Or,
s
s
P (t)dt = P (s )
0 (s s)P (s )+
s
s
dt+
s
s
[P (t)P (s )]dt (3)
P (t)dt =
s
s
So note that (3) holds because P () is decreasing. The proof is similar for the
buyer. QED.
Lemma 2 (Mirrlees, Myerson)
An incentive compatible mechanism, < p, x >, is
private valuation and the bidding strategy of their opponent. Since we dont know
this, well integrate over what we dont know.
Dene inverse functions b = y (q ) and s = x(p). Eg, a seller who oers p has valuation
x(p).
Sellers Problem (note y (p) is the
1 1
.
1 1 2
Player 1 always accepts y or more.
Player 2 always oers: y = 1
Player 2 always accepts x or less.
Note that when 2 oers y, the payos are (y, 1 y ).
Note also that as i 0, ie player i gets more impatient, their own share of the pie
shrinks. I
6
Lecture 6: September 19, 2006
6.1
More on Finitely Repeated Games
Consider the game in G-6.1. The game has two pure strategy NE of (x1, x2) and
(z 1, z 2). When we have multiple NE of the stage game, the nitely repeated game can
have more equilibria th
Economics 703: Advanced Microeconomics
Peter Cramton
Matthew Chesnes
Updated: December 12, 2006
1
Lecture 1: August 31, 2006
1.1
Introduction
Denition: Game theory is the study of mathematical models of conict and cooperation between rational (expected u