A DISa.m'ATION
?reaented to the Faculty of PrincetOll
University 1n Cand1c1aoy tor tho Decree
Phlloaollh7
of Doctor Clt
tor
~Ade4 ~ 'by 1M
~ Qt!4M'a• •1!\\Qa

oo~copt ~
of non-cooporat1.:vo
Thi:s i'Apor mtroduco:; the And
Do
dovclopa m.cJthoda tor the matb.etatlcal analya1a of pmea. :he
$Uch
to,ies and r-y-ott runctlcma detlned tor t.'l.e oombinat1ona of ?ure
atrato:;1ea.
The dlat1ncr-viCX1 botnen ooo,orative and non-oooperat1ve SlUJGa 1s
at pure
unrolatod to tho atha:at1cal doacrit'tioll by atntO&1es
::JItUla
funct10nn ot it deyer.da on t.'us ,oluJ1:;'U1tj
~e.:;-o.cr c~. ~ther.
e.::d
a1de-pr.~ta.
or 1mpo&o1b1llty of cOGllt1cma. oor:m1n1catiou. alI1
at an Gqu1l.1brlut1 pomt. a aolut1oD. a IVon& ao1utioa.,
The concopta
.~u\"tic&l. 1D:tro<1uced by matbaat1.cal d4rt1n1tiQla.
a And valuoa aN
or
And in later aectlQtl8 the 1ntGrpretation thote ooncepta 1n non-cooper-
cS1acuaa~.
.t1'VQ 1.
pmM
..
,
or at lean equ1l1brlua polat. Other· ruulta po-
OODMnl tM
pu ODe
Mtr1ca1 atruofsure of the aet of equ111brlUll point. of ....... with • 10-
1 ' ~,
~
~ lut1oA.t the ~ ~ aub-eolutd.oaa, ud the a1nenae of • •,..ta-lea1
-
~ .
.
p..
~~
.. , equ111brla po!Jrb !D •
J 11luaftt;1~ of 1;t. ,..ibU1Uea te appl1Gat;1c •• a.. . .
.le U
1ot • .!apt. tbNe 'WI poIm' lI/Idel la 1Dclu4ed.
J .
o
~
I
'-
n
,
a..,

Tabla of Contenu
!:oct!on
1. Introduction... • • • • • • • • • • • • • • • • •• 1
z. Ponal ::e£!nit!QOB and ':erminoloG:/ • • • • • • • • •• 2
3. 5
••••••••••
7
4. of Game_ • • • • • • • • • • • • • • • ••
Sj~tr1ea
.. . .. . . . . . . .
!::ol:.rtio~ ••••• 9
••••
~
6. Geametr10al of Solut1ona • • • • • • • • • • ••
F~ 13
15
7. Dom1 na n 08 and Contradiction !!ethoda • • • • • • • ••
o. 17
A Throe-Man !)o)cer Game • • • • • • • • • • • • • • • •
o. and IDterpretatlon • • • • • • • • • • •• 21
MDt1~t1on
10. Appl1oat1ona. • • • • • • • • • • • • • • • • • • •• 25
B1bl1Qcra~.......
11. • • • • • • • • • • • • •• 27
12. •••••••••••••••••••
JL~l~~

Introduction
fruitful theory
Von ~reumanu and !Jorgenstorn have developed a very
or
of two-,erson _era-sum games in their book Theor;r Games and Economio
or a type
30r..o.v1cr.~,~l~ book also contains a tr.ecry of o-:'HJrson QUOOs
wtich wo would call oco,?erativa. theo~J
Thi3 is baaed on an analyai3
or the 1ntorrelatlonahinG of tho various coalitions which cnn be ro~d
caco.
by the pla!rers or trA
Our thoory, in contradistinction. is based on the absor.oo of coo.l1-
tiona 10 thnt it is a&&umed oach nartioipant aots
~;at indo~endeot1y.
without collaboration or oommunioation with any or the othern.
Tho notion of an equilibrium. !>Oint 1a the basio lncrediont in our
This notion yiold4 a Z8%lOru1:ut1on or tho concO':lt of the solu-
'tl1.ecrJ.
It turns out that the set ot equili-
tion of a two-!)eraon sero-eu.m~.
brium points or a tiro-person :ero-aum gaos is simply the set ai' all pair.
\"r;ood strategies.-
of oppoaing
oqu1l1b~iWll
In the 1:u:ledlately followinG aectioDB &ball d.fine
'We
pointa and prow that t1nlte QOll-Cooperatlve game always has at least
Do
:re shall .lao 1ntroduoe the notiona of aolvabil1ty
one equl111lrlw.:1 poUlt.
and. UrOD; aolvabll1ty of DOD-Ooop8l'atlw g _ and prove a tbeor. 011
&
the geaMtr1cal atruotura t4 the Nt of equ1l1brium pointe at • aol_ble
ga.-.
I.e an oxomplo at the application or our theo17 wo include a solution
or ~.
a simplified three penon poker
or tho athematlcal
The motivation and 1ntorpretntion em-
COD08?U
. ployed in the theor:! are roaer'n!d for d1ecusalon in a epeoial section at
th18 paper.

For.cal and
~erlnit1ons Te~nolobY
In this section define the besic concept3 this and set
~~ o~ ~apcr
U~ :.standard tor::li:!ology and notation. I:aports:lt dol' in! tiona will 00
i~clcati~g conce~t
proceeded by cub-title tho defined. The non-coc?-
Q
~111
erati-'.-e i::!ea be implicit, rathor than oxplicit, below.
F1.o.ite Ge.ms ,
For us an n-nerson will be a set of n or ryos1tions,
C~ ~laj~rs.
strntogioa; and corros?or.dinC
each with an associatoci finite sat ~~
. Pr ' or
whioh =-PH tho set
to each player, I • a no.y-off funotion, all
n-tuplea of pure strategies into the real nu.:Dbers. Whan '?Ie uno tho term
n-tuplo wo ahall always ocan a sot oJ: n 1tema. with oach item Q~300iAted
a d1.frercnt !layer.
with
•
A .mixed strategy or player 'will be a oollection of non-D.8:;at1ve
numbera which have unit aum and are 1n one to one correspondenoe with hia
pure strate,1ca.
~c. --,
=~ C ic(. TT \\ L{. 101-
Sa with
wr1 t.
';I. 01.
0(.
c(.,
atrawO. where the
to repr...nt auob a mixed are tho pure
.
atrategl.. ot player • W. re;ard the
I
1T ; ol ' S
plG whose verticea are tr.o ThiB s1,mplex tllly be regarded
•
ot a. real veator apace. Giving
as a convex subset natural proce•• of
tl8 •
11.cear cotabinatlcn ter tho ::l1Xe4 atratet,;ie••
olJ~'~
i,j,J<..
w. shall use the .uriaea tor players and to
$;, t: _
The G~la
lnd1a.t. wr10ua pure strate,1\". o£ a pla)\"Jr.
wHl 1::dl-

-:5-
oL th pure strategy. eta.
ca.ote tllo \\ th player' a
L n-11nonrJ.
which 1& linear 1n the mi~ed stratsGY or each pla:,\"Or
Pi • writinG p;(S.,S~,.-. Sn).
This i:Xtons1on we shall alao donota by
or
·\"e shill wr! 'to ~ or ;;t::: to denote ttn n-tu,?la :nixed :ztratacios
p; (l.)
(S, I Sn)
A:=
and if then
....
4 • w111 also ~e regarded ns a ,oint ~o a 'Voctor
Such an n-tu.,le, 8':'1108,
th~ mixed stratec:ies.
For convenience we introduce the eubat1tutlon notation
CS., St.., .... s: -'.\" t: ,5;+1) ••• $ ..)
(A; t;) to stand for
S't))
-J... -==
where (S./5&/ .... • • The effect or suocessive subat1tu-
/'
~; t: ; t\" j
;i-J) ) , Gtc.
tiona ((A.;t;) we 1%Jdlcato by
Equilibrium Po1at.
(1) •
--4.. that each ~layerl a
Thua an oquilibrium point is n-tuple
all KUoh
mixed strater;y ma.x1.izd.zd his Pfll--aff if the stratecies ot the others are
held fixed. Thua oaoh ,layer' 8 stratogy 1s optimal against tho.e of the
We .hall ooo••1onally abbreviate equiliiJri\\.Ull point by Oq. pt.
others.

Si
fie say ~h8t ~a r ure strato:y -rr;~
m..-:ed strateGY if
Cl
~ C~ C ~~ ~= C$.~5t., - - ·
>0 · s \"')
?i:: •
IT;oL and If
S; thnt ~ uses
Trio(. we alao
end un 03 l'T;at •
&8y
S.~
r=; C • \". • 511) S; ,
:'rcm t!1e l!nee.rity of 1n
:1'';~.s [P{-<t;,:)] =. VV:~LP;(4 ilT;<t)]
(2) •
R~f4..) -= P.. (A IT.ac.) • Thon we obtain
;
.,4
the followinG triv!al noccsonr;,-' and auf.!'1cient cond1'tion for to be
\\..
an ?Oint:
equilib~~UQ
P;~)
(3) •
~::
(S,,, S\"\" ••• S,,)
u and
P;(4.);: ~(;ol. P,'al~) , oonaeGuentl~ for
•
IT.oI..
..4... doe. not us. 1t iD an optical
to ••y that WllOIi a
which 18
i•
pure strnteQ\" for pltt1'Jr SO we . .ite
(4) if
another neeeaaary and lutf1ciant oondition tor an equUlbn\\a point.
all
(\"'3) eq. pt. can b. Gpreaaed •• tho
Sinoe a criterion tor &A
A..
equating of two continuous tunct10na the apaco at n-tuplea the
011
ton a cloud aubaet
Oq. pta. obv1oaaly of th!a apace. Actually. thi.
v\\u-Mbs..,...
or
~
subset 1s formed frcn a piecee o£ albegraic varietl... cut out by
other a1cebraio vnr1etioa.

~'01.at8
ExiBtenco of Zqu11!briwm
!!.!!. 36
I have previously publ18hGd£PrOC.:.l!.!. (1950) 48-49J a
proof of \"t~c :-esult !;olC\";I based on l:aJ( utnr.i~ bcneralized fixed point
theoreo. t\"l'he proof b i van here uses the Brouwer \"theore.:n.
!hft ~hod la -to aet u!' Bequence of oontinUOl.lS mMpp lut;a,
t\\
whoee
!,Oi~t8
fued have an eq,ul1ibritua point 88 limit !,>oint. A limit cappinC
dlscontil:uC\"~:l, au~r
exists. but is (L.\"ld :lceO not have fixed points.
l'lU:..'O. 1, Evor:r finite garno has an equ11ibriun ?oiJlt.
Uainc our atGJldard notation. let -<l.
Proof, be an ll-tuple of mixed
tho pa,-.ott to i'lnyer
atratoglea. and 11- he usee his
;
Viot
pure at.. t.gy and the others use thoir rcspect!n m1xed strate-
...
l'~ar each 1ntet;Or A we
-<i... dGi'1ne t.ha following continuous
glee in
..4... ,
1'unctl ana of
P:cll4-) •
~~
q,.{4.)
¢i oJ..--J..J~ =: Po Y\"A. •
+
(4.) - '1 ;(-.tJ and
0(
¢itf-t),.) =- max (0) ¢jot (4~)..)J •
=y)o. >0
~ ¢~(A/.;)'-) mt-J(¢~~~~)
>
ot
<6t (-<L/-J
L ~;t(-4.7A) 1& cont1nuoua.
(l
bt.) ~) =: C< ,')..)
~
L 1T: 0(
S;I'
Define and
(.
_4.:'(.4.~ -,...) ::- (S.'7 $ ~7. S:') oc: • all the operat1oDII
£ il:c G
••
18 con-

or -<L • 15 • coll, thero auat
tinuouaJ and sinoe the apaoe n-tuplea.
A' • 4,.... ,
bo n fixed poiot for each lionce thero wUl bo a aubsoquence
-<t.* , whGre
cotmJrt:~ to 4~ in :JAi??~ .A.--tA'f.t.,/(~».
; !'i:ad under the
nOll.upp_~e~ ~1t- wore not an equUibrium point. -:hen 1£ •
P.=. (S,*, •• • s#) 5;* must be non-
soma c<=pQnont
$;* uaea ~0tXJ ~ atratoQ
opt1al acainat tho others. which moana
(if)) ,,~ . 4-J ~ that
IT:ol' which 1:1 non-opt1tal. £aoo :-hiu
P;ol~*) < C(;&-t*) which juatU\"!.oa
P;~A*) - '1;(4*) ~ - E •
I[hot f ~) <. %..
Cf: (..t.*J] <'74 a-l
[11D({4Jf) -
(-<l#-'-) - Cfj{:<lrtU -
q,- &<ttJ.) + y}.(N <. 0
p; (4) -
Addlnw whioh 1a
0(
¢;O{ (AI'\\-~ ft£I\"-l) < 0 J ..,,-~ ¢! (4,..).u.i}:ojlanoe
Ci ~ (-'..,.., 7'{,...) ) == .0 · an equation we know that
From this
A,... .dJ1De
18 ACt WMtd 1A
;r;al
A.f4 = ~ -mot Cj-:C (4.,...1 ~))
t 1Xed po.11'lt.
•
wh1ch contftd1n. our uaumptlon.
-4..~ 18 1Ddeed an equilibrium. point.
IIe.aoe

auto;:r.vJZJ?hi!~ ~
An or 8it.??:8'hr;\" of a 'Will bo a portrutatlc.t: of
condl'tionG. c1von 001011'.
p~ oortai.~
lta atrate&108 which IAtiafiea
atratocaa belont; to G~O pla~rcr ~'1fr; ~o into
tlUst
If two t-;IO
(&
¢
GtnlteGi~ ~.~ to 1:1 tho ?or::zuta-
A sin;;lo pla:1tJr. Thus!.r
<f
tion of t.'lo :m.\"\"\" atrnto::;!oa 1~ !nducoo a ~Ilt!on or the
players.
Each n-tuple at pure atratocioa 18 therotoro per::lUtod into another
X the !.nducod ~or:::utat!on
n-tuplo of pure stratobioa. 2.:/ call
\".0
S
or thea. n-tuplea. denote an A-tuple of pure atrategiea and
Let
P;~) ~r p~ S
j wbaJl the n-tuplo
to 18 __
the
lie require that it
ployea.
J=.,'f
•
then
l~rJ.
which canplatea the de.f1n1tioll at a
¢ baa a Wl1que l1DMar exte1l81011 to the JId:XIId
The penJLttat10n
nrateb1es. It
~. . aD
¢ to tho :d..u4 atn.teC1U olear17
!'he ext.uIlcm ot
X to the n-tuplM of a\\xaS a1ft~. ~
oxteMlon ot lYe ahall
~ tb1a by 7<- •
VIe det1ne a .~1o Il-'Wple ~y
ca-
.4. at &
~\"= A X ~s
to%' all
A
'beinG a penattat10n deriTed trca a .,..try
it Wlderatood that ID8a1W
¢.

-8-
~ hu a ayoaotr10 equ!llbriun !)Oint.
Any finite
'riEO. 4.
Firat we not. that
:.:lrool't
=- ~-rrj\"\" ;>rope~y (S,o)¢~
5;<> ';.;<>
haG the 'IIhore
r't
Ao :: (~.o 5\"'0 •••
tf·
J\" -- .·\"tro':that the o-tuple S'h).J
, 14
j ~ ~
X a~':Xletrio
f!;;od under an\";{ J henoe any GntlO ho.G At lOQst one :p
tuple.
It A~(S,~ .•• S,,)
Sr-;:j is so
c...une~ j:= i 'f ) hence
too because
= lS;):+ (t;)¢'\", lSI;t) ¢ ,hence ~t*\")X ~ AC!rs .
'l'hia allOW6 that tho Get at S\";:anetr1c n-tuplea 1& a convex aubaet of
tho apace of n-tuplea a1t!ce it 18 obvioual:/ cloaed.
-<t.--:,-J... /(-t. ~ A. )
A
!Iow ob.\",\"\", that for uoh t;be _pp!.ng uaed
1.D the proot of exiatenoe tbecrea • • intrinsically defined. Therefore.
f<.> .. ':
-x.\" 1M '~ '~t~ ;\"'1 \"\",'\" .-
-4.1.. ~ -<L / ( At. , \"}\\.) at the r ; -
1: and morphia
= ,4/(4;X,'A) ~.
A;J<.
wewUlhaTo • 18.,...
tt
-4.;:<- = -4. / (-4.,}'A.)
...4.:x.:: = -d. t.
-4, aDd tbaN.rcre
tr1Q •
coueqU8llt.l.7 this -pplDc apa tt» ad of n-tupl. 1.trto ita.It.
.~c
Since th1a aet 1a • cell there . .t be • •~tr1c t1xed point -J.~ •
or
.And. .. in the proot could obtain a l1nlt
the aSatAnco thGo~ _
~br10.
A-k Wh1ch would to be
po1n\\ bay.

~olutlona
1';a dof1:18 here solutions. stronG solutions, aM t;ub-solut1.ol'ls. A
no~oopE}ra:tive alw~
rJUJIJ dooa not haw a :lol:. ;tion, but whon it does
~onc u~ci&l
the aaolution l.:l unique. solutions aru noluticns with
~
S. wm denote a set of m1xad Gtrateciea of player j am!
or
a set n-tuploa of r.d..oo;cd atrnte;ietl.
GolvahUit:r:
~ ~ • or
1a solvable if ita set. equillbr1ua pointe ••tis-
A
tielS the condlt1oD
(1)
Thia 18 _lled the _ b 1 1 1 t y condit1on. The aolutloa of •
an.,£. or equilibrium point••
aol_ble guw 1a 1ta
~ sol_bUl\"~
~ ,aolvable 11' 111 bu. •
aa- aolutlou.'; • auall that
1a
A
s
tor all ;
s;

-10-
tr~t (~; S;)
;t- 16 an equi11briwa
the n-tuple
such for sane
L o~ul11~!.um point.J
i oomponent o.r SCi:lO
th
Si
point. is the
.
S; 01~ !llayer ••
the set at ocul1ibr1um. st:-nt&hies
call
710
.i
:r 1D a Gubaet of tho cot of equilibrium ,o1nta of a Game and
Q1~mal ~'U4
d rolAt,170 to
:sat1B:t1M condition (1); and if 1G ?rO-
~y t~..n ~ a Dul>Bolutlon.
call
1I8
~ S; • a.a
we dot1.no the .. tJl factor se;t.
For any atID-eolU\"\"tooion
~ coma'M ( :::t-; s:) tor
'tho :wt 0: all S : IS .UGh tt-At II . . .
;;t-.
~ue.
a aul>-aolut1on. when a aolut1on, and ita tactor
Nate that 14
are the Hta of 4Iqu1llbrium strateGi•••
uta
~•
2J aub-aolutlon. 18 the set of all n-tuples
A
1'RJJ).
S.
S;;
(S •., S1..,. _. Sft) $i E 1a the
mob 1aha'b each where
J. ~ proGu~ ot ita
o.c.tr1oally_ 18 the
; th taotor Nt ot
factor ..te.
• 57 det1n1t1on
r ft-;- ; s,-) t:-eL ·
3 ..;;C ~ ;;t;.., • • • .,;;ttl\\. ~ ftr _ch
8tIDb
t1JI1rlg the oond1tlon (1) D-l t1mee ... obtaln auccoa.1....\",y
(*.;S,;S1.;S~; .. -; S..) e-laDd the lan 18
0=ljSI;S1.)e-~, ••• )
.1Dp~ (SI/ S.., ••• S..) fj . wh1ah we neodec! 1:0 abcIIr.
!llEO. ~ The.1'aotor aota S' ., S ~ or. aubwGolut10J1
1.\" • • • Y\\
.JaO\".
or atratec'
are olosed and oonvax as aubaeta the a1ZIId
S·•
Sf e
S; aDd
•

-11-
S'i
Si* = (S: +')t)/2. S;# is
; (b) 1!
€
then
S; Si
Si\"1tf
• limit point at •
then
€.£ .
/t
Let Then we have
'tj , (I), P9- 3 for an eq.
for any by ueing the criterion of
f:- (51.1 ••• S.,) in
Adding these inequalitiea, using the linearity of
pt.
PJ (t-; $;1 ~ I) (A; \\; *;t,D
Sj • and dividing by 2. bet
'IfO
S; *:::: ($; +S; I ) /L . ::'rom t..'1is we know that
:since
pt. for any -*E~. Ii: the set of
(?t; Si7E) is an eq.
C?t; ~
C;:*) is added to
all such eq. pta. the aU(;lD8nted set
/)
ana ainee ~
cloarly aatiai'iea couciitiOll (1). 'Was to be maximal it
S; .
S:* E-
follows that
;::t-c~
s: *)
(:t;
To attack (b) note that the n-tupla , where
f+; Si)
Umit point ot the .at of n-tuplea
will be the form.
oi~
&
~ S; . sinoe Si .
'5;\"*
S; i s . limit point of
where Eut
thla aet 1a a eet ot eq. pta. and hence any point 1n ita closure 18 an
p,. trJ.
.inc. L\"ee
eq. pt., the ••t of all eq. pta. 1. cloaed There-
0-; S; ~) S;
<:
') ,=1* -
tor. 18 an eq. pt. and M.I1Oe trOll!
the . . . argument as tor •
Values,
~ be the sat of equUibrium po1l:ta ot a ga:.e.
Let ;fa define
.:;:t[P:hii]
:::'LLP:[4-J] ,
Vj+- Vi-=: •
'\"\". - \"'.-
,~+ .- v, Vi-t
'W'l\" 1t
we e 18
•
i
the upoer value to player of the game J

-12-
the value, 1.r 1t _iata.
'\"
·ialu8a will obvioualy have to exist if thero 1$ but one QC1uilibriuu
point.
One can det1ne associated ;:or a sub-solution by reatr!ot1oc
·nU.U80
~ ~ \":.. - pta. in tho aub-oolution IILd then ua1n.:; tho llama defl.niog
to
equat10rus as aboN.
is. a \"uddla l'01nta 1n
stron& t,Jolutio.~ GXiat thel~
aol'VabloJ only when

Geamotrical Form at Solut!ona
In tho two-peraon zero-sum cane 1t han boon shown thn t tho Nt of
an:r solvable ~.
equllibr..u= atratet;ica 1n
if •
lfh10b 18 ~1 t10n (S) on paGe An equivalent conait10n 1& for
(2)
5j at equU1br1ura nnW-
tom at tb4f.n
ooca1der 1m
Lft WI JlQW
*
~1o., p~ j •
SJ. ot bI aq equI 11brl\\a po1A1a,t.t.A
Let
oa1~ 1t s; Sj : •
(;t ; Sj) E.
w1ll be aD equW.brlua po1at 1t and
(A-; S;)
frca'rheo.:' We:lOW appl: ooad1t-1cma (2) to
P: a ;C
SlJJH il-11ne&r a.cd !a conatant theae ,are a set of linMr
11.,.. (5;) ~ a •
1neqtal1t1e. ot the tara Each such 1mqual1t)'
SJ or tor tho.a lying ,ide
1\" either aatl.t1ed tor all and to one
OIl
or .aae h,-perpla.be pua!Jl£ 'I'beretore. the
through the atftteg' , .bpla.

-12a-
Simple J;D.mplee
the 10ft. ate.
-3
5
Ex. 1 (to(..
,,~ 4
-4
-S bat 5
~
~-4
b
.J
..,
_.\\
1 a..at 1 :olution:
..:..x. :~n~
I..
-10 (l. ~ 10
V. =. \\1\". -::. -I
10 b~-l0
-1 b~ -1
Unaolvahle J equUlbrlUl4 JKl~ ( a 101.) ) (3) ~ (k
14.C(
Ex. 3 1
('Vi...... bf ~ , .:;(/1. + (5/1.) .:ho at.~-
,,-10
-lO and
CL
teGioa in the last caB. have maxi-min and
-10 b 0(. -10
1 b p> 1 ~!li-!:lSX pro~erties.
or .'l11xed
1 a.. Co(.. 3tro~
Ex. 4 1 ':olutiona all:>airs strate-
0<1. ~ ~1es. ~
1
v\"\\.- ::::. 0
=1 ,
\\I, += ,,~
a
1 \\0 0( '1,-:::.
o\\:)~ 0
Ex. 5 let.. c( 2
-1 q ~ -4
-4 hot -1
2b~ 1
la..Q{ 1
~.e .w1th
O C1 r.> 0
Ob~ 0 an CU1p1e at !.:stability.
o r.>
b 0

LWh!ch t1nitoJ 0: conditione r...ll all
co:1ploto aet be satisi'1ed
Ui
. ~
a~~ polyhedra~
si::lultaoeoualy on convex eubset of pla;rer J S stra-
taw· ciraplo....:. £Interaect1on of ilalf-6paoea.J
Skis the couve.x cloeu.ro
a oorollary we %!Jay concll..tde that
Ali
£'ve-r; ice aJ•
;;.t . ~ .mixed
01\" A i'inita strntocica

Doo.1 nnnco a&d contrad1ct1on !rethodD
tl-..e..t
Gal~l
,;g
X.
.
ror overy
~
':his. ~~~ ~o G&:li.nG that I a hiJlcr ~..
Si t;iVtis paJCr
or
ott than ~:.oo titrtlt8~io:J
r.atter lthat the the other plaj'Ors are.
,
To ae. whetb.or a atrateQ' S i it aufficea to conai-
dominates
):
d.r only pure atrate&ioa for tho other !,llcyera becauae of the n-11near-
P: •
it:! 01\"
It iaobvioua £rom the definitions that .a2 esuilibr1um ZOilIt ~
involve .! dom.i..oated atratoQ ~i •
than 1n
P:::a c
I
'S. • enoubh
Zhon tor a amll
-t/= ti + p(5(- Si)
prove a t81l proport1ea of tho aet at Wldominated atnte&1ea.
C.QO Call
h1 tM \\m101l at acas ooUttC'Uon at
.~ ~ &Ad 10 torMd
It ia
taoea ot the .~ a1mplex.
tor ~
The intonation obta1McS by dlaoovor1rlg dom1nano. . one
. , be ot relnano. to the othen. 1naofar ... tba el1m1natlon of olu...
u
ot Ja1xed ftrateG1e. .a poaalb1e OOJIPOIlenta ot an equU1brlla po!l*
;t 's are all
ccmo..-d. For 1:he an all
1IboM 0CIIlp0DIDta undam.i.nated
atratec1H or
that need be COJl81dered and this elhdnat1I1G acme ot the
olaaa at atratec1ee
~
one 1'la,.- .y taka posa1blo the ol1m1natlon a nGW
tar another play.-.

-18-
.l41athcr :n--occdure which =-:t. bo uaod ill locatinG ~uil1briUL'1 ,ointo
cl,)~..trurJlct!.cn-~e 01'.0 t:.s:tt.U:J::i t};at &.'1 OCiuilib~iUtl
ar..alys!t;. Eare
is tlw
e~~ ro~!.ons
ba\"linu caczponcnt at.,touioa l\"JicG ·.vithin certain
?Oint of
:wt
.!\"..l...-t1lCr coc.diJ,;:or..a
s-;:-n\"t~; :;~coc ced\\.~cc ·Hh.icl~
-:h.o a.-:d :?rocccd%! to
to 5a~~!'ed ~' ~?otllQC!.a. ~ .~.l.O. r:1~ ~~ ~.
tho aart cf be
to evo,nt'wllly obtain a oontrat.iict1on in-
t..lu-out;h &Overal
Can\" 10<1 GtD.t;OG
~t utiar~~ ~7C­
there iz no equil!.briu::1 point
dicatint; t.ho initial
thea1&.

\"
include tr..o si:npll!'iad. l'ok.er ~ ~1ven belaar. Tho
rcali=t!.c \\1110
CQ.SO
rules are aa :~11a~81
l2!!
TI1 th 8q un11 :-- ::iSJ1~.
( 1; hiGh and cards. and
':'he dec 1: is l.ar :;'6,
cons lata or one card.
a ~d
~
(:;) The ,layora :,llay 1n rotation and tho enes a.fter all ha9JQ
puaod or ai'-~er .J.t::.e ?~or llnfi opOl4eC ar.d .l;!:o ot!10ra ha-re had a cbulce
to call.
tho antGa are retrioved.
(.;) If no o&le beta
Othorwiae the pot ia <llviQed oClually amollb tha h~st l~
(5)
whioh bA711 bot.
como
!:lOre ::atiai'o.ctar:; to treat the in torma at qtaDt1-
~'Jo .r1ltd i.$.;
u. .. to:m ~ tt~
Gall \"bebn1or puw,roter.. tbau .1ll tba nonlDl
oc:e.
~ and 1Jc0D0aio ~T1or.\" In tho noral i'om repreaeatat101.1 two
atratec1u 'at a p~ ~ btt equivalent in the . . . . tbat -.oh
Jrdxed
akee the ind1vidual ohooae each a_1lable ooura8 01' .OUOD 1D each par-
t1cular altuat10n nqw.rinG l\\Ct1cn OD hi. pertt-w1th the . . . tntquenoy.
tM __ bebaY10r pattern em the pan at the 1D-
~
That 1a. thq
dlv14ual.
give the prcbabU1t1ea ot tald.nc .ach of the
~.
Be1:aT1or
~1oua poaa1ble 81tuatiolll wh1ch
vu-loua poaa1b18 acti0D8 1A each at the
l\"hwa tbq deao.r1be bebarlar pattorna.
Je1 arl.ae.
pla~
In tanaa of behavior pa.nmatera the atrategt.. of the - : be
,lm,o, that a1noe
ropruantod aa foll.owa, •• is co point 1n puainG
~here
hiGl! card at one'. laat opportun1ty to bert that thia wUl not be
with a

·.
i
0(· open~-.!!!fA
t
~ on ~ A on..2.
Gall !I t
:: Opon
J
on.!£!.
,.... call II and II:r
f
le.!!
i !~!
-zJ Call on low
Call: on
~ ~ Ct.ll !II and :: on ~
!! Opon on high
-
E Cptln on law
~ Call :l on ~
~ and
~ Opon on Ja!
....
.,....,.
e Call I -
lew
OIl
-
& Call I! on low
w. leo_to all porus1blo equ1l1br1ucL po1nta b:: t1rat ah.owing b.'t moat
at the creek ~er. mtLIl't maW:/, w1th a 11tt1.
1'IUlUh. By c;lom'na M •
oontrad1ot1o~ ~1a ~,~,
f.> 18 eli-Seated and wlth it 60
f'-, ~) ,~ A~ k
t9 .,
'by 4ca' .. nee. Thea oonVadlc\\1ODa e1Sm1nete
aDd
a? YJ •
orcter.
1ll that Thi. 1.ava ua 'With €
aM aIlS
V 0(,
DOC. at
Contr'ad1ct1011 a.nalya1a sm.. that the.. can be aero or one and
tJwa we obtain a system of ail:w:taneoUG al.pbra1c equat1ona. ThIa equatiOJla
happen to haw but one solution 'l:1th tho -var1ablea 111 tl:.a racp (0\" , ) •
~\\ s=
J5ii ljOlt\\ ':7--~c(
11_
,.J_ -
J0 •I- 4-
V\\- , ,:)+0(
E :::. ~ o(-} These Ute Id
~O~\" '1==. G\"!>I; J 'b =.~\"
•
J
ct-t t; 0(-= - 7 .....
-I

..
,
~1nco equ111br1ua point the GU28
there is onl;l valuos J theae are
OM 1'..48
- - Oq~ = 1- ~o( \" CJ.h\\da
- ' r '\"
1--.
\"
•
tioneG are zero.
I ill
Q.#\\d]I
= ~/.\" , O~.,~ ~
V~
t -=-
0{
· 0 ~11..~~ y~ ~
III' :
~= €=J value to
III.
][ ~ cI I
'fttnJU8
'h
0<-=
I : -. J'G7 ea
'ftlue to =: - ' /
nr
I ~d
h~h. louJ
- - .. _.-
~ -= -X, ,€ == 3/1J
bet pct.S\\
_ .. -_ ....
pass ~SS
If.: -.1 r~ G::: - ':.fiI. Y-
valuo to
p~d. a~n1ts.c.uxt. o~
plq be1'ore the pm 1. thia ad'ftntaGe 'becClt:lea
I ill. ']I[
ot coalition
in ti'A case whore ray open arter two pua••
I on both hl?ib and Je! but will not opea 11'
h:1d planned to pua
when

·.

..
-
Uotlvatlou Uld InterpretatIon
or
In th1a section .. shall tx-/ to oxplain the alcniflcance tho
1ntroduaed 1n th1a paper. shall try to ahow how
That la, _
COIlCepta
equilibrium polnta &.ad aolut1cma can be conneoted w1th obaervable
.- -
pMnomlaa.
tor • tlOD-COOperatlve
The b. . l0 requirements 1a that thare
pM
£ unl••• it haa
Ihould be no pre-play o~lcatlon among the pla,.ra
beering on the~. Thus. br 1mpUcatlO1l. there are no coallt1ona
I¥)
£\"par-ottJ
and no ald..pa)'Mnta. Decaun there 1a no extra-game uti11tr
~ta
tranater, the of d1ttereJl'U plaJera are .ttecti..,..l)' 1ncamparabl.J
Pi ~ a ~ Pi + b:
~ •
11 we tncatona the tuDctloDa l1nearlfl
Q;>o
where
equUlbrium po1%lta are preaenecl UDder ,UGh i:nDataa-.tiam.
We ,hall now take up the - . . .-action- 1nterpretat1011 of equilibrIum
18 mm--1A17 to uslat that thG partloipanta hne .tul1 kI¥Ar1Mce at the
tIuwch
the .bU1~ an4 1nc11mtlOJl to CO
total ItrucWre of the pa. OJ-
'aD7 oaapl_ retlaoDiDc proa..... !hR the partlo,\"peta U. IUPPO'\" to ac-
tbe~.H1a~l,\" achaDtaca.ot the ftl'loua
cumulate·..,1J1.oallntormaUOIl.OA
~ atntlllp. &11 1Ibe1zo 4!apoeal.
puN
4eta1led. we ..... that tb8n 18 • population CIA
To be IIOre 1;be
the....
nat1aUoV ot Let
of sar'lo1paDta tor ••• podt1oa of
M ...
alao . . . . . that the ·a~ p~ of the _ 1molwa particl-
WI Ii
na-
tna tbe ~
put. ae1eoted at rud_ popa1aUou. aA4 that S. •
A
.....-ac. trequeaay with 1Ib1ah . . . PUN ..,J.o,. by the
'b~ ....- 0 1a
•.-nne- ...,... ot the appropr1at. populatloa.
in
S1I1ce there 1a to be no oollaboratlOD be. . . . Wi v14ua1a p~

-22-
ca-. probability that a partioular n-
~
d.1Eterent posit1ons at the
tuple of pure Gtntebloa will be caploycd in playinc or the GIlm8
It
or
should be the product at the probabUlt1_ 1ndloat1n& the obaooe
at the Jl punt atratel.':iea to be emplcr.,91 1Jl a nmdom. p1ay1ng •
_ch
- ... _\".. \"\"_ ....-'\" - .
.
Let the probabU1ty that lTio( 11'111 be emplo;,ed 1A a nndara
• and let S; -= L. C; c( IT; l(
ot Ci ot
the gam be
play1n& •
0(
~(~.,S1., •• - ~ ..) Then the ex~ pay-01't to an 1nd1-v-
•
t th poalt1on at the game aD4 employ1Dc the pure
1dual play1Dc 1n the
p; (4; Tr;o() = P:ol (~) •
lr.o(
atrategy 14
U. ooneid.. ~
what efteate the experlenoe the pari10ipanta
let
BQW
1'0 \"CUM. .. •• did. tha't they .oo~te4 ..,1r1oa1
will produce.
evidence the pure etrate&iea at their d1apoeal 1. to uaw. 1Ulat
011
.• leuD the numb...
thoae playlnc 1n pooitlon •
the1 w1ll epl07 oal.1 optSal atntep.e••
But it they lmow thea. puN
'bIc'- ·1ricl auah tha1a
1.... 1sbo_ pun .....
•
s: atta-.. poet-
.~lJ .bIoI .xpre. . . their bebn!~
Si
~_lo~ ~ to o~ pun 11n~. ~,.... \"'
'1..,.
tor ~ to be
ISIIJ17 • equUlbrl_ poln.
Bu\" th1a 1& oOll41t1orl aD
£ ... [II-},ri:9
T1wa the ..~1oaa we .... JA tlWa \" . . . . .OUOIl\" JAMrpret10A
'0 lvatecU. re~-tiDC the a'V'8raP
tM ooaolualO1l tbat 1Ibe a1xed
lead
bebaYio.- in _ob ot the populaUOIli tana ID equ1l1br1ua pl1lrt.
tt the u8lJlllptlos. nW !:\"(t~b ! ~:
!he popalaUoDII need D01J be lar£fJ
-.

-23-
:_:'..1 hold. There an aituations in economic. or international
politics in lrh1ch, effoctl..,l)'... &rOUp of interests are involved 10
non-cooperat1va £UIIIt without be1nc -.re or it, the l1~enea.
0.
euq1l1br'1ua., a1nco tho inrOr.:a'tdOl1. ita ut1ll&at1on. aDd the atabUlty
or a~bO trequenc1u wUl be 1=partaot.
the
w. sketch another lAterprotat1on. one in which aolutlona play
110W
pla1ed but once •
~or ~
a role. and which 18 applicable to a
. . p.roceed b1 1n\".nlga1W1& the qu..tiol1& ldlat would be • \"ratiOllal-
p1\"ff41ot1on of the behavior to b. cpected ~ rat10nal plq1&lc tJ. pm8 1:1
By \\l.iD& the principles that • rational prediotion ehould be
ClueatlO11'
or
UI11que. that the abould be able to deduce and aake u••
p~ lt, and
or eaah ot What to expect
pla~ ~
tha'b such knowledp on the pert;
at oOllto:m1t1 with the
ou~
otben 150 do .hould .not lead h1a to act pre-
1. leeS to the aoaoep1I of a aolut1aD 4etlMd
d1otloa; ObI betON.
S'JS~, ••• .... the . . . ot ~ nn.tec1a
S\"
U
~ a aolwble p~ aho'leS\", \"n.....-.p
sa-. the -r&t1ocal\"
i 1ICIU1d det~ • au.d
beh&ftW Of ra1:1oml aID plq1Dc 1D. poelt1oa
a~ ~ out.\"
S; i t . experlmc1J were
S; SA
run
~ p~
1A th1a 1nterprot&UOA ... J»04 to Ji:DoJr the
the
atn1c~ of tM CMO 111 order to be able to deduce tile p*lo14Ob tor
atr~
thezuelvea. It 18 quite a ratloaal1at1a aDd w.11a1Dc inter-
p....Uoa.
ca- 1t aaa.t1aa happua that &ooci blur1aUo
In an UDaolvabl.
rel.acm a oan be tOWld tor Dal'row1ng dCWD the aet at equ1l1brium. painta to
_, tho.. 111 a 81z1c1e lub-aolutloD, wh1ch theA plqll th. role ot • aolat1on.
- /

r ... ,.
III CGnenl a aub-aolutlon 'SU:¥ be looked at ... a aet or IIltually
• coherent whole. 'the sub-
t~
oompatible equU1br1um poiJIta,
,i. -che
aolutlone apptlal' to a natural 8ubdiv1aion ofj.eet at equilibrium
·'.oa---
. pointa ~

-\"
\"
Applicat1oll8
ot fair
~10.
The atudy of n-peraOll guea tor 'Whioh the aoooptod
play imply non-oooperatl\" playing 18. of course, an obvioua d1reotlOD
in which .-to apply -this t1uJo17'a .And poker 18 the moat obvioU8 target.
-...
. ~'''-~.~--.. .'-
-'.
anal~1a of • lION rea11stic poker game than our very a1mple model
'the
Ihoul4 hi quite intereet1.ng affair.
aD
the oamplox1ty at tho mathGtatloal work needed tor a oaaplete 1D.-
'V'Mtl~t1on lncr...... rather rapidly. hawewr, with increaaiDt complex-
1'b7 at the OU-J ao that 1t . . . . that &Ilalya1a ~ a much more ooa-
gatlO
onl7 be
pla than the exuplAt Clwn here waUd toaa1bl. ua1ng appraxi-
.te oc.puta111oaa1 ..thode.
A leu obY1oua ~~ appl1cat1oD 1& to the atuq of oooporat1w
ca- we ..\". a altua'b1oa:r. ~
B1 •. ooopenl\"l.... lnvolv1nc a ••t
pMa.
ph,... pu.r. atntec1... and p~. . . uaual, but with i;h8 . .8UlDpt1on
tha, the ph,... 08D aDd 1IU1 oollaboft_ .. 1ZQ do 111 tlw WA . . .,pn
-ana the pta,... .., oanmioa1lll ..s tora
a.rxi Jlarc8AlteJ'D tbe0J7. Tbi.
u
wU1 be elltoroed b7 an *'Pint. It 1a IUi17
wh1~
oa.11_1aa8 UDbIIO. .
tl'aDltenbU;V. or ..,. 00IIpUIIb11-
natrlotlw. baIrewr. to aD7
UIUDMt
1V ot·.1;he pq-ott. Lwhloh ahou14 be ill uf;W:'7 Wl11:V to .41tt..--t
~. AIt¥ 4.1N4 tnutenbWt7 au 1M C- lUalt 1D-
be pW lDto
eztn,..ca- oollaboftt1on.
.'Mad ot ...ma1q 1t poa.1b1e 1n the
_1,.. 84)'_a1oalw apprcuh to.the
baa 4--lot*I ~ of
!M 00-
&
os-raU- ..... baNd OM
rec!uot1oa 1:0 Dat-aoopca1;1\". tar. PIG-
upoA
. . . 1»7 ~ • the \".. p'q ~1aUoA ao tbail tM
.,g]. ot
bJo\" .,... 111. larcc- J1CIIl-ooop....l . . . . . C1dWJb
a1Jepe ~ D8got1a~lon
1Dt1D1~ at pure atn:blg1_J 4cJaar1b1Dc the total altsuation.
wU1 ha.... Ul
trea\"* 1A t . . . of tM 'heor7 ~ th1a ,.,.-
Tlda lupr ,pM SA the

£ extended to 1nt1n1t. ~ u.d are are teken
it _luau obtained they
or
.. the 'ftluea the cooperatl'1e Thue tho probl_ anall=ing a
gax:G.
or
cooperatlft ,ame beoome8 the ,robl_ obtaininr; a suitable. aJld con-
-vr1no1ng. noa-oooperatlw model for the negotiation.
The writer bu. by 8uch • treatment. ob1:a1ned values tor all finite
two penon ooopC'Gt1w t;ame8, and so:ae speoial n-peraon game••

'. \"..
B1bllograpl:q
von Neumann, llor&enatern, wTheor:r of Games and Economic Behavior\".
(1)
Pr1noeton Un1.venl\\7 PHa •• 1944.
a...-.
J. F. lZuh. Jr •• -BquUibrlua l'o1nta 1n N-Per8On ?roc. N.
(2)
A. s. (1950) 48-49.
3a
jln. ~. G~ut1ona
Gal•• awl AuhA &avo valuable cri1Uciaa a£¥l
tor 1mprov1ng the expoelt1on at the material 1n th1a paper. DaYid Gale
auggeated the tnwetlgatlon at .~1c pmea. The .olution of the Poker
.,.1 . . a jolD projan s. ahapleJ uc1 tba
~ _~.
'b)' Llo)U
PlDaU7_ the author\".. auria1Ded t1DaJ3a1allJ by the Ataaio IDerg COWJa-
.1oA 1D the period 1949-60 cluriDg wb10h thia work.... doMe
-- ~