eases as zdec eases.477
Co olla y 4.3. Le z,z021
2,1⇤. Then I(z)✓I(z0)i and only i zz0.478
P oo . I is su icien o show ha ⇤(z)⇤(z0)i andonlyi zz0. Bu his ollows479
om he obse a ion ha ⇤(z)=z1+p(1z)z
2z1, which is s ic ly dec easing on 1
2,1⇤.⌅480
The ela ion be ween zand ⇤(z) is depic ed in Figu e 2. No e ha ⇤(z)>1z o all481
z21
2,1,482
5 In o ma ion Selec ion unde Ambigui y483
In his sec ion we shall e u n o he decision p oblem when zis no ixed, bu s ems om484
some compac se Z✓1
2,1⇤.28 We i s de i e he op imal s a egy o a decision make 485
wi h Gilboa-Schmeide p e e ences. A e wa ds we conside an al e na i e app oach in486
28We exclude 1
22Z o a oid some edious case dis inc ions.
19
0z
0.5

0.5
1
⇤(z)
Figu e 2: The g aph o z7! ⇤(z)
U⇤
z(gz()) >U
⇤
zg1
z() o 21
2,1.473
474
Theo em 4.1 p o es ha e en wi hou ambigui y he e will be belie s a which a decision475
make will igno e con adic ing messages. This egion, deno ed by I(z)=[0,⇤(z)] [476
[1 ⇤(z),1] inc eases as zdec eases.477
Co olla y 4.3. Le z,z021
2,1⇤. Then I(z)✓I(z0)i and only i zz0.478
P oo . I is su icien o show ha ⇤(z)⇤(z0)i andonlyi zz0. Bu his ollows479
om he obse a ion ha ⇤(z)=z1+p(1z)z
2z1, which is s ic ly dec easing on 1
2,1⇤.⌅480
The ela ion be ween zand ⇤(z) is depic ed in Figu e 2. No e ha ⇤(z)>1z o all481
z21
2,1,482
5 In o ma ion Selec ion unde Ambigui y483
In his sec ion we shall e u n o he decision p oblem when zis no ixed, bu s ems om484
some compac se Z✓1
2,1⇤.28 We i s de i e he op imal s a egy o a decision make 485
wi h Gilboa-Schmeide p e e ences. A e wa ds we conside an al e na i e app oach in486
28We exclude 1
22Z o a oid some edious case dis inc ions.
19
Thus,652
ˆ
U⇤
Z()= ˆ
TZ,⇤ˆ
U⇤
Z()>ˆ
TZ,⌧ˆ
U⌧
Z()= ˆ
U⌧
Z()653
654
as equi ed. ⌅655
Fo any non-degene a e belie , he an icipa o y u ili y o a s a egic decision make is656
highe han ha o an agen who always mechanically upda es he belie . Tha is, s a egic657
igno ance makes people happie , as long as he s eam o signals con inues.658
6.2 Final Ou come659
We ha e seen hus a ha un il he end o he signal s eam he s a egic decision make 660
is happie han he Bayesian decision make . We shall now look a wha happens when661
he e a e no new signals and a choice c2{0,1}has o be aken. The ques ion we shall662
answe is: who will make he co ec choice wi h highe p obabili y?663
Th oughou he subsec ion, we will assume ha he agen s a e acing a known signal664
z21
2,1, i.e., we a e in he si ua ion o Sec ion 4. Mo eo e , he ini ial belie is co ec ,665
ha is, Na u e d aws he ue s a e o he wo ld acco ding o . The decision make 666
hen ecei es a s eam o messages m , 2{1,...,T}whose leng h Tis geome ically667
dis ibu ed wi h pa ame e p.668
Conside he s a egy ⇤⇤
zde ined by669
⇤⇤
z()=8
>
>
>
<
>
>
>
:
(l,i) o 2[0,⇤(z)] ,
(l,l) o 2(⇤(z),1⇤(z)) ,
(i, l) o 2[1 ⇤(z),1] .
(21)670
671
By Theo em 4.1, his s a egy is op imal. Obse e ha i T1, i.e., i he e is a leas 672
one message, hen 12I(z), so ha he s a egic decision make will ne e lea n upon673
any con adic ing message. Thus, he agen exhibi s an ex eme p imacy e↵ec and will674
ha e made he choice a la es a e he i s message. This allows us o easily de i e he675
p obabili y ha he choice is co ec .676
P oposi ion 6.2. Fo ini ial belie 2[0,1], he p obabili y o a co ec decision unde 677
26
Thus,652
ˆ
U⇤
Z()= ˆ
TZ,⇤ˆ
U⇤
Z()>ˆ
TZ,⌧ˆ
U⌧
Z()= ˆ
U⌧
Z()653
654
as equi ed. ⌅655
Fo any non-degene a e belie , he an icipa o y u ili y o a s a egic decision make is656
highe han ha o an agen who always mechanically upda es he belie . Tha is, s a egic657
igno ance makes people happie , as long as he s eam o signals con inues.658
6.2 Final Ou come659
We ha e seen hus a ha un il he end o he signal s eam he s a egic decision make 660
is happie han he Bayesian decision make . We shall now look a wha happens when661
he e a e no new signals and a choice c2{0,1}has o be aken. The ques ion we shall662
answe is: who will make he co ec choice wi h highe p obabili y?663
Th oughou he subsec ion, we will assume ha he agen s a e acing a known signal664
z21
2,1, i.e., we a e in he si ua ion o Sec ion 4. Mo eo e , he ini ial belie is co ec ,665
ha is, Na u e d aws he ue s a e o he wo ld acco ding o . The decision make 666
hen ecei es a s eam o messages m , 2{1,...,T}whose leng h Tis geome ically667
dis ibu ed wi h pa ame e p.668
Conside he s a egy ⇤⇤
zde ined by669
⇤⇤
z()=8
>
>
>
<
>
>
>
:
(l,i) o 2[0,⇤(z)] ,
(l,l) o 2(⇤(z),1⇤(z)) ,
(i, l) o 2[1 ⇤(z),1] .
(21)670
671
By Theo em 4.1, his s a egy is op imal. Obse e ha i T1, i.e., i he e is a leas 672
one message, hen 12I(z), so ha he s a egic decision make will ne e lea n upon673
any con adic ing message. Thus, he agen exhibi s an ex eme p imacy e↵ec and will674
ha e made he choice a la es a e he i s message. This allows us o easily de i e he675
p obabili y ha he choice is co ec .676
P oposi ion 6.2. Fo ini ial belie 2[0,1], he p obabili y o a co ec decision unde 677
26
Thus,652
ˆ
U⇤
Z()= ˆ
TZ,⇤ˆ
U⇤
Z()>ˆ
TZ,⌧ˆ
U⌧
Z()= ˆ
U⌧
Z()653
654
as equi ed. ⌅655
Fo any non-degene a e belie , he an icipa o y u ili y o a s a egic decision make is656
highe han ha o an agen who always mechanically upda es he belie . Tha is, s a egic657
igno ance makes people happie , as long as he s eam o signals con inues.658
6.2 Final Ou come659
We ha e seen hus a ha un il he end o he signal s eam he s a egic decision make 660
is happie han he Bayesian decision make . We shall now look a wha happens when661
he e a e no new signals and a choice c2{0,1}has o be aken. The ques ion we shall662
answe is: who will make he co ec choice wi h highe p obabili y?663
Th oughou he subsec ion, we will assume ha he agen s a e acing a known signal664
z21
2,1, i.e., we a e in he si ua ion o Sec ion 4. Mo eo e , he ini ial belie is co ec ,665
ha is, Na u e d aws he ue s a e o he wo ld acco ding o . The decision make 666
hen ecei es a s eam o messages m , 2{1,...,T}whose leng h Tis geome ically667
dis ibu ed wi h pa ame e p.668
Conside he s a egy ⇤⇤
zde ined by669
⇤⇤
z()=8
>
>
>
<
>
>
>
:
(l,i) o 2[0,⇤(z)] ,
(l,l) o 2(⇤(z),1⇤(z)) ,
(i, l) o 2[1 ⇤(z),1] .
(21)670
671
By Theo em 4.1, his s a egy is op imal. Obse e ha i T1, i.e., i he e is a leas 672
one message, hen 12I(z), so ha he s a egic decision make will ne e lea n upon673
any con adic ing message. Thus, he agen exhibi s an ex eme p imacy e↵ec and will674
ha e made he choice a la es a e he i s message. This allows us o easily de i e he675
p obabili y ha he choice is co ec .676
P oposi ion 6.2. Fo ini ial belie 2[0,1], he p obabili y o a co ec decision unde 677
26
make wi h smoo h p e e ences ha e a unique an icipa o y u ili y unc ion ha is achie ed
as a ixed poin o he unc ionals T
F,ˆ
T
F:F([0,1]) !F([0,1]), de ined by
T
FU()=(1p)u()+pH ⇥q0
·()H⇥U, G0
F,0()()⇤+q1
·()H⇥U, G1
F,1()()⇤,F⇤,
ˆ
T
FU()=(1p)u()+pH ⇥q0
·()H⇥U, G0
·,0()()⇤+q1
·()H⇥U, G1
·,1()()⇤,F⇤.
We show ha hese unc ionals coincide. This con as s wi h Example 5.3 which shows
ha such a s a emen ails o hold in he case o Gilboa-Schmeidle p e e ences.
Lemma 8.1. Then o any U2F([0,1]) and 2[0,1],T
FU()= ˆ
T
FU().
I di ec ly ollows om Lemma 8.1 ha in case o ambigui y neu al smoo h p e e ences,
he an icipa o y u ili y unc ions coincide o he sophis ica ed and he nai e decision
make .
Co olla y 8.2. Fo each 2S,U
F=ˆ
U
F.
Hence, in case o ambigui y neu al p e e ences, we do no ha e o dis inguish be ween
sophis ica ed and nai e decision make s. Thus, he op imal an icipa o y u ili ies o
he nai e and he sophis ica ed decision make coincide, i.e., U⇤
F()=sup
2SU
F()=
sup2Sˆ
U
F()= ˆ
U⇤
F() o all. Analogously o P oposi ion 3.5, one can show ha
an op imal s a egy exis s. In gene al, i is no clea whe he op imal s a egies ha e a
h eshold s uc u e. In he ollowing example, a nume ical analysis sugges s ha hey do.
Example 8.3. Recall Example 5.2 whe e Z=8
10,9
10 and le Fbe he uni o m dis ibu-
ion on Z. Suppose he decision make has smoo h p e e ences and is ambigui y neu al.
The an icipa o y u ili y unc ion U⇤
Fis depic ed by he solid line in Figu e 3. The unc ion
EF[U⇤
Z(gz())] is depic ed by he do ed line. The wo unc ions ha e a unique in e sec-
ion, deno ed ⇤(F). The op imal s a egies equi e ha he agen igno es any 1-message
o <⇤(F) and upda es a e any 1-message o >⇤(F). Nume ically, we ob ain
ha ⇤(F)⇡0.2803 21
4,1
3=⇤9
10,⇤8
10.⇤
8.2 Mo e han Two S a es
The model can be gene alized o mo e han wo symme ic s a es, a message space ha
coincides wi h he s a e space, and a se o symme ic signals ha gene a e wi h p oba-
bili y z1
na co ec message and wi h p obabili y 1z
n1each o he message. While he
22
Figu e 3: An icipa o y u ili y unc ion o ambigui y neu al smoo h p e e ences.
8.2 Mo e han Two S a es
The model can be gene alized o mo e han wo symme ic s a es, a message space ha
coincides wi h he s a e space, and a se o symme ic signals ha gene a e wi h p oba-
bili y z≥1
na co ec message and wi h p obabili y 1−z
n−1each o he message. While he
exis ence o op imal s a egies can be p o en as be o e, i emains unclea whe he hey
will ha e he same h eshold s uc u e, as he p oo o Theo em 4.1, in pa icula pa
2b, does no gene alize o mo e han wo s a es.
9 Conclusion
This pape in es iga es he beha io o a decision make who ecei es a s eam o messages
o known o unknown quali y abou he ue s a e o he wo ld. A each ime pe iod, she
aces he decision whe he o lea n how he message has been gene a ed and upda e he
belie acco dingly, o o disca d he message and s ick o he p e ious con ic ion. This
“ ee disposal o in o ma ion” is inspi ed by he way use s selec i ely ead news a icles
pos ed on social media pla o ms. I he e is no ambigui y and unde he assump ion
ha he decision make is Ma ko ian, she will disca d con adic o y messages i he belie
is su icien ly s ong. This emains ue unde ambigui y i she has Gilboa-Schmeidle
p e e ences, whe e he espec i e belie h eshold depends solely on he wo s possible
signal.
When compa ing he beha io o a decision make who upda es a e each message,
23
Le U∈ C∗
z. We show ha TU sa is ies (18), so le 1 −λ∗≤λ < λ0≤1. Then,
by (18), i holds ha U(λ0)> U(λ) and, by he s ic mono onici y o gz, we ha e
U(gz(λ0)) > U (gz(λ)) as well as U(gz(λ0)) > U(λ0). Hence, we can apply Lemma A.1
wi h α=q1
z(λ0), β=q1
z(λ), x0=U(gz(λ0)), y0=U(λ0), x=U(gz(λ)), and y=U(λ)
and ge
q1
z(λ0)U(gz(λ0)) + q0
z(λ0)U(λ0)> q1
z(λ)U(gz(λ)) + q0
z(λ)U(λ)
whe e we also used 1 −q0
z(·) = q1
z(·). Toge he wi h he de ini ion o σwe ha e
(TU)(λ0)−(TU)(λ) = (1 −p)[u(λ0)−u(λ)]
+pq1
z(λ0)U(gz(λ0)) + q0
z(λ0)U(λ0)−q1
z(λ)U(gz(λ)) −q0
z(λ)U(λ)
>(1 −p)[u(λ0)−u(λ)]
= (1 −p)(λ0−λ)
as claimed.
We show ha Tp ese es (19). Fo all λ∈[0,1] de ine
(˜
TU)(λ):=








q0
z(λ)U(g−1
z(λ)) + q1
z(λ)U(λ), λ ∈[0, λ∗],
q0(λ)U(g−1
z(λ)) + q1
z(λ)U(gz(λ)) , λ ∈(λ∗,1−λ∗),
q0
z(λ)U(λ) + q1
z(λ)U(gz(λ)) , λ ∈[1 −λ∗,1] ,
and obse e ha (TU)(λ) = (1 −p)u(λ) + p(˜
TU)(λ). Thus,
(TU) (gz(λ)) −(TU)(λ) = (1 −p) [u(gz(λ)) −u(λ)] + ph(˜
TU) (gz(λ)) −(˜
TU)(λ)i
o all λ. Hence, in o de o p o e (19), i is su icien o show ha ( ˜
TU) (gz(λ)) ≥(˜
TU)(λ)
o all λ∈[λ∗,1).
1. Fi s , le λ∈[1 −λ∗,1). Then, since z > 1
2, i holds ha gz(λ)∈(1 −λ∗,1], so ha
(˜
TU) (gz(λ)) = q1
z(g(λ)) U(gz(gz(λ))) + q0
z(gz(λ)) U(gz(λ))
≥q1
z(λ)U(gz(λ)) + q0
z(λ)U(λ)
= ( ˜
TU) (λ)
30

whe e he inequali y ollows om (15), using ha U(gz(gz(λ))) ≥U(gz(λ)) ≥U(λ)
by he mono onici y o Uon [1 −λ∗,1], and q1
z(gz(λ)) ≥q1
z(λ) by he mono onici y
o q1
z.
2. Second, le λ∈(λ∗,1−λ∗). Then, g−1
z(λ)∈[0, λ∗) and gz(λ)∈(1 −λ∗,1]. As Uis
s ic ly mono one in hese a eas, U(gz(λ)) is inc easing, and U(g−1
z(λ)) is dec easing
in λ.
By Lemma 3.1, i holds ha gz1
2= 1 −g−1
z1
2, so ha by he symme y o U,
Ugz1
2=U1−g−1
z1
2=Ug−1
z1
2. Consequen ly,
U(gz(λ)) > U g−1
z(λ) o λ∈1
2,1−λ∗and
U(gz(λ)) < U g−1
z(λ) o λ∈λ∗,1
2.(20)
(a) Suppose i s ha λ∈1
2,1−λ∗. Then U(gz(gz(λ))) ≥U(gz(λ)) ≥U(g−1
z(λ))
and he e o e
(˜
TU) (gz(λ)) = q1
z(gz(λ)) U(gz(gz(λ))) + q0
z(gz(λ)) U(gz(λ))
≥U(gz(λ))
≥q1
z(λ)U(gz(λ)) + q0
z(λ)Ug−1
z(λ)
= ( ˜
TU) (λ).
(b) Finally, le λ∈λ∗,1
2. Since
U(gz(gz(λ∗))) = U(gz(1 −λ∗)) = U1−g−1
z(λ∗)=Ug−1
z(λ∗),
he mono onici y o Uon [0, λ∗) and (1 −λ∗,1] oge he wi h (20) imply ha
U(gz(gz(λ))) ≥Ug−1
z(λ)≥U(gz(λ)) .
Thus, using ha q1(gz(λ)) >0 and q1
z(λ) = q0
z(1 −λ)> q0
z(gz(λ)) o λ > λ∗,
we ob ain
(˜
TU) (gz(λ)) = q1
z(gz(λ)) U(gz(gz(λ))) + q0
z(gz(λ)) U(gz(λ))
=q1
z(gz(λ)) U(gz(gz(λ))) + q1
z(λ) + q0
z(gz(λ)) −q1
z(λ)U(gz(λ))
31
≥q1
z(gz(λ)) Ug−1
z(λ)+q1
z(λ)U(gz(λ))
+q0
z(gz(λ)) −q1
z(λ)Ug−1
z(λ)
=q1
z(λ)U(gz(λ)) + 1−q1
z(λ)Ug−1
z(λ)
= ( ˜
TU) (λ)
as equi ed. 
Le U∈ C∗
z. By (19) i holds ha
U(gz(λ)) > U(λ) o all λ∈(λ∗,1).(21)
Le λ∈(1 −λ∗,1). Then g−1
z(λ)∈(λ∗,1), so ha we ind wi h (21)
U(λ) = Ugzg−1
z(λ)> U g−1
z(λ) o all λ∈(1 −λ∗,1).(22)
Mo eo e , using he symme y o Uin (16), we ob ain by (19), o simila a gumen s as
abo e
Ug−1
z(λ)> U(λ) o all λ∈(0,1−λ∗),(23)
U(λ)> U (gz(λ)) o all λ∈(0, λ∗).(24)
Conside now he unc ional ope a o Sz:F([0,1]) → F ([0,1]) ha is de ined by
(SzU) (λ) = (1 −p)u(λ) + pq0
z(λ) max U(λ), U g−1
z(λ)+q1
z(λ) max (U(λ), U (gz(λ)))
o all U∈ F ([0,1]) and all λ∈[0,1]. One can easily check ha Szis a con ac ion on
F([0,1]), so Szhas a unique ixed poin , deno ed by ˜
U. Le ˜σbe a ixed bu a bi a y
s a egy ha sa is ies (8). By Equa ions (21)–(24) we ha e o all U∈ C∗
z ha SzU=
T˜σ
{z}U∈ C∗
z, which implies ˜
U∈ C∗
z. Mo eo e U∗
{z}=Tσ∗
{z}U∗
{z}=SzU∗
{z} o any σ∗ ha
sa is ies (6), so ha U∗
{z}=˜
U∈ C∗
z. This implies ha a s a egy σsa is ies (8) i and only
i i sa is ies (6), ha is, i and only i i is op imal. 
P oo o Co olla y 4.3.I is su icien o show ha λ∗(z)≤λ∗(z0) i and only i z≥z0.
Bu his ollows om he obse a ion ha λ∗(z) = z−1+√(1−z)z
2z−1, which is s ic ly dec easing
32
on 1
2,1.
P oo o Theo em 5.1.I Z={1}, he e is no hing o show. Hence, le Z6={1}, so ha
z < 1. Le σ∗be some op imal s a egy gi en Z, and le σbe some op imal s a egy gi en
{z}. I is su icien o show he ollowing ou (in-)equali ies, as hey imply he claim:
Uσ
{z}(λ) = U∗
{z}(λ)≥Uσ∗
{z}(λ)≥U∗
Z(λ)≥Uσ
Z(λ) = Uσ
{z}(λ).(25)
The i s equali y ollows om Theo em 4.1 and he i s inequali y by he de ini ion o
U∗
{z}. Fo he second inequali y no e ha H[U, Z]≤H[U, {z}] o e e y U∈ F ([0,1])
by (2). Hence, Tσ∗
ZU(λ)≤Tσ∗
{z}U(λ) o all U∈ F ([0,1]) by he mono onici y o H,
so ha Tσ∗
ZnU(λ)≤Tσ∗
{z}nU(λ) o all n∈N. Thus, U∗
Z(λ) = limnTσ∗
ZnU(λ)≤
limnTσ∗
{z}nU(λ) = Uσ∗
{z}(λ). The hi d inequali y is sa is ied by he de ini ion o U∗
Zin
(5).
Fo he las equali y i is su icien o show ha Uσ
{z}is a ixed poin o Tσ
Z. So, wi hou
loss o gene ali y le λ≤1
2, and ecall ha Uσ
{z}∈ C∗
z. By Theo em 4.1,σis op imal o
Uσ
{z}in he sense o Equa ion (6). Thus,
HUσ
{z}, B0
Z,σ0(λ)(λ)=Uσ
{z}g−1
z(λ)and
HUσ
{z}, B1
Z,σ1(λ)(λ)= max Uσ
{z}(gz(λ)) , Uσ
{z}(λ)
by he mono onici y p ope ies o Uσ
{z}in Co olla y 4.2. Thus, again by Co olla y 4.2,
HUσ
{z}, B0
Z,σ0(λ)=Uσ
{z}g−1
z(λ)≥max Uσ
{z}(gz(λ)) , Uσ
{z}(λ)=HUσ
{z}, B1
Z,σ1(λ).
As q0
·(λ) is weakly inc easing in z( ecall ha λ≤1
2) his means ha
Hq0
·(λ)HUσ
{z}, B0
Z,σ0(λ)(λ)+q1
·(λ)HUσ
{z}, B1
Z,σ1(λ)(λ), Z
=q0
z(λ)HUσ
{z}, B0
Z,σ0(λ)(λ)+q1
z(λ)HUσ
{z}, B1
Z,σ1(λ)(λ)
=Hq0
·(λ)HUσ
{z}, B0
Z,σ0(λ)(λ)+q1
·(λ)HUσ
{z}, B1
Z,σ1(λ)(λ),{z}.
Hence, i holds ha
Tσ
ZUσ
{z}(λ) = Tσ
{z}Uσ
{z}(λ) = Uσ
{z}(λ).
33
Thus, Uσ
{z}is a ixed poin o Tσ
Zas claimed. 
P oo o Lemma 5.5.Le Tσ
Zbe as de ined in (13). Analogously, le ˆ
Tσ
Z:F([0,1]) →
F([0,1]) be de ined by
ˆ
Tσ
ZU(λ) = (1 −p)u(λ) + pH q0
·(λ)HU, B0
{·},σ0(λ)(λ)+q1
·(λ)HU, B1
{·},σ1(λ)(λ), Z
o all λ∈[0,1]. Fo all z∈Zand any unc ion U∈ F ([0,1]),
HU, B0
Z,σ0(λ)(λ)≤HU, B0
{z},σ0(λ)(λ)and HU, B1
Z,σ1(λ)(λ)≤HU, B1
{z},σ1(λ)(λ).
By he mono onici y o H,Tσ
ZU(λ)≤ˆ
Tσ
ZU(λ) o any unc ion Uand all λ∈[0,1]. Thus,
as in he p oo o Theo em 5.1,Uσ
Z(λ) = limnTσ
ZnU(λ)≤limnˆ
Tσ
ZnU(λ) = ˆ
Uσ
Z(λ). 
P oo o Theo em 5.6.I Z={1}, he e is no hing le o show, so le Z6={1}, so ha
z < 1. Suppose i s ha σsa is ies (8). In o de o show ha U∗
Z=ˆ
U∗
Z=ˆ
Uσ
Zi is
su icien o show ha
ˆ
U∗
Z(λ)≤ˆ
Uσ
{z}(λ) = Uσ
{z}(λ) = Uσ
Z(λ)≤ˆ
Uσ
Z(λ)≤ˆ
U∗
Z(λ) (26)
o all λ∈[0,1]. As z∈Z,His he Gilboa-Schmeidle agg ega o , and σis op imal
gi en {z}, one can use he same a gumen s as in he p oo s o Theo em 5.1 and Lemma
5.5 o show ha ˆ
Uσ0
Z(λ)≤ˆ
Uσ0
{z}(λ)≤ˆ
Uσ
{z}(λ) o all σ0∈ S. Taking he sup emum o e
all σ0∈ S yields he i s inequali y.
Fo he single on se {z}, he e is no di e ence be ween sophis ica ed and nai e deci-
sion make s, as (3) and (9) coincide in his case. Hence, we a i e a he i s equali y.
The second equali y holds by Theo em 5.1, and he second inequali y by Lemma 5.5.
The las inequali y di ec ly ollows om he de ini ion o ˆ
U∗
Zin (10).
I is le o show ha ˆ
U∗
Z>ˆ
Uσ0
Z o any σ0∈ S ha does no sa is y (8). Assume ha
he e is σ0∈ S wi h ˆ
U∗
Z=ˆ
Uσ0
Z ha does no sa is y (8). Then ˆ
Uσ0
Z(λ) = ˆ
U∗
Z(λ) = U∗
{z}(λ)>
Uσ0
{z}(λ) = ˆ
Uσ0
{z}(λ)≥ˆ
Uσ0
Z(λ), whe e he i s equali y ollows om he assump ion, he
second one om (26), he s ic inequali y om Theo em 4.1, he hi d equali y holds as
o he single on {z}Equa ions (3) and (9) coincide, and he weak inequali y ollows om
he same a gumen s as abo e. Bu he o e all inequali y is impossible. 
34
P oo o Theo em 6.1.As Uρ
Z(λ)≤ˆ
Uρ
Z(λ) by Lemma 5.5 and U∗
Z(λ) = ˆ
U∗
Z(λ) by Theo-
em 5.6, i su ices o show ha ˆ
Uρ
Z(λ)<ˆ
U∗
Z(λ). So, le σ∗be an op imal s a egy gi en
Z.
We i s p o e he claim o λ∈(0, λ∗(z)) ∪(1 −λ∗(z),1). Wi hou loss o gene ali y
le λ∈(0, λ∗(z)). Then
Hhq0
·(λ)Hhˆ
U∗
Z, B0
{·},ρ0(λ)(λ)i+q1
·(λ)Hhˆ
U∗
Z, B1
{·},ρ1(λ)(λ)i, Zi
= min
z∈Zq0
z(λ)ˆ
U∗
Zg−1
z(λ)+q1
z(λ)ˆ
U∗
Z(gz(λ))
≤q0
z(λ)ˆ
U∗
Zg−1
z(λ)+q1
z(λ)ˆ
U∗
Z(gz(λ))
< q0
z(λ)ˆ
U∗
Zg−1
z(λ)+q1
z(λ)ˆ
U∗
Z(λ)
=Hhq0
·(λ)Hhˆ
U∗
Z, B0
{·},σ∗
0(λ)(λ)i+q1
·(λ)Hhˆ
U∗
Z, B1
{·},σ∗
1(λ)(λ)i, Zi.
Thus,
ˆ
Uρ
Z(λ) = ˆ
Tρ
Zˆ
Uρ
Z(λ)≤ˆ
Tρ
Zˆ
U∗
Z(λ)<ˆ
Tσ∗
Zˆ
U∗
Z(λ) = ˆ
U∗
Z(λ),
whe e we used he calcula ion abo e in he i s and he mono onici y o ˆ
Tρ
Zin he second
inequali y.
Suppose nex ha λ∈[λ∗(z),1−λ∗(z)]. Then g−1
z(λ)∈(0, λ∗(z)] and gz(λ)∈
[1 −λ∗(z),1) and a leas one o hem lies in he espec i e open in e al. By he de ini ion
o ˆ
U∗
Zi holds ha ˆ
U∗
Zg−1
z(λ)≥ˆ
Uρ
Zg−1
z(λ)and ˆ
U∗
Z(gz(λ)) ≥ˆ
Uρ
Z(gz(λ)), and, by he
i s pa o he p oo , a leas one o hese inequali ies is s ic . Hence,
Hhq0
·(λ)Hhˆ
Uρ
Z, B0
{·},ρ0(λ)(λ)i+q1
·(λ)Hhˆ
Uρ
Z, B1
{·},ρ1(λ)(λ)i, Zi
= min
z∈Zq0
z(λ)ˆ
Uρ
Zg−1
z(λ)+q1
z(λ)ˆ
Uρ
Z(gz(λ))
≤q0
z(λ)ˆ
Uρ
Zg−1
z(λ)+q1
z(λ)ˆ
Uρ
Z(gz(λ))
< q0
z(λ)ˆ
U∗
Zg−1
z(λ)+q1
z(λ)ˆ
U∗
Z(gz(λ))
=Hhq0
·(λ)Hhˆ
U∗
Z, B0
{·},σ∗
0(λ)(λ)i+q1
·(λ)Hhˆ
U∗
Z, B1
{·},σ∗
1(λ)(λ)i, Zi.
Thus,
ˆ
Uρ
Z(λ) = ˆ
TZ,ρ ˆ
Uρ
Z(λ)<ˆ
Tσ∗
Zˆ
U∗
Z(λ) = ˆ
U∗
Z(λ)
35

as equi ed. 
P oo o P oposi ion 6.2.The i s pa is clea as a s a egic decision make wi h λ∈I(z)
will no upda e he belie a all and choose he op ion ha is op imal gi en he (co ec )
p io .
I λ /∈I(z), wi h p obabili y 1−pshe will ob ain no message a all and ollow he p io ,
and wi h p obabili y pshe will ob ain a message and upda e he belie . Since necessa ily
λ1∈I(z), she will make a co ec choice i and only i he message was co ec , which
happens wi h p obabili y zand is independen o p.
P oo o Co olla y 6.3.Fo λ∈I(z) and λ∈(λ∗(z0),1−λ∗(z0)) he claim ollows im-
media ely om P oposi ion 6.2. Wi hou loss o gene ali y le λ∈(λ∗(z), λ∗(z0)). Since
λ > λ∗(z)≥1−z, we ha e
σ∗∗
z
z(λ) = pz + (1 −p)(1 −λ)>1−λ= σ∗∗
z0
z0(λ)
as equi ed. 
P oo o Theo em 6.4.We assume wi hou loss o gene ali y ha λ≤1
2. Fo any `∈N
deno e by g`
z he `- old applica ion o gz, and by g−`
z he `- old applica ion o g−1
z. Fo
any his o y hT(ρ) = (m , z, λ )T
=0 le M1=|{ :m = 1}| deno e he numbe o gene a ed
1-messages, and M0=T−M1 he numbe o gene a ed 0-messages. Since gzand g−1
z
commu e, we ha e ha λT=gM1−M0
z(λ).
Le G`=g−`
z1
2, g−(`−1)
z1
2iand obse e ha a Bayesian decision make wi h p io
λ∈G`who aces his o y hT(ρ) will ha e inal belie λT>1
2i and only i M1−M0≥l,
and he will ha e inal belie λT=1
2i and only i λ=g−(`−1)
z1
2and M1−M0=l−1.
Fo any T∈N0and κ∈Zdeno e by Z[T, κ] he p obabili y ha a leas κco ec
messages ha e been obse ed, condi ional on Tmessages ha ing been gene a ed al o-
ge he , ha is,
W[T, κ] = 








0 i κ > T,
PT
k=κT
kzk(1 −z)T−ki 0 ≤κ≤T,
1 i κ < 0.
36
The p obabili y ha he numbe o co ec messages exceeds he numbe o inco ec
messages by a leas `, condi ional on Tmessages being gene a ed, is WT, T+`+1
2.
Simila ly, he p obabili y ha he numbe o inco ec messages does no exceed he
numbe o co ec messages by (s ic ly) mo e han `−1, condi ional on Tmessages
being gene a ed, is WT, T−`
2+ 1. Hence,
R`
z(λ) = λ∞
X
T=0
(1 −p)pTWT, T+`+1
2+ (1 −λ)∞
X
T=0
(1 −p)pTWT, T−`
2+ 1
deno es he ex-an e p obabili y ha ei he he s a e is 1 and he numbe o 1-messages
exceeds he numbe o 0-messages by a leas `o he s a e is 0 and he numbe o 1-
messages does no exceed he numbe o 0-messages by mo e han `−1. I λ∈in G`,
his is exac ly he ex-an e p obabili y ha he Bayesian decision make will make a
co ec choice. We he e o e de ine `(λ) o be he unique in ege wi h λ∈G`and de ine
Rz(λ) = R`(λ)
z(λ). I λ∈in G` o some `∈N, hen ρ
z(λ) = Rz(λ).
The map Rz:0,1
2→[0,1] is con inuous in he in e io o G` o all `∈N. We p o e
ha R`
zg−`
z1
2=R`+1
zg−`
z1
2 o all `∈Nas his implies ha Rzis con inuous
e e ywhe e. To his end, we i s show ha
g−`
z1
2z`=1−g−`
z1
2(1 −z)` o all `∈N0.(27)
Indeed, his is su ely ue o `= 0. Suppose (27) is ue o some `∈N0. Then
g−(`+1)
z1
2z`+1 =g−`
z1
2z`(1 −z)z
g−`
z1
2(1 −z) + 1−g−`
z1
2z
=1−g−`
z1
2(1 −z)`(1 −z)z
g−`
z1
2(1 −z) + 1−g−`
z1
2z
= 1−g−`
z1
2(1 −z)
g−`
z1
2(1 −z) + 1−g−`
z1
2z!(1 −z)`+1
=1−g−(`+1)
z1
2(1 −z)`+1,
37
as equi ed. Nex obse e ha
WT, T+`+2
2−WT, T+`+1
2=


0 i T−`is odd,
−T
T−`
2zT−`
2(1 −z)T+`
2i T−`is e en,
and
WT, T−`−1
2+ 1−WT, T−`
2+ 1=


0 i T−`is odd,
T
T+`
2zT+`
2(1 −z)T−`
2i T−`is e en.
Since T
T+`
2=T
T−`
2, his implies
R`+1
zg−`
z1
2−R`
zg−`
z1
2
=∞
X
T=0
(1 −p)pT
1
{T+`e en}T
T+`
2zT−`
2(1 −z)T−`
2·−g−`
z1
2z`+ (1 −g−`
z1
2)(1 −z)`
= 0
by (27). Hence, Rzis con inuous.
Recall ha ρ
z(λ) = Rz(λ) o all λ∈0,1
2wi h λ6=g−`
z1
2 o all `∈N0. We nex
show ha his is also ue i λ=g−`
z1
2 o some `∈N0. To his end no e ha i he s a e
is 1 and he decision make obse es exac ly `mo e co ec han inco ec messages, he
will choose ei he ac ion wi h equal p obabili y, and he same is ue i he s a e is 0 and
he obse es exac ly `mo e inco ec han co ec messages. Thus, he ex-an e p obabili y
o a co ec choice is
g−`
z1
2"∞
X
T=0
(1 −p)pTWT, T+`
2+ 1+1
2
1
{T+`e en}T
T+`
2zT+`
2(1 −z)T−`
2#
+1−g−`
z1
2"∞
X
T=0
(1 −p)pTWT, T−`
2+ 1+1
2
1
{T−`e en}T
T−`
2zT−`
2(1 −z)T+`
2#
=g−`
z1
2"∞
X
T=0
(1 −p)pTWT, T+`+1
2−1
2
1
{T+`e en}T
T+`
2zT+`
2(1 −z)T−`
2#
+1−g−`
z1
2"∞
X
T=0
(1 −p)pTWT, T−`
2+ 1+1
2
1
{T−`e en}T
T−`
2zT−`
2(1 −z)T+`
2#
=R`
zg−`
z1
2+1
2
∞
X
T=0
(1 −p)pT
1
{T+`e en}T
T+`
2zT−`
2(1 −z)T−`
2−g−`
z1
2z`+1−g−`
z1
2(1 −z)`
38
=R`
zg−`
z1
2
by (27). Hence, ρ
z(λ) = Rz(λ) o all λ∈0,1
2.
We nex show ha ρ
z(λ)> σ∗∗
z
z(λ) o λ > λ∗(z). Since λ>λ∗(z)>1−z=g−1
z1
2
i holds ha `(λ) = 1. De ine o all T∈N0
T
z(λ) = λW T, T+2
2+ (1 −λ)WT, T+1
2 (28)
and no e ha ρ
z(λ) = Rz(λ) = R1
z(λ) = P∞
T=0(1 −p)pT T
z(λ). I is s aigh o wa d
ha 0
z(λ) = 1 −λ, 1
z(λ) = z, and 2
z(λ) = 2(1 −λ)(1 −z)z+z2≥z. We show ha
T+1
z(λ)≥ T
z(λ)> z o all T≥3. To his end, obse e i s ha
T
Z(λ) = 


WT, T+1
2i Tis odd,
WT, T
2−λT
T
2zT
2(1 −z)T
2i Tis e en.
Nex , one inds ha o all 1 ≤κ≤T
(1 −z)W[T, κ] + zW [T, κ −1] =
T
X
k=κT
kzk(1 −z)T−k+1 +
T
X
k=κ−1T
kzk+1(1 −z)T−k
=
T
X
k=κT
kzk(1 −z)T−k+1 +
T+1
X
k=κT
k−1zk(1 −z)T−k+1
=
T
X
k=κT+ 1
kzk(1 −z)T−k+1 +T
TzT+1
=W[T+ 1, κ].
Suppose i s ha Tis e en. Then
T+1
z(λ) = WT+ 1,T+2
2
= (1 −z)WT, T+2
2+zW T, T
2
= (1 −z)WT, T
2−T
T
2zT
2(1 −z)T
2+zW T, T
2
=WT, T
2−(1 −z)T
T
2zT
2(1 −z)T
2
39