---
title: "Blackwell's informativeness theorem"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Blackwell's_informativeness_theorem"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:37:27.550245+00:00"
instance: "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 ==