Galeazzi, Paolo; Madsen, Ma hias W.
A icle — Published Ve sion
Lea ning Decision C i e ia om Play
Dynamic Games and Applica ions
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Galeazzi, Paolo; Madsen, Ma hias W. (2024) : Lea ning Decision C i e ia om
Play, Dynamic Games and Applica ions, ISSN 2153-0793, Sp inge US, New Yo k, NY, Vol. 15, Iss. 3,
pp. 1037-1069,
h ps://doi.o g/10.1007/s13235-024-00595-2
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Dynamic Games and Applica ions (2025) 15:1037–1069
h ps://doi.o g/10.1007/s13235-024-00595-2
Lea ning Decision C i e ia om Play
Paolo Galeazzi1·Ma hias W. Madsen2
Accep ed: 5 Sep embe 2024 / Published online: 26 Sep embe 2024
© The Au ho (s) 2024
Abs ac
This pape in es iga es popula ion games unde ambigui y in which playe s may adop deci-
sion c i e ia di e en om one ano he . A e de ining equilib ia o hese si ua ions by
ex ending well-known decision- heo e ic c i e ia o he game- heo e ic con ex , we apply
hese concep s o examine he case o wo-pe son games played wi hin a popula ion whose
ela i e p opo ions o decision c i e ia a e unknown o he playe s. We s a e necessa y and
su icien condi ions unde which such games p omp he playe s o e eal hei decision
c i e ion h ough hei ac ions, and we show when he ela i e p opo ions may be lea ned
by obse ing he inc easingly in o med agen s play.
Keywo ds Decision c i e ia ·Lea ning ·Popula ion games
1 In oduc ion
In he las decades, he necessi y o ele a ing he analysis o he indi iduals’ beha io om
obse ed ac ions o unde lying mechanisms has eme ged in a ious ways in di e en ields,
om psychology o ecology o e olu iona y game heo y [1,9,14,16,17,19–21,23,31].
The gene al idea, as exp essed e.g. by [14], is ha
Na u al en i onmen s a e so complex, dynamic, and unp edic able ha na u al selec ion
canno possibly u nish an animal wi h an app op ia e, speci ic beha io pa e n o
e e y concei able si ua ion i migh encoun e . Ins ead, we should expec animals o
ha e e ol ed a se o psychological mechanisms which enable hem o pe o m well
on a e age ac oss a ange o di e en ci cums ances.
He e we ake his idea se iously and conside a game- heo e ic model whe e a popula ion
o agen s inhabi s an en i onmen consis ing o a mul i ude o di e en games (which we
call mul igame), and each indi idual in he popula ion is endowed wi h a “psychological
mechanism” ha p oduces a speci ic beha io o any possible game in he en i onmen . The
main esea ch ques ion ha we wan o in es iga e hen conce ns he possibili y o disce ning
he di e en unde lying mechanisms by obse ing he agen s’ exp essed beha io s only. Tha
BPaolo Galeazzi
[email p o ec ed]
1Uni e si y o Bay eu h, Bay eu h, Ge many
2Mic opsi indus ies, Be lin, Ge many
1038 Dynamic Games and Applica ions (2025) 15:1037–1069
is: Unde which condi ions is i possible o dis inguish he agen s’ unde lying mechanisms
gi en hei beha io ?
In his wo k, we p esen a case whe e he agen s’ beha io -gene a ing mechanisms a e
ep esen ed by di e en decision c i e ia and each o such c i e ia makes he agen ac in a
ce ain way when aced wi h a speci ic game in he en i onmen . The decision c i e ia ha
we conside in he ollowing a e a guably he wo main c i e ia o choice unde ambigui y
om he decision- heo e ic li e a u e, i.e., maxmin expec ed u ili y and eg e minimiza ion.
The ollowing example in oduces a simple ins ance o he popula ion model ha we s udy
in his pape . Conside a popula ion li ing in an en i onmen consis ing o he h ee games
below. The i s is a P isone ’s Dilemma (PD), he second is a S ag Hun (SH), and he hi d
is an an i-coo dina ion game (AG).
PD III
I2,2 0,3
II 3,0 1,1
SH III
I3,3 0,2
II 2,0 2,2
AG III
I1,1 2,5
II 5,2 0,0
Indi iduals om such popula ion andomly mee and play one o hese h ee possible
games. C ucially, howe e , each indi idual plays he game based on he own beha io -
gene a ing mechanism, namely, he own decision c i e ion—which we also e e o as he
indi idual’s ype. A maxmin playe by de ini ion chooses an ac ion ha gua an ees he highes
minimum payo . In he SH game, o ins ance, he minimum payo ha one can ge om
ac ion Iis 0, while he minimum payo om ac ion II is 2, and a maxmin playe would
he e o e choose ac ion II in he SH game. A eg e -minimizing playe ins ead aims o
choose an ac ion ha minimizes he eg e , de ined as he maximum amoun possibly gi en
up by playing a ce ain ac ion. Fo ins ance, he maximum eg e om ac ion Iin he SH
game is 2, which is he payo gi en up by playing ac ion Iwhen he opponen plays II.By
simila easoning, he eg e om ac ion II in he SH game is 1, which is he payo gi en
up by choosing II when he opponen chooses I. A eg e minimize would hence play II
in he SH game.
Simila compu a ions lead o he conclusion ha bo h a maxmin playe and a eg e -
minimizing playe would choose ac ion II in he PD game oo. In an en i onmen consis ing
uniquely o one o bo h o hese wo games he wo playe ypes would hus be beha io ally
indis inguishable. Looking a he AG game, howe e , one can compu e ha a maxmin playe
would play ac ion Iwhile a eg e minimize would play ac ion II. In he en i onmen
including all h ee games, he di e en ypes a e dis inguishable.
In he ollowing, we s udy he condi ions on he games in he en i onmen ensu ing ha
he ypes in he popula ion a e dis inguishable. To do ha , we i s de ine he concep s o
games wi h ambigui y on he decision c i e ia and o equilib ium in such games in Sec .2,
and hen we s a e he condi ions o a 2 ×2 game o be in o ma i e, ha is, o allow elling
di e en ypes apa , in Sec . 3. In Sec . 4, we in oduce he popula ion mul igame, we s udy
he p ope ies o he en i onmen ha gua an ee ha in o ma i e 2×2 games can always occu
wi h posi i e p obabili y, and we gene alize he esul s o he case o n×ngames. Sec ion 5
hen conside s a speci ic ins ance o popula ion mul igame and shows how o compu e he
p obabili y o in o ma i e games in ha pa icula case and ha he agen s can asymp o ically
lea n he p ecise p opo ions o ypes in he popula ion. Sec ion6ins ead conside s a a ie y o
di e en mul igames and compu es he p obabili y o s ongly in o ma i e games in di e en
cases by means o compu e simula ions. Finally, Sec .7concludes. Be o e mo ing o Sec .2,
howe e , in he nex subsec ion we say a ew wo ds on he li e a u e ela ed o he p esen
wo k.
Dynamic Games and Applica ions (2025) 15:1037–1069 1039
1.1 Rela ed Li e a u e
Al hough, as al eady men ioned abo e, he necessi y o de eloping models wi h agen s
cha ac e ized by di e en beha io -gene a ing mechanisms has been explici ly ad oca ed
especially by biologis s and ecologis s [14,21,23], he game- heo e ic li e a u e in eco-
nomics has almos always ocused on models wi h homogeneous decision c i e ia—i.e.,
models whe e all he agen s ollow he same decision c i e ion [3,22,24–28,30,33]. A
ew excep ions a e he ollowing. [2] s udy necessa y and su icien condi ions o he exis-
ence o an equilib ium in games unde ambigui y whe e he agen s can ha e e y gene al
subjec i e choice p e e ences. [10,12,13]and[18] in oduce epis emic ype spaces ha
allow he agen s o ollow di e en decision c i e ia and o eason s a egically abou each
o he ’s c i e ia, bu hei esul s a e pu ely on he epis emic side. On he e olu iona y side,
he e olu ion o p e e ences [9,11] is a b anch o e olu iona y game heo y ha s udies he
e olu iona y i ness o di e en subjec i e p e e ences, bu he ocus he e is on he playe s’
subjec i e u ili y unc ions a he han on he playe s’ decision c i e ia. Mo eo e , he e we
a e p ima ily in e es ed in he lea ning and no in he e olu ion o he playe s. The idea o
in es iga ing mul igame models has some imes appea ed in o he ields oo. [16]and[17]
s udy he e olu ion o decision c i e ia in an en i onmen simila o he one we conside
he e. [4] oo conside e olu iona y p ocesses d i en by a mul igame en i onmen , bu wi h
he di e ence ha hei agen s a e de ined by au oma a a he han by decision c i e ia. [38]
exploi s a mul igame consis ing o h ee di e en games o explain he e olu ion o ai ness,
bu he ypes he e a e decision ules speci ic o hose games and hence simple han he
decision c i e ia examined he e.
2 Games wi h C i e ion Ambigui y
In he simple example om he p e ious sec ion, he playe s we e implici ly assumed o
hold unce ain y o e he opponen s’ ac ions and o use possibly di e en decision c i e ia
o cope wi h such unce ain y and o pick an ac ion o any gi en game in he en i onmen .
In popula ion games, howe e , i is na u al o imagine ha he playe s’ unce ain y comes
om he dis ibu ion o he di e en ypes in he popula ion. In his sec ion, we o mally
in oduce popula ion games wi h c i e ion ambigui y and show how he unce ain y on he
opponen ’s ac ions is de i ed om he unce ain y on he ype dis ibu ion in he popula ion.
We in e p e hese games as modeling a si ua ion in which playe s a e d awn a andom om
a la ge and mixed popula ion whe e maxmin ypes (M) and eg e -minimizing ypes (R)
coexis in unknown p opo ions. The concep o game wi h c i e ion ambigui y and ha o
equilib ium in such games can hen be o malized as ollows.
De ini ion 1 Agame wi h c i e ion ambigui y consis s o he ollowing componen s:
•a se o wo lds ;
•a se o pa ame e s ;
•ase I={1,2,3,...,N}o playe s;
• o each playe i,
– an ac ion se Ai;
– a se o (c i e ion) ypes Ti;
– a se o signals Si;
– a c i e ion-assignmen unc ion τi:→Ti;
1040 Dynamic Games and Applica ions (2025) 15:1037–1069
– a signal unc ion ςi:→Si;
– a u ili y unc ion ui:A1×···×An×→R;
• o each λ∈, a p obabili y dis ibu ion Pλo e .
Apolicy o playe iin such a game is a unc ion σi:Ti×Si→Ai ha associa es each
decision c i e ion and signal wi h an ac ion.1
No e ha i he se is equipped wi h a p obabili y dis ibu ion known o all he playe s,
his de ini ion desc ibes a Bayesian game [29]. Howe e , we a e in e es ed in si ua ions whe e
his is no he case and he playe s hold ambigui y (i.e., non-p obabilis ic unce ain y) abou
he pa ame e λ.
Th oughou he es o he pape we deno e ypical p o iles o signals and ypes by s∈
S:= S1×S2×···×SNand ∈T:= T1×T2×···×TN, espec i ely. Fo p o iles o
ypes, signals, o policies, we use he no a ion x−i=(x1,...,xi−1,xi+1,...,xN) o he
(N−1)-dimensional ec o ha esul s by emo ing he i h coo dina e om he ec o x.
In games whe e he unce ain y only conce ns he c i e ia adop ed by he o he s, no p i a e
in o ma ion o he han hei own decision c i e ion is e ealed o he agen s. In he no a ion
abo e, his can be modeled by assuming ha he se s Sia e all single ons. To ease no a ion, we
he e o e dispense wi h he speci ica ion o he p i a e signals siin he ollowing discussion.
Fo any ixed alue o λ, he ype o a playe is a andom a iable. Hence, gi en policies
(σi)i∈I, he ac ion ai=σi( i) oo is a andom a iable, and (a1,a2,...,aN)is a andom
ec o . The u ili y o each playe is he e o e also a andom a iable, which has an expec ed
alue o any ixed λ. Since all o hese dis ibu ions depend on λ, he alue o his expec ed
u ili y is a unc ion o λ. We hen le each playe esol e he ambigui y abou his expec ed
u ili y by means o he decision c i e ion gi en by his o he ype.
In o mulas, he expec ed u ili y ha ollows om ype iusing ac ion ai, gi en ha he
o he playe s adop policies σ−i,is
Eλ[ui|ai, i,σ
−i]=
ui(ai,σ
−i(τ−i(ω)), ω) Pλ(dω| i). (1)
De ini ion 2 Le an incomple e in o ma ion game wi h c i e ion ambigui y be gi en as abo e,
wi h Ti={M,R} o all i∈I. We say ha a policy p o ile (σ∗
i)i∈Iis an equilib ium i , o
all i∈I, he policy σ∗
imaximizes he unc ion
σi→ in
λ∈Eλ[ui|σi(M), M,σ∗
−i]
and minimizes he unc ion
σi→ sup
λ∈sup
ai∈Ai
Eλ[ui|ai,R,σ∗
−i]−Eλ[ui|σi(R), R,σ∗
−i].
2.1 Popula ion Games wi h C i e ion Ambigui y
As a main sou ce o ambigui y abou he c i e ia, we conside a la ge popula ion o agen s
using di e en decision c i e ia ha andomly mee o play games. Since i is o en unlikely
o know he p ecise dis ibu ion o ypes in a la ge popula ion, we allow he agen s o hold
unmeasu able unce ain y abou he dis ibu ion o c i e ia in he popula ion hey a e pa
1The e is no consensus in game heo y abou he possibili y o using mixed ac ions (see o ins ance [32] o
a discussion). He e we ollow he posi ion o [13]and[12].
Dynamic Games and Applica ions (2025) 15:1037–1069 1041
o . In pa icula , he es o his pape is conce ned wi h a class Go wo-playe popula ion
games wi h c i e ion ambigui y. A i s , we ocus on 2 ×2 symme ic games, and we la e
show how he esul s can be gene alized o n×nsymme ic games.2
In he ollowing, we assume ha each o he wo playe s i∈{1,2}has his o he own deci-
sion c i e ion i∈{M,R} e ealed, bu holds unmeasu able unce ain y abou he opponen ’s
c i e ion. Speci ically, we assume ha
Pλ( 3−i=R| i=R)=Pλ( 3−i=R| i=M)=λ
Pλ( 3−i=M| i=R)=Pλ( 3−i=M| i=M)=1−λ
whe e λ∈[λ, λ]⊆[0,1]is subjec o unmeasu able unce ain y. In he case o 2 ×2 games,
he agen s a e equipped wi h he bina y ac ion se s A1=A2={I,II}, and hei u ili y
unc ions a e de ined in e ms o he symme ic 2 ×2 game ma ix
III
I a,a b,c
II c,b d,d
o all ω∈,whe e(a,b,c,d)∈R4. The wo playe s o his game a e hen agen s
sampled a andom om a la ge popula ion cha ac e ized by he unknown pa ame e λ.Fo
con enience, we use he ec o no a ion σi=(σi(M), σi(R)) o speci y he policy unc ion
o each agen i∈{1,2}.
Example 3 Suppose wo agen s a e andomly d awn om a popula ion consis ing o a p o-
po ion λo eg e minimize s and 1−λo maximinimize s. The exac alue o λis unknown
and subjec o ambigui y, wi h λ∈[1/5,1/2]. The wo agen s hus play he ollowing coo -
dina ion game unde ambigui y abou he alue o λ.
III
I1,1 0,0
II 0,0 2,2
Wha a e he equilib ia o his game? Each playe in his game is a eg e ype wi h
p obabili y λand a maxmin ype wi h p obabili y 1 −λ, and hei policy unc ions ake he
o m o pai s o pu e ac ions, wi h one ac ion o each o hese wo ypes. By inspec ing each
o he 16 policy p o iles, we can ejec he ones in which any o he wo playe s is no using
a bes eply.
Conside i s he case in which he ow playe aces he column policy σ2=(I,I).Gi en
such a homogeneous policy, he ac ion o he column playe is de e minis ic and hence no
subjec o unce ain y, unmeasu able o o he wise. We he e o e ind ha he unique bes eply
o his policy is σ1=(I,I). We simila ly ind ha he unique bes eply o σ2=(II,II)
is σ1=(II,II). Since he exac same a gumen holds o he column playe , i ollows ha
he policies
(σ1,σ
2)=((I,I), (I,I))
(σ1,σ
2)=((II,II), (II,II))
a e bo h equilib ia o his game, and ha he homogeneous policies (I,I)and (II,II)appea
in no o he equilib ia.
2Ha ing symme ic games only allows us o s ick wi h he single-popula ion model, as we need no conside
a di e en popula ion o each ole in he game.
1042 Dynamic Games and Applica ions (2025) 15:1037–1069
Suppose now ha he column playe uses he policy σ2=(I,II). Then he condi ional
expec ed u ili ies o he ow playe gi en λa e
Eλ[u1|(I,(I,II))]=1−λ
Eλ[u1|(II,(I,II))]=2λ
Fo he maxmin ype o he ow playe , ac ion Iis a bes eply o σ2=(I,II)since he
inequali y
min
λ∈[1/5,1/2]1−λ≥min
λ∈[1/5,1/2]2λ
educes o he ue s a emen 1/2≥2/5. Fo he eg e -minimizing ype o he ow playe ,
on he o he hand, ac ion II is a bes eply o σ2=(I,II). This ollows om he ac ha
he condi ional eg e s o ac ion Iand II gi en λa e
max
a1∈A1
Eλ[u1|(a1,(I,II))]−Eλ[u1|(I,(I,II))]=max{0,3λ−1}
max
a1∈A1
Eλ[u1|(a1,(I,II))]−Eλ[u1|(I,(I,II))]=max{1−3λ, 0},
and
max
λ∈[1/5,1/2]{max{0,3λ−1}}≥max
λ∈[1/5,1/2]{max{1−3λ, 0}}
educes o he ue s a emen 1/2≥2/5. The policy σ1=(I,II)is hus a bes eply o he
policy σ2=(I,II).
Suppose now ha he column playe uses σ2=(II,I). We hen ind ha
Eλ[u1|(I,(II,I))]=λ
Eλ[u1|(II,(II,I))]=2(1−λ)
I ollows ha ac ion II is a bes eply o he eg e ype, since
max
a1∈A1
Eλ[u1|(a1,(II,I))]−Eλ[u1|(I,(II,I))]=max{0,2−3λ}
max
a1∈A1
Eλ[u1|(a1,(II,I))]−Eλ[u1|(II,(II,I))]=max{3λ−2,0}
and
max
λ∈[1/5,1/2]{max{0,2−3λ}}≥max
λ∈[1/5,1/2]{max{3λ−2,0}}
educes o he ue s a emen 7/5≥0. Ac ion Iis hus no a eg e -minimizing eply o he
policy σ1=(II,I), and hence σ2=(II,I)canno be a bes eply o σ1=(II,I). Since we
ha e al eady uled ou he op ions σ2=(I,I)and σ2=(II,II)abo e, he only emaining
op ion is σ2=(I,II). Howe e , we ha e also seen ha (σ1,σ
2)=((I,II), (II,I)) is no
an equilib ium, and since he game is symme ic, nei he is (σ1,σ
2)=((II,I), (I,II)).
In sum, we ha e ha he h ee policy p o iles
(σ1,σ
2)=((I,I), (I,I))
(σ1,σ
2)=((II,II), (II,II))
(σ1,σ
2)=((I,II), (I,II))
a e he only equilib ia o he game.
Dynamic Games and Applica ions (2025) 15:1037–1069 1043
We a e in e es ed he e in symme ic equilib ia, i.e., equilib ia such ha σ1=σ2,as his
is he only ype o equilib ium ha can be in e p e ed as a popula ion adap i e s a egy. In
pa icula , in he case o 2 ×2 games we a e in e es ed in he policy p o iles
(σ1,σ
2)=((I,II), (I,II))
(σ1,σ
2)=((II,I), (II,I))
since a popula ion playing acco ding o any o hese p o iles allows he obse e s o in e he
decision c i e ia o he playe s om hei ac ions. In games whe e exac ly one o hese p o iles
is he sole symme ic equilib ium, he playe s necessa ily e eal hei decision c i e ion. We
hen say ha such games a e s ongly in o ma i e, and in he nex sec ion we p o ide necessa y
and su icien condi ions o a game o be s ongly in o ma i e.
3 S ongly In o ma i e Games
S ongly in o ma i e games a e he key o he lea ning o he ac ual p opo ions o decision
c i e ia in he popula ion, because only by playing s ongly in o ma i e games he playe s
unambiguously e eal hei ype. Fo any in e al [λ, λ]wi h λ < λ, i is solely he posi i e
p obabili y o a s ongly in o ma i e game o occu ha can gi e he playe s ele an in o -
ma ion abou he p opo ions in he popula ion. In his sec ion, we cha ac e ize he egion
o s ongly in o ma i e games in R4and we nex p o ide condi ions on he dis ibu ion o
possible games in he class G ha gua an ee he occu ence o s ongly in o ma i e games.
3.1 S ong In o ma i i y wi h Full Unce ain y
We i s o mula e he condi ions unde which a game (a,b,c,d)is s ongly in o ma i e
gi en ha all he playe s ha e ull unmeasu able unce ain y abou he p opo ions o decision
c i e ia, i.e., o λ∈[0,1]. As a i s s ep, we can immedia ely educe he se o s ongly
in o ma i e games by disca ding all games ha a e no an i-coo dina ion games. In he
ollowing, in o ma i i y in 2 ×2 games will be enough o mos o ou pu poses. Howe e ,
we can p o e he ollowing esul o gene al n×ngames. To ha aim, we pa i ion he class
o symme ic n×ngames in o h ee se s:
•coo dina ion games: ai∈b (ai) o all pu e ac ions ai,
•an i-coo dina ion games: ai/∈b (ai) o all pu e ac ions ai,
•mixed games: ai∈b (ai) o some ai,anda
i/∈b (a
i) o some a
i,
whe e b (ai)is he se o bes eplies o ac ion ai. Then he ollowing esul holds.
P oposi ion 4 Only an i-coo dina ion games can be s ongly in o ma i e.
P oo See Appendix A.
In he case o 2 ×2 symme ic games, le us deno e he wo e ealing policy unc ions
by σ◦:= (I,II)and σ•:= (II,I). We can hen gi e necessa y and su icien condi ions
in e ms o he u ili y alues (a,b,c,d) o he policy p o iles (σ ◦,σ◦)and (σ •,σ•) o be
s ic equilib ia in an i-coo dina ion games, and hence o a game o be s ongly in o ma i e
unde ull unmeasu able unce ain y.
1044 Dynamic Games and Applica ions (2025) 15:1037–1069
Theo em 5 Suppose ha (a,b,c,d)is an an i-coo dina ion popula ion game and ha λ∈
[0,1]. Then he policy p o ile (σ ◦,σ◦)de ines a s ic equilib ium i and only i
a>d and c −a>b−d.
The p o ile (σ •,σ•)de ines a s ic equilib ium i and only i
a<d and c −a<b−d.
P oo See Appendix A.
I is also c ucial o no e ha he p o ile o policy unc ions (σ◦,σ◦)and he dual p o ile
(σ•,σ•)canno be bo h s ic equilib ia o he same game.
Co olla y 6 A mos one o he wo p o iles (σ ◦,σ◦)and (σ •,σ•)can be a s ic equilib ium.
P oo See Appendix A.
3.2 S ong In o ma i i y wi h Pa ial Unce ain y
In his sec ion, we ind necessa y and su icien condi ions o (σ◦,σ◦)and (σ •,σ•) o be
s ic equilib ia in si ua ions whe e he agen s’ unce ain y is no desc ibed by he ull uni
in e al [0,1], bu by some non-emp y subin e al [λ, λ]⊆[0,1]. We de i e ou esul s by
showing ha a game (a,b,c,d)played in his s a e o pa ial unce ain y is equi alen o an
“inne game” played in a s a e o ull unce ain y. Fo b e i y, we w i e (a,b,c,d), [λ, λ]
o he game wi h game ma ix (a,b,c,d)played in he in o ma ion s a e desc ibed by he
unce ain y in e al [λ, λ]. No ice ha each game (a,b,c,d)and unce ain y in e al [λ, λ],
oge he wi h a policy p o ile (σi)i∈{1,2}, de ine an inne game as ollows.
De ini ion 7 Fo (a,b,c,d), [λ, λ]and policy unc ion σ3−i:{M,R}→{I,II}, he
co esponding inne game (a,b,c,d) o playe iis gi en by
a=Eλ[ui|I,M,σ
3−i]=Eλ[ui|I,R,σ
3−i]
b=Eλ[ui|I,M,σ
3−i]=Eλ[ui|I,R,σ
3−i]
c=Eλ[ui|II,M,σ
3−i]=Eλ[ui|II,R,σ
3−i]
d=Eλ[ui|II,M,σ
3−i]=Eλ[ui|II,R,σ
3−i]
Since we a e in e es ed in games whe e playe s e eal hei decision c i e ion, he ele an
inne games a e hose gi en by he e ealing policy σ◦=(I,II)and by he e ealing policy
σ•=(II,I).Fo (a,b,c,d), [λ, λ]and e ealing policy σ◦ he co esponding inne game
(a◦,b◦,c◦,d◦)is hus de ined by
a◦=a+λ(b−a)
b◦=a+λ(b−a)
c◦=c+λ(d−c)
d◦=c+λ(d−c)
Simila ly, he inne game (a•,b•,c•,d•)co esponding o σ•=(II,I)is de ined by
a•=b+λ(a−b)
b•=b+λ(a−b)
c•=d+λ(c−d)
d•=d+λ(c−d)
Dynamic Games and Applica ions (2025) 15:1037–1069 1051
Fig. 3 A map o he ou egions o (X,Y)-space on which he condi ional p obabili y densi y unc ion q(x,y)
is nonze o. Fo y<0o y>1, q(x,y)=0
We can ind he p obabili y densi y unc ion o (X,Y)by in eg a ing ou Vand Z om his
densi y unc ion, using he anges compu ed abo e:
q(x,y|A)=T(A)
q(x,y, ,z|A)dzd
= =∞
= ∗(x,y)z=z∗(x,y, )
z=0
q(x,y, ,z|A)dzd .
The alue o he uppe bound z∗depends on whe he x≤yo x>y, while he lowe bound
∗depends on whe he x+y≥1o x+y<1. Hence, his double in eg al can be spli
up in o ou subin eg als co e ing each o hese ou cases. We compu e each o hese ou
double in eg als sepa a ely, inding
q(x,y|A)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1/(3x3)x>y,x+y≥1
1/(3(1−x)3)x≤y,x+y<1
1/(3y3)x>y,x+y<1
1/(3(1−y)3)x≤y,x+y≥1
0y<0o y>1
Equi alen ly, his densi y unc ion can be w i en as
q(x,y|A)=1
31
1
/2+max{|x−1
/2|,|y−1
/2|}3
(8)
whe e (x,y)lies on he ho izon al band de ined by 0 <y<1. No e ha he ou di e en
cases a e sepa a ed by he diagonals o he uni squa e, as shown in Fig. 3.
5.2 The P obabili y o In o ma i i y in he Uni o m Case
Ha ing now ound he exac condi ional dis ibu ion o (X,Y), we can app oxima e he
condi ional p obabili y ha he in o ma i e policy p o ile (σ◦,σ◦)is he sole symme ic
equilib ium o he game. As obse ed in Sec .3.3, his e en coincides wi h he se
ϕ◦=x<0,λ+λ
2<y<λ∪0≤x<1
2,λ+λ
2<y<λ−(λ −λ)x.(9)
As p e iously discussed, a simila se ϕ•exis s o he p o ile (σ•,σ•). We can hen compu e
he condi ional p obabili y o ϕ◦gi en ha (a,b,c,d)is an an i-coo dina ion game as wo
1052 Dynamic Games and Applica ions (2025) 15:1037–1069
sepa a e in eg als:
P(ϕ◦|A)=x=0
x=−∞ y=λ
y=λ+λ
2
q(x,y|A)dydx
+x=1/2
x=0y=λ−(λ−λ)x
y=λ+λ
2
q(x,y|A)dydx.
On each o hese wo egions, qbeha es di e en ly. Since 0 <y<1, when x≤0 we can
educe he densi y exp ession q(x,y|A)in Eq. 8 o
q(x,y|A)=1
31
1−x3
.
The i s e m o P(ϕ◦|A) hus has he alue
Px<0,λ+λ
2<y<λ
A=x=0
x=−∞ y=λ
y=λ+λ
2
q(x,y|A)dydx =λ−λ
12 .
Fo 0 ≤x<1/2, we simpli y compu a ions by means o he sandwich bounds
1
3≤q(x,y|A)≤1
31
1−x3
.
By in eg a ing all h ee componen s, we ind ha
λ−λ
24 ≤P0≤x<1
2,λ+λ
2<y<λ−(λ −λ)x
A≤λ−λ
12 .
Adding up he wo e ms, we hus ha e he condi ional p obabili y
λ−λ
8≤P(ϕ◦|A)≤λ−λ
6.(10)
Since P(A)=1/4, his en ails ha he uncondi ional p obabili y sa is ies
λ−λ
32 ≤P(ϕ◦)≤λ−λ
24 .(11)
No e ha his ag ees wi h he esul in Sec .4.1 o he case o λ−λ=1. Finally, he o al
p obabili y o sampling an in o ma i e game is
P(ϕ) =P(ϕ◦∪ϕ•)=P(ϕ◦)+P(ϕ•)=2P(ϕ◦)(12)
since he se s ϕ◦and ϕ•a e symme ic and disjoin .
5.3 The Asymp o ic Speed o Lea ning
In he p e ious subsec ion, we assumed ha he unce ain y in e al was ixed and a bi a y
and compu ed some bounds on he p obabili y ha a andomly gene a ed game is s ongly
in o ma i e. In his sec ion, we will assume such andomly gene a ed games a e played
epea edly be ween andomly selec ed pai s o playe s om a mixed popula ion o maximin-
imize s and eg e minimize s, and ha he beha io o he selec ed playe s is isible o all
membe s o he popula ion.
Dynamic Games and Applica ions (2025) 15:1037–1069 1053
Unde some easonable assump ions abou he lea ning ule employed by he membe s o
he popula ion, his causes he p obabili y o obse ing a new s ongly in o ma i e game o
end o ze o. Howe e , as we also will a gue, his con e gence is slow enough o allow he
membe s o he popula ion o exac ly de e mine he ype p opo ions in he limi .
S ipula ion o lea ning ule E e y ime wo andomly selec ed playe s a e plucked om he
popula ion and play a s ongly in o ma i e game, hey bo h e eal hei ype. Upon obse ing
a sequence o s ongly in o ma i e games, one can he e o e coun he numbe o imes he
playe s e ealed hemsel es as maximinimize s o eg e minimize s and use his ally o
es ima e he p obabili y ha a andomly d awn playe will be o a ce ain ype.
Suppose ha kis he numbe o imes he playe s ha e e ealed hemsel es o be eg e
minimize s, and ha mis he o al numbe o s ongly in o ma i e games played. Fo easons
ha we will jus i y below, we assume ha any a ional obse e o his sequence o e en s
o ms he belie ha λ, he ue p opo ion o eg e minimize s, lies wi hin an unce ain y
in e al wi h bounds
λm=k
m−C
√m
λm=k
m+C
√m
whe e C>0 is a ixed bu a bi a y cons an independen o m. In o he wo ds, ou assump ion
is ha he wid h o he unce ain y in e al is
λm−λm=2C
√m(13)
a e he obse a ion o ms ongly in o ma i e games.
The wai ing- ime dis ibu ion Be ween wo s ongly in o ma i e games, a numbe o non-
in o ma i e games is played. A e he i s ms ongly in o ma i e games ha e been played,
one needs o play a ce ain numbe o games be o e he nex s ongly in o ma i e game
appea s. Since he games hemsel es a e gene a ed andomly, his wai ing ime m≥1is
i sel a andom a iable.
As we ha e seen in he p e ious sec ion, he p obabili y ha a speci ic andom game is
s ongly in o ma i e depends on he wid h o he unce ain y in e al, bu we ha e assumed
ha he unce ain y in e al emains unchanged as long as no new in o ma ion is e ealed.
The p ocess o wai ing o he nex s ongly in o ma i e game can he e o e be modeled as
he p ocess o lipping a ben coin wi h a ixed success pa ame e pmun il i comes up heads.
This means ha he andom a iable m ollows a geome ic dis ibu ion wi h pa ame e
pm. This dis ibu ion has an expec ed alue o
E[m]= 1
pm
and a a iance o (1−pm)/p2
m≤1/p2
m.
Bounds on he wai ing ime As we ha e seen abo e, he p obabili y pmdepends on he
wid h o he unce ain y in e al and sa is ies
λ−λ
16 ≤pm≤λ−λ
12 .(14)
Wi h he assump ions abo e, his is equi alen o
C
8√m≤pm≤C
6√m.(15)
1054 Dynamic Games and Applica ions (2025) 15:1037–1069
As we ha e seen, he expec ed wai ing ime be o e he (m+1) h s ongly in o ma i e game
a i es is E[m]=1/pm. We hus ha e
6√m
C≤E[m]≤8√m
C(16)
No e also ha Va [m]≤1/pm≤64m/C.
To al wai ing ime Since he wai ing ime om he m h o he (m+1) h s ongly in o ma i e
game is m, he wai ing ime un il a o al o ms ongly in o ma i e games ha e been obse ed
is a andom a iable
T=1+1+···+m
By he linea i y o expec a ions,
m
i=1
6√i
C≤E[T]≤
m
i=1
8√i
C
which can be weakened o
3
Cm3/2≤E[T]≤8
Cm3/2.
Since he wai ing imes 1,
2,...,
ma e independen , we also ha e
Va [T]≤
m
i=1
64i
C≤64
Cm2.
This a iance g ows quad a ically in m,so(T−E[T])2will end o be la ge when mis
la ge. In ela i e e ms, howe e , we ha e
Va T−E[T]
E[T]=EVa [T]
E[T]2≤64m2/C
(C/(3m3/2))2=64C
9m,
which goes o ze o as m→∞. Hence T/E[T]will con e ge o 1 as m→∞by, o
ins ance, Chebyshe ’s inequali y.
Limi -lea nabili y We can now b ie ly summa ize in in o mal e ms wha we ha e seen so
a . We ha e shown ha he o al numbe o games equi ed in o de o obse e ms ongly
in o ma i e games is on he o de o T∼m3/2. Con e sely, when one has obse ed a o al
o Tgames, one can expec he numbe o s ongly in o ma i e games among hem o be on
he o de o m∼T2/3. Since we ha e assumed ha he wid h o he unce ain y in e al is
in e sely p opo ional o √m, i will be on he o de o
λm−λm∼(T2/3)−1/2=T−1/3(17)
These esul s ob ain because we ha e assumed ha he agen s in he popula ion upda e
hei belie s when new in o ma ion is e ealed, so ha he wai ing ime be o e he nex
in o ma i e e en g adually inc eases. By con as , i he agen s did no upda e hei belie s,
hen he wai ing ime be ween s ongly in o ma i e games would emain cons an o e ime,
and mand Twould be o he same o de o magni ude.
Since T−1/3→0 o →∞, i ollows ha he playe s will ul ima ely lea n he
p opo ions o he wo agen ypes in he sense ha
λm−λm→0
Dynamic Games and Applica ions (2025) 15:1037–1069 1055
o m→∞. This is a somewha su p ising esul since i is also he case ha pm→0
o m→∞, so ha he s ongly in o ma i e games become mo e in equen as he agen s
become mo e in o med.
Lea ning ule jus i ica ion Going back o he s ipula ed lea ning ule, we s ill ha e o gi e
a jus i ica ion o he assump ion ha he wid h o he unce ain y in e al is p opo ional o
1/√ma e he obse a ion o ms ongly in o ma i e games. Ou a gumen o his choice
has a posi i e and a nega i e aspec . The posi i e pa o he a gumen shows ha i is possible
o es ima e he pa ame e o a Be noulli dis ibu ion om mobse a ions wi h an expec ed
e o o 1/√m, whe eas he nega i e pa shows ha no subs an ially lowe e o is possible.
The posi i e pa o ou a gumen consis s o he law o la ge numbe s as o mula ed, o
ins ance, by Chebyshe :
Theo em 17 Le X1,X2,X3,...be a se ies o independen and iden ically dis ibu ed an-
dom a iables wi h a sha ed mean E[Xi]=λand a ini e a iance, and le
ˆ
λm=(X1+···+Xm)/m
be he empi ical a e age o he i s m obse a ions. Then o any α>0 he e is a cons an
δ>0such ha o all m >0,
P(|λ−ˆ
λm|<δ/
√m)≥1−α.
This heo em ensu es ha he mean o a dis ibu ion is es ima ed by he empi ical mean
o a sample wi h an accu acy p opo ional o 1/√m. A p oo can be ound in [15], ch. IX
and X.
The nega i e pa o ou a gumen elies on much mo e ecen esul s abou he limi s on
he speed o lea ning. Fo he pu poses o s a ing his heo em, le λ1,λ
2∈(0,1)be he
pa ame e s o wo coin lipping dis ibu ions, and le mobse a ions be d awn om one o
hese wo dis ibu ions. A hypo hesis es is hen a unc ion ha maps a da a se o one o he
wo pa ame e alues. We say ha he wo dis ibu ions a e (m,α)-dis inguishable i he e
is a hypo hesis es ha e u ns he co ec pa ame e alue wi h p obabili y 1 −α o a da a
se o size m.We henha e:
Theo em 18 The e a e δ>0and α>0such ha o all m, i wo pa ame e s λ1,λ
2∈(0,1)
sa is y he p oximi y condi ion |λ1−λ2|<δ/
√m, hen he co esponding coin lipping
dis ibu ions a e no (m,α)-dis inguishable.
This heo em allows us o conclude ha he con idence bounds p o ided by Chebyshe ’s
inequali y a e he bes we can hope o : he wid h o he con idence in e al canno sh ink a a
a e as e han 1/√m. This esul can be p o en by using he Hellinge dis ance be ween wo
binomial dis ibu ions o lowe -bound he minimax isk o he hypo hesis es , as discussed
ex ensi ely elsewhe e [36,37].
Toge he , hese wo esul s show ha a con idence in e al o he shape
λm=ˆ
λm−C
√m(18)
λm=ˆ
λm+C
√m(19)
will con ain he ue pa ame e λwi h a p obabili y ha nei he con e ges o 0 no o 1 as
m→∞.Bycon as ,i
|λm−λm|
√m→∞
1056 Dynamic Games and Applica ions (2025) 15:1037–1069
o m→∞, he p obabili y o e o would end o 0, and i we had
|λm−λm|
√m→0
o m→∞, he p obabili y o e o would end o 1. This hence jus i ies ou assump ion
ha any a ional agen mus use con idence in e als o wid h p opo ional o 1/√mwhen
es ima ing he pa ame e o a Be noulli dis ibu ion om mobse a ions.
6 S ong In o ma i i y o Be a and Di ichle Dis ibu ions
In his sec ion, we expand he analysis om he p e ious sec ion o games wi h mo e han
wo ac ions and di e en u ili y-ma ix dis ibu ions. We empi ically es ima e he p obabili y
o encoun e ing a s ongly in o ma i e game bo h in he case whe e each cell in he u ili y
ma ix is sampled independen ly om he be a dis ibu ion and in he case whe e he en i e
u ili y ma ix is a sample om a Di ichle dis ibu ion (and hence u ili y alues a e no longe
independen ).
The be a dis ibu ion The be a dis ibu ion is a dis ibu ion o e he uni in e al pa ame e -
ized by wo pa ame e s αand β. We ocus exclusi ely on he case whe e he wo pa ame e s
a e iden ical, α=β= . In ou i s se o expe imen s, we use such be a dis ibu ions o
sample each alue o he u ili y ma ix independen ly.
The p obabili y densi y unc ions o he be a dis ibu ion o some o hese pa ame e s
a e shown in Fig. 4. As he igu e illus a es, he be a dis ibu ion is iden ical o he uni o m
dis ibu ion when α=β=1, while i comes o esemble a Be noulli dis ibu ion o
α, β →0 and an inc easingly na ow no mal dis ibu ion as α, β →∞.
The Di ichle dis ibu ion The Di ichle dis ibu ion o o de Dis a p obabili y dis ibu ion
o e he (D−1)-dimensional p obabili y simplex (i.e., o e (D−1)- ec o s wi h 0 ≤ d≤1
and d d=1). The Di ichle dis ibu ion o o de Dis pa ame e ized by a ec o o D
posi i e pa ame e s α1, ..., αD.
We once again ocus on he symme ic pa ame e ec o s, αd= o all d. Di ichle
dis ibu ions wi h such pa ame e ec o s a e equal o he uni o m dis ibu ion when =1.
They become inc easingly concen a ed a ound he co ne s o he simplex as →0and
inc easingly concen a ed a ound he cen e poin o he simplex o →∞.
Be a-dis ibu ion expe imen s In ou i s se o expe imen s, we conside symme ic n×n
game ma ices whose alues a e d awn om symme ic be a dis ibu ions. We can hus empi -
ically es ima e he p obabili y ha such andomly gene a ed games a e s ongly in o ma i e.
Fig. 4 Examples o be a dis ibu ions. Le : α=β=0.25. Cen e : α=β=1. Righ : α=β=50
Dynamic Games and Applica ions (2025) 15:1037–1069 1057
Fig. 5 F equencies o s ongly in o ma i e games o independen ly be a-dis ibu ed u ili y alues in he case
o maximum unce ain y [λ, λ]=[0,1]on he le , and in he case o educed unce ain y [λ, λ]=[0.6,0.8]
on he igh
We se he numbe o ac ions equal o
n=2,3,5,7,11
and he wo (iden ical) be a pa ame e s equal o
=0.25,0.5,0.75,1,2,3,4,5,6,8,10,20,30,40,50.
We addi ionally y wo di e en unce ain y in e als, [λ, λ]=[0,1]and [λ, λ]=[0.6,0.8].
Figu e5 abula es he es ima ed p obabili y ha a andom game will be in o ma i e o e e y
possible combina ion o hese pa ame e choices. The es ima ed p obabili ies a e based on a
sample o 105games.
A ew obse a ions a e in o de . Fi s , he obse ed equency o s ongly in o ma i e
games o he case o maximum unce ain y [λ, λ]=[0,1]wi h wo ac ions and be a pa am-
e e s α=β=1 is p ecisely 0.08296, which is wi hin he heo e ical bounds ound in
Sec .5,0.0625 ≤0.08296 ≤0.0833. This is also ue o he case o educed unce ain y
[λ, λ]=[0.6,0.8], whe e we ha e 0.0125 ≤0.0159 ≤0.0166.
Second, bo h he case o maximum unce ain y and he case o educed unce ain y display
a simila pa e n, wi h lowe equency o s ongly in o ma i e games in he bo om le co ne ,
as he numbe o ac ions inc eases and he be a pa ame e s dec ease. O e all, 2 ×2 games
u n ou o be he mos in o ma i e o all he chosen pa ame e s o he be a dis ibu ion.
Figu e6illus a es he e ec o sh inking he unce ain y in e al in wo cases, one in which
he numbe o ac ions is held ixed a n=2, and one in which he be a pa ame e s a e held ixed
a α=β=1. As expec ed, bo h ables show ha he p obabili y o encoun e ing a s ongly
in o ma i e game ends o 0 as he unce ain y in e al sh inks. Pe haps mo e in e es ingly,
he igh -hand inse shows ha he p obabili y o encoun e ing a s ongly in o ma i e game
also depends nega i ely on he numbe o ac ions a ailable o he playe s.
Di ichle -dis ibu ion expe imen s We now u n o he case o Di ichle -dis ibu ed u ili y
alues. As men ioned abo e, we now sample he en i e n×nsymme ic game ma ix om
a Di ichle dis ibu ion o o de D=n2wi h iden ical pa ame e s α1=···=αD= .We
again choose he ollowing numbe o ac ions
n=2,3,5,7,11
1058 Dynamic Games and Applica ions (2025) 15:1037–1069
Fig. 6 F equencies o s ongly in o ma i e games du ing lea ning. On he le : equencies o di e en
pa ame e s o he be a dis ibu ion du ing lea ning in 2 ×2 games. On he igh : equencies o di e en
numbe s o ac ions du ing lea ning o be a pa ame e s α=β=1
Fig. 7 F equencies o s ongly in o ma i e games o Di ichle -dis ibu ed u ili y alues in he case o maxi-
mum unce ain y [λ, λ]=[0,1]on he le , and in he case o educed unce ain y [λ, λ]=[0.6,0.8]on he
igh
in all combina ions wi h he pa ame e alues
=0.25,0.5,0.75,1,2,3,4,5,6,8,10,20,30,40,50
and in all combina ions wi h wo choices o unce ain y in e al. The esul s a e shown in
Fig.7.
The e a e some no iceable di e ences be ween Figs. 5and 7. Fi s , he highes equencies
o s ongly in o ma i e games a e oughly wice as high when u ili y alues a e Di ichle -
dis ibu ed as when u ili y alues a e be a-dis ibu ed. This holds bo h in he case o maximum
unce ain y (0.15 s 0.08) and in he case o educed unce ain y (0.016 s 0.025). Second,
he equency o s ongly in o ma i e games o 2 ×2 games is nea ly independen o he
dis ibu ion pa ame e s when he u ili y alues a e sampled independen ly om he be a
dis ibu ion, whe eas i depends hea ily on he dis ibu ion pa ame e s when he game ma ix
is sampled in i s en i e y om a Di ichle dis ibu ion.
Figu e8ins ead shows he equencies o s ongly in o ma i e games as he unce ain y
in e al sh inks. Excep o he di e ences jus obse ed, he wo g aphs o Fig.8look simila
o hose o Fig.6.
Lea ning dynamics The p obabili y o encoun e ing a s ongly in o ma i e games dec eases
as he unce ain y in e al sh inks. As we ha e seen in Sec .5, his has he consequence ha
Dynamic Games and Applica ions (2025) 15:1037–1069 1059
Fig. 8 F equencies o s ongly in o ma i e games du ing lea ning. On he le : equencies o di e en
pa ame e s o he Di ichle dis ibu ion du ing lea ning o wo ac ions. On he igh : equencies o di e en
numbe s o ac ions du ing lea ning o Di ichle pa ame e s (α1, ..., αD)=(1, ..., 1)
Fig. 9 Cumula i e numbe o s ongly in o ma i e games (y-axis) o e 10000 games andomly gene a ed by
d awing i.i.d. u ili y alues om a symme ic be a dis ibu ion (x-axis). The numbe o games be ween wo
subsequen s eps in he cu es co esponds o he wai ing ime be ween a s ongly in o ma i e game and he
nex s ongly in o ma i e game
he o al numbe o s ongly in o ma i e games o e ime g ows slowe han linea ly when
he agen s lea n as hey play. Figu es9and 10 illus a e his e ec by plo ing he cumula i e
numbe o s ongly in o ma i e games unde di e en dis ibu ional assump ions.
All he cu es plo ed a e conca e, illus a ing he ac ha he wai ing ime be o e he
nex s ongly in o ma i e game inc eases as he o al numbe o s ongly in o ma i e games
inc eases. In pa icula , he g ow h a e o he 2 ×2 games wi h uni o mly dis ibu ed u ili y
alues (α=β=1) is consis en wi h a g ow h a e o T2/3s ongly in o ma i e games a e
a o al o Tgames ha e been played (Fig. 9, op igh inse ).
1060 Dynamic Games and Applica ions (2025) 15:1037–1069
Fig. 10 Cumula i e numbe o s ongly in o ma i e games (y-axis) o e 10000 games andomly gene a ed by
d awing i.i.d. ma ices o u ili y alues om a Di ichle dis ibu ion (x-axis). The numbe o games be ween
wo subsequen s eps in he cu es co esponds o he wai ing ime be ween a s ongly in o ma i e game and
he nex s ongly in o ma i e game
Figu e9also shows ha andom 2×2 games a e mo e likely o be s ongly in o ma i e han
games wi h n>2 when he u ili ies a e d awn independen ly om a be a dis ibu ion. This
con as is s onges when αand βa e close o 0 and is ba ely de ec able when α=β=10.
As Fig.10 shows, andom 2 ×2 games a e also mo e likely o be s ongly in o ma i e
in he Di ichle -dis ibu ed case when he pa ame e s α1=···=αDa e la ge . When he
pa ame e s α1=···=αDa e close o 0 ins ead, he in o ma i i y wi h n=2 is lowe han
he in o ma i i y wi h n>2.
7 Discussion
One o he mos c ucial s eps in he de elopmen o mode n psychology was he ealiza ion
ha an exclusi e ocus on exp essed beha io had s a ed o weigh down he discipline: in
o de o explain he beha io s, he concep o a p i a e and unobse able men al p ocess
would ha e o be ein oduced. Recen wo ks in biology and e hology ha e s a ed o sugges
ha he same migh hold o animal beha io in gene al [14,21,23]. In e olu iona y game
heo y, ideas o his kind ha e gi en ise o s udies on he e olu ion o p e e ences, which
ele a e he le el o explana ion om exp essed beha io s o subjec i e u ili ies (e.g., [1,9,
11]).
The p esen pape oo can be ead as an a emp o explain obse able beha io s in e ms
o mo e gene al p ocesses, in his case di e en ules o decision making unde unce ain y.
Ra he han conside ing a ious decision c i e ia as compe ing philosophical heo ies, hey
can be in e p e ed as high-le el s a egies ha may coexis o play o agains each o he .
Dynamic Games and Applica ions (2025) 15:1037–1069 1067
A.2.3 P oo o Theo em 15
P oo Fix [λ, λ]wi h λ < λ. By P oposi ion 14, we can ind a non-emp y hype cube
(L◦,H◦)4whe e he dis ibu ion o (a◦,b◦,c◦,d◦)is absolu ely con inuous. Hence, by
epea ing he a gumen gi en in P oposi ion 12 wi hin he smalle hype cube (L◦,H◦)4we
ge
P(L◦<a◦<H◦)>0
P(L◦<d◦<a◦|a◦)>0
P(a◦<c◦<H◦|a◦)>0
P(d◦<b◦<d◦+c◦−a◦|a◦,c◦,d◦)>0
By Theo em 9, his ensu es he posi i e p obabili y o he p o ile (σ◦,σ◦)being he sole
symme ic equilib ium o [λ, λ].Thecaseo (a•,b•,c•,d•)is analogous.
A.2.4 P oo o Theo em 16
P oo Fix [λ, λ]wi h λ= λ. By Theo em 15, he e is posi i e p obabili y o sampling
u1,1,u1,2,u2,1,u2,2 ha sa is y he an i-coo dina ion inequali ies u1,1<u2,1and u2,2<
u1,2, and he inequali ies
u◦
1,1>u◦
2,2,u◦
2,1−u◦
1,1>u◦
1,2−u◦
2,2,and λ<λ
◦<λ,
whe e u◦
1,1,u◦
1,2,u◦
2,1,u◦
2,2a e he equi alen o a◦,b◦,c◦,d◦ o u1,1,u1,2,u2,1,u2,2=
a,b,c,d. Absolu e con inui y implies ha n×ngames whe e
ui,j<min{u1,j,u2,j} o 2 <i≤nand 1 ≤j≤n
ha e posi i e p obabili y oo. In such games, all ac ions di e en om Iand II u n ou o
be s ic ly domina ed and will no be chosen by ei he playe ype in any equilib ium. The
only possible equilib ia a e hus p o iles whe e only ac ions Iand II a e chosen. Bu hen
no ice ha om
u◦
1,1>u◦
2,2,u◦
2,1−u◦
1,1>u◦
1,2−u◦
2,2,and λ<λ
◦<λ,
i ollows ha he only possible equilib ium is when ype Mplays Iand yspe Rplays
II. Hence, he only equilib ium o he n×ngame is (σ ◦,σ◦), which p o es ha s ongly
in o ma i e n×ngames ha e posi i e p obabili y. The a gumen o (σ •,σ•)is analogous.
Au ho Con ibu ions The wo au ho s con ibu ed equally o he de elopmen o he esul s and o he w i ing
o he pape .
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Da a A ailabili y The Py hon sc ip ha gene a ed he esul s o Sec . 6is a ailable he e.
Decla a ions
Con lic o in e es The e a e no con lic o in e es o be disclosed.
E hical app o al No applicable.
1068 Dynamic Games and Applica ions (2025) 15:1037–1069
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