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
.