5.1 auction theory slides

8/10/2019 5.1 Auction Theory Slides

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Today

Auction Theory Basics

1. Bidding and Equilibria in IPV model

2. Revenue Equivalence

3. General Model: Affiliated Values

4. Linkage Principle: Revenue RankingAd Auctions: Sponsored Search


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Auctions: The Independent Private Values Model

Basic Auction Environment

Bidders i= 1,...,n

One object to be sold

Bidder i observes a signal Si F(), with typical realizationsi[s, s]. Assume F is continuous.

Bidders signals S1,...,Sn are independent.

Bidder is value vi(si) =si.

A set of auction rules will give rise to a game between the bidders.


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The IPV Model

Two important features of the model:

Bidder is information (signal) is independent of bidder js infor-mation (signal).

Bidder is value is independent of bidder js information (i.e. pri-vate values).


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Vickrey (Second-Price) Auction

Auction Rules:

Bidders are asked to submit sealed bids b1,...,bn.

Bidder who submits the highest bid wins the object.

Winner pays the amount of the second highest bid.

Proposition 1In a second price auction, it is a (weakly) domi-nant strategy to bid ones value, bi(si) =si.

Proof. Biddingbimeansi will win if and only if the price is below bi.Bid bi > si sometimes win at price above value.Bid bi < si sometimes lose at price below value.


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Expected Vickrey Auction Revenue

Sellers revenue equals second highest value.

Let Si:n denote theith highest ofn draws from distributionF.

Sellers expected revenue is

E

S2:n

.


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Open Ascending Auction

Auction Rules:

Prices rise continuously from zero.

Bidders have the option to drop out at any point.

Auction ends when only one bidder remains.

Winner pays the ending price.

With private values, just like a Vickrey auction!


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Sealed Bid (First-Price) Auction

Auction Rules:

Bidders submit sealed bids b1,...,bn.

Bidders who submits the highest bid wins the object.

Winner pays his own bid.

Bidders will want to shade bids below their values.

Well look at two approaches to solving for symmetric equilibrium bid-ding strategies.


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FOC Approach to First Price Auction

Suppose bidders j =i bid bj =b(sj), b() increasing.

Bidderis expected payoff:

U(bi, si) = (si bi) Pr [bj =b(Sj) bi, j =i]

Bidderichooses bito solve:

maxbi

(si bi) Fn1

b1(bi)

.

First order condition:

0 = (si bi) (n1)Fn2

b1(bi)

f

b1(bi) 1

b(b1(bi))Fn1

b1(bi)


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FOC Approach to First Price Auction

At symmetric equilibrium, bi=b(si), first order condition is:

b(s) = (s b(s)) (n 1)f(s)

F(s).

Solve using the boundary condition b(s) =s:

b(s) =s

sis

Fn1(s)ds

Fn1(s) .


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Envelope Approach to First Price Auction

Suppose there is a symmetric increasing equilibrium: bi=b(si).

Then is equilibrium payoff given signal si:

U(si) = (si b(si)) Fn1(si). (1)

As b(si) is optimal fori:

U(si) = maxbi

(si bi) Fn1(b1(bi)).

By the envelope theorem:d

dsU(s)

s=si

=Fn1(b1(b(si)) =Fn1(si)


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Envelope Approach to First Price Auction

Rewriting the envelope formula:

U(si) =U(s) +

sis

Fn1(s)ds. (2)

Asb(s) is increasing, bidder with signal snever winsU(s) = 0.

Equating the two representations ofU(si):

b(s) =s

sis

Fn1(s)ds

Fn1

(s)

.


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Equilibrium in the First Price Auction

Weve found necessary conditions for symmetric equilibrium.

Verifying that b(s)isan equilibrium isnt too hard.

Also can show that equilibrium is unique.


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Expected First Price Auction Revenue

Revenue is high bid b(s1:n); expected revenue is E

b(S1:n)

.

b(s) =s

ss

Fn1(s)ds

Fn1

(s)

= 1

Fn1

(s)

s

s

sdFn1(s) = E S1:n1|S1:n1 s .

Expected revenue is:

E

b

S1:n

= E

S1:n1|S1:n1 S1:n

= E

S2:n

.

First and second price auction yield same expected revenue.


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Revenue Equivalence Theorem

Theorem (Revenue Equivalence)Supposenbidders have valuess1,...,snidentically and independently distributed with cdfF(). Thenany equilibrium of any auction game in which(i) the bidder with thehighest value wins the object, and(ii) a bidder with value sgets zero

profits, generates the same revenue in expectation.


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Proof of the RET

Consider any auction where bidders submit bids b1,...,bn, and wherethe auction rule specifies for all i,

xi : B1 ... Bn[0, 1]

ti : B1 ... Bn R,

xi() gives probabilityiwill get the object.

ti() gives is payment as a function of (b1,...,bn).


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Proof of the RET

Bidderis expected payoff is:

Ui(si, bi) =siEbi[xi(bi, bi)] Ebi[ti(bi, bi)] .

Letbi() , bi() denote an equilibrium of the auction game.

Bidderisequilibriumpayoff is:

Ui(si) =Ui(si, b(si)) =siFn1 (si) Esi[ti(bi(si), bi(si)] ,

where because highest value wins:

Esi[xi(b(si), b(si))] =Fn1(si).


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Proof of the RET, continued

Combine the expressions for Ui(si) to get:

Esi[ti(bi, bi)] =siFn1 (si)

sis

Fn1(s)ds=

sis

sdFn1(s),

If auction satisfies(i) and(ii), is expected payment given si is:

E

S1:n1 |S1:n1 < si

= E

S2:n |S1:n =si

.

Thus expected revenue is constant:

E [Revenue] =nEsi[is expected payment| si] = E

S2:n

.


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Using the Revenue Equivalence Theorem

Many applications of the revenue equivalence theorem.

If the auction equilibrium satisfies(i) and(ii), we know:

Expected Revenue

Expected Bidder Profits (payoff equivalence)

Expected Total Surplus

We can use this to solve for auction equilibria.


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The All-Pay Auction

Bidders 1, . . ,n

Values s1,...,sn, i.i.d. with cdfF

Bidders submit bids b1,...,bn

Bidder who submits the highest bid gets the object.Every bidder must pay his or her bid.


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Solving the All-Pay Auction

Suppose all pay auction has a symmetric equilibrium with an in-creasing strategy bA(s). In this equilibrium, is expected payoff givenvaluesi will be:

U(si) =siFn1

(si) bA

(si) = si

s Fn1

(s)ds

So

bA(s) =sFn1(s)

ss

Fn1(s)ds


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Other Revenue Equivalent Auctions

1. The English (oral ascending) auction. All bidders start in the auc-tion with a price of zero. Price rises continuously; bidders may dropout at any point in time. Once they drop out, they cannot re-enter.Auction ends when only one bidder is left; this bidder pays the price

at which the second-to-last bidder dropped out.2. The Dutch (descending price) auction. The price starts at a very

high level and drops continuously. At any point in time, a biddercan stop the auction, and pay the current price. Then the auctionends.


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Common Value Auctions

Generalize the model to allow for the possibility that:

1. Learning bidder js information could cause bidder i to re-assesshis estimate of how much he values the object,

2. The information of i and jis not independent (when js estimateis high, is is also likely to be high).

Examples:

Selling natural resources such as oil or timber.

Selling financial assets such as treasury bills.

Selling a company.


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A General Model

Bidders i= 1,...,n

Signals S1,...,Snwith joint density f()

Value to bidder iis v(si, si).


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Affiliated Signals

Signals are(i) exchangeable, and(ii) affiliated.

Signals areexchangeable: s is permutation ofsf(s) =f(s).

Signals areaffiliated: For any s1 > s1, s

2 > s

2, we have

f(s1|s2)

f(s1|s2)

f(s1|s2)

f(s1|s2)

.

i.e. sj|si has monotone likelihood ratio property.

This implies, but is not equivalent to

F(s1|s2) F(s1|s

2)

for all s1. (FOSD)


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Examples of the General Model

1. The IPV model.

2. Pure common value with conditionally independent signals.

Example: Si = V +i, where 1, . . ,n are independent, andv(si, si) = E[V|s1,...,sn].

3. Interdependent values, independent signals model.

Example: vi(si, si) =si+

j=i sj, with 1, with S1,...,Snindependent.

Note: With interdependent values, i must realize that js bid is in-formative aboutis own value. This leads to the winners curse.


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Second Price Auction

Look for a symmetric increasing equilibrium bid strategy b(s).

Let si denote the highest signal of bidders j =i.

Ifbib(si), iwins and pays b(si).

Bidder is problem:

maxbi

ss

ESi

v(si, Si)| si, S

i =si b(si)

1{b(si)bi}f(s

i|si)dsi

or

maxbi

b1(bi)s

ESi

v(si, Si)| si, S

i =si b(si)

dF(si|si)


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Second Price Auction

The first order condition for this problem is:

0 = 1

b(b1(bi)) ESi

v(si, Si)| si, S

i =b1(bi)

b(b1(bi))

f(b1(bi)|si)

or simplifying:

bi=b(si) = ESi

v(si, Si)| si, max

j=isj =si


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Failure of Revenue Equivalence Theorem

The RET does not hold in this more general environment.

Basic idea: bidders profits are due to information rents. Whensignals are correlated, information rents can be reduced if moreinformation is used in setting the price.

Example: english auction generates more revenue than a secondprice auction.


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Linkage Principle Intuition

Bidder profits (rents) are derived from their private information

Statistical linkages align the price more closely to the buyers will-ingness to pay.

Rents associated with private information fall

Revenue rises


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Linkage Principle

Standard auctionsA, Bwith symmetric, increasing eqabA( ), bB( )

LetWA(s, s) denote the expected price paid by bidder 1 if he is thewinning bidder when he receives a signal s but bids as if his signalwere s.

Let WA2 (s, s) denote the partial w.r.t. the second argument. Thatis, W2 measures how much the expected price increases with in-creases in the bidders valuation, holding bidding strategy constant.

Theorem 1 Suppose that for alls,

W

A

2 (s, s) W

B

2 (s, s).Then the expected revenue inA is at least as large as the expected

revenue inB.


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Revenue Ranking: First vs Second Price Auction

For the first price (I) auction WI(s, s) =b1(s), so

WI2 (s, s) = 0.

For the second price (II) auction,

WII

2 (s, s)> 0,

because conditional on winning by bidding as if his signal is s...

having a higher signal smeans that others also have a higher signal...

so the expected price paid (the second highest bid) will be higher.

From the theorem, it follows that the second price auction yieldsgreater revenue than the first price auction.


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Linking Price to Seller Information

If the seller has private informationS0about the object that is affil-iated with the signals of the buyers, the seller can increase revenue bycommitting to a policy to reveal this information.

Theorem 2Verifiably revealing information variableS0raises theexpected price both in the first auction and second price auctions.Among all policies for full or partial revelation of information, thepolicy of full revelation maximizes the expected price.


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Linking Price to Seller Information

Interestingly, the policy of full information can be expected to emergeeven when the seller cannot commit to an information policy.

Theorem 3 If a seller must decide, after observing S0, whether

to report it, and if his report is verifiable, then in any equilibrium,he always reportsS0 regardless of the value.

Why would a seller report ifs0is low? Because of affiliation, biddersignals will be low with or without this information. Revealing willnonetheless link this information to price.

(If the seller observed s0 *and* bidder signals, then he may prefernot to reveal.)


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Royalties

When the value can be observed by all parties ex-post (even imper-fectly), expected revenue is higher when part of the price is a royaltybased on the observed value.

Again, a linkage of available information to final price.

5.1 auction theory slides

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