6
The Principle of Optimality
Denition 1 A T shot deviation from a strategy si is a strategy si such that there exists
b
T such that
for all ht 2 H with t
T:
si (ht ) = si (ht )
b
Denition 2 A one-shot deviation from a strategy si is a strategy si such th
11
Global Games
Common complaint about Nash equilirium is that there is an extreme amount of coordination built into the notion of equilibrium.
Global games is a device to model situations where coordination is di cult (and where
higher order beliefs matt
12
Adverse Selection and Insurance; The Case with a
Monopoly
Suppose that there are two typesof consumers. Call them fH; Lg
Type H has a probability of an accident given by
Type L has a probability of an accident gives by
H
L
For both types, the endowment
13
Social Choice
The fundamental question in social choice theory is the following: Given a collection of agents with
preferences over dierent alternatives ( allocations, outcomes), how should society evaluate these
alternatives. Put dierently, are there
14
Mechanism Design
The basic point with mechanism design is that it allows a distinction between the underlying
economic environment and the rules of the game We will take as given some set of possible
.
outcomes (alternatives, allocations) and some pref
10
Dynamic Games of Incomple Information
DRAW SELTEN HORSE
S
In game above there are no subgames. Hence, the set of subgame perfect equilibria
and the set of Nash equilibria coincide.
However, the sort of logic that led us to consider subgame perfection a
9
Games of Incomplete Information
9.1
Cournot Duopoly With Unknown Costs
Assume that two rms produce a homogenous good with inverse demand P (Q) = max f2
Q; 0g ;
where Q is interpreted as the total quantity of the good on the market. Also, assume that
the
1
Games on Normal Form
1.1
Examples of Normal Form Games
A Game in Normal (sometimes strategic) form consists of:
1. A set of players I; which we assume is a nite set I = f1; :; ng
2. A strategy space S =
n
i=1 Si ;
where Si is the strategy space for agen
2
Extensive Form Games
Denition 1 A nite extensive form game is an object
K = fI; (T; ) ; P; A; H; u; g ;
where
I = f0; 1; :; ng is the set of agents (0 denotes nature
)
(T; ) is the game tree
P is the player partitioning
A is the set of actions
H is the
3
Alternating Oers Bargaining
Consider the following setup following Rubinstein (1982) extension of a model by Stahl
s
(1972):
Two players bargain over a pie with size normalized to 1.
Given a division (x1 ; x2 ) with x1
0; x2
0 and x1 + x2
1; the instant
8
Mixed Strategies
Rock
Scissors
Paper
Rock
0,0
1,-1
-1,1
Scissors
1,-1
0,0
1,-1
Paper
1,-1
-1,1
0,0
Figure 1: A Game with no Pure Strategy Nash Equilibrium
Clearly, there is no pure strategy equilibrium in Rock, Scissors, and Paper
Denition 1 Let G = (n;
15
Optimal Auctions with Private Values and Independent Types
15.1
Set-up
A seller seeks to sell a single indivisible good. There are n potential buyers (aka bidders We
).
denote the set of of bidders I = f1; :; ng : Bidder i is equipped with a von Neuman