Knowledge part 3: Game theory

Web links about last lecture

• see here for a gruesome description.
• see here for a formal discussion from a philosophers perspective.
• hats!

The common knowledge sigma-fields

• the intersection of two sigma fields
• Represents everything BOTH people know

The cheating sigma-fields

• The union of two sigma

The god sigma-fields

• The algebra where everything has to be mesasureable

A game matrix

• Game of chicken
• Newlywed game
• Prisinors dilemma game

Rationality (or greedy)

• One player rationality:
  1. Pick the best action (not good enough--best action might change)
  2. Pick the best action possible (too good, they might not know which is best)
  3. Pick the best action given what they know (just right)
• Same thing as symbols: (X = random variable representing the action choosen. x is another possible action.)
  1. E(U(X)) >= E(U(x))
  2. U(X) >= U(x)
  3. E(U(X)|KnowledgeX) >= E(U(x)|KnowledgeX)
• Two players: Each is rational
  • E(U(X,Y)|KnowledgeX) >= E(U(x,Y)|KnowledgeX)
  • E(U(X,Y)|KnowledgeX) >= E(U(x,Y)|KnowledgeX)

Example distributions for PD

• Equal probabilities in PD
• Any probabilities in PD
• Concept: called dominance
• Can use U(X) >= U(x) definition of rationality

Example distributions for newlywed game

• Try equal probabilities (one row is rational other isn't)
• Try 50/50 on diagonal (each row is rational)
• Probably the "correct" solution to this game

Overall model

• Each player has his/her own sigma-algebra
• There is a common knowledge sigma-algebra
• there is a "cheatting" sigma-algebra
• There is the mother sigma-algebra
• Nash assumed:
  • Two primary algebras are independent
  • common knolwdege is trivial
  • Cheat = mother
• Mixed Nash drops common knowlwdge is trivial
• CE drops independence of two primary algebras
• Bayesian drops Cheat = mother
• These issues have lead to 30 years of fighting