Strong Forward Induction

Lemma 1

Every stable set contained in a single component of equilibria of game G has a strong forward induction equilibrium.

Proof.

The method of proof here is similar to Man (2012). Let E be a stable set contained in a single connected component of G. By definition, a closed set of Nash equilibria is stable in game G if it is minimal with respect to the following property. For any ϵ>0, there is δ0>0 such that for any (σ˜1,…,σ˜N)∈Σ˜0 and for any (δ1,…,δN) with 0<δi<δ0, the perturbed game where every s˜i is replaced by (1−δi)s˜i+δiσ˜i has an equilibrium ϵ-close to the set.

What this implies is that for any perturbation Gϵ of G, there exists NE of Gϵ, which converges to an equilibrium in E as ϵ→0. To apply this result, we first define sets of totally mixed strategies subject to the dominance and relevance conditions. Recall that for all i, R‾i0(E)=S˜i and Σ‾0=Σ˜0 is the set of totally mixed strategies. For n≥1, define

Let Σ_0(ϵ)=Σ→0(ϵ)=Σ˜0. For n≥1, define

and

Since Σ‾n−1(ϵ) is the set of totally mixed strategy profiles satisfying the relevance conditions, by construction, every strategy profiles in Σ_n(ϵ) satisfies the l-th order relevance condition and dominance condition for l≤n−1. Compared to Σ_n(ϵ), Σ→n(ϵ) also requires that each strategy profile satisfy the n-th order relevance condition. Because Σ‾n(ϵ)⊆Σ‾n−1(ϵ), it is true that Σ→n(ϵ)⊆Σ_n(ϵ).

For ϵ sufficiently small, Σ‾n(ϵ) is non-empty as long as R‾in(E) is non-empty for all i. Thus Σ_n(ϵ) is non-empty if R‾in−1(E) is non-empty for all i, and Σ→n(ϵ) is non-empty if R‾in(E) is non-empty for all i. With proper scaling we have Σ→n(ϵ′′′)⊆Σ_n(ϵ′′)⊆Σ→n−1(ϵ′)⊆Σ_n−1(ϵ) for all n≥1.

Next, let Ω_n be the set of sequences of perturbations {G_ϵ} of game G. The perturbations are constructed as follows: for every player i, replace each pure strategy s˜i∈S˜i by (1−ϵ)s˜i+ϵσ˜˜iϵ where σ˜˜iϵ∈Σ_in(ϵ). Also construct {G→ϵ} and Ω→n in a similar fashion: for every i, each s˜i is replaced by (1−ϵ)s˜i+ϵσˆˆiϵ where σˆˆiϵ∈Σ→in(ϵ). As long as R‾in(E) is non-empty for all i, Σ_in(ϵ) and Σ→in(ϵ) are non-empty and therefore, Ω→n and Ω_n are well-defined. By definition, Ω→n⊆Ω_n⊆Ω→n−1⋯⊆Ω→0⊆Ω_0.

For any game G′, denote NE(G′) as the set of Nash equilibria. Let E_0=E→0=E, and for n≥1, define

and

Since E is a stable set, by definition, there exists σ_∈E_0 (σ→∈E→0) for every sequence of perturbations in Ω_0 (in Ω→0).

We now show that first-order strong forward induction equilibrium exists. By definition of stable set, there exists σ_∈E_1 as a limit of a sequence of NE of the perturbed games {G_ϵ}∈Ω_1. Since every strategy of σ_∈E_1 satisfies the condition of first-order truly relevant strategy, R‾i1(E) is nonempty for all i. Because R‾1(E) is nonempty, Ω→1 is well-defined and Ω→1⊆Ω→0. The definition of stable set again implies that there exists σ→∈E→1.

Furthermore, every σ→∈E→1 is a first-order strong forward induction equilibrium. To see this, note that if σ→∈E→1, then there exists a sequence {σ→ϵ} as NE of a sequence of perturbed games such that limϵ→0σ→ϵ=σ→. Clearly, for all i and for all ϵ, the NE σ→ϵ of the perturbed game G→ϵ would assign zero probability to any s˜i∉Ri1(E), as by definition such s˜i can not be a best response. If we let the effective distribution resulted on S˜ from the mixed strategy profile σ→ϵ by σˆϵ, namely, for all i, σˆiϵ(s˜i)=(1−ϵ)σ→iϵ(s˜i)+ϵσˆˆiϵ(s˜i), then the sequence {σˆϵ} satisfies:

If s˜∈R‾1(E) and s˜′∉R‾1(E):

If s˜∈U(S˜) and s˜′∉U(S˜):

Thus, by definition, an equilibrium σ→∈E→1 is a first-order strong forward induction equilibrium, indicating E→1⊆F‾1(E).

Now we are in a position to show that for all n, E→n exists and in addition, E→n⊆F‾n(E). For n=1, we have already shown that E→1 is non-empty and E→1⊆F‾1(E). Suppose that for l≤n, for every sequence of perturbation in Ω→l, there exists a σ→∈E→l and E→l⊆F‾l(E). We first show that E→n+1 is non-empty. By definition, any σ→∈E→l is the limit of a sequence of NE of the perturbed games. Since Ω→n+1⊆Ω→n, there must exist σ→∈E→n+1 if R‾in+1(E) is nonempty for all i. But as Ω→n is well-defined by assumption, Ω_n+1 is also well-defined. Again by definition of stable set, there exists σ_∈E_n+1 for every sequence of perturbations in Ω_n+1. Moreover, since every strategy that is part of σ_∈E_n+1 is an (n+1)-th order truly relevant strategy, R‾in+1(E) is non-empty for all i. This in turn implies that Ω→n+1 is well-defined and the set E→n+1 is non-empty.

We have yet to show that E→n+1⊂F‾n+1(E). But similar argument that shows E→1⊆F‾1(E) would also show E→n+1⊆F‾n+1(E). Hence, Lemma 1 holds by induction.

Definition 6.1

A consistent assessment (b,μ) constitutes a weakly sequential equilibrium in the extensive form game ΓE if for all players i and for all player i’s information sets hi∈Hi that her own behavioral strategy bi does not exclude, bi is a best response to b−i given player i’s belief μ,

Definition 6.2

(DEFINITION 3.3 in GW) A pure behavioral strategy bi of player i is relevant for an outcome P if there exists a weakly sequential equilibrium (b∗,μ) giving outcome P such that bi prescribes an optimal continuation play given i’s equilibrium belief at every information set not excluded by bi.

Definition 6.3

(DEFINITION 3.5 in GW): An outcome satisfies forward induction if it results from a weakly sequential equilibrium in which at every information set that is relevant for that outcome the support of the belief of the player acting there is confined to profiles of Nature’s strategies and other players’ strategies that are relevant for that outcome.

Definition 6.4

An assessment (b,μ) constitutes a weakly quasi-perfect equilibrium in the extensive form game ΓE if there exists a sequence of consistent assessment (bk,μk)∈Ψm with limk→∞(bk,μk)=(b,μ) such that, for all players i, and at all information sets hi∈Hi not excluded by bi,

Definition 6.5

A pure behavioral strategy bi of a player i is relevant for outcome P if there exists a weakly quasi-perfect equilibrium (b∗,μ) with outcome P and a sequence of consistent assessment (bk,μk)∈Ψm with limk→∞(bk,μk)=(b∗,μ) such that at every information set h not excluded by bi,

Definition 6.6

An outcome P satisfies forward induction if it results from a weakly quasi-perfect equilibrium (b∗,μ) and there exists a sequence (bk,μk)∈Ψm with (b∗,μ)=limk→∞(bk,μk) such that all s∈R(P) and all s′∈S/R(P),

Definition 6.7

A pure behavioral strategy bi of player i is truly relevant for outcome P if: (1) there exists (b∗,μ) satisfying the dominance condition and inducing P; and (2) there is a sequence of consistent assessment (bk,μk)∈Ψm with limk→∞(bk,μk)=(b∗,μ) such that at every information set h not excluded by bi,

Definition 6.8

An outcome P satisfies strong forward induction if it results from a weakly quasi-perfect equilibrium (b∗,μ) satisfying the relevance condition. That is, there exists a sequence of consistent assessment (bk,μk)∈Ψm with limk→∞(bk,μk)=(b∗,μ) such that for all s∈R(P) and all s′∈S/R(P),