NOTE-1034: Bergemann and Morris — Information design, a unified perspective

Contribution

A survey that puts Bayesian persuasion (one receiver, an informed sender), communication in games (many players, an uninformed mediator) and the authors’ robust predictions under one problem: what can be achieved by choosing players’ information alone, in a game with uncertain payoffs. Its tool is Bayes correlated equilibrium (BCE). Its distinctive emphasis is the metaphorical reading, in which no designer exists and the set of BCE is used by the analyst to bound what any information structure could produce. It also places the problem in Forges’s (1993) taxonomy of correlated equilibria with incomplete information and discusses where the revelation principle fails.

Key insight

Control over what people know is control over a polytope of outcomes, and the polytope is computable before anyone asks who controls it. “The information design problem can be solved in two steps. First, we can identify the set of outcomes that could be induced by the information designer. Second, we can identify which of these outcomes would be preferred.” The first step is the same whether the designer is a government, a platform, or nobody.

Assumptions

• Finite players, actions and states; a common full-support prior ψ shared by the designer and players. A basic game G = ((A_i, u_i), Θ, ψ) and a prior information structure S = ((T_i), π).
• Commitment: the designer commits to a rule before the state and types are realized. This separates it from cheap talk and signalling.
• Omniscience in the leading case: the rule may condition on the state and on every player’s type. §6 drops this (with and without elicitation).
• The designer picks the equilibrium. The revelation principle needs it. §7 drops it (maxmin design).
• All information structures are feasible, information is free, there is one designer, and the setting is static.
• Bayes Nash equilibrium as the solution concept.

Key results

• Definition 1, obedience. For each i, t_i, a_i and a′_i: Σ u_i((a_i, a₋ᵢ), θ) σ((a_i, a₋ᵢ) | t, θ) π(t | θ) ψ(θ) ≥ the same with a′_i. Linear in σ.
• Proposition 1 (revelation principle). An omniscient designer can attain σ iff σ is a BCE. The proof is cited from Bergemann and Morris (2016a); re-derived in THEORY-844.
• §3.1, one uninformed receiver (Kamenica and Gentzkow’s example). Invest pays −1 in B and x ∈ (0, 1) in G, equal prior; the designer maximizes investment. Obedience is p_G ≥ p_B/x, so the optimum is p_G = 1, p_B = x: “obfuscate the states of the world in order to maximize investment”. Re-derived.
• §3.2, an informed receiver with a signal correct with probability q: p_G ≥ q − (1 − q)/x, binding once q ≥ 1/(1 + x). Prior information shrinks the BCE set, and leaves it unchanged while the signal alone would not change the decision (at x = 0.55, q = 0.6 < 1/(1 + x) ≈ 0.645). Re-derived from the Appendix’s (29)–(30).
• §3.3, two firms with extra payoff ε if both invest. For ε > 0, a public signal: p_B = r_B = (x + ε)/(1 − ε). Confirmed by linear programming. For ε < 0 the paper assumes x > ½ and |ε| ≤ x − ½ and states p_B = x + ε with r_B = 0, private signals. That rule is infeasible under those assumptions: p_B = x + ε ≥ ½ forces r_B ≥ 2p_B − 1 > 0, and the table’s “not invest, not invest” entry 1 − 2x − 2ε is negative. The linear program gives p_B = x/(1 − 2ε) with r_B = 2p_B − 1 (at x = 0.55, ε = −0.03: 0.5189, not 0.52). The paper’s rule is right when x + ε ≤ ½. The qualitative point stands: the designer minimizes correlation under substitutes.
• §4.1. Public signals when the designer wants correlated actions, private when not, for intrinsic or instrumental reasons. Under complements, correlating play relaxes obedience. A benevolent designer may obfuscate, since players need not act in their joint interest. The “at least one invests” rule, each firm told to invest with probability ½ in B and both in G, is obedient for x > ½ (conditional payoff (2/3)x − 1/3 > 0, re-derived; the extracted text renders the expression as “23 x − 12”).
• §4.2. For one player, three orders coincide: Blackwell’s “more informative”, his sufficiency (garbling) order, and an incentive order, one experiment being more incentive-constrained if its BCE set is smaller in every decision problem. For many players, Bergemann and Morris (2016a)‘s individual sufficiency is equivalent to the incentive order, and it neither implies nor is implied by sufficiency of the joint information. Cited, not proved here (THEORY-853).
• §4.3. BCE and linear programming, against concavification (Aumann–Maschler; Kamenica–Gentzkow; Mathevet, Perego and Taneva for many players), with the limits of each.
• §5.1, price discrimination (Bergemann, Brooks and Morris 2015): every pair of consumer and producer surplus with consumer surplus ≥ 0, producer surplus ≥ uniform monopoly profit and total ≤ efficient surplus is attained by some segmentation. A three-value example attains three corners with one segmentation.
• §5.2, volatility (Bergemann, Heumann and Morris 2015): with signal s_i = λε_i + (1 − λ)θ and no strategic interaction, maximal aggregate volatility is (σ_θ + √(σ_θ² + σ_ε²))²/4 at λ* = σ_θ/(2σ_θ + √(σ_θ² + σ_ε²)) < ½, which stays σ_ε²/4 as aggregate shocks vanish. Checked numerically at four parameter pairs.
• §6. Without access to players’ types the designer is held to incentive compatibility with elicitation (Definition 2, Proposition 2) or publicly feasible obedience without it (Definitions 3–4, Proposition 3). Proposition 5: without elicitation is a subset of with. Proposition 6: for one player with two actions and two states they are equal. A third action, “invest small”, makes the sets strictly nested. These map onto Forges’s five correlated equilibria.
• §7. If the worst equilibrium is played the revelation principle fails. Maxmin design recovers Kajii and Morris’s robustness: in their example the designer can make both players invest with probability up to ε/(1 − x).

Claims

id	claim	strength	support
C1	The outcomes an omniscient designer can attain are exactly the BCE, a set given by linear obedience constraints	strong	Proposition 1, proved in the authors’ 2016 paper; re-derived in THEORY-844
C2	Information design unifies Bayesian persuasion, communication in games and robust prediction	strong as a classification	the definitions; Forges’s taxonomy mapped in §6.4
C3	More prior information shrinks the BCE set; for one player the incentive order is Blackwell’s, for many it is individual sufficiency	strong for one player, cited for many	§4.2, from Bergemann and Morris (2016a); THEORY-853
C4	Public signals are optimal when the designer wants correlated actions, private ones when it wants them uncorrelated	moderate	the two-firm example; cited general results (Arieli and Babichenko; Meyer); the substitutes case misstated
C5	A benevolent designer may obfuscate when there are many players	moderate	the spillover example in §4.1
C6	Read metaphorically, BCE give predictions robust to the information structure: price discrimination’s surplus triangle, aggregate volatility’s bound	strong for the cited results	§5, summarizing the authors’ papers; the volatility bound checked
C7	Without access to types the attainable set shrinks, and elicitation helps exactly beyond binary one-player problems	strong	Propositions 2, 3, 5 and 6, with proofs of 5 and 6

Concepts

• Information designer: commits to a rule mapping states and types to messages, cannot choose actions.
• Decision rule σ: T × Θ → Δ(A), action recommendations.
• Obedience; Bayes correlated equilibrium: Definition 1; a BCE is an obedient decision rule.
• Literal and metaphorical information design: a real designer against an analyst’s device for bounding outcomes across information structures.
• Omniscient; with and without elicitation: whether the designer can condition on players’ types directly, after asking, or not at all.
• Individual sufficiency: one structure is individually sufficient for another if some combination makes each player’s signal in the first a sufficient statistic for his beliefs about the state and the others’ signals in the second.
• Maxmin information design: the designer optimizes against the worst equilibrium.

Connections

It generalizes Kamenica and Gentzkow (2011), held as LIT-1021 and THEORY-404, which is its one-player case without prior information. It extends Aumann’s correlated equilibrium (1974, 1987) and Myerson’s communication in games (1982, 1991) to a designer with information of her own, and Forges (1993). It surveys Taneva (2015), Mathevet, Perego and Taneva (2017) and Kolotilin et al. (2017), and the authors’ own BCE papers. Blackwell (1951, 1953) is the one-player case of its ordering of information structures.

Bearing on the record

• CLAIM-404. Its first route, governing an organization “through the evidence its own procedures receive”, is information design in the literal reading, and Proposition 1 gives the exact reach of that route when the organization’s procedures are Bayesian: any obedient joint distribution of states and decisions, and no other. The claim’s independence of the two routes holds in this frame. Information design fixes the payoffs, so it cannot reach the claim’s second route, the criteria by which evidence is weighed. A designer who could change u_i would be a mechanism designer (§6.3). The record’s join.
• The metaphorical reading is the tool the record lacks for “what could any arrangement of information have produced here?”. The set of BCE bounds the outcomes of every information structure, so if an observed pattern lies outside it, no story about who knew what explains it.
• THEORY-404 and LIT-1021 are the one-player, no-prior-information case. This survey adds many receivers and prior information, and says prior information is what limits a persuader.
• CLAIM-199, CLAIM-326 and THEORY-156. §4.2’s incentive order is a third face of Blackwell’s order, beside “more informative” and garbling: a better-informed receiver is harder to steer, in every decision problem. For the record’s ceiling, a garbling shrinks what the receiver can do and enlarges what a designer can make it do.
• Reflexive control (A25, LIT-1339, unread) and Mostagir et al. (LIT-1318, Unavailable): A25’s link is the right one in kind. Reflexive control is designing an adversary’s information to induce a chosen decision, and §7’s maxmin design and the many-player BCE are the formal versions. The paper assumes Bayesian players with a common prior, which reflexive control does not.
• TERM-064. What a designer chooses is what becomes legible to the players. The paper studies Bayesian, unbounded players, so its legibility is Blackwell informativeness, not usable information.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• A survey. Its central results, Proposition 1, the §4.2 orderings and the §5 applications, are proved in the authors’ other papers.
• The two-firm substitutes example is misstated in the copy read (see Key results). Whether the JEL version corrects it was not checked.
• Bayesian, common-prior players throughout. Bounded or mistaken receivers are outside it, so are self-confirming loops of the record’s kind.
• Commitment. “the commitment assumption may be problematic in many settings”. Without it this is cheap talk, which is not covered.
• Equilibrium selection. The revelation principle needs the designer to pick the equilibrium. §7 shows what changes when it cannot.
• Static, one designer, free information. Dynamic, competing and costly design are cited only.

Open questions

• The authors’: a many-player concavification as useful as the one-player one; a local approach to valuable information; dynamic design for platforms and recommender systems.
• For the record: what is the attainable set when receivers are bounded, so that a garbling can raise a receiver’s usable information (THEORY-846)? Then a designer could steer by making some distinctions easier to use, not only by withholding them.