Transcript GAME THEORY - Universidade Federal do Paran&#225

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