John Nash Ph D Thesis
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About This Presentation
La breve Tesis del premio Nobel John Nash, por bibliografía dos manuscritos, uno de los cuales es un PNAS suyo.
Slide Content
TODO RATIVE CALS
A DISCERTATION
Presented to the Paoulty of Princeton
Untrarsity in Candidacy for the Degree
at Dooter of Philosophy
Presented to the Paoulty of Princeton
Untrarsity in Candidacy for the Degree
at Dooter of Philosophy
Abstract
This paper introduces the concopt of a non-coopuratise geo and
developa methods for the nathametical antlycia of such games. The
ro M-porson gunos roxresaxted by mens of pure atrae
¿ares consicored
togies and payoff functions defined for the combinations of pure
strategies
‘The distinction between cooperative and nan-oooperative ganas 18
unvolatos to the mathamaticel dosortption by mans of pure strategies
asd peyrat? functions of a game Sather, £6 deporte on the possibility
ar Amporaibility of coalitions, oamnieation, an! elde-pajmontas
‘tua concapta of en oquilthstin point, a solution, a strong solution,
a sureciution, and valuas are introduced by nathenntioal definition.
and tn later sections the interpretation af thoce oopoapts in non-cooper
ative canes tn discussed
The main mathenationl result de the proof of the existence in any
gane of at least one equilibrium pointe Other resulte concern the geo
metrical structure of the set of equilibriun pointe of a Cane with a 00
Aution, the geonetry of scb-eolutions, and the existence of « ayunetrical
equilibrium potnt In a eymetrioal games
Aa on Sllustration af the possibilities for appliontien a treatment
staple thrwe-tan poker model 1a inoludeds
526-500 Doctor Ana cto
a
This paper introduces the concopt of a non-coopuratise geo and
developa methods for the nathametical antlycia of such games. The
ro M-porson gunos roxresaxted by mens of pure atrae
¿ares consicored
togies and payoff functions defined for the combinations of pure
strategies
‘The distinction between cooperative and nan-oooperative ganas 18
unvolatos to the mathamaticel dosortption by mans of pure strategies
asd peyrat? functions of a game Sather, £6 deporte on the possibility
ar Amporaibility of coalitions, oamnieation, an! elde-pajmontas
‘tua concapta of en oquilthstin point, a solution, a strong solution,
a sureciution, and valuas are introduced by nathenntioal definition.
and tn later sections the interpretation af thoce oopoapts in non-cooper
ative canes tn discussed
The main mathenationl result de the proof of the existence in any
gane of at least one equilibrium pointe Other resulte concern the geo
metrical structure of the set of equilibriun pointe of a Cane with a 00
Aution, the geonetry of scb-eolutions, and the existence of « ayunetrical
equilibrium potnt In a eymetrioal games
Aa on Sllustration af the possibilities for appliontien a treatment
staple thrwe-tan poker model 1a inoludeds
526-500 Doctor Ana cto
a
Table of Contents
o el
Forme seftatticca and Teratnology ee eee
Extetonco of Zguilibriun Zoints ss eee eee eee
symetrios of Game see esse reece cane
cecosteical Farm of Solutions » +++ oe
Dominazos and Contradiotion Methods » ses ++
A Throe-iin Soler Gan esse
Motivation and Interpretations ss oem...
Applications esse
o el
Forme seftatticca and Teratnology ee eee
Extetonco of Zguilibriun Zoints ss eee eee eee
symetrios of Game see esse reece cane
cecosteical Farm of Solutions » +++ oe
Dominazos and Contradiotion Methods » ses ++
A Throe-iin Soler Gan esse
Motivation and Interpretations ss oem...
Applications esse
Introduction
Von fousaon and Norgenstern have developed a wory fruttrel theory
of twomoraon seroraun ganes in their bock Theory of Canes and foononte
Jehaviors This book also contains a thocry of ueneruon ganes of a tyne
wish wo would call ecoperatives This theory le based on an analyeta
cf the interralettonahine of the variove coalitions which can be formed
by the players of tho gano.
our theory, in contradistänetion. Lo besod on the absence
étons in that { la ancuned thet osoh sartiotoant soto indovendently,
without collaboration or cosunteation mith any of the otharce
ho notion of an oquilibriva volot la the baste incrediont in our
theory. This action ylalda a generalization af the contoot of the solu
ion of a trowperson sero-mm ganes 15 turno out that the act of equtlie
Artus polata of a two-person zero-eun cane 18 oirply the set of all pairs
of oprosing "good stratent
In the innediately following sections we aball define equilibrium
points and prove that a finite nomoooperative gano always has at least
ene oquilibrius pointe Te «ball also introduce the notions of solvability
and strong solvability of a noo-ooperative came and prove a theores on
the gocmetrical structure of the ext of equilibriun points of a solmble
zu
‘As an osamplo af the application of our theory we include a solution
of a winplified three person poker cameo
‘The motivation and interpretotion of tho sathecatical comenta an
ployed tn tho theary aro reserved for Maciseion in a special section of
tata papers
Von fousaon and Norgenstern have developed a wory fruttrel theory
of twomoraon seroraun ganes in their bock Theory of Canes and foononte
Jehaviors This book also contains a thocry of ueneruon ganes of a tyne
wish wo would call ecoperatives This theory le based on an analyeta
cf the interralettonahine of the variove coalitions which can be formed
by the players of tho gano.
our theory, in contradistänetion. Lo besod on the absence
étons in that { la ancuned thet osoh sartiotoant soto indovendently,
without collaboration or cosunteation mith any of the otharce
ho notion of an oquilibriva volot la the baste incrediont in our
theory. This action ylalda a generalization af the contoot of the solu
ion of a trowperson sero-mm ganes 15 turno out that the act of equtlie
Artus polata of a two-person zero-eun cane 18 oirply the set of all pairs
of oprosing "good stratent
In the innediately following sections we aball define equilibrium
points and prove that a finite nomoooperative gano always has at least
ene oquilibrius pointe Te «ball also introduce the notions of solvability
and strong solvability of a noo-ooperative came and prove a theores on
the gocmetrical structure of the ext of equilibriun points of a solmble
zu
‘As an osamplo af the application of our theory we include a solution
of a winplified three person poker cameo
‘The motivation and interpretotion of tho sathecatical comenta an
ployed tn tho theary aro reserved for Maciseion in a special section of
tata papers
Por 2
thts soctien wo defino the des!
uo standard tormtzology and notations Important definitions will vo
conced by a eubstitie initesting the concept defined The zonccocr
erative {Jaa will be innlieit, rathor than explicit, Yelons
Faite Cass
vor un an penerson cane will be a aot of n laura, or noettions,
each with an aesociated Saito re stratogteas ant
to ach player, | „a zemoff function, P » which aapu tho eat of all
petuples of pure strategies into the real muberas when me use the tora
ptuglo wo shall always nenn a sot of a Liens, with onch ite: associated
with a different playore
atome Sit
A nized stentogy of player {will be « collection of seo-sazative
sunbera watch have unit sun and are in ono to ons correspondasso with his
pore stretectone
meme Scania min Few! at C20
ho represent much a nized atentar, share he TOS ae the zus
strategies of player | + We rogard the Sı’s as points ine sim
plax whose vortices are the Tria’s + this simplex say be regarded
ka a convex best of a real motor eneoey ciring un a natural prooees of
inser oocbination for the aized stretaciens
lo shal! uso the suffixes {j,k for players and LB, to
indioute various pure stratecton af a players the epols Si, ty
and Yj, oboe will indicate mixed strategions [[ joc WEIL indie
thts soctien wo defino the des!
uo standard tormtzology and notations Important definitions will vo
conced by a eubstitie initesting the concept defined The zonccocr
erative {Jaa will be innlieit, rathor than explicit, Yelons
Faite Cass
vor un an penerson cane will be a aot of n laura, or noettions,
each with an aesociated Saito re stratogteas ant
to ach player, | „a zemoff function, P » which aapu tho eat of all
petuples of pure strategies into the real muberas when me use the tora
ptuglo wo shall always nenn a sot of a Liens, with onch ite: associated
with a different playore
atome Sit
A nized stentogy of player {will be « collection of seo-sazative
sunbera watch have unit sun and are in ono to ons correspondasso with his
pore stretectone
meme Scania min Few! at C20
ho represent much a nized atentar, share he TOS ae the zus
strategies of player | + We rogard the Sı’s as points ine sim
plax whose vortices are the Tria’s + this simplex say be regarded
ka a convex best of a real motor eneoey ciring un a natural prooees of
inser oocbination for the aized stretaciens
lo shal! uso the suffixes {j,k for players and LB, to
indioute various pure stratecton af a players the epols Si, ty
and Yj, oboe will indicate mixed strategions [[ joc WEIL indie
eso tn player's OLEA pure strategy otor
rayer functions D, +
In Maga functions Pi
euro above, has a unique ext
which Le linear in the atsed stratecy of each playor / meliwar 7
pute extoraton we ant alae dono by» wetthan RCS Sues 59) >
ce tal write -Q or A to denote on metunlo of atxod strevertos
ana te x Hy) then PiCd) shat stun Pi (use Sn)
Such an returle, À à will alao ve regardod an a polit la a vector emo
Laso maco aol! be rain ho cul sinipinr torother the vector sraces
contatatos the alzed atzatecloas And tho sot of alt zuch note Forms,
es course, convex polytopo, the product of the atapltcos renresentins
‘the nixed otravecters
ror convenience we introduze the eubstitution notation
Cast) water Cutie)
share = use Sn) + The effect of succes:
tone (C45 €9 5%) wo tntteate by (567305)
equilibria Folatı
an neturle LA te an equilibrium point Af and only if Sor every À
= marl pasts]
Re) = MESURES
‘Thus an oquilibriwa polzt is en metuple 4 such that each player's
nized ctratery maxiaizon his payvoff Af the stratecten of the others are
hold fixede hus ouch player's strategy is optimal against those of the
others To shall cocastonally aboreviate oquiliuriu point by age pte
rayer functions D, +
In Maga functions Pi
euro above, has a unique ext
which Le linear in the atsed stratecy of each playor / meliwar 7
pute extoraton we ant alae dono by» wetthan RCS Sues 59) >
ce tal write -Q or A to denote on metunlo of atxod strevertos
ana te x Hy) then PiCd) shat stun Pi (use Sn)
Such an returle, À à will alao ve regardod an a polit la a vector emo
Laso maco aol! be rain ho cul sinipinr torother the vector sraces
contatatos the alzed atzatecloas And tho sot of alt zuch note Forms,
es course, convex polytopo, the product of the atapltcos renresentins
‘the nixed otravecters
ror convenience we introduze the eubstitution notation
Cast) water Cutie)
share = use Sn) + The effect of succes:
tone (C45 €9 5%) wo tntteate by (567305)
equilibria Folatı
an neturle LA te an equilibrium point Af and only if Sor every À
= marl pasts]
Re) = MESURES
‘Thus an oquilibriwa polzt is en metuple 4 such that each player's
nized ctratery maxiaizon his payvoff Af the stratecten of the others are
hold fixede hus ouch player's strategy is optimal against those of the
others To shall cocastonally aboreviate oquiliuriu point by age pte
e
do any that a misod strategy Si unes a pure strates Tg M
Siz Yow Wa m Cig>0 +0 (ss + Sn)
and Sy uses Tin wo also cay that une Tit +
von tha timertty of O 1a Sie
o 3% [Pesa] A
se tt Pela) = Plasma) + mere win
he fottoving ental sacovony and suitetont conattion for L to de
an equilibra ports
Pe) = "2X Peu Ca) .
Se, ore 9) Cin Ti
DAS a En consequently (3) to role we
A Pret) € EX Pre Ce)
which de to may that 2 dom not use Ti unions it in an optical
pure stratery for player À + Se ve mite
GO Tia teme ink then Pra) = mari. (dd)
‘es another necessary and suffietent condition for an equilibriva pointe
since a criterion (3) far an age pte can de esprensod as the
equating of too continuous functions on the apace of nrtunlen 2 the
age pits obviously fora a closed nubeet of this opacos Actually, thie
suveet Lo formas fran Die A places of albegralo varietian, out out by
other slgehraio mrtotions
do any that a misod strategy Si unes a pure strates Tg M
Siz Yow Wa m Cig>0 +0 (ss + Sn)
and Sy uses Tin wo also cay that une Tit +
von tha timertty of O 1a Sie
o 3% [Pesa] A
se tt Pela) = Plasma) + mere win
he fottoving ental sacovony and suitetont conattion for L to de
an equilibra ports
Pe) = "2X Peu Ca) .
Se, ore 9) Cin Ti
DAS a En consequently (3) to role we
A Pret) € EX Pre Ce)
which de to may that 2 dom not use Ti unions it in an optical
pure stratery for player À + Se ve mite
GO Tia teme ink then Pra) = mari. (dd)
‘es another necessary and suffietent condition for an equilibriva pointe
since a criterion (3) far an age pte can de esprensod as the
equating of too continuous functions on the apace of nrtunlen 2 the
age pits obviously fora a closed nubeet of this opacos Actually, thie
suveet Lo formas fran Die A places of albegralo varietian, out out by
other slgehraio mrtotions
ixistenco of Zguiltsriun Points
1 have previously published / ros» à 30 (1950) 40-497 a
proof of tro result tole based on reltsed fixed point
Encore The proof given here usos the ¿roumor Sieoran.
‘the aettod La to set un n sequence of continuous supp Lagos
234 lt) 34 ele); whose
£txod pointe have an equilibrium point ae Limit points A Liat mapping
exista, but Le diocontizucua, ant zus not lave any fixed 70
TIMO. 14 very Finito game haa an equilibrica ponte
proofs Volng our standard notation, let 4 be an ntuple of atxod
strategies, and Pal) tio papas to ployer à wes his
pure strategy Trier and the others uso their respective mixed strate
in de Yor each integor A we defino the Selloming continuous
= "Pad,
CAPR Rd HA ane
Bit = max (o, Pia (a, 0]
= spé» MRE) = ro mine
Bhan
ETS)
Cale, =
sario S/(a,N= Ze cie Dom
CAN sizco all tha operations
have preserved continuity, the eappins 4 > 4 (ED) 18 cone
1 have previously published / ros» à 30 (1950) 40-497 a
proof of tro result tole based on reltsed fixed point
Encore The proof given here usos the ¿roumor Sieoran.
‘the aettod La to set un n sequence of continuous supp Lagos
234 lt) 34 ele); whose
£txod pointe have an equilibrium point ae Limit points A Liat mapping
exista, but Le diocontizucua, ant zus not lave any fixed 70
TIMO. 14 very Finito game haa an equilibrica ponte
proofs Volng our standard notation, let 4 be an ntuple of atxod
strategies, and Pal) tio papas to ployer à wes his
pure strategy Trier and the others uso their respective mixed strate
in de Yor each integor A we defino the Selloming continuous
= "Pad,
CAPR Rd HA ane
Bit = max (o, Pia (a, 0]
= spé» MRE) = ro mine
Bhan
ETS)
Cale, =
sario S/(a,N= Ze cie Dom
CAN sizco all tha operations
have preserved continuity, the eappins 4 > 4 (ED) 18 cone
e
&tmouss and since the apace af mtuples, -L „is m cell, there must
bo a fixed point far auch A + llonco there will bo a aubsoquonme „du »
ccoworcias to 2% , where Lp do fixed untor the sepotng 24 (4,240),
sow suppose AN wore not on oquilisricn points then IE
ARE) comm component Si muet be nom
optimal against the othars, which sous SX uses cone pure strategy
id which do comoptisale Zoo (t),@9-4:7 cts none that
Pale*) < dite) wok qu
wetting Pate) - 4%) <-€
Tran centtont A da tar ends
[Crace)- ail [mes ni] AAA
aan Proc) Glan) + us CO wir
alply
Pia lens Nro) < O uiene Pi Moya
Geld) =O + Prom thie last equation we loe that
“Wie La mot used in du sims
SAND) à becuse Lu de
Fixed pointe
and aime dus 2% Mia to not ed in 2,
which contradtets our asnaytions
tence 4% 19 totoed an equilibrium points
&tmouss and since the apace af mtuples, -L „is m cell, there must
bo a fixed point far auch A + llonco there will bo a aubsoquonme „du »
ccoworcias to 2% , where Lp do fixed untor the sepotng 24 (4,240),
sow suppose AN wore not on oquilisricn points then IE
ARE) comm component Si muet be nom
optimal against the othars, which sous SX uses cone pure strategy
id which do comoptisale Zoo (t),@9-4:7 cts none that
Pale*) < dite) wok qu
wetting Pate) - 4%) <-€
Tran centtont A da tar ends
[Crace)- ail [mes ni] AAA
aan Proc) Glan) + us CO wir
alply
Pia lens Nro) < O uiene Pi Moya
Geld) =O + Prom thie last equation we loe that
“Wie La mot used in du sims
SAND) à becuse Lu de
Fixed pointe
and aime dus 2% Mia to not ed in 2,
which contradtets our asnaytions
tence 4% 19 totoed an equilibrium points
Symetries of Senor
an gutscornhiany or eme of a ge MEL be a permutation of
te pure strutecioe wish satiation certain conditions, ziren belawe
XE wo etrataciae belong to a einzle player Shey aust go Into to
atrotagion belacgios do a otage players us iz ia the porte
‘thon of the pure atzatoctos 15 tniveos a perautatton (Pat tha
payee.
Zach mtuplo of pure atretactos la tharafaro permuted Jute another
tuple of pure stratectoss o say coll X tha tofu garmstation
ef these mtuples. Let E denote an mtuple of pure strategies and
PA) tae payoir to player À mean the mtuple E da a
played. ie require that Ae
j= if am BE): RE)
isch completos the definition af a eymatrys
Toa perastation aa a Unique Linear extension to the mised
stratagtens 2
1e Ea A ER A
‘the extension of P to the nixed strategies clearly generates an
extension of X to the n-tuplan of mizel stratecion. Me shall also
denote this ty X»
Te define a symetric ntuple — of a cam by
Az te X’
At being understood that X means a permtation derived from a ayeuetry
B-
an gutscornhiany or eme of a ge MEL be a permutation of
te pure strutecioe wish satiation certain conditions, ziren belawe
XE wo etrataciae belong to a einzle player Shey aust go Into to
atrotagion belacgios do a otage players us iz ia the porte
‘thon of the pure atzatoctos 15 tniveos a perautatton (Pat tha
payee.
Zach mtuplo of pure atretactos la tharafaro permuted Jute another
tuple of pure stratectoss o say coll X tha tofu garmstation
ef these mtuples. Let E denote an mtuple of pure strategies and
PA) tae payoir to player À mean the mtuple E da a
played. ie require that Ae
j= if am BE): RE)
isch completos the definition af a eymatrys
Toa perastation aa a Unique Linear extension to the mised
stratagtens 2
1e Ea A ER A
‘the extension of P to the nixed strategies clearly generates an
extension of X to the n-tuplan of mizel stratecion. Me shall also
denote this ty X»
Te define a symetric ntuple — of a cam by
Az te X’
At being understood that X means a permtation derived from a ayeuetry
B-
+
41 hay Finite pane has a eymetric ecullibriva pointe
route iret wo nate that
= Ema ne terme (Si Sy
ro lat the tuple > Gre See, +++ Inc)
tized under any X. 5 hence any cano tao at Ianat one aymmetrio me
‘Tate shows that the set af aymetrio a-tuplee da a convex aubest of
‘tio space of n-tuplas since it is obvicusly cleceds
Non observe that for each A the mapping ¿re (EA) used
a the proof of extatance thaaren wen Intränsienliy defined. Therefore,
se 22 (2,2) am Kah esto morgen of the game
wenn 4% = -2°(2%,r) ed he me
wi AXz 4, métiers = (a) = di
consequently thie mapping mape the set of symetrie mtuples Anto itself,
Sinne this oot la a cell there mut be a aymnetris fixed point La.
And, as An the proof of the existance thooren we could obtain a Lait
point EN mio would have to be ayumetrios
41 hay Finite pane has a eymetric ecullibriva pointe
route iret wo nate that
= Ema ne terme (Si Sy
ro lat the tuple > Gre See, +++ Inc)
tized under any X. 5 hence any cano tao at Ianat one aymmetrio me
‘Tate shows that the set af aymetrio a-tuplee da a convex aubest of
‘tio space of n-tuplas since it is obvicusly cleceds
Non observe that for each A the mapping ¿re (EA) used
a the proof of extatance thaaren wen Intränsienliy defined. Therefore,
se 22 (2,2) am Kah esto morgen of the game
wenn 4% = -2°(2%,r) ed he me
wi AXz 4, métiers = (a) = di
consequently thie mapping mape the set of symetrie mtuples Anto itself,
Sinne this oot la a cell there mut be a aymnetris fixed point La.
And, as An the proof of the existance thooren we could obtain a Lait
point EN mio would have to be ayumetrios
solutions
Lo Catia boro colutloza, strong eolutlons, ant ebcolstionss &
nopecooporative ¿o done not slvaga hare a zolstien, tut whan 16 does
the solution la untquae Tsang aolvtiono ara sclsttone with octal
cpertioae Gumsolutions aie exist aot duro aay of the properties
es solutions, bub Jack unicuacesss
Sen denota à set of tama etratacion al player T ant ofp
cot of rotuplos of atsed etantagias»
Solvabsiitys
à ne Le solvable 12 ste sets ef » of muitièriun patate ati
pere
a lee at el tonel sm is.
‘hie ta called the Imerabangesbilitz condition. The solution of à
actrable ques ia tte ast, of.» of equttibrtun potntas
strong solvability:
à comm ta strony solve if ft bus a solutions of euch tat
forall is
el ot plat) ped > mdd
so thas oa cated a ru solutions
Bulibriun Stratecast
la a colmble gam 1st Sj ve the set of all sized strategies 5;
Lo Catia boro colutloza, strong eolutlons, ant ebcolstionss &
nopecooporative ¿o done not slvaga hare a zolstien, tut whan 16 does
the solution la untquae Tsang aolvtiono ara sclsttone with octal
cpertioae Gumsolutions aie exist aot duro aay of the properties
es solutions, bub Jack unicuacesss
Sen denota à set of tama etratacion al player T ant ofp
cot of rotuplos of atsed etantagias»
Solvabsiitys
à ne Le solvable 12 ste sets ef » of muitièriun patate ati
pere
a lee at el tonel sm is.
‘hie ta called the Imerabangesbilitz condition. The solution of à
actrable ques ia tte ast, of.» of equttibrtun potntas
strong solvability:
à comm ta strony solve if ft bus a solutions of euch tat
forall is
el ot plat) ped > mdd
so thas oa cated a ru solutions
Bulibriun Stratecast
la a colmble gam 1st Sj ve the set of all sized strategies 5;
ac
auch that for sone À tomate (43 Si) te en equilibrium
pointe [Si ts the Tab pont of somo ocultan potas 7
call Si the net of ocuillibrisa strotertos of player T +
subaolutfors:
LL 12 à ae of te ant of opus polite of «cane and
satisfies condition (1); and Ir L is maximal relative to this prow
party tran ve cal of a abrace»
zor any ect-aotution oL we destgo the Fin factor sat, Si vas
the aot of all Si’s auch that SL. cortasos (0355) for cone
te
Lote that a eubeeclution, whan waique, i solutions ani
una are the cote of equthibrium stratch
0e 25 à mrectutions of e da the not of all artuplan
(6151, -*"50) me that each se Si where Si te the
en cusco aot of of « Coomatrtoaliy, of de the product of ste
factor ante
‘proofs Comidas such an amtugle (Sı,-- Sn) + 27 definition
TQ te An mh that for ach T Erssiek +
‘tng the condition (1) mh tines we obtain mccossively
Gessssle’s ++ 55555555 35) ¿Dana tha Yast ia
aumly (use += sa) 6,2 » ritos we neoded to año
200. se me mator sota Sy Sa, +++ Sn of a mbetolutton
Prats It suftioes to show two thinge: (a) if
auch that for sone À tomate (43 Si) te en equilibrium
pointe [Si ts the Tab pont of somo ocultan potas 7
call Si the net of ocuillibrisa strotertos of player T +
subaolutfors:
LL 12 à ae of te ant of opus polite of «cane and
satisfies condition (1); and Ir L is maximal relative to this prow
party tran ve cal of a abrace»
zor any ect-aotution oL we destgo the Fin factor sat, Si vas
the aot of all Si’s auch that SL. cortasos (0355) for cone
te
Lote that a eubeeclution, whan waique, i solutions ani
una are the cote of equthibrium stratch
0e 25 à mrectutions of e da the not of all artuplan
(6151, -*"50) me that each se Si where Si te the
en cusco aot of of « Coomatrtoaliy, of de the product of ste
factor ante
‘proofs Comidas such an amtugle (Sı,-- Sn) + 27 definition
TQ te An mh that for ach T Erssiek +
‘tng the condition (1) mh tines we obtain mccossively
Gessssle’s ++ 55555555 35) ¿Dana tha Yast ia
aumly (use += sa) 6,2 » ritos we neoded to año
200. se me mator sota Sy Sa, +++ Sn of a mbetolutton
Prats It suftioes to show two thinge: (a) if
ur
sil eS;
ante point or Si ten SES;
Ê
Lot tel + thon we bare
x
Rss) ER bs) = Bots) PC
for any dz, by using the ortterton or (I) , 29.3 for an age
rhe Adding there Anoquli the Linearity of (5s, 2050) in
Si, and aiviaing dy 2, vo cot BGs SP) = pts)
imo Si (S45) AZ a
SI wee A
an such age pes. GCS SR) o atar to of the augmente sot
clearly zatieries condition (i), and since af me to be maxton ke
follows that SE Sr .
70 attack (9) note that the mtuple (63 SE), mere tel
WEL bo a Lintt point of ta met of metuples or tae fom (5 Si)
mers SET zone Ss mama point of Sp. aut
this eet La a set of age pie. and bance any point Sn Ste closure da an
ae pts since the ant of alt eqs pte ta cloned Zoo pg. 7 + theres
tore (tg SA) teen eae pte and tome Sie Si trom
‘the sane argunent as for SO,
pe AAA a dD dion
+ ERAN) ven] .
N we write ; e NT
‘tho upper value to player i of the gene; Vi” the lower values and
sil eS;
ante point or Si ten SES;
Ê
Lot tel + thon we bare
x
Rss) ER bs) = Bots) PC
for any dz, by using the ortterton or (I) , 29.3 for an age
rhe Adding there Anoquli the Linearity of (5s, 2050) in
Si, and aiviaing dy 2, vo cot BGs SP) = pts)
imo Si (S45) AZ a
SI wee A
an such age pes. GCS SR) o atar to of the augmente sot
clearly zatieries condition (i), and since af me to be maxton ke
follows that SE Sr .
70 attack (9) note that the mtuple (63 SE), mere tel
WEL bo a Lintt point of ta met of metuples or tae fom (5 Si)
mers SET zone Ss mama point of Sp. aut
this eet La a set of age pie. and bance any point Sn Ste closure da an
ae pts since the ant of alt eqs pte ta cloned Zoo pg. 7 + theres
tore (tg SA) teen eae pte and tome Sie Si trom
‘the sane argunent as for SO,
pe AAA a dD dion
+ ERAN) ven] .
N we write ; e NT
‘tho upper value to player i of the gene; Vi” the lower values and
there te but one eguilibriun
one can define ancociatad dues Zor a sub=eolution by restricting
to the oge ptes tn the eudvcolution and then usin; the cane defining
equations na above.
à tso-peraon sero-ou zum Le abmye solvable in the sense defined
above. The sete of equilibrium stratectos S, ami Se are simply
‘the abs où "good" atrategians Such a gina la not jonoraily otroncly
solvable; strong solutions extat only when hero ia a “saddle point” in
pure stratect
one can define ancociatad dues Zor a sub=eolution by restricting
to the oge ptes tn the eudvcolution and then usin; the cane defining
equations na above.
à tso-peraon sero-ou zum Le abmye solvable in the sense defined
above. The sete of equilibrium stratectos S, ami Se are simply
‘the abs où "good" atrategians Such a gina la not jonoraily otroncly
solvable; strong solutions extat only when hero ia a “saddle point” in
pure stratect
Gennotrical Fora of Solutions
in the twoeperson sero=sun case it han boon chow: that the oot of
nov" atratocios of a player la a convex polytedral subset of ito.
strategy opaca» ‘lo alall citala the sano raault for a player's set af
oquiltbrtun stratecies in any solwblo pense
rump. 61 tho ote Shen Sn of muilieriun stration
ses 12 a eolmble caso aro zolybodmad come: aibeata of the somoct!
ised ctratecy space
amont an mobile ad wilh be ar oquilibetia potet LE and only IE
m pla)= EX habe)
which de condition (5) on pace (Y + An equivalent oontitän a Zor
wu i and
@ PQ) -Pato BO -
Let a now cocatdar the fom of the ert Oj of equilibrio strate
ten, 55 of player j + Let ot be any ella pols ten
(0555) vin do an equilibrica pots 19 and only S55 es; «
tran Theos 2+ 0 nom apply conditions (2) t0 (75 Sj)» obtaining
O eS era te ts) Aue
simon Pts arismur ant ZE da content Shane aro « eet of Linear
Sorquisios of tha fern Flg(S5) 20 + ah eh tng
hs either satiafied for aD $ or for those lying on and to one aide
of «cum hyperplane passing through the strategy sinplaxe Therefore, the
in the twoeperson sero=sun case it han boon chow: that the oot of
nov" atratocios of a player la a convex polytedral subset of ito.
strategy opaca» ‘lo alall citala the sano raault for a player's set af
oquiltbrtun stratecies in any solwblo pense
rump. 61 tho ote Shen Sn of muilieriun stration
ses 12 a eolmble caso aro zolybodmad come: aibeata of the somoct!
ised ctratecy space
amont an mobile ad wilh be ar oquilibetia potet LE and only IE
m pla)= EX habe)
which de condition (5) on pace (Y + An equivalent oontitän a Zor
wu i and
@ PQ) -Pato BO -
Let a now cocatdar the fom of the ert Oj of equilibrio strate
ten, 55 of player j + Let ot be any ella pols ten
(0555) vin do an equilibrica pots 19 and only S55 es; «
tran Theos 2+ 0 nom apply conditions (2) t0 (75 Sj)» obtaining
O eS era te ts) Aue
simon Pts arismur ant ZE da content Shane aro « eet of Linear
Sorquisios of tha fern Flg(S5) 20 + ah eh tng
hs either satiafied for aD $ or for those lying on and to one aide
of «cum hyperplane passing through the strategy sinplaxe Therefore, the
Staple Sacplee
so to tLiuatrate the enncon
apectal phenenena whieh coeur An these ganes
ayer has tio rome letter otretocios and the pay
cu (Lar Zb,jdr9p)
Ey vas
hash
som (26)
12 PAGE PAR
coonmbln corr BB Abbe
Unsolpablen equilibrium as) bP),
EE ER Le OP ae
cas in che Aare case are mexica and
SEE propensos
2 PA
Son Eten a ete ar aad anto
"te ati wa Vo 20
Unsolmbley eq» pte (ap) (26) ama
Cat Hab, ext 0) + Homer
eqpirical testé show a tendency tomará (ax).
RGR PRE:
a
o:
v
a
a
b
v
a
>
b
a
e
o
a
a
b
e
1 (a) am (bp) wi
(OP) un maple of instansisty.
coor whe oron
ope
PAD Pi
so to tLiuatrate the enncon
apectal phenenena whieh coeur An these ganes
ayer has tio rome letter otretocios and the pay
cu (Lar Zb,jdr9p)
Ey vas
hash
som (26)
12 PAGE PAR
coonmbln corr BB Abbe
Unsolpablen equilibrium as) bP),
EE ER Le OP ae
cas in che Aare case are mexica and
SEE propensos
2 PA
Son Eten a ete ar aad anto
"te ati wa Vo 20
Unsolmbley eq» pte (ap) (26) ama
Cat Hab, ext 0) + Homer
eqpirical testé show a tendency tomará (ax).
RGR PRE:
a
o:
v
a
a
b
v
a
>
b
a
e
o
a
a
b
e
1 (a) am (bp) wi
(OP) un maple of instansisty.
coor whe oron
ope
PAD Pi
cemplote set J'unteh is finite 7 of conditions will all be catiafied
stultaneously on sao conver polyhedral subact of pleyer J‘ otra
ataplaxe intersection of hali=opucos» 7
ha a corollary we muy conclude tint Sk ia the courex closure
ste dot of mixed strutogios J vortices 7.
stultaneously on sao conver polyhedral subact of pleyer J‘ otra
ataplaxe intersection of hali=opucos» 7
ha a corollary we muy conclude tint Sk ia the courex closure
ste dot of mixed strutogios J vortices 7.
Dentnanse and Contradietion Yethods
o cay thet Si” cœnte Si (25 5:) > RES Si)
sor every À
cate snouste So caying that Sf wire player À a Higher paye
ces tian St no matter whut the strate,ies of the other players ares
20 cos whether a strategy Sí domínates Si it auffices to cousi~
der only pure atratasias far the other playera because of the melting
wy er Py «
obvious San the definitions that po aguilibetis point can
gesisated strategy Si +
se domination of ons nixed strategy by another will almya entail
other dominatiora» For suppose Si donimmten Si and t; usos all
af the pure atrategien mich have a higher ooafrioient in S; than in
Si. Than for a eli PO
4 ti + PSS)
10 a mizot strategy and ti’ dantneton ti by Linearity.
One can prove a few propertise of tho set of undoninsted stratactons
16 is alaply comeated and la farmed by the union of acne collection af
face of the strategy sinpleze
‘Tae information obtained by discovering dontnances for one player
may bo of relevance to the others, insofar as the elininstion of classes
of mixed strategies as possible componente of an equilibrium point te
comen. Por the ZÉ/S whose componente are all undoninated are all
that neod be considered and this elininating sone of the strategies of
one player may male possible the olintnation of a now clans of strateciee
for another player»
o cay thet Si” cœnte Si (25 5:) > RES Si)
sor every À
cate snouste So caying that Sf wire player À a Higher paye
ces tian St no matter whut the strate,ies of the other players ares
20 cos whether a strategy Sí domínates Si it auffices to cousi~
der only pure atratasias far the other playera because of the melting
wy er Py «
obvious San the definitions that po aguilibetis point can
gesisated strategy Si +
se domination of ons nixed strategy by another will almya entail
other dominatiora» For suppose Si donimmten Si and t; usos all
af the pure atrategien mich have a higher ooafrioient in S; than in
Si. Than for a eli PO
4 ti + PSS)
10 a mizot strategy and ti’ dantneton ti by Linearity.
One can prove a few propertise of tho set of undoninsted stratactons
16 is alaply comeated and la farmed by the union of acne collection af
face of the strategy sinpleze
‘Tae information obtained by discovering dontnances for one player
may bo of relevance to the others, insofar as the elininstion of classes
of mixed strategies as possible componente of an equilibrium point te
comen. Por the ZÉ/S whose componente are all undoninated are all
that neod be considered and this elininating sone of the strategies of
one player may male possible the olintnation of a now clans of strateciee
for another player»
another procedure which
La Sw ecotmadlettontyge aralysice Eure one secure tet en ogull
point extats having component strategies lying stthin certain rectors of
sho atratas spaces and proceeds to deduce Further condiciona white: aust
lo ans! the hypothasls ls tune Te sort of reonaning ay do
carried throug several stages to eventually obtain a contradiction tm
dlcating Hat there is no equilibriza pol sntlatying Yo initial humor
heats
La Sw ecotmadlettontyge aralysice Eure one secure tet en ogull
point extats having component strategies lying stthin certain rectors of
sho atratas spaces and proceeds to deduce Further condiciona white: aust
lo ans! the hypothasls ls tune Te sort of reonaning ay do
carried throug several stages to eventually obtain a contradiction tm
dlcating Hat there is no equilibriza pol sntlatying Yo initial humor
heats
eutton of ou
real tude tho ataplised goior ¿ase ctven voran.
rules are as 72
(2) The deck ta Juris M
a band constata of oo cards
(2) 2m chine aso weed to ante, oros, ar cali
(3) Tue ylayors play in rotation and the cano onde after all hare
passed os after ote player has opened and the cthera haw bad a chance
19 no one bets the antes are rotricvade
Ctharvive the pot ia divided equally among the bigest ind
nero cetistootary to tront the gino in terns of quests
ties wo cali "behavior parametre than in the normal fam of "Zhoory
at Garma and Commis Jebaviore" In the normal form representation tro
mized etratacies af a player may be equivalent in the sense that each
makes the initvidual choose sach available cure of action in wach pare
Keular situation requiring ection on his part with the ame Srequemsys
‘That day they represent the sane behavior pattern on the park of the tie
asriduale
Relavior parameters give the probabilities of taking each af the
various possible actions in ench of the various possible situations which
say arives Thus they describo behavior patiornas
In term of behavior parnmatere the strategies of the players my be
represented as follows, asnuing thet sine there is ro point In passing
with a high card at one's last opportunity to bat that thie will not be
real tude tho ataplised goior ¿ase ctven voran.
rules are as 72
(2) The deck ta Juris M
a band constata of oo cards
(2) 2m chine aso weed to ante, oros, ar cali
(3) Tue ylayors play in rotation and the cano onde after all hare
passed os after ote player has opened and the cthera haw bad a chance
19 no one bets the antes are rotricvade
Ctharvive the pot ia divided equally among the bigest ind
nero cetistootary to tront the gino in terns of quests
ties wo cali "behavior parametre than in the normal fam of "Zhoory
at Garma and Commis Jebaviore" In the normal form representation tro
mized etratacies af a player may be equivalent in the sense that each
makes the initvidual choose sach available cure of action in wach pare
Keular situation requiring ection on his part with the ame Srequemsys
‘That day they represent the sane behavior pattern on the park of the tie
asriduale
Relavior parameters give the probabilities of taking each af the
various possible actions in ench of the various possible situations which
say arives Thus they describo behavior patiornas
In term of behavior parnmatere the strategies of the players my be
represented as follows, asnuing thet sine there is ro point In passing
with a high card at one's last opportunity to bat that thie will not be
1
dono» cso creck lottera nro the probabilitins af the various acts.
First Lows
open en la
¿pon on 20
call 7 on low
open on high
pen on dom
all and 22 on low
open on ow
ne Acosta all osetbio Oquilibrii potnts by Fürst shoring tint moat
ce te crock paremactare must vasithe Dy dominance misly with a Little
A AOS
ant Ó ty domine Then ccctradietions slinioate ME, C,A, 5
aot D in that orders Tila leaves ua with A Í € at y+
coutrudistion aralyula shore that noce cf these can be sero or co and
sinus ve obtain a eyrtan of etaaltarsous algwtrade equtticnte The oquattons
hapoan to have but ope eolution with the mriabzos in tes rage (0,1) +
e oot
ar A ye Eat Se fac
TS
e= bach „These yield =.
a yield X=. 308,
E= -0#.
dono» cso creck lottera nro the probabilitins af the various acts.
First Lows
open en la
¿pon on 20
call 7 on low
open on high
pen on dom
all and 22 on low
open on ow
ne Acosta all osetbio Oquilibrii potnts by Fürst shoring tint moat
ce te crock paremactare must vasithe Dy dominance misly with a Little
A AOS
ant Ó ty domine Then ccctradietions slinioate ME, C,A, 5
aot D in that orders Tila leaves ua with A Í € at y+
coutrudistion aralyula shore that noce cf these can be sero or co and
sinus ve obtain a eyrtan of etaaltarsous algwtrade equtticnte The oquattons
hapoan to have but ope eolution with the mriabzos in tes rage (0,1) +
e oot
ar A ye Eat Se fac
TS
e= bach „These yield =.
a yield X=. 308,
E= -0#.
>
Stree there lo only one equilibriun point the caso has valuoo, these aro
=-.06= - as > and.
ERE -
mrostlcgction af tha conlíticn pavers yields tha following “coat
etrutorion" ani values for tha various sonl!tietz. Furammtern not nome
stonec are zoron
Taal ur
a= 34 Cae , 067%
$= €=l vive to IT: OH: Yr
i
TL ond IT versus pa
=f
re 1% S=% „es Y
as Hal valo to IT: mes Try
Te coalition menbere have the power to agree upon a zuttern of
play betere the gue ta played. This atmntace bases stenifteant only
tn the care of conltste: [TIL store TIL say op after too paseos
ten T td planned to pans on both hich and Jow but will not open If
Stree there lo only one equilibriun point the caso has valuoo, these aro
=-.06= - as > and.
ERE -
mrostlcgction af tha conlíticn pavers yields tha following “coat
etrutorion" ani values for tha various sonl!tietz. Furammtern not nome
stonec are zoron
Taal ur
a= 34 Cae , 067%
$= €=l vive to IT: OH: Yr
i
TL ond IT versus pa
=f
re 1% S=% „es Y
as Hal valo to IT: mes Try
Te coalition menbere have the power to agree upon a zuttern of
play betere the gue ta played. This atmntace bases stenifteant only
tn the care of conltste: [TIL store TIL say op after too paseos
ten T td planned to pans on both hich and Jow but will not open If
I: À to Det So he got Bio Tho values gérez ao ai
courte, Wat du atado player aceros hie his Tente” oreo
à aura dotallod veoataont of sts se Is Lolas grepared for pub
courte, Wat du atado player aceros hie his Tente” oreo
à aura dotallod veoataont of sts se Is Lolas grepared for pub
Motivation and Interpretation
In thie section vo aball try to oxplain the significance of the
concepts Introduned in this paper. That ia, we aball try to show how
equilibria points ant solutions oan be connected with observable
canas
‘the baste requirements for a noncooperative gano la that there
shoold be no prowplay comuntostion among the players unten it has
20 bearing on the game» Thun, ty implication, there are no coalitions
and no aide-paynentae Decause there is no extrongune utility [pay-off 7
teanater, the peyroffe of different players are effectively Insompersbler
Af we tranaform the pay-off functions lineeriys P:~ a; p;+ bi .
mere Qi DO ti gan will De essertially the ses Hote that
equilibria points aro proserved under such tranefermttonse
o sball now tale up the "mmanmaotion interpretation of agullibriun
pointe. In this interpretation solutions bave no rent significances 16
18 coneceesary to assume that tho yerttotpants have full kmorledge of the
‘total structure of the gano, or the ability and inoltntion to go through
say complex rensoning provensess But the participants are supposed to 40-
casulate expirical information on the relative admatages of the various
> puro strategies at ther dispose
10 be more detailed, we assume that there a a population Zin the
sarao of statiation 7 of participante for each position of the gums Let
us also assume that the “average playing" of the pane involves n partiol=
pacta aslected at runion fron the n populeticods and thet there is a stan
bla average Srequecoy with which ach pure strategy de played by the
"arerago meaber" of the appropriate populations
Since there Le to be no collaboration beben intlviduala playing ia
In thie section vo aball try to oxplain the significance of the
concepts Introduned in this paper. That ia, we aball try to show how
equilibria points ant solutions oan be connected with observable
canas
‘the baste requirements for a noncooperative gano la that there
shoold be no prowplay comuntostion among the players unten it has
20 bearing on the game» Thun, ty implication, there are no coalitions
and no aide-paynentae Decause there is no extrongune utility [pay-off 7
teanater, the peyroffe of different players are effectively Insompersbler
Af we tranaform the pay-off functions lineeriys P:~ a; p;+ bi .
mere Qi DO ti gan will De essertially the ses Hote that
equilibria points aro proserved under such tranefermttonse
o sball now tale up the "mmanmaotion interpretation of agullibriun
pointe. In this interpretation solutions bave no rent significances 16
18 coneceesary to assume that tho yerttotpants have full kmorledge of the
‘total structure of the gano, or the ability and inoltntion to go through
say complex rensoning provensess But the participants are supposed to 40-
casulate expirical information on the relative admatages of the various
> puro strategies at ther dispose
10 be more detailed, we assume that there a a population Zin the
sarao of statiation 7 of participante for each position of the gums Let
us also assume that the “average playing" of the pane involves n partiol=
pacta aslected at runion fron the n populeticods and thet there is a stan
bla average Srequecoy with which ach pure strategy de played by the
"arerago meaber" of the appropriate populations
Since there Le to be no collaboration beben intlviduala playing ia
æ
dssrerent positions of the cane, the probebility that a particular be
tuple of pure ateatacios W122 be eqployed In a playing af the gine
chould be tha product of the probabilities indiosting the chance of
each of the n pure atratecies to do exployed in a santos playtnge
Lot the probability that Tix will be omployed in a randon
playing of the gum be Cig sand let Siz Ea Tra +
del, 50 eee Sm) + Then the expected pay-off to an indie
dual playing An the ith position of the guns and exploying the pure
strategy Tia te (43 Mix) = Pra)
low let us consider wat effects the experience of the participante
ALL produces To assume, as we did, that they aooumlated empirical
evidence oa the pure strategies at thelr dispomal La to asuma that
‘hove playing in position T leam the mabes Pra EL) +
ut Af they know these they will eaploy only optimal pure strategies
Lees, those pure strategies TTiol aueh that
Piatt)= EX Praca)
consequmtly since S; expresos their tabavior Si attaches poste
‘tive ooeffioienta only to optimal yure strategies, po thet
trata in 5; => Pas XPD) >
put thie 18 ateply a condition ford to be an equilibria pol
Lund
‘Thus the aseumptions we made in this “mss-nction" interpretion
‘lead to the concluston that the sized strategies representing the average
batavior In eech of the populations form an equilibrium pointe
‘The populations nest nat be larce if the aanaytions stilt rue tt
dssrerent positions of the cane, the probebility that a particular be
tuple of pure ateatacios W122 be eqployed In a playing af the gine
chould be tha product of the probabilities indiosting the chance of
each of the n pure atratecies to do exployed in a santos playtnge
Lot the probability that Tix will be omployed in a randon
playing of the gum be Cig sand let Siz Ea Tra +
del, 50 eee Sm) + Then the expected pay-off to an indie
dual playing An the ith position of the guns and exploying the pure
strategy Tia te (43 Mix) = Pra)
low let us consider wat effects the experience of the participante
ALL produces To assume, as we did, that they aooumlated empirical
evidence oa the pure strategies at thelr dispomal La to asuma that
‘hove playing in position T leam the mabes Pra EL) +
ut Af they know these they will eaploy only optimal pure strategies
Lees, those pure strategies TTiol aueh that
Piatt)= EX Praca)
consequmtly since S; expresos their tabavior Si attaches poste
‘tive ooeffioienta only to optimal yure strategies, po thet
trata in 5; => Pas XPD) >
put thie 18 ateply a condition ford to be an equilibria pol
Lund
‘Thus the aseumptions we made in this “mss-nction" interpretion
‘lead to the concluston that the sized strategies representing the average
batavior In eech of the populations form an equilibrium pointe
‘The populations nest nat be larce if the aanaytions stilt rue tt
=.
21 holds There are situations In eccnonioa or international
politics In which, effoativly. a croup of interests aro involved in
a noncooperative game without being mare of ity the non-emrensse
helping to mabe the situation truly nen-cooperatives
Actually, of course, we can only oxpoot cone sort of spproxizate
eugilitriim, since the information, Sta utilization, and the atab{lity
of the average Srequenoles will do isparteoto.
Mo now slatch another Interpretation, one in which solutions play
a major role, and which Se applicable to a game played but onces
We procead Uy Anventigating the questions what would be a “rational
prediction of the behavior to be expected of rational playing the cane In
question? By using the principles that a rational prediction should be
unique, that the players should be able to deduce and make use of it, and
‘that such knorladge on the part of each player of what to expect the
othera to do should not lead him to act out af conformity with the pren
étotions one ja led to the conoeyt of a solution defined bafarwe
1e Si Sa, +- Sn vers the sete of equilibrium strategies
of a salvable cane, the “rational” predibiion should ber "The average
petavioe of rational men playing in position | would define a mixed
strategy Si An Si tf en cxperiamt were carried cute"
In this interpretation we nod to uesume the players know the fll
structure of the gave ln order to be able to detuce the prediction for
chanealraso Zt Ls quite strongly a rationalistie and Mdsaliatng inter
pretations
la an unsolvable cane it somatines happens that good beurtatio
reasns can be fous for narrowing dom the set of equilibria pointe to
bone in a single sub-eclution, which then playa the role of « solutions
21 holds There are situations In eccnonioa or international
politics In which, effoativly. a croup of interests aro involved in
a noncooperative game without being mare of ity the non-emrensse
helping to mabe the situation truly nen-cooperatives
Actually, of course, we can only oxpoot cone sort of spproxizate
eugilitriim, since the information, Sta utilization, and the atab{lity
of the average Srequenoles will do isparteoto.
Mo now slatch another Interpretation, one in which solutions play
a major role, and which Se applicable to a game played but onces
We procead Uy Anventigating the questions what would be a “rational
prediction of the behavior to be expected of rational playing the cane In
question? By using the principles that a rational prediction should be
unique, that the players should be able to deduce and make use of it, and
‘that such knorladge on the part of each player of what to expect the
othera to do should not lead him to act out af conformity with the pren
étotions one ja led to the conoeyt of a solution defined bafarwe
1e Si Sa, +- Sn vers the sete of equilibrium strategies
of a salvable cane, the “rational” predibiion should ber "The average
petavioe of rational men playing in position | would define a mixed
strategy Si An Si tf en cxperiamt were carried cute"
In this interpretation we nod to uesume the players know the fll
structure of the gave ln order to be able to detuce the prediction for
chanealraso Zt Ls quite strongly a rationalistie and Mdsaliatng inter
pretations
la an unsolvable cane it somatines happens that good beurtatio
reasns can be fous for narrowing dom the set of equilibria pointe to
bone in a single sub-eclution, which then playa the role of « solutions
he
a cet of mutually
In general a mub-aolution may be looked at as
forming a ocherent wholes The sub-
the
inten offeet of equilibra
compatible equilibriun point
solutions appear to give a natural subd?
points of opens
a cet of mutually
In general a mub-aolution may be looked at as
forming a ocherent wholes The sub-
the
inten offeet of equilibra
compatible equilibriun point
solutions appear to give a natural subd?
points of opens
Applications
she otuty of neperson games for which the accopted ethios of fair
play Saply non-cooperative playing in, of course, an cbvious direction
in wetoh to apply this theorys And poker is the most obvious tarcets
she ecalyule of a nore realistic poker gasa than cur very sisple model
should be quite an interesting affaire
Te complezity of the sathemtioal work needed for a complete ta
voctigation Snorenses rather rapidly, however with increasing cospleo-
lay of the paros 0 that it mms that analyais of cane mich pore son
plex then the example given bare would only de feasible veloz apprasic
mata computational method
à Lans ol type of application la to the study af sooparatine
ann. by a cooperative cise we mean a situation involving a set of
layers pure strategies, an payeeffe un venal but with the eesmption
het the players can and will collaborate a they do da the von Sama
nd Morgenstern thacrys This mans the players my comuntonte ant fora
coalitions which will be enforoed by an fapires 19 1s uneosssartly
restrictiva, however, to sens any transferebility, or even comparable
Ag of the payotte [ua should be in utility unite 7 to different
players Any deatred trenafersbility oan be put into the game Saad tor
ead of assuming At possible in the extrengune collaborations
the weiter hes developed a “dynamical” approach to the study af oo
‘operative panes bused upon reduction to non-cooperative Farm. One pro
cuota by constructing a modal of the pre-play negotiation so that the
eo of negotiation boots moves in a larger nin-oooperasive zum {whisk
822 have an Sufintty of pure stratecion 7 dasartblag the total situation.
Ta larger cane La then trosted An terms of the theory of this paper
she otuty of neperson games for which the accopted ethios of fair
play Saply non-cooperative playing in, of course, an cbvious direction
in wetoh to apply this theorys And poker is the most obvious tarcets
she ecalyule of a nore realistic poker gasa than cur very sisple model
should be quite an interesting affaire
Te complezity of the sathemtioal work needed for a complete ta
voctigation Snorenses rather rapidly, however with increasing cospleo-
lay of the paros 0 that it mms that analyais of cane mich pore son
plex then the example given bare would only de feasible veloz apprasic
mata computational method
à Lans ol type of application la to the study af sooparatine
ann. by a cooperative cise we mean a situation involving a set of
layers pure strategies, an payeeffe un venal but with the eesmption
het the players can and will collaborate a they do da the von Sama
nd Morgenstern thacrys This mans the players my comuntonte ant fora
coalitions which will be enforoed by an fapires 19 1s uneosssartly
restrictiva, however, to sens any transferebility, or even comparable
Ag of the payotte [ua should be in utility unite 7 to different
players Any deatred trenafersbility oan be put into the game Saad tor
ead of assuming At possible in the extrengune collaborations
the weiter hes developed a “dynamical” approach to the study af oo
‘operative panes bused upon reduction to non-cooperative Farm. One pro
cuota by constructing a modal of the pre-play negotiation so that the
eo of negotiation boots moves in a larger nin-oooperasive zum {whisk
822 have an Sufintty of pure stratecion 7 dasartblag the total situation.
Ta larger cane La then trosted An terms of the theory of this paper
20m
Lrextonted to infinite guns 7 and Af values are obtained they are taken
as the values af the cooperative gano. Thus the problen analysing a
cooperative gane becomes the problem of obtaining a suitable, end oon-
vineinga con-eooperative model far the negotiations
The writer has, by such a trostaont, obtained veluos for all finite
bro person cooperative panes, and cose special ¡person gant
Lrextonted to infinite guns 7 and Af values are obtained they are taken
as the values af the cooperative gano. Thus the problen analysing a
cooperative gane becomes the problem of obtaining a suitable, end oon-
vineinga con-eooperative model far the negotiations
The writer has, by such a trostaont, obtained veluos for all finite
bro person cooperative panes, and cose special ¡person gant
Biblography
von Tounean, Uorgenstern, "Theory of Canes and Eoonsaic Behavior,
‘Princeton University Press, 144s
(2) de Po ashy dre, “Equilibrium Points in YePerson Came", Zro0s Hr
"As 52 56 (1950) 4049
AnknowLedgecata
Bras Tuekor, Gale, end Au gave valuable oriticias ani cugseations
or Asproving the exonttion of the material in this paper. Devid Galo
augrested the investigation of aymetrio games The solution of the Poker
model was a joist project undertaken by Lloyd de Shapley and the authar«
Fiaaliys the author was sustained finanolaliy by the Atonio Energy Comia=
sion An the period 1949-80 during which this work was dents
von Tounean, Uorgenstern, "Theory of Canes and Eoonsaic Behavior,
‘Princeton University Press, 144s
(2) de Po ashy dre, “Equilibrium Points in YePerson Came", Zro0s Hr
"As 52 56 (1950) 4049
AnknowLedgecata
Bras Tuekor, Gale, end Au gave valuable oriticias ani cugseations
or Asproving the exonttion of the material in this paper. Devid Galo
augrested the investigation of aymetrio games The solution of the Poker
model was a joist project undertaken by Lloyd de Shapley and the authar«
Fiaaliys the author was sustained finanolaliy by the Atonio Energy Comia=
sion An the period 1949-80 during which this work was dents
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