Wang, Congcong; Wang, Yuhan; Chen, Shanshan; Luck az, Sh a an; Panse a,
B uno An onio
A icle
A no e on he wel a e and policy implica ions o a wo-
pe iod eal op ion game unde impe ec in o ma ion
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Wang, Congcong; Wang, Yuhan; Chen, Shanshan; Luck az, Sh a an; Panse a,
B uno An onio (2025) : A no e on he wel a e and policy implica ions o a wo-pe iod eal op ion
game unde impe ec in o ma ion, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 1, pp. 1-7,
h ps://doi.o g/10.3390/g16010007
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Recei ed: 3 Decembe 2024
Re ised: 18 Janua y 2025
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Published: 3 Feb ua y 2025
Ci a ion: Wang, C., Wang, Y., Chen, S.,
Luck az, S., & Panse a, B. A. (2025). A
No e on he Wel a e and Policy
Implica ions o a Two-Pe iod Real
Op ion Game Unde Impe ec
In o ma ion. Games,16(1), 7.
h ps://doi.o g/10.3390/g16010007
Copy igh : © 2025 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license
(h ps://c ea i ecommons.o g/
licenses/by/4.0/).
A icle
A No e on he Wel a e and Policy Implica ions o a Two-Pe iod
Real Op ion Game Unde Impe ec In o ma ion
Congcong Wang
1,†
, Yuhan Wang
1,†
, Shanshan Chen
1,†
, Sh a an Luck az
2,
*
,†
and B uno An onio Panse a
3,
*
,†
1School o Finance, Zhejiang Uni e si y o Finance and Economics, Hangzhou 310012, China
2School o Economics and CeDeX China, Uni e si y o No ingham Ningbo China, Ningbo 315104, China
3Depa men o Law, Economics and Human Sciences & Decisions_Lab, Uni e si y Medi e anea o Reggio
Calab ia, Via dell’Uni e si á 25, I-89124 Reggio Calab ia, I aly
*Co espondence: [email p o ec ed](S.L.); b uno.panse a@uni c.i (B.A.P.)
†These au ho s con ibu ed equally o his wo k.
Abs ac : We show ha he disc e e eal op ion game model p oposed in he ecen li -
e a u e can be ex ended o he case o impe ec in o ma ion. As a esul , he model can
co e a wide ange o applica ions. Howe e , we also obse e ha he e ec i eness o
implemen ing he subsidy is a ec ed by he impe ec in o ma ional s uc u e.
Keywo ds: wel a e; policy implica ions; wo-pe iod eal op ion game; impe ec in o ma ion
1. In oduc ion
In a ecen s udy, Wang e al. (2023) analyzed he wel a e implica ions o a disc e e
eal op ion game whe e i ms make in es men decisions sequen ially. One o hei key
indings was ha , unde equilib ium condi ions whe e no i m in es s in he i s pe iod,
he go e nmen can in oduce a subsidy in a imely manne o induce a leas one i m
o in es ea ly. Howe e , while hei analysis assumes a sequen ial decision-making
p ocess wi h a designa ed leade i m mo ing i s , many eal-wo ld scena ios in ol e i ms
ope a ing unde impe ec in o ma ion. In such con ex s, i ms lack knowledge o whe he
compe i o s ha e al eady in es ed when making hei decisions. This is pa icula ly
ele an in applica ions like Public–P i a e Pa ne ships (PPPs), whe e in o ma ional
asymme ies a e p e alen .
In his pape , we elax he assump ion o pe ec in o ma ion and econside he eal
op ion game unde impe ec in o ma ion. We explo e he esul ing wel a e implica ions
and e alua e he e ec i eness o policy in e en ions in his mo e gene al in o ma ional
amewo k. Ou indings e eal ha while he equilib ium ou comes emain unchanged,
he shi o an impe ec in o ma ional s uc u e signi ican ly impac s he e ec i eness
o subsidies.
This wo k builds on a ich li e a u e examining eal op ions and s a egic decision
making unde unce ain y. Fo example, Lamb ech and Pe audin (2003) in eg a ed pa ial
in o ma ion and an icipa o y beha io in o a model o compe i i e in es men decisions,
showing how op imal s a egies ange be ween he ze o-NPV h eshold and a monop-
olis ’s p e e ed s a egy. Simila ly, Weeds (2002) in es iga ed i e e sible in es men s
in compe i i e R&D unde unce ain p o i s wi hin a pa en sys em, highligh ing how
lack o coope a ion leads o delays in in es men due o ea s o ini ia ing a pa en ace.
Huisman and Ko (2004) explo ed dynamic ma ke s whe e i ms compe e in adop ing
new echnologies, e ealing how he likelihood o echnological inno a ion a ec s s a egic
beha io s—shi ing om p eemp ion games o wa s o a i ion. Mil e sen and Schwa
Games 2025,16, 7 h ps://doi.o g/10.3390/g16010007
Games 2025,16, 7 2 o 7
(2004) ex ended he eal op ions amewo k o include game- heo e ic dynamics o e alu-
a ing pa en -p o ec ed R&D p ojec s unde compe i ion. Pawlina and Ko (2006) examined
asymme ic in es men cos s in a duopoly, iden i ying h ee dis inc equilib ium s a egies.
Mu o and Keppo (2002) de eloped a game model whe e mul iple i ms compe e o a
single in es men oppo uni y, demons a ing he exis ence o Nash equilib ia based on
a ying assump ions abou i ms’ knowledge o compe i o s’ p ojec e alua ions.
Ou wo k also ela es o Smi and Ankum (1993), who adop ed a mic oeconomic
app oach o analyze compe ing business in es men s using eal op ions and game heo y.
They o ecas ed cash in lows om ope a ions ha use economic en s o excess p o i (see
also Smi ,2003;Smi & T igeo gis,2006), while G enadie (1996) applied a game equilib ium
model o housing de elopmen , in es iga ing how ma ke demand and asse alues shape
in es men capaci y. Ma zoukos and Zacha ias (2009) highligh ed s a egic decision
making in R&D, inco po a ing spillo e e ec s and p icing dynamics, and McGahan (1993)
examined sec o al s a egies in eg a ing eal op ions and game heo y wi h an emphasis on
collabo a i e e sus compe i i e R&D. Kula ilaka (1993) p o ided a lexible amewo k o
e alua ing in es men decisions unde unce ain y, such as swi ching be ween ope a ing
modes in a dual- uel indus ial boile p ojec . Mye s (1977) linked co po a e deb beha io
o eal op ion alues, iden i ying how isky loans can cons ain in es men s a egies and
educe i m alue. Tondji (2016) examined wel a e ou comes in R&D-in ensi e ma ke s
using Cou no and Be and compe i ion models, while Paddock e al. (1988) ex ended
inancial op ions heo y o e alua e claims on eal asse s like o sho e oil leases.
In ligh o his ex ensi e li e a u e, ou s udy o e s new insigh s in o he low pa ic-
ipa ion a es obse ed in PPP p ojec s. As no ed by Wang e al. (2023), pa icipa ion in
such ini ia i es in China emains as low as 10–20% despi e he in oduc ion o subsidies.
Ou model iden i ies wo p ima y easons o his phenomenon. Fi s , i ms ace s ong
incen i es o delay in es men due o unce ain u u e p o i s. Second, compe i ion unde
impe ec in o ma ion de e s en y, as i ms canno an icipa e whe he hey will ace a
compe i o in la e s ages o he game. This con as s wi h he sequen ial mo e case, whe e
ollowe s ha e ull isibili y o he leade ’s ac ions.
The emainde o he pape is s uc u ed as ollows: Sec ion 2in oduces he ex ended
e sion o he Wang e al. model and p esen s ou main esul s and Sec ion 3concludes.
2. A 2-Pe iod 2-S age Real Op ion Game
We conside he (Wang e al.,2023) model in an en i onmen wi h impe ec in o ma-
ion. The game is played o e wo pe iods,
=
0, 1. A
=
0, i ms make hei in es men
decisions simul aneously. As in (Wang e al.,2023), a
=
1, demand ollows a simple
binomial p ocess and depending on he in es men decisions he ma ke s uc u e could
be a duopoly o monopoly a
=
0 o
=
1 o in bo h pe iods. The game is played
unde pe ec in o ma ion ac oss he pe iod, bu impe ec in o ma ion wi hin each pe iod.
The cos o in es men is deno ed by
I
. As in (Wang e al.,2023) he demand and p o i
unc ions o i m iis gi en as ollows:
pi=a−bqi+θqj,i=1, 2 i=j,a,b>0 (1)
πi=Y (piqi−cqi)(2)
whe e
Y
, o
=
0, 1, is an exogenous shock, which can be on he supply side o on he
demand side and each i m aces a ma ginal cos o
c
. As in (Wang e al.,2023), we assume
ha Y0is de e minis ic while Y1is s ochas ic wi h he possible ealiza ion o ei he a good
Games 2025,16, 7 3 o 7
s a e deno ed by
G
o a bad s a e deno ed by
B
. Mo e o mally, he s ochas ic p ocess is
as ollows:
Y1=(uY0i G
dY0i B(3)
whe e 0
≤d<
1
<u
. Le he p obabili y
p
deno e he p obabili y ha s a e
G
is ealized.
As in (Wang e al.,2023), i we le
µ≡pu + (
1
−p)d
, he expec ed alue o
Y1
is gi en by
E(Y1)=µY0
. We deno e by
he isk ee in e es a e and we le
R≡
1
+
. As in (Wang
e al.,2023), we deno e he duopoly symme ic p o i o a i m in pe iod
by
D(1, 1)Y
so
ha om he abo e we ha e:
D(1, 1)≡ (1−θ)
b(1+θ)a−c
2−θ2!. (4)
Fu he mo e, we deno e he monopoly p o i o he i m in pe iod
by
D(1, 0)Y
so
ha we ha e:
D(1, 0)≡1
b (a−c)2
4!. (5)
When no i m in es s, each i m ob ains 0 p o i s. This is deno ed by
D(0, 0)Y
and
inally, when only one i m in es s, he i m ha did no in es ob ains 0 p o i s, which we
deno e by D(0, 1)Y . Thus, we ha e:
D(0, 0)=D(0, 1)=0 (6)
Following (Wang e al.,2023), we ha e
D(1, 0)>D(1, 1)
and he ollowing inequali-
ies hold:
D(1, 0)>D(1, 1)>D(0, 0)=D(0, 1)=0 (7)
and
D(1, 0)−D(0, 0)>D(1, 1)−D(0, 1)(8)
We ep esen he abo e 2-pe iod game as an ex ensi e game and i s comple e desc ip-
ion is gi en in he game ee below.
We now ind he pe ec Nash Equilib ium subgame. As in (Wang e al.,2023), we use
he ollowing exp essions in he p esen a ion o ou esul s. Le
E≡D(1, 1)Y0R+(1−p)d
R−p
and
F≡D(1, 0)Y0R+(1−p)d
R−p
. We assume ha he magni ude o he posi i e (nega i e)
shock i he good (bad) s a es ealizes is su icien ly high (low), ha is:
D(1, 1)dY0<E(9)
and
D(1, 1)uY0>F(10)
As a esul o he abo e assump ion, we ha e he ollowing:
D(1, 0)dY0<E<F<D(1, 1)uY0(11)
In Figu e 1, he game is ep esen ed as a game in ex ensi e o m. Each has i e
in o ma ion se s (1.1–1.5) o playe 1 and in o ma ion se s (2.1–2.2) o playe 2. A s a egy
o he
k
playe is a mapping
sk
om
{{1.i}}5
i=1→{I,NI}
, whe e
k=
1, 2. Le
S1
deno e
he se o all s a egies o playe 1 while
S2
deno e he se o all s a egies o playe 2.
Mo eo e , le
S=S1×S1
deno e he se o all s a egy p o iles o he game. We say ha
some p o ile
s≡(s1;s1)∈S
is a pe ec Nash Equilib ium (SPNE) subgame o he game
Games 2025,16, 7 4 o 7
i i s es ic ion o each subgame is a Nash Equilib ium o ha subgame. Conside he
ollowing p o iles: s∗,s∗∗,s∗∗∗, whe e:
s∗≡(s∗
1;s∗
2), (12)
s∗
1=s∗
2=(I,I,NI,I,NI), (13)
s∗∗ ≡(s∗∗
k;s∗∗
l), (14)
s∗∗
k=(I,I,NI,I,NI), (15)
s∗∗
l=(NI,I,NI,I,NI), whe e k=1, 2 and k=l(16)
s∗∗∗ ≡(s∗∗
1;s∗∗
2), (17)
s∗∗∗
1=s∗∗∗
2=(NI,I,NI,I,NI), (18)
Ou main esul cha ac e izes he se o equilib ia o he game desc ibed by Figu es 1and 2.
Figu e 1. The 2-pe iod, 2-s age, 2-playe game.
Figu e 2. The game ee o pe iod 0.
Games 2025,16, 7 5 o 7
P oposi ion 1. The subgame pe ec Nash equilib ia o he game a e gi en as ollows:
s=
s∗i D(1, 0)dY0<I ≤ E
s∗∗ i E <I ≤ F
s∗∗∗ i F <I ≤ D(1, 1)uY0
(19)
P oo .
F om inequali ies
D(1, 0)dY0<E
,
F<D(1, 1)uY0
,
I ≤ D(1, 1)uY0
and
D(1, 0)dY0<I
, we know ha each i m will in es in he
G
s a e and will no in es
in he
B
s a e in pe iod 1 unde all h ee s a egies. Using backwa d induc ion, we can hen
p oceed o pe iod 0. The educed pe iod 0 game ee is gi en in Figu e 3.
Figu e 3. The gene al case.
The e o e, we conside he ollowing h ee cases o he p oposi ion.
Case (i): D(1, 0)dY0<I ≤ E.
In Case (i), sol ing he game using backwa d induc ion again, i can be shown ha he
inequali ies ensu e ha we ha e he equilib ium pa h shown in Figu e 4.
Figu e 4. The diag am o Case 1.
Case (ii) E<I ≤ F.
In Case (ii), sol ing he game using backwa d induc ion again, i can be shown ha
he inequali ies ensu e ha we ha e he equilib ium pa h shown in Figu e 5.
Figu e 5. The diag am o Case 2.
Case (iii): F<I ≤ D(1, 1)uY0.
In Case (iii), sol ing he game using backwa d induc ion again, i can be shown ha
he inequali ies ensu e ha we ha e he equilib ium pa h shown in Figu e 6.
Games 2025,16, 7 6 o 7
Figu e 6. The diag am o Case 3.
This comple es he p oo .
P oposi ion 1shows ha he main p oposi ion o (Wang e al.,2023) emains obus
o he impe ec in o ma ion s uc u e. Thus, unde impe ec in o ma ion i he cos o
in es men is su icien ly low, bo h i ms would exe cise hei op ions in he i s pe iod,
whe eas i i is su icien ly high, hen bo h i ms would no exe cise hei op ions in he i s
pe iod. Mo eo e , he e exis s a ange o alues o he in es men cos s o which only one
i m in es s in pe iod 0.
We can also es ablish he obus ness o p oposi ions 2 and 3 o (Wang e al.,2023)
as ollows.
P oposi ion 2. Unde an impe ec in o ma ion s uc u e, he h esholds in es men le els
E
and
F a e inc easing in d and p while dec easing in R.
P oposi ion 3. Suppose ha he in o ma ion s uc u e is impe ec ,
F<I ≤ D(1, 1)uY0
holds
and
hR+qd
R−1iY0WM>I
. Then, wel a e imp o es i he go e nmen subsidizes a leas one i m
in he ini ial pe iod. In his case, he subsidy is gi en by
S=I − F+ϵ
, whe e
ϵ
is posi i e. As
a esul , a subgame pe ec Nash Equilib ium in which exac ly one i m in es s in pe iod 0 can
be achie ed.
Rema k 1. The policymake canno implemen he subsidy wi hou eso ing o some a bi a y ule
ha disc imina es be ween he wo i ms. This is in con as o (Wang e al.,2023), whe e, a he
beginning o s age 1in pe iod 0, he policymake could announce ha i would subsidize he leade i
i in es s and hen implemen he subsidy i he leade has in es ed. Unde simul aneous mo es, he
designa ions “leade ” and “ ollowe ” no longe exis , and he e o e, al hough he policymake could
s ill implemen a subsidy ha induces exac ly one i m o in es , i can only do so by a bi a ily
disc imina ing be ween i m 1and i m 2.
3. Conclusions
We ha e shown ha he p e ious esul emains unchanged unde his new in o ma-
ional s uc u e. Howe e , we ound ha he e ec i eness o he implemen a ion o he
policy is educed o he simul aneous mo e case.
Au ho Con ibu ions: Concep ualiza ion, C.W., Y.W., S.C., S.L. and B.A.P.; alida ion, C.W., Y.W.,
S.C., S.L. and B.A.P.; o mal analysis, C.W., Y.W., S.C., S.L. and B.A.P.; in es iga ion, C.W., Y.W., S.C.,
S.L. and B.A.P.; w i ing—o iginal d a p epa a ion, C.W., Y.W., S.C., S.L. and B.A.P.; w i ing— e iew
and edi ing, C.W., Y.W., S.C., S.L. and B.A.P.; supe ision, S.L. All au ho s ha e ead and ag eed o
he published e sion o he manusc ip .
Funding: This esea ch was suppo ed by he Zhejiang P o incial Na u al Science Founda ion o
China unde G an No. LMS25G030001.
Da a A ailabili y S a emen : No new da a we e c ea ed o analyzed in his s udy. Da a sha ing is
no applicable o his a icle.
Games 2025,16, 7 7 o 7
Acknowledgmen s: The au ho s exp ess hei g a i ude o he h ee anonymous e iewe s and
he academic edi o o hei use ul commen s, sugges ions, and obse a ions aimed a imp o ing
he pape .
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
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Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
