Download Extensive Form Games and Subgame Perfection

Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12
February 15, 2007
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 1
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Lecture Overview
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 2
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Example: the sharing game
q1
HH
2–0
2
q A
no AA yes
q
(0,0)
A
Aq
(2,0)
Extensive Form Games and Subgame Perfection
1–1
HH
H
q2
A
no AA yes
A
q
Aq
(0,0)
(1,1)
0–2
HH
H
HHq 2
A
no AA yes
A
Aq
q
(0,0)
(0,2)
ISCI 330 Lecture 12, Slide 3
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Example: the sharing game
q1
HH
2–0
2
q A
no AA yes
q
(0,0)
A
Aq
(2,0)
1–1
HH
H
q2
A
no AA yes
A
q
Aq
(0,0)
(1,1)
0–2
HH
H
HHq 2
A
no AA yes
A
Aq
q
(0,0)
(0,2)
Play as a fun game, dividing 100 dollar coins. (Play each partner
only once.)
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 3
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Lecture Overview
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 4
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Pure Strategies
I
In the sharing game (splitting 2 coins) how many pure
strategies does each player have?
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 5
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Pure Strategies
I
In the sharing game (splitting 2 coins) how many pure
strategies does each player have?
I
player 1: 3; player 2: 8
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 5
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Pure Strategies
I
In the sharing game (splitting 2 coins) how many pure
strategies does each player have?
I
I
player 1: 3; player 2: 8
Overall, a pure strategy for a player in a perfect-information
game is a complete specification of which deterministic action
to take at every node belonging to that player.
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 5
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Pure Strategies
I
In the sharing game (splitting 2 coins) how many pure
strategies does each player have?
I
player 1: 3; player 2: 8
I
Overall, a pure strategy for a player in a perfect-information
game is a complete specification of which deterministic action
to take at every node belonging to that player.
I
Can think of a strategy as a complete set of instructions for a
proxy who will play for the player in their abscence
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 5
Recap
at each
Perfect-Information
Extensive-Form
Perfection
choice node, regardless
of whether
or not itGames
is possible to reach thatSubgame
node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
Pure Strategies Example
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
H
(1,0)
A perfect-information game in extensive form.
What are the pure strategies for player 2?
In order to define a complete strategy for this game, each of the players must choose
an action at each of his two choice nodes. Thus we can enumerate the pure strategies
of the players as follows.
S1 = {(A, G), (A, H), (B, G), (B, H)}
S2 = {(C, E), (C, F ), (D, E), (D, F )}
It isForm
important
toSubgame
note that
we have to include the strategies (A, G) and
even
Extensive
Games and
Perfection
ISCI(A,
330H),
Lecture
12, Slide 6
Recap
at each
Perfect-Information
Extensive-Form
Perfection
choice node, regardless
of whether
or not itGames
is possible to reach thatSubgame
node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
Pure Strategies Example
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
H
(1,0)
A perfect-information game in extensive form.
What are the pure strategies for player 2?
InIorder
complete
game,
S2 to
=define
{(C,aE);
(C, Fstrategy
); (D, for
E);this
(D,
F )}each of the players must choose
an action at each of his two choice nodes. Thus we can enumerate the pure strategies
of the players as follows.
S1 = {(A, G), (A, H), (B, G), (B, H)}
S2 = {(C, E), (C, F ), (D, E), (D, F )}
It isForm
important
toSubgame
note that
we have to include the strategies (A, G) and
even
Extensive
Games and
Perfection
ISCI(A,
330H),
Lecture
12, Slide 6
Recap
at each
Perfect-Information
Extensive-Form
Perfection
choice node, regardless
of whether
or not itGames
is possible to reach thatSubgame
node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
Pure Strategies Example
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
H
(1,0)
A perfect-information game in extensive form.
What are the pure strategies for player 2?
InIorder
complete
game,
S2 to
=define
{(C,aE);
(C, Fstrategy
); (D, for
E);this
(D,
F )}each of the players must choose
an action at each of his two choice nodes. Thus we can enumerate the pure strategies
What are the pure strategies for player 1?
of the players as follows.
S1 = {(A, G), (A, H), (B, G), (B, H)}
S2 = {(C, E), (C, F ), (D, E), (D, F )}
It isForm
important
toSubgame
note that
we have to include the strategies (A, G) and
even
Extensive
Games and
Perfection
ISCI(A,
330H),
Lecture
12, Slide 6
Recap
at each
Perfect-Information
Extensive-Form
Perfection
choice node, regardless
of whether
or not itGames
is possible to reach thatSubgame
node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
Pure Strategies Example
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
H
(1,0)
A perfect-information game in extensive form.
What are the pure strategies for player 2?
InIorder
complete
game,
S2 to
=define
{(C,aE);
(C, Fstrategy
); (D, for
E);this
(D,
F )}each of the players must choose
an action at each of his two choice nodes. Thus we can enumerate the pure strategies
What are the pure strategies for player 1?
of the players as follows.
I S1 = {(B, G); (B, H), (A, G), (A, H)}
S1 = {(A, G), (A, H), (B, G), (B, H)}
I This is true even though, conditional on taking A, the choice
S2 = {(C, E), (C, F ), (D, E), (D, F )}
between G and H will never have to be made
It isForm
important
toSubgame
note that
we have
Extensive
Games and
Perfection
to include the strategies (A, G) and
even
ISCI(A,
330H),
Lecture
12, Slide 6
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Nash Equilibria
Given our new definition of pure strategy, we are able to reuse our
old definitions of:
I
mixed strategies
I
best response
I
Nash equilibrium
Theorem
Every perfect information game in extensive form has a PSNE
This is easy to see, since the players move sequentially.
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 7
o
q
,0)
yes
no
Recap
A
A
q
(0,0)
Aq
yes
no
yes Games
APerfect-InformationExtensive-Form
A
A
A
Aq
q
Aq
(1,1)
(0,0)
(0,2)
Subgame Perfection
(2,0)
Induced
Normal Form
Figure 5.1 The Sharing game.
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
H
(1,0)
A perfect-information game in extensive form.
o define a complete strategy for this game, each of the players must choose
each of his two choice nodes. Thus we can enumerate the pure strategies
rs as follows.
A, G),
(A, H),
(B,Games
G), (B,
Extensive
Form
andH)}
Subgame Perfection
ISCI 330 Lecture 12, Slide 8
o
q
,0)
yes
no
Recap
A
A
q
(0,0)
Aq
yes
no
yes Games
APerfect-InformationExtensive-Form
A
A
A
Aq
q
Aq
(1,1)
(0,0)
(0,2)
Subgame Perfection
(2,0)
Induced
Normal Form
Figure 5.1 The Sharing game.
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
H
AG
AH
BG
BH
CE
3, 8
3, 8
5, 5
5, 5
CF
3, 8
3, 8
2, 10
1, 0
DE
8, 3
8, 3
5, 5
5, 5
DF
8, 3
8, 3
2, 10
1, 0
(1,0)
A perfect-information game in extensive form.
o define a complete strategy for this game, each of the players must choose
each of his two choice nodes. Thus we can enumerate the pure strategies
rs as follows.
A, G),
(A, H),
(B,Games
G), (B,
Extensive
Form
andH)}
Subgame Perfection
ISCI 330 Lecture 12, Slide 8
q
,0)
Aq
(2,0)
q
(0,0)
Recap
Aq
(1,1)
q
(0,0)
Aq
(0,2)
Perfect-Information Extensive-Form Games
Subgame Perfection
5.1 The Sharing game.
InducedFigure
Normal
Form
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
H
AG
AH
BG
BH
CE
3, 8
3, 8
5, 5
5, 5
CF
3, 8
3, 8
2, 10
1, 0
DE
8, 3
8, 3
5, 5
5, 5
DF
8, 3
8, 3
2, 10
1, 0
(1,0)
this illustrates the lack of compactness of the normal form
Figure 5.2
I A perfect-information game in extensive form.
I games
aren’t
always
small
o define a complete strategy
for this
game, each
of thethis
players
must choose
each of his two choice
enumerate
the 16
purepayoff
strategiespairs
I nodes.
even Thus
herewewecanwrite
down
rs as follows.
instead of 5
A, G), (A, H), (B, G), (B, H)}
C, E), (C, F ), (D, E), (D, F )}
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 8
,0)
(2,0)
Recap
(0,0)
(1,1)
(0,0) Extensive-Form
(0,2) Games
Perfect-Information
Subgame Perfection
Figure 5.1 The Sharing game.
Induced Normal Form
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
H
AG
AH
BG
BH
CE
3, 8
3, 8
5, 5
5, 5
CF
3, 8
3, 8
2, 10
1, 0
DE
8, 3
8, 3
5, 5
5, 5
DF
8, 3
8, 3
2, 10
1, 0
(1,0)
while we can write any extensive-form game as a NF, we can’t
dostrategy
the reverse.
o define a complete
for this game, each of the players must choose
Figure 5.2
I A perfect-information game in extensive form.
each of his two choice
Thus we can enumerate
pure strategies
I nodes.
e.g., matching
pennies the
cannot
be written
rs as follows.
as a
perfect-information extensive form game
A, G), (A, H), (B, G), (B, H)}
C, E), (C, F ), (D, E), (D, F )}
Form
andinclude
Subgame
ant Extensive
to note that
weGames
have to
thePerfection
strategies (A, G) and (A, H), even
ISCI 330 Lecture 12, Slide 8
,0)
(2,0)
Recap
(0,0)
(1,1)
(0,0) Extensive-Form
(0,2) Games
Perfect-Information
Subgame Perfection
Figure 5.1 The Sharing game.
Induced Normal Form
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
H
AG
AH
BG
BH
CE
3, 8
3, 8
5, 5
5, 5
CF
3, 8
3, 8
2, 10
1, 0
DE
8, 3
8, 3
5, 5
5, 5
DF
8, 3
8, 3
2, 10
1, 0
(1,0)
What are the (three) pure-strategy equilibria?
Figure 5.2
I A perfect-information game in extensive form.
o define a complete strategy for this game, each of the players must choose
each of his two choice nodes. Thus we can enumerate the pure strategies
rs as follows.
A, G), (A, H), (B, G), (B, H)}
C, E), (C, F ), (D, E), (D, F )}
Form
andinclude
Subgame
ant Extensive
to note that
weGames
have to
thePerfection
strategies (A, G) and (A, H), even
ISCI 330 Lecture 12, Slide 8
,0)
(2,0)
Recap
(0,0)
(1,1)
(0,0) Extensive-Form
(0,2) Games
Perfect-Information
Subgame Perfection
Figure 5.1 The Sharing game.
Induced Normal Form
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
H
AG
AH
BG
BH
CE
3, 8
3, 8
5, 5
5, 5
CF
3, 8
3, 8
2, 10
1, 0
DE
8, 3
8, 3
5, 5
5, 5
DF
8, 3
8, 3
2, 10
1, 0
(1,0)
What are the (three) pure-strategy equilibria?
Figure 5.2
I A perfect-information game in extensive form.
I (A,for
o define a complete strategy
this(C,
game,
G),
F each
) of the players must choose
each of his two choice
Thus(C,
we F
can) enumerate the pure strategies
I nodes.
(A, H),
rs as follows.
I
(B, H), (C, E)
A, G), (A, H), (B, G), (B, H)}
C, E), (C, F ), (D, E), (D, F )}
Form
andinclude
Subgame
ant Extensive
to note that
weGames
have to
thePerfection
strategies (A, G) and (A, H), even
ISCI 330 Lecture 12, Slide 8
,0)
(2,0)
Recap
(0,0)
(1,1)
(0,0) Extensive-Form
(0,2) Games
Perfect-Information
Subgame Perfection
Figure 5.1 The Sharing game.
Induced Normal Form
at the definition contains a subtlety. An agent’s strategy requires a decision
ice node, regardless of whether or not it is possible to reach that node given
hoice nodes. In
the situationto
is straightforward—
I the
InSharing
fact, game
the above
connection
the normal form is even tighter
s three pure strategies, and player 2 has eight (why?). But now consider the
I we can “convert” an extensive-form game into normal form
n in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
H
AG
AH
BG
BH
CE
3, 8
3, 8
5, 5
5, 5
CF
3, 8
3, 8
2, 10
1, 0
DE
8, 3
8, 3
5, 5
5, 5
DF
8, 3
8, 3
2, 10
1, 0
(1,0)
What are the (three) pure-strategy equilibria?
Figure 5.2
I A perfect-information game in extensive form.
I (A,for
o define a complete strategy
this(C,
game,
G),
F each
) of the players must choose
each of his two choice
Thus(C,
we F
can) enumerate the pure strategies
I nodes.
(A, H),
rs as follows.
I
(B, H), (C, E)
A, G), (A, H), (B, G), (B, H)}
C, E), (C, F ), (D, E), (D, F )}
Form
andinclude
Subgame
ant Extensive
to note that
weGames
have to
thePerfection
strategies (A, G) and (A, H), even
ISCI 330 Lecture 12, Slide 8
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Lecture Overview
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 9
Recap
Perfect-Information Extensive-Form Games
Notice that the definition contains a subtlety. An agent’s strategy requires a decision
at each choice node, regardless of whether or not it is possible to reach that node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
Subgame Perfection
Subgame Perfection
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
I
H
(1,0)
A perfect-information game in extensive form.
order to define a complete
strategy for this
game, eachwith
of the players
choose
There’sInsomething
intuitively
wrong
themust
equilibrium
an action at each of his two choice nodes. Thus we can enumerate the pure strategies
(B, H),
(C,
E)
of the
players
as follows.
I
Why
player
S1 =would
{(A, G), (A,
H), (B,1
G),ever
(B, H)}choose to play H if he got to the
second
choice
S2 = {(C,
E), (C, Fnode?
), (D, E), (D, F )}
Iimportant
It is
note G
that dominates
we have to include
strategies
Aftertoall,
Hthefor
him(A, G) and (A, H), even
though once A is chosen the G-versus-H choice is moot.
The definition of best response and Nash equilibria in this game are exactly as they
are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example,
the perfect-information game of Figure 5.2 can be converted into the normal form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are
Multi Agent Systems, draft of September 19, 2006
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 10
Recap
Perfect-Information Extensive-Form Games
Notice that the definition contains a subtlety. An agent’s strategy requires a decision
at each choice node, regardless of whether or not it is possible to reach that node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
Subgame Perfection
Subgame Perfection
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
I
H
(1,0)
A perfect-information game in extensive form.
order to define a complete
strategy for this
game, eachwith
of the players
choose
There’sInsomething
intuitively
wrong
themust
equilibrium
an action at each of his two choice nodes. Thus we can enumerate the pure strategies
(B, H),
(C,
E)
of the
players
as follows.
I
Why
player
S1 =would
{(A, G), (A,
H), (B,1
G),ever
(B, H)}choose to play H if he got to the
second
choice
S2 = {(C,
E), (C, Fnode?
), (D, E), (D, F )}
I
HeThe
does
it oftobestthreaten
2, to
prevent
himas from
choosing
definition
response and player
Nash equilibria
in this
game are exactly
they
in for normal
form games.
Indeed, this example illustrates how every perfectFare, and
so gets
5
Iimportant
It is
note G
that dominates
we have to include
strategies
Aftertoall,
Hthefor
him(A, G) and (A, H), even
though once A is chosen the G-versus-H choice is moot.
information game can be converted to an equivalent normal form game. For example,
However, this
seems
like
threat
theIperfect-information
game of
Figure 5.2
can a
be non-credible
converted into the normal
form imageIof the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are
If player 1 reached his second decision node, would he really
follow through
and play
Multi Agent Systems,
draft ofH?
September 19, 2006
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 10
Recap
Perfect-Information Extensive-Form Games
Subgame Perfection
Formal Definition
I
Define subgame of G rooted at h:
I
I
Define set of subgames of G:
I
I
I
the restriction of G to the descendents of H.
subgames of G rooted at nodes in G
s is a subgame perfect equilibrium of G iff for any subgame
G0 of G, the restriction of s to G0 is a Nash equilibrium of G0
Notes:
I
I
since G is its own subgame, every SPE is a NE.
this definition rules out “non-credible threats”
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 11
(0,0)
(2,0)
(0,0)
(1,1)
(0,0)
Perfect-Information Extensive-Form Games
Recap
(0,2)
Subgame Perfection
Figure 5.1 The Sharing game.
Back to the Example
Notice that the definition contains a subtlety. An agent’s strategy requires a decision
at each choice node, regardless of whether or not it is possible to reach that node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
I
H
(1,0)
A perfect-information game in extensive form.
In order to define a complete strategy for this game, each of the players must choose
Whichan equilibria
from
thenodes.
example
subgame
perfect?
action at each of his
two choice
Thus we can are
enumerate
the pure strategies
of the players as follows.
S1 = {(A, G), (A, H), (B, G), (B, H)}
S2 = {(C, E), (C, F ), (D, E), (D, F )}
It is important to note that we have to include the strategies (A, G) and (A, H), even
though once A is chosen the G-versus-H choice is moot.
The definition of best response and Nash equilibria in this game are exactly as they
are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example,
the perfect-information game of Figure 5.2 can be converted into the normal form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 12
(0,0)
(2,0)
(0,0)
(1,1)
(0,0)
Perfect-Information Extensive-Form Games
Recap
(0,2)
Subgame Perfection
Figure 5.1 The Sharing game.
Back to the Example
Notice that the definition contains a subtlety. An agent’s strategy requires a decision
at each choice node, regardless of whether or not it is possible to reach that node given
the other choice nodes. In the Sharing game above the situation is straightforward—
player 1 has three pure strategies, and player 2 has eight (why?). But now consider the
game shown in Figure 5.2.
1
A
B
2
2
C
D
E
(3,8)
(8,3)
(5,5)
F
1
G
(2,10)
Figure 5.2
I
H
(1,0)
A perfect-information game in extensive form.
In order to define a complete strategy for this game, each of the players must choose
Whichan equilibria
from
thenodes.
example
subgame
perfect?
action at each of his
two choice
Thus we can are
enumerate
the pure strategies
I
I
I
of the players as follows.
(A,SG),
(C, F ) is subgame perfect
1 = {(A, G), (A, H), (B, G), (B, H)}
(B,S2H)
is an non-credible threat, so (B, H), (C, E) is not
= {(C, E), (C, F ), (D, E), (D, F )}
subgame
It is important perfect
to note that we have to include the strategies (A, G) and (A, H), even
though
onceisA is
chosennon-credible,
the G-versus-H choiceeven
is moot.though H is “off-path”
(A,
H)
also
The definition of best response and Nash equilibria in this game are exactly as they
are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example,
the perfect-information game of Figure 5.2 can be converted into the normal form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are
Extensive Form Games and Subgame Perfection
ISCI 330 Lecture 12, Slide 12