Refinements of the Nash Equilibrium Concept
In this monograph, noncooperative games are studied. Since in a noncooperative game binding agreements are not possible, the solution of such a game has to be self enforcing, i. e. a Nash equilibrium (NASH [1950,1951J). In general, however, a game may po
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Vol . 102: Anal,se Conve ,,, &1 $esApplications. Comples Rendus. Janv ;e' 1914, Edited by l .·P. Aubin. IV. 244 p.gn. 11114.
Vol. 103: D. E. Boyce..... Fa'hi. R. Weischedel, OptImal Subse, Selec1,on. Multiple ~fession. Inttl'depende nce and Opl'",al Network Algorithm s. XIII, 187 jnQes. 1117. Vol. 10.. , S. Fujino. A Neo·Ke'>'y of Maho,ds. IX. 102 P~ge$, 1915. Vol, 110 : C. Su iet>el. OpI,mal Contool of Ooscrf)le T,me Slochaslic Syslems . 1I1. 208 pages. 1975. Vol. III: V."able S l,uclure Systems W.lh AppI,cat,on 10 Economics and Sology. Proceed ings 1974. Ediled by A. Ruber~ and R R Mohl", VI, 32t pages. 197~ . Vol. 112: J. Wahelm. Objeclive. Ind Mulli·Objective De .... lizations . VII, 95 pages. 11176. VO/, 141 : MathemalicaIEconomicsand GameTh&c.y. Es ....y. in Honor 01 OSkar "'0'gen. t",n, Ediled by R. Henn and Q. MOflschlin. XIV. 703 PIg", 1911. Vol. 1_2: J. S . Lane.OnOplimai PopuI. tion P1oth •. V. 123 pag.ea. 11177. Vol. 1~3: B. Ntsluncl, An AnalySIS of Economic Siu OiSlributio",", "IN, 100 1"'9$8. 1917. Vol . 144: Convex Analysis ancll18 AppIica!i",, ". ProceOOings 1976. Edi ted by A. Auatender, VI. 219 pages . 1977. Vol. 145: J. R".....,mQllar. Extreme 126 pa9et.1977:
G ~ m es
and Thei, Sclullon • . IV,
Vct. 146 : In Search of Economic Indicator•. Ed'ted by W. H. S1fOgaL XVI, 111 9 pag .... 1977. Vol. 1 ~7: Resoufce Allocation .nd Division of Space. Pfoceed in9S. Ediled by T, F,,~ i and R. Sato. VIII. 164 1"'9'"" lyn . Vol. 148: C. E. Mandl. Simulalionslec hni~ und S imullhOnsrnodelle in den Sozial · und W inschaltsw; ..""achah....,.IX. 173 Seilan, 1977. \0:11. 149: SlallM l ra ufld $~f umplende E\evijlkerung"": Demogr.· phisches Null· und NegatiowacMtum in Osterreich. i-tet-ausgegeben von G. F~htinger . VI. 262 Sei1.en. IY77 . Vol. 150: Bauer el al.. SUP 0 there exists some
=
lim S(E), due to the fact that an E+O E. Furthermore, it is clear that a proper
equilibrium is perfect, since an E-proper equilibrium is E-perfect. From the proof of Theorem 2.2.5 we can deduce:
LEMMA 2.3.2. Let s be a proper equilibrium of a normal form game r and for E > 0, let S(E) be an E-proper equilibrium of r, such that lim S(E) = s. Then s is a best E+O reply against S(E) for all E which are sufficiently close to zero.
33 By considering the game of figure 1.5.3, we see that the properness concept is a strict refinement of the perfectness concept (namely
(Rl'~)
is a,perfect equilibrium
of this game, which is not proper). Obviously we would like a refinement of the Nash equilibrium concept to generate a nonempty set of solutions for every normal form game. It will now be shown (as in MYERSON [1978]) that this is indeed the case for the properness concept. THEOREM 2.3.3. (MYERSON [1978]). Every normal form game possesses at least one proper equilibrium. ~.
Let
r =
(~l""'~n,Rl,
••• ,Rn) be a normal form game. It suffices to show
that, for E > 0 sufficiently close to zero, there exists an E-proper equilibrium of
r.
Let E
€
(0,1). For i
€
N, defin,e n i for all k
€
€
by:
F(~i,lR)
~i'
Furthermore,
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