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Lectures in Microeconomic Theory
Fall 2010, Part 19
G.B. Asheim, ECON4230-35, #19 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)
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Industrial organization (incl. oligopoly theory)
Bargaining and auction theory
Labor market and financial economics
Macro economics
International economics
G.B. Asheim, ECON4230-35, #19 2
The players choose strategies .
Each player’s payoff depends on the strategy profile.
1
2
Demand for self
1 Demand for the
3 other
Demand 1 for self 1, 1 4, 0
Demand for the
3 other
0, 4 3, 3
Prisoners’ dilemma
If the players do not choose strictly dominated strategies, both will demand 1 for themselves.
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G.B. Asheim, ECON4230-35, #19 3
1

Solving games through iterated elimination of strongly dominated strategies
1, 1
0, 0
1, 0
0, 1
The game is solved through iter. elimi. of str. dom. strat.
 1 chooses 1 does not choose a str dom.
2
2 believes does sexes not that 1 does not choose
Battle of the
T
B
2, 1
0, 0 choose a str.
a str.
 dom.
0,
1, r strat.
0
2
 dom.
strat.
2 chooses 
&
No strategy is str. dominated. for any of the players!
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G.B. Asheim, ECON4230-35, #19 4
Are there strategies for the two players so that no player will regret his own choice when being told of the other player’s choice?
If yes, then such a strategy profile is a
Nash equilibrium .
Battle of the sexes 2, 1 0, 0
0, 0 1, 2
( T ,  ) and ( B , r ) are Nash equilibria.
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G.B. Asheim, ECON4230-35, #19 5



Players : {1, ... , i , ... , n }
For each player, a strategy set : S i
For each player, a payoff function : u i and where
G

( S
1
,  , S n
; ; u
1
,  , u n
Payoff for each player i : u i
( s
1
,
 where,
, s for n
)
 all j u i
, s
( s i j
, s
 i

S j
)
, we write s
 i

( s
1
,

, s i

1
, s i

1
,

, s n
)
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G.B. Asheim, ECON4230-35, #19 6
2

Definition u i
( s i

, s
 i
)
 u
: s i
 is strongly dominated i
( s i

, s
 i
) for all s
 i
.
by s i
 if
Iterated elimination of str. dom. strategies requires that it is commonly believed that players are rational.
Definition : ( s
1

,

, s n

) is a Nash equilibriu m
G it

( S
1
,

, S n
; u
1
,

, u n
) if, for each player i holds that u i
( s i

, s

 i
)
 u i
( s i
, s

 i
) for all s i
.
for
,
Nash equilibrium requires that players are rational and have correct conjectures conc. the opponents’ choices.
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G.B. Asheim, ECON4230-35, #19 7
1, 0
0, 3
1, 2
0, 1
0, 1
2, 0
Iterated elimination of strongly dominated strategies works well in some examples.
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0, 4
4, 0
3, 5
4, 0 5, 3
0, 4 5, 3
3, 5 6, 6 strategy is str. dom.
Even in games w/only one Nash equil., incorr. conjectures can be held.
G.B. Asheim, ECON4230-35, #19 8
Monitoring (of traffic)
Legal
Monitoring
1, 0
Illegal
1, -
I
Player 2’s best response fn monitoring
0, 0 3, 1
1
3
Player 1’s best response fn
Such games have a
Nash equilibrium in mixed strategies.
L
N 1
5
M
Interpretation as a steady state.
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G.B. Asheim, ECON4230-35, #19 9
3

Will the players coordinate, and if yes, on which

A game of trust equilibrium?
Each of you is to choose 1 or 2 indep. of each other.
A player who chooses 1 , receives 100 kr from UofO.
A player who chooses 2 , receives 200 kr from UofO if all others also choose 2 , but must pay 500 kr if at least one other player chooses 1 .
Write 1 or 2 on a piece of paper!
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G.B. Asheim, ECON4230-35, #19 10
A game of trust 1
2
Choose 1 Choose 2
Choose 1 100, 100
Choose 2
110 100 200
Will a guarantee help?
Conclusion : Some games (e.g., the battle of the sexes and the game of trust) have multiple Nash equilibria.
Will the players coordinate on a particular Nash equil.?
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G.B. Asheim, ECON4230-35, #19 11
In some games the Nash equilibria require that the players choose differently.
Chicken 2
1
Not R & D
R
Not
& D
2, 2
R & D
0, 3
R & D
3, 0 1, 1
Will the players coordinate on a particular Nash equil.?
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G.B. Asheim, ECON4230-35, #19 12
4