Source
Bergemann and Morris (2019), LIT-1313, §4.2, read in NOTE-1034, citing Bergemann and Morris (2016a) for the proofs. Blackwell (1951, 1953) for the first two orders.
What was actually shown
In the survey, by statement only:
- For one player there is a “three way equivalence between (i) the ‘more informative’ ordering; (ii) the ‘sufficiency’ ordering; and (iii) the ‘incentive constrained’ ordering”. (i) ⇔ (ii) is Blackwell’s theorem.
- For many players, “individual sufficiency” is equivalent to the incentive order. One structure is individually sufficient for another if some combination of the two makes each player’s signal in the first a sufficient statistic for his beliefs about the state and the others’ signals in the second. It “neither implies nor is implied by” joint sufficiency.
- The value of information in games is ambiguous because more information widens what players can condition on and narrows the incentive-compatible set. Common-interest games isolate the first, BCE the second.
Re-derived here, one direction for one player. Let the prior structure S be a garbling of S′: a combined structure in which t is drawn from t′ by a kernel independent of θ. Take any σ′ obedient for S′ and define σ(a | t, θ) = Σ_{t′} σ′(a | t′, θ) P(t′ | t, θ). In the combined structure t is independent of (θ, a) given t′, so for the receiver of type t told a,
E[u(a, θ) − u(a′, θ) | t, a] = Σ_{t′} P(t′ | t, a) E[u(a, θ) − u(a′, θ) | t′, a] ≥ 0,
each term non-negative by obedience under S′. So σ is obedient under S and induces the same joint law of (θ, a). Every outcome attainable when the receiver knows more is attainable when it knows less.
What could have failed: a designer facing a less informed receiver could have been unable to replicate a recommendation that used the receiver’s finer information. Simulating t′ from (t, θ), which an omniscient designer can do, is what makes it possible.
What this does not say
- The converse is not checked. That a structure not more informative admits, in some decision problem, an outcome attainable under it but not under the other, is stated, not shown here.
- The many-player equivalence is not checked.
- Not for bounded receivers. For a receiver restricted to a rule class, a garbling can raise what it can use (THEORY-846), so the better-informed receiver need not be the harder one to steer.
- Not for designers who cannot condition on the prior information. The re-derivation simulates t′ from (t, θ).
Connections
- THEORY-156 is Blackwell’s (i) ⇔ (ii). This adds (iii).
- CLAIM-199 and CLAIM-326. In the record’s language: Blackwell’s ceiling bounds what a receiver can do with a rendering, and the incentive order says the same ceiling bounds, from the other side, how far a designer can push a receiver who holds it.
- THEORY-844 gives the set this orders.