Alices Arcade and Bobs Boardgames (Inspired By Dutta Example 20.3)

Alice and Bob are back in business. They are playing the following pricing game. Alice and Bob can both choose either H, M, or L prices. Consumers may either view their products as substitutes = S or as complements = C. Alice is aware of s value, but Bob is not. Their payoffs are

Assume that Bob believes that P( = S) = p (0, 1).

1. What is a strategy for Bob in this game? For Alice? Does Alice have any strictly dominated strategies?

2. Are there any pure strategy Bayes-Nash equilibriums in which Bob plays H? If so, what are they, and under which values of p are they Bayes-Nash equilibriums?

3. Are there any pure strategy Bayes-Nash equilibriums in which Bob plays M? If so, what are they, and under which values of p are they Bayes-Nash equilibriums?

4. Are there any pure strategy Bayes-Nash equilibriums in which Bob plays L? If so, what are they, and under which values of p are they Bayes-Nash equilibriums?

5. Are there any mixed strategy Bayes-Nash equilibriums in this game? (Your answer may include either a mixed strategy BNE, or it may include an answer of Yes or No with a persuasive argument that references the ex-ante game matrix and a theorem we studied much earlier in the course...)

0 Bob Bob L L 5,5 | 6,3 || 10, 1 3,6 | 4,4 1,10 2,5 5,5 | 0,8| 0,6 8,0 | 4,4 0,6 L|6,0 6,3 3,3 Alice Alice 5,2 3,3 0 Bob Bob L L 5,5 | 6,3 || 10, 1 3,6 | 4,4 1,10 2,5 5,5 | 0,8| 0,6 8,0 | 4,4 0,6 L|6,0 6,3 3,3 Alice Alice 5,2 3,3