Mean-field ranking games with diffusion control

Anki chne , S.; Kazi-Tani, N.; Wend , J.; Zhou, C.
A icle — Published Ve sion
Mean- ield anking games wi h di usion con ol
Ma hema ics and Financial Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Anki chne , S.; Kazi-Tani, N.; Wend , J.; Zhou, C. (2024) : Mean- ield anking
games wi h di usion con ol, Ma hema ics and Financial Economics, ISSN 1862-9660, Sp inge ,
Be lin, Heidelbe g, Vol. 18, Iss. 2, pp. 313-331,
h ps://doi.o g/10.1007/s11579-024-00354-2
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315443
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h p://c ea i ecommons.o g/licenses/by/4.0/
Ma hema ics and Financial Economics (2024) 18:313–331
h ps://doi.o g/10.1007/s11579-024-00354-2
Mean- ield anking games wi h di usion con ol
S. Anki chne 1
·N. Kazi-Tani2
·J. Wend 1
·C. Zhou3
Recei ed: 30 June 2023 / Accep ed: 9 Janua y 2024 / Published online: 26 Ma ch 2024
© The Au ho (s) 2024
Abs ac
We conside a s ochas ic di e en ial game, whe e each playe con inuously con ols he di -
usion in ensi y o he own s a e p ocess. The playe s mus all choose om he same di usion
a e in e al [σ1,σ
2], and ha e indi idual andom ime ho izons ha a e independen ly d awn
om he same dis ibu ion. The playe s whose s a es a hei espec i e ime ho izons a e
among he bes p∈(0,1)o all e minal s a es ecei e a ixed p ize. We show ha in he
mean ield e sion o he game he e exis s an equilib ium, whe e he ep esen a i e playe
chooses he maximal di usion a e when he s a e is below a gi en h eshold, and he mini-
mal a e else. The symme ic n- old uple o his h eshold s a egy is an app oxima e Nash
equilib ium o he n-playe game. Finally, we show ha he mo e ime a playe has a he
disposal, he highe he chances o winning.
Keywo ds Di usion con ol ·Game ·Rank-based ewa d ·Mean ield limi ·Oscilla ing
B ownian mo ion
Ma hema ics Subjec Classi ica ion P ima y: 91A15; seconda y: 91A06 ·91A10 ·91A16 ·
93E20
BS. Anki chne
[email p o ec ed]
N. Kazi-Tani
nabil.kazi- ani@uni -lo aine.
J. Wend
[email p o ec ed]
C. Zhou
[email p o ec ed]
1Ins i u e o Ma hema ics, Uni e si y o Jena, E ns -Abbe-Pla z 2, 07743 Jena, Ge many
2Ins i u Elie Ca an de Lo aine, Uni e si é de Lo aine, UFR MIM, 3 ue Augus in F esnel, 57073
Me z Cedex 03, F ance
3Depa men o Ma hema ics and Risk Managemen Ins i u e, Na ional Uni e si y o Singapo e, 10
Lowe Ken Ridge Road, 119076 Singapo e, Singapo e
123
314 Ma hema ics and Financial Economics (2024) 18:313–331
1 In oduc ion
We conside he ollowing s ochas ic di e en ial game: each playe can con ol he luc ua ion
in ensi y o he own s a e p ocess, up o an indi idual andom ime ho izon. The con ols o a
playe a e indi idually de ined as a se o p og essi ely measu able p ocesses, wi h espec o
a il a ion modeling he playe ’s in o ma ion low, wi h alues in a bounded in e al [σ1,σ
2],
whe e 0 <σ
1<σ
2a e he same o all. The playe s whose e minal s a es a e among he
highes p∈(0,1) ecei e a ixed p ize, se o be equal o one. The o he playe s do no
ecei e any hing.
The game models in s ylized o m compe i ions whe e only he bes pe o ming agen s
ecei e a ixed ewa d and whe e e e y agen can choose be ween isky and sa e ac ions.
The game hus allows o analyze he impac o ank-based ewa ds on he isk appe i e o he
compe i o s. The game has mul iple in e p e a ions. We e e o Sec . 6 o mo e de ails.
As usual, we all back on he concep o Nash equilib ia o p edic ing he playe s’ beha io .
Gi en he discon inuous ewa ds, i u ns ou o be di icul o compu e explici equilib ia.
Mo eo e , i seems al eady di icul o p o e exis ence by abs ac means. A way ou o
games wi h many playe s is o all back on he game’s mean ield e sion and o de i e an
app oxima e equilib ium. The equilib ium o he mean ield game he e consis s o a con ol
and he espec i e s a e dis ibu ion a he e minal ime. Unde egula i y condi ions on he
ewa ds and on he s a e equa ion coe icien s, exis ence o equilib ia in mean ield games
wi h di usion con ol and common noise is p o ed in [2] using a elaxed o mula ion o
he p oblem and he heo y o second o de BSDEs. O he con ibu ions conside mean ield
games wi h di usion con ol, such as [8] in a P incipal-Agen se ing, wi h applica ions o
op imal ene gy demand managemen o [6,7] in he case o ex ended mean ield games wi h
con ol in e ac ions.
In his pape , we a oid second o de BSDEs and show, using a di ec a gumen , ha
he e exis s an equilib ium wi h a h eshold con ol ha consis s in choosing he maximal
di usion a e σ2when he s a e is below a gi en h eshold, and he minimal a e σ1else. The
co esponding equilib ium dis ibu ion is he dis ibu ion o an oscilla ing B ownian mo ion
a he e minal ime.
The game bea s simila i ies wi h he di usion con ol game s udied in [1]. In con as
o [1], howe e , we allow he e he agen s o be he e ogeneously in o med. Each playe is
assumed o obse e he own s a e p ocess, bu he assump ion on how much he agen s know
beyond his emains gene al. Some o he playe s may know, e.g., he ime ho izons, and some
no . Whe he a playe knows he own ime ho izon o he ime ho izons o he opponen s
u ns ou o ha e no e ec on he equilib ium. Indeed, in he many playe game he empi ical
dis ibu ion o he ime ho izons is close o T. Thus knowing Tis su icien o implemen ing
an app oxima e equilib ium con ol. Simila ly, i has no e ec on he equilib ium whe he
he agen s can obse e he s a e p ocesses o he opponen s o no . Fo implemen ing he
h eshold con ol o he equilib ium each agen needs only o obe e he own p ocess.
By de ining he s a e dynamics in e ms o solu ions o con olled ma ingale p oblems and
choosing con ols o open loop ype, we ob ain a model amewo k ha allows o co e gene al
in o ma ion s uc u es, in pa icula si ua ions whe e a playe has only pa ial knowledge
abou he o he playe s’ s a es and abou he indi idual andom ime ho izons. In he game
o [1] he s a e dynamics a e desc ibed in e ms o s ochas ic di e en ial equa ions and he
playe s’ con ols a e modeled as closed loop con ols.
In he app op ia e mean ield e sion o he game he ep esen a i e playe knows he
dis ibu ion To he ime ho izon, bu he ac ual ime ho izon a i es unp edic ed. The
123
Ma hema ics and Financial Economics (2024) 18:313–331 315
h eshold desc ibing he equilib ium con ol s ongly depends on he dis ibu ion T.Fo he
cases whe e Tis an exponen ial dis ibu ion o a uni o m dis ibu ion, we cha ac e ize he
h eshold o he equilib ium con ol as he unique oo o a simple equa ion.
The a icle is o ganized as ollows: in Sec .2, we in oduce he game model in mo e de ail.
Fo de i ing a candida e o an app oxima e Nash equilib ium, we s udy he co esponding
mean ield game in Sec .3and show ha an equilib ium con ol is gi en by a h eshold
con ol. In Sec .4, we show ha he n- uple consis ing o he mean ield equilib ium con ols
is an O(n−1/2)-Nash equilib ium o he n-playe game. This means, in he app oxima e Nash
equilib ium playe s only use in o ma ion abou hei own s a e and choose maximal di usion
in ensi y below he op imal h eshold and minimal di usion in ensi y abo e. All addi ional
in o ma ion abou he opponen s’ s a es o he andom imes is i ele an . In Sec .5,we
analyze he winning p obabili y o a gi en playe depending on he ime ho izon Tand
in Sec .6, we p o ide a pa icula applica ion o ou model se ing o online compe i ions.
Finally, we conside in Sec . 7an ex ension o he game o mo e gene al ewa d unc ions
ha a e con inuous, ha e exponen ial g ow h, sa is y a symme y condi ion, and a e con ex
below a ce ain h eshold and conca e abo e. We show ha he same uple consis ing o he
mean ield equilib ium con ols is also an app oxima e Nash equilib ium o hese ewa d
unc ions.
2 Game model
We desc ibe he playe s’ s a es by means o con olled ma ingale p oblems. To his end,
le R+:= [0,∞),andC(R+,Rn)deno e he space o con inuous unc ions :R+→Rn
equipped wi h he me ic
d(ω1,ω
2):= ∞

n=1
1
2n
sup ∈[0,n]|ω1( )−ω2( )|
1+sup ∈[0,n]|ω1( )−ω2( )|,ω
1,ω
2∈C(R+,Rn).
Le B(C(R+,Rn))deno e he co esponding Bo el σ-algeb a on C(R+,Rn)and deno e by
X=(X1,...,Xn) he canonical p ocess on C(R+,Rn), i.e., Xi
(ω) := ωi( ), ≥0, o
i=1,...,nand ω∈C(R+,Rn). We e e o [18], Sec ion 1.3, o mo e de ails on he
cons uc ion o his measu able space and i s p ope ies.
Le Tbe a p obabili y measu e on R+, equipped wi h he usual Bo el σ-algeb a B(R+),
ha desc ibes he dis ibu ion o he playe s’ andom imes. We suppose ha Tsa is ies:
Assump ion 2.1
∞
0
1
√ T(d )<∞.
Le BRn
+be he Bo el σ-algeb a on Rn
+and de ine he measu e Tn=n
i=1Ton Rn
+,
i.e., Tnis he n- old p oduc o T.Wede ineτ=(τ1,...,τ
n)as canonical map on Rn
+,
i.e., τ(ω) =(τ1(ω), . . . , τn(ω)) =(ω1,...,ω
n) o any ω∈Rn
+. No e ha τ1,...,τ
na e
independen and iden ically dis ibu ed unde Tnby de ini ion and he law o τiis gi en by
T.
We de ine a common measu able space by se ing:
(i) := C(R+,Rn)×Rn
+,
(ii) F:= B(C(R+,Rn))⊗BRn
+, i.e., he smalles σ-algeb a con aining all se s o he
o m A×B o A∈B(C(R+,Rn)),B∈BRn
+.
123
316 Ma hema ics and Financial Economics (2024) 18:313–331
We ex end he de ini ions o Xand τ o by se ing X(ω1,ω
2):= X(ω1)and τ(ω1,ω
2):=
τ(ω2) o (ω1,ω
2)∈. We de ine on (, F) he il a ion (F ) ≥0by
F =σ(τ1,...,τ
n)∨σ(Xs:0≤s≤ ), ≥0.
The il a ion (F ) ≥0desc ibes he o e all in o ma ion low including he in o ma ion abou
he alues o he ime ho izons. Mo eo e , o Playe i, we in oduce he il a ion (Fi
) ≥0
desc ibing he p i a e in o ma ion o Playe i.Weassume ha σXi
s:0≤s≤ ⊆Fi
⊆F
o all ≥0.
Le Aideno e he se o all (Fi
) ≥0-p og essi ely measu able α:×R+→[σ1,σ
2].
We e e o elemen s o Aias admissible con ols o s a egies and w i e An o he p oduc
A1×... ×An. We cha ac e ize he law o he s a e p ocesses by means o ma ingale
p oblems, in oduced by S oock and Va adhan. We e e o he monog aph [18] o mo e
de ails on ma ingale p oblems.
De ini ion 2.2 Le α=(α1,...,αn)∈An. Then, a p obabili y measu e Pαon (, F)is
called a easible s a e dis ibu ion i
(i) Pα◦X−1
0=δ0,
(ii) Pα(C(R+,Rn)×B)=Tn(B) o all B∈BRn
+,
(iii) o all ∈C2
c(Rn,R), i.e., o all wice con inuously di e en iable :Rn→Rwi h
compac suppo , he p ocess M ,de inedby
M
s:= (Xs)− (0)−1
2s
0
n

j=1αj
2
∂jj (X )d ,s≥0,(1)
is an (F ) ≥0-ma ingale unde Pα.
We deno e by Q(α) he se o all easible s a e dis ibu ions Pα.
Rema k 2.3 The assump ions on he uple αin he p e ious de ini ion do no exclude ha
Q(α) is emp y. Fo a uple α o be a Nash equilib ium i is necessa y, howe e , ha Q(α) =∅
(see De ini ion 2.8 below).
Rema k 2.4 No e ha condi ion (ii) implies ha each andom ime τihas dis ibu ion T.
Mo eo e , condi ion (iii) yields ha each s a e p ocess Xiis a local (F ) ≥0-ma ingale and
Xi,Xj =
0(αi
s)2ds,i i=j,
0,else, ≥0.
The s a e p ocesses a e e en ue ma ingales ha a e squa e in eg able, i.e., EPα(Xi
)2<
∞, ≥0, because he con ols a e bounded and hus, EPαXi,Xi <∞ o all ≥0
(see, e.g., [17], Sec ion II.6, Co olla y 3, p.73).
Rema k 2.5 The de ini ion o he il a ions (Fi
) ≥0is a he gene al. Each il a ion (Fi
) ≥0
can con ain in o ma ion abou he o he playe s’ s a es and andom imes. The e o e, he
game se ing co e s he cases whe e playe s can o canno obse e each o he , and ha e
knowledge abou he andom ime ho izons.
I , e.g., Fi
=F , each playe can obse e he s a e p ocesses o he opponen s and make
he s a egy depend on he opponen s’ s a e ajec o ies. Mo eo e , each playe has p io
knowledge o he andom ime ho izons. I , howe e , Fi
=σ(Xi
s:0≤s≤ ), hen each
playe can only obse e he own s a e p ocess. Nei he in o ma ion abou he opponen s’
s a es no abou he andom imes is a ailable.
123

Ma hema ics and Financial Economics (2024) 18:313–331 317
Despi e his qui e gene al in o ma ion s uc u e, we show in Theo em 4.1 ha an app oxi-
ma e Nash equilib ium is gi en by a uple o h eshold con ols depending only on he posi ion
o he single playe ’s s a e p ocess.
Rema k 2.6 The s a e p ocesses can be equi alen ly desc ibed as weak solu ions o an n-
dimensional SDE (o ia a s ochas ic in eg al w. . . some B ownian mo ion). Indeed, o some
con ol α∈Anwi h Q(α) =∅and P∈Q(α), he e exis s an n-dimensional B ownian
mo ion Won (, F,(F ) ≥0,P)such ha he s a e p ocesses X1,...,Xnsa is y P-a.s.
Xi
=
0
αi
sdWi
s, ≥0,i=1,...,n,(2)
and (, F,(F ) ≥0,P,X,W)is a weak solu ion o (2) (see, e.g., [11], P oposi ion 5.4.6).
We use his connec ion in he mean ield game p esen ed in Sec .3, and hus, cha ac e ize
he s a e p ocesses ia s ochas ic in eg als.
Rema k 2.7 Fo all measu able eedback unc ions a:R+×Rn→[σ1,σ
2]n, he e exis s
a solu ion Pa o he ma ingale p oblem (1) wi h (αs)s≥0=a(s,X1
s,...,Xn
s)s≥0.This
ollows, e.g., om [13], Theo em 2.6.1, and [11], P oposi ion 5.4.11. I he il a ion (Fi
) ≥0
con ains he in o ma ion abou all s a es, i.e. i σ(Xs:0≤s≤ )⊂Fi
, hen he con ol
a(s,X1
s,...,Xn
s)s≥0is con ained in Aiand {Pa}⊆Q(α). This pa icula ly holds o
con ol uples whe e each en y is a h eshold con ol, i.e., each en y is gi en by he eedback
unc ion
mb(x)=σ2,i x≤b,
σ1,i x>b,(3)
whe e b∈R. In his case, he solu ion o he ma ingale p oblem (1) is e en unique: Rema k
3.1 below implies ha he SDE (7) wi h m=mbhas a unique s ong solu ion and [11],
Co olla y 5.4.9, hen implies ha uniqueness o he ma ingale p oblem (1) holds.
We suppose ha each playe aims a maximizing he p obabili y o he own s a e a he
e minal andom ime o be g ea e han he empi ical (1−p)-quan ile o all s a es a he
indi idual andom imes. Mo e p ecisely, le
μn=1
n
n

i=1
δXi
τi
be he empi ical dis ibu ion o he playe s’ s a es a he e minal imes. We de ine he empi -
ical (1−p)-quan ile by q(μn,1−p)=in { ∈R:μn((−∞, ])≥1−p}.No e ha
Xi
τi>q(μn,1−p)i and only i he s a e o Playe iis among he bes npplaye s a he
e minal imes.
I is s anda d o p edic o explain he playe s’ beha io in e ms o (app oxima e) Nash
equilib ia, which a e he e de ined as ollows.
De ini ion 2.8 Le ε≥0. A uple α=(α1,...,α
n)∈Anwi h Q(α) =∅is called ε-Nash
equilib ium o he n-playe game i o all i∈{1,...,n}and Pα∈Q(α),weha e
Pα(Xi
τi>q(μn,1−p)) +ε≥sup
β∈Ai
sup
P∈Q(α−i,β)
P(Xi
τi>q(μn,1−p)), (4)
whe e (α−i,β)=(α1,...,α
i−1,β,α
i+1,...,α
n)and sup ∅=−∞.
123
318 Ma hema ics and Financial Economics (2024) 18:313–331
No e ha o ε=0, he uple αin De ini ion 2.8 is a Nash equilib ium in he usual sense.
In he case ε>0, he uple αis also called an app oxima e Nash equilib ium.
We do no assume ha he solu ions o he ma ingale p oblem (1) a e unique by limi ing
he se o con ols. Hence, we equi e ha (4) holds o all solu ions o he ma ingale p oblem
(1). We emphasize ha o an a bi a y con ol uple α∈An, uniqueness o solu ions o he
ma ingale p oblem (1) can ail. Fo example, i n≥3, hen i was shown in [15] ha he e
exis s a di usion coe icien , and hence, an ope a o such ha he co esponding ma ingale
p oblem does no ha e a unique solu ion. Equi alen ly, he e exis s a di usion coe icien
σ:Rn→Rn×n ha is uni o mly ellip ic and such ha o he co esponding SDE uniqueness
in law ails (see also [5], Example 1.24). Thus, one can in e p e (4) as ollows: Playe ihas
no incen i e o change he s a egy om α o β, no ma e which dis ibu ion om Q(α−i,β)
is chosen. We s ess, howe e , ha o he app oxima e Nash equilib ia de i ed om he
co esponding mean ield game, uniqueness is always sa is ied.
We do no compu e an exac Nash equilib ium o he n-playe game. As in [1], we compu e
an app oxima e Nash equilib ium o la ge games by conside ing he mean ield limi o he
game. We show ha a mean ield equilib ium s a egy is gi en by a h eshold con ol.
Rema k 2.9 Le T>0andτ1=... =τn=T, i.e., T=δT.Se Fi
=σ(Xs:0≤s≤ ),
≥0, i=1,...,n, and conside only con ols o he o m α =a( ,X1
,...,Xn
) o some
a:R+×Rn→[σ1,σ
2]. Then, he abo e model is equi alen o he game model p esen ed
in he a icle [1].
3 Mean ield game
In his sec ion we desc ibe he mean ield e sion o he game in oduced in Sec .2.
Le 0 <σ
1<σ
2and (, F,(F ) ≥0,P)be a comple e il e ed p obabili y space sa is y-
ing he usual condi ions. We suppose ha (, F,(F ) ≥0,P)suppo s an (F ) ≥0-B ownian
mo ion (W ) ≥0and an R+- alued andom a iable τ. We assume ha (W ) ≥0and τa e
independen and
Eτ−1
2<∞,(5)
see Assump ion 2.1 abo e. Le ˜
Abe he se o all p ocesses α:×R+→[σ1,σ
2] ha
a e (F ) ≥0-p og essi ely measu able. Gi en an agen chooses he con ol α∈˜
A, hes a e
p ocess is de ined by
Xα
:= 
0
αsdWs, ≥0.(6)
Rema k 3.1 All eedback con ols wi h a eedback unc ion m:R→[σ1,σ
2]o bounded
a ia ion a e con ained in ˜
A. Indeed, he SDE
dX =m(X )dB
,X0=0,(7)
has a weak solu ion because o Theo em 2.6.1 in [13], and pa hwise uniqueness applies
acco ding o esul s in [16]. Hence, he e exis s a unique s ong solu ion Xm o (7)(c .
Sec ion 5.3 in [11]) and he con ol (m(Xm
)) ≥0is con ained in ˜
A.
Le p∈(0,1)and deno e by q(μ, 1−p) he (1−p)-quan ile o some p obabili y
measu e μ∈P(R), i.e., o a Bo el p obabili y measu es μon R,q(μ, 1−p)=in { ∈
123
Ma hema ics and Financial Economics (2024) 18:313–331 319
R:μ((−∞, ])≥1−p}.I μ=Law(Xα
τ), i.e., μis he law o Xα
τ o a con ol α∈˜
A,we
w i e q(Xα
τ,1−p) o q(μ, 1−p). In he mean ield game ha co esponds o he n-playe
game desc ibed in Sec .2, a single playe wan s o maximize he p obabili y o being la ge
han he popula ion (1−p)-quan ile o e all admissible con ols. In he mean ield game,
we de ine equilib ia in he ollowing sense:
De ini ion 3.2 A uple (μ∗,α∗)∈P(R)ט
Ais called mean ield equilib ium i
(i) o all α∈˜
A,
PXα∗
τ>q(μ∗,1−p)≥P(Xα
τ>q(μ∗,1−p)),
(ii) i holds μ∗=Law(Xα∗
τ).
Lemma 3.3 Le b ∈R.Then
P(Xmb
τ>b)=max
α∈˜
A
P(Xα
τ>b), (8)
whe e mbis de ined in Eq.(3).
P oo The esul ollows om he di usion con ol p oblem s udied by McNama a [14]. In
mo e de ail, le α∈˜
A. Assume, wi hou loss o gene ali y, ha he e exis s a egula condi-
ional p obabili y Q :R+×F→[0,1] o Fgi en τ. Because τand Wa e independen ,
Wis also a B ownian mo ion unde Q( ,·) o Pτ-a.e. ∈R+. Hence, we see ha
Q( ,{Xα
τ>b})=Q( ,{Xα
>b})≤Q( ,{Xmb
>b}), o Pτ-a.e. ∈R+,
using ei he [14], Rema k 8, o [19], P oposi ion C.5. Finally,
P(Xα
τ>b)=∞
0
Q( ,Xα
>b)Pτ(d )≤∞
0
Q( ,Xmb
>b)Pτ(d )=P(Xmb
τ>b).
We e e o [11], Sec ion 5.3.C, o [10], Sec ion 1.3, o mo e de ails on egula condi ional
p obabili ies. 
Lemma 3.3 implies ha he op imal con ol o (8) is he h eshold con ol mb.In he
ollowing, we jus w i e Xb o he s a e Xmb. The p ocess Xbis a so-called oscilla ing
B ownian mo ion (OBM) wi h h eshold b, in oduced in [12]. In mo e de ail, OBM is de ined
as ollows.
De ini ion 3.4 Le b∈R. We call he solu ion Xbo he SDE
dX =mb(X )dB
,X0=0,(9)
oscilla ing B ownian mo ion (OBM) wi h h eshold band ini ial alue 0.
The e exis s indeed a unique s ong solu ion o he SDE (9) because o Rema k 3.1.
Fo OBMs, one can explici ly calcula e he p obabili y densi y unc ion and he cumula i e
dis ibu ion unc ion. Fo he eade ’s con enience, we ecall he ollowing esul on OBMs.
P oposi ion 3.5 Le b ∈Rand Xbbe an OBM wi h h eshold b and ini ial alue 0. Then,
o all >0, he andom a iable Xb
has a p obabili y densi y unc ion p( ,−b,·−b)wi h
123
320 Ma hema ics and Financial Economics (2024) 18:313–331
espec o he Lebesgue measu e, whe e
p( ,x,y)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
2σ1
σ2(σ1+σ2)
1
√2π e−(x
σ1−y
σ2)21
2 ,i x ≥0,y<0,
2σ2
σ1(σ1+σ2)
1
√2π e−(y
σ1−x
σ2)21
2 ,i x <0,y≥0,
1
σ1√2π e−(y−x)2
2σ2
1 +σ2−σ1
σ1+σ2e−(y+x)2
2σ2
1 ,i x ≥0,y≥0,
1
σ2√2π e−(y−x)2
2σ2
2 +σ1−σ2
σ1+σ2e−(y+x)2
2σ2
2 ,i x <0,y<0,
o all x,y∈R. Mo eo e , he cumula i e dis ibu ion unc ion o Xb
is gi en by
Fb
(x)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
σ2√ −σ2−σ1
σ1+σ2x−2b
σ2√ ,i x <b,b≥0,
2σ2
σ1+σ2x−b1−σ1
σ2
σ1√ −σ2−σ1
σ1+σ2,i x ≥b,b≥0,
2σ1
σ1+σ2x−b1−σ2
σ1
σ2√ ,i x <b,b<0,
x
σ1√ −σ2−σ1
σ1+σ22b−x
σ1√ ,i x ≥b,b<0.
(10)
The p oo o P oposi ion 3.5 ollows ei he om [12], Theo em 1, o [19], P oposi ion
B.2 and P oposi ion B.4. Fo mo e de ails on OBMs, we e e o [12]and[19], Appendix
B. Using he independence o he OBM Xband he andom ime τ, one can de i e om
P oposi ion 3.5 he cumula i e dis ibu ion unc ion o Xb
τ.
Lemma 3.6 Le b ∈Rand Xbbe an OBM wi h h eshold b and ini ial alue 0. Then he
cumula i e dis ibu ion unc ion o he andom a iable Xb
τ, deno ed by Fb
τ, is gi en by
Fb
τ(x)=∞
0
Fb
(x)Pτ(d ), x∈R.
We e e o P oposi ion B.8 in [19] o he p oo o Lemma 3.6. One can show ha he
unc ions Fb
and Fb
τa e Lipschi z con inuous in he s a e a iable as well as in he h eshold
b(see Appendix B in [19] o mo e de ails).
The s anda d app oach o sol e mean ield games is o conside mappings om p obabili y
dis ibu ions o he dis ibu ions o op imally con olled s a es and ind hei ixed poin s, he
so-called equilib ium measu es (see, e.g., [3]and[4]). Howe e , Lemma 3.3 allows o s udy
he dis ibu ions o OBMs only, which can be pa ame e ized by he eal- alued h eshold
b∈R. Fo iden i ying equilib ia, i su ices o show ha he unc ion
:R→R,b→ q(Xb
τ,1−p),
has a unique ixed poin . Indeed, i (b)=b, henb=q(Xb
τ,1−p). Lemma 3.3 u -
he implies ha P(Xb
τ>q(Xb
τ,1−p)) =maxβ∈MP(Xβ
τ>q(Xb
τ,1−p)); hence,
(Law(Xb
τ), mb)is an equilib ium. The main esul o his sec ion is he ollowing:
Theo em 3.7 The e exis s a unique b∗∈Rsuch ha q Xb∗
τ,1−p=b∗. The uple
(Law(Xb∗
τ), mb∗)is a mean ield equilib ium.
123
Ma hema ics and Financial Economics (2024) 18:313–331 327
No ice ha (31) implies ha he h eshold le el b∗o Theo em 3.7 is posi i e. We deno e
he mean ield equilib ium dis ibu ion by μ∗. Recall ha μ∗is he law o he OBM wi h
h eshold b∗a an independen andom ime wi h dis ibu ion T.
Now suppose ha nis la ge and ha in he game wi h nplaye s e e yone con ols hei
s a es wi h he h eshold con ol mb∗. We selec one playe and assume ha he ealized ime
ho izon is . Then he p obabili y o his pa icula playe o be among he bes pa he end
o he game is app oxima ely gi en by
w( ):= PXb∗
>qμ∗,1−p=1−Fb∗
(b∗)=2σ2
σ1+σ21−b∗
σ2√ .
We e e o he unc ion was he winning p obabili y. No e ha he winning p obabili y is
con inuous and inc easing in . Mo eo e , we ha e
lim
↓0w( )=0.
Thus, i he ac ual ime ho izon is small, hen he winning p obabili y is close o ze o. This
is plausible, since he s a e p ocess s a s in ze o and is s opped ea ly, and hence a ains he
posi i e le el b∗wi h a small p obabili y only.
Nex obse e ha
lim
→∞w( )=σ2
σ1+σ2
.
Thus, he winning p obabili y is bounded by σ2
σ1+σ2, and he bound is almos a ained o
la ge ime ho izons . The bound co esponds o he expec ed a e age ime ha an OBM is
spending abo e he h eshold in he long un. Indeed, i espec i e o he h eshold b, one can
show ha
lim
→∞ P(Xb
≥b)=σ2
σ1+σ2
.(32)
The bound allows also o a con ol heo e ical in e p e a ion: o his end conside he e godic
con ol p oblem wi h a ge unc ional
J(α) := lim in
T→∞
1
TET
0
1{Xα
≥b}d ,α∈˜
A,
whe e ˜
Ais de ined as in Sec .3and Xαis de ined as in (6). One can show ha mbis an
op imal con ol (see Rema k 8 in [14]) and hence, using (32), supα∈˜
AJ(α) =σ2
σ1+σ2.
6 An applica ion o online compe i ions
Ou game o mula ion is gene ic and can co espond o a a ie y o p ac ical si ua ions. Fo
example, he game applies o manage s o mu ual unds s i ing o hei unds o be among
he bes pe o ming. This applica ion is desc ibed, o homogeneous manage s, in de ail in
Sec ion 7 o [1].
We he e p o ide an al e na i e applica ion o online compe i ions in which, usually, eams
ha e o collabo a e in o de o sol e a p oblem and p o ide a solu ion wi hin a limi ed ime
ame. Hacka ons a e examples o such compe i ions, bu also da a science and machine lea n-
ing ela ed compe i ions o e ed on some pla o ms such as Kaggle, D i enDa a, AIC owd
e c. Du ing hese compe i ions, a sco e, based on a gi en e alua ion me ic, can be calcula ed
123

328 Ma hema ics and Financial Economics (2024) 18:313–331
by each pa icipan and a "public leade boa d" displays he ela i e anks du ing he whole
leng h o he compe i ion.
In his con ex , he p i a e s a e Xi
o playe iis in e p e ed as he sco e displayed a ime
. The numbe o eams in ol ed in online compe i ions can each se e al ens o housands,
enough o conside he mean ield app oxima ion.
Le el o isk.
We in e p e he di usion con ol as he possibili y o con ol he le el o isk aken. In he
online con es ’s se ing, eams can indeed choose o y and use well es ablished me hods,
whose obus ness is al eady s udied and o which e o s can be mo e easily and quickly
co ec ed. We in e p e his as low isk and di usion coe icien σ1. On he o he hand, he
choice o di usion coe icien σ2is in e p e ed as ying new echniques, o which he e is
less o no expe ience. Ou esul s igo ously show in his con ex ha i sub asks a e going
well, playe s will play sa e, whe eas i a gi en eam is poo ly pe o ming on he e alua ion
me ic, i has an incen i e o play isky and y less common s a egies.
Obse abili y.
Pa icipan s o online compe i ions can submi a solu ion, and in ha case hei cu en
sco e is calcula ed and displayed o all pa icipan s. Howe e , i a eam ob ains a solu ion
and es s i o line on he p o ided da a se , hen he associa ed sco e is no isible du ing he
ime i is no submi ed on he pla o m. A gi en eam can choose o e eal i s solu ion and
sco e only owa ds he end o he submission pe iod.
Mo eo e , i is possible o design online compe i ions wi h only pa ial obse abili y: one
could easily imagine ha eams only obse e he bes sco e, o a gi en quan ile o he sco es
dis ibu ion, o assess hei ela i e pe o mance.
Ou esul s show ha i he numbe o playe s is la ge, obse abili y does no ma e , a
leas o he pa icula ype o discon inuous c i e ia ha we conside , which a e common
in hese compe i ions, whe e a ixed cash p ize is o e ed o he bes pe o ming eam. This
esul also holds o con inuous unc ions o he ank sa is ying he symme y condi ion gi en
in Assump ion 7.1.
Te minal ime.
We conside wo cases. Fi s ly, he case whe e he ime ho izons o all playe s a e cons an
equal o T∈(0,∞), in e p e ed as he da e a which a inal assessmen o he e alua ion
me ic is made by he con es o ganize s. Secondly, he ime ho izon τio eam iquan i ies he
esou ces i can pu in o he compe i ion, e.g. he numbe o wo king hou s. Fo example, τi
can be se p opo ional o he de e minis ic assessmen da e and he numbe o eam membe s.
Sec ion5 e eals ha la ge eams ha e an ad an age compa ed o smalle eams.
7 Ex ensions
In his sec ion, we discuss mo e gene al ewa d c i e ia o he n-playe game. In pa icula ,
we conside ewa ds a he andom ime ho izons ha a e gi en by measu able unc ions
g:R×R→Ro he s a e and he popula ion quan ile, ins ead o he “all-o -no hing”
payo gi en by he unc ion (x,q)→ 1(q,∞)(x)be o e. Fi s , we show ha he mean
ield equilib ium con ol o Sec .3is also an equilib ium o he ewa d unc ions g,i g
sa is ies a symme y and con exi y condi ion. Then, we p o e ha his equilib ium p o ides
an app oxima e Nash equilib ium o he n-playe game.
123
Ma hema ics and Financial Economics (2024) 18:313–331 329
7.1 Mean ield game
Assume ha we a e in he se ing o Sec .3. The whole analysis in Sec . 3depends on he
op imali y o he h eshold con ol mb o he pa icula choice o he ewa d 1(b,∞)(Lemma
3.3). Resul s o McNama a [14] imply ha mbis no only op imal o his ewa d bu also
o mo e gene al ewa d unc ions ha a e con inuous, ha e exponen ial g ow h, and sa is y
a con exi y condi ion and a symme y condi ion. In mo e de ail, we can gene alize Theo em
3.7 o measu able unc ions g:R×R→Rsa is ying:
Assump ion 7.1 (i) g(·,q)is con inuous and has exponen ial g ow h o any q∈R,
(ii) g(·,q)is con ex on (−∞,q]and conca e on [q,∞) o any q∈R,
(iii) o all x≥0andq∈Ri holds
σ2g(σ1x+q,q)+σ1g(−σ2x+q,q)=(σ1+σ2)g(q,q).
P oposi ion 7.2 Le b∗be gi en by Theo em 3.7 and g sa is y Assump ion 7.1. Then,
(Law(Xb∗
τ), mb∗)is also a mean ield equilib ium o he ewa d unc ion g, i.e.,
EgXb∗
τ,qXb∗
τ,1−p=sup
α∈˜
A
EgXα
τ,qXb∗
τ,1−p.
P oo As in Lemma 3.3, one can show o ixed b∈R ha
EgXb
τ,b=sup
α∈˜
A
EgXα
τ,b,
using ei he [14], Theo em 6, o [19], Theo em C.5. The assump ions on ggua an ee ha
hese heo ems apply. Mo eo e , Theo em 3.7 implies he exis ence o a unique ixed poin
b∗o he map b→ qXb
τ,1−p. Fo his ixed poin b∗,wesee ha
EgXb∗
τ,qXb∗
τ,1−p=sup
α∈˜
A
EgXα
τ,qXb∗
τ,1−p,
i.e., (Law(Xb∗
τ), mb∗)is an equilib ium o he ewa d unc ion g.
7.2 App oxima e Nash equilib ium in he n-playe game
Now, in he se ing o Sec .2, we show ha he n- uple wi h each en y equal o he mean
ield equilib ium s a egy p o ides an app oxima e Nash equilib ium o he n-playe game
wi h ewa d g.
De ini ion 7.3 Le ε>0. A uple α=(α1,...,α
n)∈Anwi h Q(α) =∅is called ε-Nash
equilib ium o he n-playe game i o all i∈{1,...,n}and Pα∈Q(α)
EPαgXi
τi,q(μn,1−p)+ε≥sup
β∈Ai
sup
P∈Q(α−i,β)
EPgXi
τi,q(μn,1−p),
whe e (α−i,β)=(α1,...,α
i−1,β,α
i+1,...,α
n)and sup ∅=−∞.
Wi h some addi ional assump ions on he e minal ewa d g,wecanshow:
P oposi ion 7.4 Le g sa is y Assump ion 7.1. In addi ion, assume ha g is uni o mly
bounded, con inuous, and g(x,·)is mono onically dec easing o all x ∈R.Le α∗∈Anbe
de ined as in Theo em 4.1. Then, he e exis s a sequence εn≥0wi h limn→∞ εn=0such
ha he con ol uple α∗=(α1,∗,...,αn,∗)is an εn-Nash equilib ium o he n-playe game.
123
330 Ma hema ics and Financial Economics (2024) 18:313–331
P oo All de ails o he p oo can be ound in P oposi ion 3.2.25 in [19]. Le i∈{1,...,n}.
Mo eo e , le β∈Aisuch ha Q(α−i,∗,β)=∅and choose P∈Q(α∗),˜
P∈Q(α−i,∗,β).
Using he empi ical quan iles A(n)and D(n)de ined in (16)and(18), espec i ely, we ind
ha
˜
Eg(Xi
τi,q(μn,1−p))≤˜
Eg(Xi
τi,D(n)),
Eg(Xi
τi,q(μn,1−p))≥Eg(Xi
τi,A(n)),
because o he mono onici y o g(x,·). We use he no a ion Eand ˜
E o he expec a ion w. . .
Pand ˜
P, espec i ely. De ine he unc ion
G(q):= E[g(Xi
τi,q)]=∞
0∞
−∞
g(x,q)p( ,−q,x−q)dxP
τ(d ), q∈R,
whe e pdeno es he p obabili y densi y unc ion o he OBM de ined in P oposi ion 3.5.
No e ha
Eg(Xi
τi,A(n))=E[G(A(n))],
because X1
τ1,...,Xn
τna e independen unde P. Simila o S, one can show ha
˜
Eg(Xi
τi,D(n))≤˜
E[G(D(n))]=E[G(D(n))],
simila o S ep 2 in he p oo o Theo em 3.2.16 in [19]. The uppe bound ollows om
Theo em 6 in [14]: he con ol maximizing he le -hand side is he h eshold con ol wi h
h eshold D(n), i condi ioned on D(n). We conclude ha
˜
Eg(Xi
τi,q(μn,1−p))−Eg(Xi
τi,q(μn,1−p))≤E[G(D(n))]−E[G(A(n))].
No e ha Gis bounded and con inuous, and he igh -hand side only depends on he dis i-
bu ion o n−1 independen OBMs wi h h eshold b∗. Mo eo e , A(n)and D(n)con e ge
o b∗in p obabili y (see Lemma 3.2.20 in [19]) and hence, also in dis ibu ion. This means
lim
n→∞|E[G(D(n))]−E[G(A(n))]|=0.
The e o e, we can ind a sequence (εn)n∈Nwi h he desi ed p ope ies. The sequence (εn)n∈N
is independen o i∈{1,...,n},β∈A,P∈Q(α∗),and ˜
P∈Q(α−i,∗,β) because he
dis ibu ions o A(n)unde Pand o D(n)unde ˜
Pa e unique. 
Acknowledgemen s We hank wo anonymous e e ees o ca e ully eading he a icle and o p o iding
many sugges ions o imp o emen s. Suppo om he Ge man Resea ch Founda ion h ough he p ojec AN
1024/5-1 is g a e ully acknowledged. Nabil Kazi-Tani’s esea ch is suppo ed by he ANR p ojec DREAMES
ANR-21-CE46-0002-03.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence,
and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he
a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is
no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y
egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde .
To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/.
123
Ma hema ics and Financial Economics (2024) 18:313–331 331
Re e ences
1. Anki chne , S., Kazi-Tani, N., Wend , J., Zhou, C.: La ge anking games wi h di usion con ol. Ma h.
Ope . Res. (2023) (in p ess)
2. Ba asso, A., Touzi, N.: Con olled di usion mean ield games wi h common noise and Mckean–Vlaso
second o de backwa d sdes. Theo y P obab. Appl. 66(4), 613–639 (2022)
3. Ca mona, R., Dela ue, F.: P obabilis ic heo y o mean ield games wi h applica ions. I, olume 83 o
P obabili y Theo y and S ochas ic Modelling. Sp inge , Cham. Mean ield FBSDEs, con ol, and games
(2018)
4. Ca mona, R., Dela ue, F.: P obabilis ic heo y o mean ield games wi h applica ions. II, olume 84 o
P obabili y Theo y and S ochas ic Modelling. Sp inge , Cham. Mean ield games wi h common noise
and mas e equa ions (2018)
5. Che ny, A., Engelbe , H.-J.: Singula s ochas ic di e en ial equa ions. Lec u e No es in Ma hema ics,
ol. 1858. Sp inge , Be lin (2005)
6. Dje e, M.F.: Ex ended mean ield con ol p oblem: a p opaga ion o chaos esul . Elec on. J. P obab. 27,
1–53 (2022)
7. Dje e, M.F.: Mean ield games o con ols: on he con e gence o nash equilib ia. Ann. Appl. P obab.
33(4), 2824–2862 (2023)
8. Elie, R., Hube , E., Mas olia, T., Possamaï, D.: Mean- ield mo al haza d o op imal ene gy demand
esponse managemen . Ma h. Financ. 31(1), 399–473 (2021)
9. Folland, G.B.: Fou ie Analysis and I s Applica ions. Wadswo h & B ooks/Cole Ad anced Books &
So wa e, Paci ic G o e (1992)
10. Ikeda, N., Wa anabe, S.: S ochas ic di e en ial equa ions and di usion p ocesses, olume 24 o No h-
Holland Ma hema ical Lib a y, 2 edn. No h-Holland Publishing Co., Ams e dam; Kodansha, L d., Tokyo
(1989)
11. Ka a zas, I., Sh e e, S. E.: B ownian mo ion and s ochas ic calculus, olume 113 o G adua e Tex s in
Ma hema ics, 2nd edn. Sp inge , New Yo k (1991)
12. Keilson, J., Wellne , J.A.: Oscilla ing B ownian mo ion. J. Appl. P obab. 15(2), 300–310 (1978)
13. K ylo , N.V.: Con olled di usion p ocesses, olume 14 o S ochas ic Modelling and Applied P obabili y.
Sp inge , Be lin. T ansla ed om he 1977 Russian o iginal by A. B. A ies, Rep in o he (1980) edi ion
(2009)
14. McNama a, J.M.: Op imal con ol o he di usion coe icien o a simple di usion p ocess. Ma h. Ope .
Res. 8(3), 373–380 (1983)
15. Nadi ash ili, N.: Nonuniqueness in he ma ingale p oblem and he Di ichle p oblem o uni o mly
ellip ic ope a o s. Annali della Scuola No male Supe io e di Pisa. Classe di Scienze. Se ie IV 24(3),
537–549 (1997)
16. Nakao, S.: On he pa hwise uniqueness o solu ions o one-dimensional s ochas ic di e en ial equa ions.
Osaka J. Ma h. 9(3), 513–518 (1972)
17. P o e , P.E.: S ochas ic in eg a ion and di e en ial equa ions, olume 21 o S ochas ic Modelling and
Applied P obabili y, 2 edn. Sp inge , Be lin, 2005. Ve sion 2.1, Co ec ed hi d p in ing (2005)
18. S oock, D.W., Va adhan, S.R.S.: Mul idimensional di usion p ocesses. In: Classics in Ma hema ics.
Sp inge , Be lin. Rep in o he (1997) edi ion (2006)
19. Wend , J.: Di usion con ol and games. Ph.d hesis, F ied ich-Schille -Uni e si ä Jena (2023)
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and
ins i u ional a ilia ions.
123