8.1. Independent Firms
The downstream firm’s problem is
max
x
(a?bx)x?wx.
The upstream firm’s problem is
max
x
(a?2bx)x?cx.
The output is
y
?
=
a?c
4b
.
8.2. Integrated Firm
Suppose now that the two firms merge into one firm. This firm’s problem is
max
y
(a?by)y?cy.
The output is
ˉy =
a?c
2b
.
The integrated firm produces twice as much. Why?
In general, an integrated monopolist will always produce more than an upstream-
downstream pair of monopolists. With an upstream-downstream pair, the upstream
monopolist raises its price above its MC and then the downstream monopolist raises its
price again above its already inflated MC.
2—16
Chapter 3
Game Theory
Micro Theory, 2005
The materials are from MWG (1995), Chapters 7—9, 219—305.
1. Two Game Forms
1.1. Extensive-Form Game
The extensive form is a game tree.
Example 3.1. Matching Pennies (B). Two players. Player 1 puts a penny down first.
Then, after seeing player 1’s choice, player 2 puts her penny down. If the two pennies
match, player 1 pays $1 to player 2; otherwise player 2 pays $1 to player 1. Write down
the extensive-form game.
Example 3.2. Matching Pennies (C). This game is just like version B except that
when player 1 puts her penny down, she keeps it covered with her hand. Write down the
extensive-form game.
Definition 3.1. Agameinextensive form consists of the following items:
1. Sets. A finite set of nodes X, a finite set of possible actions A, a finite set of players
N = {1,...,n}, and a collection of information sets H.
2. Sequence. A game starts from a single node. Except the initial node, each node
follows from a single immediate predecessor node. The set of terminal nodes is T.
All other nodes in X are called decision nodes.
3—1
3. Information Structure. Each node belongs to one and only one information set.
Denote H(x) as the information set that contains node x. When H(x) is a singleton,
i.e., H(x)={x}, we often refer to H(x) as node x. Each information set is followed
by a few branches. Each branch represents a possible action taken by the player who
is to make a decision upon observing that information set, i.e., when the play reaches
that information set. For H ∈ H, let A(H)={all the branches following H}. If
it is player i’s turn to make a move at an information set H, we call H player i’s
information set. Each information set belongs to one and only one player, including
a special player called nature.
4. Nature. Sometimes nature is included. Let H
0
be the information set where nature
makes a move. A function ρ : A(H
0
) → [0, 1] assigns probabilities to actions at
information set H
0
. Nature is like a player in the model, except that it does not have
apayo? function and it does not optimize its choices.
5. Payo?s. A collection of payo? functions U = {u
1
(·),...,u
n
(·)} assigns utilities to
the players at each terminal node, u
i
: T→ R.
Thus, a game in extensive form is specified by the collection
Γ
E
= {X,A,N,H,U, ρ(·),A(·),H(·)}.
Agameisofperfect information if each information set contains a single decision
node. Otherwise, it is a game of imperfect information.
Thegamestructureiscommon knowledge, meaning that all players know the
structure of the game, know that their rivals know it, know their rivals know that they
know it, and so on.
1.2. Normal-Form Game
A strategy is a complete contingent plan that specifies how a player will act in every
possible distinguishable circumstance. Thus, a player’s strategy is a specification of how
he plans to move at each of his information set.
Definition 3.2. A(pure)strategy for player i is a function s
i
: H
i
→ A such that
s
i
(H) ∈ A(H) for all H ∈H
i
.
Denote S
i
as the strategy space of player i, which contains all the possible strategies
of player i.
3—2
Example 3.3. Strategies for Matching Pennies (B). Write down the strategies.
Example 3.4. Strategies for Matching Pennies (C). Write down the strategies.
Denote a profile of strategies as s =(s
1
,...,s
n
), where s
i
∈ S
i
is a strategy from
player i. The normal form or strategic form of a game is to specify a game in terms of
strategies and their associated payo?s.
Definition 3.3. For a game with n players, the normal form of a game specifies for
each player i a set of strategies S
i
and a payo? function u
i
(s
1
,...,s
n
). Formally, we
write the game as Γ
N
=[N,{S
i
},{u
i
}].
Example 3.5. The Normal Form of Matching Pennies (B). Write down the
normal-form game.
Example 3.6. The Normal Form of Matching Pennies (C). Write down the
normal-form game.
Inthenormalform,thegamelookslikeasimultaneous-movegame.
For any extensive form of a game, there is a unique normal form. However, the
converse is not true, because the condensed representation of a game in the normal form
generally omits some of the details present in the extensive form.
1.3. Mixed Strategy
Definition 3.4. Given a normal-form game Γ
N
=[N,{S
i
},{u
i
}], for S
i
=
{s
1i
,...,s
n
i
i
}, we denote a mixed strategy as σ
i
=(σ
1i
,...,σ
n
i
i
), where σ
ki
is the
probability that s
ki
is taken. That is, a mixed strategy σ
i
is a probability distribution
over the pure strategies in S
i
. Denote the mixed extension of S
i
as
null(S
i
)=
+
(σ
1i
,...,σ
n
i
i
) ≥ 0
n
i
[
k=1
σ
ki
=1
,
.
We sometimes denote σ
ki
as σ
i
(s
ki
), i.e., σ
i
(s
ki
) is the probability that the mixed
strategy σ
i
assigns to the pure strategy s
ki
∈S
i
.
For an extensive-form game, there is a simpler way that a player can randomize.
She could randomize separately over the possible actions at each of her information sets
H ∈H
i
. This is called a behavior strategy.
3—3
Definition 3.5. GivenanextensiveformgameΓ
E
, a behavior strategy for player
i specifies, for every information set H ∈ H
i
and action a ∈ A(H), a probability
λ
i
(a,H) ≥ 0, with
S
a∈A(H)
λ
i
(a,H)=1 for all H ∈H
i
.
For games of perfect recall, the two types of randomization are equivalent. Because of
this, we typically use behavior strategies for extensive-form games and mixed strategies
for normal-form games. In fact, we will refer to behavior strategies as mixed strategies.
2. Simultaneous-Move Games
We study simultaneous-move games using the normal form. We introduce four equi-
librium concepts: Nash equilibrium, dominant-strategy equilibrium, and trembling-hand
NE.
The expected utility function is
u
i
(σ
i
,σ
?i
)=
n
1
[
k
1
=1
···
n
n
[
k
n
=1
σ
k
1
1
···σ
k
n
n
u
i
(s
k
1
1
,...,s
k
n
n
)
=
[
s
i
∈S
i
,i∈N
σ
1
(s
1
)···σ
n
(s
n
)u
i
(s
1
,...,s
n
).
2.1. Nash Equilibrium
Definition 3.6. Astrategyprofile s
?
=(s
?
1
,...,s
?
n
) is a Nash equilibrium (NE) in
Γ
N
=[N, {S
i
}, {u
i
}] if for every i ∈N,u
i
(s
?
i
,s
?
?i
) ≥ u
i
(s
i
,s
?
?i
) for all s
i
∈S
i
.
Definition 3.7. A mixed strategy profile σ
?
=(σ
?
1
,...,σ
?
I
) is a Nash equilibrium
(NE) in Γ
N
=[N, {null(S
i
)}, {u
i
}] if for every i ∈ N,u
i
(σ
?
i
,σ
?
?i
) ≥ u
i
(σ
i
,σ
?
?i
) for all
σ
i
∈null(S
i
).
Let S
i
= {s
1i
,s
2i
,...,s
n
i
i
}. Given a NE σ
?
, let S
+
i
be the set of pure strategies that
player i plays with a positive probability σ
?
ki
> 0 under σ
?
, and let S
0
i
be the set of
pure strategies that player i plays with probability zero σ
?
ki
=0 under σ
?
.
Proposition 3.1. Strategy profile σ
?
=(σ
?
1
,...,σ
?
I
) is a NE in Γ
N
=[N, {null(S
i
)}, {u
i
}]
i? for each i ∈N,
(i) u
i
(s
ki
,σ
?
?i
)=u
i
(s
ji
,σ
?
?i
) for all s
ki
,s
ji
∈S
+
i
, (2.1)
(ii) u
i
(s
ki
,σ
?
?i
) ≥ u
i
(s
ji
,σ
?
?i
) for all s
ki
∈S
+
i
and s
ji
∈S
0
i
. (2.2)
3—4