Microeconomics II
Extensive Form Games II
Levent Koçkesen
Koç University
Levent Koçkesen (Koç University)
Extensive Form Games II
1 / 51
Extensive Form Games with Imperfect Information
In extensive form games with perfect information every player
observes the previous moves made by all the players
What happens if some of the previous moves are not observed?
We cannot apply backward induction algorithm anymore
Consider the following modified entry game
Player 1 doesn’t know whether Player 2
fights or accommodates
1
We cannot determine the optimal action
for Player 1 at that information set
O
... and therefore cannot apply backward
induction algorithm
0, 20
2
F
−5, −5
Extensive Form Games II
A
1
F
Levent Koçkesen (Koç University)
E
A
F
5, 15 15, 5
A
10, 10
2 / 51
Subgame Perfect Equilibrium
We will introduce another solution concept: Subgame Perfect Equilibrium
For any extensive form game Γ = (N, H, ι, βc , I, (ui )) and history
h ∈ H \ Z define the restriction of H to histories that start with h as
follows
H|h = {h} ∪ {h′′ ∈ H : ∃h′ s.t. h′′ = (h, h′ )}
H|h contains all the histories in Γ that start with h.
We similarly define N |h , ι|h , βc |h , I|h , ui |h , si |h , Si |h , Ui |h
Levent Koçkesen (Koç University)
Extensive Form Games II
3 / 51
Subgame Perfect Equilibrium
Definition (Subgame)
Let Γ = (N, H, ι, βc , I, (ui )) be an extensive form game and take any
h ∈ H \ Z. We say that Γ|h = (N |h , H|h , ι|h , βc |h , I|h , (ui |h )) is a
subgame of Γ if
1
2
3
it starts at a singleton information set: I(h) is singleton
it contains every history that follows h. If there exists h′ ∈ H such
that (h′ , h, h′′ ) ∈ H then (h, h′′ ) ∈ H|h
whenever it contains a node in an information set, it contains all the
nodes in that information set. If h′ ∈ H|h, then I(h′ ) ⊆ H|h
The game itself is a subgame
We call subgames other than the game itself proper subgames
Levent Koçkesen (Koç University)
Extensive Form Games II
4 / 51
N = {1, 2}
1
H = {a0 , O, E, (E, F ), (E, A), (E, F, F ), (E, F, A), (E, A, F )(E, A, A)}
O
E
I = {{a0 }, {E}, {(E, F ), (E, A)}}
2
ι(a0 ) = ι({(E, F ), (E, A)}) = 1, ι(E) = 2
0, 20
F
A
1
F
A
−5, −5
F
A
5, 15 15, 5
10, 10
2
F
N |E = {1, 2}
A
H|E = {E, (E, F ), (E, A), (E, F, F ), (E, F, A), (E, A, F )(E, A, A)}
1
I|E = {{E}, {(E, F ), (E, A)}}
F
−5, −5
A
F
5, 15 15, 5
A
ι|E ({(E, F ), (E, A)}) = 1, ι|E (E) = 2
10, 10
Levent Koçkesen (Koç University)
Extensive Form Games II
5 / 51
Subgame Perfect Equilibrium
Definition
A strategy profile in an extensive form game is a subgame perfect
equilibrium (SPE) if it induces a Nash equilibrium in every subgame of the
game. More formally, s∗ ∈ S is a SPE of Γ if s∗ |h ∈ S|h is a Nash
equilibrium of Γ|h for all h ∈ H \ Z.
Levent Koçkesen (Koç University)
Extensive Form Games II
6 / 51
Subgame Perfect Equilibrium
Consider the following game
There are two subgames: the game
itself and the one after history E.
1
Strategic form of the game
O
E
2
2
0, 20
F
OF
OA
1
EF
EA
A
1
F
A
−5, −5
F
A
−1, 8 8, −1
10, 10
Levent Koçkesen (Koç University)
F
0, 20
0, 20
−5, −5
−1, 8
A
0, 20
0, 20
8, −1
10, 10
Set of Nash equilibria
N(Γ) = {(OF, F ), (OA, F ), (EA, A)}
Extensive Form Games II
7 / 51
Subgame Perfect Equilibrium
Do those Nash equilibria induce a Nash equilibria in the other subgames?
The only proper subgame: Γ|E
2
Strategic form of Γ|E
F
A
2
1
F
−5, −5
A
F
1
A
−1, 8 8, −1
F
A
F
−5, −5
−1, 8
A
8, −1
10, 10
10, 10
Set of Nash equilibria
N(Γ|E ) = {(A, A)}
We had
N(Γ) = {(OF, F ), (OA, F ), (EA, A)}
Therefore SPE(Γ) = {(EA, A)}
Levent Koçkesen (Koç University)
Extensive Form Games II
8 / 51
Subgame Perfect Equilibrium
In practice you may use an algorithm similar to backward induction:
1
Find the Nash equilibria of the “smallest” subgame(s)
2
Fix one for each subgame and attach payoffs to its initial node
3
Repeat with the reduced game
Levent Koçkesen (Koç University)
Extensive Form Games II
9 / 51
Subgame Perfect Equilibrium
The “smallest” subgame
Its strategic form
2
2
F
A
1
F
A
−5, −5
F
−1, 8 8, −1
A
10, 10
1
F
A
F
−5, −5
−1, 8
A
8, −1
10, 10
Unique Nash equilibrium of the
subgame is (A, A)
Reduced subgame is
1
O
0, 20
E
10, 10
Its unique Nash equilibrium is (E)
Therefore the unique SPE of the game is ((E, A), A)
Levent Koçkesen (Koç University)
Extensive Form Games II
10 / 51
Subgame Perfect Equilibrium
Proposition
Let Γ be an extensive form game with perfect information and s∗ be a
subgame perfect equilibrium of Γ. Then s∗ is a backward induction
equilibrium of Γ.
Clearly every SPE is a NE but not conversely. In that sense we say that
SPE is a refinement of NE.
Levent Koçkesen (Koç University)
Extensive Form Games II
11 / 51
One-Deviation Property
In complicated extensive form games checking whether a strategy
profile is a SPE could be quite difficult.
One-deviation property simplifies this process tremendously.
Definition
Let Γ be an extensive form game with perfect information. For any
si |h ∈ Si |h in a subgame Γ|h and an action ai ∈ A(h) such that
ai 6= si |h (h) define a one-deviation (or one-shot deviation) sod
i |h ∈ Si |h as
follows:
(
′
ai ,
h =h
′
sod
i |h (h ) =
si |h (h′ ), otherwise
Levent Koçkesen (Koç University)
Extensive Form Games II
12 / 51
One-Deviation Property
Proposition (One-Deviation Property)
Let Γ be a finite horizon extensive form game with perfect information. A
strategy profile s∗ is a SPE of Γ if, and only if, for every player i ∈ N and
every h ∈ H \ Z for which ι(h) = i we have
∗
Ui |h (s∗i |h , s∗−i |h ) ≥ Ui |h (sod
i |h , s−i |h )
for every one-deviation sod
i |h .
There are versions of ODP for games with imperfect information and
infinite horizon games.
Levent Koçkesen (Koç University)
Extensive Form Games II
13 / 51
One-Deviation Property: Example
Consider the following (centipede) game
1
C
S
2
S
1, 0
0, 2
C
1
S
3, 1
C
C
2
S
2, 4
C
1
4, 6
S
5, 3
Is s1 = SSS, s2 = SS a SPE?
Is SSS a best response to SS?
◮
◮
Possible deviations: SSC, SCS, SCC, CSS, CSC, CCS, CCC
ODP says you only need to check against CSS
⋆
⋆
SSS → 1
CSS → 0
Is SS a best response to S?
◮
◮
Possible deviations: SC, CS, CC
ODP says you only need to check against CS
⋆
⋆
SS → 3
CS → 2
Is S a best response?
Levent Koçkesen (Koç University)
Extensive Form Games II
14 / 51
Subgame Perfect Equilibrium
What is the pure strategy SPE of the following game?
1
N
M
2
−1/2, −1/2
H
T
1
H
T
1, −1
H
−1, 1 −1, 1
T
1, −1
We have to introduce mixed strategies.
Levent Koçkesen (Koç University)
Extensive Form Games II
15 / 51
Mixed vs. Behavioral Strategies
There are two ways through which a player may randomize:
1 randomize over pure strategies in the entire game: mixed strategy
2 randomize separately at each information set: behavioral strategy
For any i ∈ N a mixed strategy σi is a probability distribution over Si .
The set of mixed strategies for player i
Σi = ∆(×I∈Ii A(I))
σi (si ): probability assigned to si
A behavioral strategy of player i ∈ N is a collection of independent
probability distributions βi = (βi (.|I))I∈Ii , where βi (.|I) is a
probability distribution over A(I).
The set of behavioral strategies for player i
Bi = ×I∈Ii ∆(A(I))
βi (a|I) denotes the probability assigned to a ∈ A(I)
Levent Koçkesen (Koç University)
Extensive Form Games II
16 / 51
Mixed vs. Behavioral Strategies
A mixed strategy:
1
σ1 (N H) = 1/2
N
M
σ1 (M H) = 1/2
2
−1/2, −1/2
A behavioral strategy:
H
T
1
β1 (N |{a0 }) = 1/2
β1 (H|{(M, H), (M, T )}) = 1/2
H
1, −1
Levent Koçkesen (Koç University)
T
H
−1, 1 −1, 1
Extensive Form Games II
T
1, −1
17 / 51
Strategies and Outcomes
A mixed strategy profile σ ∈ Σ and Nature’s probability distribution
βc induces an outcome P σ
◮
◮
A probability distribution over the set of terminal histories Z
Let P σ (z) denote the probability of z ∈ Z under σ
For any i ∈ N ∪ {c}, I ∈ Ii , and h ∈ I define
si (h) = si (I)
βi (.|h) = βi (.|I)
Take any non-empty history h = (a0 , a1 , . . . , ak ). Fix any i ∈ N ∪ {c}
and si ∈ Si . We say that si is consistent with h if for all
hl = (a0 , a1 , . . . , al ), l < k, for which ι(hl ) = i
si (hl ) = al+1
Note that every si is consistent with a0 and any history h in which i
does not move
Levent Koçkesen (Koç University)
Extensive Form Games II
18 / 51
Strategies and Outcomes
For any sc ∈ Sc define
Y
σc (sc ) =
βc (sc (h)|h)
h∈Hc
Denote the set of all si consistent with h by Sih . Let
X
σi (si )
P σi (h) =
si ∈Sih
Take any mixed strategy profile σ. Probability of reaching history h
under σ is given by
Y
P σi (h)
P σ (h) =
i∈N ∪{c}
Levent Koçkesen (Koç University)
Extensive Form Games II
19 / 51
Strategies and Outcomes
Similarly for a behavioral strategy profile β ∈ B we can define an
outcome P β
Take any non-empty history h = (a0 , a1 , . . . , ak ) and a behavioral
strategy profile β. For any l < k, let hl = (a0 , a1 , . . . , al ) and define
P β (h) =
k−1
Y
βι(hl ) (al+1 |hl )
l=0
We assume that players’ preferences can be represented by expected
payoffs:
X
P σ (z)ui (z)
Ui (σ) =
z∈Z
Ui (β) =
X
P β (z)ui (z)
z∈Z
Levent Koçkesen (Koç University)
Extensive Form Games II
20 / 51
Strategies and Outcomes: Example
Suppose the mixed strategy profile is
1
σ1 (N H) = 1/2, σ1 (M H) = 1/2, σ2 (H) = 1/3
N
M
2
−1/2, −1/2
and we want to calculate
H
T
1
P σ (M, T, H)
H
1, −1
T
H
−1, 1 −1, 1
T
1, −1
Pure strategies of player 1 that are consistent with (M, T, H) must have
s1 (a0 ) = M, s1 (M, T ) = H
Therefore
(M,T,H)
S1
P
σ1
= {M H}
X
((M, T, H)) =
σ1 (s1 ) = σ1 (M H) = 1/2
(M,T,H)
s1 ∈S1
Levent Koçkesen (Koç University)
Extensive Form Games II
21 / 51
Strategies and Outcomes: Example
Do the same with player 2. We have
(M,T,H)
S2
X
P σ2 ((M, T, H)) =
= {T }
σ2 (s2 ) = σ2 (T ) = 2/3
(M,T,H)
s2 ∈S2
So,
Y
P σ ((M, T, H)) =
P σi ((M, T, H)) = 1/3
i∈{1,2}
Similarly,
P σ1 (N ) = σ1 (N H) + σ1 (N T ) = 1/2
P σ2 (N ) = 1
P σ (N ) = 1/2
Verify that:
P σ (M, H, H) = 1/6,
Levent Koçkesen (Koç University)
P σ (M, H, T ) = P σ (M, T, T ) = 0
Extensive Form Games II
22 / 51
Strategies and Outcomes: Example
Let I0 = {a0 }, I1 = {M }, I2 = {(M, H), (M, T )}
and suppose the behavioral strategy profile is
1
N
β1 (N |I0 ) = 1/2, β1 (H|I2 ) = 1, β2 (H|I1 ) = 1/3
M
2
−1/2, −1/2
H
T
Then
1
P β (N ) = β1 (N |a0 ) = 1/2
H
P β (M, H, H) = β1 (M |a0 )β2 (H|M )β1 (H|(M, H)) = 1/6
1, −1
T
H
−1, 1 −1, 1
T
1, −1
P β (M, H, T ) = β1 (M |a0 )β2 (H|M )β1 (T |(M, H)) = 0
P β (M, T, H) = β1 (M |a0 )β2 (T |M )β1 (H|(M, T )) = 1/3
P β (M, T, T ) = β1 (M |a0 )β2 (T |M )β1 (T |(M, T )) = 0
In this example the mixed strategy and the behavioral strategy profiles
induce the same outcome.
Levent Koçkesen (Koç University)
Extensive Form Games II
23 / 51
Equivalent Mixed and Behavioral Strategies
Definition
We say that σi ∈ Σi and βi ∈ Bi are outcome equivalent if for all pure
strategy profiles of the other players they lead to the same probability
distribution over the terminal histories, i.e., for any s−i ∈ S−i
P (σi ,s−i ) (z) = P (βi ,s−i ) (z)
Levent Koçkesen (Koç University)
Extensive Form Games II
for all z ∈ Z
24 / 51
Equivalent Mixed and Behavioral Strategies
Let p ∈ (0, 1) and suppose the behavioral
strategy is
1
L
β(L|{a0 , (a0 , R)}) = p
R
1
L
P β (L) = β1 (L|a0 ) = p
R
P β (R, L) = β1 (R|a0 )β1 (L|(a0 , R)) = (1 − p)p
P β (R, R) = β1 (R|a0 )β1 (R|(a0 , R)) = (1 − p)2
What is the outcome equivalent mixed strategy?
Any mixed strategy is a probability distribution over S1 = {L, R}
Only L and (R, R) can receive positive probability
There is no outcome equivalent mixed strategy
Levent Koçkesen (Koç University)
Extensive Form Games II
25 / 51
Equivalent Mixed and Behavioral Strategies
1
Suppose the mixed strategy is
L
σ1 (Ll) = 1/2, σ1 (Rr) = 1/2
What is the outcome equivalent behavioral
strategy?
β1 (L|{a0 }) =?
R
1
l
r
l
r
β1 (l|{L, R}) =?
Under σ1 decisions at the two information sets are correlated
prob(l|L) = 1, prob(l|R) = 0
In this game no behavioral strategy can reproduce this correlation
◮
Player 1 cannot condition her decision at the second information set on
what she has done before, because she has forgotten what she has done
The previous two games are games with imperfect recall.
Levent Koçkesen (Koç University)
Extensive Form Games II
26 / 51
Perfect Recall
A game has perfect recall if no player forgets what she did or what she
knew. More formally
Definition (Perfect Recall)
Suppose Γ is an extensive form game, x ∈ H and a ∈ A(x) such that
y = (x, a, h) ∈ H for some sequence h and ι(x) = ι(y). Γ has perfect
recall if w ∈ I(y) implies that there exists z ∈ I(x) such that
w = (z, a, h′ ) for some sequence h′ .
x
1
x
L
R
y
L
1
L
1
R
1
w
R
x = a0 , y = (a0 , R), ι(x) = ι(y).
x ∈ I(y) but ∄z ∈ I(x) : x = (z, R, h′ )
Levent Koçkesen (Koç University)
l
r
y
l
r
x = a0 , y = (a0 , R), ι(x) = ι(y).
(a0 , L) ∈ I(y) but ∄z ∈ I(x) : (a0 , L) = (z, R, h′ )
Extensive Form Games II
27 / 51
Equivalent Mixed and Behavioral Strategies
Kuhn proved that in games with perfect recall there is an equivalent
behavioral strategy for every mixed strategy and conversely.
Theorem (Kuhn 1953)
In every finite extensive form game with perfect recall there is an outcome
equivalent behavioral strategy for every mixed strategy and an outcome
equivalent mixed strategy for every behavioral strategy.
Therefore, in extensive form games with perfect recall we are free to work
with either mixed strategies or behavioral strategies; nothing is lost or
added with this choice.
For example existence issue is more easily dealt with using mixed strategies
whereas calculating equilibria is easier with behavioral strategies.
Levent Koçkesen (Koç University)
Extensive Form Games II
28 / 51
Equivalent Mixed and Behavioral Strategies
Given a behavioral strategy βi we can define an outcome equivalent mixed
strategy σi as follows. For any si ∈ Si let
Y
σi (si ) =
βi (si (I)|I)
I∈Ii
There may be other outcome equivalent mixed strategies. Above definition
gives just one.
Levent Koçkesen (Koç University)
Extensive Form Games II
29 / 51
Equivalent Mixed and Behavioral Strategies
In the other direction, suppose we have a mixed strategy σi and we want
to derive an outcome equivalent behavioral strategy βi .
Definition
A pure strategy si is consistent with an information set I if there exists a
history h ∈ I with which si is consistent.
Iisi : set of information sets of i with which si is consistent
Iiσi : set of information sets of i with which some pure strategy in the
support of σi is consistent.
For any I ∈ Ii and a ∈ A(I) define the set of all pure strategies that
specify a at information set I as follows
Si (a) = {si ∈ Si : si (I) = a}
and those that are consistent with I as
Si (I) = {si ∈ Si : I ∈ Iisi }
Levent Koçkesen (Koç University)
Extensive Form Games II
30 / 51
Equivalent Mixed and Behavioral Strategies
For any I ∈ Ii and a ∈ A(I) we define an outcome equivalent βi as
follows:
P

σi (si )



i (a)
 si ∈Si (I)∩S
P
, if I ∈ Iiσi
σi (si )
βi (a|I) =

si ∈Si (I)

P

if I ∈
/ Iiσi
si ∈Si (a) σi (si ),
Levent Koçkesen (Koç University)
Extensive Form Games II
31 / 51
Suppose σ1 (N H) = 1/2, σ1 (M H) = 1/2
1
What is the equivalent behavioral strategy?
N
M
2
S1 = {N H, N T, M H, M T }
−1/2, −1/2
H
supp(σ1 ) = {N H, M H}
T
1
I1σ1 = {{a0 }, {(M, H), (M, T )}}
H
1, −1
T
H
−1, 1 −1, 1
T
1, −1
Since {a0 } ∈ I1σ1
β1 (N |{a0 }) =
P
σ1 (s1 )
s1 ∈S1 ({a0 })∩S1 (N )
P
σ1 (s1 )
s1 ∈S1 ({a0 })
S1 ({a0 }) = {N H, N T, M H, M T },
S1 (N ) = {N H, N T }
σ1 (N H) + σ1 (N T )
1
β1 (N |{a0 }) =
=
σ1 (N H) + σ1 (N T ) + σ1 (M H) + σ1 (M T )
2
Levent Koçkesen (Koç University)
Extensive Form Games II
32 / 51
σ1 (N H) = 1/2, σ1 (M H) = 1/2
1
{(M, H), (M, T )} ∈ I1σ1
N
M
S1 ({(M, H), (M, T )}) = {M H, M T }
2
−1/2, −1/2
S1 (H) = {N H, M H}
H
T
1
H
1, −1
β1 (H|{(M, H), (M, T )}) =
T
H
−1, 1 −1, 1
T
1, −1
σ1 (M H)
=1
σ1 (M H) + σ1 (M T )
Therefore, the equivalent behavioral strategy is
β1 (N |{a0 }) = 1/2, β1 (H|{(M, H), (M, T )}) = 1
Levent Koçkesen (Koç University)
Extensive Form Games II
33 / 51
The equivalent mixed strategy to β1 (N |{a0 }) = 1/2, β1 (H) = 1 is given
by
σ1 (N H) = β1 (N |{a0 })β1 (H|{(M, H), (M, T )}) = 1/2
σ1 (N T ) = β1 (N |{a0 })β1 (T |{(M, H), (M, T )}) = 0
σ1 (M H) = β1 (M |{a0 })β1 (H|{(M, H), (M, T )}) = 1/2
σ1 (M T ) = β1 (M |{a0 })β1 (T |{(M, H), (M, T )}) = 0
Levent Koçkesen (Koç University)
Extensive Form Games II
34 / 51
Extensive Form Game Equilibria
Definition
A Nash equilibrium in mixed strategies of an extensive form game
Γ = (N, H, ι, βc , I, (ui )) is a mixed strategy profile σ ∗ ∈ Σ such that for
every player i ∈ N
∗
∗
Ui (σi∗ , σ−i
) ≥ Ui (σi , σ−i
) for all σi ∈ Σi
A mixed strategy profile σ ∗ is a Nash equilibrium of an extensive form
game Γ if and only if it is a Nash equilibrium of its strategic form
GΓ = (N, (Si ), (Ui )).
A mixed strategy profile σ ∗ is a Nash equilibrium of an extensive form
game Γ if and only if no player has a profitable pure strategy
deviation.
Every finite extensive form game has a mixed strategy Nash
equilibrium.
Levent Koçkesen (Koç University)
Extensive Form Games II
35 / 51
Extensive Form Game Equilibria
Definition
A Nash equilibrium in behavioral strategies of an extensive form game
Γ = (N, H, ι, βc , I, (ui )) is a behavioral strategy profile β ∗ ∈ B such that
for every player i ∈ N
∗
∗
Ui (βi∗ , β−i
) ≥ Ui (βi , β−i
) for all βi ∈ Bi
In extensive form games with perfect recall there is an equivalent
behavioral strategy for every mixed strategy and conversely. This implies
the following.
A behavioral strategy profile β ∗ is a Nash equilibrium of an extensive
form game Γ with perfect recall if and only if no player has a
profitable pure strategy deviation.
Every finite extensive form game with perfect recall has a behavioral
strategy Nash equilibrium.
Levent Koçkesen (Koç University)
Extensive Form Games II
36 / 51
Extensive Form Game Equilibria
Definition
Let Γ = (N, H, ι, βc , I, (ui )) be an extensive form game. A mixed strategy
profile σ ∗ ∈ Σ is a subgame perfect equilibrium in mixed strategies of Γ if
σ ∗ induces a Nash equilibrium in every subgame of Γ.
Definition
Let Γ = (N, H, ι, βc , I, (ui )) be an extensive form game. A behavioral
strategy profile β ∗ ∈ B is a subgame perfect equilibrium in behavioral
strategies of Γ if β ∗ induces a Nash equilibrium in every subgame of Γ.
Every finite extensive form game has a mixed strategy subgame
perfect equilibrium.
Every finite extensive form game with perfect recall has a behavioral
strategy subgame perfect equilibrium.
Levent Koçkesen (Koç University)
Extensive Form Games II
37 / 51
Extensive Form Game Equilibria: Example
Strategic form of the game
2
1
N
NH
NT
1
MH
MT
M
2
H
T
1
1, −1
T
−1/2, −1/2
−1/2, −1/2
−1, 1
1, −1
No pure strategy equilibrium.
−1/2, −1/2
H
H
−1/2, −1/2
−1/2, −1/2
1, −1
−1, 1
T
H
−1, 1 −1, 1
T
N H and N T are never best
response and therefore strictly
dominated.
1, −1
Unique mixed strategy Nash equilibrium is
σ1 (M H) = σ1 (M T ) = 1/2, σ2 (H) = 1/2
Levent Koçkesen (Koç University)
Extensive Form Games II
38 / 51
Extensive Form Game Equilibria: Example
The unique Nash equilibrium is also the unique SPE.
We could have applied the backward induction like algorithm to
calculate SPE
◮
◮
Unique Nash equilibrium of the ‘smallest’ subgame is
σ1 (H) = σ2 (H) = 1/2 with payoff vector (0, 0).
Given that, the unique Nash equilibrium of the reduced game is M
1
1
N
M
N
M
2
−1/2, −1/2
−1/2, −1/2
H
0, 0
T
1
H
1, −1
T
H
−1, 1 −1, 1
Levent Koçkesen (Koç University)
T
1, −1
Extensive Form Games II
39 / 51
Extensive Form Game Equilibria: Example
We can also calculate the SPE in behavioral strategies.
In the subgame that follows M the unique NE in behavioral strategies
is
β1 (H|{(M, H), (M, T )}) = 1/2, β2 (H|{M }) = 1/2
Given that we must have β1 (M |{a0 }) = 1
Unique SPE in behavioral strategies is
β1 (M |{a0 }) = 1, β1 (H|{(M, H), (M, T )}) = 1/2, β2 (H|{M }) = 1/2
It is also the unique Nash equilibrium in behavioral strategies
Not surprisingly, it is outcome equivalent to
σ1 (M H) = σ1 (M T ) = 1/2, σ2 (H) = 1/2
Levent Koçkesen (Koç University)
Extensive Form Games II
40 / 51
An Application: Strategic Delegation in Cournot
The game has the following structure:
1 Firm 1 offers a contract to her manager w : R2 → R , that specifies
+
+
a monetary payment to her as a function of market outcome
2 Manager accepts (y) or rejects (n) the offer
3 We assume that Firm 2 observes the contract and the manager’s
decision
4 If the manager rejects the offer, Firm 1 an Firm 2 play the standard
Cournot game and manager receives his outside option um = 0
5 If the manager accepts the offer, manager and Firm 2 simultaneously
choose output levels q1 , q2 ≥ 0
The profit function of firm i is given by
(
(a − c − b(q1 + q2 ))qi , q1 + q2 ≤ a/b
πi (q1 , q2 ) =
−cqi ,
q1 + q2 > a/b
We assume 5c > a > 2c ≥ 0 and b > 0
Levent Koçkesen (Koç University)
Extensive Form Games II
41 / 51
An Application: Strategic Delegation in Cournot
Payoff functions are given by
(
π1 (q1 , q2 ) − w(q1 , q2 ), if θ = y
u1 (w, θ, q1 , q2 ) =
π1 (q1 , q2 ),
if θ = n
(
w(q1 , q2 ), if θ = y
um (w, θ, q1 , q2 ) =
0,
if θ = n
u2 (w, θ, q1 , q2 ) = π2 (q1 , q2 )
Denote the subgame that follows history (w, y) by Γwy and the one
that follows (w, n) by Γwn
Levent Koçkesen (Koç University)
Extensive Form Games II
42 / 51
An Application: Strategic Delegation in Cournot
The revenue function
Ri (q1 , q2 ) =
(
(a − b(q1 + q2 ))qi ,
0,
q1 + q2 ≤ a/b
q1 + q2 > a/b
For simplicity we will limit the analysis to contracts of the form
wαβ (q1 , q2 ) = β + (1 − α)π1 (q1 , q2 ) + αR1 (q1 , q2 )
where α ∈ [0, 1]
◮
Therefore, a contract can be completely characterized by (α, β).
So, the manager’s payoff function after accepting a contract is
(
β + R1 (q1 , q2 ) − (1 − α)cq1 , q1 + q2 ≤ a/b
um ((α, β), y, q1 , q2 ) =
β − (1 − α)cq1 ,
q1 + q2 > a/b
◮
except for the constant β, this is equivalent to the profit function of
firm 1 with cost parameter (1 − α)c
Levent Koçkesen (Koç University)
Extensive Form Games II
43 / 51
An Application: Strategic Delegation in Cournot
The strategies:
(α, β) ∈ [0, 1] × R, sm (α, β) ∈ {y, n}
q1 ((α, β), y), q2 ((α, β), y) ∈ R+
q1 ((α, β), n), q2 ((α, β), n) ∈ R+
We want to find the pure strategy SPE of this game.
Any SPE must induce a Nash equilibrium in every subgame.
In particular q1 ((α, β), n), q2 ((α, β), n) must be a NE of the standard
Cournot game:
a−c
q1 ((α, β), n) = q2 ((α, β), n) =
, for all (α, β)
3b
Therefore, the equilibrium payoffs in the subgame Γ(α,β)n are as
follows:
(a − c)2
, um ((α, β), n) = 0
u1 ((α, β), n) = u2 ((α, β), n) =
9b
Levent Koçkesen (Koç University)
Extensive Form Games II
44 / 51
An Application: Strategic Delegation in Cournot
NE of the subgame Γ(α,β)y
β has no effect on optimal output level for the manager
Therefore, he acts like a firm 1 with cost parameter (1 − α)c
So, the best response correspondences are
(
a−(1−α)c−bq2
, q2 <
2b
Bm (q2 ) =
0,
q2 ≥
(
a−c−bq1
, q1 < a−c
2b
b
B2 (q1 ) =
0,
q1 ≥ a−c
b
a−(1−α)c
b
a−(1−α)c
b
Therefore, we have
q1 ((α, β), y) =
Levent Koçkesen (Koç University)
a − c + 2αc
a − c − αc
, q2 ((α, β), y) =
3b
3b
Extensive Form Games II
45 / 51
An Application: Strategic Delegation in Cournot
NE payoffs in the subgame Γ(α,β)y
(a − c + 2αc)2
+β
9b
u1 ((α, β), y) = π1 ((α, β), y) − um ((α, β), y)
um ((α, β), y) =
u2 ((α, β), y) =
(a − c − αc)2
9b
where
(a − c − αc)(a − c + 2αc)
9b
The manager will accept (reject) if
π1 ((α, β), y) =
um ((α, β), y) > (<)um ((α, β), n) = 0
and is indifferent in case of equality.
Levent Koçkesen (Koç University)
Extensive Form Games II
46 / 51
An Application: Strategic Delegation in Cournot
Firm 1’s contract decision
The highest payoff that Firm 1 can get by making an offer that the
manager accepts is found by solving the following problem
max{π1 ((α, β), y) − um ((α, β), y)}
(α,β)
s.t. um ((α, β), y) ≥ 0
At the solution the constraint must hold with equality. Substituting
the constraint and the expression for π1 ((α, β), y) in the objective
function the problem becomes
max
(α,β)
(a − c − αc)(a − c + 2αc)
9b
Objective function does not depend on β. If optimal α is in the
interior of [0, 1], then the first order condition must hold with equality
ac − c2 − 4c2 α = 0
which is solved as α =
Levent Koçkesen (Koç University)
a−c
4c ,
which is indeed in the interior of [0, 1].
Extensive Form Games II
47 / 51
An Application: Strategic Delegation in Cournot
Therefore, Firm 1 can get arbitrarily close to the following payoff by
offering a contract that the manager accepts
(a − c)2
8b
The payoff she gets by a contract that the manager rejects is
(a − c)2
9b
This implies that in any SPE the manager must be accepting the
equilibrium contract
Levent Koçkesen (Koç University)
Extensive Form Games II
48 / 51
An Application: Strategic Delegation in Cournot
Since in any SPE in which the manager accepts α = (a − c)/4c, the
following must hold in any SPE
um ((α, β), y) =
(a − c)2
+β ≥ 0
4b
Is it possible that
(a − c)2
+β >0
4b
No, because Firm 1 could decrease β slightly, still get the manager to
accept, and increase payoff. Therefore, in any SPE
β=−
Levent Koçkesen (Koç University)
(a − c)2
4b
Extensive Form Games II
49 / 51
An Application: Strategic Delegation in Cournot
Proposition
The unique SPE is given by
a − c (a − c)2
(α, β) = (
,−
)
4b
( 4c
2
+
β≥0
y, (a−c+2αc)
9b
sm (α, β) =
n, otherwise
a − c − αc
a − c + 2αc
, q2 ((α, β), y) =
q1 ((α, β), y) =
3b
3b
a−c
q1 ((α, β), n) = q2 ((α, β), n) =
3b
Proof.
We have already shown that any SPE must be as in the proposition. It is
easy to verify that this strategy profile is indeed a SPE, completing the
proof.
Levent Koçkesen (Koç University)
Extensive Form Games II
50 / 51
An Application: Strategic Delegation in Cournot
In the unique SPE
q1 =
a−c
a−c
, q2 =
2b
4b
These are exactly the Stackelberg output levels when Firm 1 is the
leader
Firm 1 uses delegation strategically, i.e., as a commitment device
Levent Koçkesen (Koç University)
Extensive Form Games II
51 / 51