Week 3
Games in extensive form with perfect information
Reading: Ch 5, 6, 7
Introduction
De nitions
Game in extensive form: CN.A.X.E.u.luilien
Set of playersN 11,2 nl
Set of actions A
Set of nodes including the initial node
Any nodeocean.az
amresults from this sequence of actions
The set of actions available at x ACAEA
Set of terminal notes EwhereAla0
for ee
The player choosing action at node x is cXIE N
i’s payo function
A game is perfect recall if each player remembers what they did in prior moves, and each player
remembers everything that they knew before
T is nite if A and X are nite
Assumptions
• Common knowledge of rationality
• Common knowledge of game
• Perfect recall
Strategies
Pure strategy:si X A satisfyings ibdeAG
Mixed strategy:
Behavioural strategy: Oi X s a satisfyingsupploiculcAG
Behavioural strategy pro le: osi.osil.n.losn.si
Randomisation between actions ⫫ at each node
SPNE: a PS pro le, s*, is an SPNE of a game of perfect information if and only if it is a backward
induction strategy pro le of this game
Game tree
Example a'b
N 1,2
d BD16,41 Coal
3,11 2 o.L.R.CR.al.CR.bs.CR.b.d.CR.b.pl Be16,41s3
X
16,4
is 3
I D
cog
PS ke.LD.rc.ro
o Cocoaonions
Games in extensive form with perfect information
Reading: Ch 5, 6, 7
Introduction
De nitions
Game in extensive form: CN.A.X.E.u.luilien
Set of playersN 11,2 nl
Set of actions A
Set of nodes including the initial node
Any nodeocean.az
amresults from this sequence of actions
The set of actions available at x ACAEA
Set of terminal notes EwhereAla0
for ee
The player choosing action at node x is cXIE N
i’s payo function
A game is perfect recall if each player remembers what they did in prior moves, and each player
remembers everything that they knew before
T is nite if A and X are nite
Assumptions
• Common knowledge of rationality
• Common knowledge of game
• Perfect recall
Strategies
Pure strategy:si X A satisfyings ibdeAG
Mixed strategy:
Behavioural strategy: Oi X s a satisfyingsupploiculcAG
Behavioural strategy pro le: osi.osil.n.losn.si
Randomisation between actions ⫫ at each node
SPNE: a PS pro le, s*, is an SPNE of a game of perfect information if and only if it is a backward
induction strategy pro le of this game
Game tree
Example a'b
N 1,2
d BD16,41 Coal
3,11 2 o.L.R.CR.al.CR.bs.CR.b.d.CR.b.pl Be16,41s3
X
16,4
is 3
I D
cog
PS ke.LD.rc.ro
o Cocoaonions
