Game Theory with Applications to Finance and Marketing

a homogeneous good must compete in quantity given that the inverse
market demand is (in the relevant region) p = 1 − ∑ N
i=1 q i . This game
again exhibits the unique-NE property, and hence has a unique CPE.
8. The last equilibrium concept we shall go over is rationalizability (Bernheim,
1984). Define Σ 0 i = Σ i. For all natural numbers n, define
Σ n i
= {σ i ∈ Σ n−1
i
: ∃σ −i ∈ Π j≠i co(Σj
n−1 ), u i (σ i , σ −i ) ≥ u i (σ i, ′ σ −i ) ∀σ i ′ ∈ Σ n−1
i }.
We call elements in ⋂ +∞
n=0 Σ n i rationalizable strategies. Intuitively, rational
players will never use strategies which are never best responses.
Rationalizability extends this idea to fully utilize the assumption that
rationality of players is their common knowledge.
9. Let us now develop the notion of rationalizability in detail. Given a
game Γ in normal form with I players, consider sets H i ⊂ Σ i for all
i = 1, 2, · · · , I. We shall adopt the following definitions.
• Let H i (0) ≡ H i and define inductively
H i (t) ≡ {σ i ∈ H i (t − 1) : ∃σ −i ∈ Π j≠i co(H j (t − 1))
∋: u i (σ i , σ −i ) ≥ u i (σ ′ i , σ −i) ∀σ ′ i ∈ H i(t − 1)},
where co(A) is the smallest convex set containing A, called the
convex hull generated by A. Define
∞⋂
R i (Π I i=1 H i) ≡ H i (t).
t=1
• A I-tuple of sets (A 1 , A 2 , · · · , A I ) has the best response property
if for all i, A i ⊂ Σ i and for all i, for all σ i ∈ A i , there exists
σ −i ∈ Π j≠i co(A j ) such that σ i is a best response for i against σ −i .
• A i ⊂ Σ i has the pure strategy property if for all σ i ∈ A i , for all
s i ∈ S i such that σ i (s i ) > 0, s i ∈ A i .
• A profile σ ∈ Σ is rationalizable, if σ i ∈ R i (Σ) for all i.
With these definitions, we have
Lemma 2. Suppose that the I-tuple of sets (A 1 , A 2 , · · · , A I ) is such
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