Source
Bergemann and Morris (2019), Journal of Economic Literature 57(1), LIT-1313, read in NOTE-1034 from Cowles Discussion Paper 2075R3: Definition 1, Proposition 1 (whose proof the survey cites from Bergemann and Morris 2016a) and §§3.1–3.3. The example is Kamenica and Gentzkow’s (2011), LIT-1021. The re-derivations are the record’s.
What was actually shown
A decision rule σ(a | t, θ) is obedient if for every player i, type t_i, recommended a_i and alternative a′_i,
Σ_{a₋ᵢ, t₋ᵢ, θ} [u_i((a_i, a₋ᵢ), θ) − u_i((a′_i, a₋ᵢ), θ)] σ((a_i, a₋ᵢ) | t, θ) π(t | θ) ψ(θ) ≥ 0.
If. Let the designer draw a ∼ σ(· | t, θ) and tell each player a_i privately. Given that the others obey, player i of type t_i told a_i compares a_i with a′_i using exactly the conditional law in the inequality. So obeying is a Bayes Nash equilibrium that induces σ.
Only if. Let a communication rule φ(m | t, θ) and an equilibrium b induce σ(a | t, θ) = Σ_m φ(m | t, θ) Π_j b_j(a_j | t_j, m_j). Wherever b_i(a_i | t_i, m_i) > 0, a_i is a best reply to (t_i, m_i). Multiply that inequality by b_i(a_i | t_i, m_i) and sum over m_i. The result is the obedience inequality for (t_i, a_i): relabelling messages by the actions they lead to loses no incentive.
The constraints are linear in σ, so the set is a polytope, and maximizing any designer objective V(σ) = Σ v(a, θ) σ π ψ over it is a linear program.
The example. One receiver, states B and G equally likely, invest pays −1 in B and x ∈ (0, 1) in G, not investing pays 0. A decision rule is (p_B, p_G), the investment probabilities. Obedience after “invest” is ½p_B(−1) + ½p_G x ≥ 0, that is p_G ≥ p_B/x. After “not invest” the constraint is slack because x < 1. So the most investment is p_G = 1, p_B = x: invest is recommended in the good state always and in the bad state with probability x, which leaves the receiver exactly indifferent. Total investment (1 + x)/2, against 0 without information and ½ with full information.
Prior information. If the receiver sees a signal correct with probability q, the “not invest” constraint for the good signal binds once q ≥ 1/(1 + x), and with p_B = 0 it gives p_G ≥ q − (1 − q)/x, which rises to 1 as q → 1. Below 1/(1 + x) the set is the uninformed one. So prior information shrinks what the designer can do, and only once it would change the receiver’s decision by itself.
Two receivers, with ε added if both invest, solved as a linear program over the full joint rule. For ε > 0 the optimum is public: p_B = r_B = (x + ε)/(1 − ε) (0.7222 at x = 0.55, ε = 0.1, as the paper says). For ε < 0 it minimizes joint investment in B. The paper’s p_B = x + ε, r_B = 0 is right when x + ε ≤ ½. For x + ε > ½, which the paper’s own assumption x > ½, |ε| ≤ x − ½ gives, that rule is not a probability distribution, and the optimum is p_B = x/(1 − 2ε) with r_B = 2p_B − 1.
What could have failed: an indirect rule with richer messages could have attained outcomes no recommendation rule can. The “only if” direction shows it cannot, as long as the designer may pick the equilibrium.
What this does not say
- Not when the worst equilibrium is played. The revelation principle then fails, and the attainable set is smaller (§7).
- Not without commitment. A designer who chooses messages after seeing the state is in cheap talk, which this does not cover.
- Not for designers who cannot condition on players’ types. §6 gives the smaller sets, with and without elicitation.
- Not for bounded or non-Bayesian receivers. Obedience is computed with Bayes’s rule and a common prior.
- Not that a designer exists. Read metaphorically the set bounds all information structures, with no designer.
Connections
- THEORY-404 is the one-receiver, no-prior case: concavification and BCE give the same optimum in the example.
- CLAIM-404. Its evidential route has exactly this reach for Bayesian procedures. Its evaluative route, changing how evidence is weighed, changes u_i, which information design holds fixed.
- THEORY-853 orders information structures by the size of this set.