Lecture ??: 04/18/2007
Recall:
Finite strategic form games, PSNE, mixed strategies, Nash equilibrium with possibly mixed
strategies, examples.
Recall definition of Nash equilibrium : A (possibly mixed) strateg profile (σ
1
*
, σ
2
*
,
…, σ
n
*
)= σ
*
is a NE
for every I and every mixed stratigey σ
i
є
Ai
, the following holds:
u
i
(σ
1
*
,…, σ
i-1
*
, σ
i
*
, σ
I+1
*
,…, σ
n
*
)>=u
i
(
ie. σ
i
*
is a best reply
or best response
for player i to the “opposition strategy profile”
σ
-i
*
= (σ
1
*
, σ
2
*
,…, σ
i-1
*
, σ
i+1
*
,…, σ
n
*
)
ie. “σ
*
is a NE” just means “For all I, σ
i
*
is a best rply to the opposition choices in σ
*
”
Defintion:
Given σ
i
є
Ai
, the support
of σ
i
(supp(σ
i
)) is the set of pure actions in A
i
to which σ
i
assigns
positive
probability.
Fact:
If σ
*
is a NE, then, for every I, every action in supp(σ
i
*
) is a best reply to the opposition strategy
profile (σ
1
*
,…, σ
i-1
*
, σ
i+1
*
,…, σ
n
*
)
(Nb.:
actions in A
i
<--> strategies in
Ai
assigning all probabilities to
one action)
Idea of proof:
u
i
(σ
*
)=a convex combination of terms of the form u
i
(σ
1
*
,…, σ
i-1
*
, a
i
*
, σ
I+1
*
,…, σ
n
*
) where a
i
is a member of
supp(σ
i
*
).
If any such a
i
is not
a best reply to σ-
i
*