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Unformatted text preview: there exists si� , si�� ∈ Si such that si� , si�� ∈ Bi (s−i )
and λsi� + (1 − λ)si�� ∈ Bi (s−i ). In other words,
λui (si� , s−i ) + (1 − λ)ui (si�� , s−i ) > ui (λsi� + (1 − λ) si�� , s−i ).
But this violates the concavity of ui (si , s−i ) in si [recall that for a
concave function f (λx + (1 − λ)y ) ≥ λf (x ) + (1 − λ)f (y )].
Therefore B (s ) is convex-valued. 4. The proof that B (s ) has a closed graph is identical to the previous
proof. 24 Game Theory: Lecture 5 Existence Results Remarks Nash’s theorem is a special case of this theorem: Strategy spaces are
simplices and utilities are linear in (mixed) strategies, hence they are
concave functions of (mixed) strategies.
Continuity properties of the “Nash equilibrium set”:
Consider strategic form games with ﬁnite pure strategy sets Si and
utilities ui (s , λ), where ui is a continuous function of λ.
Let G (λ) = �I , (Si ), (ui (s , λ))� and let E (λ) denote the Nash
correspondence that associates with each λ, the set of (mixed) Nash
equilibria of G (λ).
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
- Spring '10