Download Lecture Notes on Game Theory

Lecture Notes on Game Theory
Levent Koçkesen
1
Bayesian Games
So far we have assumed that all players had perfect information regarding the elements of
a game. These are called games with complete information. A game with incomplete
information, on the other hand, tries to model situations in which some players have private
information before the game begins. The initial private information is called the type of the
player. For example, types could be the privately observed costs in an oligopoly game, or
privately known valuations of an object in an auction, etc.
1.1
Preliminaries
A Bayesian game is a strategic form game with incomplete information. It consists of:
² a set of players, N = f1; : : : ; ng ;
and for each i 2 N
² an action set, Ai ; (A = £i2N Ai )
² a type set, £i ; (£ = £i2N £i )
² a probability function,
² a payo¤ function,
pi : £i ! 4 (£¡i )
ui : A £ £ ! R:
The function pi summarizes what player i believes about the types of the other players given
her type. So, pi (µ¡i jµi ) is the conditional probability assigned to the type pro…le µ¡i 2 £¡i :
Similarly, ui (ajµ) is the payo¤ of player i when the action pro…le is a and the type pro…le is µ:
In summary, we can de…ne a Bayesian game with the tuple
GB = (N; (Ai ) ; (£i ) ; (pi ) ; (ui )) :
We call a Bayesian game …nite if N; Ai and £i are all …nite, for all i 2 N: A pure
strategy for player i in a Bayesian game is a function which maps player i’s type into her
action set
ai : £i ! Ai :
1
In many situations it is easier to work with in…nite type spaces. In this case the probability
distributions pi must be de…ned on all measurable subsets of £¡i , instead of just individual
elements of £¡i :
We say that beliefs (pi )i2N in a Bayesian game are consistent if there is some common
prior distribution over the set of type pro…les £ such that each player’s beliefs given her
type are just the conditional probability distribution that can be computed from the prior
distribution by Bayes’s rule. In the …nite case, beliefs are consistent if there exists some
probability distribution P in 4 (£) such that,
P (µ)
s¡i 2£¡i P (s¡i; µ i )
pi (µ¡i jµi ) = P
for all µ 2 £ and for all i 2 N:
A mixed strategy for player i is
®i : £i ! 4 (Ai )
so that ®i (ai jµi ) is the probability assigned by ®i to action ai by type µi of player i:
1.2
Bayesian Equilibrium
De…nition 1 A Bayesian equilibrium of a Bayesian game (N; (Ai ) ; (£i ) ; (pi ) ; (ui )) is a
mixed strategy pro…le ® = (®i )i2N ; such that for every player i 2 N and every type µi 2 £i ;
we have
®i (:jµi ) 2 arg max
°24(Ai )
X
µ¡i 2£¡i
pi (µ¡i jµi )
X
a2A
0
@
Y
j2Nnfig
1
®j (aj jµj )A ° (ai ) ui (ajµ) :
Remark 1 Type, in general, can be any private information that is relevant to the player’s
decision making, such as the payo¤ function, player’s beliefs about other players’ payo¤ functions, her beliefs about what other players believe her beliefs are, and so on.
Remark 2 Notice that, in the de…nition of a Bayesian equilibrium we need to specify strategies
for each type of a player, even if in the actual game that is played all but one of these types are
non-existent. This is because, given a player’s imperfect information, analysis of that player’s
decision problem requires us to consider what each type of the other players would do, if they
were to play the game.
Example 2 (Battle of Sexes with incomplete information). Suppose player 2 has perfect
information and two types l and h: Type l loves going out with player 1 whereas type h hates
it. Player 1 has only one type and does not know which type is player 2. Her beliefs place
2
probability 1=2 on each type. The following tables give the payo¤s to each action and type
pro…le:
B
S
B
S
B 2,1 0,0
B 2,0 0,2
S 0,0 1,2
S 0,1 1,0
type l
type h
We can represent this situation as a Bayesian game:
² N = f1; 2g
² A1 = A2 = fB; Sg
² £1 = fxg ; £2 = fl; hg
² p1 (ljx) = p1 (hjx) = 1=2; p2 (xjl) = p2 (xjh) = 1:
² u1 ; u2 are given in the tables above.
Let us …nd the Bayesian equilibria of this game by analyzing the decision problem of each
player of each type:
Player 2 of type l : Given player 1’s strategy ®1 ; his expected payo¤ to
² action B is ®1 (Bjx) ;
² action S is 2 (1 ¡ ®1 (Bjx))
so that his best response is to play B if ®1 (Bjx) > 2=3 and to play S if ®1 (Bjx) < 2=3:
Player 2 of type h : Given player 1’s strategy ®1 ; his expected payo¤ to
² action B is (1 ¡ ®1 (Bjx)) ;
² action S is 2®1 (Bjx)
so that his best response is to play B if ®1 (Bjx) < 1=3 and to play S if ®1 (Bjx) > 1=3:
Player 1: Given player 2’s strategy ®2 (:jl) and ®2 (:jh) ; her expected payo¤ to
² action B is
1
1
®2 (Bjl) (2) + ®2 (Bjh) (2) = ®2 (Bjl) + ®2 (Bjh) ;
2
2
² action S is
1
®2 (Bjl) + ®2 (Bjh)
1
(1 ¡ ®2 (Bjl)) (1) + (1 ¡ ®2 (Bjh)) (1) = 1 ¡
:
2
2
2
3
Therefore, her best response is to play B if ®2 (Bjl) + ®2 (Bjh) > 2=3 and to play S if
®2 (Bjl) + ®2 (Bjh) < 2=3:
Let us …rst check if there is a pure strategy equilibrium in which both types of player 2
play B; i.e. ®2 (Bjl) = ®2 (Bjh) = 1: In this case player 1’s best response is to play B
as well to which playing B is not a best response for player 2 type h: Similarly check that
®2 (Bjl) = ®2 (Bjh) = 0 and ®2 (Bjl) = 0 and ®2 (Bjh) = 1 cannot be part of a Bayesian
equilibrium. Let’s check if ®2 (Bjl) = 1 and ®2 (Bjh) = 0 could be part of an equilibrium. In
this case player 1’s best response is to play B: Player 2 type l’s best response is to play B and
that of type h is S: Therefore,
(®1 (Bjx) ; ®2 (Bjl) ; ®2 (Bjh)) = (1; 1; 0)
is a Bayesian equilibrium.
Clearly, there is no equilibrium in which both types of player 2 mixes. Suppose only type
l mixes. Then, ®1 (Bjx) = 2=3; which implies that ®2 (Bjl) + ®2 (Bjh) = 2=3: This, in turn,
implies that ®2 (Bjh) = 0: Since ®2 (Bjh) = 0 is a best response to ®1 (Bjx) = 2=3; the
following is another Bayesian equilibrium of this game
(®1 (Bjx) ; ®2 (Bjl) ; ®2 (Bjh)) = (2=3; 2=3; 0) :
As an exercise show there is one more equilibrium given by
(®1 (Bjx) ; ®2 (Bjl) ; ®2 (Bjh)) = (1=3; 0; 2=3) :
Example 3 (Cournot Duopoly with incomplete information) The pro…t functions are given
by
ui = qi (µi ¡ qi ¡ qj ) :
Firm 1 has one type µ1 = 1; but …rm 2 has private information about its type µ 2 : Firm 1
believes that µ 2 = 3=4 with probability 1=2 and µ2 = 5=4 with probability 1=2; and this belief is
common knowledge.
We will look for a pure strategy equilibrium of this game. Firm 2 of type µ2 ’s decision
problem is to
max q2 (µ 2 ¡ q1 ¡ q2 )
q2
which is solved at
µ2 ¡ q1
:
2
Firm 1’s decision problem, on the other hand, is
q2¤ (µ 2 ) =
max
q1
½
¾
1
1
q1 (1 ¡ q1 ¡ q2¤ (3=4)) + q1 (1 ¡ q1 ¡ q2¤ (5=4))
2
2
4
which is solved at
q1¤ =
2 ¡ q2¤ (3=4) ¡ q2¤ (5=4)
:
4
Solving yields,
1
11
5
q1¤ = ; q2¤ (3=4) = ; q2¤ (5=4) = :
3
24
24
As an exercise, show that this is the unique equilibrium.
Example 4 (Public Good Contribution) Two players simultaneously decide whether to contribute towards producing a public good. The good is produced if at least one of them contributes. The bene…t derived from the public good is 1 and the cost to player i of contributing
is ci : The payo¤s are depicted in the following table
C
1 ¡ c1 ; 1 ¡ c2
1; 1 ¡ c2
C
N
N
1 ¡ c1 ; 1 :
0; 0
The bene…ts are common knowledge, but each player’s cost is private information. However, both players believe it is common knowledge that each ci is drawn independently from a
uniform distribution on [0; 2]:
Player 1’s decision problem: Action C yields 1 ¡ c1 whereas action N yields
Z
2
1
®2 (Cjc2 ) dc2 :
2
0
Thus, ®1 (Cjc1 ) = 1 if
c1 < 1 ¡
Z
2
0
1
®2 (Cjc2 ) dc2 ´ c¤1 :
2
Similarly for player 2, ®2 (Cjc2 ) = 1 if
c2 < 1 ¡
Since, at c1 =
c¤1 ;
Z
0
2
1
®1 (Cjc2 ) dc2 ´ c¤2 :
2
player 1 is indi¤erent between contributing and not contributing we have
c¤1
=1¡
Z
c¤2
0
Z 2
1
1
c¤
(1)dc2 ¡ ¤ (0)dc2 = 1 ¡ 2 :
2
2
c2 2
Similarly,
c¤2 = 1 ¡
Solving yields,
c¤1
:
2
2
c¤1 = c¤2 = :
3
Notice that in the complete information version with costs ci 2 (2=3; 1) at least one of the
players contribute in equilibrium, whereas in the incomplete information version no player
contributes. Why is the outcome di¤erent in incomplete information version?
5
1.3
Puri…cation of Mixed Strategy Equilibria
Interpreting equilibria in mixed strategies is di¢cult. Not only the idea of individuals ‡ipping
coins to determine their actions is counter-intuitive, but also it is di¢cult to provide reasons
for individuals choosing the exact equilibrium probabilities when they are indi¤erent between
actions.
As a response to these di¢culties Harsanyi (1973) showed that mixed strategy equilibria
can be interpreted as limits of Bayesian equilibria of perturbed games in which each player is
almost always choosing his uniquely optimal action. Harsanyi perturbs the games by slightly
perturbing the payo¤s, calculates the Bayesian equilibria of the games parametrized on the
perturbations and then take the perturbations to zero. (A single sequence of perturbed games
can be used to purify all the mixed strategy equilibria of the limit game.
In this section, rather than delving into the technical details of the argument, we will
demonstrate the idea using an example. Consider the following two-player game
T
B
L
R
0; 0 0; ¡1 :
1; 0 ¡1; 3
In the unique Nash equilibrium of this game, player 1 plays T with probability 3=4 and player
2 plays L with probability 1=2: Now, consider the following perturbed game
T
B
L
"µ1 ; "µ 2
1; "µ2
R
"µ 1 ; ¡1
¡1; 3
where " 2 (0; 1) ; and types µ1 ; µ2 are independently, identically and uniformly distributed
over [0; 1] :
Let’s …rst calculate the Bayesian equilibria of this game. For player 1 expected payo¤ of
playing T is "µ1 ; whereas that of B is
Z
1
®2 (Ljµ2 ) (1) dµ 2 +
0
so that
1
0
(1 ¡ ®2 (Ljµ 2 )) (¡1) dµ2
R1
®2 (Ljµ2 ) dµ2
´ x =) ®1 (T jµ1 ) = 1;
"
R1
¡1 + 2 0 ®2 (Ljµ2 ) dµ2
<
´ x =) ®1 (T jµ1 ) = 0:
"
µ1 >
µ1
¡1 + 2
Z
0
Similarly for player 2,
µ2
R1
®1 (T jµ1 ) dµ 1
´ y =) ®2 (Ljµ2 ) = 1;
"
R
3 ¡ 4 01 ®1 (T jµ1 ) dµ 1
<
´ y =) ®2 (Ljµ2 ) = 0:
"
µ2 >
3¡4
0
6
At µ1 = x player 1 is indi¤erent, i.e.,
"x = ¡1 + 2
½Z
y
0
(0) dµ 2 +
= ¡1 + 2 (1 ¡ y) ;
Z
y
1
(1) dµ2
¾
and at µ2 = y player 2 is indi¤erent, i.e.,
"x = 3 ¡ 4
½Z
x
0
(0) dµ1 +
= 3 ¡ 4 (1 ¡ y) :
Z
1
x
(1) dµ 1
¾
Solving for x and y; we get
2+"
8 + "2
4¡"
y =
:
8 + "2
x =
So, the Bayesian equilibrium satis…es,
2+"
=) ®1 (T jµ1 ) = 1;
8 + "2
2+"
<
=) ®1 (T jµ1 ) = 0;
8 + "2
µ1 >
µ1
and
4¡"
=) ®2 (Ljµ2 ) = 1;
8 + "2
4¡"
<
=) ®2 (Ljµ2 ) = 0:
8 + "2
µ2 >
µ2
®1 (:jµ1 ) and ®2 (:jµ2 ) can be chosen arbitrarily in the zero probability events that µ1 =
4¡"
and µ2 = 8+"
2 : So, the equilibrium is almost always unique and is a pure strategy
equilibrium. Notice that as " goes to zero, the equilibrium converges to the unique mixed
strategy equilibrium of the original game.
Therefore, one can interpret randomization by players as depending on minor factors that
have been omitted from the description of the game. The players almost always play pure
strategies, but they condition these strategies upon un-modeled random events which have
very slight e¤ects on the game.
2+"
8+"2
1.4
Auctions
Auctions are means of allocating goods such as works of art, government bonds, in which the
individual who declares to pay the highest price (i.e., the highest bidder) obtains the good.
7
The bids may be called sequentially or may be submitted in sealed envelopes, and the price
paid by the winner may be equal to the highest bid or some other price.
Important questions to ask in analysis of auctions are: Is the auction e¢cient? If not, does
there exist a more e¢cient auction design? Which auction design raises more revenue? The
following two examples provides an introduction to the subject.
Previously, we have looked at two forms of auctions, namely First Price and Second Price
Auctions, in a complete information framework in which each bidder knew the valuations
of every other bidder. In this section we relax the complete information assumption and
revisit these two form of auctions. In particular, we will assume that each bidder knows only
her own valuation, and the valuations are independently distributed random variables whose
distributions are common knowledge.
The following elements de…ne the general form of an auction that we will analyze:
² Set of bidders, N = f1; 2; : : : ; ng ;
and for each i 2 N
² a type set (set of possible valuations), £i = [v; v¹] ; v ¸ 0:
² an action set, Ai = R+ (actions are bids)
² a belief function: player i believes that her opponents’ valuations are independent draws
from a distribution function F that is strictly increasing and continuous on [v; v¹] :
² a payo¤ function, which is de…ned for any a 2 A; v 2 £ as follows
ui (a; v) =
8
<
vi ¡P (a)
;
m
: 0;
if aj · ai for all j 6= i; and jfj : aj = ai gj = m
if aj > ai for some j 6= i
where P (a) is the price paid by the winner if the bid pro…le is a: Notice that in the case
of a tie the object is divided equally among all winners.
1.4.1
Second Price Auctions
In this design, highest bidder wins and pays a price equal to the second highest bid. Although
there are many Bayesian equilibria of second price auctions, bidding own valuation vi is weakly
dominant for each player i: To see this let x be the highest of the other bids and consider
bidding a0i < vi ; vi ; and a00i > vi : Depending upon the value of x; the following table gives the
payo¤s to each of these actions by i
a0i
vi
a00i
a0i < x < vi
x · a0i
win/tie;pay x lose
win; pay x
win; pay x
win; pay x
win; pay x
x = vi
lose
tie; pay vi
win; pay vi
8
vi < x · a00i
lose
lose
win/tie;pay x
a00i < x
lose
:
lose
lose
1.4.2
First Price Auctions
In …rst price auctions, the highest bidder wins and pays her bid. Let us denote the bid of
player with type vi by ¯ i (vi ) and look for symmetric equilibria, i.e. ¯ i (v) = ¯ (v) for all
i 2 N: First, although we will not attempt to do so here, it is possible to show that strategies
¯ i (vi ) ; and hence ¯ (v) ; are strictly increasing and continuous on [v; v¹] :(see Fudenberg and
Tirole, 1991). So, let’s assume that they are, and check if they are once we locate a possible
equilibrium.
Let G¯ (b) be the probability that the bid, under strategy ¯; is at most b; i.e.,
G¯ (b) = prob (¯ (v) · b)
³
´
= prob v · ¯ ¡1 (b)
³
´
= F ¯ ¡1 (b) :
(1)
(we know that ¯ is invertible since it is strictly increasing).
The expected payo¤ of player with type v who bids b is given by
(v ¡ b) prob (highest bid is b) = (v ¡ b) (G¯ (b))n¡1 :
(2)
(Because vi are independently distributed).
Now, let Bv (¯) be the best response of player with type v when all the other players play
according to ¯: Then, it has to satisfy the FOC:
¡ (G¯ (Bv (¯)))n¡1 + (v ¡ Bv (¯)) (n ¡ 1) (G¯ (Bv (¯)))n¡2 G0¯ (Bv (¯)) = 0:
(3)
So, let (¯ ¤ ; : : : ; ¯ ¤ ) be an equilibrium. Then,
Bv (¯ ¤ ) = ¯ ¤ (v) for all v 2 [v; v¹] :
Therefore, by (1), we have
³
´
G¯ ¤ (Bv (¯ ¤ )) = F ¯ ¤¡1 (¯ ¤ (v)) = F (v)
and, for any ¯
³
´³
³
´;
G0¯ (b) = F 0 ¯ ¡1 (b)
=
³
´
F 0 ¯ ¡1 (b)
¯ 0 ¯ ¡1 (b)
´0
¯ ¡1 (b)
by the fact that ¯ is almost everywhere di¤erentiable (since it is strictly increasing), and by
the inverse function theorem. Therefore, substituting ¯ ¤ (v) for b
G0¯ ¤ (¯ ¤ (v)) =
=
³
´
F 0 ¯ ¤¡1 (¯ ¤ (v))
³
´
¯ ¤0 ¯ ¤¡1 (¯ ¤ (v))
F 0 (v)
:
¯ ¤0 (v)
9
Therefore, from (3). we get
¯ ¤0 (v) (F (v))n¡1 + (n ¡ 1) ¯ ¤ (v) (F (v))n¡2 F 0 (v) = (n ¡ 1) v (F (v))n¡2 F 0 (v)
which is a di¤erential equation in ¯ ¤ : Integrating both sides, we get
¤
n¡1
¯ (v) (F (v))
=
Z
v
v
(n ¡ 1) x (F (x))n¡2 F 0 (x) dx
= v (F (v))n¡1 ¡
Solving for ¯ ¤ (v) ;
Rv
v
¤
¯ (v) = v ¡
Z
v
(F (x))n¡1 dx:
v
(F (x))n¡1 dx
(F (v))n¡1
:
One can easily show that ¯ ¤ is continuous and strictly increasing in v as we hypothesized.
Furthermore, notice that ¯ ¤ (v) = v; but ¯ ¤ (v) < v for v > v: That is, except the player with
the lowest valuation, everybody bids less than her valuation.
As an exercise, calculate ¯ ¤ assuming F is uniform on [0; 1]:
2
Correlated Equilibrium
Suppose players can condition their actions on random outcomes. (Say, before the game they
can agree, in a non-binding way, to use certain actions if they observe certain signals sent by
a random signal generator).
Aumann (1974, J. Math. Econ.) introduced the notion of correlated equilibrium to capture
this idea. He used the following example:
U
D
L
R
5; 1 0; 0
4; 4 1; 5
This game has two pure Nash equilibria: (U; L) and (D; R) and a mixed strategy equilibrium in which players play their two actions with equal probabilities. The mixed strategy
equilibrium has the payo¤ pro…le (2:5; 2:5):
Now, suppose they agree to ‡ip a coin and if it is heads player 1 agrees to play U and
player 2 agrees to play L; whereas if it is tails, player 1 agrees to play D and player 2 agrees
to play R: They then play the above game. Suppose now that the coin ‡ip results in heads.
Given that player 2 plays L; it is a best response for player 1 to play U ; and given that player
1 plays U it is a best response for player 2 to play L: Similarly, if it is tails, then neither player
has an incentive to deviate from the pre-play (non-binding!) agreement. Therefore, outcome
(U; L) with probability 1=2 and (D; R) with probability 1=2 is a Nash equilibrium outcome of
10
this extended game. (Note that there is no independent randomizations by players which can
generate this outcome): The expected payo¤ at this equilibrium is (3; 3) :
Remark 3 Any payo¤ pro…le in the convex hull of the Nash equilibrium payo¤ pro…les can
be obtained if players can condition their strategies on publicly observable random events.
Furthermore, only those outcomes can be obtained.
Exercise 2.1 Prove the above claim.
However, players can do even better by using a device that sends di¤erent but correlated
signals. To illustrate, suppose there three equally likely states: A, B, and C. Also let player 1’s
information partition to be ffAg; fB; Cgg and that of player 2 to be ffA; Bg; fCgg: Suppose
they agree on the following:
Player 1 :
fAg ! U; fB; Cg ! D
Player 2 :
fA; Bg ! L; fCg ! R:
It is easy to verify that this is a Nash equilibrium: if state A occurs player 1’s best response
is to play U: If state B or C occurs, then player 1 does not know which one occurred. She can
calculate the conditional probability that player 2 observed fA; Bg as
prob (fA; Bg \ fB; Cg)
prob (fB; Cg)
1=3
=
1=3 + 1=3
1
=
:
2
prob (fA; Bg j fB; Cg) =
Therefore, she assesses the probability that player 2 will play L as 1=2 to which D is a best
response. Verify for player 2 yourself.
Notice that:
prob (U; L) = prob (fAg \ fA; Bg) = 1=3
prob (U; R) = prob (fAg \ fCg) = 0
prob (D; L) = prob (fB; Cg \ fA; Bg) = 1=3
prob (D; R) = prob (fB; Cg \ fCg) = 1=3
which leads to an expected payo¤ of 3 13 for each player, which is outside the convex hull of
Nash equilibrium payo¤s.
Remark 4 Original Nash equilibria are still equilibria of the extended game because if you
ignore your signal ignoring mine is a best response.
11
De…nition 5 A correlated equilibrium of a strategic form game (N; (Ai ) ; (ui )) consists of
² a …nite probability space (­; ¼) where ­ is a set of states, and ¼ is a probability measure
on ­;
² for each i; a partition Pi of ­ (this is called player i’s information partition)
² for each i; a function
¾ i : ­ ! Ai
with ¾ i (!) = ¾ i (! 0 ) whenever ! 2 Pi and ! 0 2 Pi for some Pi 2 Pi : (¾ i is player i’s
strategy)
such that for every i 2 N and every function ¿ i : ­ ! Ai for which ¿ i (!) = ¿ i (! 0 )
whenever ! 2 Pi and ! 0 2 Pi for some Pi 2 Pi we have
X
!2­
¼ (!) ui (¾ ¡i (!) ; ¾ i (!)) ¸
X
!2­
¼ (!) ui (¾ ¡i (!) ; ¿ i (!)) :
Example 6 Let us formally express the correlated equilibria of the opening example:
The …rst equilibrium is given by
² ­ = fH; T g ; ¼ (H) = ¼ (T ) = 1=2;
² P1 = P2 = ffHg ; fT gg ;
² ¾ 1 (H) = U; ¾ 1 (T ) = D;
¾ 2 (H) = L; ¾ 1 (T ) = R:
The second equilibrium is given by
² ­ = fA; B; Cg ; ¼ (A) = ¼ (B) = ¼ (C) = 1=3;
² P1 = ffAg ; fB; Cgg ; P2 = ffA; Bg ; fCgg ;
² ¾ 1 (A) = U; ¾ 1 (B) = ¾ 1 (C) = D;
¾ 2 (A) = ¾ 2 (B) = L; ¾ 2 (C) = R:
The following is an alternative de…nition of correlated equilibrium which is sometimes more
useful in calculating the correlated equilibria of particular games.
12
De…nition 7 A correlated equilibrium is any probability distribution p over A such that,
for every i 2 N we have
X
a2A
or
p (a) ui (a¡i ; ai ) ¸
X
a¡i 2A¡i
X
a2A
p (a) ui (a¡i ; ai ) ¸
p (a) ui (a¡i ; di (ai )) for all di : Ai ! Ai ;
X
a¡i 2A¡i
p (a) ui (a¡i ; a0i ) for all ai ; a0i 2 Ai :
Common Values and Winner’s Curse: the example I gave in the class.
13