Lecture 40: Auctions as Bayesian Games –Sealed Bid First Price


Hello, welcome to another module in this online
course Strategy,An Introduction to Game Theory. So, let us continue our discussion on auctions,
let us start by looking at a sealed bid first price auction. So, let us begin our discussion on auctions
by looking at a sealed bid first price auction. So, we consider I have two
player auctions. So, let us say we consider a two player auction,
let us call theseplayer 1 and player 2, these are the two players participating in the auctionand
both these players submit their bids for the object being auction, let say their bids are
b 1 and b 2. So, these players submit their bids So, p 1 comma p 2 they submit their individualbids, b 1 comma b 2 respectively
for the object being auction and these bids are submitted in a sealed envelope,these bids
are sealed or these bids submitted in sealed envelope. So, these bids are sealed therefore, the bidof
each player is not known to the other player. So, the bid of each player is not known to
the other player implies p 1 does not knows the bid b 2 of player p 2.And p 2 similarly
does not know the bid b 1 of player p 1therefore, each playerdoes not know the bid of the other
player that is p 1 does not know b 2 of player 2 and similarly p 2 player 2 does not know
b 1 of player 1 are basically p 2. So, p 2 p 1 and p 2 both the players they
submit their bid b 1 and b 2 in a, in sealed envelope. So, which means they are not disclosing
their bids to the other players.Therefore, p 1 does not know the bid b 2 of player 2
and p2 that is the second player also does not know the bid b 1 of player 1 and the auction
mechanism is as follows.The player with the highest bid wins the auction, this is known
as the first price auction. So, the player with the highest bid wins the
auction and he pays an amount equal to his bid to get the object, this is known as the
first price auction. So, player with highest bid wins the auction and pays an amount equal
to his bid to get the object being auction. For instance, let us take an example, if b
1 is greater than or equal to b 2, if b 1 is greater than that is bid of player 1 is
greater than or equal to bid b 2 of player 2, then this implies then p 1 or player 1
wins the auction and pays his bid which is b 1 to get theobject.However, player 2 with
his bid b 2who has lostthe auction does not pay anything.The other player who have lost
auction that is player 2who has lost the auction does not pay anything. Similarly, if b 2 is greater than p 1 on the
other hand, if b 2 is greater than b 1 then player 2that is p 2 wins the auction that
is he wins the auction and pays his bid amount that is b 2 to get the object. So, if b 2
is greater than b 1 that is player 2has bid higher than player 1, then player 2 gets the
object and he pays his bid amount that is b 2 and player 1 who has last theauction does
not pay anything. So, this is known as the first price auction that is basically we are
saying that is first price auction is an auction in which the player with the highest bid wins
the auction and pays an amount equal to his bid value. So, this is known as the first price auction,
which is basically player with the highest bid wins the auction and pays an amount equal
to his bid value to getauction, he pays an amount equal to his bid valuethat is if b
1 is greater than or equal to b 2, then player 1 wins the auction and he pays b1 to get the
auction, well player 2 does not pay anything.On the other hand, if b 2 is greater than b 1
then player 2 wins the auction and pays his bid amount b 2 to get the object and player
1 does not get anything, this is known as this auction format or this auction mechanism
is also known as a first price auction.Now, in addition to this bids b 1 and b 2, in an
auction each player has a private valuation for the objects. So, in addition to their bids each player has a privatevaluation for the
object.What is the valuation? Thevaluation basically is how much the bidder value is
the object,the valuation is I simply put a valuation is what value the player or the
bidder player assigns to the objects. So, each player has a private valuation and this
valuation is basically nothing but,the value that the player assigns to the object and
this can be different from the bid, this need not be equal to the bid that is the player might
bid differently and he might have a different value that he attachesto the object.
And ultimately his aim is to get secure the object or secure the object at a bid that
is much lower than the value he places on it, sothat he can make a profit out of it,so
let us call these different private valuations, valuations of player 1 and player 2. Let say player 1has a valuation v 1 and player
2has a valuation v 2. What are v 1 and v 2?V1 and v 2 these are the valuations,so v 1 and
v 2 are the valuations of player 1 and player 2 for the object being auctioned. So, v 1
comma v 2 are the valuations of the player 1 comma player 2
respectively for the object, these are the valuations of player 1 and player 2 respectively
for the object being auctioned.And these valuations are private, which means player 1 does not
know the valuation v 2 of player 2 and player 2 does not know the valuation v 1 of player
1. So, therefore, these valuations are private,
implies player 1 does not know thevaluation v 2 of player 2,further
player 2 does not know the valuation of player 1. So, these each player does not know the
valuation of the other. So, player 1 does not know v 2 which is the valuation of player
2 and player 2 does not know v 1 which is the valuation of player 1 therefore, these
are known as private valuations.And; however, what is known is let say some information
related to these valuations,some statistical information.For instance, let say with these
valuations are distributed uniformly in the interval 0 to 1. So, what we are going to assume as that these
private valuations are what is known is that these private valuations
are distributed uniformly in the interval 0 to 1.That is probability density of both
v 1 and v 2 is uniform in the interval 0 to 1, this is what we are already seen, we are
already seen an example of auniform random variable in the interval 0 to 1 that is a
random variable which is distributed uniformly in the interval 0 to 1.And what we are saying
now is, we are using this uniform random variable in the interval 0 to 1 to characterized the
valuations v 1 and v 2. We are saying that the densities f of v 1
of v 1 and f of v 2 of v 2 at both follow I uniform random variable which is distributed
uniformly in the interval 0 to 1. So, we are saying these valuations v 1 and v 2 are distributed uniformly in the interval 0 to 1and what we wish to find is now the Nash equilibrium of
this auction game or this the Bayesian auction game. Now,you can clearly see why this auction
game is Bayesian nature, because there is uncertainty in this game that is uncertainly
regarding the valuations. So, each bidder or each player does not knowthe
valuation of the other player, but here certain probability, a certain statistically information
about the valuation all that type of the other. So, this context of an auction if you want
to think of it as a Bayesian game, you can think of the valuation of the other player
as the type of the other player. So, each type of the other way player has the different
valuation and you can see there is a infinite number of types, because the valuation can
take any real number between 0 to 1. So, this is the Bayesian games, since there is uncertainty
regarding the valuations of the other place. So, this game is this game of this auction
game is Bayesian in nature, since there is uncertainty
regarding the valuation of the other player.And therefore,
we now have to analyze this gamesimilar to the other games that we have consider before
we have to find what is the bidding strategy of each player of each time.Therefore, we
have to come up with the Nash equilibrium bidding strategy of each player participating
in the auction. So, you would like to analysis this game to find the Nash equilibrium bidding
strategy of each player. So, we wish to analyze this game to find the
Nash equilibrium bidding strategy of each player, at Nash equilibrium what is the strategy
employed by each bidderor each of this players in this first prize auction.And towards this
what we are going to demonstrate is that we are going to demonstrate that the bidding
strategy. So, we will demonstrate that the bidding strategy b 1 equals half
v 1 and b 2 equals half v 2 is the Nash equilibrium biddingstrategy for this game that is b 1
equals half b 1, b 2 equals half v 2 of the Nash equilibrium bids to each player in this
first price auction that is with each player is bidding half his for her valuation is the
Nash equilibrium bidding strategy.therefore half his valuation
is the Nash equilibrium therefore, each player bidding half his valuation is the Nash equilibrium
bidding strategy.So; however, we are going to demonstratives, again similar to what we
have done in the context of previous game that is by showing that each player is playing
is or her best response. That is we want to demonstrate that is Nash equilibrium b 1 equals
half v 1 is the best response to bid b 2 equals half v 2 of player 2.
And similarly for player 2 b 2 equals half b 2 is the best response tob 1equals half
v 1 of player 1. So, we want to demonstrate that each of the players is playing his or
her best response that is we wished to demonstrate that b 1 and b 2 that is b 1equals half v
1 and v 2 equals half v 2 are the best responses to each other. So, we wish we wish to demonstrate that b 1equals half v 1 and b 2 equals half v 2
are best responses to each other. Now, let us start by considering that player 2 is bidding
b 2 equals half b 2 and let us find, what is the best response b 1 of player 1. So, let us assume indeed player 2 is bidding
b 2 equals half b 2.What is the bestresponse bid b of player 1?To find the best response
bid b of player 1 we have to find the valuation or we have to findthe payoff to player 1 as
the function of the bid band then we have to find the value of b or the bid b at which
his payoff is maximized. So, first we have to find the payoff as a function of b and
we have to choose that particular value of b for which this payoff is maximized and that
is his best response bid b to the bid b 2 equals half b 2 of player 2.So, let us find
the best response,so now, first let us start by findingthe payoff of player 1 orthe payoff
to player 1 as a function of his bid b. So, let us denote this quantity by pi of b
that is pi of b denotes payoffto player 1 as
a function ofthe bid b.And remember the payoff to the bid b payoff to player 1 as a function
of bid b depends on one of two scenarios that is either player 1 wins the auction, if player
1 wins the auction which means is bid b is higher than the bid b 2. So, let us considered
the first scenario that is player 1,if player 1 wins the auction and this corresponds to the scenario that
is bid b beinghigher thanequal to b 2 when his payoff is equal to, because if he wins
the auction with bid b then he pays an amount equals to the bid b.
And therefore, he loses his bid b; however, he games in terms out of the valuation, because
now he is going to get the object which has a valuation to which he attaches a value v
1. So, his net payoff is v 1 minus the amount paid which is the bid b,sois net payoff is
v 1 that is his valuation minus the bid amount b So, net payoff equals v 1 minus b, v 1 is
this valuation while bid bis the bid paid on wining the auction. So, as paid an amount
b to get the object and he has the value of v 1 for the object. So, his net profit or
his net payoff is v 1 minus v if he wins the auction on the other hand if he loses the
auction that is if b is less than or equal to bid b 2 of player 2 then his net payoff
is 0,because it does not pay anything and need a does he get the object,sohis net payoff
is 0. So, if he loses the auction if player 1 loses
then hispayoff is 0,because he does not pay anything and neither does he get the object.
So, if player 1 loses the auction which occurs when his bid b is less than the bid b 2 of
player 2,then he loses the auction he does not pay anything. So, the amount paid by him
0 neither does he get the object,so therefore, his net payoff is 0. So, therefore, the average payoff of player
1 is given as probability winning the probability he wins the auction times v 1 minus b.Because,
his payoff if he wins the auction is v 1 minus b plus the probability that he loses the auction
the probability of lose times 0,because if he loses the auction his net payoff is 0.
So, what we are saying, we aresaying that if he wins the auction then his payoff is
v 1 minus b that is valuation b 1 minus bid amount b paid.
And therefore, we are multiplying that by the probability of winning probability of
winning times v 1 minus b plus the probability of loss times 0.Because, we loses the auction
then he does not get the objectand either does he payanything. So, the net profit and
the loss of auction is 0,which means the average profit or the average payoff is probability
of winning times v 1 minus p plus probability of lose times 0 which can be simplified as
probability of win times v 1 minus b. So, net payoff pi b to player b equals the
probability of winning the auction times v 1 minus b. So, what is pi of b payoff to player
1 as a function of bid b,sopi of b which is thepayoff player 1 as a function of the bid
b is equal to the probability of winning times v 1 minus v.It now remains to find what this
quantity probability of winning this that isas a function of the bid b, what is the probability
that he wins the auction. What is pr win that is the probabilityof wining
the auction, remember to win the auction the bid b must be greater than equal to bid b
2 of player 2, but b 2 of player 2 remember we are assumed b 2 is equal to half v 2. So,
to win the auctionbid b that is to win we must have to win or for player 1 to win we
must have b greater than or equal to b 2 equals half of v 2. So, we must have forplayer 1
to win b must be greater than or equal to half v
2 which implies that v 2 less than or equal to 2 b.
So, to win the auction bid b must be greater than equal to half v2 which means v 2 must
be less than or equal to twice B.Therefore, it means that v 2 which is remembered the
valuations are uniformly distributed in the interval 0 to 1. Therefore, it means that v 2 must lie in the
interval 0 to 2 v, this is the valuationv 2 is
distributed uniformly in 0 to 1 it we must have v 2 in or v 2 in 0 to 2b.Therefore, if
v 2 is less than or equal to 2 b that is for player 1 to win the auction, it must be the
case that v 1 lies in 0 to 2 b. And what is the probability that v 2 lies
in 0 to 2b the probability, rememberprobability v 2 lies in 0 to 2b is equal to the integral
0 to 2b f of v 2 that is the probability density of f of v 2 integrated between 0 to 2 b.But,between
0 to 2bsince this integral is contain in the interval 0 to 1 the probability density is
1therefore, we contrivesthis as 0 to 2 b1 times d v 2 which is v 2 integrated between the limits
0 to 2 b which is equal to 2 b. And therefore, what we have shown is that
the probability that the valuation v 2 of player 2 lies in the interval 0 to 2 b is
equal to 2 b and therefore, this is the probability that player 1 wins the auction. Before the probability of win for player 1 as a function ofhis bid
b 2 b and therefore, now if we go back to this expression over here, where we are characterizing
the payoff we now the probability of win, the probability of win is 2 b as a function
of his bid b.Therefore, the net payoff pi of b to player 1 as a function of his bid
b is pi of b equals the probability of winning 2 b times v 1 minus b which is the payoffor
bidding, which is equal to 2 b v 1minus 2 b square. So, pi of b which is the payoff
to player 1 as a function of his bid b is 2 b v1 minus 2 b square.
Now, we have the payoff as a function of b we have to find the b for which the payoff
is maximum for this purpose we have to differentiate thisfunction of b and set it equal to 0 and
there by solve it to find the value the best response b. So, duo by b we have pi of b equals 2 b v
1 minus 2 b square therefore, dou pi of b by dou b equals twice v 1 minus 4 times b
which we equateto 0 which implies gives us the resultthat b equals halfv 1 that is the
best response bid b star is equal to half v 1. So, what are we shown, we have shown
that if b2 is bidding player 2 is bidding b 2 equals half v 2 then the bid b star equals
half v 1 is the best response bid of player 1 So, if b 2 equals half v 2 then the bid b 1
equals half v 1 is the best response of player 1.Similarly, by symmetry it can be shown that
if player 1 is bidding b 1equals half v1 b2equals half v2 is the best response for player 2
using a similar procedure, if I repeating the same procedure. Using a similar procedure it can be shown
that if b 1 equals half v 1 that is player 1 is bidding b1 equals half v 1 then b 2 equals
half v 2 is the best response of player 2. So,what we have shown is that if player 2
isbidding b 2 equals half v2 then b 1equals half v 1 is the best response of player 1.Similarly,
we can show that if player 1 is bidding b 1 equals half v1 then b 2 equals half v2 is
the best response bid of player 2. And therefore, since these bids b 1equals
half v1 andb 2 equals half v2 are best responses to each other, this is the Nash equilibrium
of this first price auction here.Therefore, we have b1 equals half v 1 comma b 2 equals
half v 2 are best responses to each other,therefore, b1 equals half v 1 andb2 equals half v 2 is
the Nash equilibrium of this first price auctions. So, this is the Nash equilibrium of our sealed bid.
Therefore, this is the Nash equilibrium of our sealed bid first price auction that is
what we are saying is, in the sealed bid first price auction in which the two players that
is the two players v 1 and v 2 are also in this bidders submit their bids in sealed envelope
that is the bid of each player is unknown to the other with private valuations v 1 and
v 2 distributed uniformlyin the interval 0 to 1 the Nash equilibrium for this Bayesian
sealed bid first price auction is b 1 equals half v 1 and b 2 equals half b 2.So, let us
stop with thisderivation of the Nash equilibriumhere and we will continueour discussion on this
auction format in the next module. Thank you.

5 comments

Your videos are great! Thank you for them. Though, they are long, just a positive critique, if you just write when you talk instead of explaining and then writing; because i understand writing takes longer. Otherwise one can always type in a notepad/word. Thank you!

Hi.Thank you for this excellent presentation.I have watched all the videos and find them super useful.Unfortunately, there is 1 exercise about game theory that I can't solve.Pleaseee could you help me? Pleasee
This is the exercise:

 
B.  Consider a sealed‐bid first‐price auction. There are two bidders, 1 and 2, with valuations for the good
of v1 and v2, respectively. Each buyer’s valuation is his/her own private information. v1 is uniformly
distributed on include (0,1) and v2 is uniformly distributed on include (0,2). So player 1, for instance, knows  v1 exactly
but regarding v2 he/she knows only that it is uniformly distributed between 0 and 2. 
1. Define the Bayesian equilibrium for this game.
2. Find a Bayesian equilibrium. 
3. Discuss uniqueness.

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