Our main result states that short-run players asymptotically reach consensus when they act according to a MPBE strategy profile. We discuss the implications of the results presented here in Section 4.4.1.
Theorem 4.3. Let σ be a MPBE. For all i, j ∈ N, σi,t−σj,t →0, P-almost surely,
as t goes to infinity.
Proof. We letS denote the smallest sub sigma-algebra ofBthat makes the mapping
ω 7→ s(ω) = (s1(ω), . . . , sn(ω)) measurable, and let Hσ∞ =
Wn
i=1H
σ
i,∞. Since the
information available to the players in any stage of the game is no more than the information jointly contained in their private signals, Hσ
∞ ⊆ S. Therefore, σi,t is
measurable with respect to S for all i and t, so σi,t(ω) = σi,t(ω0) whenever s(ω) =
letting σi,t(s) = σi,t(ω(s)), where ω(s) is a selection of Ω(s) = {ω ∈ Ω :s(ω) = s}.
The statement of the theorem is therefore equivalent to the following: σi,t(s)−
σj,t(s)→0 for all s∈S with P(s) = P(Θ× {s})>0.
Suppose to the contrary that there exists some neighboringi, j ∈ N, somes0 ∈S with P(s0) > 0, and a divergent sequence {k0t}t∈N such that |σi,k0t(s0)−σj,k0t(s0)|
is uniformly bounded away from zero. Since S is countable, there exists an enu-
meration s1, s2, . . . of S. Since A is a compact metric space, there exists a further subsequence {k1t}t∈N of {k0t}t∈N such that the sequence {σk1t(s1)}t∈N is convergent.
Likewise, there exists a further subsequence {k2t}t∈N of {k1t}t∈N such that the se-
quence {σk2t(s2)}t∈N is convergent, and by induction, for m ∈ N, there exists a fur-
ther subsequence{km+1,t}t∈N of {kmt}t∈N such that the sequence {σkm+1,t(sm+1)}t∈N
is convergent. Construct the sequence {lt}t∈N by letting lt = ktt. For all s ∈ S,
as t goes to infinity σlt(s) converges to some σ∞(s) ∈ A with σi,∞(s0) 6= σj,∞(s0).
With slight abuse of notation, define the measurable mapping σ∞ : Ω → A by let-
ting σ∞(ω) = σ∞(s(ω)). Since ui is continuous and A and Θ are compact, by the
dominated convergence theorem,
E ui(σlt, θ) →E ui(σ∞, θ). Define the Hσ
i,t-measurable mapping σi,t0 : Ω → Ai as follows: σi,01 = σi,1, σi,l0 t+1 =
σj,lt, and σ 0
i,τ =σi,τ0 −1 for allτ /∈ {1} ∪
S
t∈N{lt}. This mapping constitutes a feasible
strategy for player i according to which he imitates the actions chosen by player j
in periods {lt}. By construction, (σi,l0 t, σ−i,lt)→(σj,∞, σ−i,∞) for all ω ∈Ω. Thus,
Eui(σ0i,lt, σ−i,lt, θ)
→Eui(σj,∞, σ−i,∞, θ)
Since σ is an equilibrium, Eui(σlt, θ) ≥Eui(σ0i,lt, σ−i,lt, θ) for all t∈N, so E[ui(σi,∞, σ−i,∞, θ)]≥E[ui(σj,∞, σ−i,∞, θ)]. (4.1) By a similar argument, E[uj(σj,∞, σ−j,∞, θ)]≥E[uj(σi,∞, σ−j,∞, θ)]. (4.2)
Letu(ai;aj, a−ij, θ) denote the utility of a player in role iwhen he choosesai, player
j choosesaj, and other players choose a−ij. By the symmetry assumption, the payoff
of a player in rolej when player j choosesai, player ichooses aj, and others choose
a−ij is also equal to u(ai;aj, a−ij, θ). Equations (4.1) and (4.2) thus can be written
as
E[u(σi,∞;σj,∞, σ−ij,∞, θ)]≥E[u(σj,∞;σj,∞, σ−ij,∞, θ)],
E[u(σj,∞;σi,∞, σ−ij,∞, θ)]≥E[u(σi,∞;σi,∞, σ−ij,∞, θ)].
Summing the above equations,
E[u(σi,∞;σj,∞, σ−ij,∞, θ) +u(σj,∞;σi,∞, σ−ij,∞, θ)]≥
E[u(σi,∞;σi,∞, σ−ij,∞, θ) +u(σj,∞;σj,∞, σ−ij,∞, θ)]. (4.3)
On the other hand, since u is strictly supermodular, for all ai ∈Ai and aj ∈Aj,
u(ai;aj, a−ij, θ) +u(aj;ai, a−ij, θ)≤u(ai;ai, a−ij, θ) +u(aj;aj, a−ij, θ), (4.4)
σj,∞ for P-almost all ω, contradicting the assumption that σi,∞(s0)6= σj,∞(s0) and
P(s0)>0.
An immediate corollary of consensus in strategies is asymptotic consensus in payoffs.
Corollary 4.4. Let σ be a MPBE. For all i, j ∈ N, ui(σt, θ)−uj(σt, θ) → 0, P-
almost surely, as t goes to infinity.
Proof. Define σi,t : S → Ai as in the proof of Theorem 4.3. It is sufficient to show
that ui(σt(s), θ)−uj(σt(s), θ)→ 0 for all θ ∈Θ and s ∈S with P(s)>0. Suppose
to the contrary that there exists some neighboring i, j ∈ N, some θ0 ∈ Θ and
s0 ∈S withP(s0)>0, and a divergent sequence{k0t}t∈Nsuch that
ui(σk0t(s0), θ0)−
uj(σk0t(s0), θ0)
is uniformly bounded away from zero. As in the proof of Theorem 4.3,
we can construct a further subsequence {lt}t∈N of{k0t}t∈N such that for alls∈S, as
tgoes to infinity,σlt(s) converges to someσ∞(s)∈A. Furthermore, by Theorem 4.3,
σi,∞(s) = σj,∞(s) for all i, j ∈ N and s ∈ S. Therefore, since ui is continuous and
symmetric, ui(σlt(s0), θ0) −uj(σlt(s0), θ0) → 0 for all i, j ∈ N, contradicting the
assumption that ui(σk0t(s0), θ0)−uj(σk0t(s0), θ0)
is uniformly bounded away from
zero for some i, j.
The above result also implies ex ante consensus in the expectation of big players’ asymptotic payoffs. Prior to the start of the game, players in all roles expect their successors to asymptotically achieve similar payoffs. In Section 4.4.1, we show by means of an example that players might disagree in their conditional expected payoffs even when they are receiving the same payoffs.