In the mathematical subjects of information theory and decision theory, Blackwell's informativeness theorem is an important result related to the ranking of information structures, or experiments. It states that there is an equivalence between three possible rankings of information structures: one based in expected utility, one based in informativeness, and one based in feasibility. This ranking defines a partial order over information structures known as the Blackwell order, or Blackwell's criterion.[1] [2]
The theorem states equivalent conditions under which any expected utility maximizing decision maker prefers information structure
\sigma
\sigma'
A decision maker faces a set of possible states of the world
\Omega
A
\omega\in\Omega
a\inA
u(\omega,a)
\omega
p:\Omega → [0,1]
\sum\omegau(a,\omega)p(\omega)
Given such prior
p
a\inA
V(p)=\underset{a\inA}\operatorname{max}\sum\omegau(a,\omega)p(\omega)
We refer to the data
(\Omega,A,u,p)
An information structure (or an experiment) can be seen as way to improve on the utility given by the prior, in the sense of providing more information to the decision maker. Formally, an information structure is a tuple
(S,\sigma)
S
\sigma:\Omega → \DeltaS
\sigma(s|\omega)
s\inS
\omega
By observing the signal
s
\omega
\pi(\omega|s)=
p(\omega)\sigma(s|\omega) | |
\pi(s) |
where
\pi(s):=\sum\omega'p(\omega')\sigma(s|\omega')
s
(S,\sigma)
V(\pi,s)=\underset{a\inA}\operatorname{max}\sum\omegau(a,\omega)\pi(\omega|s)
and the "expected value of
(S,\sigma)
W(\sigma)=\sumsV(\pi,s)\pi(s)
If two information structures
(S,\sigma)
(S,\sigma')
\sigma
\sigma'
\sigma'
\sigma
S
\Gamma:S → S
\sigma'=\Gamma\sigma
Intuitively, garbling is a way of adding "noise" to an information structure, such that the garbled information structure is considered to be less informative.
A mixed strategy in the context of a decision making problem is a function
\alpha:S → \DeltaA
s\inS
\alpha(a|s)
A
(S,\sigma)
\alpha
\alpha\sigma(a|\omega)
\omega
\omega\mapsto\alpha\sigma(a|\omega)=\sums\alpha(a|s)\sigma(s|\omega)\in\DeltaA
That is,
\alpha\sigma(a|\omega)
a\inA
\omega\in\Omega
(S,\sigma)
\alpha(a|s)
\sigma(s|\omega)
\alpha\sigma(a|\omega)
(S,\sigma)
Given an information structure
(S,\sigma)
\Phi\sigma=\{\alpha\sigma(a|\omega)
\alpha:S → \DeltaA\}
be the set of all conditional probability overactions (i.e., strategies) that are feasible under
(S,\sigma)
Given two information structures
(S,\sigma)
(S,\sigma')
\sigma
\sigma'
\Phi\sigma'\subset\Phi\sigma
Blackwell's theorem states that, given any decision making problem
(\Omega,A,u,p)
\sigma
\sigma'
W(\sigma')\leqW(\sigma)
\sigma
\sigma'
\Gamma
\sigma'=\Gamma\sigma
\sigma'
\sigma
\Phi\sigma'\subset\Phi\sigma
\sigma
\sigma'
Blackwell's theorem allows us to construct a partial order over information structures. We say that
\sigma
\sigma'
\sigma'\preceqB\sigma
The order
\preceqB
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.[6] [7]