Part 1: Optimal Multi-Item Auctions - PowerPoint Presentation

Slide1

Part 1: Optimal Multi-Item Auctions

Constantinos DaskalakisEECS, MIT

Reference:

Yang

Cai

, Constantinos Daskalakis and Matt Weinberg:

An Algorithmic Characterization of Multi-Dimensional

Mechanisms

,

STOC 2012

.

http://eccc.hpi-web.de/report/2011/172

/Slide2

Auctions

Motivating Question for Parts 1&2: Of all possible auctions, which one optimizes the auctioneer’s revenue?We really mean “of all:” want to choose the best among all possible protocols setting up a bidder interaction, in the end of which an allocation of items and pricing is decided.

spectrum allocation

sponsored search

selling itemsSlide3

Single-Item Auctions

Optimal Auction?

[Myerson’81]:

The optimal single-bidder auction prices item at

[Myerson’81]:

Single item, multiple bidders whose values are

i.i.d

. from F: optimal auction is second price auction with reserve r(F

). *Slide4

Myerson’s Auction [1981]

[Myerson’81]: The optimal auction is a virtual welfare maximizer:Collects bids b

1

,…,

b

m

from biddersFor all i:

(i’s “

ironed virtual bid”)Allocates item to bidder with highest positive (if any)

Bidders are priced according to the “payment identity,” ensuring that it’s in their best interest to report .

…

1

i

m

…

independent biddersSlide5

Beyond Single-Item Auctions?Large body of work in Economics:

e.g. [Laffont-Maskin-Rochet’87], [McAfee-McMillan’88], [Wilson’93], [Armstrong’96], [Rochet-Chone’98], [Armstrong’99],[Zheng’00], [Basov’01], [Kazumori’01], [Thanassoulis’04],[Vincent-Manelli ’06,’07], [Figalli-Kim-McCann’10], [Pavlov’11], [Hart-Nisan’12],…Progress slow. No general approach.Challenge already with 1 bidder, 2 independent items.

1

2

???Slide6

Example 1: Two IID Uniform Items

Optimal auction:



The optimal mechanism need not sell items separately.

Bundling items increases revenue.

$3

expected revenue: 3



¾ = 2.25

Obvious approach:

run Myerson for each item separately

price each item at 1

each bought with probability 1

expected revenue: 2



1 = 2Slide7

Example 2: Two ID Uniform Items

Optimal auction:



The optimal mechanism may not only bundle items, but also use randomization.

$4

$2.50

This item with probability ½

expected revenue: $2.625Slide8

Beyond Single-Item Auctions?Large body of work in Economics:

e.g. [Laffont-Maskin-Rochet’87], [McAfee-McMillan’88], [Wilson’93], [Armstrong’96], [Rochet-Chone’98], [Armstrong’99],[Zheng’00], [Basov’01], [Kazumori’01], [Thanassoulis’04],[Vincent-Manelli ’06,’07], [Figalli-Kim-McCann’10], [Pavlov’11], [Hart-Nisan’12],…Progress slow. No general approach.Challenge already with 1 bidder, 2 independent items.Recent algorithmic work: Constant Factor Approximations[Chawla-Hartline-Kleinberg ’07], [

Chawla

et al’10], [Bhattacharya et al’10], [Alaei’11], [Hart-Nisan ’12], [Kleinberg-Weinberg ’12]Slide9

The Menu

Motivation

Auctions from Linear Programs

-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational RemarksSlide10

The Menu

Motivation

Auctions from Linear Programs

-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational RemarksSlide11

Bidders are additive

(for Part 1)each bidder i is characterized by some vector his value for subset S of items is:

Bayesian assumption:

bidder types (

t

1

,…,

tm

) drawn from product distn’ ’ s

are known is supported on set T

i

which is assumed

finite

INPUT:

m

,

n

,

T

1

,…,

T

m

, GOAL: Find auction optimizing revenue.

Multi-item Auctions

maximize revenue

…

1

j

n

…

1

i

m

…

…Slide12

…

1

j

n

…

1

i

m

…

…

Commits to an auction design, specifying possible bidder actions, the allocation and the price rule

Asks bidders to choose actions

Implements the

promised allocation and price rule

Goal:

Optimize revenue

Auction in Action

Auctioneer:

Each Bidder

i

:

Uses as input

:

the auction specification, her own type

t

i

and

Chooses action

Goal:

optimize her own utility

expected revenue:

over bidder types

t

1

, …,

t

m

, the randomness in the auction (if any), and the randomness in the bidders’

strategic behavior

given their types

payment made by bidder

i

to the auctioneer

Bayesian Nash EquilibriumSlide13

Simplification:

Direct Auctions

Focus on Direct Auctions (

wlog

)

huge universe of possible auctions: what bidders can do, and how to allocate items and charge bidders when they do it

The direct revelation principle:

“Any auction has an equivalent one where the bidders are only asked to report their type to the auctioneer, and it is best for them to truthfully report it. Such auctions are called

direct

.”equivalent ?

point-wise

w.r.t

. : the two auctions result in the same allocation, the same payments, and the same bidder utilities

upshot:

mechanism design reduces to computing functions:

: probability (over randomness in auction) that item

j

is allocated to bidder

i

when the reported types by bidders are

: expected price that bidder

i

pays when reports are

called the auction’s

ex-post allocation and price ruleSlide14

Finding Optimal Direct Auction

FindSuch that:Feasible: It is in every bidder’s “best interest” to truthfully report his type.

Captured by Bayesian Incentive Compatibility (BIC) constraint:

for all

i

,

and types :

The expected revenue is maximized

Actually an LP, but of the “laundry-list” kind…

number of variables: vs input size

Incentive Compatibility (IC)

ditto, but point-wise

w.r.t

.

(i.e. without expectation over ; just the randomness in the mechanism)Slide15

The Menu

Motivation

Auctions from Linear Programs

-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational RemarksSlide16

The Menu

Motivation

Auctions from Linear Programs

-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational RemarksSlide17

the interim rule

of an auctiona.k.a. the reduced form :

Example: Suppose 1 item, 2 bidders

Consider auction that allocates item preferring A to C to B to D, and charges $2 dollars to whoever gets the item.

Then

: probability item

j

is allocated to bidder

i

conditioning on his type being

t

i

(over the randomness in the other bidders’ types, and the randomness in the auction)

: expected price paid by bidder

i

conditioning on his type being

t

i

bidder 1

A

B

½

½

bidder 2

C

D

½

½Slide18

Variables:

Constraints:

Maximize:

- the expected revenue

Mechanism Design with Reduced Form

Truthfulness:

Need: (

i

) ability to check feasibility of interim allocation rules

(ii) efficient map from

feasible interim rules

to

ex-post allocation rules

(optimal feasible reduced form is useless in itself)

the reduced form of sought auction

expected value of bidder

i

of type for being given

exists auction with this interim ruleSlide19

Feasibility of Reduced Forms (example)

easy setting:

single item

, two bidders with types uniformly distributed in

T

1

={

A,

B, C} and T2

={

D, E, F

} respectively

Question:

Is the following interim allocation rule feasible?

( A, D/E/F)



A wins.

(B/C, D)



D wins.

so infeasible !

bidder 1

A

B

⅓

⅓

C

⅓

bidder 2

D

E

⅓

⅓

F

⅓

(B, F)



B wins.

(C, E)



E wins.

(B, E)



B needs to win

w.p

. ½, E needs to win

w.p

. ⅔

✔

✔Slide20

Feasibility of Reduced Forms

[Border ’91, Border ’07, Che-Kim-Mierendorff ’11]: Exist linear constraints characterizing feasibility of

single-item

reduced forms.

Problem:

Single-item, and exponentially many inequalities.

[Cai-Daskalakis-Weinberg’12]: -many inequalities suffice.

([Alaei et al’12]:

polynomial-time algorithm for feasibility)Still only single-item reduced forms.Slide21

Feasibility of

Multi-Item Reduced FormsCan view

Denote feasible interim allocation rules by

How does look geometrically?Slide22

Claim 1:

Feasibility of

Multi-Item Reduced Forms

set of feasible interim allocation rules

Proof: Easy.

If feasible, exists (ex-post) allocation rule

M

with interim rule .

M

is a distribution over deterministic feasible allocation rules, of which there is a finite number. So: , where is deterministic.

Easy to see:

So

Slide23

Extreme Points of Polytope?Slide24

Extreme Points of Polytope?

interpretation

:

virtual

value

derived by bidder

i

when given item

j

,

if his type is

A

expected

virtual welfare

achieved by allocation rule with interim rule

interim rule of virtual welfare maximizing allocation rule with virtual functions

f

1

,…, f

mSlide25

Claim 1:

Feasibility of

Multi-Item Reduced Forms

set of feasible interim allocation rules

Claim 2:

Every vertex of the

polytope

is the interim rule of a virtual welfare maximizing allocation rule for some virtual functions

f

1

,…,

f

m

.



Any

interim rule is implementable by a convex combination of (

i.e

randomization over) virtual-welfare

maximizers

.Slide26

An Example

1 item, 2 bidders, each with uniform type in {A, B}consider following (somewhat funky) allocation rule M

:

If types are equal, give item to bidder 1

Otherwise, give item to bidder 2

Can

M

be implemented as a distribution over virtual-welfare maximizing allocation rules?A: No

Proof: Suppose M was distn

’ over virtual welfare max. alloc. rules.

If reported types are (

t

1

=

A

,

t

2

=

A

), or (

t

1

=B

, t2=B

) then bidder 1 gets the item with probability 1. So all virtual welfare maximizing allocation rules in the support of the

distn

’ have virtual value functions f

1 and

f2

satisfying: f

1(A)>f

2(A) and f1

(B)>f2

(B). (*)Likewise, all virtual rules in the support need to satisfy:

f2(A)>

f1(B) and f

2(B)>f

1(A). (**)

can’t hold simultaneouslySlide27

1 item, 2 bidders, each with uniform type in {A,

B}consider following (somewhat funky) allocation rule M:If types are equal, give item to bidder 1Otherwise, give item to bidder 2

Can

M

be implemented as a distribution over virtual-welfare maximizing allocation rules?

A:

No

OK, what’s the interim rule of M?

A: Can this be implemented as a distribution over virtual-welfare maximizing allocation rules?A: yes, use the following distn

’ over virtual functions f1,

f

2

:

f

1

(A)=

f

1

(B)=1,

f

2

(A)=

f2

(B)=0, w/ prob. ½ f

1

(A)=f

1(B)=0, f

2(A)=

f2

(B)=1, w

/ prob. ½

An ExampleSlide28

The Menu

Motivation

Auctions from Linear Programs

-the interim allocation rule

Multi-Item Auction Setting

Characterization of Optimal Multi-item Auctions

Computational RemarksSlide29

Variables:

Constraints:

Maximize:

- the expected revenue

Truthfulness:

the reduced form of sought auction

Mechanism Design with Reduced Form

Two auctions with same interim allocation rule have same revenueSlide30

Characterization of Optimal Multi-Item Auctions

[Cai-Daskalakis-Weinberg’12]:

For every multi-item auction, there exists an auction with the same interim rule, which is a distribution over virtual welfare

maximizers

.

Corollary:

Optimal multi-item auction has the following structure:

Bidders submit types (t

1,…,tm

) to auctioneer.Auctioneer samples virtual transformations f

1

,…,

f

m

Auctioneer computes virtual types

Virtual welfare maximizing allocation is chosen.

Namely, each item is given to bidder with highest virtual value for that item (if positive)

Prices are charged to ensure truthfulnessSlide31

Characterization of Optimal Multi-Item Auctions

Bidders submit types (t

1

,…,t

m

) to auctioneer.

Auctioneer samples virtual transformations

f1

,…, fmAuctioneer computes virtual types Virtual welfare maximizing allocation is chosen.

Namely, each item is given to bidder with highest virtual value for that item (if positive)Prices are charged to ensure truthfulness

Exact same structure as Myerson

in Myerson’s theorem: virtual function = deterministic

here,

randomized

(and they must be)Slide32

The Menu

Motivation

Auctions from Linear Programs

-the interim allocation rule

Multi-Item Auction Setting

Characterization of Optimal Multi-item Auctions

Computational RemarksSlide33

Variables:

Constraints:

Maximize:

- the expected revenue

Truthfulness:

the reduced form of sought auction

Mechanism Design with Reduced Form

To solve need: (

i

) ability to check feasibility of interim allocation rules

(ii) efficient map from

feasible interim rules

to

ex-post allocation rules

(optimal feasible reduced form is useless in itself)Slide34

Poly-time Feasibility and Implementation

[Grötschel-Lovász-Schrijver ’80/Papadimitriou-Karp’80]:

Linear Optimization



Separation

What this means for us is: suffices to be able to find in polynomial-time, the extreme interim allocation rule in an arbitrary direction .

But we know that is virtual welfare

maximizer

for some f1, f

2,…,fm

Hence:

Can be found in polynomial time.

✔

Need separation oracle for:Slide35

Variables:

Constraints:

Maximize:

- the expected revenue

Truthfulness:

the reduced form of sought auction

Mechanism Design with Reduced Form

To solve need: (

i

) ability to check feasibility of interim allocation rules

(ii) efficient map from

feasible interim rules

to

ex-post allocation rules

(optimal feasible reduced form is useless in itself)

✔

✔Slide36

Summary

Compared to Single-Item auctions, optimal multi-item auctions: have richer structureare computationally more challengingUnderstanding Interim allocation rule allowed us to characterize the structure of optimal multi-item auctions for additive bidders:“The revenue optimal auction is a virtual-welfare

maximizer

.”

Difference to Myerson: virtual transformation randomized.

Finding Optimal Auction: polynomial-time solvable

Up next:

Yang: Beyond additive bidders/trivial allocation constraints

Matt: Beyond revenue objective