Machine Learning for Mechanism
Design and Pricing Problems
Avrim Blum
Carnegie Mellon University
Joint work with Maria-Florina Balcan,
Jason Hartline, and Yishay Mansour
[Informs 2009]
Auctions/pricing
Designing auction/pricing mechanisms esp for complex
markets: challenging problems at the intersection of
CS and Economics
Auction mechanisms for selling digital goods
Software,
movies,
information
access
Auctions/pricing
Designing auction/pricing mechanisms esp for complex
markets: challenging problems at the intersection of
CS and Economics
Ad-auctions
Auctions/pricing
Designing auction/pricing mechanisms esp for complex
markets: challenging problems at the intersection of
CS and Economics
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?
Auctions/pricing
Designing auction/pricing mechanisms esp for complex
markets: challenging problems at the intersection of
CS and Economics
 Even if all customers’ preference information, how
much they would be willing to pay, etc. is known upfront, 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.
Auction/Pricing Problems
One Seller, Multiple Buyers with Complex Preferences.
Seller’s Goal: maximize profit.
CS / optimization
Economics
Version 1: Seller knows
the true values.
Version 2: values given by
selfish agents.
Algorithm Design
Problem (AD)
Incentive Compatible
Auction (IC)
Previous Work on IC : specific mechanisms for restricted
settings.
Auction/Pricing Problems
One Seller, Multiple Buyers with Complex Preferences.
Seller’s Goal: maximize profit.
CS / optimization
Economics
Version 1: Seller knows
the true values.
Version 2: values given by
selfish agents.
Algorithm Design
Problem (AD)
Incentive Compatible
Auction (IC)
Our Work: Generic Reduction using ML
Previous Work on IC : specific mechanisms for restricted
settings.
How is this related to Machine Learning?
Simple version: basic digital good auction problem.
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.
How is this related to Machine Learning?
Simple version: basic digital good auction problem.
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.
 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?
How is this related to Machine Learning?
Simple version: basic digital good auction problem.
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.
 Could ask people for their valuations and use this to set a
price as before, but people will low-ball (not incentivecompatible…)
How is this related to Machine Learning?
Simple version: basic digital good auction problem.
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.
Random sampling auction:
Ask buyers to submit bids bi.
Randomly partition bidders into
S1
two sets S1, S2.
Find best price over bids in S1…
and use it as offer price on S2!
S2
(& vice versa).
How is this related to Machine Learning?
More interesting version: combinatorial auctions
You’re Sperizon-mobile. Want to price various services.
-Basic service
-Extra lines
-Data package
-TV features, …
 People have potentially nonlinear valuations over subsets.
 Might also have known info about customers (current usage,
demographics,…). (Combinatorial Attribute Auction)
 Want to perform nearly as well as best (simple) pricing
function over known info.
How is this related to Machine Learning?
More interesting version: combinatorial auctions
You’re Sperizon-mobile. Want to price various services.
-Basic service
-Extra lines
-Data package
-TV features, …
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.
S1
S2
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?
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.
S1
S2
Generic Setting
• S set of n bidders.
• Bidder i: privi , pubi , bidi
• Space of legal offers/pricing functions G.
• g 2 G maps the pubi to pricing over the outcome space.
• g is “take it or leave it” offer, so any fixed g is IC.
• 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?
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.
S1
S2
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.
 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.
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.
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 if hard to directly bound
# possible outputs
valuations
Use covering arguments:
• find G’ that covers G ,
attributes
• show that all functions in G’ behave well
 # functions in G.
 # functions in G the alg could possibly output over splits S1,S2 +1.
 Multiplicative L1 cover size.
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.
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’|.
 # functions in G.
 # functions in G the alg could possibly output over splits S1,S2 +1.
 Multiplicative L1 cover size.
Attribute Auctions, Linear Pricing Functions
Assume X=Rd. N= (n+1)(1/) ln h.
d+1
|G’| · N
x
x
valuations
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
attributes
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.
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 unitdemand valuations.
 # functions in G.
 # functions in G the alg could possibly output over splits S1,S2 +1.
 Multiplicative L1 cover size.
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.
$1 $1 $1 $1 $1 $h $1 $1 $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.
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.