8.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Blackwell's informativeness theorem | 2/2 | https://en.wikipedia.org/wiki/Blackwell's_informativeness_theorem | reference | science, encyclopedia | 2026-05-05T14:37:27.550245+00:00 | kb-cron |
=== Feasibility === A mixed strategy in the context of a decision-making problem is a function
α
:
S
→
Δ
A
{\displaystyle \alpha :S\rightarrow \Delta A}
which gives, for every signal
s
∈
S
{\displaystyle s\in S}
, a probability distribution
α
(
a
|
s
)
{\displaystyle \alpha (a|s)}
over possible actions in
A
{\displaystyle A}
. With the information structure
(
S
,
σ
)
{\displaystyle (S,\sigma )}
, a strategy
α
{\displaystyle \alpha }
induces a distribution over actions
α
σ
(
a
|
ω
)
{\displaystyle \alpha _{\sigma }(a|\omega )}
conditional on the state of the world
ω
{\displaystyle \omega }
, given by the mapping
ω
↦
α
σ
(
a
|
ω
)
=
∑
s
∈
S
α
(
a
|
s
)
σ
(
s
|
ω
)
∈
Δ
A
{\displaystyle \omega \mapsto \alpha _{\sigma }(a|\omega )=\sum _{s\in S}\alpha (a|s)\sigma (s|\omega )\in \Delta A}
That is,
α
σ
(
a
|
ω
)
{\displaystyle \alpha _{\sigma }(a|\omega )}
gives the probability of taking action
a
∈
A
{\displaystyle a\in A}
given that the state of the world is
ω
∈
Ω
{\displaystyle \omega \in \Omega }
under information structure
(
S
,
σ
)
{\displaystyle (S,\sigma )}
– notice that this is nothing but a convex combination of the
α
(
a
|
s
)
{\displaystyle \alpha (a|s)}
with weights
σ
(
s
|
ω
)
{\displaystyle \sigma (s|\omega )}
. We say that
α
σ
(
a
|
ω
)
{\displaystyle \alpha _{\sigma }(a|\omega )}
is a feasible strategy (or conditional probability over actions) under
(
S
,
σ
)
{\displaystyle (S,\sigma )}
. Given an information structure
(
S
,
σ
)
{\displaystyle (S,\sigma )}
, let
Φ
σ
=
{
α
σ
(
a
|
ω
)
{\displaystyle \Phi _{\sigma }=\{\alpha _{\sigma }(a|\omega )}
|
α
:
S
→
Δ
A
}
{\displaystyle \alpha :S\rightarrow \Delta A\}}
be the set of all conditional probabilities over actions (i.e., strategies) that are feasible under
(
S
,
σ
)
{\displaystyle (S,\sigma )}
. Given two information structures
(
S
,
σ
)
{\displaystyle (S,\sigma )}
and
(
S
,
σ
′
)
{\displaystyle (S,\sigma ')}
, we say that
σ
{\displaystyle \sigma }
yields a larger set of feasible strategies than
σ
′
{\displaystyle \sigma '}
if
Φ
σ
′
⊂
Φ
σ
{\displaystyle \Phi _{\sigma '}\subset \Phi _{\sigma }}
== Statement == Blackwell's theorem states that, given two information structures
σ
{\displaystyle \sigma }
and
σ
′
{\displaystyle \sigma '}
, the following are equivalent:
W
(
σ
′
)
≤
W
(
σ
)
{\displaystyle W(\sigma ')\leq W(\sigma )}
for any decision-making problem
(
Ω
,
A
,
u
,
p
)
{\displaystyle (\Omega ,A,u,p)}
: that is, the decision maker attains a higher expected utility under
σ
{\displaystyle \sigma }
than under
σ
′
{\displaystyle \sigma '}
for any utility function. There exists a stochastic map
γ
{\displaystyle \gamma }
such that
σ
′
(
s
′
|
w
)
=
∑
s
∈
S
γ
(
s
,
s
′
)
σ
(
s
|
w
)
{\displaystyle \sigma '(s'|w)=\sum _{s\in S}\gamma (s,s')\sigma (s|w)}
: that is,
σ
′
{\displaystyle \sigma '}
is a garbling of
σ
{\displaystyle \sigma }
.
Φ
σ
′
⊂
Φ
σ
{\displaystyle \Phi _{\sigma '}\subset \Phi _{\sigma }}
: that is,
σ
{\displaystyle \sigma }
yields a larger set of feasible strategies than
σ
′
{\displaystyle \sigma '}
.
== Blackwell order ==
=== Definition === Blackwell's theorem allows us to construct a partial order over information structures. We say that
σ
{\displaystyle \sigma }
is more informative in the sense of Blackwell (or simply Blackwell more informative) than
σ
′
{\displaystyle \sigma '}
if any (and therefore all) of the conditions of Blackwell's theorem holds, and write
σ
′
⪯
B
σ
{\displaystyle \sigma '\preceq _{B}\sigma }
. The order
⪯
B
{\displaystyle \preceq _{B}}
is not a complete one, and most experiments cannot be ranked by it. More specifically, it is a chain of the partially-ordered set of information structures.
=== Applications === The Blackwell order has many applications in decision theory and economics, in particular in contract theory. For example, if two information structures in a principal-agent model can be ranked in the Blackwell sense, then the more informative one is more efficient in the sense of inducing a smaller cost for second-best implementation.
== References ==