Updating Awareness and Information Aggregation

Appendix A

Proof of Proposition 1.

For the first claim, let j ≠ i be the trader who makes the announcement at t − 1 . By definition, S t − 1 i ( s * ) ∧ S t − 1 j ( s * ) < S t i ( s * ) ∧ S t − 1 j ( s * ) implies S t − 1 i ( s * ) < S t i ( s * ) and i updates her awareness at t.

Conversely, suppose that at t trader i updates her awareness, so that S t − 1 i ( s * ) < S t i ( s * ) ≡ S , but it is not the case that S t − 1 i ( s * ) ∧ S t − 1 j ( s * ) < S t i ( s * ) ′ S t − 1 j ( s * ) . Note that S t − 1 i ( s * ) < ¯   S   t i ( s * ) implies S t − 1 i ( s * ) ∧ S t − 1 j ( s * ) < ¯ S t i ( s * ) ∧ S t − 1 j ( s * ) . By the definition of a lattice, if A   < ¯   B but not A < B , then A = B . We therefore have S t − 1 i ( s * ) ∧ S t − 1 j ( s * ) = S t i ( s * ) ∧ S t − 1 j ( s * ) = S ∧ S t − 1 j ( s * ) ≡ S ′ .

We next show that P t − 1 j ( s S * ) = P t − 1 j ( s S ′ * ) . Since S ′ < ¯ S and from Projections Preserve Ignorance, we have S t − 1 j ( s S * ) ≻ ¯ S t − 1 j ( s S ′ * ) . Also, S t − 1 j ( s * ) ≻ ¯ S t − 1 j ( s S * ) and S ≻ ¯ S t − 1 j ( s S * ) imply S ′ = S t − 1 j ( s * ) ∧ S ≻ ¯ S t − 1 j ( s S * ) ∧ S = S t − 1 j ( s S * ) . Moreover, S ∧ S t − 1 j ( s * ) = S ′ implies that S t − 1 j ( s * ) ≻ ¯ S ′ . Projections Preserve Awareness implies that S t − 1 j ( s S ′ * ) = S ′ . However, S t − 1 j ( s S ′ * ) = S ′ ≻ ¯ S t − 1 j ( s S * ) , therefore S t − 1 j ( s S ′ * ) = S ′ = S t − 1 j ( s S * ) . Stationarity and S t − 1 j ( s S * ) = S t − 1 j ( s S ′ * ) = S ′ imply P t − 1 j ( s S * ) = P t − 1 j ( s S ′ * ) .

From Generalized Reflexivity, we have that s S t − 1 i ( s * ) * ∈ P t − 1 i ( s * ) = P 0 i ( s * ) S t − 1 i ( s * ) ∩ F t − 2 y ( S t − 1 i ( s * ) ) and s S t i ( s * ) * ∈ P t i ( s * ) = P 0 i ( s * ) S t i ( s * ) ∩ ​ F t − 1 y ( S t i ( s * ) ) . Moreover, i updates her awareness from t − 1 to t because ( P 0 i ( s ∗ ) ) S t − 1 i ( s ∗ ) ∩ ​ F t − 1 y ( S t − 1 i ( s ∗ ) ) = ϕ .

Because S ≻ ¯ S t − 1 i ( s * ) ≻ ¯ S ′ we have S t − 1 j ( s S * ) ≻ ¯ S t − 1 j ( s S t − 1 i ( s * ) * ) ≻ ¯ S t − 1 j ( s S ′ * ) . Hence, S t − 1 j ( s S * ) = S t − 1 j ( s S t − 1 i ( s * ) * ) = S t − 1 j ( s S ′ * ) and P t − 1 j ( s S * ) = P t − 1 j ( s S t − 1 i ( s * ) * ) = P t − 1 j ( s S ′ * ) . This implies that both s S t − 1 i ( s * ) * and s S t i ( s * ) * describe the same information and awareness about j. Since s S t i ( s * ) * ∈ F t − 1 y ( S t i ( s * ) ) , it means that s S t i ( s * ) * can rationalise announcement y t − 1 . But then this means that s S t − 1 i ( s * ) * can also rationalise y t − 1 , so that s S t − 1 i ( s * ) * ∈ F t − 1 y ( S t − 1 i ( s * ) ) , contradicting that P 0 i ( s * ) S t − 1 i ( s * ) ∩ F t − 1 y ( S t − 1 i ( s * ) ) = ϕ .

For the second claim, by construction j does not update her awareness at t, as she is the one making the announcement at t − 1 . If i also does not update her awareness, then the result is true. Suppose now that at time t, trader i updates her awareness, so that S t − 1 i ( s * ) < S t i ( s * ) ≡ S , but S t − 1 i ( s * ) ∨ S t − 1 j ( s * ) < S t i ( s * ) ∨ S t j ( s * ) . We will show that there exists S ′ such that S t − 1 i ( s * ) < ¯ S ′ < ¯ S , s S ′ * rationalizes j's announcement and S t − 1 i ( s * ) ∨ S t − 1 j ( s * ) = S ′ ∨ S t − 1 j ( s * ) . These three conditions imply that i updating her awareness to S t i ( s * ) is not minimal.

Define S ′ ≡ ( S ′ S t − 1 j ( s * ) ) ∨ S t − 1 i ( s * ) . Since S ≻ ¯ ( S ′ S t − 1 j ( s * ) ) and S ≻ ¯ S t − 1 i ( s * ) , we have S ≻ ¯ S ′ . It is also straightforward that S t − 1 i ( s * ) < ¯ S ′ . From Lemma 6.1 in Davey and Priestley (1990) we have S ′ = ( S ′ S t − 1 j ( s * ) ) ∨ S t − 1 i ( s * ) ≻ ¯ ( S ∨ S t − 1 i ( s * ) ) ∧ ( S t − 1 j ( s * ) ∨ S t − 1 i ( s * ) ) = S ′ ( S t − 1 j ( s * ) ∨ S t − 1 i ( s * ) ) ≻ ¯ S t − 1 j ( s * ) ∨ S t − 1 i ( s * ) . Hence, S ′ ∨ S t − 1 j ( s * ) ≻ ¯ S t − 1 i ( s * ) ∨ S t − 1 j ( s * ) . It is straightforward that S ′ ∨ S t − 1 j ( s * ) ≻ ¯ S t − 1 i ( s * ) ∨ S t − 1 j ( s * ) . These two relations imply that S ′ ∨ S t − 1 j ( s * ) = S t − 1 i ( s * ) ∨ S t − 1 j ( s * )

We next show that S t − 1 j ( s S * ) = S t − 1 j ( s S ′ * ) and P t − 1 j ( s S * ) = P t − 1 j ( s S ′ * ) . Since S ′ < ¯ S and from Projections Preserve Ignorance, S t − 1 j ( s S * ) ≻ ¯ S t − 1 j ( s S ′ * ) . Also, S t − 1 j ( s ∗ ) ≻ ¯ S t − 1 j ( s S ∗ ) implies S t − 1 j ( s * ) ∧ S ≻ ¯ S t − 1 j ( s S * ) ′ S = S t − 1 j ( s S * ) and S ′ = ( S t − 1 j ( s * ) ′ S ) ∨ S t − 1 i ( s * ) ≻ ¯ S t − 1 j ( s S * ) ∨ S t − 1 i ( s * ) ≻ ¯ S t − 1 j ( s S * ) . Again by Projections Preserve Ignorance, we have S t − 1 j ( s S ' * ) ≻ ¯ S t − 1 j ( s S t − 1 j ( s S * ) * ) = S t − 1 j ( s S * ) . The last equality holds from Generalized Reflexivity and Stationarity. Finally, Stationarity and S t − 1 j ( s S * ) = S t − 1 j ( s S ′ * ) imply P t − 1 j ( s S * ) = P t − 1 j ( s S ′ * ) .

The last equality implies that both s S * and s S ′ * describe the same information and awareness about j. Since s S * = s S t i ( s * ) * ∈ F t − 1 y ( S t i ( s * ) ) , it means that s S t i ( s * ) * can rationalise announcement y t − 1 . But then this means that s S ′ * can also rationalise y t − 1 , so that s S ′ * ∈ F t − 1 y ( S ′ ) . We know that F t − 1 y ( S t − 1 i ( s * ) ) ≠ ϕ , as it rationalizes all announcement up to y t − 2 . Because S t − 1 i ( s * ) < ¯ S ′ , we also have F t − 1 y ( S ′ ) ≠ ϕ , therefore contradicting that P 0 i ( s * ) S ′ ∩ F t − 1 y ( S ′ ) = ϕ and that updating to S t i ( s * ) = S is minimal.

Proof of Theorem 1.

Note that for all S and t, F t + 1 y ( S ) ⊆ F t y ( S ) . Although for some state space S and period t, we can have F t y ( S ) = ϕ , by construction for the full state space S ¯ and any t we have that F t y ( S ¯ ) ≠ ϕ . Since the collection of all states Σ is finite, there exists t such that F t y ( S ) = F t ′ y ( S ) for all S ∈ S and t ′ ≥ t .

Consider a state space S and time t such that F t ′ y ( S ) ≠ ϕ for all t ′ ≥ t and S ′ < S implies F t y ( S ′ ) = ϕ . Such S and t exist because of the finiteness of Σ , the fact that F t y ( S ¯ ) ≠ ϕ for all t and the fact that S is a lattice. Confinement and the fact that there is no less expressive state space than S imply that for all s ∈ F t y ( S ) and i ∈ I , P t i ( s ) ⊆ F t y ( S ) ⊆ S . That is, F t y ( S ) is partitioned by each P t i .

Note that each i announces the conditional expectation of X given her private information at s * and the public information at each t ' . Because the public information does not update after t, i repeats the same announcement y ′ for all t ′ ≥ t . However, since each state s ∈ F t ′ y ( S ) , t ′ ≥ t , is not excluded, it must be that i's announcement given each s ∈ F t ′ y ( S ) is also y ′ . We therefore have that E π ′ [ X | P t ′ i ( s ) ] = y ′ for all s ∈ F t ′ y ( S ) . Integrating over F t ′ y ( S ) , we have that E π ′ [ X ] = y ′ , where π ′ is the Bayesian update of the common generalized prior given the public information at each state space. Using the same arguments, it cannot be that another trader announces y ″ ≠ y ′ at each s ∈ F t ′ y ( S ) , because this would imply E π ′ [ X ] = y ″ , a contradiction. Hence, after t all traders agree on their announcement and y t = y t ′ for all t ′ ≥ t .

For the second claim, note that we have established in the proof of the first claim that for some t ′ , for all t ≥ t ′ , each trader i makes the same announcement y, where E π ′ [ X | P t i ( s 1 * ) ] = y for all s 1 * ∈ F t y ( S ¯ ) , and π ′ is the generalized prior which is the Bayesian update of π given F t y ( S ) for each S ∈ S t .

Suppose that the true state is s 1 * . If for all s * ∈ S ¯ with π ′ ( s * ) > 0 we have X ( s * ) = v , there is information aggregation at s 1 * . Suppose that for some s * ∈ S ¯ with π ′ ( s * ) > 0 we have X ( s * ) ≠ v = y . Since the security X is separable and setting y = v , condition (ii) of the definition of non-separability specifies that for some s 1 * ∈ F t y ( S ¯ ) , which is the support of π ′ on S ¯ , we have E π ′ [ X | P t i ( s 1 * ) ] ≠ v = y . But this contradicts the result of the previous paragraph, that all states in F t y ( S ¯ ) specify that all traders announce y = v .