Econ 206

# Econ 206 - Memorandum to Students EC 206 Microeconomic...

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Memorandum to Students EC 206 Microeconomic Theory II Prof. Glenn C. Loury March 16, 2007 Students: I recognize that you have had a lot of material thrown at you lately, without proper motivation or organization. Because I don’t want our time and efforts to be wasted, I am providing this note to give an overview of the material. You should study it carefully. In our study of Game Theory, there were three examples I introduced in class that I wish now to emphasize. I will refer to these as: (1) the Reputation Game; (2) the Investment- Hiring Game (introduced at the end of last class); and (3) the Partnership Game (from the homework.) I will discuss the first two of these in this note. Analyzing the Partnership Game is one of this week’s homework problems. (1) The Reputation Game models the interaction between two antagonists, Bob and John. It is a dynamic game of incomplete information. John can be one of two types – soft or hard. Bob chooses whether to Attack or Not; John responds, choosing whether to Fight of Give. The payoffs are such that both types of John much prefer Not to be attacked. If Attacked, a soft John wants to Give, but a Hard John wants to fight. Also, Bob does not wants to Attack if he knows he will be Fought. So when this interaction between Bob and John occurs only once, Subgame Perfect Nash Equilibrium requires that, if Attacked, the soft type of John Gives and the hard type Fights. This leads to the conclusion that Bob Attacks only if the prior probability that John is hard is not too great. Notice that, in this case, incomplete information has no strategic consequences : Bob just computes the average of his payoffs over the two different terminal nodes that would be reached if he were to Attack, and decides on that basis. The two types of John react naturally. The game is trivial. But, when the same players interact two times in succession, the game becomes much more interesting. Now, John’s action in the first stage of the game potentially serves as a signal about his type , which affects the way the second stage is played. John knows this, Bob knows that John knows it, John knows that Bob knows that he knows…, etc. Given this structure, we have the following two preliminary results: Lemma 1: If avoiding Attack is important enough to John, then there can be no Bayesian Perfect Equilibrium of the “two-interactions-between-the-same-players” game in which the soft type of John Gives with probability 1 if Attacked at the first stage . There can also be no such equilibrium in which soft John Gives with probability zero if attacked at the first stage .

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Memo to Students, page 2 Proof: If soft John were always to Give when first attacked, then because hard John always Fights, John’s first stage action would be a perfect signal of his type: Fight-1 (at the first stage) would mean he’s hard, Give-1 would means he’s soft. Thus, Bob’s best response to this strategy is to Attack-2 (at the second stage) if and only if John was observed to Give-1. But, avoiding Attack is
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## This note was uploaded on 07/06/2009 for the course ECON 206 taught by Professor G.loury during the Spring '07 term at Brown.

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Econ 206 - Memorandum to Students EC 206 Microeconomic...

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