Game Theory in Finance

Gano TLoory In FInanco
Lric Iasmnscn
Jnnc 2ò, 1991 orali. In¦lislco: J|· A·u 1n|j:n· 1.·í.·nn:j ·] '·n·j
nnn 1.nnn··. coiico ¦y Joln Laiwcll, Mnrray Milgaic, ano Icicr Ncwman.
Ncw York: Siockion Ircss, 1992.
¹h:í:n·í
Drafl: 4.ò. (Drafl 1.1, July 1000).
2000: Lric Iasmuson, Irofossor of Lusinoss Lconomics ano IuLlic Iolicy ano
Sanjay SuLlooar Facully Follow, Inoiana Univorsily, Kolloy Sclool of Lusinoss, LU
4ò6, 1800 L 10ll Slrool, Lloominglon, Inoiana, 4740ò-1701. Omco: (812) 8òò-
0210. Fax: 812-8òò-88ò4. Lrasmuso^inoiana.oou. Ilp.inoiana.oou,orasmuso.
Gamc ilcory is a mooclling approacl wlicl orops pcrlcci compciiiion`s
assnmpiion ilai inoivionals arc pricc-iakcrs ano insicao rcqnircs ilcm io
¦clavc siraicgically, iaking inio acconni ilai ilcir aciions will alicr ilc
¦clavionr ol ilc rcsi ol ilc markci. Jlis may ¦c in a conicxi in wlicl iwo
playcrs conscionsly cloosc aciions ilai ancci cacl oilcr, as in onopoly, ¦ni in
nnancc ii is morc common lor onc playcr io iry io manipnlaic ilc ¦clavionr
ol compciing playcrs on ilc oilcr sioc ol ilc markci. In ciilcr casc, gamc
ilcory aoorcsscs siraicgic ¦clavionr ¦y ocnning ilc playcrs in ilc gamc, ilc
payon lnnciions ilcy arc maximizing, ano ilc siraicgics availa¦lc io ilcm.
Ii is crncial io oclincaic carclnlly ilc orocr ol aciions ano ilc inlormaiion
availa¦lc io cacl playcr: prccommiimcni ano inlormaiion iransmission arc
ilc iwo pillars ol moocrn gamc ilcory.
Gamc ilcory`s mosi imporiani coniri¦niion io nnancc is a vcry olo onc:
ilc ilcory ol cxpccico niiliiy, wlicl was ociailco in ilc sccono coiiion ol
Von Ncnmann ano Morgcnsicrn`s J|··:j ·] Gnn·: nnn 1··n·n.· 1·|n.·:
in 1947. Iniiing ilai asioc, gamc ilcory las ¦ccn mosi nsclnl in ilc conicxi
ol asymmciric inlormaiion, wlicl las incrcasco in rcscarcl imporiancc as
rcinrns lavc oiminislco in ilc cconomics ol nnccriainiy. Financc, pcrlaps
cvcn morc ilan oilcr sn¦jccis, is amcna¦lc io gamc ilcory`s approacl. In n-
nancial markcis inlormaiional aovaniagcs maiicr, cvcnis arc clcanly ocnnco,
ano ilc imporiani pariicipanis arc cxpcricncco playcrs wiil cnongl ai siakc
io jnsiily carclnl ilongli. Jlns, ilc siylizco moocls ano soplisiicaico raiio-
naliiy ol gamc ilcory may apply ¦ciicr io markcis lor corporaiions ilan lor
canialonpcs.
Jlis ariiclc will aiicmpi no morc ilan io convcy ilc navonr ol gamc
ilcory in nnancc. For pariicnlar icclniqncs, scc Ircps (1990) or Iasmnscn
(1989): lor rclcrcnccs, scc Harris ano Iaviv (lorilcoming). Iailcr ilan
snrvcy ilc liicrainrc, I will lcrc convcy iis navonr nsing ilrcc iypical moocls,
ol signalling, commiimcni, ano inccniivc ocsign.
1rnn¡|· 1. o.jnn||.nj .n J·nn·: U]·::: 1|j 1· J·nn·: U]·:: o·n·-
í.n·: 1n.| A major nsc ol gamc ilcory is io lormalizc ininiiion, o¦iain
1
a com¦inaiion ol ininiiivc ano connicrininiiivc rcsnlis, ano ilcn rcnnc ilc
ininiiion ¦y mooilying ilc moocl nniil ilc rcsnlis ¦ccomc rcalisiic ano ilcir
origins nnocrsiooo. A scqncncc ol moocls ol icnocr oncrs illnsiraics ilis
niccly. Il a ¦ioocr wlo can incrcasc a iargci nrm`s valnc ¦y : makcs a conoi-
iional icnocr oncr ilai is r < : a¦ovc ilc cnrrcni siock pricc, no slarclolocrs
will scll, cvcn ilongl ilcy wonlo joinily ¦cncni. Jlc slarclolocrs arc in a
prisoncr`s oilcmma: ii is ¦ciicr (¦y : r) io ¦c a lolooni ilan a icnocrcr
il ilc oncr snccccos ano no worsc il ii lails, so cvcry slarclolocr lolos oni
(Grossman ano Hari [1980[). Ly ilis argnmcni, icnocr oncrs slonlo ncvcr
occnr, ¦ni Sllcilcr ano Vislny (1980) poini oni ilai il ilc ¦ioocr ¦cgins wiil
a siakc ol c in ilc company, icnocr oncrs can ¦c pronia¦lc. Jlc ¦ioocr can
proni on lis original slarcs cvcn il lc oncrs r : ano loscs on ilc icnocrco
slarcs.
Hirsllcilcr ano Jiiman (1990) cxplain wly oncrs sncccco somciimcs,
¦ni noi always. Nainrc clooscs ilc ¦ioocr`s syncrgy valnclis iypc¨ io
¦c : # (0, :[. Jlc ¦ioocr oncrs a prcminm ol r lor cacl ol proporiion .
ol ilc slarcs, ano cacl ol a coniinnnm ol slarclolocrs occiocs wlcilcr io
acccpi or rcjcci ilc oncr. Il ovcr (.ò c) acccpi, ilc payons arc r lor ilosc
ilai acccpi ano : lor ilosc ilai rclnsc: oilcrwisc, all payons arc zcro.
Jwo kinos ol cqnili¦ria arc possi¦lc. Onc kino is a scparaiing cqnili¦-
rinm¨ in mixco siraicgics¨: cacl iypc ol ¦ioocr ¦clavcs oincrcnily ano ilc
slarclolocrs ranoomizc ilcir ¦clavionr. Jlc ¦ioocr oncrs r = : lor . = .ò
ol ilc slarcs, ano ilc slarclolocrs ranoomizc ilcir acccpianccs so ilai wiil
pro¦a¦iliiy (r,:)

ilc oncr snccccos. Jlc ligl-: ¦ioocr will noi oncr a
lowcr r ¦ccansc ilc oncr wonlo morc likcly lail ano lc wonlo losc ilc poicn-
iial gain on lis iniiial c slarcs. A sccono kino ol cqnili¦rinm, lcss plansi¦lc
lcrc, is a pooling cqnili¦rinm,¨ in wlicl oincrcni iypcs ol ¦ioocrs ¦clavc
ilc samc. In onc ol ilc pooling cqnili¦ria, r = 0 lor any :, ano oncrs always
lail ¦ccansc any posiiivc oncr wonlo ¦c rcjccicoilc slarclolocrs wonlo all
lolo oni nnocr ilc oni-ol-cqnili¦rinm ¦clicl¨ ilai il r 0, ilcn : = :.
Iooling cqnili¦ria arc rnlco oni lcrc ¦y ilc rcasona¦lc oni-ol-cqnili¦rinm
¦clicl ilai a liglcr ¦io signals a liglcr valnc ol :, in wlicl casc a low-:
2
¦ioocr conlo pronia¦ly ocviaic lrom ilc pooling cqnili¦rinm ¦y oncring a
low r ano ilc slarclolocrs wonlo acccpi lis oncr.
Jlis is an cxamplc ol a signalling¨ moocl, wiil a low prcminm sig-
nalling low syncrgy. Wlai makcs a signal crcoi¦lc is ilai ii ¦c morc cxpcn-
sivc lor onc iypc ol playcr ilan lor anoilcr, as ilc low-prcminm oncr (wiil
iis lowcr pro¦a¦iliiy ol sncccss) is lor ilc low-syncrgy playcr. Signalling moo-
cls olicn lavc mnliiplc cqnili¦ria, ano mncl rcscarcl cnori las ¦ccn ocvoico
io rcnning ilc cqnili¦rinm conccpi io rconcc ilc nnm¦cr ol cqnili¦ria (scc
Clapicr xxx ol Ircps [1990[).
1rnn¡|· ´. ·n¡.ín| oí:&·í&:· n: 1:···nn.ín·ní: ·nn 1.j| 1·hí 1·|¡
1&:.n·:: Ilaycrs olicn nnocriakc aciions io rcsirici ilcir aciions or inlor-
maiion laicr in a gamc. An cxamplc is ilc lollowing capiial sirncinrc moocl
¦asco on Lranocr ano Lcwis (1980). Jlc playcrs arc iwo nrms in ilc samc
markci. In ilc nrsi movc, ilc nrms simnlianconsly cloosc oc¦i lcvcls, ano in
ilc sccono ilcy simnlianconsly cloosc onipni lcvcls ¡
Q
ano ¡

. Nainrc ilcn
clooscs ilc lcvcl ol a ranoom ocmano slock : ano pronis arc rcalizco. Ii is
assnmco ilai nrm i`s proni, H

(¡

, ¡

, :), is occrcasing in ¡

ano incrcasing in
:, ano ilai ilc nn:j.nn| proni (cH

,c¡

) is incrcasing in ilc slock :. Wlcn
: is largc, a nrm`s pronis arc liglcr, cspccially il ii las closcn a ligl onipni
lcvcl.
Il ¦oil nrms cloosc zcro oc¦i, ilis is ilc Conrnoi gamc wiil nnccr-
iainiy: ilc nrms iraoc on ilc aovaniagc ol ligl onipni wlcn : is largc
againsi ilc oisaovaniagc wlcn : is small. A nrm wiil lcavy oc¦i, lowcvcr,
wonlo go ¦ankrnpi il : wcrc low in any casc, ano iis slarclolocrs oo noi
carc a¦oni ilc oisaovaniagc ol ligl onipni in ilai siaic, ilanks io limiico
lia¦iliiy. Jlcy oo ¦cncni lrom ligl onipni wlcn : is ligl, so lcavy oc¦i is
an inccniivc lor ligl onipni.
A scllcr in a Conrnoi onopoly wonlo likc io ¦c a¦lc io commii io ligl
onipni, ¦ccansc ilis inonccs lis rival io cloosc a lowcr onipni. Dc¦i is a lorm
ol prccommiimcni. Wlcn nrm i incnrs oc¦i, nrm , knows ilai i`s inccniivc
io prooncc ligl onipni las incrcasco, so , will cni ¦ack. Il ¦oil nrms incnr
3
oc¦i, lowcvcr, wlicl is ilc cqnili¦rinm lcrc, ¦oil ol ilcir inccniivcs lor ligl
onipni lavc incrcasco, ano comparco io zcro oc¦i ¦oil onipnis arc grcaicr
ano ¦oil pronis lowcr.
Unlikc Lxamplc 1, ilis is a gamc ol symmciric inlormaiion, wlcrc ilc lo-
cns is on commiimcni, noi inlormaiion iransmission. Lacl nrm ocli¦craicly
risks ¦ankrnpicy io crcaic a connici ol inicrcsi ¦ciwccn oc¦i ano cqniiy ilai
incrcascs iis aggrcssivcncss in sccking markci slarc. Jlc onicomc is worsc
lor ilc nrms ilan il ilcy joinily avoio oc¦i, ¦ccansc oc¦i lowcrs inonsiry
pronis wlilc lclping ilc nrm ilai nscs ii as a commiimcni ioolanoilcr
cxamplc ol ilc prisoncr`s oilcmma.
1rnn¡|· ·. 1n··ní.· 1·:.jn: 1|j 1:· 1.nnn·.n| 1ní·:n·n.n:.·: In
somc moocls, ilc playcrs ¦cgin wiil symmciric inlormaiion, ¦ni ilcy know
ilai ccriain playcrs will laicr acqnirc an aovaniagc. In Diamono`s 1984
moocl ol nnancial inicrmcoiarics, ` risk-ncniral invcsiors wisl io nnancc A
risk-ncniral cnircprcncnrs. Lacl cnircprcncnr las a projcci ilai rcqnircs 1
in capiial ano yiclos 1 , wlcrc 1 is iniiially nnknown io anyonc. Il 1 < 1,
lc loncsily cannoi rcpay ilc invcsiors, ¦ni ilc pro¦lcm is ilai only lc,
noi ilc invcsiors, will o¦scrvc 1 , so ilcy cannoi valioaic lis claim ilai
1 < 1. Jlcy mnsi rcly on onc ol iwo ilings io clicii ilc irnil: moniioring
or an inccniivc coniraci. Unocr moniioring, cacl invcsior pays 1 io o¦scrvc
1 , wlicl makcs ii a coniracii¦lc varia¦lc, on wlicl ilc rcpaymcni can ¦c
maoc coniingcni. Unocr ilc inccniivc coniraci, ilc cnircprcncnr snncrs a
oissipaiivc pnnislmcni c(:) il lc rcpays :. Jlc cosi ol moniioring is `1,
wlilc ilc cxpccico cosi ol an inccniivc coniraci is 1c, so in ilc a¦scncc ol
an inicrmcoiary ilc inccniivc coniraci is prclcrrco il 1c < `1.
Jlc pnrposc ol a nnancial inicrmcoiary is io climinaic rconnoancy ¦y
rcplacing ilc ` inoivional moniiors wiil a singlc ccniral moniior. Jlc
inicrmcoiary iiscll rcqnircs an inccniivc coniraci, ai cosi 1c, ano ii mnsi
sprcao ilis cosi ovcr scvcral cnircprcncnrs io makc iis cxisicncc worilwlilc.
Oilcrwisc, il A = 1, ilc inicrmcoiary incnrs a cosi ol 1 lor moniioring ano
1c lor iis own inccniivc, wlcrcas a oircci invcsior-cnircprcncnr coniraci
4
wonlo cosi only 1c.
Jlc nsc ol ilis moocl is io slow ilai (a) an inicrmcoiary lclps only
il ilcrc arc ¦oil many invcsiors ano many cnircprcncnrs, ano (¦) inccniivc
coniracis lavc cconomics ol scalc comparco io moniioring. Jlc moocl is
an cxamplc ol ilcory-¦asco insiiiniional cconomics: ilc insiiiniion iakcs iis
pariicnlar lorm io avoio inlormaiion pro¦lcms ¦y coniraciing wlilc inlorma-
iion is siill symmciric.
Lacl ol ilc ilrcc moocls oiscnssco a¦ovc is iypical ol a liicrainrc, ano
oilcr liicrainrcs in nnancc (c.g., cxccniivc compcnsaiion, markci microsirnc-
inrc) also nsc gamc ilcory. Jlc nnm¦cr ano iniricacy ol ilc moocls can ¦c
oanniing, ano ilcy lavc ¦ccn criiicizco lor ilc oimcnliy ol cmpirically vcri-
lying ilc assnmpiions ano lor scnsiiiviiy io sccmingly minor clangcs in as-
snmpiions a¦oni wlai inlormaiion is availa¦lc or wlo movcs nrsi. Wlcilcr
ilcsc criiicisms apply ocpcnos on ilc moocl, ¦ni ilcy may apply wiiloni
¦cing irnly o¦jcciiona¦lc. Ii is nnlorinnaic il imporiani varia¦lcs arc laro
io mcasnrc, ¦ni ilai oocs noi oiminisl ilcir imporiancc: railcr, casc-¦y-
casc vcrincaiion mnsi rcplacc ilc rcgrcssion-rnnning ilai las oominaico
cconomisis` cmpirical work. Scnsiiiviiy io assnmpiions is noi a oraw¦ack
¦ni a ··ní:.h&í.·n ol gamc ilcory, poiniing oni ilc imporiancc ol wlai wcrc
oncc ilongli io ¦c insignincani ociails ol ilc worlo. Jnsi as marginalism
is morc ilan ilc applicaiion ol calcnlns io olo pro¦lcms in cconomics, so
gamc ilcory is as imporiani lor clanging ilc agcnoa as lor inirooncing ncw
icclniqncs.
ò
LILLIOGIAIHY.
Lranocr, J. ano Lcwis, J. 1980. Oligopoly ano nnancial sirncinrc: Jlc
limiico-lia¦iliiy cncci. ¹n·:.·nn 1··n·n.· 1·.·u 70: 9ò0-970.
Diamono, D. 1984. Financial inicrmcoiaiion ano oclcgaico moniioring.
1·.·u ·] 1··n·n.· oí&n.·: ò1: 393-414.
Grossman, S. ano Hari, O. 1980. Jakcovcr ¦ios, ilc lrcc-riocr pro¦lcm,
ano ilc ilcory ol ilc corporaiion. 1·|| J·&:nn| ·] 1··n·n.·: 11: 42-04.
Harris, M. ano Iaviv, A. Forilcoming. Financial coniraciing ilcory.
In ¹nnn··: .n 1··n·n.· J|··:j: o.rí| 1·:|n ··nj:·::, coiico ¦y Jcan-
Jacqncs Lanoni, Cam¦riogc: Cam¦riogc Univcrsiiy Ircss.
Hirsllcilcr, D. ano Jiiman, S. 1990. Slarc icnocring siraicgics ano ilc
sncccss ol losiilc iakcovcr ¦ios. J·&:nn| ·] 1·|.í.·n| 1··n·nj 98: 29ò-324.
Ircps, D. 1990. ¹ ··&::· .n '.·:····n·n.· J|··:j. Irinccion: Irincc-
ion Univcrsiiy Ircss.
Iasmnscn, L. 1989. Gnn·: nnn 1n]·:nní.·n. Oxloro: Lasil Llackwcll.
Sllcilcr, A. ano Vislny, I. 1980. Largc slarclolocrs ano corporaic
conirol. J·&:nn| ·] 1·|.í.·n| 1··n·nj 94: 401-88.
Von Ncnmann, J. ano Morgcnsicrn, O. 1947. J|· J|··:j ·] Gnn·: nnn
1··n·n.· h·|n.·&:. Sccono Loiiion. Ncw York: Wilcy.
0