game theory lecture notes introduction - muhamet yildiz

14.12 G am e Theory L ecture N otes
Introd uction
M u h a met Yild iz
(Lecture 1)
Gam e Theor y is a misnomer for Multiperson Decision Theory, analyzing the decision-
making process when there are more than one decision-m akers where eac h agent’s pa yo ff
possibly depend s on the actions taken b y the other agen ts. Since an agen t’s preferences
on his actions depend on whic h actions the other parties take, his action depends on his
beliefs about what the oth ers do . Of course, wh at the others do depends on their beliefs
about what eac h agent does. In this w a y, a player’s action, in principle, depends on the
actions available to eac h agent, eac h agent’s preferences on the outcom es, each player’s
beliefs a bout w hich actions are available to eac h pla yer and h ow each player ranks th e
outcomes, and further his beliefs about eac h player ’s beliefs, ad infinitum.
Under perfect com petition, there are also more than one (in fact, infinitely many )
decision makers. Yet, their decision s are assum ed to be decentra lized. A consumer tries
to choose the best consumption bundle that he can afford, giv en the prices — without
pa ying atten tion what the other consumers do. In reality, the future prices are not
know n. C onsu mers’ decisions depend on their expectations about the future prices. A n d
the futu re p rices depend on con sumers’ d ecision s today. O nce again, even i n perfectly
competitive enviro nment s , a consum er’s decisio ns are affected by their beliefs about
what other c onsumers do — in a n aggregate level.
Wh en agen ts think through what the other players w ill do, taking wh at the other
players think about them into accoun t, they may findaclearwaytoplaythegame.
Consid er the follow ing “game”:
1
1 \ 2L m R
T (1, 1) (0, 2) (2, 1)
M (2, 2) (1, 1) (0, 0)
B (1, 0) (0, 0) (−1, 1)
Here, Players 1 has strategies, T, M, B and Player 2 has strateg ies L, m, R. (They
pick their strategies simultaneously.) The pa y offs for players 1 and 2 are indicated by

the numbers in pa rent heses, the first one for player 1 an d the secon d one for player 2.
For instance, if P layer 1 plays T and Player 2 pla ys R, then Player 1 gets a payo ff of 2
and Playe r 2 gets 1. L et’s assum e that each pla yer kno w s that these are the strategies
and the payo ffs, each pla yer know s that eac h pla ye r kno w s this, each pla ye r kno w s that
each player kn ows that each player kno w s this, ad infinitum.
Now, player 1 looks at his payoffs, and realizes that, no matter what the other player
plays, it is better for him to play M rather than B. Th at is, if 2 play s L, M gives 2 and
B gives 1; if 2 play s m, M gives 1, B gives 0; an d if 2 pla y s R, M gives 0, B gives -1.
Therefor e, he realizes that he sho uld not play B.
1
NowhecomparesTandM.Herealizes
that, i f Player 2 plays L or m, M i s better than T, but if she p lay s R, T is definitely
better than M . Would P la y er 2 pla y R? W hat would she pla y? To find an answer to
these questions, Player 1 looks at th e g am e f rom Player 2’s point o f view . He realizes
that, for P layer 2, there is no s tr ategy that is outright better th an any other strategy.
For instance, R is the best strategy if 1 plays B, but oth er wise it is strictly worse t han
m. Wou ld Player 2 think that Player 1 wou ld play B? We ll, she kno w s that Pla ye r 1 i s
trying to maximize his expected payoff,givenbythefirst en tries as everyone know s. She
m u st then de duce that P la yer 1 will no t play B. T herefore, Play er 1 concludes, she will
not pla y R (as it is w orse than m in t his case). Ruling out the possibilit y that Play er 2
play s R, Player 1 looks at his pa yoffs, and sees that M is now better than T, no matter
what. On the other side, Player 2 goes through similar reasoning, and concludes that 1
mu st play M , and therefore pla ys L.
This kind of reasoning does not always yield suc h a c lear prediction. Imagine that
y ou w ant to meet with a friend in one of two places, about which y ou both are indifferent.
Unfortunately, you cannot commun icate with eac h other un til you meet. This situation
1
After all, he cannot have any belief about what Player 2 plays that would lead him to play B when
M is available.
2

is forma lized in the following game, wh ich is called pure coor dination gam e:
1
\ 2Left Right
Top (1,1) (0,0)
Bottom (0,0) (1,1)
Here, Pla ye r 1 c h ooses bet ween Top and Bottom rows, wh ile Player 2 c h ooses between
Left and Right column s. In each box, the first and the second num bers denote the von
Neumann-Mo rgenstern utilities of players 1 and 2, respectively. Note that Player 1
prefers Top to Bottom if he kno ws that Player 2 pla ys Left; he prefers Bottom if he
knows that Player 2 plays Right. He is indifferen t if he t hinks that the other p la yer is
lik ely to play either strategy with equal probab ilities. Similarly, Player 2 prefers Left if
she knows that player 1 pla ys Top. There is n o c lear p rediction about the outcome o f
this game.
One ma y look for the stable outcomes (strategy profiles) in the sense that no player
has incentiv e to deviate if he knows that the other pla y ers pla y the prescribed strategies.
Here, Top-Left and Bottom-Righ t are suc h outcomes. But Bottom-Left and Top-Right
are not stable in this sense. Fo r instan ce, if Bottom -L eft is kno wn to be played, eac h
player would like to deviate — as it is sho wn in the following figure:
1
\ 2Left Right
Top (1,1) ⇐⇓(0,0)
Bottom (0,0) ⇑=⇒ (1,1 )
(Here, ⇑ means pla yer 1 deviates to Top , etc.)
Un like in this gam e , m ostly players have differen t p references on the outcomes, in-
ducing conflict. In the follo w ing gam e, which is known as the Battle of Sexes,conflict
and the need for coordination are present together.
1
\ 2Left Right
Top (2,1) (0,0)
Bottom (0, 0) (1 ,2)

Here, once again play ers would like to coordinate on Top-Left or Bottom-Righ t, but
now Pla yer 1 prefers to coordinate on Top-L eft, wh ile Player 2 prefers to coordinate on
Bottom -Right. The stable outcomes are again Top-Left and Bottom - Right .
3
2
1
2
TB
L LRR
(2,1) (0,0) (0,0) (1,2)
Figure 1:
No w, in the Battle of S exes, imagine th at Pl a y er 2 kno ws wha t Pl a y er 1 does w hen
she takes her action. This can be form alized via the follow ing tree:
Here, Pla ye r 1 chooses between Top a n d Bottom , t h en (knowing what Player 1 has
c ho sen ) Playe r 2 ch ooses between Left a nd Right. Clearly, now Pla ye r 2 w o uld choose
Left if Player 1 p lay s Top, and c hoose Right if Player 1 play s B o ttom. K nowing this,
Player 1 w ould play Top. Therefore, one can argue that the only reasonable outcome of
this game is To p-L eft. (This kind of reasoning is called backward indu ction.)
W hen Player 2 is to ch eck wh at the other pla yer d oes, he gets only 1, while Player 1
gets 2. (In the previous game, two outcomes were stable, in whic h Pla yer 2 wou ld get 1
or 2.) That is, Player 2 prefers that P layer 1 has infor m ation about what Player 2 does,
rather than she herself has inform ation about what pla yer 1 does. When it is common
knowledgethataplayerhassomeinformationornot,theplayermayprefernottohave
that inform a tion — a robust fact that we will see in various contexts.
Exercise 1 Clearly, this is generated by the fact that Player 1 know s that Player 2
will know what Player 1 does when she moves. Consid er the situation that Player 1
thinks that Play er 2 will k n ow what P la yer 1 does o n ly with p robability π<1,andthis
prob ability does not depend on what Player 1 does. What w ill happe n i n a “re asonable”
equilibrium? [By the end of this course, hopefully, you will be able to formalize this
4

situatio n, and compu te the equilibr ia .]
Anoth er in terp reta tion is tha t Player 1 can co m municate to Player 2, w h o cannot
comm unicate to play er 1. This enables pla yer 1 to commit to his actions, pro viding a
strong position in the relation.
Exercise 2 Consider the following version of t he last game: after knowing what Player
2 does, Player 1 gets a chance to change his action; then, the game ends. In other words,
Player 1 chooses b etween Top and Bottom ; knowing P layer 1’s choice, P layer 2 chooses
between Left and Right; knowing 2’s choice, Player 1 d ecides whether to stay w here he
is or to change his position. What is the “reasonable” outcome? What wou ld happen if
changin g his action would cost player 1 c utiles?
Imagin e that, before playin g the Battle of Sexes, Player 1 has the o p tion of e xitin g,
in wh ich c a se e a ch p layer will get 3 / 2, or playing the B attle o f S exes. When asked to
pla y, Player 2 will know that Pla yer 1 c ho se to play the Battle of Sexes.
Ther e are two “reasonable” equilibria (or stab le outco m es). One is that Player 1
exits, thinking that, if he plays the Battle of Sexes, they will p lay the Bottom -R ight
equilibrium of the Battle of Sexes, yielding only 1 for pla yer 1. The second one is
that Player 1 c hooses to P lay the Battle o f Sexes, a nd in the B attle of Sexes they play
To p-Left equilib rium.

2
1
Left Right
Top (2,1) (0,0)
Bottom (0,0) (1,2)

1
Play
Exit
(3/2,3/2)
Some would argue that the first outcome is n ot really reasonable? Beca use, when

askedtoplay,Player2willknowthatPlayer1haschosentoplaytheBattleofSexes,
forgo in g the pa yoff of 3/2. She must therefore realize t hat Pla ye r 1 ca nn ot possibly be
5
planning to play Bottom, which yields the pay off of 1 max. That is, when asked to pla y,
Player 2 s hould understand t hat Pla yer 1 is planning t o play Top, a nd thus she should
play L eft. Ant icipating this, Player 1 s hould c hoose to pla y the Battle of Sexes game,
in whic h they play Top-Left. Therefor e, the second outcome is the only reasonab le one.
(This kind of reasoning is called Forwar d Induction.)
Here are some more examples of gam es:
1. Prisoners’ Dilem ma:
1 \ 2 C onfess Not Confess
Confess (-1, -1) (1, -10)
Not Co nfess (-10, 1) (0, 0)
This is a well known game that most of yo u kno w . [It is also discussed in Gibbons.]
In this game no matter what the other pla yer does, eac h pla yer w ould like to
confess, yielding (-1,-1), which is do m ina ted by (0,0).
2. Ha w k-D ove game
1
\ 2Hawk Dove
Ha wk
¡
V −C
2
,
V −C
2
¢
(V , 0)
Dov e (0,V ) (
V

2
,
V
2
)
This is a generic biological game, but is also quite similar to many games in
econom ics and political science. V is the v alue of a resource that one of the pla yers
will enjo y. If th ey shar ed the resource, th eir values are V/2. Hawk stands for
a “ tou gh” strategy, w h ereby the player does not give up the resource. H owever,
if the other play er is also playing haw k, th ey end up fighting, and incur t he cost
C/2 each . O n the other hand, a Hawk player gets the whole resource for itself
when pla ying a Dov e. When V>C, w e ha ve a P risoners’ Dilemma game, where
we would observe figh t.
When w e have V<C,sothatfigh ting is costly, this game is similar to another
well-k nown game, inspired by the m ovie Rebel W itho ut a C a use , named “C hicken”,
where t wo players driving towa rds a cliff have to decid e wheth er to stop or continue.
The one who s tops first l oses face, but may save his life. More generally, a class
of games called “wars of attrition” are used to model this type of situations. In
6
this case, a player would like to play Ha wk i f his opponent play s Dove , an d play
Do v e if hi s opponen t plays Ha wk.
7
14.12 G am e Theory L ecture N otes
Theory of Choice
M u h a met Yild iz
(Lecture 2)
1 The basic theory of c hoice
We consider a set X of altern atives. Alternative s are mutually exclu sive in th e sense
that one cannot choose t wo distinct alternatives at the same time. We also take the set
of feasible alterna tive s exhaustive so that a pla yer’s cho ices w ill always be defined. Note

that this is a ma tter of modeling. For instanc e, if we have options Coffee and Tea, we
define alternatives as C = Coffee but no Tea, T = Tea but no Coffee, CT = Coffee and
Te a, and NT = no Co ffee and no Tea.
Take a relation º on X. Note that a relation on X is a subset of X × X.Arelation
º is said to be complete if and on ly i f, give n any x, y ∈ X,eitherx º y or y º x.A
relation º is said to be transitive if and only if, given any x, y, z ∈ X,
[x º y an d y º z] ⇒ x º z.
Arelationisapreference relation if and only if it is complete and transitiv e. Giv en any
preference relation º,wecandefin e strict preference  by
x  y ⇐⇒ [x º y
and y 6º x],
and the indifference ∼ by
x ∼ y ⇐⇒ [x º y and y º x].
Apreferencerelationcanberepresen ted by a ut ility functio n u : X → R in the
follow in g sense:
x º y ⇐⇒ u(x) ≥ u(y) ∀x, y ∈ X.
1
The follo w ing theorem states further that a relation needs to be a preference relation in
order to be represented by a utility function.
Theorem 1 Let X be finite. A re latio n ca n be presented by a utility functio n if and only
if it is co mple te and tra n s itiv e. Moreover, if u : X → R represents º,andiff : R → R
is a stric tly increasin g function, then f ◦ u also represents º.
By the last statement, we call such utility functions ordin al.
In order to u se th is ord inal theor y of ch oic e, w e sh ould k now the agent’s preferenc es on
the alternatives. As w e have seen in the previous lecture, in game theory, a player ch ooses
between his strategies, and his preferences on his strategies depend on the strategies
pla yed b y the other players. Typ ically, a player does not know which strategies the
other pla yers pla y. Therefore, we need a theory of decisio n-m akin g under uncertainty.
2 Decision-making under uncertainty
We con sider a finite set Z of prizes, and the set P of all probability distrib u tion s p : Z →

[0, 1] on Z,where
P
z∈Z
p(z)=1. We call these prob ability distr ibu tions lotteries. A
lottery can be dep icted b y a tree. For examp le, in Figure 1, Lottery 1 depicts a situation
in which if head the pla yer gets $10, and if t ail, he gets $0.
Lottery 1
1/2
1/2
10
0
Figure 1:
Unlike th e situation w e just described, in game theory and more broadly when agen ts
make th eir d ecision und er u ncertainty, we do n ot have the lotteries as in casinos where the
probabilities are generated by so m e mac h ines or given. Fortuna tely, it h as been show n
by Sa vage (1954) under certain conditions that a player’s beliefs can be represented by
2
a (unique) probability distribution . Usin g these probabilities, w e can represent our acts
b y lotteries.
We wo uld like to have a theory that constr ucts a playe r’s preferen ces on the lotteries
from his preferen ces on the prizes. There are many of them. Th e most well-kno wn–a nd
the most canonical and the m ost useful–on e is the theo ry of expected utility maxim iz a-
tion by Von Neum a nn a nd Morgenstern. A preference relation º on P is said to be
represented by a vo n Neuman n-Morge nst ern utility function u : Z → R if and only if
p º q ⇐⇒ U(p) ≡
X
z∈Z
u(z)p(z) ≥
X
z∈Z

u(z)q(z) ≡ U(q) (1)
for each p, q ∈ P .NotethatU : P → R represents º in ordinal sense. That is, the agent
acts as if he wants t o m a x im ize t he ex pected value of u. For in stance, the expected
utilit y of Lottery 1 for our agent is E(u(Lottery 1)) =
1
2
u(10) +
1
2
u(0).
1
The necessary and sufficient conditio ns for a representation as in (1) are as follow s:
Axiom 1 º is comple te and transitive.
This is necessary by Theorem 1, for U represents º in ordinal sense. The seco nd
condition is c alled independence axiom, stating t ha t a player’s preference between t wo
lotteries p and q does not chan ge if we toss a coin and giv e him a fixed lottery r if “tail”
come s up.
Axiom 2 For any p, q, r ∈ P ,andanya ∈ (0, 1], ap +(1− a)r  aq +(1− a)r ⇐⇒
p  q.
Let p and q be the lotteries depicted in Figure 2. Then, the lotteries ap +(1− a)r
and aq +(1− a)r canbedepictedasinFigure3,wherewetossacoinbetweenafixed
lottery r and our lotteries p and q. Axiom 2 stipulates that the agent w ould not change
his mind after the coin toss. Therefore, our axiom can be tak en as an axiom of “dynam ic
consistency ” in this sense.
The third condition is purely technica l, a nd called co n tin uity axiom. It states that
there are no “infinitely good” or “infinitely bad” prizes.
Axiom 3 For any p, q, r ∈ P ,ifp  r, then there exist
a, b ∈ (0, 1) suc h that ap +(1−
a)r  q  bp +(1− r)r.
1

If Z were a continuum, like R, we would compute the expected utility of p by
R
u(z)p(z)dz.
3
³
³
³
³
³
³
³
³
³
P
P
P
P
P
P
P
P
P
d ³
³
³
³
³
³
³
³

³
P
P
P
P
P
P
P
P
P
d
pq
Figure 2: Two lotteries
¡
¡
¡
¡
¡
@
@
@
@
@
d
a
1 − a
³
³
³
³

³
³
³
³
³
P
P
P
P
P
P
P
P
P
d
p
r
ap +(1− a)r
¡
¡
¡
¡
¡
@
@
@
@
@
d
a

1 − a
³
³
³
³
³
³
³
³
³
P
P
P
P
P
P
P
P
P
dq
r
aq +(1− a)r
Figure 3: Two compound lotteries
4
-
6
¡
¡
¡
¡

¡
¡
¡
¡
¡
¡
δ
z
0
p(z
2
)
p(z
1
)
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@

@
@
@
@
@
1
1
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H

H
H
H
p
p
0
β
α
q
q
0
l
l
0
Figure 4: Indifference curv es on the s pa ce of lotteries
Axioms 2 and 3 imply t hat, given any p, q, r ∈ P and an y a ∈ [0, 1],
if p ∼ q,thenap +(1− a) r ∼ aq +(1− a)r. (2)
This has two implications:
1. The indifference curv es on the lotteries are straight lines.
2. The indifferen ce curves, whic h are straigh t lines, are parallel to each other.
To illustra te these facts, consider th ree prizes z
0
,z
1
,andz
2
,wherez
2
 z
1

 z
0
.
A lotter y p canbedepictedonaplanebytakingp (z
1
) as the first coordinate (on
the h or izontal axis), and p (z
2
) as the secon d coordinate (on the vertica l axis). p (z
0
)
is 1 − p (z
1
) − p (z
2
). [See Figure 4 for the illustration .] Given any two lotteries p
and q, the convex combinations ap +(1− a) q with a ∈ [0, 1] form the line segm ent
connecting p to q.Now,takingr = q, we can deduce from (2) that, if p ∼ q,then
5
ap +(1−a) q ∼ aq +(1− a)q = q for each a ∈ [0, 1]. Thatthis,thelinesegment
connecting p to q is an indifference curv e. Moreover, if the lines l and l
0
are parallel,
then α/β = |q
0
| / |q|,where|q| and |q
0
| are the distances of q and q
0
to the origin,

respectiv ely. Hence, taking a = α/β,wecomputethatp
0
= ap +(1−a) δ
z
0
and q
0
=
aq +(1−a) δ
z
0
,whereδ
z
0
is the lo ttery at th e o rigin , and gives z
0
with pro ba bility 1.
Therefor e, b y (2 ), if l is an indifferen ce curve, l
0
is also an indifference curve, sho w ing
that the indifference curves are parallel.
Line l can be defined by equation u
1
p (z
1
)+u
2
p (z
2
)=c for some u

1
,u
2
,c∈ R.Since
l
0
is parallel to l,thenl
0
can also be defined by equation u
1
p (z
1
)+u
2
p (z
2
)=c
0
for some
c
0
. Since the indifference curv es are defined by equality u
1
p (z
1
)+u
2
p (z
2
)=c for variou s

values of c, the preferences are repr esented b y
U (p)=0+u
1
p (z
1
)+u
2
p (z
2
)
≡ u(z
0
)p(z
0
)+u(z
1
)p (z
1
)+u(z
2
)p(z
2
),
where
u (z
0
)=0,
u(z
1
)=u

1
,
u(z
2
)=u
2
,
giving the desired representation .
This is true in general, as stated in the next theorem:
Theorem 2 Arelationº on P c an be represented by a von Neumann-Mo rgenstern
utility func tion u : Z → R as in (1) i f and o nly if º s a tis fies A xioms 1-3. More over, u
and ˜u re present the same preference relation if and only if ˜u = au + b for some a>0
and b ∈ R.
By the last statement in our theorem, this represen tation is “unique up to affine
transformations”. That is, an agent’s preferences do not c hange when we change his
v o n Neum an n-Mor gen ster n (VNM ) utility function by m u ltiply ing it with a positive
n umber, or adding a constant to it; but they do c hange when we transform it through a
non-linear transforma tion . In t his sense, t h is representation is “cardinal”. Reca ll that,
in ordinal representatio n, the preferences w ould n’t cha nge even if the transforma tion
6
w ere non-linear, so long as it w as increasing. For instance, under certain ty, v =
√
u and
u w ould represent the same preference relation, while (when there is uncerta inty) the
VN M utility fun ctio n v =
√
u represents a ver y different set of preferences on the lotteries
than tho se are repr esented by u. Because, in card inal representation, the curvature of
the function also matters, m easuring the agent’s attitudes t o wards risk.
3 Attitudes Tow ards R isk

Suppose ind ividu al A has utility function u
A
. How do we determine whether he dislik es
risk or not?
Theanswerliesinthecardinalityofthefunctionu.
Let us first d efine a fair gamb le, as a lottery that has expected value
equal to 0. For
instance, lottery 2 belo w is a fair gamble if and only if px +(1− p)y =0.
Lottery 2
p
1-p
x
y
We define an agent as Risk-Neutral if and only if he is indifferen t between accepting
and r eject in g all fair g a mbles. Thus, an agent with u tility fun ct io n u is risk neutral if
and only if
E(u(lottery 2)) = pu(x)+(1− p)u(y)=u(0)
for all p, x,andy.
This can only be true for all p, x,andy if and only if the ag ent is m a ximizin g t h e
expected valu e, that is, u(x)=ax + b. There fore, we nee d the u tility fun ction to be
linear
.
Therefore, an agen t is risk-neutral if and only if he has a linear Von-Neumann-
Mo rgenst ern utility function.
7
An agent is strictly risk-a verse if and only if he rejects all fair gamb les:
E(u(lottery 2)) <u(0)
pu(x)+(1− p)u(y) <u(px +(1− p)y) ≡ u(0)
Now , recall that a function g(·) is strictly conca ve if and only if w e have
g(λx +(1− λ)y) >λg(x)+(1− λ)g(y)

for all λ ∈ (0, 1). Therefore, strict risk-a version is equ ivalen t to havin g a strictly conca ve
utility function. We will call an agent risk-averse iff he has a concave utilit y function,
i.e., u(λx +(1− λ)y) >λu(x)+(1− λ)u(y) for each x, y
,andλ.
Similar ly, a n agent is said to be (strictly) risk seeking iff he has a (strictly) convex
utility function.
Consider Figure 5. The cord A B is the utility difference that this risk-a verse agent
wou ld lose b y taking the gamb le that give s W
1
with pr ob a bility p an d W
2
with prob ab ility
1 − p. BC is the maxim um am oun t that she w ould pa y in order to avoid to take the
gamble. Su ppose W
2
is her wealth level and W
2
−W
1
is the v alue of her house and p is
the probabilit y that the ho use burns down. Th us in the absence of fire insurance this
individua l will ha ve utility given by EU(gamb le), which is lower than the utility of the
expected va lue of the gamb le.
3.1 R isk sharing
Consid er an agent with utility funct io n u : x 7→
√
x. He has a (risky) asset th at gives
$100 w ith probabilit y 1/2 and g ives $0 with probability 1/2. The expected u tilit y of
our agent from this asset is EU
0

=
1
2
√
0+
1
2
√
100 = 5. Now consider an other agen t
who is identical to our a gen t, i n the sense that he has t he same utility function a nd an
asset that pay s $100 with proba bility 1/2 and g ive s $0 with probab ility 1/2. We assume
througho ut that what an asset pays is statistically independent from wha t the other
asset pays. Imagin e that ou r agents fo rm a mutual fund by pooling their a ssets, eac h
agent owning half o f the m utual fund. This mutual f und gives $200 the probability 1/4
(when both assets yield high dividends), $100 with probability 1/2 (when only one on the
assets gives high dividend), and gives $0 with probab ility 1/4 (when both assets yield low
dividends). Thus, each agent’s share in the mutual fun d yields $100 with probabilit y
8

E
U
u
u
(
pW
1
+(1-
p
)
W

2
)
EU
(Gamble)
W
1
pW
1
+(1-
p
)
W
2
W
2
B
C
A
Figure 5:
9
1/4, $50 with probabilit y 1/2 , and $0 with probability 1/4. Therefo re, his expected
utilit y from the share in this m utual fund is EU
S
=
1
4
√
100 +
1
2

√
50 +
1
4
√
0=6.0355.
This is clearly la rger than his expecte d utility from his own asset. Therefore, our a gents
gain from sharing the risk in their assets.
3.2 Insurance
Imagine a wo rld where in addition to one of the agents abo ve (with utilit y function
u : x 7→
√
x and a risky asset that giv es $100 with probabilit y 1/2 and gives $0 with
probabilit y 1/2), we ha ve a risk-neutral agent with lots of money. We call this new agent
the insurance company. The insurance company c an insure the agent ’s asset, by giving
him $100 if his asset happens to yield $0. Ho w much prem iu m , P , our risk averse agent
w ould be willing to p a y to get this insurance? [A premium i s an a mount t hat is to be
paid to insurance company regardless of the outcome.]
If the r isk-averse agent p ay s p rem ium P and buys the insurance h is wealth w ill be
$100 − P for s u re. If he does not, then h is w ea lth will be $100 with proba b ility 1/2
and $0 with proba bility 1/2. Therefore, he will be willin g to pay P in order to g et the
insurance iff
u (100 − P ) ≥
1
2
u (0) +
1
2
u (100)
i.e., iff

√
100 − P ≥
1
2
√
0+
1
2
√
100
iff
P ≤ 100 − 25 = 75.
On the o ther hand, if the i nsurance com pan y sells the insurance for premiu m P ,itwill
get P for sure and pa y $100 w it h probability 1/2. Therefore it is willing to take th e deal
iff
P ≥
1
2
100 = 50.
Therefor e, both parties would gain, if the insurance company insures the asset for a
premium P ∈ (50, 75), a deal both parties are willing to accept.
Exercise 3 Now consider t he case that we have t wo identical risk-averse a gents as
above, and the insuranc e company. Insurance com pany is to charge the same premium
10
P for each agent, and the risk-averse agents have an opti on of form ing a m utual fund.
Wh at is th e ra n ge of premiu m s that a re acceptable to a ll parties?
11
14.12 G am e Theory L ecture N otes
∗
Lectures 3-6

M u h a met Yild iz
†
We will form ally define the g ames and s ome solution concepts, such as Nash E qui-
librium, and discuss the assumptions behind these solution concepts.
In order to analyze a game, w e need to know
• who the players are,
• whic h actions are available to them ,
• how much each player values each outcome,
• what eac h player know s.
Notice that w e need to specify not only what eac h play er knows about external
parameters, suc h as the payoffs, but also about what they know about the other pla yers’
know ledge a nd beliefs about these parameters, etc. In the first half of this course, we
will confine ourselves to the gam es of complete information, where everything that is
know n by a player is common knowledge.
1
(We s ay that X is comm o n know ledge if
∗
These notes are somewhat incomplete – they do not include some of the topics covered in the
class.
†
Some parts of these notes are based on the notes by Professor Daron Acemoglu, who taught this
course before.
1
Know ledge is defined as an operator on the propositions satisfying the following properties:
1. if I know X, X must be true;
2. if I know X, I know that I know X;
3. if I don’t know X, I know that I don’t know X;
4. if I know something, I know all its logical implications.
1
ev ery one knows X, and every one know s that every one kno ws X, and everyone kno ws

that everyone kno ws that e veryone know s X, ad infinitum.) In the second ha lf, we will
relax this assum ption and allo w pla yer to have asymm etric inform a tion, focusing on
informational issues.
1Representationsofgames
The gam es can be represen ted in tw o form s:
1. The norm al (strategic) form,
2. The extensive form.
1.1 N ormal form
Definition 1 (Norm al form) A n n-player game is any list G =(S
1
, ,S
n
; u
1
, ,u
n
),
where, for each i ∈ N = {1, ,n}, S
i
is the set of all strategies that are available to
player i,andu
i
: S
1
× × S
n
→ R is player i’s von Neum an n-M orgenstern utility
function.
Notice that a p layer’s utility depends n ot only on his own stra tegy but also on the
strategies pla yed by other players. Mo reover, each p layer i tries t o m a xim ize the ex pected

value of u
i
(where the expected va lues are computed with respect to his o w n beliefs); in
other word s, u
i
is a von N eumann -M org enstern u tility function . We will sa y that player
i is rational iff hetriestomaximizetheexpectedvalueofu
i
(given his beliefs).
2
It is also assumed that it is common kno wledge that the pla y ers are N = {1, ,n},
that the set of strategies available to eac h pla yer i is S
i
,andthateachi tries to maximize
expected va lue of u
i
given his beliefs.
Wh en there are only t wo p la yers, we can represent the (normal form) game by a
bimat rix (i.e., by t wo ma t r i c es):
1\2leftright
up 0,2 1,1
do wn 4,1 3,2
2
We have also made another very strong “rationality” assumption in defining knowledge, by assuming
that, if I kno w something, then I know all its logical consequences.
2
Here, Player 1 has strategies up and down, and 2 has the strategies left and righ t. In
each bo x the first number is 1’s payoff and the second one is 2’s (e.g., u
1
(up,left)=0,

u
2
(up,left)=2.)
1.2 E xtensiv e form
The extensiv e form contains all the information about a g ame, b y defining w h o moves
when , what e ach player k n ow s when he moves, what moves are a vailable to him, and
whereeachmoveleadsto,etc.,(whereasthenormalformismoreofa‘summary’repre-
sen tation ). We first in troduce some formalisms.
Definition 2 A tree
is a set of nodes and directed edges c onn ecting these nodes such
that
1. there is an initial node, fo r which there is no i ncoming edge;
2. for e ve ry other nod e , there is one incoming ed ge ;
3. for a ny t wo nodes, ther e is a unique path that conne ct these t wo nodes.
Imagine the branches of a tree arising from the trunk . For example,
.
.
.
.
.
.
.
is a tree. On the other hand,
3
A
B
C
is not a tree because there are two alternativ e paths through whic h point A can be
reached(viaBandviaC).
A

B
C
D
is not a tree either since A and B are not connected to C and D.
Definition 3 (Extensive form)AGame consists of a set of players, a tree, an al-
location of each node of the tree (except the end nodes) to a player, an informational
partition, and pay offs for ea ch player at each end n ode.
The set of players will include the agents taking part in the game. Ho wever, in man y
games there is room for c hance, e.g. the thro w of dice in backgammon or the card draws
in poker. More broadly, we need to consider “c h an ce” wh en ever there is uncertainty
about some relevan t fact. To represen t these possibilities we introduce a fictional playe r:
Nature. There is no payo ff for Nature at end nodes, and every time a node is allocated
to Natu re, a prob ab ility distribution ove r th e br anch e s that follow needs to be specified,
e.g., Tail with probabilit y of 1/2 and Head with probabilit y of 1/2.
An informatio n set is a collection of poin ts (nodes) {n
1
, ,n
k
} such that
1. the same pla ye r i i s to move at eac h of these nodes;
2. the same mo ves are available at each of t hese nodes.
4
Here the play er i, who is to mo ve at the inform ation set, is assumed to be un ab le to
distinguish bet we en th e poin ts in t he in formation set, bu t able to distingu ish between
the poin ts outside the informa tion set from those in it. For instance, consider the game
in Figure 1. Here, Player 2 kno ws that Playe r 1 has tak en action T or B and not action
X; but P layer 2 cannot know for sure wheth er 1 h as t aken T or B. The sam e ga m e is
depicted in Fi gure 2 slightly differently.

1

B
T
x
2
L
R
R
L
Figure 1:
1 x
T
B
2
L
R
L
R
Figure 2:
An informa tion pa rtitio n is an allocation of each node of the tree (except th e starting
and end-nodes) to an information set.
5
To sum up: at any node, we know: which player is to move, which moves are available
to the player, and which informatio n set contains the node, summa rizing the player’s
information at the node. Of course, if t wo nodes are in the same information set,
the available moves in these nodes m ust be the same, for otherwise the player could
distingu ish the nodes b y the available c h oice s. Again, all these are assumed to be
common knowledge. For instance, in the game in Figure 1, pla yer 1 knows that, if
player 1 tak es X, pla yer 2 will know this, but if he takes T or B, pla yer 2 w ill not kno w
which of t h ese two actions has been tak en. (She will know that either T or B will have
been taken.)

Definition 4 Astrategyof a player is a com p lete contin gent-plan dete rm in in g whic h
action he will take at each information set he is to move (including the information sets
that will not be r eached according to t his s trate gy).
Fo r certain pu rposes it migh t suffice to look at the reduced-form s trateg ies. A reduced
form strategy is defined as an incomplete contin gent plan that determines which action
the agent will tak e at each informa tion set h e is to move and that has not been precluded
b y this plan. But for man y other purposes we need to look at all the strategies. Let us
now consider some exam ples:
Gam e 1: M atching Pennies with Perfect Information
1
Head
2
Head
Tail
Tail
2
Head
Tail
O
O
(-1, 1)
(1, -1)
(1, -1)
(-1, 1)
The tree co nsists of 7 nodes. The first one is allocated t o player 1, a n d the next
t wo to pla yer 2. T he four end-nodes have p a yoffs attach ed to them . Since there are
6
two players, payoff vecto rs have tw o elemen ts. The first number i s the payoff of player
1 and the second is the payoff of pla yer 2. These payo ffs are von Neumann-M orgenstern
utilities so that w e can tak e expectations ove r them and calculate expected utilities.

The informational partition is very simple; all nodes are in their ow n information set.
In other wo rd s, all informatio n sets are singletons
(have only 1 element). This implies
that there is no u ncertainty regarding the previous play (history) in the g ame. A t this
poin t recall that in a tree, eac h node is reached through a unique path. Therefore, if all
information sets are singletons, a pla yer can con struct the history
of the game perfectly.
For instance in this game, play er 2 kno ws whether pla y er 1 c hose Head or Tail. And
player 1 knows that when he plays Head or T ail, Player 2 will know what player 1 has
played. (Games in which all information sets a re singletons are called gam es o f perfect
information .)
In this g am e, the set of strategies fo r player 1 is {H ead , Tail}. A strategy of play er
2 determines w hat to do depending on what player 1 does. So, his strategies are:
HH = Head if 1 plays H ead , and Head if 1 pla y s Tail;
HT = Head if 1 plays Head, and Tail if 1 plays Tail;
TH = Tail if 1 plays Head, and Head if 1 pla ys Tail;
TT = Tail if 1 pla ys Head, and Tail if 1 pla ys Tail.
Wh at are the payoffs genera ted by each strategy pair? If player 1 plays Head and 2
pla y s HH, then the outcome is [1 c hooses H ead and 2 chooses Head] an d th u s the payoffs
are (-1,1). If pla yer 1 pla ys H ead and 2 plays H T, the outcome is the same, hence t he
pa y offs are (-1,1). If 1 plays Ta il and 2 pla ys HT , then the outcome is [1 chooses Ta il
and 2 chooses Tail] and thu s the payoffs are once again ( -1,1). However, if 1 plays Tail
and 2 pla ys HH, then the outcome i s [1 chooses Tail and 2 chooses Head] a nd thus the
pa y offs are (1,-1). One can compute the payo ffs for the other strategy pairs similarly.
Ther efor e, the norm a l or the strategic form game correspondin g to this gam e is
HH HT TH TT
Head -1,1 -1,1 1,-1 1,-1
Tail 1,-1 -1,1 1,-1 -1,1
Inform ation sets are very important! To see this, consider the follo w in g gam e.
7