Infinite horizon case

Pick a sequence ξ = (ξ₁, ξ₂, . . .) such that (1) ξm is a transition probability from H_m−1 toM(Xm) for any m≥1, and (2) ξ_m(A_m(h_m−1)|hm−1) = 1 for anym≥1 and h_m−1 ∈H_m−1. Denote the set of all such ξ as Υ.

Fix any t ≥ 1, define correspondences Ξ^t_t and ∆^t_t as follows: in the subgame ht−1,

Ξ^t_t(h_t−1) =M(At(h_t−1))⊗λt, and

∆^t_t(h_t−1) =M(At(h_t−1))⊗f_t0(h_t−1).

For anym₁ > t, suppose that the correspondences Ξ^m_t ¹⁻¹and ∆^m_t ¹⁻¹have been de-fined. Then we can define correspondences Ξ^m_t ¹:H_t−1 → M

Q

t≤m≤m1(Xm×Sm) and ∆^m_t ¹:H_t−1 → M

Q

t≤m≤m1(Xm×Sm)

as follows:

Ξ^m_t ¹(ht−1) ={g(ht−1)⋄(ξm1(ht−1,·)⊗λm1) :

g is a Borel measurable selection of Ξ^m_t ¹⁻¹, ξ_m1 is a Borel measurable selection ofM(Am1)}, and

∆^m_t ¹(ht−1) ={g(ht−1)⋄(ξm1(ht−1,·)⊗fm10(ht−1,·)) : g is a Borel measurable selection of ∆^m_t ¹⁻¹, ξ_m1 is a Borel measurable selection ofM(Am1)},

where M(Am1) is regarded as a correspondence from H_m1−1 to the set of Borel probabilities on Xm₁. For any m₁ ≥ t, let ρ^m_(h¹

t−1,ξ) ∈ Ξ^m_t ¹ be the probability on Q

t≤m≤m1(Xm×Sm) which is induced by {λm}t≤m≤m1 and {ξm}t≤m≤m1, and

̺^m_(h¹

t−1,ξ) ∈ ∆^m_t ¹ be the probability on Q

t≤m≤m1(Xm×Sm) which is induced by {fm0}t≤m≤m1 and {ξm}t≤m≤m1. Then Ξ^m_t ¹(h_t−1) is the set of all such ρ^m_(h¹

t−1,ξ), and ∆^m_t ¹(h_t−1) is the set of all such ̺^m_(h¹

t−1,ξ). Notice that ̺^m_(h¹

t−1,ξ) ∈∆^m_t ¹(h_t−1) if and only if ρ^m_(h¹

t−1,ξ) ∈Ξ^m_t ¹(h_t−1), and ̺^m_(h¹

t−1,ξ) and ρ^m_(h¹

t−1,ξ) can be both regarded as probability measures on Hm₁(h_t−1).

Similarly, let ρ_(ht−1,ξ) be the probability on Q which implies that ̺^t_(h

t−1,ξ)=ϕ_t0(h_t−1,·)◦ρ^t_(h measurable and sectionally continuous onQ

t≤k≤m−1Xk. By Lemma4,ϕm0(ht−1,·) can be extended to instead of ´ϕm0(ht−1,·), to denote the above extension. Similarly, we can work with a suitable extension of the payoff functionuas needed.

= compact valued, and sectionally continuous on X^t−1 for anym1 ≥t.

2. For any t≥1, the correspondence ∆t is nonempty and compact valued, and sectionally continuous on X^t−1.

Proof. (1) We first show that the correspondence Ξ^m_t ¹ is nonempty and compact valued, and sectionally continuous on X^t−1 for any m₁ ≥t

Consider the case m1 =t≥1, where

Ξ^t_t(ht−1) =M(At(ht−1))⊗λt.

Since Ati is nonempty and compact valued, and sectionally continuous on X^t−1, Ξ^t_t is nonempty and compact valued, and sectionally continuous onX^t−1.

Now suppose that Ξ^m_t ² is nonempty and compact valued, and sectionally continuous onX^t−1 for somem₂≥t≥1. Notice that

Ξ^m_t ²⁺¹(h_t−1) ={g(ht−1)⋄(ξm₂+1(h_t−1,·)⊗λ_(m2+1)) : g is a Borel measurable selection of Ξ^m_t ²,

ξm₂+1 is a Borel measurable selection ofM(Am₂+1)}.

Define a correspondence A^t_t from H_t−1 ×St to Xt as A^t_t(h_t−1, st) = At(h_t−1).

ThenA^t_tis nonempty and compact valued, sectionally continuous onXt−1, and has a B(X^t×S^t)-measurable graph. For any (s₁, . . . , st), since H_t−1(s₁, . . . , s_t−1) is compact and A_t(·, s1, . . . , s_t−1) is continuous and compact valued,A^t_t(·, s1, . . . , s_t) has a compact graph by Lemma3 (6). For anyh_t−1 ∈H_t−1 and τ ∈Ξ^t_t(h_t−1), the marginal of τ on St isλt andτ(Gr(A^t_t(h_t−1,·))) = 1.

For any m1 > t, suppose that the correspondence A^m_t ¹⁻¹:Ht−1× Y

t≤m≤m1−1

Sm→ Y

t≤m≤m1−1

Xm

has been defined such that

1. it is nonempty and compact valued, sectionally upper hemicontinuous on X_t−1, and has aB(X^m1−1×S^m1−1)-measurable graph;

2. for any (s₁, . . . .sm1−1), A^m_t ¹⁻¹(·, s₁, . . . .sm1−1) has a compact graph;

3. for any ht−1 ∈ Ht−1 and τ ∈ Ξ^m_t ¹⁻¹(ht−1), the marginal of τ on Q

t≤m≤m1−1Sm is ⊗t≤m≤m1−1λm and τ(Gr(A^m_t ¹⁻¹(h_t−1,·))) = 1.

We define a correspondence A^m_t ¹: H_t−1 ×Q

t≤m≤m1Sm → Q

t≤m≤m1Xm as follows:

A^m_t ¹(h_t−1, s_t, . . . , s_m1) ={(xt, . . . , x_m1) :

xm₁ ∈Am₁(h_t−1, xt, . . . , xm₁−1, st, . . . , sm₁−1), (xt, . . . , xm₁−1)∈A^m_t ¹⁻¹(h_t−1, st, . . . , sm₁−1)}.

It is obvious that A^m_t ¹ is nonempty valued. For any (s₁, . . . , sm₁), since A^m_t ¹⁻¹(·, s1, . . . .sm1−1) has a compact graph and Am1(·, s1, . . . , sm1−1) is continu-ous and compact valued,A^m_t ¹(·, s1, . . . .sm₁) has a compact graph by Lemma3(6), which implies thatA^m_t ¹ is compact valued and sectionally upper hemicontinuous on X_t−1. In addition, Gr(A^m_t ¹) = Gr(Am1)×Sm1, which isB(X^m1×S^m1)-measurable.

For any h_t−1 ∈ H_t−1 and τ ∈ Ξ^m_t ¹(h_t−1), it is obvious that the marginal of τ on Q

t≤m≤m1Sm is ⊗t≤m≤m1λm and τ(Gr(A^m_t ¹(ht−1,·))) = 1.

By Lemma 13, Ξ^m_t ²⁺¹ is nonempty and compact valued, and sectionally continuous onX^t−1.

Now we show that the correspondence ∆^m_t ¹ is nonempty and compact valued, and sectionally continuous onX^t−1 for any m₁ ≥t.

Given s^t−1 and a sequence{x^k₀, x^k₁, . . . , x^k_t−1} ∈H_t−1(s^t−1) for 1≤k≤ ∞. Let

h^k_t−1 = (s^t−1,(x^k₀, x^k₁, . . . , x^k_t−1)). It is obvious that ∆^m_t ¹ is nonempty valued, we first show that ∆^m_t ¹ is sectionally upper hemicontinuous on X^t−1. Suppose that

̺^m1

(h^k_t−1,ξ^k)∈∆^m_t ¹(h^k_t−1) for 1≤k <∞and (x^k₀, x^k₁, . . . , x^k_t−1)→(x^∞₀ , x^∞₁ , . . . , x^∞_t−1), we need to show that there exists some ξ^∞ such that a subsequence of ̺^m1

(h^k_t−1,ξ^k)

weakly converges to̺^m_(h¹∞

t−1,ξ^∞) and̺^m_(h¹∞

t−1,ξ^∞) ∈∆^m_t ¹(h^∞_t−1).

Since Ξ^m_t ¹ is sectionally upper hemicontinuous on X^t−1, there exists someξ^∞ such that a subsequence ofρ^m1

(h^k_t−1,ξ^k), say itself, weakly converges toρ^m_(h¹∞

t−1,ξ^∞) and ρ^m_(h¹∞

t−1,ξ^∞)∈Ξ^m_t ¹(h^∞_t−1). Then ̺^m_(h¹∞

t−1,ξ^∞)∈∆^m_t ¹(h^∞_t−1).

For any bounded continuous functionψ on Q

t≤m≤m1(Xm×Sm), let χk(xt, . . . , xm₁, st, . . . , sm₁) =

ψ(xt, . . . , xm₁, st, . . . , sm₁)· Y

t≤m≤m1

ϕ_m0(h^k_t−1, xt, . . . , x_m−1, st, . . . , sm).

Then {χk}is a sequence of functions satisfying the following three properties.

1. For each k,χk is jointly measurable and sectionally continuous onX.

2. For any (st, . . . , sm1) and any sequence (x^k_t, . . . , x^k_m1)→(x^∞_t , . . . , x^∞_m1) in X, χk(x^k_t, . . . , x^k_m1, st, . . . , sm₁)→χ_∞(x^∞_t , . . . , x^∞_m1, st, . . . , sm₁) as k→ ∞.

3. The sequence{χk}_1≤k≤∞ is integrably bounded.

By Lemma14, ask→ ∞, which implies that ̺^m1

(h^k_t−1,ξ^k) weakly converges to ̺^m_(h¹∞

t−1,ξ^∞). Therefore, ∆^m_t ¹ is sectionally upper hemicontinuous on X^t−1. If one chooses h¹_t−1 = h²_t−1 = · · · = h^∞_t−1, then we indeed show that ∆^m_t ¹ is compact valued.

In the argument above, we indeed proved that if ρ^m1

(h^k_t−1,ξ^k) weakly converges to ρ^m_(h¹∞

t−1,ξ^∞), then̺^m_(h¹k

t−1,ξ^k) weakly converges to ̺^m_(h¹∞ t−1,ξ^∞).

The left is to show that ∆^m_t ¹ is sectionally lower hemicontinuous on X^t−1.

Suppose that (x^k₀, x^k₁, . . . , x^k_t−1)→(x^∞₀ , x^∞₁ , . . . , x^∞_t−1) and ̺^m_(h¹∞ Ξ^m_t ¹ is sectionally lower hemicontinuous on X^t−1, there exists a subsequence of {(x^k₀, x^k₁, . . . , x^k_t−1)}, say itself, and ρ^m1 Therefore, ∆^m_t ¹ is nonempty and compact valued, and sectionally continuous on X^t−1 for any m₁≥t.

(2) We show that ∆t is nonempty and compact valued, and sectionally continuous onX^t−1.

It is obvious that ∆_t is nonempty valued, we first prove that it is compact valued.

Givenh_t−1 and a sequence{τ^k} ⊆∆t(h_t−1), there exists a sequence of{ξ^k}k≥1

such thatξ^k= (ξ₁^k, ξ₂^k, . . .)∈Υ andτ^k=̺_(ht−1,ξ^k₎ for each k.

By (1), Ξ^t_tis compact. Then there exists a measurable mappinggtsuch that (1) g^t= (ξ₁¹, . . . , ξ_t−1¹ , gt, ξ¹_t+1, . . .)∈Υ, and (2)ρ^t_(h

t−1,ξ^k)weakly converges toρ^t_(h

t−1,g^t). Note that{ξ^k_t+1}is a Borel measurable selection ofM(At+1). By Lemma13, there is a Borel measurable selectiong_t+1 of M(At+1) such that there is a subsequence of {ρ^t+1_(h

t−1,ξ^k)}, say itself, which weakly converges to ρ^t+1_(h

t−1,g^t+1), where g^t+1 = (ξ₁¹, . . . , ξ¹_t−1, gt, g_t+1, ξ_t+2¹ , . . .)∈Υ.

Repeat this procedure, one can construct a Borel measurable mapping g such that ρ_(ht−1,g) is a convergent point of {ρ_(ht−1,ξ^k₎}. Thus, ̺_(ht−1,g) is a convergent point of{̺_(ht−1,ξ^k₎}.

The sectional upper hemicontinuity of ∆_t follows a similar argument as above.

In particular, given s^t−1 and a sequence {x^k₀, x^k₁, . . . , x^k_t−1} ⊆ H_t−1(s^t−1) for k ≥ 0. Let h^k_t−1 = (s^t−1,(x^k₀, x^k₁, . . . , x^k_t−1)). Suppose that (x^k₀, x^k₁, . . . , x^k_t−1) → (x⁰₀, x⁰₁, . . . , x⁰_t−1). If{τ^k} ⊆ ∆t(h^k_t−1) for k≥ 1 and τ^k → τ⁰, then one can show thatτ⁰ ∈∆t(h⁰_t−1) by repeating a similar argument as in the proof above.

Finally, we consider the sectional lower hemicontinuity of ∆_t. Suppose that τ⁰ ∈ ∆t(h⁰_t−1). Then there exists some ξ ∈ Υ such that τ⁰ = ̺_(h⁰ which implies that ∆t is sectionally lower hemicontinuous.

Define a correspondence Q^τ_t:H_t−1→Rⁿ++ as follows:

Lemma 19. For anyt, τ ≥1, Q^τ_t is bounded, measurable, nonempty and compact valued, and essentially sectionally upper hemicontinuous on X^t−1.

Proof. We prove the lemma in three steps.

Step 1. Fix t > τ. We will show thatQ^τ_t is bounded, nonempty and compact valued, and sectionally upper hemicontinuous onX^t−1.

The boundedness and nonemptiness of Q^τ_t are obvious. We shall prove that Q^τ_t is sectionally upper hemicontinuous on X^t−1. Given s^t−1 and a sequence {x^k₀, x^k₁, . . . , x^k_t−1} ⊆ H_t−1(s^t−1) for k ≥ 0. Let h^k_t−1 = (s^t−1,(x^k₀, x^k₁, . . . , x^k_t−1)).

Suppose thata^k ∈Q^τ_t(h^k_t−1) fork ≥1, (x^k₀, x^k₁, . . . , x^k_t−1) →(x⁰₀, x⁰₁, . . . , x⁰_t−1) and a^k →a⁰, we need to show that a⁰ ∈Q^τ_t(h⁰_t−1).

By the definition, there exists a sequence {ξ^k}k≥1 such that a^k=

Since the game is continuous at infinity, there exists a positive integer M ≥t such thatw^m < ¹₅δ for any m > M.

For each j > M, by Lemma4, there exists a measurable selectionξ_j^′ ofM(Aj) such that ξ_j^′ is sectionally continuous on X^j−1. Let µ: HM → Q

m>M(Xm × Sm) be the transition probability which is induced by (ξ_M^′ ₊₁, ξ_M^′ ₊₂, . . .) and

{f_(M+1)0, f_(M+2)0, . . .}. By Lemma 9,µ is measurable and sectionally continuous

Then V_M is bounded and measurable. In addition, V_M is sectionally continuous on X^M by Lemma14.

for k ≥ K1, where K1 is a sufficiently large positive integer. In addition, there exists a positive integer K₂ such that |a^k−a⁰|< ¹₅δ for k≥ K₂. Combining the inequalities above, we prove inequality (3), which implies that Q^τ_t is sectionally upper hemicontinuous onX^t−1 fort > τ.

Furthermore, to prove that Q^τ_t is compact valued, we only need to consider the case that{x^k₀, x^k₁, . . . , x^k_t−1}={x⁰₀, x⁰₁, . . . , x⁰_t−1}for anyk≥0, and repeat the above proof.

Step 2. Fix t > τ, we will show thatQ^τ_t is measurable.

Fix a sequence (ξ^′₁, ξ₂^′, . . .), whereξ_j^′ is a selection ofM(Aj) measurable ins^j−1

and continuous in x^j−1 for each j. For anyM ≥t, let W_M^M(h_t−1, xt, . . . , x_M, st, . . . , s_M) = (Z

Q

m>M(Xm×Sm)

u(h_t−1, xt, . . . , xM, st, . . . , sM, x, s)̺_{(ht−1,xt,...,xM,st,...,sM,ξ}^′₎(d(x, s)) )

.

By Lemma9,̺_{(ht−1,xt,...,xM,st,...,sM,ξ}^′₎is measurable fromHM toM Q

m>M(Xm×Sm) , and sectionally continuous onX^M. Thus,W_M^M is bounded, measurable, nonempty, convex and compact valued. By Lemma14,W_M^M is sectionally continuous onX^M. Suppose that for somet≤j ≤M,W_M^j has been defined such that it is bounded, measurable, nonempty, convex and compact valued, and sectionally continuous on X^j. Let

W_M^j−1(h_t−1, xt, . . . , x_j−1, st, . . . , s_j−1) =

Z

Xj×Sj

w^j_M(h_t−1, x_t, . . . , x_j, s_t, . . . , s_j)̺^j_(h

t−1,xt,...,x_j−1,st,...,s_j−1,ξ)(d(x_j, s_j)) :

̺^j_(h

t−1,xt,...,x_j−1,st,...,s_j−1,ξ) ∈∆^j_j(h_t−1, xt, . . . , x_j−1, st, . . . , s_j−1), w_M^j is a Borel measurable selection ofW_M^j .

Let ˇS_j =S_j. Since Z

Xj×Sj

W_M^j (ht−1, xt, . . . , xj, st, . . . , sj)̺^j_(h

t−1,xt,...,x_j−1,st,...,s_j−1,ξ)(d(xj, sj))

= Z

Sj

Z

Xj×Sˇj

W_M^j (h_t−1, x_t, . . . , x_j, s_t, . . . , s_j)ρ^j_(h

t−1,xt,...,x_j−1,st,...,s_j−1,ξ)(d(x_j,sˇ_j))

·ϕ_j0(h_t−1, xt, . . . , x_j−1, st, . . . , sj)λj(dsj), we have

W_M^j−1(h_t−1, xt, . . . , x_j−1, st, . . . , s_j−1) =

Z

Sj

Z

Xj×Sˇj

w_M^j (ht−1, xt, . . . , xj, st, . . . , sj)ρ^j_(h

t−1,xt,...,x_j−1,st,...,s_j−1,ξ)(d(xj,sˇj))

·ϕ_j0(h_t−1, xt, . . . , x_j−1, st, . . . , sj)λj(dsj) : ρ^j_(h

t−1,xt,...,xj−1,st,...,sj−1,ξ)∈Ξ^j_j(h_t−1, xt, . . . , x_j−1, st, . . . , s_j−1), w_M^j is a Borel measurable selection ofW_M^j .

Let

Wˇ_M^j (h_t−1, xt, . . . , x_j−1, st, . . . , sj) =

Z

Xj×Sˇj

w^j_M(h_t−1, x_t, . . . , x_j, s_t, . . . , s_j)·ρ^j_(h

t−1,xt,...,x_j−1,st,...,s_j−1,ξ)(d(x_j,sˇ_j)) : ρ^j_(h

t−1,xt,...,xj−1,st,...,sj−1,ξ) ∈Ξ^j_j(h_t−1, xt, . . . , x_j−1, st, . . . , s_j−1), w_M^j is a Borel measurable selection ofW_M^j .

Since W_M^j (ht−1, xt, . . . , xj, st, . . . , sj) is continuous in xj and does not depend on ˇ

sj, it is continuous in (xj,sˇj). In addition,W_M^j is bounded, measurable, nonempty, convex and compact valued. By Lemma 10, ˇW_M^j is bounded, measurable, nonempty and compact valued, and sectionally continuous onX^j−1.

It is easy to see that

W_M^j−1(h_t−1, xt, . . . , x_j−1, st, . . . , s_j−1) = Z

S_j

Wˇ_M^j (h_t−1, xt, . . . , x_j−1, st, . . . , sj)ϕ_j0(h_t−1, xt, . . . , x_j−1, st, . . . , sj)λj(dsj).

By Lemma 5, it is bounded, measurable, nonempty and compact valued, and sectionally continuous onX^j−1.

Let W = ∪M≥tW_M^t−1. That is, W is the closure of ∪M≥tWM, which is measurable due to Lemma 3.

First, W ⊆Q^τ_t because W_M^t−1 ⊆Q^τ_t for each M ≥t andQ^τ_t is compact valued.

Second, fixh_t−1 and q ∈Q^τ_t(h_t−1). Then there exists a mapping ξ ∈Υ such that q=

Z

Q

m≥t(Xm×Sm)

u(h_t−1, x, s)̺_(ht−1,ξ)(d(x, s)).

ForM ≥t, let

VM(ht−1, xt, . . . , xM, st, . . . , sM) = Z

Q

m>M(Xm×Sm)

u(h_t−1, xt, . . . , xM, st, . . . , sM, x, s)̺_{(ht−1,xt,...,xM,st,...,sM,ξ)}(x, s) and

q_M = Z

Q

t≤m≤M(Xm×Sm)

V_M(h_t−1, x, s)̺^M_(h

t−1,ξ)(d(x, s)).

Because the dynamic game is continuous at infinity, q_M → q, which implies that q∈W(h_t−1) andQ^τ_t ⊆W.

Therefore, W =Q^τ_t, and hence Q^τ_t is measurable fort > τ.

Step 3. For t≤τ, we can start with Q^τ_τ+1. Repeating the backward induction in Section 5.3.1, we have that Q^τ_t is also bounded, measurable, nonempty and compact valued, and essentially sectionally upper hemicontinuous onX^t−1.

The following three lemmas show that Q^∞_t (ht−1) = Φ(Q^∞_t+1)(ht−1) =Et(ht−1)

forλ^t−1-almost allh_t−1 ∈H_t−1.²⁸

Lemma 20. 1. The correspondenceQ^∞_t is bounded, measurable, nonempty and compact valued, and essentially sectionally upper hemicontinuous on X^t−1. 2. For any t≥1, Q^∞_t (h_t−1) = Φ(Q^∞_t+1)(h_t−1) for λ^t−1-almost all h_t−1 ∈H_t−1. Proof. (1) It is obvious thatQ^∞_t is bounded. By Lemma 3(2),Q^∞_t is measurable.

It is easy to see that ifτ₁ ≥τ₂, thenQ^τ_t¹ ⊆Q^τ_t². SinceQ^τ_t is nonempty and compact valued, Q^∞_t is nonempty and compact valued.

Fix any s^t−1 ∈ S^t−1 such that Q^τ_t(·, s^t−1) is upper hemicontinuous on Ht−1(s^t−1) for any τ. By Lemma 3 (7), Q^∞_t (·, s^t−1) is upper hemicontinuous on H_t−1(s^t−1). SinceQ^τ_t is essentially upper hemicontinuous onX^t−1 for eachτ,Q^∞_t is essentially upper upper hemicontinuous on X^t−1.

(2) For any τ, Φ(Q^∞_t+1)(h_t−1) ⊆ Φ(Q^τ_t+1)(h_t−1) ⊆ Q^τ_t(h_t−1), and hence Φ(Q^∞_t+1)(ht−1)⊆Q^∞_t (ht−1).

The space {1,2, . . .∞}is a countable compact set endowed with the following metric: d(k, m) = |_k¹ − _m¹| for any 1 ≤ k, m ≤ ∞. The sequence {Q^τ_t+1}1≤τ≤∞

can be regarded as a correspondence Qt+1 from Ht× {1,2, . . . ,∞} to Rⁿ, which is measurable, nonempty and compact valued, and essentially sectionally upper hemicontinuous on X^t× {1,2, . . . ,∞}. The backward induction in Section 5.3.1 shows that Φ(Q_t+1) is measurable, nonempty and compact valued, and essentially sectionally upper hemicontinuous onX^t× {1,2, . . . ,∞}.

Since Φ(Qt+1) is essentially sectionally upper hemicontinuous onX^t×{1,2, . . . ,∞}, there exists a measurable subset ˇS^t−1 ⊆ S^t−1 such that λ^t−1( ˇS^t−1) = 1, and Φ(Q_t+1)(·,·,sˇ^t−1) is upper hemicontinuous for any ˇs^t−1 ∈ Sˇ^t−1. Fix ˇs^t−1 ∈ Sˇ^t−1. For h_t−1 = (x^t−1,ˇs^t−1) ∈ H_t−1 and a ∈ Q^∞_t (h_t−1), by its definition, a∈Q^τ_t(h_t−1) = Φ(Q^τ_t+1)(h_t−1) for τ ≥t. Thus, a∈Φ(Q^∞_t+1)(h_t−1).

In summary, Q^∞_t (ht−1) = Φ(Q^∞_t+1)(ht−1) for λ^t−1-almost allht−1∈Ht−1. Though the definition of Q^τ_t involves correlated strategies for τ < t, the following lemma shows that one can work with mixed strategies in terms of equilibrium payoffs, due to the combination of backward inductions in multiple steps.

Lemma 21. Ifc_tis a measurable selection ofΦ(Q^∞_t+1), thenc_t(h_t−1)is a subgame-perfect equilibrium payoff vector for λ^t−1-almost allh_t−1 ∈H_t−1.

28The proofs for Lemmas 20 and 22 follow the standard ideas with some modifications; see, for example,Harris(1990),Harris, Reny and Robson(1995) andMariotti(2000).

Proof. Without loss of generality, we only prove the caset= 1. Suppose thatc₁ is a measurable selection of Φ(Q^∞₂ ). Apply Proposition 5recursively to obtain Borel measurable mappings {fki}i∈I for k ≥ 1. That is, for any k ≥ 1, there exists a Borel measurable selectionc_k ofQ^∞_k such that forλ_k−1-almost allh_k−1 ∈H_k−1,

1. f_k(h_k−1) is a Nash equilibrium in the subgameh_k−1, where the action space is Aki(h_k−1) for playeri∈I, and the payoff function is given by

Z

Sk

c_k+1(h_k−1,·, sk)f_k0(ds_k|hk−1).

2.

ck(h_k−1) = Z

A_k(hk−1)

Z

S_k

c_k+1(h_k−1, xk, sk)f_k0(dsk|h_k−1)fk(dxk|h_k−1).

We need to show that c₁(h₀) is a subgame-perfect equilibrium payoff vector for λ₀-almost allh₀ ∈H₀.

First, we show that {fki}i∈I is a subgame-perfect equilibrium. Fix a player j and a strategy gj = {gkj}k≥1. By the one-step deviation principle, it suffices to show that for any t^′≥1,λ^t′−1-almost allht^′−1, and any δ >0,

Z

Q

m≥t′(Xm×Sm)

uj(h_t^′₋₁, x, s)̺_{(ht′−1,f)}(d(x, s))

>

Z

Q

m≥t′(Xm×Sm)

uj(ht^′−1, x, s)̺_{(ht′−1,(f−j,˜gj))}(d(x, s))−δ, where ˜gj = (g1j, . . . , gt^′j, f_(t^′_+1)j, f_(t^′_+2)j, . . .).

Since the game is continuous at infinity, there exists a positive integer M > t^′ such that w^m < ¹₃δ for any m ≥ M. By Lemma 20, c_k(h_k−1) ∈ Φ(Q^∞_k+1)(h_k−1) = Q^∞_k (h_k−1) = ∩_τ≥1Q^τ_k(h_k−1) for λ_k−1-almost all h_k−1 ∈ H_k−1. Since Q^τ_k = Φ^τ−k+1(Q^τ_τ+1) for k ≤ τ, c_k(h_k−1) ∈ ∩τ≥1Φ^τ−k+1(Q^τ_τ+1)(h_k−1) ⊆ Φ^M−k+1(Q^M_M+1)(h_k−1) for λ_k−1-almost allh_k−1 ∈H_k−1.

Thus, there exists a Borel measurable selectionwofQ^M_M+1and a strategy profile ξ such that for λ_M−1-almost allh_M−1 ∈H_M−1,

1. f_M(h_M−1) is a Nash equilibrium in the subgame h_M−1, where the action space is AM i(hM−1) for playeri∈I, and the payoff function is given by

Z

SM

w(hM−1,·, sM)f_M0(dsM|hM−1).

2. and the payoff function is given by

Z

Therefore, for λ^t′−1-almost allh_t^′₋₁∈H_t^′₋₁, Z

Q

m≥t′(Xm×Sm)

uj(ht^′−1, x, s)̺_{(ht′−1,(f−j,fj))}(d(x, s))

>

Z

Q

m≥t′(Xm×Sm)

uj(h_t^′₋₁, x, s)̺_{(ht′−1,(f−j,˜gj))}(d(x, s))−δ, which implies that {fki}i∈I is a subgame-perfect equilibrium.

Then we show that for λ0-almost allh0 ∈H0, c₁(h₀) =

Z

Q

m≥1(Xm×Sm)

u(h₀, x, s)̺_(h0,f)(d(x, s)).

As shown in (1), for any positive integerM, there exists a strategy profileξ such that forλ₀-almost allh₀ ∈H₀,

c₁(h₀) = Z

Q

m≥1(Xm×Sm)

u(h₀, x, s)̺_(h0,f^′₎(d(x, s)),

wheref_k^′ isf_kifk≤M, andξ_kifk≥M+1. Since the game is continuous at infini-ty,R_Q

m≥1(Xm×Sm)u(h0, x, s)̺_(h0,f)(d(x, s)) andR_Q

m≥1(Xm×Sm)u(h0, x, s)̺_(h0,f^′₎(d(x, s)) are arbitrarily close whenM is sufficiently large. Thus, forλ₀-almost all h₀ ∈H₀,

c₁(h₀) = Z

Q

m≥1(Xm×Sm)

u(h₀, x, s)̺_(h0,f)(d(x, s)).

This completes the proof.

For t ≥1 and ht−1 ∈ Ht−1, recall that Et(ht−1) is the set of payoff vectors of subgame-perfect equilibria in the subgameh_t−1. Then we shall show the following lemma.

Lemma 22. For anyt≥1,Et(h_t−1) =Q^∞_t (h_t−1)forλ^t−1-almost allh_t−1 ∈H_t−1. Proof. (1) We will first prove the following claim: for any t and τ, if E_t+1(ht) ⊆ Q^τ_t+1(h_t) for λ^t-almost all h_t ∈ H_t, then E_t(h_t−1) ⊆Q^τ_t(h_t−1) for λ^t−1-almost all h_t−1∈H_t−1. We only need to consider the case thatt≤τ.

By the construction of Φ(Q^τ_t+1) in Subsection 5.3.1, there exists a measurable subset ´S^t−1 ⊆ S^t−1 with λ^t−1( ´S^t−1) = 1 such that for any ct and ht−1 = (x^t−1,s´^t−1)∈H_t−1 with ´s^t−1 ∈S´^t−1, if

1. ct = R

At(ht−1)

R

Stq_t+1(h_t−1, xt, st)f_t0(dst|h_t−1)α(dxt), where q_t+1(h_t−1,·) is measurable andq_t+1(h_t−1, xt, st)∈Q^τ_t+1(h_t−1, xt, st) for λt-almost allst∈St

and xt∈At(ht−1);

2. α ∈ ⊗i∈IM(Ati(h_t−1)) is a Nash equilibrium in the subgameh_t−1 with payoff R

Stqt+1(ht−1,·, st)ft0(dst|ht−1) and action spaceQ

i∈IAti(ht−1), thenc_t∈Φ(Q^τ_t+1)(h_t−1).

Fix a subgame h_t−1 = (x^t−1,´s^t−1) such that ´s^t−1 ∈ S´^t−1. Pick a point ct ∈ Et(´s^t−1). There exists a strategy profile f such that f is a subgame-perfect equilibrium in the subgameht−1 and the payoff is ct. Let ct+1(ht−1, xt, st) be the payoff vector induced by{fti}i∈I in the subgame (ht, xt, st) ∈Gr(At)×St. Then we have

1. ct=R

At(ht−1)

R

Stc_t+1(h_t−1, xt, st)f_t0(dst|ht−1)ft(dxt|ht−1);

2. ft(·|ht−1) is a Nash equilibrium in the subgame h_t−1 with action space A_t(h_t−1) and payoff R

Stc_t+1(h_t−1,·, st)f_t0(ds_t|ht−1).

Since f is a subgame-perfect equilibrium in the subgame h_t−1,c_t+1(h_t−1, xt, st) ∈ Et+1(ht−1, xt, st) ⊆Q^τ_t+1(ht−1, xt, st) for λt-almost all st ∈ St and xt ∈ At(ht−1), which implies that ct∈Φ(Q^τ_t+1)(h_t−1) =Q^τ_t(h_t−1).

Therefore, E_t(h_t−1)⊆Q^τ_t(h_t−1) for λ^t−1-almost allh_t−1 ∈H_t−1.

(2) For anyt > τ,Et⊆Q^τ_t. Ift≤τ, we can start withE_τ+1 ⊆Q^τ_τ+1and repeat the argument in (1), then we can show thatEt(h_t−1)⊆Q^τ_t(h_t−1) for λ^t−1-almost all ht−1 ∈Ht−1. Thus, Et(ht−1)⊆Q^∞_t (ht−1) for λ^t−1-almost allht−1 ∈Ht−1.

(3) Suppose that ct is a measurable selection from Φ(Q^∞_t+1). Apply Proposi-tion 5 recursively to obtain Borel measurable mappings {fki}i∈I for k ≥ t. By Lemma21,ct(h_t−1) is a subgame-perfect equilibrium payoff vector forλ^t−1-almost all h_t−1 ∈ H_t−1. Consequently, Φ(Q^∞_t+1)(h_t−1) ⊆ Et(h_t−1) for λ^t−1-almost all ht−1∈Ht−1.

By Lemma 20, Et(h_t−1) = Q^∞_t (h_t−1) = Φ(Q^∞_t+1)(h_t−1) for λ^t−1-almost all h_t−1∈H_t−1.

Therefore, we have proved Theorem 1 and Proposition 1.

Im Dokument Dynamic Games with Almost Perfect Information (Seite 39-53)