Representing
p
g games
g
Lectures in Game Theory
Fall 2011, Part 1
24.07.2011

G.B. Asheim, ECON3/4200-1
1
Game theory studies multi-person decision
problems, and analyzes agents that
 are rational (have well-defined preferences)
 reason strategically
i ll (take
( k into
i
account their
h i
knowledge and beliefs about what others do)

Applications
Industrial organization (incl. oligopoly theory)
Bargaining
g
g and auction theoryy
Labor market and financial economics
Macro economics
International economics
24.07.2011
G.B. Asheim, ECON3/4200-1
2
1
Classification

Non-cooperative game theory
Studies the outcome of individual actions in
a situation without external enforcement.

Contract and cooperative game theory.
Studies the outcome of joint actions in a
situation with external enforcement.
Seeks to develop solution concepts,
prescriptions or predictions about the outcomes of games
24.07.2011
G.B. Asheim, ECON3/4200-1
3
Major tensions of strategic interaction

The conflict between individual and group interests.

Strategic uncertainty.

The specter of inefficient coordination
24.07.2011
G.B. Asheim, ECON3/4200-1
4
2
Representing games
A game can be analyzed both in
the extensive form
and
the normal form.
Stay out
Entrant
IIncumbent
b t
Accept Fight
0, 2
Fight
- 1, - 1
Enter
Incumbent
Inc
mbent
Accept
24.07.2011
Enter 1, 1 - 1, - 1
Entrant
Stay out 0, 2 0, 2
1, 1
5
G.B. Asheim, ECON3/4200-1
The extensive form specifies



Players: {1, ... , i, ... , n}
What actions an actingg player
p y can choose
among, what an acting player knows.
Payoff for each of the players as a function
of the actions that are realized.
2 H
1
Decision node
(initial node)
24.07.2011
H
L
1, 2
1, 1
L 1
2 H 2, 1
Decision nodes L
Payoffs assigned
to players 1 and 2
at terminal nodes
0, 0
G.B. Asheim, ECON3/4200-1
6
3
Information sets
Dynamic
2 H 1, 2
game
H
1
L 1, 1
2 H 2, 1
L
L 0, 0
H
1
L
Static game
2 H 1, 2
L
2 H
1, 1
2, 1
L
0, 0
Definition : An informatio n set for player i is
a set of decision nodes that satisfies
 at all decision nodes in the info. set, player i moves,
 when the info. set is reached, i does not know which
of the set' s decision nodes has been reached.
24.07.2011
7
G.B. Asheim, ECON3/4200-1
Strategy
Definition : A strategy for player i is a plan of action that,
for each of i' s info. sets, specifies a feasible action.
2 H 1,
2 H 1,
1 2
1 2
H
H
1
1
L 1, 1
L 1, 1
2 H 2, 1
2 H 2, 1
L
L
L 0, 0
L 0, 0
HH HL LH LL
H 1, 2 1, 2 1, 1 1, 1
H
H 1, 2
L 2, 1 0, 0 2, 1 0, 0
L 2, 1 0, 0
24.07.2011
G.B. Asheim, ECON3/4200-1
L
1, 1
8
4
The normal form specifies
Players: {1, ... , i, ... , n}
 For each p
player, a strategy set: Si
 For each player, a payoff function: ui

G  ( S1 ,, Sn ; u1 ,, un )
Payoff for each player i depends on the strategy profile :
ui ( s1 , , sn )  ui ( si , si )
where, for all j , s j  S j ,
and where we write si  ( s1 , , si 1 , si 1 , , sn )
24.07.2011
9
G.B. Asheim, ECON3/4200-1
Classic normal form games

Matching Pennies

Coordination

Prisoners’ Dilemma

Pareto Coordination

Battle of the Sexes

Stag Hunt

Hawk-Dove/Chicken

Pigs
24.07.2011
G.B. Asheim, ECON3/4200-1
10
5
Application of the normal form
Stay out
Entrant
E t
Enter
Incumbent
Accept Fight
Enter 1, 1 - 1, - 1
Entrant
Fight
g - 1,
1 -1
Stay out 0, 2 0, 2
0, 2
Incumbent
Accept
1, 1
Fight
Stay out
Accept
Entrant
Enter
24.07.2011
Does a normal form represent dynamic interaction in an adequate way?

Or should a normal form only be used
for the analysis of static interaction?

Notice that different extensive forms
may have the same normal form.
0, 2
0, 2
Fight - 1, - 1
Incumbent
Accept

1, 1
G.B. Asheim, ECON3/4200-1
11
Beliefs, mixed strategies, and exp. utility

Strategic uncertainty (uncertainty about opponent
choice) leads to beliefs about opponent behavior.

If payoffs are von Neumann-Morgenstern utility, then:
Expected payoff for player i :
ui ( si ,  i )   s
i S i
 i ( si )ui ( si , si )
where  i is a p
prob. distr. over opponent
pp
str. p
profiles.

A mixed strategy is a probability distribution over the
player’s own strategies. Interpretation: (a) The player
randomizes. (b) Her opponents are uncertain.
24.07.2011
G.B. Asheim, ECON3/4200-1
12
6