Representing games
Lectures in Game Theory
Spring 2011, Part 1
10.12.2010

G.B. Asheim, ECON4240-1
1
Game theory studies multi-person decision
problems, and analyzes agents that
 are rational (have well-defined preferences)
 reason strategically (take into account their
knowledge and beliefs about what others do)

Applications
Industrial organization (incl. oligopoly theory)
Bargaining and auction theory
Labor market and financial economics
Macro economics
International economics
10.12.2010
G.B. Asheim, ECON4240-1
2
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
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1
Major tensions of strategic interaction

The conflict between individual and group interests.

S
Strategic
i uncertainty.
i

The specter of inefficient coordination
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G.B. Asheim, ECON4240-1
Representing games
A game can be analyzed both in
and
the normal form.
the extensive form
Stay out
Entrant
0, 2
Fight
Enter
Incumbent
Accept
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Incumbent
Accept Fight
Enter 1,
- 1,
1 -1
1 1 - 1,
1 -1
Entrant
Stay out 0, 2 0, 2
1, 1
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G.B. Asheim, ECON4240-1
The extensive form specifies



Players: {1, ... , i, ... , n}
What actions an acting player can choose
among, what an acting player knows.
Payoff for each of the players as a function
off the
h actions that
h are realized.
l d
2 H
1
Decision node
(initial node)
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H
L
1, 2
L 1, 1
2 H 2, 1
Decision nodes L
Payoffs assigned
to players 1 and 2
at terminal nodes
0, 0
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2
Information sets
Dynamic
2 H
game
H
1
L
2 H
L
L
1, 2
1, 1
2, 1
H
1
L
0, 0
Static game
2 H 1, 2
L
2 H
1, 1
2, 1
L
0, 0
Definition : An information 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.
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G.B. Asheim, ECON4240-1
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
2 H 1, 2
H
H
1
1
L 1, 1
L 1, 1
2 H 2, 1
2 H 2,
2 1
L
L
0,
0
L 0, 0
L
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
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L
1, 1
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The normal form specifies
Players: {1, ... , i, ... , n}
For each 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 )
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Classic normal form games

Matching Pennies

Coordination

Prisoners’ Dilemma

Pareto Coordination

Battle of the Sexes

Stag Hunt

Hawk-Dove/Chicken

Pigs
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Application of the normal form
Stay out
Entrant
Enter
Incumbent
Accept Fight
Enter 1, 1 - 1, - 1
Fight - 1, - 1 Entrant
Stay out 0, 2 0, 2
0, 2
Incumbent
Accept
1, 1
Fight
Stay out
Accept
Entrant
Enter
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0, 2
Fight - 1, - 1
Incumbent
Accept

Does a normal form represent
p
dynay
mic 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
1, 1
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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
p
payoff
p
y for p
player
y i:
ui ( si ,  i )   s
i S i
 i ( si )ui ( si , si )
where  i is a prob. distr. over opponent str. profiles.

A mixed strategy is a probability distribution over the
player’s own strategies. Interpretation: (a) The player
randomizes. (b) Her opponents are uncertain.
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