Machine Learning for Mechanism

Design and Pricing Problems

Avrim BlumCarnegie Mellon University

Joint work with Maria-Florina Balcan, Jason Hartline, and Yishay Mansour

[Informs 2009]

Auctions/pricing

Software, movies, information access

Designing auction/pricing mechanisms esp for complex markets: challenging problems at the intersection of CS and Economics

Auction mechanisms for selling digital goods

Auctions/pricing

Ad-auctions

Designing auction/pricing mechanisms esp for complex markets: challenging problems at the intersection of CS and Economics

Auctions/pricing

Combinatorial Auctions

Selling many different kinds of items. Buyers with complex preferences over bundles:

“I only want the hotel room if I get the flight too…”

Some items or services that overlap, others only good if have something else too. How should you set prices to make the most profit?

Designing auction/pricing mechanisms esp for complex markets: challenging problems at the intersection of CS and Economics

Auctions/pricing

Even if all customers’ preference information, how much they would be willing to pay, etc. is known up-front, setting prices to maximize revenue can be a challenging algorithmic problem.

But in addition, incentive constraints: customers won’t give you the (correct) information if (possibly) not in their best interest.

Designing auction/pricing mechanisms esp for complex markets: challenging problems at the intersection of CS and Economics

Auction/Pricing Problems

Version 1: Seller knows the true values.

Version 2: values given by selfish agents.

One Seller, Multiple Buyers with Complex Preferences.

CS / optimizationCS / optimization EconomicsEconomics

Algorithm Design Problem (AD)

Incentive Compatible Auction (IC)

Previous Work on IC : specific mechanisms for restricted settings.

Seller’s Goal: maximize profit.

Auction/Pricing Problems

Version 1: Seller knows the true values.

Version 2: values given by selfish agents.

One Seller, Multiple Buyers with Complex Preferences.

CS / optimizationCS / optimization EconomicsEconomics

Algorithm Design Problem (AD)

Incentive Compatible Auction (IC)

Previous Work on IC : specific mechanisms for restricted settings.

Seller’s Goal: maximize profit.

Our Work: Generic Reduction using ML

How is this related to Machine Learning?

You’ve developed a cool new software tool & want to sell it. - n potential buyers. Buyer i has valuation vi.

- Can potentially sell to all of them, but buyer i will

only purchase if priced below vi.

- Unfortunately, you don’t know the vi.

Simple version: basic digital good auction problem.

How is this related to Machine Learning?

You’ve developed a cool new software tool & want to sell it.

Simple version: basic digital good auction problem.

Classic econ model: buyers “types” (valuations) chosen iid from known distribution D. In this case, just

set sales price pD to maximize expected profit.

But what if don’t want to assume this?

- n potential buyers. Buyer i has valuation vi.

- Can potentially sell to all of them, but buyer i will

only purchase if priced below vi.

- Unfortunately, you don’t know the vi.

How is this related to Machine Learning?

You’ve developed a cool new software tool & want to sell it.

Simple version: basic digital good auction problem.

Could ask people for their valuations and use this to set a price as before, but people will low-ball (not incentive-compatible…)

- n potential buyers. Buyer i has valuation vi.

- Can potentially sell to all of them, but buyer i will

only purchase if priced below vi.

- Unfortunately, you don’t know the vi.

How is this related to Machine Learning?

You’ve developed a cool new software tool & want to sell it.

Simple version: basic digital good auction problem.

Ask buyers to submit bids bi.

Randomly partition bidders into

two sets S1, S2.

Find best price over bids in S1…

and use it as offer price on S2! (& vice

versa).

Random sampling auction:

SS11

SS22

- n potential buyers. Buyer i has valuation vi.

- Can potentially sell to all of them, but buyer i will

only purchase if priced below vi.

- Unfortunately, you don’t know the vi.

How is this related to Machine Learning?

You’re Sperizon-mobile. Want to price various services.-Basic service

-Extra lines-Data package-TV features, …

More interesting version: combinatorial auctions

People have potentially nonlinear valuations over subsets.

Might also have known info about customers (current usage, demographics,…).

Want to perform nearly as well as best (simple) pricing function over known info.

(Combinatorial Attribute Auction)

How is this related to Machine Learning?

You’re Sperizon-mobile. Want to price various services.-Basic service

-Extra lines-Data package-TV features, …

More interesting version: combinatorial auctions

Random sampling auction: Split randomly into S1, S2.

Apply optimization alg A on S1,

perhaps with penalty term. Use A(S1) on S2 and vice-versa.

SS11

SS22

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

Interesting issues: What quantities to use for , ?

What kind of regularization makes sense?

SS11

SS22

Random sampling auction: Split randomly into S1, S2.

Apply optimization alg A on S1,

perhaps with penalty term. Use A(S1) on S2 and vice-versa.

Generic Setting• S set of n bidders.

• Space of legal offers/pricing functions G.

• g is “take it or leave it” offer, so any fixed g is IC.

• g 2 G maps the pubi to pricing over the outcome

space.

• Bidder i: privi , pubi, bidi

• Goal: Incentive Compatible mechanism to do nearly as well as the best g 2 G.

• Assume max profit h per bidder.

Unlimited supply

Profit of g: sum over bidders.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

Interesting issues: What quantities to use for , ?

What kind of regularization makes sense?

SS11

SS22

Random sampling auction: Split randomly into S1, S2.

Apply optimization alg A on S1,

perhaps with penalty term. Use A(S1) on S2 and vice-versa.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

What should be large?

# bidders? But bidders of valuation 0 don’t help very much.

Instead: OPT profit.

Even if assume all valuations ¸ 1, bounds will be loose.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

What should be large?

# bidders? But bidders of valuation 0 don’t help very much.

Instead: OPT profit.

As a function of what?

# functions in G.

Even if assume all valuations ¸ 1, bounds will be loose.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

What should be large?

# bidders? But bidders of valuation 0 don’t help very much.

Instead: (OPT profit)/h.

As a function of what?

# functions in G.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

What should be large?

# bidders? But bidders of valuation 0 don’t help very much.

Instead: (OPT profit)/h.

As a function of what?

# functions in G.

# functions in G the alg could possibly output over splits S1,S2 +1.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

As a function of what?

# functions in G.

# functions in G the alg could possibly output over splits S1,S2 +1.

Multiplicative L1 cover size.

E.g., digital-good auction. Algorithm uses S1 to choose price to offer for S2 and vice-versa.

Can discretize to powers of (1+). Get |G| = (log h)/

Or use fact that alg will only output a bid value. |G| · n+1.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

# functions in G.

# functions in G the alg could possibly output over splits S1,S2 +1.

Multiplicative L1 cover size.

What if hard to directly bound # possible outputs

Use covering arguments: • find G’ that covers G , • show that all functions in G’ behave well

attributes

valuations

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

# functions in G.

# functions in G the alg could possibly output over splits S1,S2 +1.

Multiplicative L1 cover size.

G’ -covers G wrt to S if for all g exists g’ 2 G’ s.t.

i |g(i)-g’(i)| · g(S). [g(i) ´ profit made from bidder i]

Theorem (roughly):

If G’ is -cover of G, then the previous bounds hold with |G| replaced by |G’|.

Attribute Auctions, Linear Pricing Functions

Assume X=Rd. N= (n+1)(1/) ln h.

|G’| · Nd+1

attributes

valuations

xx

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x

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xx

xx

xx

xx

xx

xx

xx

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x xx

xx

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x

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

# functions in G.

# functions in G the alg could possibly output over splits S1,S2 +1.

Multiplicative L1 cover size.

For combinatorial auctions with m items, G = class of item-pricings, to get ¸ (1-)OPT, sufficient to have:

OPT = Õ(hm2/2) for general valuation functions.

OPT = Õ(hm/2) for unit-demand valuations.

First results for general case, factor m savings over GH01 for unit-demand valuations.

Goal

If is large as a function of , then the random sampling auction (perhaps regularized) performs nearly as well as best pricing function in class G.

Regularization/SRM: Can do SRM as usual, penalizing higher-complexity function classes.

But even individual functions can have different complexity levels! E.g., digital-good auction. Say S1 has 1 bid of value h and h-1 bids of value 1.

So, {1,h} are both optimal prices. But much better stats for 1.

Allows to replace “h” with “price used by OPT” in previous bounds.

$1 $1 $1 $1 $1 $h $1 $1 $1

Summary• Explicit connection between machine learning

and mechanism design. Use ideas of MLT to analyze when random sampling auction will do well.

• This application brings out interesting twists on usual ML issues. What has to be large as a function of what? SRM.

• Challenges:•Loss function discontinuous and

asymmetric.•Range of valuations large.

Challenges/Future Directions

• Apply similar techniques to limited supply.

• Online Setting.

• How big a “focus group” do you need for other kinds of pricing/allocation/decision-making problems.