Game Theory with Applications to Finance and Marketing

11. Because of proposition 2, we can show that
Proposition 3. Every NE, denoted σ, is rationalizable.
Proof. The I-tuple of sets { {σ 1 }, {σ 2 }, · · ·, {σ I } } satisfies the best
response property and σ i ∈ {σ i }, ∀i, so that proposition 2 implies that
for all i, σ i ∈ R i (Σ).
12. An important connection between the rationalizable set of profiles and
the profiles surviving the iterated strict dominance is now given. In
general, the former is contained in the latter.
Proposition 4. In two-person finite games, the two concepts coincide.
Proof. Suppose that σ i is not a best response to any element of Σ j ;
i.e. for each σ j ∈ Σ j there exists b(σ j ) ∈ Σ i such that
u i (b(σ j ), σ j ) > u i (σ i , σ j ).
Call the original game Γ, and construct a zero-sum game Γ 0 as follows.
The new game has the same set of players and pure strategy spaces,
but the payoffs are defined as
for all (σ ′ i, σ j ) ∈ Σ, and
u 0 i (σ ′ i, σ j ) ≡ u i (σ ′ i, σ j ) − u i (σ i , σ j )
u 0 j(σ ′ i, σ j ) = −u 0 i (σ ′ i, σ j ).
This game has an NE in mixed strategy. Let it be (σ ∗ i , σ ∗ j ). For any
σ j ∈ Σ j , we have
u 0 i (σ∗ i , σ j) ≥ u 0 i (σ∗ i , σ∗ j ) ≥ u0 i (b(σ∗ j ), σ∗ j )
> u 0 i (σ i, σ ∗ j ) = 0,
proving that σ i is strictly dominated by σ ∗ i . Thus a strategy for player
i that can never be a best response against player j’s strategy must be
strictly dominated from player i’s point of view. Define for the purpose
of iterated deletion of strictly dominated strategies S 0 i = S i , Σ 0 i = Σ i ,
and for all t ∈ Z + ,
S t i ≡ {s i ∈ S t−1
i
: ∀σ i ∈ Σi
t−1 , ∃s −i ∈ S−i t−1 , u i(s i , s −i ) ≥ u i (σ i , s −i )},
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