Pure knowledge: When the conditional probability is zero or one

Definition of E(X|F)

• We know that E(X|F) is a random variable with the following two properties:
  • Further E(X|F) must be measurable with respects to F
  • Further if I is an indicator of a set in F, then E(IX) = E(IE(X|F))
• These two properties define a conditional expectation

Practical definition of E(X|F)?

• Find a partition that generates F
• Compute E(X|S) for each S in F

Relative amounts of knowledge

• If A is contained in B, then B knows more.
• In other words, from the random variable E(X|B) we can compute E(X|A)
• But not the other way around
• Note: E(E(X|B)|A) = E(X|A) and E(E(X|A)|B) = E(X|A).
• So in sequential expectations it is the weaker one that wins.

The common knowledge sigma-fields

• the intersection of two sigma fields
• Represents everything BOTH people know
• Also represents E(E(E(E(E(E(E(...E(X|A)....B)A)B)A)B)A)B)

The cheating sigma-fields

• The union of two sigma

Baby hats

• at least 1 person has a red hat. (start with 3 people--give 1 a red everyone else another color.)
• Do you konw if your hat is red? (Yes, IF everyone else doesn't have a red hat).
• After I ask the red-hat person, even the other people can figure out their hat color
• Notice: No probability. Only sigma-algebras.

More hats

• at least 1 person has a red hat.
• Try 2 red
• Try 3 red

Even more hats--One for everyone

• at least 1 person has a red hat.
• Try 1 red out of all hats
• Try 5 red

The unfaithful wifes

• see here for a gruesome description.
• see here for a formal discussion from a philosophers perspective.
• Simple version
  • 20 couples in town. It is traditional to set a husbands shoes out in the street if he is found to be unfaithful. This constitutes a divorce.
  • Every woman knows the status of everyone else's husband, but not their own.
  • This is common knowledge (what pray tell does that mean?)
• Easy case 1:
  • Chief (who knows everything) says, 1 of your husbands have been unfaithful.
  • The next morning that woman sets out her husbands shoes
  • (identical to the "one red hat" case previously discussed)
• easy case 2:
  • Chief says, 2 of your husbands have been unfaithful.
  • The next morning two women set out their husbands
  • (identical to the "two red hats")
• medium case:
  • Chief says, one or two of your your husbands have been unfaithful.
  • The first morning every, anyone who knows everyone else in not faithful will set out her husband. If this doesn't occur, then we know that she sees at least one unfaithful husband.
  • The second morning, if no one has set out their husband, both wifes figure it out and divorce them.

hats!