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.

Thank you so much for the clear explanation! 😀

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!

God Bless you sir!

why did you pick b1=0.5v1 in the first place?

is it just an example?

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.