Microeconomics II
Extensive Form Games I
Levent Koçkesen
Koç University
Levent Koçkesen (Koç University)
Extensive Form Games I
1 / 32
Extensive Form
The strategies in strategic form games are specified so that each
player chooses an action (or a mixture of actions) once and for all.
In games that involve a sequential character this specification may
not be suitable as players may find it profitable to reassess their
strategies as the events in the game unfold.
Extensive form provides an explicit description of a strategic
interaction by specifying
◮
◮
◮
who moves when
to do what
with what information
It provides a richer environment to study interesting question such as
◮
◮
◮
◮
commitment
repeated interaction
signaling
reputation building, etc.
A given game can be represented in both a strategic form and an
extensive form
◮
The choice depends on the questions that we ask
Levent Koçkesen (Koç University)
Extensive Form Games I
2 / 32
Extensive Form Games
An extensive form game is defined by
1
2
N : a finite set of players
H: a set of histories (or nodes) such that
i. a0 ∈ H (called the initial node or empty history)
ii. For any integer k ≥ 0, (a0 , a1 , . . . , ak ) ∈ H whenever
(a0 , a1 , . . . , ak+1 ) ∈ H
iii. If (ak )k=0,... satisfies (ak )k=0,...,K ∈ H for every integer K, then
(ak )k=0,... ∈ H
⋆
(ak )k=0,... is an infinite history
◮
Let (a0 , a1 , . . . , ak ) ∈ H and h′ = (b1 , b2 , . . . , bl ) so that
(a0 , a1 , . . . , ak , b1 , b2 , . . . , bl ) ∈ H. We write
◮
An history h is said to be terminal if (h, a) ∈
/ H for any a or if it is
infinite.
◮
Length of a history h = (a0 , a1 , . . . , ak ) is |h| = k
For any h ∈ H \ Z, A(h) = {a : (h, a) ∈ H} denotes the set of actions
available after history h
(h, h′ ) = (a0 , a1 , . . . , ak , b1 , b2 , . . . , bl )
⋆
◮
Z: the set of terminal histories
Levent Koçkesen (Koç University)
Extensive Form Games I
3 / 32
Extensive Form Games
3
ι : H \ Z → N ∪ {c}: the player function
◮
◮
◮
ι(h) = i: i moves immediately after history h and chooses an action
from the set A(h)
If ι(h) = c, then chance determines the action taken after history h
Set of histories after which i ∈ N ∪ {c} moves:
Hi = {h ∈ H : ι(h) = i}
◮
Set of actions available to i ∈ N ∪ {c} in the entire game:
[
Ai =
A(h)
ι(h)=i
4
βc : chance’s probability distribution
◮
◮
◮
For any h ∈ Hc , βc (.|h) is a probability distribution over A(h)
βc (a|h) denotes the probability assigned to a ∈ A(h)
We sometimes use “nature” instead of chance
Levent Koçkesen (Koç University)
Extensive Form Games I
4 / 32
Extensive Form Games
5
An information partition I of H \ Z such that for all I ∈ I and all
h, h′ ∈ I we have
i. ι(h) = ι(h′ )
ii. A(h) = A(h′ )
◮
◮
◮
Each element of I is an information set
The information set containing h is denoted I(h)
Set of information sets of i ∈ N ∪ {c}
Ii = {I(h) : ι(h) = i, for some h ∈ H \ Z}
◮
6
We may write A(I) instead of A(h), ι(I) instead of ι(h), and βc (a|I)
instead of βc (a|h) for any h ∈ I
For each i ∈ N , a von Neumann-Morgenstern payoff function
ui : Z → R
Levent Koçkesen (Koç University)
Extensive Form Games I
5 / 32
Extensive Form Games
An extensive form game is given by
Γ = (N, H, ι, βc , I, (ui )i∈N )
An extensive form is given by
Γ = (N, H, ι, βc , I)
If H is finite we call the game a finite extensive form game
If |h| is finite for all h ∈ H the game has a finite horizon
If all information sets in Γ are singletons, we say that the game is
with perfect information, and omit the information partitions in its
definition. Otherwise we say that the game is with imperfect
information.
If there exist h ∈ Hc , two distinct actions a, a′ ∈ A(h), and I ∈ I
such that (a, h′ ), (a′ , h′′ ) ∈ I for some h′ , h′′ we say that the game is
with incomplete information. Otherwise we say that the game is with
complete information
Levent Koçkesen (Koç University)
Extensive Form Games I
6 / 32
The Dating Game
We can use game trees to represent extensive form games
This is a game with incomplete information
2, 1
1, 1
m
m
1
q
f
d
d
0, 0
(1/3)
−1, 0
S
2
2
chance
2, 0
(2/3)
−1, 0
D
m
m
q
f
1
d
d
0, 1
−3, 1
Levent Koçkesen (Koç University)
Extensive Form Games I
7 / 32
2, 1
1, 1
m
m
1
q
f
d
d
0, 0
(1/3)
−1, 0
S
2
2
chance
2, 0
(2/3)
−1, 0
D
m
m
q
1
f
d
d
0, 1
1
2
3
−3, 1
N = {1, 2}
H = {a0 , S, D, (S, f ), (D, f ), (S, q), (D, q), (S, f, m), (S, f, d),
(D, f, m), (D, f, d), (S, q, m), (S, q, d), (D, q, m), (D, q, d)}
Z = {(S, f, m), (S, f, d), (D, f, m), (D, f, d), (S, q, m),
(S, q, d), (D, q, m), (D, q, d)}
We may omit a0 from the description of any other history
|(S, f, d)| = 3
ι(a0 ) = c, ι(S) = ι(D) = 1, ι(S, f ) = ι(D, f ) = ι(S, q) = ι(D, q) = 2
A(a0 ) = {S, D}, A(S) = A(D) = {f, q}
A(S, f ) = A(D, f ) = A(S, q) = A(D, q) = {m, d}
Levent Koçkesen (Koç University)
Extensive Form Games I
8 / 32
2, 1
1, 1
m
m
1
q
f
d
d
0, 0
(1/3)
−1, 0
S
2
2
chance
2, 0
(2/3)
−1, 0
D
m
m
q
f
1
d
d
0, 1
−3, 1
4
βc (S|a0 ) = 1/3, βc (D|a0 ) = 2/3
5
I = {{a0 }, {S}, {D}, {(S, f ), (D, f )}, {(S, q), (D, q)}}
6
u1 (S, f, m) = 1, u1 (S, f, d) = −1, etc.
u2 (S, f, m) = 1, u2 (S, f, d) = 0, etc.
Levent Koçkesen (Koç University)
Extensive Form Games I
9 / 32
Sequential Battle of the Sexes
This is a game with perfect information
chance
x
1
B
S
2
2
B
S
B
S
2, 1
0, 0
0, 0
1, 2
Levent Koçkesen (Koç University)
Extensive Form Games I
10 / 32
chance
x
1
B
S
2
1
2
3
4
5
6
2
B
S
B
S
2, 1
0, 0
0, 0
1, 2
N = {1, 2}
H = {a0 , x, (x, B), (x, S), (x, B, B), (x, B, S), (x, S, B), (x, S, S)}
Z = {(x, B, B), (x, B, S), (x, S, B), (x, S, S)}
ι(a0 ) = c, ι(x) = 1, ι(x, B) = ι(x, S) = 2
A(a0 ) = {x}, A(x) = {B, S}, A(x, B) = A(x, S) = {B, S}
βc (x|a0 ) = 1
I = {{a0 }, {x}, {(x, B)}, {(x, S)}}
u1 (B, B) = 2, u1 (B, S) = 0, u1 (S, B) = 0, u1 (S, S) = 1
Levent Koçkesen (Koç University)
Extensive Form Games I
11 / 32
Sequential Battle of the Sexes: A Game with Perfect
Information
1
B
S
2
2
B
S
B
S
2, 1
0, 0
0, 0
1, 2
Levent Koçkesen (Koç University)
Extensive Form Games I
12 / 32
1
B
S
2
2
B
S
B
S
2, 1
0, 0
0, 0
1, 2
Since the game is with perfect information we may omit information
partition I from the definition
Since chance’s role is trivial, we may omit anything about it from the
definition
1
2
3
4
N = {1, 2}
H = {a0 , B, S, (B, B), (B, S), (S, B), (S, S)}
Z = {(B, B), (B, S), (S, B), (S, S)}
ι(a0 ) = 1, ι(B) = ι(S) = 2
u1 ((B, B)) = 2, u1 ((B, S)) = 0, u1 ((S, B)) = 0, u1 ((S, S)) = 1
Levent Koçkesen (Koç University)
Extensive Form Games I
13 / 32
A Game with Imperfect and Complete Information
1
B
S
2
1
2
3
4
5
2
B
S
B
S
2, 1
0, 0
0, 0
1, 2
N = {1, 2}
H = {a0 , B, S, (B, B), (B, S), (S, B), (S, S)}
ι(a0 ) = 1, ι(B) = ι(S) = 2
I = {{a0 }, {B, S}}
u1 ((B, B)) = 2, u1 ((B, S)) = 0, u1 ((S, B)) = 0, u1 ((S, S)) = 1
Levent Koçkesen (Koç University)
Extensive Form Games I
14 / 32
Extensive Form Strategies
A pure strategy: For any i ∈ N ∪ {c}
si : Ii → Ai with si (I) ∈ A(I) for all I ∈ Ii
The set of pure strategies for i ∈ N ∪ {c}
Si = ×I∈Ii A(I)
Set of pure strategy profiles:
S = ×i∈N Si
Note that we also define pure strategies for Nature, although it is a
non-strategic “player”. We do this only for notational convenience in
defining outcomes and expected payoffs of players in N .
Players may randomize in extensive form games. Later on, we will
introduce different ways through which they may do so.
Levent Koçkesen (Koç University)
Extensive Form Games I
15 / 32
Extensive Form Strategies
Dating Game:
sc : {{a0 }} → {S, D}
s1 : {{S}, {D}} → {q, f }
s2 : {{(S, f ), (D, f )}, {(S, q), (D, q)}} → {m, d}
Sequential Battles of the Sexes:
s1 : {{a0 }} → {B, S}
s2 : {{B}, {S}} → {B, S}
Set of pure strategies:
S1 : {B, S}
S2 : {BB, BS, SB, SS}
Levent Koçkesen (Koç University)
Extensive Form Games I
16 / 32
Extensive Form Strategies
Consider the following (centipede) game
1
S
C
2
C
S
1, 0
1
C
2, 4
S
0, 2
3, 1
S1 = {SS, SC, CS, CC}
S2 = {S, C}
An extensive form strategy has to specify how a player will act after any
history she may be called upon to move, including those that are precluded
by the previous play of the same player.
Levent Koçkesen (Koç University)
Extensive Form Games I
17 / 32
Extensive Form Strategies and Payoffs
A pure strategy profile s ∈ S and Nature’s probability distribution βc
induces an outcome P s
◮
◮
A probability distribution over the set of terminal histories Z
Let P s (z) denote the probability of z ∈ Z under s ∈ S
We assume that players’ preferences can be represented by expected
payoffs:
X
Ui (s) =
P s (z)ui (z)
z∈Z
For example in sequential BoS game if s1 = B and
s2 (B) = S, s2 (S) = B then the outcome is (B, S) with probability 1:
P s (B, S) = 1
In the dating game if s1 (S) = f , s1 (D) = q,
s2 ({(S, f ), (D, f )}) = m, s2 ({(S, q), (D, q)}) = d, then
P s (S, f, m) = 1/3, P s (D, q, d) = 2/3
Next we will start discussing solution concepts for extensive form
games. We begin with a conceptually simpler form: extensive form
games with perfect information
Levent Koçkesen (Koç University)
Extensive Form Games I
18 / 32
Extensive Form Games with Perfect Information
Each player observes the previous moves of every player (including
nature)
◮
Every information set is a singleton
Consider the centipede game again
C
1
C
2
S
1
S
1, 0
C
2, 4
S
0, 2
3, 1
How do you think rational players will play this game?
We have just applied backward induction algorithm
Backward induction equilibrium of the centipede game is given by
s1 (a0 ) = S, s1 (C, C) = S, s2 (C) = S
We could write s1 = SS, s2 = C when the meaning is clear
Backward induction outcome is (S)
Levent Koçkesen (Koç University)
Extensive Form Games I
19 / 32
Backward Induction
1
s1 (C, C) = S
C
S
1, 0
1
s2 (C) = S
C
C
1
2, 4
S
0, 2
S
3, 1
C
2
3, 1
S
1, 0
1
s1 (a0 ) = S
C
2
S
0, 2
C
0, 2
S
1, 0
Backward induction equilibrium: s1 = SS, s2 = S
Levent Koçkesen (Koç University)
Extensive Form Games I
20 / 32
Nash Equilibrium
Definition
A (pure strategy) Nash equilibrium of an extensive form game with perfect
information Γ = (N, H, ι, βc , I, (ui )) is a strategy profile s∗ ∈ S such that
for every player i ∈ N
Ui (s∗i , s∗−i ) ≥ Ui (si , s∗−i )
for all si ∈ Si
Nash equilibrium of an extensive form game can also be defined as the
Nash equilibrium of its strategic form.
Definition
The strategic form of the extensive form game with perfect information
Γ = (N, H, ι, βc , I, (ui )) is the strategic form game GΓ = (N, (Si ), (Ui )).
Levent Koçkesen (Koç University)
Extensive Form Games I
21 / 32
Nash Equilibrium: Centipede Game
Strategic Form of the centipede game:
1
1
C
2
C
1
N = {1, 2}
2
S1 = {SS, SC, CS, CC}, S2 = {S, C}
3
Payoffs are given in the following
bimatrix:
SS
SC
CS
CC
S
1, 0
1, 0
0, 2
0, 2
S
1, 0
S
0, 2
C
2, 4
S
3, 1
C
1, 0
1, 0
3, 1
2, 4
BI(Γ) = {(SS, S)}
N(Γ) = {(SS, S)(SC, S)}
BIE and NE are outcome equivalent in this game
This is not always the case
Levent Koçkesen (Koç University)
Extensive Form Games I
22 / 32
Entry Game
1
BI(Γ) = {(E, A)}
Strategic Form of the Game
O
E
2
O
E
0, 4
F
−1, −1
A
F
0, 4
−1, −1
A
0, 4
2, 2
N(Γ) = {(E, A), (O, F )}
2, 2
(O, F ) equilibrium sustained by an incredible threat
Backward induction rules incredible threats or promises out
NE requires rationality whereas BIE requires sequential rationality
◮
Players must play optimally at every point in the game
Levent Koçkesen (Koç University)
Extensive Form Games I
23 / 32
Proposition
Every finite extensive form game with perfect information has a backward
induction equilibrium.
Proposition
If s∗ is a backward induction equilibrium of a finite extensive form game
with perfect information Γ, then s∗ is a Nash equilibrium of Γ.
Proposition
Every finite extensive form game with perfect information has a Nash
equilibrium.
Levent Koçkesen (Koç University)
Extensive Form Games I
24 / 32
Example: Stackelberg Duopoly
Remember Cournot Duopoly model?
◮
◮
Two firms simultaneously choose output (or capacity) levels
What happens if one of them moves first?
⋆
or can commit to a capacity level?
The resulting model is known as Stackelberg oligopoly
◮
After the German economist Heinrich von Stackelberg in Marktform
und Gleichgewicht (1934)
The model is the same except that, now, Firm 1 moves first
Payoff function of each firm is given by
(
(a − c − b(q1 + q2 ))qi , q1 + q2 ≤ a/b
ui (q1 , q2 ) =
−cqi ,
q1 + q2 > a/b
Levent Koçkesen (Koç University)
Extensive Form Games I
25 / 32
Nash Equilibrium of Cournot Duopoly
Best response correspondence of firm i 6= j is given by
( a−c−bq
j
, qj < a−c
2b
b
Bi (qj ) =
0,
qj ≥ a−c
b
Unique Nash equilibrium:
q1c = q2c =
a−c
3b
Equilibrium profits:
π1c = π2c =
Levent Koçkesen (Koç University)
(a − c)2
9b
Extensive Form Games I
26 / 32
Stackelberg Model
The game has two stages:
1
2
Firm 1 chooses a capacity level q1 ≥ 0
Firm 2 observes Firm 1’s choice and chooses a capacity q2 ≥ 0
1
N = {1, 2}
H = {a0 } ∪ R+ ∪
Z = R2+
R2+
q1
2
ι(a0 ) = 1, ι(q1 ) = 2, for all q1 ∈ R+
(
(a − c − b(q1 + q2 ))qi , q1 + q2 ≤ a/b
ui (q1 , q2 ) =
−cqi ,
q1 + q2 > a/b
q2
u1 , u2
q1 ∈ R+
q2 : R+ → R+
Levent Koçkesen (Koç University)
Extensive Form Games I
27 / 32
Backward Induction Equilibrium of Stackelberg Game
Sequential rationality of Firm 2 implies that for any q1 it must play a
best response:
(
a−c−bq1
, q1 < a−c
2b
b
q2 (q1 ) =
0,
q1 ≥ a−c
b
Firms 1’s strategy has to maximize:
(
(a − c − b(q1 + q2 (q1 )))q1 , q1 + q2 (q1 ) ≤ a/b
u1 (q1 , q2 (q1 )) =
−cq1 ,
q1 + q2 (q1 ) > a/b
Any q1 ≥ (a − c)/b leads to non-positive payoff whereas
0 < q1 < (a − c)/b gives positive payoff
Therefore, Firm 1 will choose q1 to maximize
a − c − bq1
)]q1
[a − c − b(q1 +
2b
Since q1 = 0 cannot be the solution, FOC must hold with equality
a−c
q1 =
2b
Levent Koçkesen (Koç University)
Extensive Form Games I
28 / 32
Backward Induction Equilibrium of Stackelberg Game
Backward Induction Equilibrium
q1s =
a−c
2b
q2s (q1 ) =
(
a−c−bq1
,
2b
0,
q1 <
q1 ≥
a−c
b
a−c
b
Backward Induction Outcome
a−c
a−c
>
= q1c
2b
3b
a
−
c
a
−
c
q2s =
<
= q2c
4b
3b
Equilibrium Profits
Firm 1 commits to an aggressive
strategy
q1s =
(a − c)2
(a − c)2
>
= π1c
8b
9b
2
2
(a − c)
(a − c)
π2s =
<
= π2c
16b
9b
π1s =
Levent Koçkesen (Koç University)
There is first mover advantage
Extensive Form Games I
29 / 32
Cournot vs. Stackelberg
q2
a−c
b
BR1 (q2 )
a−c
2b
a−c
3b
a−c
4b
Cournot
Stackelberg
BR2 (q1 )
q1
a−c
3b
Levent Koçkesen (Koç University)
a−c
2b
Extensive Form Games I
a−c
b
30 / 32
Example: Ultimatum Bargaining
A
Two players, A and B, bargain
over a cake of size 1
Player A makes an offer
x ∈ [0, 1] to player B
If player B accepts the offer
(Y ), agreement is reached
◮
◮
x
B
Y
N
A receives x
B receives 1 − x
If player B rejects the offer (N )
both receive zero
Levent Koçkesen (Koç University)
x, 1 − x
Extensive Form Games I
0, 0
31 / 32
Backward Induction Equilibrium of Ultimatum Bargaining
B’s optimal action
◮
◮
1
x < 1 → accept
x = 1 → accept or reject
Suppose in equilibrium B accepts any offer x ∈ [0, 1]
◮
◮
What is the optimal offer by A? x = 1
The following is a BIE
x∗ = 1
s∗B (x) = Y for all x ∈ [0, 1]
2
Now suppose that B accepts if and only if x < 1
◮
What is A’s optimal offer?
⋆
⋆
x = 1?
x < 1?
Unique BIE
x∗ = 1, s∗B (x) = Y for all x ∈ [0, 1]
Levent Koçkesen (Koç University)
Extensive Form Games I
32 / 32