GAME THEORY - Universidade Federal do Paraná
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Transcript GAME THEORY - Universidade Federal do Paraná
GAME THEORY
STRATEGIC
DECISION MAKING
Strategic (Normal)
Form Games
Static Games of
Complete and
Imperfect Information
What is a Normal Form Game?
A normal (strategic) form game consists of:
Players: list of players
Strategies: all actions available to all
players
Payoffs: a payoff assigned to every
contingency (every possible strategy
profile as the outcome of the game)
Prisoners’ Dilemma
Two suspects are caught and put in different
rooms (no communication). They are offered the
following deal:
– If both of you confess, you will both get 5 years in
prison (-5 payoff)
– If one of you confesses whereas the other does not
confess, you will get 0 (0 payoff) and 10 (-10 payoff)
years in prison respectively.
– If neither of you confess, you both will get 2 years in
prison (-2 payoff)
Easy to Read Format of
Prisoner’s Dilemma
Prisoner 2
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
Assumptions in Static Normal
Form Games
All players are rational.
Rationality is common knowledge.
Players move simultaneously. (They do not
know what the other player has chosen).
Players have complete but imperfect
information.
Solution of a Static Normal
Form Game
Equilibrium in strictly dominant strategies
– A strictly dominant strategy is the one that
yields the highest payoff compared to the
payoffs associated with all other strategies.
– Rational players will always play their strictly
dominant strategies.
Solution of a Static Normal
Form Game
Iterated elimination of strictly dominated
strategies
– Rational players will never play their
dominated strategies.
– Eliminating dominated strategies may solve
the game.
Solution of a Static Normal
Form Game (cont.)
Nash Equilibrium (NE):
– In equilibrium neither player has an incentive
to deviate from his/her strategy, given the
equilibrium strategies of rival players.
– Neither player can unilaterally change his/her
strategy and increase his/her payoff, given the
strategies of other players.
Definition of Nash Equilibrium
A strategy profile is a list (s1, s2, …, sn) of the
strategies each player is using.
If each strategy is a best response given the other
strategies in the profile, the profile is a Nash
equilibrium.
Why is this important?
– If we assume players are rational, they will play Nash
strategies.
– Even less-than-rational play will often converge to
Nash in repeated settings.
An Example of a Nash Equilibrium
Column
a
b
a
1,2
0,1
b
2,1
1,0
Row
(b,a) is a Nash equilibrium.
To prove this:
Given that column is playing a, row’s best response is b.
Given that row is playing b, column’s best response is a.
Finding Nash Equilibria –
Dominated Strategies
What to do when it’s not obvious what the
equilibrium is?
In some cases, we can eliminate dominated
strategies.
– These are strategies that are inferior for every
opponent action.
In the previous example, row = a is
dominated.
Example
A 3x3 example:
Column
a
b
57,42
c
a
73,25
66,32
b
80,26
35,12
32,54
c
28,27
63,31
54,29
Row
Example
A 3x3 example:
Column
a
b
57,42
c
a
73,25
66,32
b
80,26
35,12
32,54
c
28,27
63,31
54,29
Row
c dominates a for the column player
Example
A 3x3 example:
Column
a
b
57,42
c
a
73,25
66,32
b
80,26
35,12
32,54
c
28,27
63,31
54,29
Row
b is then dominated by both a and c for the row player.
Example
A 3x3 example:
Column
a
b
57,42
c
a
73,25
66,32
b
80,26
35,12
32,54
c
28,27
63,31
54,29
Row
Given this, b dominates c for the column player –
the column player will always play b.
Solution of Prisoners’ Dilemma
Dominant Strategy Equilibrium
Prisoner 2
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
Solution of Prisoners’ Dilemma
Iterated Elimination Procedure
Prisoner 2
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
Solution of Prisoners’ Dilemma
Cell-by-cell Inspection
Prisoner 2
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
NE of Prisoners’ Dilemma
The strategy profile {confess, confess} is
the unique pure strategy NE of the game.
In equilibrium both players get a payoff of
–5.
Inefficient equilibrium; (don’t confess,
don’t confess) yields higher payoffs for
both.
A Pricing Example
Firm 2
High
Price
Low
Price
High Price 100, 100 -10, 140
Low Price 140, -10
0, 0
3x3 Game
Using Iterated Elimination
Player 2
Left
Center
Right
Top
1, 0
1, 3
3, 0
Middle
0, 2
0, 1
3, 0
Bottom
0, 2
2, 4
5, 3
A Coordination Game
Battle of the Sexes
Husband
Opera
Movie
Opera
2, 1
0, 0
Movie
0, 0
1, 2
Battle of the Sexes:
After 30 Years of Marriage
Husband
Opera
Movie
Opera
3, 2
0, 0
Movie
0, 0
1, 2
Mixed strategies
Unfortunately, not every game has a pure strategy
equilibrium.
– Rock-paper-scissors
However, every game has a mixed strategy Nash
equilibrium.
Each action is assigned a probability of play.
Player is indifferent between actions, given these
probabilities.
Mixed Strategies
In many games (such as coordination games) a
player might not have a pure strategy.
Instead, optimizing payoff might require a
randomized strategy (also called a mixed strategy)
Wife
football
shopping
football
2,1
0,0
shopping
0,0
1,2
Husband
A Strictly Competitive Game
Matching Pennies
Player 2
Heads
Heads
Tails
Tails
1, -1
-1, 1
No NE in pure
strategies
-1, 1
1, -1
Extensive Form Games
Dynamic Games of
Complete and Perfect
Information
What is a Game Tree?
Player 1
Right
Left
Player 2
Player 2
A
B
C
D
P11
P12
P13
P14
P21
P22
P23
P24
An Advertising Example
Migros
Normal
Aggressive
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680
730
700
800
-50
0
400
0
Assumptions in Dynamic
Extensive Form Games
All players are rational.
Rationality is common knowledge
Players move sequentially. (Therefore, also
called sequential games)
Players have complete and perfect information
– Players can see the full game tree including the
payoffs
– Players can observe and recall all previous moves
Solution of an Extensive Form
Game
Subgame Perfect Equilibrium: For an
equilibrium to be subgame perfect, it has to
be a NE for all the subgames as well as for
the entire game.
– A subgame is a decision node from the
original game along with the decision nodes
and end nodes.
– Backward induction is used to find SPE
Advertising Example:
3 proper subgames
Migros
Wal-Mart
Wal-Mart
680
730
700
800
-50
0
400
0
Solution of the Advertising
Game
Subgame 1
Subgame 2
Wal-Mart
Wal-Mart
Enter
Stay out
Enter
Stay out
680
730
700
800
-50
0
400
0
Solution of the Advertising
Game (cont.)
Migros
Aggressive
Normal
730
700
0
400
SPE of the game is the strategy profile:
{aggressive, (stay out, enter)}
Properties of SPE
The outcome that is selected by the
backward induction procedure is always a
NE of the game with perfect information.
SPE is a stronger equilibrium concept than
NE
SPE eliminates NE that involve incredible
threats.
Suppose WM threatens to enter no
matter what Migros does. Is this a
credible threat?
Migros
Normal
Aggressive
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680
730
700
800
-50
0
400
0