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/

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

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selling items

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). *

Myerson’s Auction [1981]

[Myerson’81]: The optimal auction is a virtual welfare maximizer:1. Collects bids b1,…, bm from bidders2. For all i: (i’s “ironed virtual bid”)3. Allocates item to bidder with highest positive (if any)4. Bidders are priced according to the “payment identity,” ensuring

that it’s in their best interest to report .

…

1

i

m

…

independent bidders

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

???

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 = 2

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: $2.625

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]

The Menu

Motivation

Auctions from Linear Programs-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational Remarks

The Menu

Motivation

Auctions from Linear Programs-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational Remarks

- 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 (t1,…,tm) drawn from product distn’ - ’ s are known- is supported on set Ti which is assumed finite

- INPUT: m, n, T1,…,Tm , - GOAL: Find auction optimizing revenue.

Multi-item Auctionsmaximize revenue

…

1

j

n

…

1

i

m

… …

…

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 ti and- Chooses action- Goal: optimize her own utility

expected revenue:

over bidder types t1, …, tm, 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 Equilibrium

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 rule

Finding Optimal Direct Auction► Find

► Such that:1. Feasible:

2. 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 :

3. 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)

The Menu

Motivation

Auctions from Linear Programs-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational Remarks

The Menu

Motivation

Auctions from Linear Programs-the interim allocation rule

Multi-Item Auction Setting

Characterization of Multi-item Auctions

Computational Remarks

the interim rule of an auction► a.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 ti (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 ti

bidder 1A

B

½

½bidder 2

C

D

½

½

• 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 rule

Feasibility of Reduced Forms (example)

► easy setting: single item, two bidders with types uniformly distributed in T1={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

⅓F

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

✔✔

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.

Feasibility of Multi-Item Reduced Forms

► Can view

► Denote feasible interim allocation rules by

► How does look geometrically?

Claim 1:

Feasibility of Multi-Item Reduced Formsset 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

Extreme Points of Polytope?

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 f1,…, fm

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 f1,…, fm.

Any interim rule is implementable by a convex combination of (i.e randomization over) virtual-welfare maximizers.

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 (t1=A, t2=A), or (t1=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 f1 and f2 satisfying:► f1(A)>f2(A) and f1(B)>f2(B). (*)

Likewise, all virtual rules in the support need to satisfy:► f2(A)>f1(B) and f2(B)>f1(A). (**)

can’t hold simultaneously

► 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► 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, f2:

f1(A)=f1(B)=1, f2(A)=f2(B)=0, w/ prob. ½

f1(A)=f1(B)=0, f2(A)=f2(B)=1, w/ prob. ½

An Example

The Menu

Motivation

Auctions from Linear Programs-the interim allocation rule

Multi-Item Auction Setting

Characterization of Optimal Multi-item Auctions

Computational Remarks

• 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 revenue

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 (t1,…,tm) to auctioneer.

► Auctioneer samples virtual transformations f1,…, fm

► 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 truthfulness

Characterization of Optimal Multi-Item Auctions

► Bidders submit types (t1,…,tm) to auctioneer.

► Auctioneer samples virtual transformations f1,…, fm

► 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 truthfulness

► Exact same structure as Myerson►in Myerson’s theorem: virtual function = deterministic►here, randomized (and they must be)

The Menu

Motivation

Auctions from Linear Programs-the interim allocation rule

Multi-Item Auction Setting

Characterization of Optimal Multi-item Auctions

Computational Remarks

• 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)

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, f2,…,fm

Hence:

Can be found in polynomial time.

✔

Need separation oracle for:

• 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)

✔✔

Summary

► Compared to Single-Item auctions, optimal multi-item auctions: have richer structure are computationally more challenging

► Understanding 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