Chapter 6
Extensive Games, perfect info
• Detailed description of the sequential
structure of strategic situations
– as opposed to Strategic Games
• Players perfectly informed of occurred
events
• Initially decisions are not made at the same
time, no randomness
Example 91.1 / Def. 89.1
Extensive game: Players to share
two objects
– Conventional definition with trees as
primitives:
1
(2,0)
(1,1)
2
yes
no
2,0
0,0
yes
1,1
(0,2)
2
no
yes
0,0
0,2
no
0,0
– Definition with players´ actions as primitives:
• G = (N, H, P, (i))
• A Set of Players
– N = {1,2}
• A Set of possible histories (sequences, finite or infinite)
– H = {, (2, 0), (1, 1), (0, 2),
((2, 0), yes), ((2, 0), no), ((1, 1), yes), ((1, 0), no), ((0, 2), yes), ((0, 2),
no)}
• Terminal histories
– Z = {((2, 0), yes), ((2, 0), no), ((1, 1), yes), ((1, 1), no), ((0, 2), yes), ((0,
2), no)}
1
(2,0)
(1,1)
2
yes
no
2,0
0,0
yes
1,1
(0,2)
2
no
yes
0,0
0,2
no
0,0
• A player function that assigns a player to each non
terminal history
– P() = 1 and P(h) = 2 for every non terminal h  
• A preference relation for each player on Z:
– i: ((2, 0), yes) >1 ((1, 1), yes) >1 ((0, 2), yes) ~1 ((2, 0),
no) ~1 ((1, 1) ~1 yes) ~1((1, 1), no)
and
((2, 0), yes) >2 ((1, 1), yes) >2 ((0, 2), yes) ~2 ((2, 0), no)
~2 ((1, 1) ~2 yes) ~2((1, 1), no)
1
(2,0)
(1,1)
2
yes
no
2,0
0,0
yes
1,1
(0,2)
2
no
yes
0,0
0,2
no
0,0
Def. 92.1
Strategies
• A strategy of player i is a function that assigns an
action to each nonterminal history
– Even for histories that, if strategy is followed, are never
reached
– Player 1 below has AE, AF, BE, BF
• The outcome O(s) of strategy profile s = (si)iN
yields the terminal history when each player i
follows si
1
2
1
E
a
C
A
B
d
D
F
b
c
Def. 93.1
Nash Equilibrium
• Nash Equilibrium for an extensive game with
perfect info is a strategy profile s* such that for
every player iN we have
– O(s*-i, s*i) i O(s*-i, si) for every strategy si of player i
– (If other players follows s* you would better follow s*
too... )
• Alternatively it is the Nash Equilibrium of a
strategic game derived from the extensive game
Equivalent strategic games
1
2
1
E
a
C
d
D
c
F
b
Extensive
Game
C
D
AE
a
c
AF
b
BE
BF
B
A
C
D
AE
a
c
c
AF
b
c
d
d
B
d
d
d
d
Equivalent
Strategic Game
Equivalent
Strategic Game
Reduced form
Example 95.2
1
2
L
0, 0
L
R
A
0,0
2,1
B
1,2
1,2
B
A
R
1, 2
2, 1
• Given that player 2 chooses L it is optimal for
player 1 to choose B
• The Nash equilibrium (B,L) lacks plausibility
since P2 wouldn’t choose L after A.
Def. 97.1 Subgame
(h) = (N, H|h, P|h, (i)|h)
is the subgame to
 = (N, H, P, (i))
that follows the history h
h
(h)
Def. 97.2
Subgame Perfect Equlibrium
• A subgame perfect equilibrium is a strategy
profile s* such that for any history h the
strategy profile s*|h is a Nash equlibrium of
the subgame (h)
OR?
Example 95.2 again
1
2
L
0, 0
L
R
A
0,0
2,1
B
1,2
1,2
B
A
R
2, 1
1, 2
(A)
• (B,L) is a Nash equilibrium
• Is (B,L) a subgame perfect equilibrium?
– The strategy profile s*|h = (B,L)|A in the subgame (A)
is for instance no Nash Equilibrium
– Player 2 wouldn’t chose L given that player 1 has
chosen A
-
L
R
0,0
2,1
Prop. 99.2
Kuhn´s Theorem
• Every finite extensive game with perfect
info has a subgame perfect equilibrium.
– E.g chess is draw once a position is repeated
three times => chess is finite
Two Extensions to Extensive
Games with perfect info
• Exogenous uncertainty
– The Player function P(h) has a probability that
chance determines the action after the history h
– Definition of a subgame perfect equilibrium
and Kuhn’s theorem still OK
• Simultaneous moves
– The Player function P(h) assigns a set of
players that make choices after the history h
6.5.1 The Chain Store Game
• Multitude of Nash equilibria
– Every terminal history which the outcome in any period
is either Out or (In,C)
– Intuitively unappealing for small K
• Unique Subgame Perfect Equilibrium
– Always (In, C)
– Not that appealing for large K
k
CS
F
0, 0
Out
In
C
2, 2
5 ,1
Ex. 110.1 BoS with an outside
option
• Elimination of dominated actions yields: (B, B)
• Interpretation
– BB >1 Book >1 SS >1 BS ~1 SB
– Player 2 knows that if player 1 selects concert he would
choose Bach otherwise he would better stay home
reading the book
– Thus player 1 can select B knowing that player 2 also
selects B
1
B
Concert
Book
2, 2
B
S
B
3,1
0,0
S
0,0
1,3
S
Book 2,2
B
3,1
2,2
S
1,3
0,0
0,0
Ex. 111.1 Burning money
– Elimination of dominated actions yields: (0B, BB)
– Interpretation:
• P2 thinks that if P1 spends D then he wants to go Bach
otherwise he would loose compared to not spending D => P2
chooses B if P1 chooses D
• P1 knows this and can expect a payoff of 2 by choosing DB
• P2 knows that the rationality of P1 choosing 0 is that he
expects to gain better than 2 (by choosing DB)
• Thus P1 can choose 0B and gain 3
– Authors think that this example is implausible
1
D
0
B
S
B
3,1
0,0
S
0,0
1,3
B
S
B
2,1
-1,0
S
-1,0 0,3