Announcements No class next Tuesday CS Department Research - - PowerPoint PPT Presentation

1

Announcements

ØNo class next Tuesday ØCS Department Research Symposium (10/08, next Tuesday)

CS6501: T

• pics in Learning and Game Theory

(Fall 2019) Optimal Auction Design for Single-Item Allocation

(Part II)

Instructor: Haifeng Xu

3

Outline

Ø Recap: Mechanism Design Basics Ø Optimal Auction Design for Independent Bidders

4

Single-Item Allocation

5

Single-Item Allocation

ØFor convenience, think of 𝑤" ∼ 𝑔

" independently

ØObjective: maximize revenue ∑"∈[(] 𝑞"

Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣, 𝑔⟩ where:

Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø𝑊 = 𝑊

7× ⋯× 𝑊 ( is the set of all possible value profiles

Ø𝑌 = {0,1, ⋯ , 𝑜} is the set of winners Ø𝑣 = (𝑣7, ⋯ , 𝑣() where 𝑣" = 𝑤"𝑦" − 𝑞" is the utility function of 𝑗

for any (randomized) allocation 𝑦 ∈ Δ(@7 and payment 𝑞"

Ø𝑔 is the public prior on buyer values 𝑤 ∈ 𝑊

6

The Design Space – Mechanisms

ØThat is, we will design ⟨𝐵, 𝑕⟩ ØPlayers’ utility function will be fully determined by ⟨𝐵, 𝑕⟩ ØWe want to maximize revenue at the Bayes Nash equilibrium of

this resulting game A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØA = 𝐵7× ⋯× 𝐵( where 𝐵" is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to outcome = [an allocation

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏7, ⋯ , 𝑏() ∈ 𝐵

7

The Design Space – Mechanisms

Example: second-price auction Ø 𝐵" = ℝ@ for all 𝑗 Ø 𝑕 𝑏 allocates the item to the buyer 𝑗∗ = arg max

"

𝑏" and asks 𝑗∗ to pay max2" 𝑏", and all other buyers pay 0 A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØA = 𝐵7× ⋯× 𝐵( where 𝐵" is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to outcome = [an allocation

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏7, ⋯ , 𝑏() ∈ 𝐵

Ø Truthful bidding is a dominant-strategy equilibrium, thus also a BNE Ø Thus expect truthful bidding (i.e., 𝑏" = 𝑤"); Revenue will be 𝔽P∼Q max2" 𝑤"

8

Incentive Compatible Mechanisms

• Definition. A mechanism ⟨𝐵, 𝑕⟩ is a direct revelation mechanism

if 𝐵" = 𝑊

" for all 𝑗. In this case, the mechanism is described by 𝑕.

ØA stronger notion of IC is dominant-strategy IC (DIC) ØA DIC mechanism is also BIC ØExample: second-price auction is DIC

• First price auction can be “modified” to be BIC
• Definition. A direct revelation mechanism 𝑕

is Bayesian incentive-compatible (a.k.a., truthful or BIC) if truthful bidding forms a Bayes Nash equilibrium in the resulting game

ØIn DR mechanism, we only need to design 𝑕

9

The Revelation Principle

ØProof idea: let the auctioneer to simulate the strategic behaviors

• n behalf of bidders, so they only need to react honestly
• Theorem. If there is a mechanism that achieves revenue 𝑆 at a

Bayes Nash equilibrium [resp. dominant-strategy equilibrium], then there is a direct revelation, Bayesian incentive-compatible [resp. DIC] mechanism achieving revenue 𝑆.

10

The Revelation Principle

ØProof idea: let the auctioneer to simulate the strategic behaviors

• n behalf of bidders, so they only need to react honestly
• Theorem. If there is a mechanism that achieves revenue 𝑆 at a

Bayes Nash equilibrium [resp. dominant-strategy equilibrium], then there is a direct revelation, Bayesian incentive-compatible [resp. DIC] mechanism achieving revenue 𝑆. Optimal Mechanism Design for Single-Item Allocation Given instance ⟨𝑜, 𝑊, 𝑌, 𝑣, 𝑔⟩, design the allocation function 𝑦: 𝑊 → 𝑌 and payment 𝑞: 𝑊 → ℝ( such that truthful bidding is a BNE in the following Bayesian game: 1. Solicit bid 𝑐7 ∈ 𝑊

7, ⋯ , 𝑐( ∈ 𝑊 (

2. Select allocation 𝑦 𝑐7, ⋯ , 𝑐( ∈ 𝑌 and payment 𝑞(𝑐7, ⋯ , 𝑐()

11

Optimal Bayesian Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

max

T,U

𝔽P∼Q ∑"V7

(

𝑞"(𝑤7, ⋯ , 𝑤()

• s. t.

𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑤", 𝑤\" − 𝑞" 𝑤", 𝑤\" ≥ 𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑐", 𝑤\" − 𝑞" 𝑐", 𝑤\" , ∀𝑗 ∈ 𝑜 , 𝑤", 𝑐" ∈ 𝑊

"

∑"V_

(

𝑦"(𝑤) = 1, ∀𝑤 ∈ 𝑊 𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑤", 𝑤\" − 𝑞" 𝑤", 𝑤\" ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤" ∈ 𝑊

"

𝑦" 𝑤 ≥ 0, ∀𝑤 ∈ 𝑊, ∀𝑗 = 0,1 ⋯ , 𝑜

12

Optimal Bayesian Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

max

T,U

𝔽P∼Q ∑"V7

(

𝑞"(𝑤7, ⋯ , 𝑤()

• s. t.

𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑤", 𝑤\" − 𝑞" 𝑤", 𝑤\" ≥ 𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑐", 𝑤\" − 𝑞" 𝑐", 𝑤\" , ∀𝑗 ∈ 𝑜 , 𝑤", 𝑐" ∈ 𝑊

"

∑"V_

(

𝑦"(𝑤) = 1, ∀𝑤 ∈ 𝑊 𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑤", 𝑤\" − 𝑞" 𝑤", 𝑤\" ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤" ∈ 𝑊

"

BIC constraints Individually rational (IR) constraints 𝑦" 𝑤 ≥ 0, ∀𝑤 ∈ 𝑊, ∀𝑗 = 0,1 ⋯ , 𝑜

13

Optimal Bayesian Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

ØIf 𝑊 has finite support, this is an LP with variables 𝑦" 𝑤 , 𝑞" 𝑤

",P

max

T,U

𝔽P∼Q ∑"V7

(

𝑞"(𝑤7, ⋯ , 𝑤()

• s. t.

𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑤", 𝑤\" − 𝑞" 𝑤", 𝑤\" ≥ 𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑐", 𝑤\" − 𝑞" 𝑐", 𝑤\" , ∀𝑗 ∈ 𝑜 , 𝑤", 𝑐" ∈ 𝑊

"

∑"V_

(

𝑦"(𝑤) = 1, ∀𝑤 ∈ 𝑊 𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑤", 𝑤\" − 𝑞" 𝑤", 𝑤\" ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤" ∈ 𝑊

"

BIC constraints Individually rational (IR) constraints 𝑦" 𝑤 ≥ 0, ∀𝑤 ∈ 𝑊, ∀𝑗 = 0,1 ⋯ , 𝑜

14

Optimal Bayesian Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

ØIf 𝑊 has finite support, this is an LP with variables 𝑦" 𝑤 , 𝑞" 𝑤

",P

Ø Drawbacks of this algorithmic approach:

(1) Support of 𝑊 may be extremely large in which case LP is large (2) Do not reveal any structure about the optimal auction – do not know what it is like except that it is a solution to an LP

ØNext, will look at continuous 𝑊 and solve out for the optimal

function 𝑦 𝑤 , 𝑞 𝑤

• This will also lead to an elegant form of the optimal auction

15

Outline

Ø Recap: Mechanism Design Basics Ø Optimal Auction Design for Independent Bidders

• That is, will assume 𝑤" ∼ 𝑔

" independently

16

The Optimal Auction (Myerson’1981)

Theorem (informal). For single-item allocation with prior distribution 𝑤" ∼ 𝑔

" independently, the following auction is BIC and optimal:

1. Solicit buyer values 𝑤7, ⋯ , 𝑤( 2. Transform 𝑤" to “virtual value” 𝜚"(𝑤") where 𝜚" 𝑤" = 𝑤" −

7\a[(P[) Q[(P[)

3. If 𝜚" 𝑤" < 0 for all 𝑗, keep the item and no payments 4. Otherwise, allocate item to 𝑗∗ = arg max

"∈[(] 𝜚"(𝑤")

and charge him the minimum bid needed to win, i.e., 𝜚"

\7 max max cd"∗ 𝜚c(𝑤c) , 0

; Other bidders pay 0.

17

Stages of a Bayesian Game

ØStages of a Bayesian game of mechanism design:

• Ex-ante: Before players learn their types
• Interim: A player learns his own type, but not the types of others
• Ex-post: All players types are revealed

ØInterim stage is when players make decisions

• The interim allocation for buyer 𝑗 tells us what 𝑗’s probability of winning

is as a function of his bid 𝑐", in expectation over others’ truthful report e 𝑦" 𝑐" = 𝔽PZ[∼QZ[𝑦" 𝑐", 𝑤\"

• Similarly, the interim payment is

e 𝑞" 𝑐" = 𝔽PZ[∼QZ[𝑞" 𝑐", 𝑤\"

• Expected bidder utility of bidding 𝑐"

𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑐", 𝑤\" − 𝑞" 𝑐", 𝑤\" = 𝑤" e 𝑦" 𝑐" − e 𝑞" 𝑐"

• If BIC, expected revenue

𝔽P∼Q ∑"V7

(

𝑞"(𝑤7, ⋯ , 𝑤() = ∑"V7

(

𝔽P∼Q 𝑞"(𝑤7, ⋯ , 𝑤() = ∑"V7

(

𝔽P[∼Q[ e 𝑞" (𝑤")

18

Stages of a Bayesian Game

ØStages of a Bayesian game of mechanism design:

• Ex-ante: Before players learn their types
• Interim: A player learns his own type, but not the types of others
• Ex-post: All players types are revealed

ØInterim stage is when players make decisions

• The interim allocation for buyer 𝑗 tells us what 𝑗’s probability of winning

is as a function of his bid 𝑐", in expectation over others’ truthful report e 𝑦" 𝑐" = 𝔽PZ[∼QZ[𝑦" 𝑐", 𝑤\"

• Similarly, the interim payment is

e 𝑞" 𝑐" = 𝔽PZ[∼QZ[𝑞" 𝑐", 𝑤\"

• Expected bidder utility of bidding 𝑐"

𝔽PZ[∼QZ[ 𝑤"𝑦" 𝑐", 𝑤\" − 𝑞" 𝑐", 𝑤\" = 𝑤" e 𝑦" 𝑐" − e 𝑞" 𝑐"

• If BIC, expected revenue

𝔽P∼Q ∑"V7

(

𝑞"(𝑤7, ⋯ , 𝑤() = ∑"V7

(

𝔽P∼Q 𝑞"(𝑤7, ⋯ , 𝑤() = ∑"V7

(

𝔽P[∼Q[ e 𝑞" (𝑤")

19

Examples

Assume two buyers, 𝑤7, 𝑤f ∼ 𝑉[0,1] independently Second-price auction Ø 𝑦7 𝑐7 = 𝔽Ph∼Q

h𝑦7 𝑐7, 𝑤f = 𝑐7

Ø 𝑞7 𝑐7 = 𝔽Ph∼Q

h𝑞7 𝑐7, 𝑤f = ∫

_ jk 𝑤f 𝑔 f 𝑤f 𝑒𝑤f = 𝑐7 f/2

Ø 𝑦f 𝑐f , 𝑞f 𝑐f have the same form

20

Examples

Assume two buyers, 𝑤7, 𝑤f ∼ 𝑉[0,1] independently Second-price auction Ø 𝑦7 𝑐7 = 𝔽Ph∼Q

h𝑦7 𝑐7, 𝑤f = 𝑐7

Ø 𝑞7 𝑐7 = 𝔽Ph∼Q

h𝑞7 𝑐7, 𝑤f = ∫

_ jk 𝑤f 𝑔 f 𝑤f 𝑒𝑤f = 𝑐7 f/2

Ø 𝑦f 𝑐f , 𝑞f 𝑐f have the same form Modified first-price auction (Recall: truthful bidding is an BNE) Ø 𝑦7 𝑐7 = 𝔽Ph∼Q

h𝑦7 𝑐7, 𝑤f = 𝑐7

Ø 𝑞7 𝑐7 = 𝔽Ph∼Q

h𝑞7 𝑐7, 𝑤f = ∫

_ jk jk f ⋅ 𝑔 f 𝑤f 𝑒𝑤f = 𝑐7 f/2

Ø 𝑦f 𝑐f , 𝑞f 𝑐f have the same form

From now on we will write 𝑦" 𝑐" = e 𝑦"(𝑐") to avoid cumbersome notation

21

Myerson’s Monotonicity Lemma

• Lemma. Consider single-item allocation with prior distribution 𝑤" ∼ 𝑔

"

• independently. A direct-revelation mechanism with interim allocation

𝑦 and interim payment 𝑞 is BIC if and only if for each buyer 𝑗: 1. 𝑦"(𝑐") is a monotone non-decreasing function of 𝑐" 2. 𝑞"(𝑐") is uniquely determined as follows, with 𝑞" 0 = 0, 𝑞" 𝑐" = 𝑐" ⋅ 𝑦" 𝑐" − ∫

jV_ j[ 𝑦" 𝑐 𝑒𝑐 .

22

Myerson’s Monotonicity Lemma

• Lemma. Consider single-item allocation with prior distribution 𝑤" ∼ 𝑔

"

• independently. A direct-revelation mechanism with interim allocation

𝑦 and interim payment 𝑞 is BIC if and only if for each buyer 𝑗: 1. 𝑦"(𝑐") is a monotone non-decreasing function of 𝑐" 2. 𝑞"(𝑐") is uniquely determined as follows, with 𝑞" 0 = 0, 𝑞" 𝑐" = 𝑐" ⋅ 𝑦" 𝑐" − ∫

jV_ j[ 𝑦" 𝑐 𝑒𝑐 .

23

Myerson’s Monotonicity Lemma

• Lemma. Consider single-item allocation with prior distribution 𝑤" ∼ 𝑔

"

• independently. A direct-revelation mechanism with interim allocation

𝑦 and interim payment 𝑞 is BIC if and only if for each buyer 𝑗: 1. 𝑦"(𝑐") is a monotone non-decreasing function of 𝑐" 2. 𝑞"(𝑐") is uniquely determined as follows, with 𝑞" 0 = 0, 𝑞" 𝑐" = 𝑐" ⋅ 𝑦" 𝑐" − ∫

jV_ j[ 𝑦" 𝑐 𝑒𝑐 .

24

Interpretation of Myerson’s Lemma

ØThe higher a player bids, the higher the probability of winning ØFor each additional 𝜗 of winning probability, pay additionally at a

rate equal to the current bid

ØProof: see the reading material on course website

25

Corollaries of Myerson’s Lemma

Corollaries.

1.

Interim allocation uniquely determines interim payment

2.

Expected revenue depends only on the allocation rule

3.

Any two auctions with the same interim allocation rule at BNE have the same expected revenue at the same BNE

Therefore, second-price and first-price auction (and its modified version) all have the same revenue in previous two bidder i.i.d example

26

Revenue as Virtual Welfare

ØDefine the virtual value of player 𝑗 as a function of his value 𝑤":

𝜚" 𝑤" = 𝑤" − 1 − 𝐺"(𝑤") 𝑔

"(𝑤")

• Lemma. Consider any BIC mechanism 𝑁 with interim allocation 𝑦

and interim payment 𝑞, normalized to 𝑞" 0 = 0. The expected revenue of 𝑁 is equal to the expected virtual welfare served ∑"V7

(

𝔽P[∼Q[ 𝜚" 𝑤" 𝑦"(𝑤")

27

Revenue as Virtual Welfare

ØDefine the virtual value of player 𝑗 as a function of his value 𝑤":

𝜚" 𝑤" = 𝑤" − 1 − 𝐺"(𝑤") 𝑔

"(𝑤")

• Lemma. Consider any BIC mechanism 𝑁 with interim allocation 𝑦

and interim payment 𝑞, normalized to 𝑞" 0 = 0. The expected revenue of 𝑁 is equal to the expected virtual welfare served ∑"V7

(

𝔽P[∼Q[ 𝜚" 𝑤" 𝑦"(𝑤")

ØThis is the expected virtual value of the winning bidder ØProof is an application of Myerson’s monotonicity lemma, plus

algebraic calculations

ØRecall the expected revenue is ∑"V7

(

𝔽P[∼Q[ 𝑞"(𝑤")

28

Revenue as Virtual Welfare

ØDefine the virtual value of player 𝑗 as a function of his value 𝑤":

𝜚" 𝑤" = 𝑤" − 1 − 𝐺"(𝑤") 𝑔

"(𝑤")

• Lemma. Consider any BIC mechanism 𝑁 with interim allocation 𝑦

and interim payment 𝑞, normalized to 𝑞" 0 = 0. The expected revenue of 𝑁 is equal to the expected virtual welfare served ∑"V7

(

𝔽P[∼Q[ 𝜚" 𝑤" 𝑦"(𝑤")

ØThis is the expected virtual value of the winning bidder ØProof is an application of Myerson’s monotonicity lemma, plus

algebraic calculations

ØRecall the expected revenue is ∑"V7

(

𝔽P[∼Q[ 𝑞"(𝑤")

29

Proof

𝔽P[∼Q[ e 𝑞" 𝑤" = r

P[

𝑤" ⋅ 𝑦" 𝑤" − r

jV_ P[

𝑦" 𝑐 𝑒𝑐 𝑔

" 𝑤" 𝑒𝑤"

By Myerson’s monotonicity lemma Assumed bidder 𝑗 bids truthfully

30

Proof

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

r

jV_ P[

𝑦" 𝑐 𝑔

" 𝑤" 𝑒𝑐 𝑒𝑤"

𝔽P[∼Q[ e 𝑞" 𝑤" = r

P[

𝑤" ⋅ 𝑦" 𝑤" − r

jV_ P[

𝑦" 𝑐 𝑒𝑐 𝑔

" 𝑤" 𝑒𝑤"

Rearrange terms

31

Proof

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

r

jV_ P[

𝑦" 𝑐 𝑔

" 𝑤" 𝑒𝑐 𝑒𝑤"

𝔽P[∼Q[ e 𝑞" 𝑤" = r

P[

𝑤" ⋅ 𝑦" 𝑤" − r

jV_ P[

𝑦" 𝑐 𝑒𝑐 𝑔

" 𝑤" 𝑒𝑤"

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

r

P[sj

𝑦"(𝑐) 𝑔

" 𝑤" 𝑒𝑤"𝑒𝑐

Exchange of integral variable order

32

Proof

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

r

jV_ P[

𝑦" 𝑐 𝑔

" 𝑤" 𝑒𝑐 𝑒𝑤"

𝔽P[∼Q[ e 𝑞" 𝑤" = r

P[

𝑤" ⋅ 𝑦" 𝑤" − r

jV_ P[

𝑦" 𝑐 𝑒𝑐 𝑔

" 𝑤" 𝑒𝑤"

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

r

P[sj

𝑦"(𝑐) 𝑔

" 𝑤" 𝑒𝑤"𝑒𝑐

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

𝑦"(𝑐)(1 − 𝐺"(𝑐)) 𝑒𝑐

Since ∫

P[sj 𝑔 " 𝑤" 𝑒𝑤" = 1 − 𝐺"(𝑐)

33

Proof

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

r

jV_ P[

𝑦" 𝑐 𝑔

" 𝑤" 𝑒𝑐 𝑒𝑤"

𝔽P[∼Q[ e 𝑞" 𝑤" = r

P[

𝑤" ⋅ 𝑦" 𝑤" − r

jV_ P[

𝑦" 𝑐 𝑒𝑐 𝑔

" 𝑤" 𝑒𝑤"

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

r

P[sj

𝑦"(𝑐) 𝑔

" 𝑤" 𝑒𝑤"𝑒𝑐

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

𝑦"(𝑐)(1 − 𝐺"(𝑐)) 𝑒𝑐 = r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

𝑦" 𝑤" 1 − 𝐺" 𝑤" 𝑒𝑤"

34

Proof

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

r

jV_ P[

𝑦" 𝑐 𝑔

" 𝑤" 𝑒𝑐 𝑒𝑤"

𝔽P[∼Q[ e 𝑞" 𝑤" = r

P[

𝑤" ⋅ 𝑦" 𝑤" − r

jV_ P[

𝑦" 𝑐 𝑒𝑐 𝑔

" 𝑤" 𝑒𝑤"

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

r

P[sj

𝑦"(𝑐) 𝑔

" 𝑤" 𝑒𝑤"𝑒𝑐

= r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r j

𝑦"(𝑐)(1 − 𝐺"(𝑐)) 𝑒𝑐 = r

P[

𝑤" ⋅ 𝑦" 𝑤" 𝑔

" 𝑤" 𝑒𝑤" − r P[

𝑦" 𝑤" 1 − 𝐺" 𝑤" 𝑒𝑤" = r

P[

𝑦" 𝑤" ⋅ 𝑤"𝑔

" 𝑤" − 1 − 𝐺" 𝑤"

𝑒𝑤" = r

P[

𝑦" 𝑤" ⋅ 𝑔

"(𝑤") 𝑤" − 1 − 𝐺" 𝑤"

𝑔

"(𝑤")

𝑒𝑤" = 𝔽P[∼Q[ 𝜚" 𝑤" 𝑦(𝑤")

35

The Optimal Auction

ØRevenue of any BIC mechanism equals ∑"V7

(

𝔽P[∼Q[ 𝜚" 𝑤" 𝑦(𝑤") Q: how to extract the maximum revenue then?

36

The Optimal Auction

ØRevenue of any BIC mechanism equals ∑"V7

(

𝔽P[∼Q[ 𝜚" 𝑤" 𝑦(𝑤") Q: how to extract the maximum revenue then?

1.

Solicit buyer values 𝑤7, ⋯ , 𝑤( and calculate virtual values 𝜚"(𝑤")

2.

If 𝜚" 𝑤" < 0 for all 𝑗, keep the item and no payments (why?)

3.

Otherwise, allocate item to 𝑗∗ = arg max

"∈[(] 𝜚"(𝑤")

4.

How much to charge? Myerson’s lemma says there is a unique interim payment

• Charging minimum bid needed to win 𝜚"

\7 max max cd"∗ 𝜚c(𝑤c) , 0

works.

The optimal auction

37

1.

Solicit buyer values 𝑤7, ⋯ , 𝑤( and calculate virtual values 𝜚"(𝑤")

2.

If 𝜚" 𝑤" < 0 for all 𝑗, keep the item and no payments (why?)

3.

Otherwise, allocate item to 𝑗∗ = arg max

"∈[(] 𝜚"(𝑤"), charge him the

minimum bid needed to win 𝜚"

\7 max max cd"∗ 𝜚c(𝑤c) , 0

; others pay 0

The Optimal Auction

Observations.

ØThe allocation rule maximizes virtual welfare point-point, thus also

maximizes expected virtual welfare

ØBy previous lemma, this is the maximum possible revenue ØPayment satisfies Myerson’s lemma (check it)

Are we done?

38

A Wrinkle

ØOne more thing – Myerson lemma requires the interim allocation to

be monotone

ØWhen 𝜚" 𝑤" = 𝑤" −

7\a[(P[) Q[(P[) is monotone in 𝑤", allocation is monotone

ØFortunately, most natural distributions will lead to monotone VV

function (e.g., Gaussian, uniform, exp, etc.)

• Such a distribution is called regular
• Conclusion. When values are drawn from regular distributions

independently, the VV maximizing auction (aka Myerson’s

• ptimal auction) is a revenue-optimal BIC mechanism!

Can be extended to non-regular distributions via ironing (won’t cover here)

39

Remark 1

ØThe optimal auction just so happens to be DIC

• Think of each bidder’s bid as bidding the virtual value instead
• It is effectively a second-price auction with reserve price 0, but in the

virtual value space

ØFor single-item auction, optimal BIC mechanism achieves the

same revenue as optimal DIC mechanism

• Not true for selling multiple items (even two items to two bidders)

40

Remark 2

ØWhen buyers’ values are i.i.d., optimal auction has an even

simpler format

• Assume regular distribution, allocate the item to largest 𝜚" 𝑤" = 𝜚(𝑤")
• Regularity implies monotonicity of 𝜚, so really just allocate to largest

𝑤"

• Payment is the minimum bid to win, which is max max2 𝑤" , 𝜚\7 0

.

• This is a second price auction with reserve 𝜚\7 0

41

Remark 3

ØApplies to “single parameter” problems more generally

• Intuitively, each bidder’s value can be captured by a single parameter

ØFor example, sell many copies of the same item to buyers

• Can even have allocation constraints, e.g., if bidder 1 gets 1 copy then

bidder 2 is not allowed to get one

Thank You

Haifeng Xu

University of Virginia hx4ad@virginia.edu