00:00:00
a bayesian game is a list
00:00:03
uh s1 s2 up to sn so
00:00:06
each si is basically set of strategy for
00:00:10
player i so if each player has a
00:00:13
discrete or finite strategy
00:00:15
then that means si is a finite strategy
00:00:18
set
00:00:18
or it can be an infinite strategy set so
00:00:21
these are strategy sets for each player
00:00:24
and then t1 t2 up to tn
00:00:27
uh well for example in the uh double
00:00:30
ocean
00:00:31
i'm sorry a second price auction example
00:00:33
i previously mentioned
00:00:34
so each si was a zero infinity interval
00:00:38
so t1 t2 up to tn so these are the
00:00:42
typeset for each player i and in the
00:00:45
previous example
00:00:46
in the simplest environment remember uh
00:00:49
each buyer could have three potential
00:00:51
types
00:00:52
so the the ti is you know the typeset
00:00:55
for each player
00:00:56
i i'm sorry
00:01:00
and then u1 u2 up to u n these are the
00:01:02
payoff vectors
00:01:04
i'm sorry payoff functions um well
00:01:06
obviously the payoff function
00:01:08
depends on not only on player eyes
00:01:12
strategy but it also depends on all the
00:01:15
others players
00:01:16
strategy and here those strategies
00:01:20
are not sort of discriminated for
00:01:23
types because i write those strategies
00:01:27
as
00:01:27
function which i will describe in a
00:01:29
moment
00:01:30
and the payoff also depends on the type
00:01:33
of player i
00:01:34
right the different types may have
00:01:36
different payoffs
00:01:37
so in the previous example again if
00:01:40
you are the buyer with valuation 110
00:01:44
your different your your payoff function
00:01:46
was is different than
00:01:48
if you are a buyer with type valuation
00:01:51
90
00:01:52
because that buyer's valuation is 90
00:01:55
minus
00:01:56
price not 110 minus price all right so
00:01:59
the different types
00:02:00
actually has different payoffs which
00:02:03
basically derives
00:02:05
different optimal strategies for
00:02:06
different types all right
00:02:08
so one more additional thing which is
00:02:11
not
00:02:11
uh described here is the p the
00:02:14
probability the player's
00:02:16
belief about other types all right so we
00:02:19
denote it by
00:02:20
pi it's a conditional probability
00:02:23
t sub minus i which means the type of
00:02:27
all the other players except player i
00:02:30
conditional on player i's type well this
00:02:34
notation allows correlated types
00:02:37
so maybe your type and the other guy's
00:02:39
types are correlated
00:02:40
which is very well possible for example
00:02:44
maybe uh so if it is for example a a
00:02:47
football game
00:02:48
um all right and so the types can be
00:02:52
uh determined whether uh uh the the
00:02:55
weather is
00:02:56
rainy or snowy or or shining those or
00:02:59
maybe the sun is shining so therefore
00:03:01
if this is the case if this is how the
00:03:03
types are determined
00:03:04
well then maybe your type and your
00:03:07
opponent's type may be correlated
00:03:09
depending on the weather you see what i
00:03:10
mean so this notation allows
00:03:13
correlation in that sense so if for
00:03:15
example
00:03:16
this probably is equal to the
00:03:19
multiplication of
00:03:20
the individual types probabilities p
00:03:24
j t j j is different than i if this is
00:03:27
the case
00:03:27
well that means independent types
00:03:29
remember if 2
00:03:31
probability of a given b for example um
00:03:35
so if this is uh equal to probability of
00:03:38
a well then we say that
00:03:41
you know a and b are independent types
00:03:44
this is exactly what's happening here
00:03:46
all right okay so what else
00:03:51
i already mentioned the payoff function
00:03:53
for player i
00:03:54
it depends on the strategies of all the
00:03:56
players because that is the essence of
00:03:59
strategic environment and also depends
00:04:01
on the type of player i
00:04:03
well here uh what if
00:04:06
the uh the player eyes other types
00:04:10
can they affect my payoff well maybe yes
00:04:13
but we
00:04:14
usually ignore this correlation because
00:04:17
it complicates the game so here the
00:04:19
strategy is
00:04:21
important is a function s-i-t-i
00:04:25
which specifies what strategy player i
00:04:28
would pick if he is type ti
00:04:31
so therefore a strategy you can think of
00:04:34
strategy
00:04:36
of player i as a function which maps
00:04:39
each ti
00:04:40
to some strategy in the set of
00:04:43
strategies
00:04:44
as i all right again in my previous
00:04:46
example i said it is b1
00:04:48
1 b1 2 b1 3
00:04:52
all right so basically this is what
00:04:55
b1 type i is so if
00:04:58
if for example b1 type one
00:05:02
so if if you're type one your bid is
00:05:04
gonna be b11 this is how i denoted it
00:05:07
so they're not equal i'm sorry for my
00:05:09
notational
00:05:10
uh mix up um b1
00:05:14
t2 is equal to b12 b1
00:05:17
t3 is equal to b13 all right
00:05:20
so this is just another way of
00:05:24
notating the strategy of player 1 as a
00:05:27
triple
00:05:28
all right you can just represent it as a
00:05:30
function
00:05:31
all right so you can either write b1
00:05:34
parenthesis t1 t2 t3
00:05:36
or b 1 1 b 1 2 b 1 3 this is
00:05:40
totally up to you i mean you'll see in
00:05:42
some examples
00:05:43
i'm going to use those notations in some
00:05:45
other i'm going to use
00:05:46
that notation so uh depending on the
00:05:49
question uh the you know some notations
00:05:52
are easier to follow
00:05:53
some notations are more and you know
00:05:58
uh making the notation hard to follow
00:06:01
and so it's totally up to you which
00:06:02
notation you want
00:06:04
you would like to prefer so i just
00:06:05
wanted to give you another so this is a
00:06:08
more uh sort of a neat way of
00:06:11
defining a strategy this strategy is a
00:06:14
function which maps each type
00:06:16
to a strategy from the set of strategies
00:06:20
of that player i obviously all right so
00:06:22
what is bayesian nash equilibrium
00:06:24
it's nothing but a nash equilibrium of
00:06:28
the bayesian game all right so the
00:06:30
bayesian game is given by easily by the
00:06:32
set of strategies
00:06:33
so instead of writing s1 s2 s3 i just
00:06:36
wrote the
00:06:38
cartesian product of the strategy sets
00:06:41
type
00:06:42
space so these are spaces so that means
00:06:46
s is nothing but s1 cross s2 cross all
00:06:50
the way cross
00:06:51
s answers the cartesian product so s1 is
00:06:54
a set
00:06:55
capital s is a space all right so the
00:06:58
set and space are two different things
00:07:00
uh why is that well set doesn't really
00:07:03
have
00:07:04
i mean they may be the same thing
00:07:06
obviously but set
00:07:07
is just you know finite or infinite just
00:07:10
one dimensional
00:07:11
set the capital s however is an n
00:07:14
dimensional
00:07:15
set all right so we call it space t
00:07:19
symmetrically t1 cross t2 cross all the
00:07:22
way up to tn so this
00:07:24
uh type space p is the beliefs
00:07:27
right you know how those types are
00:07:30
distributed
00:07:31
according to what probability this is
00:07:33
given by a p
00:07:34
it may be a correlated times
00:07:36
uncorrelated types
00:07:38
depending on the uh problem or the
00:07:40
strategic environment we're analyzing
00:07:42
and finally the payoff vector
00:07:44
so once you're given this this is the
00:07:46
bayesian game
00:07:49
the bayesian nash equilibrium of this
00:07:51
game is the nash equilibrium
00:07:53
of this game all right so bayesian nash
00:07:55
is not something new in fact
00:07:57
a strategy profile s star is
00:08:00
based in nash uh for that means
00:08:04
for every player and for every type
00:08:07
remember you have to check best
00:08:08
responding thing
00:08:09
for each player and for each type of
00:08:12
that player
00:08:13
so uh the si star ti so this is the
00:08:17
strategy of type i
00:08:19
uh given that fixing the other's
00:08:21
strategy
00:08:23
should be a best response meaning it has
00:08:25
to the expected utility of this thing
00:08:28
uh this this profile has to be greater
00:08:30
than the expected utility
00:08:32
of playing some other strategy si again
00:08:36
fixing the others are playing according
00:08:37
to this strategy profile
00:08:39
so this inequality should be true for
00:08:42
all s i element of
00:08:46
capital s i all right remember as
00:08:49
player i or type i you're allowed to
00:08:51
choose
00:08:52
any strategy in this set so if any
00:08:54
strategy you pick here
00:08:56
the expected utility should be less than
00:08:58
or equal to the expected utility
00:09:01
of playing si star ti if this is the
00:09:04
case then i'm gonna call that
00:09:05
this is the best response for type i
00:09:08
well
00:09:09
if this star strategy is the best
00:09:11
response for each player and for
00:09:13
each type well then we actually got
00:09:17
one nash equilibrium all right obviously
00:09:19
there can be a bunch of other nash
00:09:21
equilibria
00:09:22
but this is how we check or verify if
00:09:24
something is nash equilibrium or not
00:09:27
well here obviously it is important how
00:09:29
we write this expected payoff
00:09:31
okay so important question is what is
00:09:34
that expected utility expected payoff
00:09:37
so uh i'll let me give you a generic
00:09:40
uh formula for it because the
00:09:43
idea well if you get this uh generic
00:09:46
formula i think
00:09:48
of well your life is going to be very
00:09:50
very simple
00:09:52
but obviously it's not so easy and
00:09:54
straightforward
00:09:55
so try to uh picture this formula
00:09:58
every in every exercise you solve all
00:10:01
right so that's very very important once
00:10:03
you solve an exercise come back to this
00:10:06
formula
00:10:06
and see how it fits this is how you can
00:10:10
really understand this formula and again
00:10:12
once you understand this trust me
00:10:15
finding basic nash equilibrium
00:10:17
is just piece of cake so how do we write
00:10:20
the expected payoff
00:10:22
uh given that player i plays
00:10:25
some strategy as i and given that other
00:10:28
players are playing according to s star
00:10:31
minus i
00:10:32
all right well first of all this is
00:10:34
going to be sum
00:10:36
right sum of payoffs multiplied by
00:10:38
probabilities
00:10:40
so this is how we calculate the expected
00:10:43
utility
00:10:43
of a lottery p remember so if p is for
00:10:46
example
00:10:47
p1 p2 all the way up to pn
00:10:51
so what we were doing is you know p1
00:10:53
times u1
00:10:54
p2 times u2 so these are the payoffs in
00:10:58
each
00:10:58
outcome and these are the probabilities
00:11:00
of these events
00:11:01
so plus p n times u n so uh
00:11:05
which is equivalent to saying you know p
00:11:07
i times u i
00:11:08
i mean here j i'm sorry and obviously j
00:11:12
from one to n so here i am summing
00:11:15
through different type profiles
00:11:18
except player i so what is t sub minus i
00:11:23
well t sub minus i is the type profile
00:11:26
vector so it includes type one of player
00:11:30
one
00:11:30
type two i'm sorry type of player one
00:11:33
type of player two
00:11:34
type of player i minus one i plus one
00:11:38
so every other player's types except
00:11:41
type of player i alright so it's not
00:11:43
going to be in
00:11:45
a part of this vector all right why well
00:11:48
because
00:11:48
it is here the type i's player i's type
00:11:51
is
00:11:52
ti is on this conditional part so what
00:11:55
is the probability
00:11:56
that the others type is given this given
00:12:00
that
00:12:00
player i's type is ti so this
00:12:02
probability
00:12:04
remember in our previous example it was
00:12:06
sort of independent one third one third
00:12:08
one third so it was
00:12:09
life was very simple there and so here
00:12:12
is this
00:12:13
ui u1 u2 thing well
00:12:16
obviously as you change this type
00:12:18
profile this payoff will
00:12:20
change all right so that's that's very
00:12:22
important oh
00:12:25
well i am sorry because this notation is
00:12:28
not
00:12:28
100 percent true but now i'm going to
00:12:31
make it
00:12:32
true 100 s i
00:12:35
comma s i plus one star
00:12:38
t i plus one comma dot dot dot
00:12:42
comma okay now the payoff
00:12:45
of player i given that he is playing
00:12:48
s i and his opponents are playing s
00:12:51
minus i
00:12:52
so i just open this now all right so
00:12:55
expected utility this expected part
00:12:58
comes with
00:12:59
you know multiplication of this
00:13:00
probability well what about this utility
00:13:02
part
00:13:03
well utility is depend on
00:13:06
player 1's strategy player 1 of type 1's
00:13:09
strategy player 2 and his type
00:13:13
strategy player type player
00:13:16
i minus 1 and his type all right player
00:13:19
eyes
00:13:20
strategy which is s i player i plus
00:13:23
once as uh type uh i plus
00:13:27
i'm sorry uh player i plus one and his
00:13:30
type
00:13:30
and that strategy and so on and so forth
00:13:34
this is the player n and and the type uh
00:13:37
his type and his strategy so
00:13:39
as you change this type profile that
00:13:42
means you're changing this
00:13:43
tease inside the parenthesis and so
00:13:47
therefore you automatically change the
00:13:49
strategies so there's only one thing
00:13:51
that is going to be
00:13:52
fixed which is s i all right so
00:13:56
the thing is this s i is fixed
00:13:59
as you change this type profile this
00:14:02
probability will change
00:14:03
this payoff will probably change but
00:14:06
here
00:14:06
what is changing here is that you're
00:14:08
fixing si
00:14:10
and you're only changing those
00:14:11
strategies
00:14:13
simply because you are changing those
00:14:15
type
00:14:16
profiles all right so this is how we
00:14:18
calculate the expected payoff
00:14:20
um where we used it
00:14:24
to calculate the best response of
00:14:27
a type of a player
