Knowledge can be represented by sigma-algebras

Example of information

• Suppose I tell you the denomination of the card? Any takers?
• Suppose I tell you the color (red/black) of the card? Any takers?
• Detail: The cards are the 4 aces and 6 other clubs.
• This is called an option. Payment is max(E(Win|know)-.3,0).
• We need to konw what the expected value of this option is to decide.
• Three cases:
  • trivial knowledge: E(Win|know) = E(win). Option worth zero.
  • know Number: E(Win|know) = .25 or 0. Option worth zero.
  • know color: E(Win|know) = .5 or 0. Option worth 4 cents.
• So what you know matters.
• Options of this sort are studied in finance

Example of two players

• Suppose 3 cards are in a "deck" (2 black and 1 red)
• Each of two players get one card.
• Call the remaining card the "kitty"
• Computing some probabilities:
  • What is the expected probability that the kitty is black?
  • Conditional on first player seeing his card? (100%/50%)
• Learning from actions:
  • Suppose the second player sees the first player bet that the kitty will be black?
  • What is her conditional probability that the kitty is black?
  • Suppose the second player sees the first player NOT bet that the kitty will be black?
  • What is her conditional probability that the kitty is black?
  • So the second player then KNOWS the color of the card.
• Knowledge about actions
  • What is "her" probability that "he" will bet on black?

Representing all of this

• Each players knowledge is represented by a sigma-field
• Sample space contains 6 items: draw it.
• Draw each sigma-field.
• After we see the first player bet, the sigma-field is updated.