By Ewald C.-O.
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Additional resources for Games, Fixed Points and Mathematical Economics (2004)(en)(134s)
1. It is also due to Nash. 1. Nash (general Version) : Given an N -player game as above. , sn is fixed but arbitrary is convex and continuous. Then there exists an NCE in Gn . We have to build up some general concepts, before we go into the proof. It will follow later. One should mention though, that we do not assume that the game in question is the extended version of a game with only finitely many strategies and also the biloss operator does not have to be the bilinearly extended version of a biloss operator for such a game.
I lhc ∀i ⇒ i i γi uhc and compact valued. γi lhc 3. γi has open graph ∀i ⇒ i γi has open graph Proof. 2. 14. ( Convex Hull of a Correspondence ) Let γ : X →→ Y be a correspondence and Y be convex. then we define the convex hull of γ as co(γ) : X →→ Y x → co(γ(x)). 10. Let γ : X →→ Y be a correspondence and Y be convex. Then 1. γ uhc at x and compact valued ⇒ co(γ) is uhc at x 2. γ lhc at x ⇒ co(γ) is lhc at x 3. γ has open graph ⇒ co(γ) has open graph. Proof. Exercise ! 11. Let X ⊂ Rm ,Y ⊂ Rk and F be a polytope.
This however means that the suppliers produce exactly the amount of commodities the consumer want to consume and furthermore the suppliers make maximum profit. A price vector p which satisfies 0 ∈ E(p) is called a Walrasian equilibrium. The question of course is, does such an equilibrium always exists ? We will answer this question later. As in the case of the non cooperative equilibrium for two player games in chapter 2, this has to do with fixed points. But this time fixed points of correspondences.