In this appendix, we show that the set of approximate equilibria is invariant under type-splitting, i.e., payoff-irrelevant subdivisions of types do not substantively change the set of ε-equilibria, for any ε ≥ 0 (cf. Myerson, 1998, Th. 4).
First we need some more notation. Let Λ be a nonempty, finite set with complete order Λ. Let g ∈ G be a network with vertex set V (g). An extended network associated with g (given Λ) can be constructed by assigning to each vertex i ∈ V (g) a vertex state λ ∈ Λ. Let Ξ(g) be the set of extended networks associated with g, and define
Ge(n) := [
g∈G(n)
Ξ(g),
and
G :=e [
n∈N
Ge(n).
We refer to the members of eG as extended networks. Note that while there may be multiple extended networks associated with a given network g ∈ G, there is a unique network g ∈ G for each extended network ˜g ∈ eG such that ˜g is an extended network associated with g.
λ1 λ1 λ2 λ2 λ1 λ2
λ1 λ2
g g˜1 g˜2 g˜3 g˜4
Figure A.1: When Λ = {λ1, λ2}, the network g ∈ G corresponds to the extended networks
˜
g1, . . . , ˜g4, i.e., Ξ(g) = {˜g1, . . . , ˜g4}, and η(˜g`) = g for ` = 1, . . . , 4.
Denote this network by η(˜g). For each ˜g ∈ eG, for each i ∈ V (η(˜g)), let κ(i, ˜g) ∈ Λ be the vertex state of i in the extended network ˜g. An illustration is provided in Figure A.1.
Let ˜F be the σ-field generated by the set of singletons of G, and let ˜e M denote the set of all probability measures on ( eG, ˜F ). For ˜µ ∈ ˜M, the probability space ( eG, ˜F , ˜µ) is an extended network belief system.
Each vertex in an extended network is thus characterized by his connectivity and his vertex state. Define
P := {(t, λ) | t ∈ N ∪ {0}, λ ∈ Λ}˜
to be the set of all ordered pairs (t, λ). Since Λ is endowed with the complete order Λ, there exists a complete order on ˜P . We define a complete order on ˜P by
∀(t, λ), (t0, λ0) ∈ ˜P : (t, λ) (t0, λ0) ⇐⇒ t > t0 or t = t0 and λ Λλ.
For t > 0 and λ ∈ Λ, let
Ωe(t,λ)K :=(λ, (θ1, λ1), . . . , (θt, λt) ∈ Λ × ˜Pt| (θ1, λ1) . . . (θt, λt) . For t = 0, define eΩ(t,λ)K = {(0, λ)} for all λ ∈ Λ. Let
ΩetK := [
λ∈Λ
Ωe(t,λ)K , and
ΩeK := [
t∈N∪{0}
ΩetK.
Let ˜FK be the σ-field generated by the set of singletons of eΩK.
We can now define the extended local profile of a vertex in an extended network, a concept akin to the neighbor connectivity profile of a vertex in a network (see Section 2).
For ˜g ∈ eG and i ∈ V (η(˜g)) such that Di(η(˜g)) = 0, define ˜Ki(˜g) := (0, κ(i, ˜g)). Otherwise, define
N˜1 := Ni(η(˜g)),
j(1) := max{j ∈ ˜N1 | (Dj(η(˜g)), κ(j, ˜g)) (Dk(η(˜g)), κ(k, ˜g)) for all k ∈ ˜N1}, K˜i,1(˜g) := (Dj(1)(η(˜g)), κ(j(1), ˜g)),
and for ` = 2, . . . , Di(η(tg)):
N˜` := N˜`−1\ {j(` − 1)},
j(`) := max{j ∈ N` | (Dj(η(˜g)), κ(j, ˜g)) (Dk(η(˜g)), κ(k, ˜g)) for all k ∈ ˜N`}, K˜i,`(˜g) := (Dj(`)(η(˜g)), κ(j(`), ˜g)).
Then, ˜Ki(˜g) := ( ˜Ki,1(˜g), . . . , ˜Ki,Di(η(tg))(˜g)) is the extended local profile of i in ˜g.
Let Γ = hT, A, (G,F , µ), (vt)t∈Ti be a network game of incomplete information. An extension of Γ (given Λ) is a tuple
Γ = h ˜˜ T , A, ( eG, ˜F , ˜µ), (˜vt)t∈ ˜Ti
defined as follows. First, a network g ∈ G is drawn according to (G,F , µ). An extended network ˜g ∈ Ξ(g) is then created by assigning to each vertex i ∈ V (g) a vertex state λ ∈ Λ with probability pµ˜(λ), independently of the other vertices, where pµ˜(λ) ≥ 0 for all λ ∈ Λ andP
λ∈Λpµ˜(λ) = 1. A player is associated with each vertex of the extended network. Each player is endowed with the action set A and is informed of the number of neighbors he has in the extended network and the vertex state of the vertex he is associated with. That is, his extended type is a pair (t, λ) ∈ ˜P . Hence, the extended type set is ˜T = ˜P . For each (t, λ) ∈ ˜T , the payoffs to a player with extended type (t, λ) are given by a function ˜v. For each (t, λ) ∈ ˜T , t > 0, the function ˜v(t,λ) maps A × At× ˜Tt to R in the following way. For each a ∈ A, a(t) ∈ At, ((θ1, λ1), . . . , (θt, λt)) ∈ ˜Tt,
˜
v(t,λ) a, a(t), (θ1, λ1), . . . , (θt, λt)) := vt a, a(t), (θ1, . . . , θt))
are the payoffs to a player with extended type (t, λ) of action a when his neighbors have extended types (θ1, λ1), . . . , (θt, λt) and play according to the action profile (a1, . . . , at). The payoffs to a player with an extended type (t, λ) ∈ ˜T with t = 0 are given by ˜v(t,λ)(a) = v0(a) for each a ∈ A. Hence, for each (t, λ) ∈ ˜T , each player with extended type (t, λ) in the extension ˜Γ of Γ has the same payoff function as a player of type t in the original game Γ.
For each (t, λ) ∈ ˜T , let ˜σ(t,λ) be a real function defined on A which satisfies
˜
σ(t,λ)(a) ≥ 0
for all a ∈ A, and
X
a∈A
˜
σ(t,λ)(a) = 1,
with ˜σ(t,λ)(a) the probability that a player with extended type (t, λ) chooses action a. Recall that the set of all probability distributions on A is denoted by Σ. Then, a profile of functions
˜
σ ∈ ΣT˜ is referred to as an extended strategy function.
The difference between the games Γ and ˜Γ is thus that the types in Γ have been “splitted”, with the subdivisions of types being payoff-irrelevant. The interest in these “extended” games lies in the fact that now players with the same connectivity may choose different probability distributions over actions if they differ in their vertex state.15 We want to know whether equilibria change substantively when we allow for such payoff-irrelevant subdivisions.
To analyze the equilibria of these extended games, we first need to specify players’ beliefs.
Players need to form beliefs about the extended type of their neighbors, given their own extended type. Recall that C is the class of all isomorphism classes of G. Let t ∈ N, and let θ = (λ, (θ˜ 1, λ1), . . . , (θt, λt)) ∈ eΩtK be an extended local profile of a player with connectivity t. Let C ∈C . The expected number of players with extended local profile ˜θ in an extended network ˜g ∈S
g∈CΞ(g) derived from a network in C is given by pµ˜(λ)pµ˜(λ1) · · · pµ˜(λt)nC {(θ1, . . . , θt)},
where we recall that for F ∈ FK, nC(F ) is the number of players in a network g ∈ G with neighbor type profile F . Then, the expected number of players with extended type (t, λ) ∈ ˜T in an extended network derived from a network in C is
X
λ1∈λ
pµ˜(λ1) . . .X
λt∈λ
pµ˜(λt)pµ˜(λ)nC {(θ1, . . . , θt)} = pµ˜(λ)nC {(θ1, . . . , θt)}.
For each (t, λ) ∈ ˜T , the conditional beliefs of a player with extended type (t, λ) such that qµ(t) > 0 and pµ˜(λ) > 0 that his extended local profile is ˜θ ∈ eΩtK are then
˜
qµ˜(˜θ|(t, λ)) :=
P
C∈C pµ˜(λ) · pµ˜(λ1) · . . . pµ˜(λt) · nC {(θ1, . . . , θt)} P
C∈C pµ˜(λ) · nC ΩtK ,
= pµ˜(λ1) · · · pµ˜(λt) · qµ(θ|t).
We can now define expected payoffs. The extended expected payoff to a player with extended type (t, λ) ∈ ˜T such that qµ(t) > 0 and pµ˜(λ) > 0 of an action a ∈ A when other players
15Note that for each network game of incomplete information Γ, there is an extension of Γ that is essentially equivalent to Γ: when there is a λ ∈ Λ such that pµ˜(λ) = 1, the extension of Γ is strategically equivalent to Γ.
play according to the extended strategy function ˜σ ∈ ΣT˜ is given by
Definition A.1. Let ˜Γ be an extension of some network game of incomplete information, and let ε ≥ 0. An extended ε-equilibrium of ˜Γ is an extended strategy profile ˜σ ∈ ΣT˜ such
for all b ∈ A. An extended 0-equilibrium is an extended equilibrium. / By an argument similar to that used in the proof of Proposition 3.1, an extended equilibrium exists in an extension of a network game of incomplete information with bounded payoffs.
The next two propositions establish that the set of approximate equilibria remains es-sentially unchanged when we allow for payoff-irrelevant type-splitting.
Proposition A.1. Let ε ≥ 0. Let Γ be a network game of incomplete information, and let Γ be an extension of Γ. If the strategy function σ ∈ Σ˜ T is an ε-equilibrium of Γ, then the extended strategy function ˜σ ∈ ΣT˜ defined by
˜
σ(t,λ)(a) = σt(a) for all (t, λ) ∈ ˜T , a ∈ A is an extended ε-equilibrium of ˜Γ.
Proof. The proof follows directly from the definitions. For each (t, λ) ∈ ˜T and each a ∈ A,
˜
Proposition A.2. Let ε ≥ 0. Let Γ be a network game of incomplete information, and let ˜Γ be an extension of Γ. If the extended strategy function ˜σ ∈ ΣT˜ is an extended ε-equilibrium of ˜Γ, then the strategy function σ ∈ ΣT defined by
σt(a) =X
λ∈Λ
pµ˜(λ)˜σ(t,λ)(a) for all t ∈ T, a ∈ A
is an ε-equilibrium of Γ.
Proof. Again, the proof is a direct consequence of the definitions. For each t ∈ T and each a ∈ A,
Since the equilibrium definitions in network games of incomplete information and their ex-tensions differ, the set of equilibria in a network game of incomplete information does not coincide with the set of equilibria in its extension. However, Proposition A.1 and A.2 es-tablish the important result that allowing for payoff-irrelevant characteristics that might affect players’ behavior does not substantively change the set of equilibria. The intuition is that for each equilibrium in a network game of incomplete information, we can find an
extended strategy function such that an extended type in the extension faces the same dis-tribution over opponents’ actions as the corresponding type in the original game under the equilibrium. Conversely, for each extended equilibrium in an extension of a network game of incomplete information, we can find a strategy function in the original game such that a type in that game faces the same distribution over opponents’ actions as a corresponding extended type under the extended equilibrium. In that sense, there is essentially no loss of generality in assuming that all players with the same payoff function independently imple-ment the same strategy, as long as these payoff-irrelevant characteristics do not provide a player with information about his opponents given his connectivity.