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Unformatted text preview: strategy existence in inﬁnite games.
Theorem
(Debreu, Glicksberg, Fan) Consider a strategic form game
�I , (Si )i ∈I , (ui )i ∈I � such that for each i ∈ I
1 Si is compact and convex; 2 ui (si , s−i ) is continuous in s−i ; 3 ui (si , s−i ) is continuous and concave in si [in fact quasiconcavity
suﬃces]. Then a pure strategy Nash equilibrium exists. 20 Game Theory: Lecture 5 Existence Results Deﬁnitions Suppose S is a convex set. Then a function f : S → R is concave if
for any x , y ∈ S and any λ ∈ [0, 1], we have
f ( λx + (1 − λ )y ) ≥ λf (x ) + (1 − λ )f (y ) . concave function not a concave function 21 Game Theory: Lecture 5 Existence Results Proof
Now deﬁne the best response correspondence for player i ,
Bi : S−i � Si ,
�
�
Bi (s−i ) = si� ∈ Si  ui (si� , s−i ) ≥ ui (si , s−i ) for all si ∈ Si .
Thus restriction to pure strategies.
Deﬁne the set of best response correspondences as
B (s ) = [Bi (s−i )]i ∈I .
and
B : S � S. 22 Game Theory: Lecture 5 Existence Results Proof (continued)
We will again apply Kakutani’s theorem to the best response
correspondence B : S � S by showing that B (s ) satisﬁes the
conditions of Kakutani’s theorem.
1 S is compact, convex, and nonempty.
By deﬁnition
S= ∏ Si i ∈I since each Si is compact [convex, nonempty] and ﬁnite product of
compact [convex, nonempty] sets is compact [convex, nonempty].
2 B (s ) is nonempty.
By deﬁnition, Bi (s−i ) = arg max ui (s , s−i ) s ∈ Si where Si is nonempty and compact, and ui is continuous in s by
assumption. Then by Weirstrass’s theorem B (s ) is nonempty.
23 Game Theory: Lecture 5 Existence Results Proof (continued)
3. B (s ) is a convexvalued correspondence.
This follows from the fact that ui (si , s−i ) is concave [or quasiconcave]
in si . Suppose not, then there exists some i and some s−i ∈ S−i such
that Bi (s−i ) ∈ arg maxs ∈Si ui (s , s−i ) is not convex.
This implies that...
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 Spring '10
 AsuOzdaglar

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