1. Games of Incomplete Information. These games drop the assumption that players know each other’s preferences. Complete info: players know each other’s preferences Incomplete info: some players do not know some of the other’s preferences (types) Although the analysis is more complicated, we still impose fair amount of structure: 1)Players have types 2)Probability distributions associated with types 3)An extra player, Nature, is included to resolve uncertainty about types Each player is typically assumed to observe his or her own preferences (or type) Assume a player’s preferences are determined by the realization of some random variable Other players don’t observe the realization of the RV but the ex ante probability distribution of the RV is assumed to be common knowledge

2. Probability distribution of types. Nature moves first in the extensive form game and determines players types

3. Consider a modified version of chicken: -player (1)’s payoffs are common knowledge (maybe he’s a famous chicken contestant) -player (2)’s payoffs are known only to (2) Assume (2) can be one of two types – mean or not If (2) is mean, - he hates to lose even more than he hates death If (2) is not mean, - he does not care about winning or not - he just does not want to die. - he is happiest when both turn. Chicken w/ Incomplete Information. 2121 turn don’t turn turn2, 31, 2 don’t turn 4, 20, 0 2121 turn don’t turn turn2, 01, 4 don’t turn 4, -10, 0

4. Game tree with probabilities. Now consider the probability distribution over types Suppose (2) is mean with probability μ and not mean with probability (1- μ) (1) doesn’t know whether (2) is mean or wimpy (2) knows he is wimpy but doesn’t know what (1) has done (2) knows he is mean, but doesn’t know what (1) has done

5. More contingencies. A strategy for (2) is now a contingent plan that takes natures move into account. (2)’s possible strategies are: turn if mean,turn if not turn if mean, not if not not if mean, turn if not not if mean, not if not (1)’s strategy set is just like in the case of complete information: Incomplete information adds contingencies to strategies - remember, that strategies describe what to do on and off the path

6. Bayesian Game.

7. Bayesian Nash Equilibrium.

8. How to compute a Bayesian NE.

9. Solving for the Bayesian NE. In the Incomplete Information Chicken Game If (2) is mean (2) has a strictly dominant strategy – don’t turn If (2) is not (2) has a strictly dominant strategy – to turn So (2)’s optimal strategy is – not if mean – turn if not mean Player (1)s best response depends on μ (μ is the probability that (2) is mean)